beg-`kg †kÖwY : D”PZi MwYZ  350

beg Aa¨vq
m~P K xq I j Mvwi`gxq d vskb John Napier
¯ ‹wUk MwYZwe` Rb †bwcqvi (1550Ñ1671) †K
¯ ^vfvweK jMvwi`‡gi RbK ejv nq| ¯^vfvweK
Exponential & Logarithmic Functions jMvwi`g‡K †bwcqvb jMvwi`gI ejv nq| wZwb cÖ_g
myk„•Ljfv‡e decimal point e¨envi ïiæK‡ib|

A b yk xj b x 9 .1
cvV m¤úwK©Z MyiæZ¡c~Y© welqvw`
ev¯Íe msL¨v : mKj gyj` msL¨v Ges Ag~j` msL¨v‡K ev¯Íe msL¨v n −n
ejv nq| ev¯Íe msL¨vi †mU‡K R Øviv cÖKvk Kiv nq| m~Î 7 : a < 0 Ges n  N, n > 1, n we‡Rvo n‡j, a = |a|
p n m n
g~j` msL¨v : p I q c~Y©msL¨v Ges q  0 n‡j q AvKv‡ii msL¨v‡K m~Î 8 : a > 0, m  Z Ges n  N, n >1 n‡j, ( a) = am
m p
g~j` msL¨v ejv nq| m~Î 9 : hw` a > 0 Ges n = q nq, †hLv‡b m, p  Z Ges
p
Ag~j` msL¨v : †h msL¨v‡K q AvKvi cÖKvk Kiv hvq bv, †hLv‡b p, q n q
n, q  N, n > 1, q >1 Z‡e, am = ap
c~Y©msL¨v Ges q 0 †m msL¨vK Ag~j` msL¨v e‡j| Abywm×všÍ : hw` a > 0 Ges n, k  N, n >1 nq, Z‡e
c~Y©msL¨v : k~b¨mn mKj abvZ¥K I FYvZ¥K ALÊ msL¨vmg~n‡K n nk
c~Y©msL¨v ejv nq| c~Y©msL¨vi †mU‡K Z Øviv cÖKvk Kiv nq| a= ak
¯^vfvweK msL¨v : 1, 2, 3, 4 ............ BZ¨vw` mvaviYZ MYbvg~jK g~j` fMœvsk m~PK
msL¨v¸‡jv‡K ¯^vfvweK msL¨v ejv nq| ¯^vfvweK msL¨v‡K abvZ¥K 1
n
ALÊ msL¨v ejv nq| msÁv : a  R Ges n  N, n >1 n‡j, an = a hLb a > 0 A_ev
a < 0 Ges we‡Rvo|
¯^vfvweK msL¨vi †mU‡K N Øviv cÖKvk Kiv nq|
1
m
m
m~PKxq ivwk : m~PK I wfwË m¤^wjZ ivwk‡K m~PKxq ivwk ejv nq|
msÁv : a > 0, m  Z Ges n  N, n >1 n‡j (6) a n
= an
m~PK m¤úwK©Z m~Î (Laws of Exponent) : m m
m~Î 1 : a  R Ges n  N n‡j, a1 = a, an + 1= an.a msÁv : a n = ( n a) =
n
am †hLv‡b, a > 0, m  Z, n  N, n > 1
m~Î 2 : a  R Ges m, n  N n‡j, am. an = am + n m p
m~Î 3 : a  R, a  0 Ges m, n  N, m  n n‡j, myZivs p  Z, q  Z, n > 1 hw` Ggb nq †h, = nq, Z‡e
n q
m p
a hLb m > n
m−n
m
a
= 1
m~Î-9 †_‡K †`Lv hvq †h, a n = aq
an hLb m < n m~Î 10 : a > 0, b > 0 Ges r, s  Q n‡j,
an−m
n ar s
m~Î 4 : a  R Ges m, n  N n‡j, (am) = amn (K) ar.as = ar + s (L) as = ar − s (M) (ar) = ars
m~Î 5 : a, b  R Ges n  N n‡j, (a.b)n = an.bn ar
(a )
r
m~Î 6 : a  0, b  0 Ges m, n  Z n‡j, (N) (ab)r = arbr (O) b = br
(K) am.an = am + n K‡qKwU cÖ‡qvRbxq Z_¨ :
am
(L) an = am − n (i) hw` ax = 1 nq, †hLv‡b a > 0, Ges a  1, Zvn‡j x = 0
n (ii) hw` ax = 1 nq, †hLv‡b a > 0 Ges x  0, Zvn‡j a = 1
(M) (am) = amn
(N) (ab)n = an.bn (iii) hw` ax = ay nq, †hLv‡b a > 0 Ges a  1, Zvn‡j x = y
n n a
(iv) hw` ax = bx nq, †hLv‡b b > 0 Ges x  0, Zvn‡j a = b
(a) = ba
(O) b n

Abykxjbxi cÖkœ I mgvavb

 beg-`kg †kÖwY : D”PZi MwYZ  351
p 3 2 3 −3 3
− − 2

cÖkœ \ 1 \ cÖgvY Ki †h, (a ) = a

m
n
mp
n
; †hLv‡b m, p  Z Ges n  N
=
( a ) + 2.a .a
2
+ (a ) − 1
2 2 2

3 3
m p 1 m m 1 −2
mgvavb : (a ) = {(a ) } [ a = (a ) ]
P m 2
n n n n a +a +1
3 3 2
−
= a ( )
1 mp
n m n
[(a ) = a m n]
=
( a +a ) −1
2 2

3 3
mp
2
−2+1
n
=a a +a
mp 3 3 3 3
m p
− −
 an ( ) ==a n
(cÖgvwYZ)
1 =
( a + a + 1) (a + a − 1)
2 2 2 2

3 3

cÖkœ \ 2 \ cÖgvY Ki †h, a ( ) =a

1
m
n 1
mn
, †hLv‡b m, n  N, m 
3
(a + a− + 1)
−3
2 2

0, n  0
= a2 + a 2 − 1 = Wvbcÿ
mgvavb : g‡b Kwi, m1 = x Ges 1n = y −3
2 2 
− 3
a3 + a 3 + 1
∴ mx = 1 ∴ ny = 1  3 −3 = a + a − 1 (†`Lv‡bv n‡jv)
1
1 2 2
a +a +1
GLb, (a ) = (a )
n
m x y
n
cÖkœ \ 5 \ mij Ki :
= axy [∵ (am) = amn] a
a+b
 1 a − b 
2 2
mxny 11

=a
mn
=a
mn
[gvb ewm‡q]   a − b 
1 K. xa 
mn a
=a
 1 a − b a + b
2 2
1
 
1
n 1  a a − b 
myZivs a  = amn (cÖgvwYZ) x 
m
mgvavb : 
m m m a
 1 (a +(ab)(a 
− b) a + b
 1 a+ba+b
a
cÖkœ \ 3 \ cÖgvY Ki †h, (ab) n = an bn; †hLv‡b m  Z, n  N
 a − b)    
mgvavb : g‡b Kwi, n = x
m
= x   = xa  
1
m a
a
m =x = x1 = x (Ans.)
GLb, evgcÿ = (ab) = (ab) n x
[
n
= x] 3
2
= ax.bx a + ab − a
L.
m m ab − b3 a−b
n n
= a . b = Wvbcÿ 3 1
m m m a2 + ab − a a.a2 + ab − a
 (ab) n an bn (cÖgvwYZ) mgvavb : =
= ab − b 3
a − b b(a − b )
2
a−b
1
cÖkœ \ 4 \ †`LvI †h, 2
a(a + b) − a
=
1 1 2 1 1 2
b(a − b2) a−b
(
K. a − b 3 ) (a + a b + b ) = a − b
3 3 3 3 3
a( a + b) − a
mgvavb : =
1 1 2 1 1 2 b{( a)2 − b2} a−b
evgcÿ = a3 − b3( ) (a + a b + b ) 3 3 3 3
=
a( a + b) − a
  b( a + b) ( a − b) a−b
  
1 1 1 2 1 1 1 2
=(
a − b ) a  + a b + b  
3 3 3 3 3 3
a − a
1 3 1 3 =
= 
a3
 3 3 3 b( a − b) a−b
− b  = a3− b3 = a − b = Wvbcÿ
a. a − b a
=
1 1 2 1 1 2 b( a − b)
(
 a −b 3 ) (a + a b + b ) = a − b (†`Lv‡bv n‡jv)
3 3 3 3 3
=
a ( a − b)
=
a
(Ans.)
−3
  b( a − b)
−3
3 b
3
= a + a − 1
a +a +1 2 2
L. 3 −3
2 2
a +a +1
-3
a3 + a−3 + 1 a3 + 2 + a − 1
mgvavb : evgcÿ = 3 −3 = 3 −3
2 2 2 2
a +a +1 a +a +1
 beg-`kg †kÖwY : D”PZi MwYZ  352
a a 1 1 1
a−b a − ba − b c2

2 2
a+b xa xb
M.
( ) b  a  =
x
1
b2
 1
2
 1
2
= 1 (Ans.)
b b x xc xa
a−b a − b a−b

a+b
( ) b  a 
P.
a
(a2 − b−2) (a − b−1)b−a
a a (b2 − a −2)b (b + a−1)a − b
a−b a − b a−b
 (a2 − b−2)a (b − b−1)b − a
a+b

mgvavb :
( b )  a  mgvavb : 2 −2 b
(b − a ) (b + a−1)a − b
b b
a b−a
a−b a − ba − b a2 − 12 a − 1b
( )

a+b
( )
b  a   b
= b a−b
b2 − 12 b + 1a
( )
 a
a b a b
a−b −a−b −a−b
a − ba − b

a+b
= ( )
b  a   1 1 
(a + b) (a − b) (a − b)
1 a b−a

a− b a−b  
=
a−b a − ba − b  1 1  1 b a−b
 (b + a ) (b − a) (b + a)
a+b
=( ) b  a   

a − b
1 1 1 1a 1 b−a a

 ( a + ) (a − ) (a − )
a+b
=(
b )  a  b b b
=
a + b a − b a2 − b2 1 1b 1 a−b b

b
= 
a
=
ab
(Ans.) ( a ) ( a) ( a )
b + b − b +

N.
1 1 1 1 1 b−a+a
a
−m −m + −n −n + −p
1 + a bn + a cp 1 + b cp + b am 1 + c am + c−pbn
=
( a + ) (a − )
b b
mgvavb : b a−b+b
1 1 1
(b − 1a) (b + 1a)
+ +
−m −m −n −n −p
1 + a bn + a cp 1 + b cp + b am 1 + c am + c−pbn 1 a 1 b ab + 1 a ab − 1b
1 1 1 =
( a + ) (a − )
b b
=
( b )  b 
= + +
bn cp cp am am bn 1 b 1 a
ab − 1b ab + 1 a
1+ m+ m 1+ n+ n 1+ p + p
a a b b c c ( a) ( a )  a  ( a )
b − b +
a ab − 1 a b

1 1 1 ab + 1 a
=(
ab + 1)  b
 
= + +
am + bn + cp bn + cp + am cp + am + bn b ab − 1
am bn cp a a a b a a+b
=( ) ( ) =( ) (Ans.)
b b b
= 1      
am bn cp
p + 1 m p + 1 m
 a + b + c   a + b + c   a + bn + cp
m p n cÖkœ \ 6 \ †`LvI †h,
a m
b c n p K. hw` x = aq + r bp, y = ar + pbq, z = ap + q br nq, Z‡e xq − r.
= + +
am + bn + cp am + bn + cp am + bn + cp y r − p.zp − q = 1
am + bn + cp mgvavb : †`Iqv Av‡Q, x = aq + rbp
= m = 1 (Ans.) y = ar + pbq
a + bn + cp
b c a
z = ap + qbr
bc x
c ca x
a ab x
b evgcÿ = xq − r.yr − p.zp − q
O. c  a  b = (aq + r.bp)q − r.(ar + p.bq)r − p.(ap + q.br)p − q
x
b
x
c
x
a
= a(q + r)(q − r)bpq − pr.a(r + p)(r − p).bqr − pq.a(p + q)(p − q).bpr − qr
2 2 2 2 2 2
b c a = aq − r .ar − p .ap − q .bpq − pr.bqr − pq.bpr − qr
bc c ca a ab b 2 2 2 2 2 2

mgvavb :
x x x = aq − r + r − p + p − q .bpq − pr + qr − pq + pr − qr
c a  b
b c a = a0b0 = 1.1 = 1 = Wvbcÿ
x x x
b 1 c 1 a 1
 xq − r. y r − p.zp − q = 1 (†`Lv‡bv n‡jv)
c  bc a  ca b  ab L. hw` ap = b, bq = c Ges cr = a nq, Z‡e pqr = 1
x x x
= c 1  a 1  b 1 mgvavb : †`Iqv Av‡Q, ap = b, bq = c Ges cr = a
   ab
x
b bc
x
c ca a
x GLv‡b, ap = b
ev, (cr)p = b
ev, cpr = b
ev, (bq)pr = b
 beg-`kg †kÖwY : D”PZi MwYZ  353
ev, bpqr = b1 1 −1
 pqr = 1 (†`Lv‡bv n‡jv) M. hw` a = 23 + 2 3 nq, Z‡e †`LvI †h, 2a3 − 6a = 5
1 −1
M. hw` ax = p, ay = q Ges a2 = (pyqx)z nq, Z‡e xyz = 1 mgvavb : †`Iqv Av‡Q, a = 2 + 2 3 3
mgvavb : 1 1
†`Iqv Av‡Q, ax = p, ay = q Ges a2 = (pyqx)
z
( − ) [Dfqcÿ‡K Nb K‡i]
ev, a3 = 23 + 2 3
3

GLv‡b, a2 = (pyqx)z 1 3
−
1 3 1
−
1 1 1
ev, a = (2 ) + ( 2 ) + 3.2 .2 (2 + 2 )
3 3 3 3 3 −
3 3
ev, a2 = {(ax)y(ay)x}z
1 1
z −
ev, a2 = (axy.axy) ev, a3 = 2 + 2 + 3.23 . 2
−1 3
.a
ev, a2 = a2xyz 1
ev, 2 = 2xyz ev, a = 2 + 2 + 3.1.a
3

 xyz = 1(†`Lv‡bv n‡jv) ev, a3 =

4 + 1 + 6a
3 3 3
2
cÖkœ \ 7 \ K. hw` x a + y b + z c = 0 Ges a2 = bc nq, Z‡e ev, a3 = 5 + 6a
†`LvI †h, ax3 + by3 + cz3 = 3axyz  2a3 − 6a = 5 (†`Lv‡bv n‡jv)
mgvavb : 2
−
2

3 3 3
N. hw` a2 + 2 = 3 + 3
3 3 Ges, a  0 nq, Z‡e †`LvI †h,
†`Iqv Av‡Q, x a + y b + z c = 0 3a3 + 9a = 8
3 3 3 mgvavb :
ev, x a + y b = − z c 2
3 − 23
( 3
) (
ev, x a + y b = − z c
3 3
) 3
[Dfqcÿ‡K Nb K‡i] †`Iqv Av‡Q, a2 + 2 = 33 + 3
2 2 3
−3
ev, (x a) + (y b)
3
3
3
3
ev, (a2 + 2)3 = 33 + 3 ( ) [ Dfqcÿ‡K Nb K‡i]
2 3
3 3
+ 3.x a.y b x a + y b = − z3c ( 3 3
) 3 2
ev, (a2) + 3(a2) 2 + 3.a2.22 + 23 = 33 ( )
ev, x3a + y3b + 3xy ab −z c = − z3c
3
( 3
) +
2 3
( 3 ) + 3. 3 .3 (3 + 3 )
−
3
2
3
−
2
3
2
3
−
2
3
3
ev, x3a + y3b − 3xyz abc = − z3c 1 + 23 − 3
2

3 ev, a6 + 6a4 + 12a2 + 8 = 32 + 3− 2 + 3 (a2 + 2)

ev, x3a + y3b + z3c = 3xyz abc 1
3 ev, a6 + 6a4 + 12a2 + 8 = 9 + 9 + 3(a2 + 2)
ev, ax + by + cz = 3xyz a.a [ a = bc]
3 3 3 2 2
1
ev, ax3 + by3 + cz3 = 3axyz ev, a6 + 6a4 + 12a2 + 8 = 9 + 9 + 3a2 + 6
 ax3 + by3 + cz3 = 3axyz (†`Lv‡bv n‡jv) 1
1 1 ev, a6 + 6a4 + 9a2 = 7 + 9
L. hw` x = (a + b) + (a − b) Ges a2 − b2 = c3 nq, Z‡e
3 3
63 + 1
†`LvI †h, x3 − 3cx − 2a = 0 ev, (a3)2 + 2.a3.3a + (3a)2 = 9
mgvavb : 64
1 1 ev, (a3 + 3a)2 = 9
†`Iqv Av‡Q, x = (a + b) + (a − b) 3 3
8
1 1 3
ev, a3 + 3a = 3 [Dfqcÿ‡K eM©g~j K‡i]
ev, x 3
= {(a + b) + (a−b) } 3 3
[Dfqcÿ‡K Nb [ a  0 †m‡nZz ïay abvZ¥K gvb wb‡q]
K‡i]  3a + 9a = 8 (†`Lv‡bv n‡jv)
3
1 1 1 1
3
3 {
ev, x = (a + b) } + { (a − b) } + 3 (a + b)
3 3 3 3
O. hw` a2 = b3 nq, Z‡e †`LvI †h, b () ()a
3
2
+
b
a
2
3
1
= a2 + b
−
1
3
1 1 1
(a − b) {(a + b) + (a − b) } 3 3 3
mgvavb :
1 1 †`Iqv Av‡Q, a2 = b3
ev, x = a + b + a − b + 3 {(a + b) (a − b) } x
3 3 3 3 2 1 1
a 2 b 3  a 32  b 23

ev, x3 = 2a + 3x(a2− b )
1
2 3
evgcÿ = () +() ()
b a
=
 b
 + 
  a ( ) 
1 1 1 1
1
=
a32 b23 = a22 + b33
2 3 2
ev, x = 2a + 3x(c ) 3 3
[ a − b = c ] [ b3 = a2]
b3 +  a  a  b 
3 2 2 3

ev, x3 = 2a + 3x.c 1 1 1 1
ev, x3 = 2a + 3cx = (a3 − 2)2 + (b2 − 3)3 = a2 + (b−1)3
 x3 − 3cx − 2a = 0 (†`Lv‡bv n‡jv) 1
−
1
= a2 + b 3
= Wvbcÿ
 beg-`kg †kÖwY : D”PZi MwYZ  354
3 2 1 1 b
−
a 2 b 3  x = ya
A_©vr ( ) +( ) =a +b
b
2
a
1
2 3
(†`Lv‡bv n‡jv)
Avevi, zc = yb
b
P. hw` b = 1 + 3 + 3 nq, Z‡e †`LvI †h, b − 3b − 6b − 4 = 0
3 3 3 2
 z = yc
mgvavb : GLb, xyz = 1
2 1 b b
†`Iqv Av‡Q, b = 1 + 33 + 33 ev, ya.y. yc = 1
2 1 b b
3
ev, (b − 1) = 3 + 3
3 (
) [ Dfqcÿ‡K Nb K‡i]
3 3 ev, ya + 1 + c = 1
bc + ac + ab
2 1 2 1 2 1
ev, y = y
3 3
ev, b − 3b + 3b − 1 = (3 ) + (3 ) + 3.3 .3 (3 + 3 )
ac
3 2 3 3 3 3 3 3
bc + ac + ab
1 + 23 +13 ev, ac
=0
ev, b − 3b + 3b − 1 = 3 + 3 + 3
3 2 2
.(b−1)
3 +2 +1  bc + ac + ab = 0 (Ans.)
y x
ev, b3 − 3b2 + 3b − 1 = 9 + 3 + 3 3 (b − 1) M. hw` 9x = (27) nq, Zv n‡j y Gi gvb KZ?
ev, b3 − 3b2 + 3b − 1 = 12 + 9 (b − 1) mgvavb :
ev, b3 − 3b2 + 3b − 1 = 12 + 9b − 9 †`Iqv Av‡Q, 9x = (27)y
 b3 − 3b2 − 6b − 4 = 0 (†`Lv‡bv n‡jv) ev, (32)x = (33)y
−3
1 1 ev, 32x = 33y
[we: `ª: cvV¨ eB‡qi cÖ‡kœ 3 Gi ¯’‡j 3 n‡e] 3
ev, 2x = 3y
x 3
Q. hw` a + b + c = 0 nq, Z‡e †`LvI †h,  = (Ans.)
y 2
1
+
1
+
1
=1
cÖkœ \ 9 \ mgvavb Ki :
xb + x− c + 1 xc + x− a + 1 xa + x− b + 1 (K) 32x + 2 + 27x + 1 = 36
mgvavb : mgvavb :
1 1 1
evgcÿ = xb + x− c + 1 + xc + x− a + 1 + xa + x− b + 1 32x + 2 + 27x + 1 = 36
ev, 32x + 2 + 33x + 3 = 36
1 1 1
= + + ev, 32x.32 + 33x.33 − 36 = 0
1 b 1 + x c + xb + c a 1
x + c+1
x
x + b+1
x ev, (3x)2.32 + (3x)3.33− 36 = 0
[ a + b + c = 0  b + c = −a] ev, a2.9 + a3.27 − 36 = 0 [3x = a a‡i]
xc 1 xb ev, 27a3 + 9a2 − 36 = 0
= b+c + b+c +
c
1+x + x 1+x + xc
x a+b
+ 1 + xb ev, 9(3a3 + a2 − 4) = 0
c
x 1 xb ev, 3a3 − 3 + a2 −1 = 0
= c b+c + c b+c +
1+x +x 1+x +x x + xb +1
-c
ev, 3(a3 − 1) + a2 −1 = 0
c
x 1 xb ev, 3(a − 1) (a2 + a + 1) + (a − 1) (a + 1) = 0
= c b+c + c b+c + 1
1+x + x 1+x + x
xc
+ xb +1 ev, (a − 1) (3a2 + 3a + 3 + a + 1) = 0
xc 1 xb.xc ev, (a − 1) (3a2 + 4a + 4) = 0
= c
1+x + x b+c + c
1+x + x b+c +
1+ xc + xb+c nq, a − 1 = 0 A_ev, 3a2 + 4a + 4 = 0
c
x +1+x b+c c
1+x +x b+c
−4  42 − 4.3.4
= = = 1 = Wvbcÿ (†`Lv‡bv n‡jv) ev, a = 1 a=
1+xc+xb+c 1+xc + xb+c 2.3
cÖkœ \ 8 \ K. hw` ax = b, by = c Ges cz = 1 nq, Z‡e xyz = KZ ? −4  16 − 48
ev, 3x = 30 [gvb ewm‡q] =
6
mgvavb :
−4  − 32
†`Iqv Av‡Q, ax = b, by = c Ges cz = 1 x=0 =
6
GLv‡b, cz = 1
GLv‡b −32 Aev¯Íe| myZivs GwU MÖnY‡hvM¨ bq|
ev, (by)z = 1 [ by = c]
wb‡Y©q mgvavb x = 0
ev, {(a ) } = 1
x y z
[ ax = b]
(L) 5x + 3y = 8
ev, {a } = 1
xy z
− −
5x 1 + 3y 1 = 2
ev, axyz = a x y

 xyz = 0 (Ans.) mgvavb : 5 + 3 = 8........... (i)

x−1 y−1
L. hw` xa = yb = zc Ges xyz = 1 nq, Z‡e ab + bc + ca = KZ ? 5 + 3 = 2....... (ii)
mgvavb : (ii) bs mgxKiY †_‡K cvB,
x −1 y −1
†`Iqv Av‡Q, xa = yb 5 .5 + 3 .3 = 2
 beg-`kg †kÖwY : D”PZi MwYZ  355
5
x
3
y
 x = −2
ev, 5 + 3 = 2 x Gi gvb (iii) bs mgxKi‡Y ewm‡q cvB,
x y 2(−2) − y + 2 = 0
3.5 + 5.3
ev, 15
=2 ev, −4−y + 2 = 0
x ev, y = −2
ev, 3.5 + 5.3y = 30 ........ (iii)
 y = −2
(iii)  1 − (i)  3 n‡Z cvB,
y
wb‡Y©q mgvavb : x = − 2, y = − 2
2.3 = 6
y (N) 22x + 1.23y + 1 = 8
ev, 3 = 3 2x + 2.2y + 2 = 16
y=1 mgvavb :
y Gi gvb (i) bs mgxKi‡Y ewm‡q cvB,
x 1
22x + 1.23y + 1 = 8 ...........(i)
5 +3 =8 2x + 2.2y + 2 = 16........... (ii)
(i) bs mgxKiY n‡Z cvB,
x
ev, 5 = 8 − 3
x 22x + 1 + 3y + 1 = 23
ev, 5 = 5
ev, 2x + 3y + 2 = 3
x=1
wb‡Y©q mgvavb : x = 1, y = 1 ev, 2x + 3y = 3−2
 2x + 3y =1 ....... (iii)
3y−2 x+y x + 2y 2x+1
(M) 4 = 16 ; 3 =9 (ii) bs mgxKiY n‡Z cvB,
mgvavb : 43y−2 =16x + y............ (i) 2x + 2 + y + 2 = 24
3x + 2y = 92x + 1.......... (ii) ev, x + y + 4 = 4
(i) bs mgxKiY n‡Z cvB, ev, x + y = 0
43y−2 = (42)x + y  y = − x .............. (iv)
3y−2 2x+2y (iv) Gi gvb (iii) bs-G ewm‡q cvB,
ev, 4 = 4
2x + 3(−x) = 1
ev, 3y − 2 = 2x + 2y ev, 2x −3x = 1
ev, 2x − y + 2 = 0.......... (iii) ev, −x =1
(ii) bs mgxKiY n‡Z cvB,
x + 2y 2x + 1
x=−1
3 = (32) x Gi gvb (iv) bs-G ewm‡q,
ev, 3 x + 2y
= 34x + 2 y = −(−1)
ev, x + 2y = 4x + 2 y=1
ev, 3x−2y + 2 = 0 ........... (iv) wb‡Y©q mgvavb : x = − 1, y = 1
(iii)  2 − (iv)  1 n‡Z cvB,
x+2=0

¸iæZ¡c~Y© enywbe©vPwb cÖ‡kœvËi

3 5. hw` x, y, z  0, px = qy = rz nq Z‡e, wb‡Pi †KvbwU mwVK?
3 z z y z
3
1. 729 Gi gvb KZ?  q=r
y
L r=q
y
Mq=r
z
N p=q
y
1 2 1
9 9 3 6. a > 0, m  Z, n  N Ges n > 1 n‡j-
K3 3 M3 N3
3
3
3
3
3
3
3
3
i. ( n a) = n a
m
m ii. ( a n) = ( a m)
m m

e¨vL¨v : 729 = 93 = 9
iii. ( a) = a
m
2 1 2
n m n
3 
3 3 3 9
= 3 =32
=3 wb‡Pi †KvbwU mwVK?
15
x10 i L ii M iii N i, ii I iii
x8 7. k~‡b¨i m~PK k~b¨ n‡j Zvi gvb KZ?
2. x4 Gi mij gvb †KvbwU?
1 K0 L1 M Amxg  AmsÁvwqZ
K x15 Lx x
15
N1 8. a  1 n‡j ax = am n‡e, hw` Ges †Kej hw` wb‡Pi †KvbwU?
3. al = b, bm = c, cn = a n‡j, lmn Gi gvb KZ? K a=x L a=m  x=m N x=±m
l wb‡Pi Z‡_¨i Av‡jv‡K 9 I 10 bs cÖ‡kœi DËi `vI :
K abc L l N −l
abc 1 1 1
4. ax = b, by = c Ges cz = a n‡j, xyz = KZ? + + ......... GKwU Amxg aviv|
z + 1 (z + 1)2 (z + 1)3
K −1 L0 1 N2
 beg-`kg †kÖwY : D”PZi MwYZ  356
9. wb‡Pi †Kvb k‡Z© avivwUi AmxgZK mgwó _vK‡e? K z < − 2 Ges z < 0 L z < − 2 Ges z > 0
K |r| < − 1  |r| < 1 M |r| > 1 N |r| > − 1  z < − 2 Ges z > 0 N z > − 2 Ges z < 0
10. z-Gi †Kvb gv‡bi Rb¨ avivwUi AmxgZK mgwó wbY©q Kiv hvq?

AwZwi³ enywbe©vPwb cÖ‡kœvËi

1
91 : g~j` I Ag~j` m~PK a L0 M N a−1
a
21. a Ges n n‡j, a n+1
= KZ? (mnR)
mvaviY enywbe©vPwb cÖ‡kœvËi an
n
K a +a L a −a
n n
 a .a N
11. mKj g~j` I Ag~j` msL¨vi †mU †KvbwU? (mnR) a
K  Mℤ NQ 22. a Ges m, nIN n‡j, a .an = KZ? m
(mnR)
12. ¯^vfvweK msL¨vi †mU wb‡`©k K‡i †KvbwU? (mnR)
 am + n L a-(m + n) M am−n
am
N n
KN LR MQ NZ a
e¨vL¨v : mKj ¯^vfvweK msL¨vi †mU N| mKj ev¯Íe msL¨vi 23. †KvbwU m~P‡Ki †gŠwjK m~Î? (mnR)
†mU R| mKj gyj` msL¨vi †mU, Q| K a1 = a  am + n = am.an
13. ( 3) m~PKxq ivwki wfwË KZ?
7 (mnR) o
M a =1 N (ab)n = an.an
7 24. hw` a, bN Ges nN nq Z‡e (a. b)n = KZ? (mnR)
K7 L 7 3 N 3 1
14. a  0 Ges n abvZ¥K c~Y© msL¨v n‡j an Kx wb‡`©k K‡i? (ga¨g)
n
 a .b n
L an. n n
M a +b n
N a −b
n n
b
K a †K n evi †hvM L a †K n evi we‡qvM 25. aR Ges m, nN n‡j, (am)n = KZ? (mnR)
 a †K n evi ¸Y N a †K n evi fvM a n

enyc`x mgvwßm~PK enywbe©vPwb cÖ‡kœvËi

 amn L am−n M am + an N () m
26. nN, n > 1 Ges aR n‡j, x †K a Gi nZg g~j ejv n‡e hw`Ñ (mnR)
K ax = n nq L nn = 1 nq
15. ( 23) 4
Gi †ÿ‡ÎÑ
 x = n nq
n
N an = 1 nq

i. wfwË 3
2
ii. gvb
16 27. 2 Ges − 2 DfqB 16 Gi KZZg g~j? (mnR)
181
K 32 Zg g~j L 16 Zg g~j M 4 Zg g~j  4 Zg g~j
iii. m~PK 4
28. −27 Gi Nbg~j wb‡Pi †KvbwU? (mnR)
wb‡Pi †KvbwU mwVK? (mnR) K9 L3  −3 N −9
K i I ii  i I iii M ii I iii N i, ii I iii 29. 0 Gi nZg g~j KZ? (mnR)
16. ev¯Íe msL¨vi †ÿ‡ÎÑ 1
i. mKj c~Y© msL¨vi †mU
Kn 0 − N −1
M
2
ii. Q mKj g~j` msL¨vi †mU 30. cÖ‡Z¨K abvZ¥K ev¯Íe msL¨v a Gi GKwU Abb¨ abvZ¥K n
iii. mKj ev¯Íe msLvi †mU Zg g~j i‡q‡Q| G‡K wb‡Pi †Kvb cÖZxKwU Øviv cÖKvk Kiv
wb‡Pi †KvbwU mwVK? (mnR) nq? (mnR)
K i I ii L i I iii  ii I iii N ii I iii n a
 a L n M an N an
17. †mU cÖKv‡ki ixwZ AbyhvqxÑ
31. a FYvZ¥K ev¯Íe msL¨v Ges n we‡Rvo ¯^vfvweK msL¨v n‡j,
i. Z n‡jv c~Y© msL¨vi †mU
a Gi GKwU Abb¨ FYvZ¥K nZg g~j i‡q‡Q| G‡K Kx cÖZxK
ii. R n‡jv ev¯Íe msL¨vi †mU
iii. Q n‡jv g~j` msL¨vi †mU Øviv cÖKvk Kiv nq? (mnR)
wb‡Pi †KvbwU mwVK? (ga¨g) n a n n
K a L n M a − a
 i I ii L i I iii M ii I iii N i, ii I iii
enyc`x mgvwßm~PK enywbe©vPwb cÖ‡kœvËi
92 : m~PK m¤úwK©Z m~Î 32. a, b > o n‡jÑ
mvaviY enywbe©vPwb cÖ‡kœvËi i. ax = 1 Ges x  0 n‡j a = 1
18. a cÖZxKwU‡Z a †K Kx ejv nq?
m
(mnR) ii. ax = ay Ges a  1 n‡j x = y
 wfwË L m~PK M kw³ N AbycvZ iii. ax = bx Ges x  0 n‡j x = a
19. mKj ¯^vfvweK msL¨v ev abvZ¥K c~Y© msL¨vi †mU wb‡Pi wb‡Pi †KvbwU mwVK? (KwVb)
†KvbwU? (mnR)  i I ii L i I iii M ii I iii N i, ii I iii
K R LZ MQ N 33. ax = by = cz n‡jÑ
20. a n‡j, a1 = KZ? (mnR) y
x
i. a = b
 beg-`kg †kÖwY : D”PZi MwYZ  357
z m p
ii. b = c
y 45. hw` a > 0 Ges n = q nq †hLv‡b m, pZ Ges n, qN, n
y
z > 1, q > 1 Z‡e wb‡Pi †KvbwU mwVK? (KwVb)
iii. c = b
wb‡Pi †KvbwU mwVK? (ga¨g) K
m
an =
n
am 
n
am =
q
ap
K i I ii L i I iii M ii I iii  i, ii I iii m n

34. i. a †K a Gi m NvZ ev kw³ e‡j

m
M ( n a) = ( q a) N
n
am =
m
a
ii. am †K a NvZ m cov nq 46. wb‡Pi †KvbwU mwVK? (KwVb)
iii. n GKwU ev¯Íe msL¨v 3
5 = 11.665 L 4=2
wb‡Pi †KvbwU mwVK? (mnR)
3 3
 i I ii L i I iii M ii I iii N i, ii I iii M5 = 12.089 N 27 = − 3
35. i. mKj ev¯Íe msL¨vi †mU 47. a > o n‡j, mKj x R Gi Rb¨ wb‡Pi †KvbwU mwVK? (mnR)
ii. mKj g~j` msL¨vi †mU Q K ax < o  ax > o M ax  o N ax = o

iii. mKj c~Y© msL¨vi †mU Z

48. hw` x < y nq Zvn‡j a > 1 Gi Rb¨ wb‡Pi †KvbwU mwVK?
(ga¨g)
wb‡Pi †KvbwU mwVK? (mnR)
 ax < ay L ax > ay M ax = ay N axy = ay
K i I ii L i I iii M ii I iii  i, ii I iii
49. hw` x < y nq, Zvn‡j o < a < 1 Gi Rb¨ wb‡Pi †KvbwU mZ¨?
36. a  R Ges a  0 n‡j Ñ (KwVb)
i. a−n. an = 1  a >a x y
L a <a x y
M a a
x y
N a a
x y
ii. a0 = 0 1 1 1
1
iii. a−n = n 50. hw` a = b = c Ges abc = 1 nq Zvn‡j x + y + z = KZ? (KwVb)
x y z
a
K −3 L −2 M1 0
wb‡Pi †KvbwU mwVK? (mnR) 51. a > 0 n‡j †KvbwU mwVK? (mnR)
K i I ii  i I iii M ii I iii N i, ii I iii
n n n n
 a>0 L a<0 M a0 N a0
Awfbœ Z_¨wfwËK enywbe©vPwb cÖ‡kœvËi 52. 3 Zg g~j‡K Kx ejv nq? (mnR)
ax= = Ges = ac nq|
by cz b2 K eM© L eM©g~j  Nbg~j N wØNvZ
Dc‡ii Z‡_¨i Av‡jv‡K 37Ñ39 bs cÖ‡kœi DËi `vI : enyc`x mgvwßm~PK enywbe©vPwb cÖ‡kœvËi
37. a = KZ? (ga¨g)
x y
53. mKj a  Gi Rb¨
K a=y L a= by M a= by  a= bx i. a1 = 0
38. c = KZ? (ga¨g) ii. a1 = a
y z x n msL¨K
 c= bz L c= by M c= by N c=b
yz
iii. an = a.a.a.....a [n , n>1]
39. b2 = ac n‡j b2 = wb‡Pi †KvbwU? (KwVb) wb‡Pi †KvbwU mwVK? (mnR)
y x y y y xy K i I ii L i I iii  ii I iii N i, ii I iii
+ +
K b2 = bx L b2 = by z  b2 = b x z N b2 = byz 54. i. a Gi ciggvb |a|
ii. a < 0 n‡j, |a| = −a
93 : g~j Gi e¨vL¨v iii. a < 0 n‡j, |a| = a
mvaviY enywbe©vPwb cÖ‡kœvËi wb‡Pi †KvbwU mwVK? (ga¨g)
 i I ii L i I iii M ii I iii N i, ii I iii
40. a > o n‡j, wb‡Pi †Kvb m¤úK©wU mwVK †hLv‡b a ? (ga¨g)
Awfbœ Z_¨wfwËK enywbe©vPwb cÖ‡kœvËi
a a n n
K n>0 L n<0  a>0 N a0 1 1 1
n ax = by = cz = k Ges abc = 1
41. a < o Ges n , n>1, n we‡Rvo n‡j, a KZ ? (ga¨g)
Dc‡ii Z‡_¨i Av‡jv‡K 55Ñ 57bs cÖ‡kœi DËi `vI :
n n n n
 − |a| L |a| M  |a| N a 55. wb‡Pi †KvbwU mwVK? (mnR)
42. a > o Ges a  1 n‡j, ax = ay n‡e hw` I †Kej hw`Ñ (ga¨g) x 1 y
y z x
K n  y nq  x = y nq M n > y nq N xy = o nq  a=b L c=k M a=b N abc = k
43. a > o, b > o Ges x  o n‡j, a = b n‡e hw` I †Kej 56. abc wb‡Pi †KvbwUi mgvb?
x x
1
(ga¨g)
2
hw`Ñ (ga¨g)
K ab = c2 L k+3  kx + y + z Nk
x
+y+
z
 a = b nq L ab = o nq M a − b < 0 nq N a  b nq 57. x + y + z = KZ? (mnR)
44. wb‡Pi †KvbwU mwVK? (ga¨g) 2 1
K1 0 M k +1 N
K 4 = −2  4=2 M 27 = −3 N 36 = − 6 k
 beg-`kg †kÖwY : D”PZi MwYZ  358
a < 0 Ges n  , n > 1 p

71.  n = KZ? †hLv‡b, m, p Ges n

m
Dc‡ii Z‡_¨i Av‡jv‡K 58 I 59bs cÖ‡kœi DËi `vI| (mnR)
a
58. n we‡Rvo msL¨v n‡j g~jwU †Kgb n‡e? (mnR) mp
n
n
np
mp
a
m
n
+p
K abvZ¥K  FYvZ¥K M eM©g~j N g~j` a La Ma Na
59. n †Rvo msL¨v n‡j a Gi n Zg g~j KqwU? (ga¨g) 12
1 L 16 M 26 N 72. a8 a6 a4 Gi mijgvb KZ? (ga¨g)
K a12 L a4 a N1
94 : g~j` fMœvsk m~PK 12
12 4
2 12
e¨vL¨v : a
8
a6 a 4 = a
8
a6. a = a
8
a6. a2
mvaviY enywbe©vPwb cÖ‡kœvËi 12
8 12 8 4 12
12
12 = 12
= a a8 = a .a = a a =a
a 1
a b 73. a < o Ges n , n >1 Ges we‡Rvo n‡j an = KZ? (ga¨g)
60. hw` a = b nq Zvn‡j
b a
() b
Gi gvb KZ? (KwVb) 1
n
−1
n
 −|a| M −|−a|
n n
a a a L a N |a|
−1 −1 +1
a b Lb b Mb b N1 74. 92m = 3x + 1 n‡j x = KZ? (ga¨g)
61. a = p, a = q Ges a = (p q ) n‡j xyz Gi gvb KZ? (ga¨g)
x y 2 y x z
2 1 2
K  M −3 N−
1 3 3 3
K0 L 1 N2
2 n
p
q
62. a = KZ? (mnR)
75. (ba) Gi gvb wb‡Pi †KvbwU (mnR)
an a an
K L  N1
q 1 a 1 b bx bn
q 2 p p
 ap L a M aq N a 76. (am)nGi gvb wb‡Pi †KvbwU? (ga¨g)
1 1 K am L amn M0 1
63. hw` a = b = c Ges b = ac nq Z‡e wb‡Pi †KvbwU
x y z 2
+
x z
3
Gi gvb? (KwVb) 77. − 27 Gi gvb wb‡Pi †KvbwU? (mnR)
2 2 y z K9 L3  −3 N −9
K  M N
z y z x a
a
64.
3
(a3b5)3 = KZ? (mnR)
78. hw` ab = ba nq Zvn‡j () b
b
Gi gvb wb‡Pi †KvbwU? (KwVb)
a
K a9b5 L a8b3  a3b5 N a5b3 a
aa aab ab −1
3
3 5 3
1 K b L a M a  ab
e¨vL¨v : (a a ) = {(a3b5)3}3 b
bb bb
1
= {(a3)3 (b5)3}3 a
a
=
1
(a9b15) 3
1 1
79. ab = ba n‡q Z‡e () b
b
KZ?
9 15 a a a b
= a 3.b 3 −1 −1 +1 −1
= a3b5  ab L bb M ab N aa
x 3
65. hw` (16)x = (64)y n‡j y = KZ? (KwVb) 80. −8 Gi gvb KZ?
2 4 3 3 3
K L  N0 K 8 L 8 − 8 N− 83
3 3 2
x x x
1
x
1
y x 81. x = (x x) n‡j, x Gi gvb KZ?
66. (16) = (64) n‡j y = KZ? (KwVb) 2 3 9 27
K L  N
2 4 3 3 2 4 8
 L M N0 x x
3 3 2 e¨vL¨v : xx = (x x)
x

a a x x
−
1 3
a b b 1 ev, (xx)
x
= x.x2 ( ) = (x ) 2
67. ( ) =a
b
Ges a = 3b n‡j b = KZ? (mnR)
ev, (xx)= (xx) 2
x 3

K1  3 M4 N9 3
ev, x = 2
68. ( 3)5 m~PKxq ivwki wbavb ev wfwË KZ? (ga¨g)
9
5 ∴x=
K5 L 3 M 3 4
2 1 1 2 1 1 2
3 −1 −1
69. {1 − (1 − x ) } = KZ? (KwVb) 82. (a − b ) (a + a . b
3 3 3 3 3+b3 ) Gi gvb †KvbwU?
1 1 1
1 1 1 2−x 3
Ka+b  a−b M a3 − b 3 N (a−b) 3
K 3+1  1− 3 M N
x x 1 + x3 1 + x2 1 1 2 1 1 2

e¨vL¨v : (a3 − b 3) (a3 + a 3. b 3 + b 3 )

70. -8 Gi Nbg~j KZ? (ga¨g) 1 1 1 2
1 1 1 2

 −2 L −1 M2 N4 = ( 3
a −b
3
) {( ) a
3 3
+a .b +
3
( )}b
3
 beg-`kg †kÖwY : D”PZi MwYZ  359
3 3
1 1
() ()3 3
3
−8 =  2
= a − b
=a−b iii.
83. (a2b3)5 Gi gvb wb‡Pi †KvbwU? wb‡Pi †KvbwU mwVK? (ga¨g)
 a10.b15 L a25b125 M (ab)30 N a3b2  i I ii L i I iii M ii I iii N i, ii I iii
e¨vL¨v : a, b  R GwU nN n‡j (a, b)n = an.bn
 (a2b3)5 = (a2)5. (b3)5
88. i. hw` ax = 1 nq, †hLv‡b a > 0 Ges a  1 Zvn‡j x = 0
= a2  5.b3  5 ii. hw` ax = 1 nq, †hLv‡b a > 0 x  0, Zvn‡j a = 1
= a10.b15
a b iii. hw` ax = ay nq, †hLv‡b a > 0 Ges a  1, Zvn‡j x = y
a a
84. ( ) ( ) = KZ?
b

b wb‡Pi †KvbwU mwVK?
K i I ii L i I iii
ab a+b a a −b a+b
K (ba) (2ba) L M
b
 (ba) M ii I iii  i, ii I iii
Awfbœ Z_¨wfwËK enywbe©vPwb cÖ‡kœvËi
a b
a a
e¨vL¨v : ( ) ( )
b b
a
=( )
a−b
GKwU m~PKxq mgxKiY Ges 2x = y
4x − 3.2x + 2 + 25 = 0
b
Dc‡ii Z‡_¨i Av‡jv‡K 89 Ñ 91 bs cÖ‡kœi DËi `vI :
enyc`x mgvwßm~PK enywbe©vPwb cÖ‡kœvËi 89. y2 − 12y = KZ? (KwVb)
n  −32 L −36
85. i. a (1) = a1 ; †hLv‡b a > o, n
n
n
n
M −48
90. y-Gi gvb KZ?
N − 52
(ga¨g)
b b
ii. ( ) = ; †hLv‡b a, b , b> 0 Ges n
n K 3, 2 L 1, 4  4, 8 N −2, 0
a a
n am 91. x = KZ? (ga¨g)
iii. (am) = ; †hLv‡b a Ges n 3
an  2, 3 L 1, 9 M 3, 4 N −2, −
wb‡Pi †KvbwU mwVK? (ga¨g) 2
x x
 i I ii L i I iii M ii I iii N i, ii I iii x = (x x)x
86. i. 2 Zg g~j‡K eM©g~j e‡j Dc‡ii Z‡_¨i Av‡jv‡K 92 I 93bs cÖ‡kœi DËi `vI :
ii. −27 Gi Nbg~j 3 92. x Gi gvb KZ? (ga¨g)
iii. 0 Gi n Zg g~j 0 2 3 2 5
K  M− N
wb‡Pi †KvbwU mwVK? (ga¨g) 3 2 3 2
K i I ii  i I iii M ii I iii N i, ii I iii 93. x Gi gvb wb‡Pi †KvbwU? (KwVb)
87. i. a2 = a hLb a > 0 K−
2
L
3

9
N
29
3 2 4 8
ii. a2 = −a hLb a < 0

wbe©vwPZ enywbe©vPwb cÖ‡kœvËi

3 1
94. hw` ( 3)x + 5 = ( 3)2x + 5 Gi gvb KZ? 99. (2 − x) 3 2 n‡j x Gi gvb KZ?
5 −5 K 6 −6
K 25  5 M N
7 4 M 0 N−7
y y 100. cÖ‡Z¨K abvZ¥K ev¯Íe msL¨v a Gi GKwU Abb¨ abvZ¥K x
95. y = (y y )y nq n‡e y Gi gvb wb‡Pi †KvbwU? Zg g~j i‡q‡Q| G‡K wb‡Pi †Kvb cÖZxK Øviv cÖKvk Kiv
3 4 7 9
K
2
L
9
M
4

4
hvq?
x m x n x a
96. () ()
y

y
m
Gi gvb †KvbwU? 
M ax
a L
N ax
x

m+n 101. a R Gi x  N n‡j a x + 1 = KZ?

K
(xy) n

m−n

() x
y
n−m
K ax + a L ax − a
ax
M
(xy) N
() x
y
 ax. a

24
N
a
x x
97. x = (x x) x n‡j, x Gi gvb KZ? 102. a8 a6 a4 Gi mij gvb KZ?
7 8 9 1
K 4 L M 
2 3 4
K a12 L a12
98. 3mx −1 = 3amx − 2 ; a > 0, a  3 I m  0 n‡j, x Gi gvb
KZ?  a N1

m 2 103. a  R , a  0 n‡j,
K 
2 m
i. a = 1
M 2m N 2m
 beg-`kg †kÖwY : D”PZi MwYZ  360
1 M i I iii N i, ii I iii
n
ii. a−n
=a
n mn
hw` x = a nq Z‡e
n

iii. (am) = a
Dc‡ii Z‡_¨i Av‡jv‡K 105 I 106 bs cÖ‡kœi DËi `vI :
wb‡Pi †KvbwU mwVK?
K i I ii L ii I iii 105. n = 5 nq n‡j, wb‡Pi †KvbwU mwVK?
 i I iii N i, ii I iii K x = a5 L x= a
104. am  an = am + n n‡j, wb‡Pi †Kvb k‡Z© GwU mwVK? 5 n
 x= a N x=a
i. a  R. a = 0
106. DÏxcKwU wb‡Pi †Kvb k‡Z© mwVK n‡e?
ii. m, n  N, m > n
K a  R, n R  a  R, n  N
iii. a  R, m, n  N
M n  N, n  1 N a  R, n < 1
wb‡Pi †KvbwU mwVK?
K i I ii  ii I iii

¸iæZ¡c~Y© m„Rbkxj cÖkœ I mgvavb

cÖkœ-1  ax = by = cz, †hLv‡b a  b  c. z z
p p GLb, c2 = ab = cx. cy
K. hw` pp = (p p) nq, Z‡e p Gi gvb wbY©q z z
Ki| 2 +y
1 1 2 ev, c2 = cx .
L. hw` ab = c2 nq, Z‡e cÖgvY Ki †h, x + y = z 4 z z
ev, 2 = x + y
1 1 1 3
M. abc = 1 n‡j, cÖgvY Ki †h, x3 + y3 + z3 = xyz 4 1 1
 1bs cª‡kœi mgvavb  ( )
ev, z x + y = 2
1 1 2
p p p  + = (cÖgvwYZ)
K. kZ©g‡Z, p = (p p) x y z
1 M. †`Iqv Av‡Q, ax = by = cz
ev, p ( ) = (p )
1+ 1 p
p 2 1+
2 awi ax = by = cz = k
1
3 3
ev, p
p2
=p
2p  ax = k ev, a = kx
1
3
ev,
32
p = p ax = k ev, b = ky
2 1
3
p 2 cz = k ev, c = kz
3
ev, p = 2 GLb, abc = 1
3 1 1 1
−1 3
ev, p
2
=2 ev, kx, ky, kz = k
1 1 1
1 + +
ev, p =
2 3 ev, kx y z
= k
2 1 1 1
9 ev, x + y + z = 0
 p = (Ans.)
4
1 1 1 3
L. †h‡nZz ax = cz
z
(
ev, x + y + z = 0 )
1 1 1 3
ev, a = c x ev, x3 + y3 + z3 − xyz = 0
Avevi, by = cz 1 1 1 3
z  3+ 3+ 3=
x y z xyz
(cÖgvwYZ)
ev, b = cy

A byk xj b g ~j K K v‡R i A v‡j v‡K m„R b kxj cÖk œ I mg vavb

œ 2  a  R Ges m, n  N n‡j, (am)n = amn
cÖk-
 beg-`kg †kÖwY : D”PZi MwYZ  361
K. n = 1 Gi Rb¨ evK¨wUi mZ¨Zv hvPvB Ki| 2 myZivs m, n  N Gi Rb¨ evK¨wU mZ¨|
L. MvwYwZK Av‡ivn c×wZ‡Z †`LvI †h, (am)n = amn 4  n = 1 Gi Rb¨ (i) mZ¨
M. a  0 Ges m  N I n  Z n‡j, †`LvI †h, (am)n = amn 4 GLb awi, n = k Gi Rb¨ (i) mZ¨|
A_©vr am. ak = am + k ......... (ii)
 2bs cª‡kœi mgvavb  Zvn‡j, am.ak + 1 = am(ak . a) [ m~Î 1]
= (am. ak) a [¸‡Yi mn‡hvRb]
K. mN †K wbw`©ó K‡i Ges n †K PjK a‡i †Lvjv evK¨
= am + k.a [Av‡ivn Kíbv]
(am)n = ........... (i) we‡ePbv Kwi|
amn = am+ k + 1 [1bs m~Î]
(i) G n = 1 ewm‡q †`Lv hvq, A_©vr n = k + 1, Gi Rb¨ (i) mZ¨|
evgcÿ = (am)l = am myZivs MvwYwZK Av‡ivn c×wZ Abyhvqx mKj n N Gi Rb¨
Wvbcÿ = am . 1 = am (i) mZ¨|
 †h‡Kv‡bv m, n  N Gi Rb¨ am. an = am+n
 n = 1 Gi Rb¨ (i) mZ¨|
L. n = 1 Gi Rb¨ (i) mZ¨| [ÔKÕ n‡Z cvB]
(†`Lv‡bv n‡jv)
awi, n = k Gi Rb¨ (i) mZ¨ M. (i) m > 0 Ges n < 0
A_©vr (am)k = amk........................(ii) awi, n = −k †hLv‡b k  N
GLb, (am)k +1 = (am)k . (am) [ an+ 1 = an. a] Ges m  N
= amk.am [(ii) bs n‡Z] am. an = am.a−k [cÖwZ¯’vcb]
1 1
= amk+ m = am(k +1) = am. k [  a −n = n ]
a a
 n = k + 1 Gi Rb¨I (i) mZ¨|
am
myZivs MvwYwZK Av‡ivn c×wZ Abymv‡i mKj n  N Gi = k = am−k
a
Rb¨ (i) mZ¨| (†`Lv‡bv n‡jv) 1 1
M. ÔLÕ †_‡K cvB, (am)n = amn ................ (i) wKš‘, ak −m = a − (k − m) = am −k [ a−n = an ]
GLv‡b, a  0 Ges m  N I n Z  mKj †ÿ‡ÎB am. an = am − k = am + ( −k)
cÖ_‡g g‡b Kwi, n > 0 G‡ÿ‡Î L †_‡K (i) Gi mZ¨Zv = am + n [gvb ewm‡q]
¯^xKvi K‡i †bIqv n‡q‡Q| (mZ¨Zv hvPvB Kiv n‡jv)
GLb g‡b Kwi, n = 0 G‡ÿ‡Î (am)n = (am) = a= 1 Ges ii) m < 0 Ges n < 0
amn = a0 = 1 awi, m = − p, n = − q †hLv‡b p, q  N
 (i) bs mZ¨| am. an = a −p. a −q
Avevi g‡b Kwi, n < 0 Ges n = − k †hLv‡b k N 1 1 1
1 1 = p q [ a −n = n ]
a a a
G‡ÿ‡Î (am)n = (am)−k = (am)k = amk = a− mk = am(−k) = amn
1
 a  0 Ges m  N I n  Z Gi Rb¨ (am)n = amn (†`Lv‡bv n‡jv) = p+q [ am  an = am + n]
a
œ 3  a  0 Ges m, n  Z Gi Rb¨ am. an = am + n
cÖk- =a
−(p + q)
=a
−p−q
=a
− p + (− q)

K. n = 1 Gi Rb¨ evK¨wUi mZ¨Zv hvPvB Ki| 2 =a m + n [ gvb ewm‡q ] (mZ¨Zv hvPvB Kiv n‡jv)
L. MvwYwZK Av‡ivn c×wZ‡Z †`LvI †h, m, nN Gi œ 4  KwZcq m~PK mgwš^Z ivwk ay 1− p, by 1 − q, cy 1 − r Ges
cÖk-
Rb¨ evK¨wU mZ¨| 4 ay 1 −p = by1−q = cy 1 − r = x|
M. (i) m > 0 Ges n < 0 (ii) m < 0 Ges n < 0 Gi
K. a, b I c Gi gvb x, y Gi gva¨‡g cÖKvk Ki| 2
Rb¨ evK¨wUi mZ¨Zv hvPvB Ki| 4
L. aq − r  b r − p  c p − q Gi gvb wbY©q Ki| 4
 3bs cª‡kœi mgvavb  pa a2 + ab + b2 pb b2 + bc + c2
K. n =1 n‡j,
M. †`LvI †h, pb ()  ()
pc
evgcÿ = am.an = am.al = am.a = am +1 pc c2 + ca + a2

Wvbcÿ = am + n = am + 1 ()
 a
p
= aq − r  b r − p  cp − q 4
myZivs n = 1 Gi Rb¨ evK¨wU mZ¨|  4bs cª‡kœi mgvavb 
L. ÔK' n‡Z m = n = 1 Gi Rb¨ evK¨wU mZ¨|
myZivs m = n = k Gi Rb¨ mZ¨ n‡e K. †`Iqv Av‡Q, ay1 − p = by 1 − q = cy1− r = x
 ay 1 −p = x
 ak.ak = ak + k x
= a 2k ............. (i) ev, a = y1 −p
m = n = k + 1 Gi Rb¨ evK¨wU mZ¨ n‡e hw` I †Kej hw`
 a = xyp −1
ak + 1. ak + 1 = ak + 1 + k + 1
= a2k + 2 Avevi, by1 − q = x
= a2(k+1) ......................(ii) x
ev, b = y 1 − q = xyq −1
(i) I (ii) n‡Z †`Lv hvq k Gi Rb¨ evK¨wU mZ¨ n‡j k + 1
Gi Rb¨ evK¨wU mZ¨| Ges cy1 − r = x
 beg-`kg †kÖwY : D”PZi MwYZ  362
x 12
ev, c = y1 − r = xyr −1 = (a8) a6 a4  [1 − 1 {1 − (1 −
x3)−1}−1]−1
 a = xyp −1, b = xyq−1, c = xyr −1
= a  [1 − 1 {1 − (1 − x3)−1}−1]−1
L. ÔKÕ †_‡K cvB, a = xyp − 1, b = xyq − 1 Ges c = xyr − 1 −1
1 −1
= a  1 − 1 1 −
 aq − r.br − p.cp − q = (xyp − 1)q − r.(xyq−1)r − p.(xyr − 1)p − q 

= xq − ry(p − 1)(q − r).xr − py(q − 1) (r − p).xp − qy(r −1)(p − q)
  1−x 3
−1
− x3 − 1−1
= a  1 − 1 
= xq − r + r − p + p − q.y pq − pr − q + r + qr − pq − r + p + pr − qr − p + q  1

= x0.y0   1−x   3
−1
= 1  1 = 1 (Ans.)  − x3 
= a  1 − 1  
M. ÔLÕ n‡Z cvB, Wvbcÿ = aq−r  br − p  cp − q = 1  1 − x3
2 2 2 2 2 2 −1
1 − x3−1
= a  1 − 
a + ab + b b + bc + c c + ca + a
pa pb pc
evgcÿ = ()
p b 
p ()
c 
pa ()   −x   3
−1 −1
p  p − a) (c + ca + a ) 1 − x3 x3 +1 − x3
2 2 2 2 2 2

= a  1 + =a
(a−b) (a + ab + b ) (b−c) (b + bc + c ) (c
=p
=p a3
− b 3
p b 3
− c3
p c3
− a3
 x 
3
 x3 
3 3 −1
= pa − b + b −c + c − a
3 3 3 3
1
= p = 1 = Wvbcÿ
0 =a [ ] = ax = Wvbcÿ
x3
3

 1g ivwg  2q ivwk = ax3 (†`Lv‡bv n‡jv)

2 2 2 2 2 2
pa a + ab + b pb b + bc + c pc c + ca + a
 () pb pc
 () 
pa () M. GLv‡b, 1g ivwk  2q ivwk  [x − {x−1 + (a−1 − x)−1}−1]
= aq−r  br − p  cp − q (†`Lv‡bv n‡jv) −1 −1
= ax3  x −  +
1 1  
œ 5
cÖk-
12
(a8) a6 a4, [1 − 1 {1 − (1 − x3)−1}]−1 `yBwU ivwk|
 x a − x   ( ) [ÔLÕ †_‡K]
1 − ax−1−1
= ax3  x −  + 
1
K. cÖ_g ivwki mij gvb KZ? 2  x  a   
L. †`LvI †h, 1g ivwk  2q ivwk = ax3 4 a −1
= ax3  x −  +
1
M. 1 ivwk  2q ivwk  [x − {x−1 + (a−1 − x)−1}−1]  x 1 − ax 
1 − ax + ax−1
Gi gvb wbY©q Ki| 4 = ax3  x − 
  x (1 − ax)  
 5bs cª‡kœi mgvavb 
 1 −1
= ax3  x − 
12 12 12  x − ax2 
K. (a8) a6 a4 = (a8) a6.a2 = (a8) a8
= ax3  [x −{x − ax2}]
= ax3  [x − x + ax2]
12 12 12
= (a8) (a4)2 = a8.a4 = a8 + 4
12
1 = ax3  ax2
12 12
= a12 = (a ) = a ax3
wb‡Y©q mij gvb a = 2 = x (Ans.)
ax
12
L. ÔKÕ †_‡K cvB, (a8) a6 a4 = a
Zvn‡j evgcÿ = 1g ivwk  2q ivwk

AwZwi³ m„R bkxj cÖkœ I mgvavb

1 1 1 1 1 1
œ 6  xb + x−c +1, xc + x −a + 1 Ges xa + x−b + 1 wZbwU
cÖk- = xb + x−c + 1 + xc +x−a + 1 + xa + x−b + 1
m~PKxq ivwk| =
xc
+
1
+
1
1 + xc + xb+c 1 1
K. Z…Zxq ivwkwUi mij Ki| 2 xc + a
+ 1 x b +1
xa x
L. ivwk wZbwUi †hvMdj wbY©q Ki| 4 x c 1 1
M. †`LvI †h, (a + b + c) = 0 n‡j ivwk wZbwUi †hvMdj = + +
1 + xc + xb+c xa.xc + 1 + xa xa.xb + 1 + xb
1. 4 xa xb
 6bs cª‡kœi mgvavb  xc xa xb
=1 + xc + xb+c + 1 + xa + xa + c + 1 + xb + x a + b (Ans.)
1 1 1 xb
K. xa + x−b + 1 = 1
= a b
x .x + 1 + xb = 1 + xb + x a + b (Ans.) M. †h‡nZz, a + b + c = 0
xa b + 1 ev, b + c = − a
x xb
L. ivwk wZbwUi †hvMdj  ivwk wZbwUi †hvMdj
 beg-`kg †kÖwY : D”PZi MwYZ  363
1 1 1
= xb + x−c + 1 + xc +x−a + 1 + xa + x−b + 1  MvwYwZK Av‡ivn wewa Abymv‡i mKj n Gi Rb¨ (a.b)n
xc 1 1 = an.bn (†`Lv‡bv n‡jv)
= + +
1 + xc + xb+c 1 1 n
xc +
a

x c
xa
1
+ 1 x b +1
x
1
M. GLv‡b, a ( ) = a1 ............ (i)
1
n

= + + 1
1 + xc + xb+c xa.xc + 1 + xa xa.xb + 1 + xb
xa xb cÖ_g avc : n = 1 Gi Rb¨ (i) Gi evgcÿ = a ( ) = 1a
1

xc xa xb 1 1
=1 + xc + xb+c + 1 + xa + xa + c + 1 + xb + x a + b Wvbcÿ = =
a1 a
xc 1 xb  n = 1 Gi Rb¨ (i) evK¨wU mZ¨|
= + +
1 + xc + xb+c 1 + xc + x b+ c 1+ xb + x− c
wØZxq avc : awi, n = k Gi Rb¨ (i) evK¨wU mZ¨|
[∵ a + b + c = 0, ∵ a + b = − c] k

=
x c
+
1
1 + x c + xb + c 1 + x c + xb + c
+
xb
1
( ) = a1
A_©vr, a
1
k

1 + xb + c k+1 k
x 1 1 .1
=
xc
+
1
+
xb+c
1 + xc + xb +c 1 + xc + x b+c 1 + xc + xb+c
GLb, n = k + 1 n‡j, a () =
a ()
a
1 .1 1 1
1 + xc +xb +c = k = k = k+1
= a a a .a a
1 +xc + xb+c k+1
=1
 a + b + c = 0 n‡j cÖ`Ë ivwk wZbwUi †hvMdj 1. (†`Lv‡bv n‡jv)  (1a) =
1
ak+1
œ 7  a, b  N Ges an, n N n‡j MvwYwZK Av‡ivn c×wZ‡Z
cÖk-  n = k + 1 Gi
Rb¨ (i) evK¨wU mZ¨|
†`LvI †h,  MvwYwZK Av‡ivn c×wZ Abymv‡i mKj n Gi Rb¨
m n
K. (a ) = a mn
2 n

L. (a.b)n = anbn
n
4 (1) = a1
myZivs a n (†`Lv‡bv n‡jv)
1 1 1
M. (1a) = a1 †hLv‡b, a > 0
n œ 8
4 cÖk- + +
1 + a−mbn+a−mcp 1 + b−ncp + b−nam 1 + c−pam + c−pbn
 7bs cª‡kœi mgvavb  K. cÖ`Ë ivwki cÖ_g As‡ki mijxKiY Ki| 2
m n
K. GLv‡b, (a ) = amn L. cÖ`Ë ivwki mij gvb †ei Ki| 4
b b+c c+a
cÖ_g avc : (i) bs G n = 1 ewm‡q cvB, M. †`LvI †h, cÖ`Ë ivwki mij gvb  c
x
  a
xc
evgcÿ = (am)1 = am x  x 
Wvbcÿ = am.1 = am a a+b
 xb Gi mij gv‡bi mgvb| 4
 n = 1 Gi Rb¨ (i) bs evK¨wU mZ¨| x 
wØZxq avc : awi, n = k Gi Rb¨ (i)bs evK¨wU mZ¨|  8bs cª‡kœi mgvavb 
k
 (am) = amk 1
k+1 k K. cÖ`Ë ivwki cÖ_g Ask = 1 + a−mbn + a−mcp
GLb, (am) = (am) am
am(k +1) = amk + m = am(k + 1) am
=
 n = k + 1 Gi Rb¨ (i) bs evK¨wU
mZ¨|
m
a (1 + a−mbn + a−mcp)
m
a
 MvwYwZK Av‡ivn c×wZ Abymv‡i mKj n Gi Rb¨ =
am + am.a−mbn + am.a−m.cp
(am)n = amn (†`Lv‡bv n‡jv) am
= m (Ans.)
a + bn + cp
L. GLv‡b, (a.b)n = an.bn .................. (i)
L. ÔK' n‡Z cvB,
cÖ_g avc : n = 1 n‡j (i) ev‡K¨i evgcÿ = (a.b)1 = a.b
am
Wvbcÿ = a1.b1 = a.b cÖ`Ë ivwki cÖ_g As‡ki mij gvb = am + bn + cp
 n = 1 Gi Rb¨ (i) evK¨wU mZ¨| bn
Abyiƒcfv‡e wØZxq As‡ki mij gvb = am + bn + cp
wØZxq avc : awi, n = k Gi Rb¨ (i) evK¨wU mZ¨|
A_©vr, (a.b)k = ak.bk .................... (ii) cp
Ges Z…Zxq As‡ki mij gvb = am + bn + cp
GLb, (a.b)k + 1 = (a.b)k.(a.b)1
cÖ`Ë ivwk,
= ak.bk.a1.b1 1 1 1
= ak + 1.bk + 1 + +
1+ a−mbn + a−mcp 1+ b−ncp + b−nam 1+c−pam + c−pbn
 n = k + 1 Gi Rb¨ (i) evK¨wU mZ¨|
 beg-`kg †kÖwY : D”PZi MwYZ  364
am bn cp 1
+y+z
1 1
= p+ m p+ m x
a + b + c a + b + c a + bn + cp
m n n
ev, k = k0
am + bn + cp 1 1 1
= m n p = 1 (Ans.)  + + =0
x y z
(cÖgvwYZ)
a +b +c
1 1 1
M. ÔLÕ n‡Z cvB cÖ`Ë ivwki mij gvb 1. Avevi, x + y + z = 0
b b+c c+a a+b 1 1 1
GLb, xc
x
  a
xc
 xb
xa ev, x + y = − z
  x    3 3
1 1 1
= (xb −
2
c b+c
2
)
2
 (xc − a)c + a  (xa − b)a + b
2 2 2
( ) ( )
ev, x + y z
[Nb K‡i]
= −
= xb − c  xc − a  xa − b 1 1 1 1 1 1 1
2 2
= x b −c + c − a + a − b
2 2 2 2
x y ( )
ev, 3 + 3 + 3 . x . y x + y = − 3
z
= x0 = 1 hv cÖ`Ë ivwki mij gv‡bi mgvb|(†`Lv‡bv n‡jv) 1 1 1 1 1
œ 9  ax = by = cz; †hLv‡b a  b  c.
cÖk-
( )
ev, 3 + 3 + 3 . xy − z = − 3
x y z
1 1 1 1
y y ev, x3 + y3 − 3 . xyz + z3 = 0
y z z−1
K. b = z Ges c = y n‡j †`LvI †h, ( ) =y
z
2 1 1 1
x y z
3
 3 + 3 + 3 = xyz (†`Lv‡bv n‡jv)
L. a, b Ges c ci¯úi wZbwU abvZ¥K ALÊ msL¨v n‡j
1 1 2 3 3 3
cÖgvY Ki †h, x + y = z œ 10  x a + y b + z c = 0 Ges a2 = bc.
4 cÖk-
1 1 1 3 3
M. abc = 1 n‡j †`LvI †h, x + y + z = 0 Ges y a− c
K. a  0 Ges x + y + z = 0 n‡j †`LvI †h, z = 2
3 3
1 1 1
+ + =
3
4 b− a
x3 y3 z3 xyz L. †`LvI †h, ax3 + by3 + cz3 = 3axyz 4
 9bs cª‡kœi mgvavb  1
3
−
1
3
M. a = 2 + 2 Ges xyz = 1 n‡j †`LvI †h,
K. b = z Ges c = y n‡j
6(by3 + cz3) = (2a3 − 5)(3 − x3) 4
cÖ`Ë kZ©g‡Z, zy = yz .............. (i)
y
y y y y  10bs cª‡kœi mgvavb 
z z z z
Zvn‡j, z (y) = (y) = (y)z
y
z
1
y z
=
y
1
z z
=
y
y1
3
K. †`Iqv Av‡Q, x a + y b + z c = 0 ........ (i)
3 3

(z) (z ) (y ) Ges x + y + z = 0 ........... (ii)

y y
−1 (ii) bs mgxKiY †_‡K cvB, x = − (y + z)
 z = (y) z
(†`Lv‡bv n‡jv)
yz
(i) bs mgxKi‡Y x Gi gvb ewm‡q cvB,
L. †`Iqv Av‡Q, a = by = cz
x
3 3 3
− (y + z) a + y b + z c = 0
g‡b Kwi, ax = by = cz = k
3 3 3 3
1 ev, − y a − z a + y b + z c = 0
Zvn‡j, x
a = k ...................... (i) 3 3 3 3
1 ev, y b − y a = z a − z c
y
b = k ...................... (ii) 3 3 3 3
1 ev, y( b − a) = z( a − c)
z
c = k ...................... (iii) 3 3
y a− c
GLb †h‡nZz a, b Ges c wZbwU abvZ¥K ALÊ msL¨v z= (†`Lv‡bv n‡jv)
3 3
 b2 = ac b− a
1 1 1
2
L. †`Iqv Av‡Q,
ev, (k ) =k . k
y x z
3 3 3
2 1 1 x a+y b+z c=0
y x +z
ev, k = k 3 3 3
2 1 1
ev, (x a + y b) = −z c .................... (i)
ev, y = x + z 3 3 3 3 3
ev, (x a + y b) = (−z c) [Nb K‡i]
1 1 2
 + = (cÖgvwYZ) 3 3 3
x z y ev, x3a + y3b + 3xy ab (x a + y b) = −z3c
M. cÖ`Ë kZ©, abc = 1 ev, x3a + y3b + z3c + 3xy ab (−z c) = 0 [(i) †_‡K]
3 3
1 1 1
x y z 3
ev, k .k .k = 1 ev, x3a + y3b + z3c + 3xyz (− abc) = 0
 beg-`kg †kÖwY : D”PZi MwYZ  365
3 ev, x3 = 2a + 3x.c
ev, ax3 + by3 + cz3 − 3xyz a.a2 = 0
 x3 − 3cx − 2a = 0 (†`Lv‡bv n‡jv)
3
ev, ax3 + by3 + cz3 − 3xyz a3
a
3
2 b
2
3  a
1
32 b
1
23
 ax + by + cz = 3axyz (†`Lv‡bv n‡jv)
3 3 3 M. evgcÿ = b () () () +
a
=
 b
 +
  a ( ) 
1 1
M. †`Iqv Av‡Q,
=  3 +  2 
 a 2  b 3
3 2

1
−
1 b  a 
3 3 1 1
a=2 +2 ................ (i)
=  2 +  3 [ a2 = b3]
 a 2  b 3
3 2
1 1
−
3
ev, a3 = (2 + 2 ) 3 3
[Nb K‡i] a  b 
1 1 1 1
1 1 1 1 1 1 −
− − − −2 2
3 3 3 3
ev, a3 = (2 ) + (2 ) + 3.2 .2 (2 + 2 ) 3 3 3 3 = (a3 ) + (b2 − 3)3 = a2 + b 3

1
ev, a3 = 2 + 2−1 + 3.20.a [(i) †_‡K] =a +
2 1
1 = a+
1
= Wvbcÿ
1 3
ev, a = 2 + 2 + 3a
3 3 b
b
ev, 2a3 = 4 + 1 + 6a  evgcÿ = Wvbcÿ (cÖgvwYZ)
ev, 2a3 = 5 + 6a œ 12  GKwU m~PKxq ivwk we‡ePbv Ki,
cÖk-
ev, 6a = 2a3 − 5
 3 3  3 3 3 3
1 1 2 1 1 2
2a3 − 5
a= 6
a − b  a + a . b + b ; a, b > 0
K. ivwkwUi mv‡_ b †hvM K‡i mijxKiY Ki| 2
ÔLÕ bs †_‡K cvB, 2
3
ax3 + by3 + cz3 = 3axyz L. ÔKÕ †_‡K cÖvß mijgvbwUi eM© mgvb −2 + 3 +
ev, ax3 + by3 + cz3 = 3a.1 [ xyz = 1] −
2
3
ev, by + cz = 3a − ax
3 3 3 3 n‡j †`LvI †h, 3a3 + 9a − 8 = 0 4
2 1
ev, by3 + cz3 = a (3 − x3) 3 3
2a − 5 2a − 5
M. ÔKÕ †_‡K cÖvß mijgvbwU 1 + 3 + 3 Gi mgvb
ev, by3 + cz3 = 6 (3 − x3)  a = 6 
3 3
n‡j †`LvI †h, a3 − 3a2 − 6a − 4 = 0 4
 6(by3 + cz3) = (2a3 − 5)(3 − x3) (†`Lv‡bv n‡jv)  12bs cª‡kœi mgvavb 
1 1 K. cÖ`Ë ivwkwUi mv‡_ b †hvM Ki‡j `uvovq,
œ 11  a > 0 Ges a  0, x = (a + b) + (a − b) Ges a2 = b3
cÖk- 3 3
1 1 2 1 1 2
3 3 3 3 3 3
K. †`LvI †h, a0 = 1 2 (a − b ) (a + a .b + b ) + b
1 1 1 1 1 1
L. hw` a2 − b2 = c3 nq, Z‡e †`LvI †h, x3 − 3cx − 2a = 0 4 3 3 3 3 3 3
= (a − b ) {(a )2 + a .b + (b )2} + b
3 2 1 1
a 2 b 3 1
M. cÖgvY Ki †h, () () b
+
a
= a+
3
b
4
=a−b+b
3
= (a )3 − (b )3 + b
3

 11bs cª‡kœi mgvavb  = a (Ans.)

L. ÔKÕ †_‡K cÖvß gvb a
1−1
K. a = a
0
2 −2
3 3
1 −1
= a .a [m~P‡Ki †gŠwjK m~Î a m + n m
= a .a n
]  a = −2 + 3 + 3
2

 2  
1 1 2
1 a − 1 −1
= a. = = 1 ev, a2 = 3  + 3  − 2.33.3
3 3 3
a a
 a =1 (†`Lv‡bv n‡jv)
0
 3 − 32
1 1

ev, a = 3 − 3 
2
L. †`Iqv Av‡Q, 1 1
1 1 −
3 3 ev, a = 33 − 3 3
[eM©g~j K‡i]
x = (a + b) + (a − b) ............... (i)
 
1 1 3
1 1 −
ev, a3 = 3 − 3 
3 3
ev, x3 = {(a + b)3 + (a − b)3}3 [Nb K‡i] [Nb K‡i]
   − 33 3 3
− 3
1 1 1 1 1 3 1 −1 1 1 1

ev, a = 3    − 3.3 .3 3 − 3 
3 3
3
ev, x = (a + b) + (a − b) + 3(a + b) (a − b) {(a + b) + (a − b) }
3 3 3 3 3
− 3
1 1 −1
1
ev, x = 2a + 3(a − b ) .x [(i) †_‡K]
3 2 2 3 ev, a = 3 − 3 − 3.a [ a = 33 − 3
3 3
]
1
3 3
ev, 3a3 = 9 − 1 − 9a
ev, x3 = 2a + 3x(c ) [ a2 − b2 = c3]
 3a3 + 9a − 8 = 0 (†`Lv‡bv n‡jv)
 beg-`kg †kÖwY : D”PZi MwYZ  366
M. ÔKÕ †_‡K cÖvß gvb a ev, y − 2 = 1 + 2.1 y −9 + y −9
2 1
3 3 ev, y −2 −y + 9 −1 = 2 y −9
a=1+3 +3
ev, 6 = 2 y −9
 3 33
2 1

ev, (a − 1) = 3 + 3  [cÿvšÍi Kivi ci Nb K‡i]

3 ev, y −9 = 3
 33  33 32 31  3 3
2 1 2 1 ev, ( y −9 )2 = 9
ev, a −1−3a + 3a = 3  + 3  + 3.3 .3 .3 + 3 
3 2
ev, y −9 = 9
2 1  y = 18
ev, a3 − 3a2 + 3a − 1 = 32 + 31 + 3.31.(a − 1) [ a − 1 = 33 + 33] ev, 2x2 + 5x = 18 [y Gi gvb ewm‡q]
ev, a3 − 3a2 + 3a − 1 = 9 + 3 + 9a − 9 ev, 2x2 + 5x −18 = 0
 a3 − 3a2 − 6a − 4 = 0 (†`Lv‡bv n‡jv) ev, 2x2 + 9x − 4x − 18 = 0
4x + 7 2x + 7 ev, x(2x + 9) −2(2x + 9) = 0
œ 13 
cÖk- ( 4) 5
= ( 11
64 ) Ges 2x2 + 5x −2 − ev, (2x + 9) (x −2) = 0
2x2 + 5x −9 =1, `yBwU mgxKiY ev, 2x = − 9 A_ev x − 2 = 0
−9
K. 1g mgxKiYwU‡K am = an AvKv‡ii cÖKvk Ki| 2 ev, x = 2 ev, x = 2
L. 2q mgxKiYwU mgvavb Ki| 4 −9
M. mgxKiY؇qi †Kv‡bv mvaviY g~j Av‡Q wKbv Zv wb‡Y©q mgvavb x = 2 , 2
wba©viY Ki| 4 4x + 7 6x + 21

 13bs cª‡kœi mgvavb  M. ÔKÕ n‡Z cvB, 4 5

= 4 11
4x + 7 6x + 21
4x + 7 2x + 7 ev, 5 = 11
K. ( 4)
5
= ( 11
)
64
ev, 44x + 77 = 30x + 105
1 2x + 7

ev, (4 )
1 4x + 7
5 = {(64) } 11 ev, 44x − 30x = 105 − 77
ev, 14x = 28
4x + 7 6x + 21
ev, 4 5
=4 11
x=
28
 am = an AvKv‡i †`Lv‡bv n‡jv| 14
=2
L. 2x2 + 5x −2 − 2x2 + 5x −9 =1 wb‡Y©q mgvavb x = 2
ev, y −2 − y −9 = 1 [2x2 + 5x = y a‡i]  mgxKiY؇qi g‡a¨ GKwU mvaviY g~j Av‡Q Ges Zv n‡”Q x
ev, y −2 = 1 + y −9 =2
ev, ( y −2 )2 = (1 + y −9 )2 [eM© K‡i]

wbe©vwPZ m„Rbkxj cÖkœ I mgvavb

œ 14 
cÖk- awi, xa = bb = zc = k
K. hw` ax = b, by = c Ges cz = 1 nq, Z‡e xyz =  xa = k
1
KZ? 2 x = ka ...............(1)
L. hw` xx = yb = zc Ges xyz = 1 nq, Z‡e ab+bc+ca yb = k
= KZ? 4 1

x ev, y = kb ...............(2)
M. hw` 9x = (27)y nq, Zvn‡j y
Gi gvb KZ? 4 zc = k
1
 14bs cª‡kœi mgvavb  ev, z = kc ...............(3)
K. †`Iqv Av‡Q, ax = b ........................(1) (1)  (2)  (3) n‡Z cvB
by = b ........................(2) 1 1 1
cz = 1 .........................(3) xyz = ka. kb. kc
(i) n‡Z cvB, ax = b 1 + 1 +1

ev, (ax)y = (b)y ev, 1 = ka b c

ab + bc + ca
ev, (ax)y = (b)y ev, k = k abc
ev, axy = c [(2) n‡Z] ab + bc + ca
ev, axy = cz ev, 0 = abc
ev, axyz = a  ab + bc + ca = 0
 xyz = 0 M. †`Iqv Av‡Q, 9x = (27)y
L. †`Iqv Av‡Q, xa = yb = zc Ges xyz = 1 ev, (32)x = (33)y
 beg-`kg †kÖwY : D”PZi MwYZ  367
ev, 32x = 33y x=2
ev, 2x = 3y hLb, x = 2
x 3 ZLb y2 = 22
 =
y 2 ev, y2 = 4
a2−b2
a y=2
 1 a+b
Avevi, hLb, x = −2
œ 15  GKwU m~PKxq ivwk we‡ePbv Kwi,
cÖk-
 a
 x () a−b 
 ZLb, y−2 = (−2)2
K. ivwkwU‡K mijxKiY Ki| 2 1 1
1 1 ev, y2 = 4 [a−m = am]
−
L. cÖ`Ë ivwkwU 23 + 2 3 n‡j Z‡e †`LvI †h, 2x3 − 1
6x = 58. 4 ev, y2 = 4
1 1 1
M. cÖ`Ë ivwkwU (a + b)3 + (a − b)3 Ges a2 − b2 = c3 y=
2
Z‡e †`LvI †h, 2x − 6cx = 4a Ges a I c Gi
3

†Kvb gv‡bi Rb¨ L I M †_‡K cÖvß mgxKiY GKB

1
wb‡Y©q mgvavb (x, y) = (2, 2), (2, − 2), −2 2 ( ) (−2 −21)
mgxKiY wb‡`©k K‡i| 4 M. †`Iqv Av‡Q, wØZxq mgxKiY †RvU,
 15bs cª‡kœi mgvavb  yx = 4 ...................... (iii)
a
y2 = 2x .................... (iv)
 1 a2−b2 (iv) bs n‡Z cvB,
a+b
K. DÏxc‡K cÖ`Ë ivwkwU n‡jv, ()
 a
 x
a−b 

a
y2 = 2x
ev, (y2)x = (2x)x [Dfqc‡ÿi NvZ x G DbœxZ K‡i]
 1  (a−b) (a+b)a+b ev, y2x = 2x
2
 a (a−b) 
= x  ev, (yx)2 = 2x
2
1 a
a  (a+b)  a+b ev, (4)2 = 2x [(iii) bs n‡Z yx Gi gvb ewm‡q]
2
=x
ev, 16 = 2x
2
= x (Ans.)
1 −1 ev, 2x = 24
2

L. cÖ`Ë ivwkwU, x = 23 + 2 3 ; †`Lv‡Z n‡e †h, 2x3 − 6x = 5 ev, x2 = 4 [am = an n‡j m = n]

Abykxjbx 9.1 Gi 7(M) cÖ‡kœvËi `ªóe¨| x=2
1 1 (iii) bs G x Gi gvb ewm‡q cvB,
M. cÖ`Ë ivwk, x = (a + b)3 + (a − b)3 Ges a2 − b2 = c3, †`Lv‡Z hLb, x = 2, ZLb y2 = 4
n‡e †h, 2x3 − 6cx = 4a y=2
Gici : Abykxjbx 9.1 Gi 7(L) cÖ‡kœvËi `ªóe¨| Avevi hLb, x = −2 ZLb
ÔLÕ n‡Z cÖvß mgxKiY 2x3 − 6x = 5 I 2x3 − 6cx = 4a mgxKiY y−2 = 4
5 1
GKB n‡e hw` c = 1 Ges 4a = 5 ev, a = 4 nq| (Ans). ev, y2 = 4
1
yx = x2 yx = 4 ev, y2 = 4
} }
œ 16  x2x = y4 Ges y2 = 2x , y  1 `yBwU `yB PjKwewkó
cÖk-
1
m~PKxq mgxKiY| y=
2
K. m~PK mgxKiY Kv‡K e‡j? 2 1 1
L. cÖ_g mgxKiY †Rv‡Ui mgvavb wbY©q Ki| 4
wb‡Y©q mgvavb (x, y) = (2, 2), (2, −2), −2 2 −2 −2 ( )( )
M. †`LvI †h, wØZxq mgxKiY †Rv‡Ui mgvavb cÖ_g myZivs, wØZxq mgxKiY †Rv‡Ui mgvavb cÖ_g mgxKiY †Rv‡Ui
mgxKiY †Rv‡Ui mgvav‡bi mgvb| 4 mgvav‡bi mgvb| (†`Lv‡bv n‡jv)
 16bs cª‡kœi mgvavb  1
−
1 2
−
2

cÖkœ-17  a = 23 + 2 3 Ges b2 + 2 = 33 + 3 3, b  0.
K. m~PK mgxKiY : m~PK I wfwË m¤^wjZ mgxKiY‡K m~PK 1
−
1

yx = 4 K. wØZxq mgxKiY †_‡K †`LvI †h, b = 33 − 3 3. 2

mgxKiY e‡j| †hgb : y2 = 2x , y  1 } L. cÖgvY Ki †h, 3b3 + 9b = 8 4
L. †`Iqv Av‡Q, cÖ_g mgxKiY †RvU, M. cÖ_g mgxKiY †_‡K †`LvI †h, 2a3 − 6a = 5. 4
yx = x2 .................... (i)
x2x = y4 ................... (ii)  17bs cª‡kœi mgvavb 
(ii) bs n‡Z cvB, 2
−
2

x2x = y4 K. wØZxq mgxKiY, b2 + 2 = 33 + 3 3.

ev, (x2)x = y4 2
−
2

ev, (yx)x = y4 [(i) bs n‡Z x2 Gi gvb ewm‡q] ev, b2 = 33 + 3 3 − 2

1 2 −1 2 1 1
ev, yx2 = y4 = 3( ) + (3 ) − 2.3 . 3
3 3 3
−
3

ev, x2 = 4 [ am = an n‡j m = n]
 beg-`kg †kÖwY : D”PZi MwYZ  368
1
−
1
= xq − r − 1 + r − p −1 + p − q − 1. y(p − 1) (q − r − 1) + (q − 1) (r − p − 1) + (r − 1) (p − q − 1)
(
= 3 −3
3
)
3
= x−3.ypq − pr − p − q + r + 1 + qr − pq − q − r + p + 1 + pr − qr − r − p + q + 1
1
−
1
= x−3.y3−(p + q + r) [ p + q + r = 3]
b=3 −3
3 3
(†`Lv‡bv n‡jv) = x−3.y3 − 3 = x−3.y0
1
−
1 = x−3 = Wvbcÿ (†`Lv‡bv n‡jv)
L. ÔKÕ n‡Z cvB, b = 33 − 3 3 [ b  0 †h‡nZz abvZ¥K gvb wb‡q] M. †`Iqv Av‡Q, p + q + r = 3
1 1 pq + qr + rp = 3
= (3 − 3 )
3 − 3
ev, b3
3
[Dfqcÿ‡K Nb K‡i] a−2 b−2 c−2
1 3 1 3 1 1 1 1
cÖ`Ë ivwk = ap + 1 bq + 1 cr + 1
ev, b3 = 3 ( ) − (3 ) − 3.3 .3 (3 −3 )
3
−
3 3
−
3 3
−
3

=
(xyp − 1)−2  (xyq − 1)−2  (xyr − 1)−2
(xyp − 1)p + 1  (xyq − 1) q + 1  (xyr − 1)r + 1
[ (a − b)3 = a3 − b3 − 3ab (a − b)]
x−2.y−2p + 2.x−2. y−2q + 2.x−2.y−2r + 2
ev, b3 = 3−3−1 − 3.30.b [(a − b)3 = a3 − b3 − 3ab (a − b)] = p + 1 p2 − 1 q + 1 q2 − 1 r + 1 r2 − 1
x .y x y x .y
2 2
1 = x−2 − 2 −2 − p − 1 − q − 1 − r − 1y−2p + 2 − 2q + 2 − 2r + 2 − p + 1 − q + 1 − r + 1
2
ev, b3 = 3 − 3 − 3b 2 + q2 + r2)
=x −9 − (p + q + r) y 9 − 2(p + q + r) − (p
2
ev, b3 + 3b = 3
8 = x−9−3  y9 − 2.3 − {(p + q + r) − 2(pq + qr + rp)} [ p + q + r = 3]
[ p + q + r = 3 Ges pq + r + rp = 3]
2 − 2.3}
=x y
−12 9 − 6 − {(3)

 3b3 + 9b = 8 (cÖgvwYZ) = x−12 . y3 − (9 − 6)

1
−
1 = x−12.y0
M. †`Iqv Av‡Q, a = 23 + 2 3 = x−12
a−2.b−2.c−2
 p + 1 q + 1 r + 1 = x−12 (Ans.)
1 1 3
ev, a3 = (23 + 2 3)
−
[Dfqcÿ‡K Nb K‡i] a .b .c
1 3 1 3 1 1 1 1 a a2 + ab + b2 b b2 + bc + c2 pc c2 + ca + a2
ev, a3 = (2 ) + (2 ) + 3.2 .2 (2 + 2 )
− − −
3 3 3 3 3 3
cÖkœ-19  (pp ) b , (pp ) c pa ()
[ (x + y)3 = x3 + y3 + 3xy (x + y)] 2 x −y 2 y −z 2 z −x
p(x + y)  p(y + z)  p(z +x) 
ev, a3 = 21 + 2−1 + 320a  xy  ,  yz  ,  zx 
 p   p   p 
1 1 1 1

[ 2 .2 3
−
3
= 23
1
−
1
3
= 20
−
Ges 23 + 2 3 = a ] K. 1g I 4_© ivwki gvb wbY©q Ki| 2
pa a2 + ab + b2 pb b2 + bc + c2 pc c2 + ca + a2
ev, a3 = 2 + 2 + 3a
1 L. () pb
 c
p
() (p ) a

4 + 1 + 6a Gi gvb wbY©q Ki| 4

ev, a3 = x −y y −z z −x
2  (x + y) 2   (y + z) 2
  (z +x) 2 
p p p
ev, 2a3 = 4 + 1 + 6a M. †`LvI †h,  pxy  .  pyz  .  pzx  = 1 4
 2a3 − 6a = 5 (†`Lv‡bv n‡jv)
 19bs cª‡kœi mgvavb 
œ 18  a = xyp − 1, b = xyq − 1 Ges c = xyr − 1 nq, Zvn‡jÑ
cÖk-
K. †`Iqv Av‡Q,
3
K. p + q + r = 3 n‡j †`LvI †h, abc = x 2 a a2 + ab + b2
L. †`LvI †h, aq − r − 1. br − p − 1.cp − q − 1 = x−3 hLb p + q + r = 3 4
1g ivwk = pb (p )
a−2b−2c−2
M. p + q + r = 3, pq + qr + rp = 3 n‡j ap + 1 bq + 1cr + 1
2 + ab + b2
= (pa − b)a
2 2)
Gi gvb wbY©q Ki| 4 = p(a − b) (a + ab + b
 18bs cª‡kœi mgvavb 
3 3
= Pa −b (Ans.)
p  (x + y)2 x − y px + 2xy + y 
2 2 x−y
K. †`Iqv Av‡Q, a = xyp−1, b = xyq − 1 Ges 4_© ivwk =  pxy  = 
 pxy 
Ges c = xyr − 1 2 + 2xy − xy + y2) (x −y)
 abc = xyp −1. xyq − 1. xyr − 1 = p(x
= x 1 + 1 + 1. y P + q + r − 1 − 1 − 1
2 + xy + y2) (x − y)
= p (x
= x3.y(p + q + r) − 3 = px
3 − y3 (Ans.)
= x3. y3 − 3 [p + q + r = 3]
pa a2 + ab + b2 pb b2 + bc + c2 2
pc c + ca + a
2
= x3.y0
= x3.1
L. () pb
 () pc
 ()
p a

= x3 = (pa − b)(a + ab + b )
2 2
 (p b − c)(b + bc + c ) 
2 2
(pc − a)(c
2
+ ca + a2)

ev, abc = x3 = p(a − b) (a + ab + b ) p(b − c) (b

2 2 2
+ bc + c2) p(c − a) (c + ca + a )
2 2

3 3 − b3 3 − c3 3 − a3
 abc = x (†`Lv‡bv n‡jv) = pa  pb  pc
L. evgcÿ = aq − r − 1. br − p − 1.cp − q − 1
3 3 3 3 3 3
= pa − b + b −c +c −a

= (xyp − 1)q − r − 1. (xyq − 1)r − p − 1. (xyr−1)p−q−1 = p0

= xq − r − 1. y(p − 1) (q − r − 1). xr − p − 1. y(q − 1)(r − p − 1). xp − q − 1. y(r − 1) (p − q − 1) = 1 (Ans.)
 beg-`kg †kÖwY : D”PZi MwYZ  369
p  (x + y)2 x − y 1 1 1
M. ÔKÕ n‡Z cvB,  pxy  = px
3 − y3 ev, x + y + z = 0

p  (y + z)2 y−z 3 3
ev, x−1 + y−1 + z−1 = 0 (†`Lv‡bv n‡q‡Q)
Abyiƒcfv‡e,  pyz  = py −z
1 1 1
Avevi, x + y + z = 0
2 z−x
p(z + x)  1 1 1
Ges  pzx  = pz3 − x3 ev, x + y = −z

(1) (1) + 31x1y (1x + 1y) = −1z

3 3
2 x−y 2 y−z 2 z−x
p(x + y)  p(y + z)  p(z + x)  ev, x + y [Nb K‡i]
  xy    yz    zx  3
 p   p   p 
−1
ev, x3 + y3 + 3xy  z  = − z3
1 1 1 1 1 1 1
= px
3 − y3
 py
3 − z3
 pz
3 − x3 [ + = − ]
x y z
= px − y + y − z + z − x = p0 = 1 1 1 1 3
3 3 3 3 3 3
ev, x3 + y3 + z3 = xyz
p(x + y)  p(y + z)  p(z + x) 
2 x−y 2 y−z 2 y−z
A_©vr  pxy   pyz   pzx  = 1 (†`Lv‡bv n‡jv)  x−3 + y−3 + z−3 = 3(xyz)−1 (†`Lv‡bv n‡jv)
œ 20  hw` = = †hgb a  b  c Ges
cÖk- ax by c2, n‡j, cÖkœ-21  ax = bx = cz, †hLv‡b, a, b I c abvZ¥K I ci¯úi
92R = 3R + 1 [[

K. R Gi gvb wbY©q Ki| 2 Amgvb Ges x, y, z  N.

a 32 b 23 1 K. 92x = 3x + 1 n‡j x Gi gvb KZ? 2
L. x = 2 Ges y = 3 nq Z‡e †`LvI †h, b + a = a +
3
4 () ()
L. b2 = ac n‡j, cÖgvY Ki †h, x−1 + z−1 = 2y−1
b
4
1 1 1 1 1 1 3
M. abc = 1 n‡j †`LvI †h, x−1 + y−1 + z−1 = 0 Ges M. abc = 1 n‡j, †`LvI †h, x + y + z = 0 Ges x3 + y3 + z3 = xyz 4
x−3 + y−3 + z−3 = (3xyz)−1 4  21bs cª‡kœi mgvavb 
 20bs cª‡kœi mgvavb  K. †`Iqv Av‡Q, 92x = 3x + 1
K. GLv‡b, = 92R 3R + 1 ev, (32)2x = 3x + 1
ev, (32)2R = 3R + 1 ev, 34x = 3x + 1
ev, 34R = 3R + 1 ev, 4x = x + 1
ev, 4R = R + 1 ev, 4x − x = 1
ev, 3R = 1 ev, 3x = 1
1
1 x= (Ans.)
 R = (Ans.) 3
3
L. Aby-9 Gi D`vniY 11 bs `ªóe¨| c„ôv-184|
L. †`Iqv Av‡Q, ax = by = cx M. awi, ax = by = cz = k
GLv‡b, x = 2, y = 3 n‡j cvB, a2 = b3 GLv‡b, ax = k
 a = b3/2, b = a2/3 1 1 1
 a = kx Abyiƒcfv‡e, b = ky Ges c = kz
a 3
( ) ( ) = (aa ) + bb 
2 3 2
b
evgcÿ = b 2 + a 3
2/3
2
3
3
[gvb ewm‡q] 1
 abc = kx  ky  kz
1 1

2 1 1 1
+ +
1
3 2 ev, 1 = kx y z [ abc = 1]
= a ( ) + (b ) = a + 1
3
2 1
3
3 1
2
1
1 1 1
+ +
ev, k0 = kx y z
b3 1 1 1
1  + + = 0 (†`Lv‡bv n‡q‡Q)
x y z
= a+ = Wvbcÿ 1 1 1
3
b GLb, x + y + z
3 2
a b 1 1 1 1
 b () ()= 2
+
a
3
a+ (†`Lv‡bv n‡jv) ev, x + y = −z
3
b −1 3
ev, x + y =  z  [Nb K‡i]
1 1 3
M. †`Iqv Av‡Q, ax = by = cz †hLv‡b, a  b  c ( )
awi, ax = by = cz = k 1 1 11 1 1 1
 ax = k by = k cz = k
ev, x3 + y3 + 3.x y x + y = −z3 [(i) ( ) e¨envi K‡i]
1 1 1 1 1 1 1
a=k x
b=k y
c=k z ev, x3 + y3 − 3xyz = −z3
GLb, abc = 1 1 1 1 3
1 1 1  3+ 3+ 3=
x y z xyz
(†`Lv‡bv n‡jv)
ev, kx  ky  kz = 1 [gvb ewm‡q]
1 1 1
+ +
ev, kx y z
= k0
 ................................................................................ cÖ_g Aa¨vq  A_©bxwZ cwiPq..................................................................................370

m„R bkxj cÖkœe¨vsK DËimn

1
−
1 2 −2 L. x I y -Gi gvb †ei Ki| 4
œ 22  a = 23 + 2 3 Ges b2 + 2 = 33 + 3 3 , b > 0
cÖk- −
1
+
1
−1 2 3 x
1 M. 4a − 3a = 3 2 −22a−1 n‡j,
a †`LvI †h, a = x A_ev a = 2 4
K. wØZxq mgxKiY †_‡K †`LvI †h, b = 33 2
−33,b>0
L. cÖgvb Ki †h, 3b2 + 9b = 8 4 DËi : L. (x, y) = (3, 8), (2, 4)
M. cÖ_g mgxKiY †_‡K †`LvI †h, 2a3 − 6a = 5 4 1
−
1 2
−
2
œ 23  (i) 3x − 9y (ii) 5x+y+1 = 25xy (iii) 8yx − y2x = 16, 2x = y2
cÖk- 2
œ 27  a = 23 + 2 3 Ges a + 2 + 33 = 3 3, b > 0
cÖk-
K. (i) n‡Z x †K y Gi gva¨‡g †`LvI| 2 1 1
L. (i) I (i) n‡Z (x, y) wbY©q Ki| 4 K. wØZxq kZ© †_‡K †`LvI †h, b = 33 − 3− 3 2
M. (iii) bs †K mgvavb K‡i wØPjK wØNvZ wKbv Zv eywS‡q `vI| 4
DËi : K. x = 2y ; L. cÖgvY Ki †h, 3b + 9b = 8
3 4
M. cÖ _ g kZ© †_‡K †`LvI †h, 2a3 − 6a = 5 4
L. (2, 1), −1 2 4
(−1
; ) m n
œ 28  a .a = (a ) Ges m, n  0
cÖk- m n
− 2
(
M. 2  2
2
) , − 2 2 ( , wØPjK wØNvZ| ) K. †`LvI †h, m + n − mn = 0 2
1 1 2 1 1 2 L. cÖ gvY Ki †h, m(n − 2) + n(m − 2) = 0 4
cÖk- (
œ 24  x − y
3 3
)(
3
x +x y +y
3 3 3
)
: x.y > 0 GKwU m~PKxq M. †`LvI †h, m(n − 2) + n(m − 2) = 0 mgxKiYwU wm× n‡e
ivwk| Gi mvnv‡h¨ wb‡Pi mgm¨v¸‡jvi mgvavb Ki| hw` I †Kej hw` m = n = 2 nq| 4
K. ivwkwUi mv‡_ y †hvM Ki‡j mij dj KZ n‡e? 2
2 −2 œ 29  a = b , a = b, b = c Ges c = a
cÖk- b a p q r

L. ÔKÕ n‡Z cÖvß mij gvbwUi eM© mgvb 3 − 3 − 2 n‡j 3 3 a a

−1
†`LvI †h, 3x3 + 9x = 8 4 K. ab = ba n‡j †`LvI †h, a b = ab 2 ()
2 1 b
M. ÔKÕ n‡Z cÖvß mij gvbwU 1 + 33 + 33 n‡j †`LvI †h, x3 − 3x2 − 6x = 4. 4 L. cÖgvY Ki †h, pqr = 1 4
DËi : K. m„Rbkxj cÖkœ 14 Gi Abyiƒc| M. ax = p, ay = q Ges az = (pyqx)z nq Z‡e xyz = 1 cÖgvY Ki| 4
œ 25  a = xyp−1; b = xyq−1; c = xyr −1
cÖk- 2
−
2
œ 30  a = xyp−1, b = xyq−1, c = xyr − 1 Ges z2 + 2 = 33 + 3 3
cÖk-
K. p + q + r = 3 n‡j abc = KZ? 2 †hLv‡b, z  0|
L. †`LvI †h, aq−r.br−p.cp−q = 1 4 K. p + q + r = 3 n‡j abc = KZ? 2
M. p + q + r = 3, pq + qr + rp = 3 n‡j, a .b .c = KZ? 4 L. †`LvI †h, aq − r. br − p. cp − q = 1.
p+1 q+1 r+1
4
DËi : K. abc = x3; L. x6 M. cÖgvY Ki †h, 3z + 9z − 8 = 0
3 4
x x
œ 26  y = 2 Ges 4 − 3.2
cÖk-
x +2 5
+ 2 = 0 n‡j DËi : K. x 3

K. cÖgvY Ki y2 − 12y + 32 = 0 2

A b yk xj b x 9 .2
cvV m¤úwK©Z MyiæZ¡c~Y© welqvw`
 jMvwi`g : Logos Ges arithmas bvgK `ywU wMÖK kã n‡Z jMvwi`g kãwUi DrcwË| Logos A_© Av‡jvPbv Ges arithmas A_© msL¨v
A_©vr, we‡kl msL¨v wb‡q Av‡jvPbv|
msÁv : hw` ax = b nq, †hLv‡b a > 0 Ges a  1, Z‡e x †K ejv nq b Gi a wfwËK jMvwi`g, A_©vr, x = logab
AZGe, ax = b  x = logab
wecixZµ‡g, hw` x = logab  ax = b n‡e|
G‡ÿ‡Î b msL¨vwU‡K wfwË a Gi mv‡c‡ÿ x Gi cÖwZjM (anti-log arithm) e‡j Ges Avgiv wjwL b = anti loga x
hw` loga = n nq, Z‡e a †K n Gi cÖwZjM ejv nq A_©vr, loga = n n‡j a = anti log n.
 jMvwi`‡gi m~Îvewj
1. logaa = 1 Ges loga1 = 0 2. loga(M  N) = logaM + logaN 5. logaM = logbM  logab
3. loga(M)N = N loga M 4. loga N = logaM − logaN (M)
 ciggvb : GKwU ivwk abvZ¥K A_ev FYvZ¥K hvB †nvK bv †Kb abvZ¥K wPýhy³ gvb‡K H ivwki ciggvb ejv nq| †hgb : †h
†Kv‡bv ev¯Íe msL¨v x Gi gvb k~b¨, abvZ¥K ev FYvZ¥K wKš‘ x Gi ciggvb memgqB k~b¨ ev abvZ¥K| x Gi ciggvb‡K |x|Øviv
cÖKvk Kiv nq| ciggvb wbgœwjwLZfv‡e msÁvwqZ Kiv hvq|
 beg-`kg †kÖwY : D”PZi MwYZ  371
x hLb x > 0

| x | =  0 hLb x = 0
−x hLb x < 0
†hgb: |0| = 0, |3| = 3, |−3| = −(−3) = 3
ciggvb dvskb : hw` xR nq, Z‡e
x hLb x > 0
= −x hLb x < 0

y = (x) = |x| †K ciggvb dvskb ejv nq|
 †Wv‡gb = R Ges †iÄ R = [0, ]
dvsk‡bi †Wv‡gb I †iÄ wbY©q :
†h‡nZz cÖ‡Z¨K dvskb GKwU Aš^q| myZivs dvsk‡bi †Wv‡gb Ges †iÄ ej‡Z Aš^‡qi †Wv‡gb Ges †ićKB †evSv‡e|
AZGe y = (x) dvsk‡bi (x,y) µ‡gv‡Rvo¸‡jvi x Gi Gi gb‡K †Wv‡gb Ges y Gi gvb‡K †iÄ e‡j|
weKí c×wZ‡Z dvsk‡bi †iÄ wbY©q :
mvaviYfv‡e †Wv‡gb wbY©q AwaKZi mnR| †Kv‡bv dvsk‡bi †Wv‡gb I †iÄ h_vµ‡g wecixZ dvsk‡bi †iÄ I †Wv‡gb|
A_©vr, g~j dvsk‡bi †Wv‡gb = wecixZ dvsk‡bi †iÄ
Avevi, g~j dvsk‡bi †iÄ = wecixZ dvsk‡bi †Wv‡gb|

Abykxjbxi cÖkœ I mgvavb

evgcÿ = logkbn + logkcn + logkan
a an bn cn
 1 a − b a − b      
2 2

  a + b 
= logk n n n
n n n
a b c
1. xa  Gi mijgvb †KvbwU? b c a 
K0 L1 Ma x = logk1 = 0 = Wvbcÿ (†`Lv‡bv n‡jv)
2. hw` a, b, p > 0 Ges a  1, b  1 nq, Z‡eÑ
i. logaP = logbP  logab (a) (b)
(L) logk(ab)logk b + logk(bc)logk c + logk(ca)logk a = 0 ( c)
ii. loga a  logb b  logc c Gi gvb 2 mgvavb :
iii. x
logay
=y
logax

Dc‡ii Z‡_¨i Av‡jv‡K wb‡Pi †KvbwU mwVK?

( a)
evgcÿ = logk(ab)logk b + logk(bc)logk c + logk(ca)logk a (b) (c)
= (logka + logkb)(logka − logkb) +
K i I ii L ii I iii  i I iii N i, ii I iii
3 - 5 bs cÖ‡kœi DËi `vI hLb x, y, z  0 Ges ax = by = cz (logkb + logkc)(logkb − logkc) +
3. †KvbwU mwVK? (logkc + logka)(logkc − logka)
y z z = (logka)2 − (logkb)2 + (logkb)2 −
b2
Ka=b
z L a=c
y a=c
x N a
c
(logkc)2 + (logkc)2 − (logka)2
z = 0 = Wvbcÿ (†`Lv‡bv n‡jv)
e¨vL¨v : ax = cz a=c
x
(M) log b  log c  log a = 8
b2 b2 a b c
†bvU : a  m¤úKwUI mZ¨; KviY, a, Gi mgvb bq|
c c
mgvavb :
4. wb‡Pi †KvbwU ac Gi mgvb?
y y y z y z z y
evgcÿ = log b  log c  log a
+ + a b c
x z x y x y y z
 b .b L b .b Mb Nb = log ( b)2  log ( c)2  log ( a)2
5. b2 = ac n‡j wb‡Pi †KvbwU mwVK? a b c
1 1 2 1 1 2 = 2 log b  2 log c  2 log a
 + = L + = a b c
x z y x y z
b  log c  log
1 1 2
M + =
y z x
1 1 z
N + =
x y 2
= 8 log
a ( b c
a
)
= 8 log b  log a
cÖkœ \ 6 \ †`LvI †h, a b

(K) logk n + logk n  + logk n = 0

n n n
a b c = 8 log a
b c a a

mgvavb : = 8.1 [ logaa = 1]

= 8 = Wvbcÿ (†`Lv‡bv n‡jv)
 beg-`kg †kÖwY : D”PZi MwYZ  372
 a b ev, (x + y) logkc = p (x − y) (x + y)
(N) loga loga logaaa =b x+y
ev, logkc = p(x2 − y2) ............... (iii)
mgvavb : GLb, (i) + (ii) + (iii) n‡Z cvB,
y+z z+x x+y
 a b ev, logka logkb logkc = p(y2 − z2 + z2 − x2 + x2 − y2)
evgcÿ = loga loga logaaa  y+z z+x
ev, logk(a . b . c ) = p.0
x -y

ab ( ) y+z z+x x+y

= loga loga a logaa [ loga xr = r logax] ev, logk(a . b . c ) = 0
y+z z+x x+y
= loga(a ) logaa  1 [ logaa = 1]
b ev, logk(a . b . c ) = logk1
y+z z+x x+y
= b logaa  1 a b c = 1 (†`Lv‡bv n‡jv)
y2 + yz + z2 z2 + zx + x2 x2 + xy + y2
=b1 2. a .b .c =1
=b mgvavb :
= Wvbcÿ (†`Lv‡bv n‡jv) logka logkb logkc
g‡b Kwi, y − z = z − x = x − y = p
log a log b log c
cÖkœ \ 7 \ (K) hw` b −kc = c −ka = a −kb nq, Z‡e †`LvI †h, aabbcc = 1 logka
Zvn‡j, y − z = p
mgvavb :
ev, logka = p(y − z)
logka logkb logkc
g‡b Kwi, = = =p ev, (y2 + yz + z2) logka = p(y − z)(y2 + yz + z2)
b−c c−a a−b 2 2
ev, logka y + yz + z
= p(y3 − z3) ................ (i)
 logka = p(b − c)
logkb
ev, a logka = pa(b − c) [Dfqcÿ‡K a Øviv ¸Y K‡i] Avevi, z − x = p
ev, logkaa = p (ab − ac) ...... (i) ev, logkb = p(z − x)
ev, logkb = p(c − a) ev, (z2 + zx + x2) logkb = p(z − x)(z2 + zx + x2)
2 zx 2
 b logkb = pb(c − a) [Dfqcÿ‡K b Øviv ¸Y K‡i] ev, logkbz + + x = p (z3 − x3) .............. (ii)
ev, logkbb = p(bc − ab) .................... (ii) logkc
Ges x − y = p
logkc = p(a − b)
ev, logkc = p(x − y)
 c logkc = pc(a − b) [Dfqcÿ‡K c Øviv ¸Y K‡i]
ev, (x2 + xy + y2) logkc = p (x − y) (x2 + xy + y2)
ev, logkcc = p(ac − bc) ...................... (iii) x2 + xy + y2
 log kc = p (x3 − y3) ............... (iii)
GLb, (i) + (ii) + (iii) n‡Z cvB,
GLb, (i) + (ii) + (iii) n‡Z cvB,
ev, logkaa + logkbb + logkcc = p(ab − ca + bc − ab + ca − bc) x2 + xy + y2
y2 + yz + z2 z2 + zx + x2
ev, logkaabbcc = 0 logka + logkb + logkc
 aabbcc = k0 = 1 (†`Lv‡bv n‡jv) = p(y3 − z3) + p(z3 − x3) + p(x3 − y3)
ev, logk (a y +yz + z .b z + zx +x .c x + zy + y ) = p(y3 − z3 + z3 − x3 + x3 −y3)
2 2 2 2 2 2

logka logkb logkc y2 + yz + z2 z2 + zx + x2 x2 + xy + y2

(L) hw` y − z = z − x = x − y nq, Z‡e †`LvI †h, ev, logk (a .b .c ) = p.0 = 0
y2 + yz + z2 z2 + zx + x2 x2 + xy + y2
1. ay + zbz + xcx + y = 1 ev, logk (a .b .c ) = logk1
y2 + yz + z2 z2 + zx + x2 x2 + xy + y2
mgvavb : a .b .c = 1 (†`Lv‡bv n‡jv)
logka logkb logkc
g‡b Kwi, y − z = z − x = x − y = p logk(1 + x) 1+ 5
(M) hw` logkx
= 2 nq, Z‡e †`LvI †h, x =
2
logka
Zvn‡j, y − z = p logk(1+x)
mgvavb : †`Iqv Av‡Q, log x = 2
ev, logka = p(y − z) k

ev, (y + z) logka = p(y − z)(y + z) ev, logk(1 + x) = 2 logkx

ev, logkay + z = p(y2 − z2)   (i) ev, logk(1 + x) = logkx2
logkb ev, 1 + x = x2
Avevi, z − x = p
ev, x2 − x = 1
ev, logkb = p(z − x) 2
1
ev, (z + x) logkb = p (z − x) (z + x)
z+x
ev, (x)2 − 2.x.2 + 2 ( ) − 41 = 1
1

ev, logkb = p(z2 − x2) ............... (ii)

1 2 1
logkc
Ges x − y = p
ev, x − 2( ) =1+
4

ev, logkc = p(x − y)

 beg-`kg †kÖwY : D”PZi MwYZ  373
2
( 1) = 45
ev, x − 2 (b)
ev, 2xlogk a = 2logka
b
ev, (x − 2) =  2   xlog (a) = log a (†`Lv‡bv n‡jv)
2 2
1 5
k k

1 5 a−1 b−1 c−1

ev, x − 2 =  2 (P) hw` xy = p, xy =q Ges xy =r nq, Z‡e †`LvI †h,
(b − c) logkp + (c − a) logkq + (a − b) logkr = 0
1 5
ev, x = 2  2 mgvavb : †`Iqv Av‡Q, xy = p
a −1

a −1
1 5 ev, logkxy = logkp [Dfq cv‡k logk wb‡q]
ev, x = 2 a−1
ev, logkx + logky = logkp
1+ 5 1− 5
ev, x = 2 A_ev, 2  logkx + (a − 1) logky = logkp ......... (i)
b −1
1− 5 Avevi, xy = q
GLv‡b x = 2
MÖnY‡hvM¨ bq| KviY x Gi FYvZ¥K
b −1
gv‡bi Rb¨ logx Gi †Kv‡bv gvb †bB| ev, logkxy = logkq
b−1
1+ 5 ev, logkx + logky = logkq
 x = 2 (†`Lv‡bv n‡jv)
ev, logkx + (b − 1) logky = logkq ............ (ii)
x − x2 − 1 c−1
Ges, xy = r
(N) †`LvI †h, log = = 2log (x − x2 − 1)
x+ x −1 2
c−1
ev, logkxy = logkr
mgvavb : c−1
x− x2 − 1
ev, logkx + logky = logkr
evgcÿ = log  logkx + (c − 1) logky = logkr ........... (iii)
x+ x2 − 1
(x − x − 1)(x −
2
x − 1)
2 GLb, evgcÿ = (b − c)logkp + (c − a)logkq + (a − b)logkr
= log = (b − c) {logkx + (a −1) logky} + (c − a) {logkx +
(x + x2 − 1)(x − x2 − 1)
[je I ni‡K (x − x2−1 Øviv ¸Y K‡i] (b −1) logky} + (a − b) {logkx + (c −1) logky}
= (b−c) logkx + (b − c)(a−1) logky + (c − a) logkx
(x − x2 − 1)2
= log + (c − a)(b −1) logky + (a − b) logkx + (a − b)(c−1) logky
x2 − ( x2 − 1) 2
= (b−c)logkx + (c−a)logkx + (a−b)logkx +
(x − x2 − 1)2
= log (b−c)(a−1)logky + (c−a)(b−1)logky + (a−b)(c−1)logky
x2 − x 2 + 1
2 = (b − c + c − a + a − b)logkx + (ab − b − ac + c +
= log (x − x2 − 1)
bc − c − ab + a + ac − a − bc + b) logky
= 2log (x − x−2 − 1) = 0  logkx + 0  logky
= Wvbcÿ (†`Lv‡bv n‡jv) =0
3 − x 5x 5 + x 3x = Wvbcÿ (†`Lv‡bv n‡jv)
(O) hw` a b =a b nq, Z‡e †`LvI †h, xlogk a = logka (b) ab logk(ab) bc logk(bc) ca logk(ca)
mgvavb : (Q) hw` a+b
=
b+c
=
c+a
nq, Z‡e †`LvI
3 − x 5x 5 + x 3x
†`Iqv Av‡Q, a b =a b †h, a = b = c
a b c

b 5x
a5+x ab logk(ab) bc logk(bc) ca logk(ca)
ev, b3x = mgvavb : a + b = b + c = c + a = p (awi)
a3 − x
5x − 3x
5 + x−3 + x ab logk(ab)
ev, b =a Zvn‡j, a + b = p
2x 2 + 2x
ev, b = a ev, ab logk(ab) = p(a + b)
2x 2 2x
ev, b = a .a p(a + b)
2x ev, logk(ab) = ab
b 2
ev, 2x =a p(a + b)
a  logka + logkb = ab .......... (i)
2x
(b) = a
ev, a
2
p(b + c)
Abyiƒcfv‡e, logkb + logkc = bc .................. (ii)
b 2x p(c + a)
ev, log (a) = log a [Dfq cv‡k log wb‡q] Abyiƒcfv‡e, logkc + logka = ca ................... (iii)
2
k k k

GLb (i) + (ii) + (iii) K‡i cvB,

 beg-`kg †kÖwY : D”PZi MwYZ  374
logka + logkb + logkb + logkc + logkc + logka p logkz
p(a + b) p(b + c) p(c + a)
ev, x + y − z = z .................... (iii)
= + +
ab bc ca GLb, (i) + (ii) + (iii) †_‡K cvB,
ev, 2(logka + logkb + logkc) p logkx p logky p logkz
y+z−x+z+x−y+x+y−z= + +
p(ca + bc) + p(ab + ca) + p(bc + ab) x y z
= p logkx p logky p logkz
abc
ev, x + y + z = x + y + z ...(iv)
ev, 2(logka + logkb + logkc) =
p(ca + bc + ab + ca + bc + ab) GLb (iv) bs †_‡K (i) bs we‡qvM K‡i cvB,
(x + y + z) − (y + z − x) =  
abc p logkx p logky p logkz p logkx
p(2ab + 2bc + 2ca)  x + y + z − x
ev, 2(logka + logkb + logkc) = abc p logky p logkz
ev, x + y + z − y − z + x = y + z
2p(ab + bc + ca)
ev, 2(logka + logkb + logkc) = abc pz logky + py logkz
ev, 2x = yz
p(ab + bc + ca)
ev, logka + logkb + logkc = abc
... (iv) ev, 2xyz = p logkyz + p logkzy
GLb,(iv) bs †_‡K (i) we‡qvM K‡i cvB, ev, 2xyz = p(logkyz + logkzy)
logka + logkb + logkc − logka − logkb = 2xyz
ev, p = logkyz + logkzy
p(ab + bc + ca) p(a + b)
− 2xyz
 p = logk(yz.zy) ............ (v)
abc ab
p(ab + bc + ca) − p(ca + bc) Avevi, (iv)-(ii) †_‡K cvB,
ev, logkc = abc p logkx p logky p logkz p logky
p(ab + bc + ca − ca − bc) x + y + z − z − x + y= + + −
ev, logkc = abc
x y z y
p logkx p logkz
pab ev, 2y = x + z
ev, logkc = abc
pz logkx + px logkz
p ev, 2y =
ev, logkc = c zx
ev, 2xyz = p(logkxz + logkzx)
ev, clogkc = p 2xyz
ev, p = logkxz + logkzx
 logkcc = p ................ (v)
2xyz
Avevi, (iv) bs †_‡K (ii) bs we‡qvM K‡i Abyiƒcfv‡e cvB,  p = logk(xz.zx) .................. (vi)
 logkaa = p ..................... (vi) Avevi, (iv) - (iii) bs †_‡K cvB,
Avevi, (iv) bs †_‡K (iii) bs we‡qvM K‡i Abyiƒcfv‡e cvB, p logkx p logky p logkz p logkz
−
x + y + z − x−y + z = + +
 logkbb = p ................. (vii) x y z z
p logkx p logky
GLb (v), (vi) I (vii) bs mgxKiY Zzjbv K‡i cvB, ev, 2z = x + y
logkcc = logkaa = logkbb py logkx + px logky
ev, 2z =
 aa = bb = cc (†`Lv‡bv n‡jv) xy
ev, 2xyz = p(logkxy + logkyx)
x(y + z − x) y(z + x − y) z(x + y − z)
(R) hw` logkx
=
logky
=
logkz
nq, Z‡e 2xyz
ev, p = logkxy + logkyx
†`LvI †h, xyyz = yzzy = zxxz 2xyz
mgvavb :  p = logk(xy.yx) .................. (vii)
x(y + z − x) y(z + x − y) z(x + y − z) GLb, (v), (vi) I (vii) bs Zzjbv K‡i cvB,
g‡b Kwi, logkx
=
logky
=
logkz
=p
logk(yz.zy) = logk(xz.zx) = logk(xy.yx)
x(y + z − x) ev, yz.zy = xz.zx = xyyx
Zvn‡j, log x = p
k ev, xyyx = yzzy = zxxz (†`Lv‡bv n‡jv)
ev, x(y + z − x) = p logkx [we: `ª: cvV¨eB‡q xyyz Gi cwie‡Z© xyyx n‡e]
p logkx
ev, y + z − x = x ....................... (i) cÖkœ \ 8 \ ÔjM mviwYÕ (gva¨wgK exRMwYZ `ªóe¨) e¨envi K‡i P
y(z + x − y) Gi Avmbœ gvb wbY©q Ki †hLv‡b,
Avevi, log y = p l
k (K) P = 2 g
†hLv‡b   3.1416, g = 981 Ges l = 25.5
p logky
ev, (z + x − y) = y .................... (ii) l
mgvavb : †`Iqv Av‡Q, p = 2 g
z(x + y − z)
Ges log z = p 25.5
k ev, p = 2  3.1416  981
 beg-`kg †kÖwY : D”PZi MwYZ  375
25.5 1
ev, p = 6.2832  ev, log p = 100 log 10 + 2 log e
981
1
ev, log p = log 6.2832 
25.5  ev, log p = 100 log 10 + 2 log 2.718
981 
1
1 ev, log p = 100  1 + 2  0.434249452 [log mviwY n‡Z]
25.5 2
ev, log p = log 6.2832  ( )
981 ev, log p = 100 + 0.217124726
1  log p = 100.217124726
ev, log p = log 6.2832  2 (log 25.5 − log 981) .. (i)
 ln p = 2.3026  100.217124726
GLb, log mviwY n‡Z cvB, = 230.76 (cÖvq) (Ans.)
1
log p = 0.79818 + (1.40654 − 2.99167) cÖkœ \ 10 \ †jLwPÎ A¼b Ki :
2
ev, log p = 0.79818 + 0.70327 − 1.495835 (K) y = 3x
ev, log p = 1.50145 − 1.495835 mgvavb : cÖ`Ë dvskb y = 3x
ev, log p = 0.005615 cÖ`Ë dvsk‡bi †jLwPÎ A¼‡bi Rb¨ x Ges y Gi gvb¸‡jvi
ev, p = anti log 0.005615 ZvwjKv ˆZwi Kwi :
 P = 1.01302 x −2 −1 0 1 2
myZivs P = 1.01302 (cÖvq) (Ans.) y 0.11 0.33 1 3 9
(L) p = 10000  e0.05t †hLv‡b e = 2.718 Ges t = 13.86 QK KvM‡Ri XOX eivei x Aÿ Ges YOY eivei y Aÿ
mgvavb : †`Iqv Av‡Q, p = 10000  e0.05t AuvwK| x Aÿ eivei ¶z`ªZg e‡M©i cuvP evûi ˆ`N©¨‡K GKK
ev, p = 10000  (2.718)0.05  13.86 Ges y Aÿ eivei ¶z`ªZg e‡M©i wZb evûi ˆ`N©¨‡K GKK a‡i
ev, log p = log {10000  (2.718)0.05  13.86 (−2, 0.11), (−1, 0.33), (0, 1), (1, 3), (2, 9) we›`y¸‡jv QK
ev, log p = log 10000 + log (2.718)0.05  13.86
KvM‡R ¯’vcb Kwi Ges mvejxjfv‡e †hvM K‡i †jLwPÎ A¼b
ev, log p = log 10000 + (0.05  13.86) log 2.718
Kwi|
ev, log p = 4 + 0.693  0.4342495 [log mviwY n‡Z]
ev, log p = 4 + 0.300934903
ev, log p = 4.300934903
ev, p = antilog 4.300934903
myZivs p = 19995.62 (cÖvq) [atilog mviwY n‡Z] (Ans.)
cÖkœ \ 9 \ lnP  2.3026  logP m~Î e¨envi K‡i lnP Gi Avmbœ
gvb wbY©q Ki, hLb − (K) P = 10000; (L) P = 0.00le2 (M) P =
10100  e
(K) p = 10000
mgvavb : †`Iqv Av‡Q, p = 10000
ev, log p = log 10000
ev, log p = 4 [log mviwY n‡Z]
GLb, lnp = 2.3026  4 = 9.2104 (cÖvq) (Ans.)
(L) p = 0.00le2
mgvavb : †`Iqv Av‡Q, p = 0.00le2 (L) y = − 3x
ev, log p = log 0.001e2 mgvavb : cÖ`Ë dvskb y = − 3x
ev, log p = log 0.001 + 2log 2.718 [ e  2.718]
cÖ`Ë dvsk‡bi †jLwPÎ A¼‡bi Rb¨ x Ges y Gi gvb¸‡jvi
ev, log p = − 3 + 2  0.434249452 [log mviwY n‡Z] ZvwjKv ˆZwi Kwi :
ev, log p = −3 + 0.868498904 x −1 0 1 2
 log p = −2.131501095 y −0.33 −1 −3 −9
 lnp = 2.3026  (−2.131501095) QK KvM‡Ri XOX eivei x Aÿ Ges YOY eivei y Aÿ
= − 4.90799 (cÖvq) (Ans.) AuvwK| x Aÿ eivei ¶z`ªZg e‡M©i cuvP evûi ˆ`N©¨‡K GKK
(M) p = 10100  e Ges y Aÿ eivei ¶z`ªZg e‡M©i wZb evûi ˆ`N©¨‡K GKK a‡i
(−1, −0.33), (0, −1), (1, −3), (2, − 9) we›`y¸‡jv QK KvM‡R
mgvavb : †`Iqv Av‡Q, p = 10100  e
¯’vcb Kwi Ges mvejxjfv‡e †hvM K‡i dvskbwUi †jLwPÎ
ev, log p = log (10100  e) A¼b Kwi|
ev, log p = log 10100 + log e
1
ev, log p = 100 log 10 + log e2
 beg-`kg †kÖwY : D”PZi MwYZ  376
QK KvM‡R ¯’vcb Kwi Ges mvejxjfv‡e †hvM K‡i dvskbwUi
†jLwPÎ A¼b Kwi|

x+1
(M) y = 3
mgvavb : cÖ`Ë dvskb y = 3x + 1 −x + 1
cÖ`Ë dvsk‡bi †jLwPÎ A¼‡bi Rb¨ x I y Gi gvb¸‡jvi (O) y = 3
−x + 1
ZvwjKv ˆZwi Kwi : mgvavb : cÖ`Ë dvskb y = 3
x −2 −1 0 1 2 cÖ`Ë dvsk‡bi †jLwPÎ A¼‡bi Rb¨ x I y Gi gvb¸‡jvi
y 0.33 1 3 9 27 ZvwjKv ˆZwi Kwi :
QK KvM‡R XOX eivei x Aÿ Ges YOY eivei y Aÿ x −2 −1 0 1 2
AuvwK| x Aÿ eivei ¶z`ªZg e‡M©i cuvP evûi ˆ`N©¨‡K GKK y 27 9 3 1 0.33
Ges y Aÿ eivei ¶z`ªZg e‡M©i cÖwZevûi ˆ`N©¨‡K GKK a‡i QK KvM‡Ri XOX eivei x Aÿ Ges YOY eivei y Aÿ
(−2, 0.33), (−1, 1), (0, 3), (1, 9) I (2, 27) we›`y¸‡jv QK AuvwK| x Aÿ eivei ¶z`ªZg e‡M©i cuvP evûi ˆ`N©¨‡K GKK
KvM‡R ¯’vcb K‡i mvejxjfv‡e †hvM K‡i dvskbwUi †jLwPÎ Ges y Aÿ eivei cÖwZ evûi ˆ`N©¨‡K GKK a‡i (−2, 27),
A¼b Kwi| (−1, 9), (0, 3), (1, 1), (2, 0.33) we›`y¸‡jv QK KvM‡R ¯’vcb
Kwi Ges mvejxjfv‡e †jLwPÎ A¼b Kwi|

(N) y = − 3x + 1
x−1
mgvavb : cÖ`Ë dvskb y = − 3x + 1 (P) y = 3
cÖ`Ë dvsk‡bi †jLwPÎ A¼‡bi Rb¨ x I y Gi gvb¸‡jvi mgvavb : cÖ`Ë dvskb y = 3
x−1

ZvwjKv ˆZwi Kwi :

cÖ`Ë dvsk‡bi †jLwPÎ A¼‡bi Rb¨ x I y Gi gvb¸‡jvi
x −2 −1 0 1 2
ZvwjKv ˆZwi Kwi :
y −0.33 −1 −3 −9 −27
x −1 0 1 2
QK KvM‡Ri XOX eivei x Aÿ Ges YOY eivei y Aÿ
y 0.11 0.33 1 3
AuvwK| x Aÿ eivei ¶z`ªZg e‡M©i cuvP N‡ii ˆ`N©¨‡K GKK
Ges y Aÿ eivei ¶z`ªZg e‡M©i cÖwZ evûi ˆ`N©¨‡K GKK a‡i QK KvM‡Ri XOX eivei x Aÿ Ges YOY eivei y Aÿ
(−2, −0.33), (−1, −1), (0, −3), (1, −9), (2, −27) we›`y¸‡jv AuvwK| QK KvM‡Ri ¶z`ªZg e‡M©i `k evûi ˆ`N©¨‡K GKK
a‡i (−1, 0.11), (0, 0.33), (1, 1), (2, 3) we›`y¸‡jv QK KvM‡R
 beg-`kg †kÖwY : D”PZi MwYZ  377
¯’vcb Kwi Ges mvejxjfv‡e †hvM K‡i dvskbwUi †jLwPÎ
A¼b Kwi|

cÖkœ \ 11 \ wb‡Pi dvsk‡bi wecixZ dvskb †jL Ges †jLwPÎ

0
A¼b K‡i †Wv‡gb I †iÄ wbY©q Ki| hLb x = 0 ZLb y = 1 − 2 = 1 − 1 = 0, Kv‡RB †jL †iLvwU
(K) y = 1 − 2
x (0, 0) we›`yMvgx|
x hLb x →  ZLb y → 1
mgvavb : cÖ`Ë dvskb, y = 1− 2
x
hLb, x → − ZLb y → −
ev, 2 = 1 − y  †Wv‡gb, D = (, − )
x
ev, 1 − y = 2 I †iÄ, R = (1 − ) (Ans.)
ev, log2(1 − y) = x
(L) y = log10x
ev, x = log2(1 − y) mgvavb : cÖ`Ë dvskb, y = log10x
ev, x = log2(1 − y)  x = 10y
ev, x = log21 + log2(1 − y) [ log21 = 0] -1
ev, f (y) = 10y
ev, x = log2 (11 − y) = log2 (1 − y) -1
 f (x) = 10
x

1
ev, − (y) = log2(1 − y) †jLwPÎ A¼b : cÖ`Ë dvsk‡bi †jLwPÎ A¼‡bi Rb¨ x I y
1 Gi gvb¸‡jvi ZvwjKv ˆZwi Kwi :
  (x) = log2 (1 − x)
−

x
x 0.5 1 2 3 4 5
†jLwPÎ A¼b : y = 1 − 2 Gi †jLwPÎ A¼‡bi Rb¨ x I y y −0.3 0 0.3 0.5 0.6 0.7
Gi gvb¸‡jvi ZvwjKv ˆZwi Kwi :
x −2 −1 0 1 2 3
y 075 05 0 −1 −3 3, −7
QK KvM‡Ri XOX eivei x Aÿ Ges YOY eivei y Aÿ Ges 0
g~jwe›`y| QK KvM‡Ri ¶z`ªZg e‡M©i Pvi evûi ˆ`N©¨‡K GKK
a‡i (−2, 075), (−1, 05), (0, 0), (1, −7), (2, −3)(3, −7)
we›`y¸‡jv QK KvM‡R ¯’vcb Kwi Ges mvejxjfv‡e †hvM K‡i
dvskbwUi †jLwPÎ A¼b Kwi| †jLwPÎ †_‡K †`Lv hvq †h,

g‡b Kwi, QK KvM‡Ri XOX eivei x Aÿ Ges YOY

eivei y Aÿ AuvwK Ges 0 g~jwe›`y| QK KvM‡Ri x Aÿ eivei
¶z`ªZg e‡M©i cÖwZ cuvP evûi ˆ`N©¨‡K GKK Ges y Aÿ eivei
cÖwZ `k evûi ˆ`N©¨‡K GKK a‡i (0.5, −0.3), (1, 0), (2, 0.3),
(3, 0.5), (4, 0.6), (5, 0.7) we›`y¸‡jv QK KvM‡R ¯’vcb Kwi
Ges mvejxjfv‡e †hvM K‡i dvskbwUi †jLwPÎ A¼b Kwi|
 beg-`kg †kÖwY : D”PZi MwYZ  378
†h‡nZz jMvwi`g ïay abvZ¥K ev¯Íe msL¨vi Rb¨ msÁvwqZ nq Avevi, awi, y = (x) = ln(x − 2)
Ges k~b¨‡Z AmsÁvwqZ| ev, ey = x − 2
 †Wv‡gb, D = (0, ) ev, x − 2 = ey
Avevi, †jLwPÎ n‡Z †`Lv hvq, ev, x = ey + 2
hLb, x → 0 ZLb y →  y Gi mKj ev¯Íe gv‡bi Rb¨ ey ev¯Íe|
hLb, x →  ZLb y →  d‡j, x = ey + 2 ev¯Íe|
 †iÄ, R = (− + )  †iÄ, R = R (Ans.)
1−x
(M) y = x2, x > 0 cÖkœ \ 13 \ (x) = ln 1 + x dvskbwUi †Wv‡gb Ges †iÄ wbY©q Ki|
mgvavb : cÖ`Ë dvskb, y = x , x > 0
2
mgvavb : Avgiv Rvwb, jMvwi`g ïay abvZ¥K ev¯Íe msL¨vi
awi, y = (x) = x2 Rb¨ msÁvwqZ nq|
ev, x = y; [x >0 nIqvq FYvZ¥K gvb MÖnY‡hvM¨ bq|] 1−x
 1 + x > 0 hw`
ev,  (y) = y
−1
(i) 1 − x > 0 Ges 1 + x > 0 nq|
ev, −1(x) = x A_ev, (ii) 1 − x < 0 Ges 1 + x < 0 nq|
†jLwPÎ A¼b : cÖ`Ë dvsk‡bi †jLwPÎ A¼‡bi Rb¨ x Ges ev, 1 > x Ges x > −1
y Gi gvb¸‡jvi ZvwjKv ˆZwi Kwi : ev, x < 1 Ges x > −1
x 1 2 3 4 ev, x >−1 Ges x < 1
y 1 4 9 16  †Wv‡gb = {x : x > − 1}  {x : x < 1}
= { − 1, }  { − , 1} = ( − 1, 1)
(ii) ev, 1 < x Ges x < − 1
ev, x < − 1 Ges x > 1
 †Wv‡gb = {x : x < − 1}  {x : x > 1} = 
 cÖ`Ë dvsk‡bi †Wv‡gb, D = (i) I (ii) †ÿ‡Î cÖvß
†Wv‡g‡bi ms‡hvM = (−1, 1)   = (−1, 1)
1−x
†iÄ, y = ln 1 + x
1−x
ev, ey = 1 + x
ev, 1 − x = ey + xey
ev, xey + ey = 1 − x
ev, xey + x = 1 − ey
1 − ey
g‡b Kwi, QK KvM‡Ri XOX eivei x Aÿ Ges YOY ev, x = 1 + ey
eivei y Aÿ AuvwK Ges 0 g~jwe›`y| x Aÿ eivei ¶z`ªZg e‡M©i y Gi mKj ev¯Íe gv‡bi Rb¨ x Gi gvb ev¯Íe nq|
cuvP evûi ˆ`N©¨‡K GKK Ges y Aÿ eivei ¶z`ªZg e‡M©i cÖwZ  cÖ`Ë dvsk‡bi †iÄ, R = R.
evûi ˆ`N©¨‡K GKK a‡i (1, 1), (2, 4), (3, 9), (4, 16) cÖkœ \ 14 \ †Wv‡gb, †iÄ D‡jøLmn †jLwPÎ A¼b Ki|
we›`y¸‡jv QK KvM‡R ¯’vcb Kwi Ges mvejxjfv‡e †hvM K‡i K. (x) = |x| hLb −5  x  5
dvskbwUi †jLwPÎ A¼b Kwi| mgvavb :
†h‡nZz y = x2, x > 0 †m‡nZz 0 e¨ZxZ mKj ev¯Íe gv‡bi Rb¨ (x) = |x| hLb −5  x  5
dvskbwU msÁvwqZ| + x 0  x  5
 †Wv‡gb D = (0, + ) Ges
= { −x −5  x  0
†iÄ R = (0, + ) †Wv‡gb : GLv‡b −5  x  5 mxgvi g‡a¨ x Gi cÖwZwU ev¯Íe
gv‡bi Rb¨ (x) Gi cÖwZ”Qwe i‡q‡Q|
cÖkœ \ 12 \ (x) = ln (x − 2) dvskbwUi D I R wbY©q Ki : dvsk‡bi †Wv‡gb n‡jv D = [−5, 5]
mgvavb : †iÄ : −5  x  5 mxgvi g‡a¨ x Gi abvZ¥K ev FYvZ¥K
Avgiv Rvwb, jMvwi`g ïay abvZ¥K ev¯Íe msL¨vi Rb¨ Dfq gv‡bi Rb¨ (x) abvZ¥K, Avi x = 0 n‡j (0) = 0
msÁvwqZ| myZivs dvsk‡bi †iÄ, R = [0, 5]
 (x) = ln (x − 2) Gi gvb ev¯Íe n‡e hw` (x) = |x| Gi †jLwPÎ A¼b :
x−2>0 g‡b Kwi, y = (x) = |x|
ev, x > 2 nq| −5 †_‡K 5 Gi g‡a¨ K‡qKwU gvb wb‡q mswkøó y Gi gvb
 †Wv‡gb, D = {x : x > 2} = (2, ) (Ans.) wb‡Pi Q‡K †`Lv‡bv n‡jv Ñ
x −3 0 3
 beg-`kg †kÖwY : D”PZi MwYZ  379
y 3 0 3  hLb x  0 |x|
GLb QK KvM‡R myweavgZ X Aÿ XOX Ges Y Aÿ YOY (M) (x) =  x
AuvwK| X Aÿ eivei ¶z`ªZi 2 eM©Ni = 1 GKK Ges Y Aÿ 0 hLb x = 0
eivei ¶z`ªZi 5 eM©Ni = 1 GKK a‡i (x, y) we›`y¸‡jv cvZb | x | hLb x  0
mgvavb : cÖ ` Ë dvskb  (x) =  x
Kwi| we›`y¸‡jv‡K mnRfv‡e eµ‡iLvq hy³ K‡i y =(x) Gi
0 hLb x = 0
†jL cvIqv hvq|
GLv‡b, x Gi cÖwZwU ev¯Íe gv‡bi Rb¨ (x) Gi cÖwZ”Qwe
−− Y
i‡q‡Q e‡j dvsk‡bi †Wv‡gb n‡jv ev¯Íe msL¨vi †mU R
(−3, 3) (3, 3)  †Wv‡gb, D = R
hLb, x = 0 ZLb (x) = 0
x
hLb, x > 0 ZLb (x) = x = 1
−x
hLb, x < 0 ZLb (x) = x = −1
myZivs dvsk‡bi †iÄ n‡jv, R = {− 1, 0, 1} †hLv‡b †Kej
O (0, 0)
wZbwU Dcv`vb i‡q‡Q|
X X †jLwPÎ A¼b :
 hLb x  0
|x|
awi, y = (x) =  x
Y 0 hLb x = 0
cÖ`Ë dvsk‡bi †jLwPÎ A¼‡bi Rb¨ x Ges y Gi gvb¸‡jvi
L. (x) = x + | x | hLb −2  x  2
ZvwjKv ˆZwi Kwi|
mgvavb : GLv‡b, −2 ≤ x ≤ 2 mxgvi g‡a¨ x Gi cÖwZwU
x −2 −1 0 1 2
ev¯Íe gv‡bi Rb¨ (x) cÖwZ”Qwe i‡q‡Q|
y −1 −1 0 1 1
 dvsk‡bi †Wv‡gb, D = [−2, 2]
hLb x = 0 ZLb (0) = 0 + | 0 | = 0 QK KvM‡R XOX eivei x Aÿ Ges YOY y Aÿ Ges 0
hLb x = −2 ZLb (−2) = −2 + |−2| = −2 + 2 = 0 g~jwe›`y| ¶z`ªZg e‡M©i wZb evûi ˆ`N©¨‡K GKK awi,
(−2, −1), (−1, −1), (0, 0), (1, 1), (2, 1) we›`y¸‡jv ¯’vcb K‡i
hLb x = 2 ZLb (2) = 2 + | 2 | = 2 + 2 = 4
myZivs dvsk‡bi †iÄ, R = [0, 4] dvskbwUi †jLwPÎ A¼b Kwi|
†jLwPÎ A¼b :
cÖ`Ë dvskb (x) = x + |x| hLb −2  x  2
cÖ`Ë dvsk‡bi †jLwPÎ A¼‡bi Rb¨ x Ges y Gi gvb¸‡jvi
ZvwjKv ˆZwi Kwi :
x −2 −1 0 1 2
y 0 0 0 2 4
QK KvM‡R XOX eivei x Aÿ Ges YOY eivei y Aÿ Ges
0 g~jwe›`y| ¶z`ªZg e‡M©i Pvi evûi ˆ`N©¨‡K GKK a‡i (−2,
0), (−1, 0), (0, 0), (1, 2), (2, 4) we›`y¸‡jv ¯’vcb K‡i
dvskbwUi †jLwPÎ A¼b Kwi|

cÖkœ \ 15 \ †`Iqv Av‡Q,

2x y−1
2 .2 = 64 ............................... (i)
y−2
6
Ges 6x. 3 = 72 ..................... (ii)
K. (i) I (ii) †K x I y PjKwewkó mij mgxKi‡Y cwiYZ Ki|
L. mgxKiYØq mgvavb K‡i ï×Zv hvPvB Ki|
M. x I y gvb hw` †Kv‡bv PZzf©y‡Ri mwbœwnZ evûi ˆ`N©¨ nq
†hLv‡b evû؇qi AšÍf©y³ †KvY 90| Z‡e PZzf©yRwU AvqZ bv
eM© D‡jøL Ki Ges Gi †ÿÎdj I K‡Y©i ˆ`N©¨ wbY©q Ki|
 beg-`kg †kÖwY : D”PZi MwYZ  380
mgvavb : mgvavb :
2x y−1
K. †`Iqv Av‡Q, 2 .2 = 64 ............ (i) log(1 + x)
K. †`Iqv Av‡Q, logx = 2
y
x 6 −2 ev, 2 logx = log(1 + x)
Ges 6 . 3 = 72 ................... (ii) 2
2x + y − 1 6 ev, logx = log(1 + x)
(i) n‡Z cvB, 2 =2 2
ev, x = 1 + x
ev, 2x + y − 1 = 6
x −x−1=0
2
ev, 2x + y = 6 + 1 2
wb‡Y©q wØZxq mgxKiY, x − x − 1 = 0
 2x + y = 7 2
L. ÔKÕ †_‡K cvB, x − x − 1 = 0
n‡Z cvB, 6 − = 72  3
x+y 2
(ii)
(−1)2  (−1)2−41(−1) 1 1 +4
ev, 6 − = 216
x+y 2 ev, x = 21
=
2
x+y−2
ev, 6 =6
3
1 5
x=
ev, x + y − 2 = 3 2
1+ 5
x+y=5 ïw× cixÿv : x = n‡j,
2
 mijxK…Z mgxKiYØq n‡jv, 2x + y = 7
log 1 +
1 + 5
log 
3 + 5
L.
x+y=5
ÔKÕ n‡Z cvB, 2x + y = 7 ............. (iii) log(1 + x)  2   2 
evgcÿ = = =
log
1 + 5
log
x + y = 5 ............. (iv) logx 1 + 5
(iii) n‡Z (iv) we‡qvM K‡i cvB, 2x + y − x − y = 7 − 5  2   2 
x=2 = 2 (K¨vjKz‡jUi e¨envi K‡i]
x Gi gvb (iv) bs mgxKi‡Y ewm‡q cvB, = Wvbcÿ
2+y=5
 y=3 1− 5
wb‡Y©q mgvavb, (x, y) = (2, 3) Avevi, x = 2
n‡j,
ïw× cixÿv :
log 1 −
1 + 5
log 
3 − 5
x = 2, y = 3 n‡j (iii) bs mgxKi‡Yi evgcÿ = 2  2 + 3 = 7  2   2 
= Wvbcÿ evgcÿ = =
log
1 − 5
log
1 − 5
Avevi, x = 2, y = 3 n‡j (iv) mgxKi‡Yi evgcÿ = 2 + 3 = 5  2   2 
= Wvbcÿ
Gi ev¯Íe gvb cvIqv m¤¢e bq| KviY 
1 − 5
 cÖvß mgvavb mwVK| 2 
FYvZ¥K|
M. Avevi FYvZ¥K msL¨vi jMvwi`‡gi m¤¢e gvb †bB| myZivs x Gi
D C gvb †Kej GKwU gvb mgxKiYwU‡K wm× K‡i| (†`Lv‡bv n‡jv)
2
M. ÔLÕ n‡Z cvB, g~jØq h_vµ‡g,
1+ 5 1− 5
x1 =
2
Ges x2 = 2
......... (i)
90 2
 x12 = 
A B 1 + 5 1
2  4
= (1 + 5 + 2 5)
3
GLv‡b, ABCD PZzf‚‡R©i `yBwU mwbœwnZ evû 1 3 1 1 5
AB = y = 3 = (6 + 2 5) = + ( 5) = 1 + +
4 2 2 2 2
AD = x = 2
†h‡nZz AB  AD Ges AB = DC, AD = BC 1+ 5
 x12 = 2 + 1 = x1 + 1
myZivs ABCD PZzfz©RwU GKwU AvqZ| 2

Avevi, x22 =  2  = 4 (1 − 2 5 + 5)
 †ÿÎdj = xy = 2  3 eM© GKK = 6 eM© GKK (Ans.) 1− 5 1
2 2
Ges e‡M©i ˆ`N©¨ = AB + BC GKK 1 3 5
2 2
= 3 +2 GKK = (6 − 2 5) = −
4 2 2
= 9 +4 GKK = 13 GKK (Ans.)
=1+
1 5 1 − 5
=1+ −
log (1 + x) 2 2  2 
cÖkœ \ 16 \ †`Iqv Av‡Q, log x
=2  x22 = 1 + x2
K. cÖ`Ë mgxKiYwU‡K x PjK msewjZ
GKwU wØNvZ mgxKi‡Y myZivs g~j؇qi cÖwZwUi eM© Zvi ¯^xq gvb A‡cÿv 1 †ewk (cÖgvwYZ)
cwiYZ Ki| 1+ 5 1− 5
L. cÖvß mgxKiYwU‡K mgvavb Ki Ges †`LvI †h, x Gi †Kej GLb, x1 = 2
= 1.618 Ges x2 = 2
= −0.618
GKwU exR mgxKiYwU‡K wm× K‡i| QK KvM‡R ¶z`ªZg e‡M©i cuvP evûi ˆ`N©¨‡K GKK a‡i,
M. cÖgvY Ki †h, g~j؇qi cÖwZwUi eM© Zvi ¯^xq gvb A‡cÿv 1 (1.618, 0) Ges (−0.618, 0) we›`y w`‡q y A‡ÿi mgvšÍivj
(GK) †ewk Ges Zv‡`i †jLwPÎ ci¯úi mgvšÍivj| K‡i †jL‡iLv `yBwU A¼b Kwi|
 beg-`kg †kÖwY : D”PZi MwYZ  381
x
y = 2 Gi ˆewk󨸇jv wbgœiƒc :
(i) †jLwPÎwU (0, 1) we›`yMvgx
x
(ii) †jLwPÎwU EaŸ©Mvgx; x Gi gvb evovi mv‡_ mv‡_ 2 Gi
gvbI evo‡e|
x
(iii) x → − n‡j y = 2 → 0+
(iv) x Gi †h †Kvb gv‡bi Rb¨ y abvZ¥K|
x
M. †`Iqv Av‡Q, y = 2
ev, x = log2y
1
Avgiv Rvwb, y = (x) n‡j, − (y) = x
1
 − (y) = log2y
†jL n‡Z †`Lv hvq †iLvØq ci¯úi mgvšÍivj| 1
 − (x) = log2x
x
cÖkœ \ 17 \ †`Iqv Av‡Q, y = 2  cÖ`Ë dvsk‡bi wecixZ dvskb, (x) = log2x
K. cÖ`Ë dvskbwUi †Wv‡gb Ges †iÄ wbY©q Ki| awi, x1R Ges x2R
L. dvskbwUi †jLwPÎ A¼b Ki Ges Gi ˆewk󨸇jv †jL| 1
Zvn‡j, − (x1) = log2x1
M. dvskbwUi wecixZ dvskb wbY©q K‡i GwU GK-GK wKbv Zv 1
Ges − (x2) = log2x2
wba©viY Ki Ges wecixZ dvskbwUi †jLwPÎ AuvK| 1 1
GLb, − (x1) = − (x2)
mgvavb : ev, log2x1 = log2x2
x 0
K. †`Iqv Av‡Q, y = 2 hLb x = 0 ZLb y = 2 = 1 ev, x1 = x2
 wecixZ dvskbwU GK-GK|
Avevi, x Gi FYvZ¥K †h †Kv‡bv gv‡bi Rb¨ y Gi gvb †Kv‡bv
wecixZ dvskbwUi †jLwPÎ A¼b Ki‡Z n‡e A_©vr y = log2x
mgq (0) k~‡b¨i LyeB KvQvKvwQ †cuŠQvq wKš‘ k~b¨ nq bv| Gi †jLwPÎ A¼b KivB h‡_ó|
+
A_©vr x → −, y → 0 x
†h‡nZz y = log2x n‡j y = 2 Gi wecixZ dvskb|
GKBfv‡e, x Gi †h †Kv‡bv abvZ¥K gv‡bi Rb¨ y Gi gvb y = x †iLvi mv‡c‡ÿ m~PK dvsk‡bi cÖwZdjb jMvwi`wgK
µgvš^‡q Wvbw`‡K (Dc‡i) e„w× †c‡Z _vK‡e ev  w`‡K dvskb wbY©q Kiv n‡q‡Q, hv y = x †iLvi mv‡c‡ÿ m`„k|
avweZ n‡e| 0
Avevi, 2 = 1 Kv‡RB y = log21 = 0
A_©vr, x → −, y → −
myZivs †iLvwU (1, 0) we›`yMvgx|
myZivs †Wv‡gb, D = (−, ) hLb x → − ZLb y → 0
Ges †iÄ R = (0, )  y = log2x †iLvwU e„w×cÖvß|
x
L. y = 2 Gi †jLwPÎ A¼b : wb‡P †iLvwUi †jLwPÎ A¼b Kiv n‡jv|
x
cÖ`Ë dvskb y = 2
cÖ`Ë dvsk‡bi †jLwPÎ A¼‡bi Rb¨ x Ges y Gi gvb¸‡jvi
wbgœiƒc ZvwjKv ˆZwi Kwi|
x −3 −2 −1 0 1 2
y 0.125 0.25 0.5 1 2 4
QK KvM‡R XOX eivei x Aÿ I YOY eivei y Aÿ Ges
g~jwe›`y O| ¶z`ªZg e‡M©i cÖwZ Pvi evûi ˆ`N©¨‡K GKK a‡i
(−3, 0.125), (− 2, 0.25), (−2, 0.5), (0, 1), (1, 2), (2, 4)
x
we›`y¸‡jv ¯’vcb K‡i mvejxjfv‡e †hvM K‡i, y = 2 Gi
†jLwPÎ A¼b Kiv n‡jv|
 beg-`kg †kÖwY : D”PZi MwYZ  382

¸iæZ¡c~Y© enywbe©vPwb cÖ‡kœvËi

1. log 16 2 = KZ? 7. log42 + log6 6 = KZ?
2
1 3
K2 2 L4 M8 9 K 1 M N −2
2 2
2. M = 1 + logp qr n‡j, pM = KZ?
K p + qr L 1 + qr  pqr N qr 8. p = logab + logcc nq Z‡e 1 + p = KZ?
3. ax = y n‡j, wb‡Pi †KvbwU mwVK? K1 L 1 + bc
K loga x = y L log y = x  loga y = x N x log a = y  loga abc N abc loga1
1 9. hw` ax = b nq, hLb a > 0, n  N; ZLb−
4. log5 ( )
25
Gi gvb KZ? i. logab = x ii. logaab = b
K5 L −5 M2  −2 iii. loga b = log5b loga5
5. hw` a, b, p > 0 Ges a  1, b  1 nq Z‡e- wb‡Pi †KvbwU mwVK?
p p a 4 1 K i I ii L i I iii M ii I iii  i, ii I iii
i. logb = loga  logb ii. logb b=
4 10. 400 Gi−
1
iii. loga a  logb b  log c = i. gvb (2 5)4 Gi mgvb ii. jM 4 n‡j wfwË 2 5
2
wb‡Pi †KvbwU mwVK? iii. 2 5 wfwËK jM 4
 i I ii L i I iii M ii I iii N i, ii I iii wb‡Pi †KvbwU mwVK?
6. loga loga loga (aa)a Gi gvb KZ? K i I ii L i I iii M ii I iii  i, ii I iii
K0 1 Ma N −1

AwZwi³ enywbe©vPwb cÖ‡kœvËi

 `yBwU L wZbwU M PviwU N cuvPwU
96 : jMvwi`g 1
20. log3 gvb †KvbwU? 81
= Gi (KwVb)
mvaviY enywbe©vPwb cÖ‡kœvËi K −1 L −2 M −3  −4
11. abvZ¥K msL¨vi N Gi mvaviY jMvwi`g‡K KqwU As‡ki 1 1
e¨vL¨v : log3 81 = log3 34 = log33−4 = − 4log33 = 41 = −4
mgwó w`‡q cÖKvk Kiv hvq? (mnR)
21. b = anti log3 x wK wb‡`©k K‡i? (ga¨g)
K GKwU  `yBwU M wZbwU N PviwU
 b msL¨vwU‡K wfwË a‡i a Gi mv‡c‡ÿ x Gi cÖwZjM
12. a > 0 Ges a  1 hw` ax = y nq Z‡e x †K ejv nq y Gi a L a msL¨vwU‡K wfwË a‡i x Gi mv‡c‡ÿ b Gi cÖwZjM
wfwËKÑ (mnR) M x msL¨vwU‡K wfwË a‡i b Gi mv‡c‡ÿ a Gi cÖwZjM
 jMvwi`g L m~PK M NvZ N AskK
N x msL¨vwU‡K wfwË a‡i a Gi mv‡c‡ÿ b Gi cÖwZjM
13. a > 0 Ges a1 Ges y > 0 n‡j y Gi Abb¨ a wfwËK jMvwi`g‡K 22. a > 0, a  1 n‡j, ax = b Gi †ÿ‡Î x †K Kx ejv nq? (ga¨g)
wb‡Pi †KvbwU Øviv cÖKvk Kiv nq? (mnR)  b Gi a wfwËK jMvwi`g L a Gi b wfwËK jMvwi`g
K loga L logya  logay N logey
M a Gi n wfwËK jMvwi`g N b Gi e wfwËK jMvwi`g
14. logay = x hw` I †Kej hw`Ñ (mnR) e¨vL¨v : ax = b †hLv‡b a > 0 Ges a  1 nq, Z‡e x †K ejv nq b Gi a
1
x
wfwËK jMvwi`g ax = bev x = logab
 ax = y nq L ao = x nq M ay = x nq N a = y nq 23. logab = x n‡j wb‡Pi †KvbwU mZ¨? (mnR)
1
15. log5 ( ) Gi gvb KZ?
25
(ga¨g)  ax = b L a = b
24. log16256 Gi gvb †KvbwU?
M xa = b N x = b
(KwVb)
K0 L −1  −2 N −3 K 1 2 M3 N4
16. log64256 Gi gvb KZ? (KwVb) e¨vL¨v : log16256 = log16162 = 2log1616 = 21 = 2
4 2 3 1 25. 4x = 16 n‡j x Gi gvb KZ? (mnR)
 L M N
3 3 4 2
K 1 2 M4 N8
17. log101000 Gi gvb KZ? (ga¨g)
K2 3 M4 N 1.001 enyc`x mgvwßm~PK enywbe©vPwb cÖ‡kœvËi
18. ¯^vfvweK jMvwi`g logey †K wb‡Pi †Kvb cÖZxK Øviv cÖKvk 26. a, m, n, x PjK n‡j−
Kiv nq? (mnR) am
1 i. n = am − n n‡e, hw` m < n nq
 ln y L y ln y M ly a N log y
y
ii. x ( R, a ( 0 n‡j, a(n = eq \f(1,an)
19. cÖ‡Z¨K abvZ¥K msL¨vi jMvwi`‡gi KqwU Ask _v‡K? (mnR)
 beg-`kg †kÖwY : D”PZi MwYZ  383
1 K2 L8 M 16  32
iii. log x=1 n‡j, x = 4 1 10
8 3 e¨vL¨v : log x = 3 3 = 3
wb‡Pi †KvbwU mwVK? (mnR) 8
10 3 10
K i I ii L i I iii M ii I iii  i, ii I iii
ev, x = ( 8) =
3
()
2
2 3
= 25 = 32
27. i. e wfwËK jMvwi`g n‡jv ¯^vfvweK jMvwi`g
ii. e¨envwiK MwY‡Z mvaviYZ e wfwËK jMvwi`g e¨eüZ nq enyc`x mgvwßm~PK enywbe©vPwb cÖ‡kœvËi
iii. weªMwmqvb jMvwi`g 10 wfwËK jMvwi`g 41. i. log28 = 3
wb‡Pi †KvbwU mwVK? (mnR) ii. log381 = 4
K i I ii  i I iii M ii I iii N i, ii I iii iii. log416 = 2

Awfbœ Z_¨wfwËK enywbe©vPwb cÖ‡kœvËi wb‡Pi †KvbwU mwVK? (mnR)

K i I ii L i I iii M ii I iii  i, ii I iii
wb‡Pi Z‡_¨i Av‡jv‡K 28Ñ31 bs cÖ‡kœi DËi `vI : 42. hw` a > 0 Ges a  1 nq, Z‡eÑ
loga logb logc i. loga1 = 0
= = =k
y−z z−x x−y ii. logaa = 1
28. logax Gi gvb KZ? (ga¨g) iii. loga1 = 1
 k(xy − zx) L k(zx − xy) wb‡Pi †KvbwU mwVK? (mnR)
M k(yz − zx) N k(xy − yz)
 i I ii L i I iii M ii I iii N i, ii I iii
29. logax + logby + logcz = KZ? (ga¨g)
43. i. log25 + log27 + log23 = log235
0 L xyz M −1 N logaabc
ii. log564 = 6log52
30. ax.by.cz = KZ? (ga¨g) 1
K0 L2 1 Nk iii. log764 = log74
3
31. x = a, y = b Ges z = c n‡j logax + logby + logcz = KZ? wb‡Pi †KvbwU mwVK? (ga¨g)
(mnR) K i I ii  ii I iii Mi I iii N i, ii I iii
K −1 0 M1 N AmsÁvwqZ 44. a > 0, a  1 n‡jÑ
i. logaMr = rlogaM
97 : jMvwi`‡gi m~Îvejx ii. logaMN = logaM + logaN
mvaviY enywbe©vPwb cÖ‡kœvËi iii. logaM = logbM  logaN
wb‡Pi †KvbwU mwVK? (mnR)
32. hw` a  1 nq, Z‡e a1 = a Zvn‡j, logaa = KZ? (mnR)
K i I ii L i I iii M ii I iii  i, ii I iii
1
1 L0 M3 N
a Awfbœ Z_¨wfwËK enywbe©vPwb cÖ‡kœvËi
33. logab  logba = 1 n‡j, logap = KZ? (KwVb)
logbp wb‡Pi Z‡_¨i Av‡jv‡K 45 Ñ 47 bs cÖ‡kœi DËi `vI :
1

logba
L logp(ab) M logbp N loga
p () x
GKwU dvskb (x) = | x | Øviv msÁvwqZ Ges x
34. logap  logpq  logqr  logrb = KZ? (KwVb) 45. (0) = KZ? (mnR)
K loga L logb  logab N logba
K0 L1  AmsÁvwqZ N 2
35. †hLv‡b 10x = y Ges y > 0 n‡j, y Gi mvaviY jMvwi`g
wb‡Pi †KvbwU? (mnR) 46. (x) Gi †Wv‡gb KZ? (ga¨g)
K L  M = {1}  − {0}
K x = log (1y) L x = log10x 47. (x) Gi †iÄ KZ? (KwVb)
M x = logky  x = log10y K {1} L {−1}  {−1, 1} N 
36. mvaviY jMvwi`g log10y †K mPivPi wfwË 10 Dn¨ †i‡L wb‡Pi Z‡_¨i Av‡jv‡K 48 Ñ 50 bs cÖ‡kœi DËi `vI :
wb‡Pi †KvbwU cÖKvk Kiv nq? (mnR) log abc = x,log abc = y, log abc = z
a b c
1 1
K log L log M log x  log y 1 1
y x 48. x + z = KZ? (ga¨g)
37. hw` log a = n nq Z‡e a †K n Gi Kx ejv nq? (mnR) 1 1
K jM ZvwjKv L cÖwZjM M Anti log  L+M K0 L1 M −
y y
38. logk n + logk n + logk n Gi gvb KZ?
an bn cn 49. hw` xyz = 1 nq, Z‡e xy + yz + zx = KZ? (KwVb)
(ga¨g)
b c a K0 1 M logaabc N abc
K −1 0 M1 N2
1
b 50. = KZ? (ga¨g)
39. loga loga loga aa ( ) Gi gvb KZ? b
(KwVb)
1+x
K 1 + logabc L logabc M0  logaa bc 2

Ka b M1 N aa
1
40. log x = 3 3 n‡j x Gi gvb †KvbwU?
8
(KwVb) 97 : m~PKxq, jMvwi`gxq I ciggvb dvskb
 beg-`kg †kÖwY : D”PZi MwYZ  384
mvaviY enywbe©vPwb cÖ‡kœvËi  32 L 23 M
10
N
3
3 10
51. loga loga loga aa ( b ) Gi gvb KZ? (KwVb)
70. ciggvb dvskb (x) = |x| Gi †Wv‡gb KZ?
 L  M {0} N (0, )
(mnR)
b
Ka b M1 N aa 71. ciggvb dvskb (x) = |x| Gi †iÄ KZ? (ga¨g)
52. log2 5 400 =x n‡j x Gi gvb KZ? (KwVb) K (−, ) L (0, ) M (, 0)  [0, )
K −1 L1 M2 4 5+x
53. logarithm kãwU G‡m‡Q †Kvb kã †_‡K? (mnR) 72. y = ln
5−x
dvskbwU‡Z x→5 n‡j, y Gi gvb KZ? (mnR)
K j¨vwUb L cZ©ywMR  wMÖK N divwm K0  M1 N 10
54. log264 = KZ? (ga¨g) 73. wb‡Pi †KvbwU x- †K b Gi a wfwËK jMvwi`g ejv nq?
K2 6 M8 N 64 K b = log ax L b = log xb
55. log864 = KZ? (ga¨g)  x = log ab N b = log ba
2 L4 M8 N 16 74. y = 3x Gi †iÄ KZ?
56. loga(M  N) = KZ? (mnR) K (−, 0)  (0, () M (0, (() N (((, ()
 logaM + logaN L logMa + logaN 75. y = 3x Gi wecixZ dvsk‡bi †iÄ KZ?
L (−, 0)
M loga (MN) N logaM − logaN K (0 )
a+x
 (−, ) N (−1. 1)

57. y = 2x GB dvsk‡bi †iÄ KZ? (ga¨g) 76. y = 1n

a−x
dvskbwUi †Wv‡gb KZ?
 (0, ) L (−, ) K (−1, 1) L (−, )  (−a a) N (a, −a)
M (−, 0) NR 77. f(x) = x + |x| hLb −2  x  2 Gi †Wv‡gb KZ?
58. y = 2x †iLvwUÑ (ga¨g)
 (−2, 2) L (0, −2) M (0 2) N (0. 3)
K g~jwe›`yMvgx  (0, 1) we›`yMvgx
78. f(x) = x + |x| hLb − 2  x  2 Gi †iÄ KZ?
M (0, 2) we›`yMvgx N (0, 3) we›`yMvgx
K (2, 2) L (0, 2) M (0, 3)  (0, 4)
59. wb‡Pi †KvbwU jMvwi`wgK dvskb? (mnR)
K y = 2x L y = x2 + 3x + 2 enyc`x mgvwßm~PK enywbe©vPwb cÖ‡kœvËi
2+x
 y = ln N y=2 2x
79. a > 0 nIqvq mKj x Gi Rb¨ ax > 0 Ges y  0 n‡jÑ
2−x
i. y Gi a wfwËK †Kv‡bv jMvwi`g †bB
60. y = 3x Gi †Wv‡gb KZ? (ga¨g)
ii. y Gi a wfwËK jMvwi`g Av‡Q
K (−, 0) L (0, )
iii. a Gi y wfwËK jMvwi`g †bB
M [0, )  (−, )
61. y = 3x Gi wecixZ dvsk‡bi †Wv‡gb KZ? wb‡Pi †KvbwU mwVK? (mnR)
K (−, )  (0, ) M (0, −] N (0, 1] i L ii M i I iii N i I iii
x 80. logay = x hw` I †Kej hw` ax = y nqÑ
62. (x) = |x| GKwUÑ (mnR)
i. loga(ax) = x
 dvskb L m~PK dvskb ii. alogay = y
M jMvwi`wgK dvskb N wecixZ dvskb 1
x iii. alogay = xyx
63. (x) = |x| Gi †Wv‡gb KZ? (ga¨g)
wb‡Pi †KvbwU mwVK? (ga¨g)
K L − 204  − {0} N (−, 0] K i I ii  i I iii M ii I iii N i, ii I iii
x
64. (x) = |x| Gi †iÄ KZ? (ga¨g) 81. x > 0, y > 0 Ges a  1 n‡j, x = y n‡e hw`Ñ
K L − {0} M {1, 1}  {−1, 1}
i. logax > 0
a+x ii. logax = logay
65. y = ln a − x dvskbwUi †iÄ KZ? (KwVb) iii. logay > 0
K − {a} L  − {a} N wb‡Pi †KvbwU mwVK? (KwVb)
− |x|
K i I ii L i I iii M ii I iii  i, ii I iii
66. (x) = e 2 ; −2 < x < 0 GB dvsk‡bi †Wv‡gb KZ? (ga¨g)
82. P = logabc n‡j 1 − p = KZ?
K (−1, 0) L (−1, 0]  (−2, 0) N (2, 0)
i. 1 − logabc
67. y = ax, a > 1 Z‡e Gi †Wv‡gb KZ? (ga¨g)
ii. logaa − logabc
K (−, ] L (−, 0] M (0, )  (−, )
a
68. logxx x x = KZ?
3
(ga¨g) ()
iii. loga
bc
3 5 4 11 wb‡Pi †KvbwU mwVK? (mnR)
K L M 
2 6 6 6 K i I ii L i I iii M ii I iii  i, ii I iii
69. log x = 10 n‡j, x = KZ? (ga¨g) 83. i. logaPQ = logaP + logaQ
2
ii. logaPQ = logaP.logaQ
 beg-`kg †kÖwY : D”PZi MwYZ  385
Awfbœ Z_¨wfwËK enywbe©vPwb cÖ‡kœvËi
(QP ) = log P + log (Q1 )
iii. loga a a

wb‡Pi †KvbwU mwVK? (ga¨g) wb‡Pi Z‡_¨i Av‡jv‡K 88 I 89 bs cÖ‡kœi DËi `vI :
K i I ii L ii I iii i I iii N i, ii I iii y = x2; x > 0
84. i. log512 = 2log52 3 88. dvskbwUi †Wv‡gb KZ? (ga¨g)
ii. log53 log35 = 1 K  (0, ) M − {0} N N
iii. xlogay = ylogax 89. dvskbwUi †iÄ KZ? (ga¨g)
wb‡Pi †KvbwU mwVK? (ga¨g)  (0, ) L − {0} M − {2} N (−, 9[
 i I ii L i I iii M ii I iii N i, ii I iii (x) = x + |x| hLb −2  x < 2
85. y = (x) = e−x; 2 < e < 3 Dc‡ii eY©bv n‡Z 90 Ñ 92 bs cÖ‡kœi DËi `vI :
i. G‡ÿ‡Î x →  n‡j, y → 0 + nq
90. dvskbwU GKwUÑ (mnR
ii. GwU (0, 1) we›`yMvgx
K jMvwi`wgK dvskb  ciggvb dvskb
iii. x → − n‡j, y →  nq
M m~PK dvskb N wecixZ dvskb
wb‡Pi †KvbwU mwVK? (ga¨g)
91. cÖ`Ë dvsk‡bi †Wv‡gb KZ? (ga¨g)
K i I ii L i I iii M ii I iii  i, ii I iii
K (−2, 2) L [−2, 2] M (−2, 2]  [−2, 2)
86. a, b > 0 Ges a  b n‡jÑ 92. cÖ`Ë dvsk‡bi †iÄ KZ? (ga¨g)
i. (ap)qr = a n‡j, pqr = 1 K (0,4) L (0, 4] M {0, 4}  (0, 4)
2
ii. (axy)(axy)z = a2 n‡j, xyz = 1 logka logkb logkc
iii. logk n + logk n + logk n = 0
n n n = =
a b c y−z z−x x−y
b  c  a 
Dc‡ii ivwk n‡Z 93 Ñ 95 bs cÖ‡kœi DËi `vI :
wb‡Pi †KvbwU mwVK? (KwVb)
93. axbycz = KZ? (ga¨g)
K i I ii L ii I iii  i I iii N i, ii I iii
1
87. (x) = 2x n‡jÑ K0 L xyz 1 N
xyz
i. (x) Gi †Wv‡gb = (−, )
2 2 2 2 2 2
94. a y + yz + z
.a z + zx + x
.a x + xy + y
= KZ? (KwVb)
ii. (x) Gi †iÄ = (0, ) K0 1 M logka N 
iii. −1(x) = log2x 95. ay + z.bz + x.cx + y = KZ? (KwVb)
wb‡Pi †KvbwU mwVK? (mnR) K0 L z−x M y −z 2 2
1
K i I ii L ii I iii M i I iii  i, ii I iii

wbe©vwPZ enywbe©vPwb cÖ‡kœvËi

96. y = 1n (x − 2) n‡j wb‡Pi †KvbwU mwVK? 103.
logk (1 + 3x)
= 2 n‡j Gi wØNvZ mgxKiY wb‡Pi †KvbwU?
K (x − 2)c
=y  =x−2
ey logkx
M ex − 2 = y N e−y = x − 2 K x2 + 3x + 1 = 0 L x2 − 3x + 1 = 0
97. log 0 Gi gvb KZ? M x2 + 3x − 1 = 0  x2 − 3x − 1 = 0
K0  bvB M N1 104. log10 (999 + x) = 3 n‡j, x Gi gvb KZ?
98. F(x) = 2x G x →  n‡j y = F(x) Gi gv‡bi †ÿ‡Î †KvbwU K 0  1 M 2 N3

mwVK? 105. a > 0, a  1 n‡jÑ

r i. logaM = rlogaM
 y→ L y→0 M y=0 N y→
ii. loga(MN) = logaM + logaN
99. y = 1 − 3−x wecixZ dvskb †KvbwU? M logaM
iii. loga =
 log3  
1 N logaN
K log3 (1 − y)
1 − x wb‡Pi †KvbwU mwVK?
M 1 − 3x N 3x − 1
 i I ii L i I iii M ii I iii N i, ii I iii
100. hw` a > 1 Ges 0 < x < 1 nq Z‡eÑ
106. f(x) = 2x
 logax < 0 L logax > 0
i. f(x) Gi †Wv‡gb (− , )
M loga x = 0 N loga a = 0
x ii. f (x) Gi †iÄ (0, )
101. f(x) = |x| dvsk‡bi †iÄ KZ? iii. f−1(x) = lgo2x
 {−1, 1} L {0, 1} M {0, −1} N {0, 0} wb‡Pi †KvbwU mwVK?
x K i I ii L i I iii M ii I iii  i, ii I iii
102. f(x) = |x| Ges x ev¯Íe msL¨v n‡j, f(0) = KZ?
107. f(x) = 3x
K0 L1  AmsÁvwqZ N −1 i. GKwU m~PK dvskb
 beg-`kg †kÖwY : D”PZi MwYZ  386
ii. GKwU GK-GK dvskb K0 1 M3 N9
iii. Gi wecixZ dvskb log3x 109. Dc‡iv³ dvskbwUi †Wv‡gb KZ?
wb‡Pi †KvbwU mwVK?  [0, ] L [−, 0] MN NR
Ki L ii 110. dvskbwUi wecixZ dvsk‡bi †Wv‡gb KZ?
Mi I ii  i, ii I iii  [0, ] L [0, ] M [−, ] Nd

wb‡Pi Z‡_¨i Av‡jv‡K 108 − 111 bs cÖ‡kœi DËi `vI : 111. dvskbwUi †iÄ nq?
K [−, 0] L [−, ] MR  R+
(x) = 3x2 GKwU m~PKxq dvskb, †hLv‡b x  R
108. −1(3) = KZ?

G A a¨v‡q i c vV mg wš^Z e nyw be©v Pwb cÖ‡ kœv Ëi

enyc`x mgvwßm~PK enywbe©vPwb cÖ‡kœvËi i. logaMr = r logaM ii. logaMN = logaM + logaN
logbM
iii. logaM =
112. wb‡Pi mgxKiY¸‡jv jÿ Ki : logba
i. 16x = 4x+2 wb‡Pi †KvbwU mwVK? (ga¨g)
ii. 2x = 8 K i I ii L i I iii M ii I iii  i, ii I iii
iii. x − 4 + 2 = x + 12 118. i. x  0, a > 0, b > 0 Ges ax = bx n‡j a = b
wb‡Pi †KvbwU mwVK? (ga¨g) ii. am = an Ges a  0 n‡j m = n
 i I ii L ii I iii Mi I iii N i, ii I iii iiilogba  logab = 1
113. i. 3x2 = 34 n‡j x = ± 2 wb‡Pi †KvbwU mwVK? (ga¨g)
ii. 22 = 9 n‡j, y = ± 3 K i I ii L i I iii M ii I iii  i, ii I iii
iii. 2.3y = 18 n‡j, y = 2 ar
119. i. (aP) = a n‡j pqr = 0
wb‡Pi †KvbwU mwVK? (ga¨g) ii. {(axy) (axy)}z = a2 n‡j xyz = 1
K i I ii L i I iii M ii I iii  i, ii I iii n n n
114. i. am.an = am+n
am
iii. (ba ) + log (bc ) + log (ca ) = 0
n k n k n

ii. n = am−n wb‡Pi †KvbwU mwVK? (KwVb)

a
K i I ii L i I iii  ii I iii N i, ii I iii
†hLv‡b a  0
iii. (am)n = amn
wb‡Pi †KvbwU mwVK? (ga¨g) Awfbœ Z_¨wfwËK enywbe©vPwb cÖ‡kœvËi
K i I ii L i I iii M ii I iii  i, ii I iii
wb‡Pi Z_¨ †_‡K 120 I 121 bs cÖ‡kœi DËi `vI:
115. i. a  0 n‡j a = 1
a3−x b5x = a5+x b3x
1 1
ii. a−1 = iii. an = b2x
a a−(−n) 120. a2x = KZ? (ga¨g)
wb‡Pi †KvbwU mwVK? (ga¨g) 1
 i I ii L i I iii M ii I iii N i, ii I iii Ka  a2 M a N
a2
116. i. logaa = 1, a > 0, a  1 121. logka Gi gvb wb‡Pi †KvbwU− (KwVb)
am l an l
ii. ( ) ( )
an = m
a K logk () a
b
L logk () b
a
iii. loga1 = 0, a > 0, a  1
a b
wb‡Pi †KvbwU mwVK? (ga¨g) M x log ( )
k  x log ( )
k
b a
K i I ii  i I iii M ii I iii N i, ii I iii
117. a > 0, a  1 n‡j−

¸iæZ¡c~Y© m„Rbkxj cÖkœ I mgvavb

œ 1  p = xya − 1, q = xyb − 1, z = xyc − 1
cÖk- wbY©q Ki| 4
a a
−1
 1bs cª‡kœi mgvavb 
a b
K. a = b n‡j †`LvI †h,
b a
() b
=b b 2 K. †`Iqv Av‡Q, ab = ba
p q a a
L. cÖgvY Ki †h, (b + a) log q +(c + b) log r + a
†`Lv‡Z n‡e †h, b b = ab − 1
r
()
(a + c) log = 0
p
4
M. (b − c) logp + (c − a) log q + (a − b) log r Gi gvb
 beg-`kg †kÖwY : D”PZi MwYZ  387
a a a p q r
a  (b + a) log + (c + b) log + (a + c) log = 0 (cÖgvwYZ)
a ab ab ab
evgcÿ = () b
b= a= 1=
b
bb (ba)b (a )b
1 [ ab = ba ]
M.
q r
†`Iqv Av‡Q, p = xya − 1, q = xyb − 1, r = xyc − 1
p

a
a
(b − c)log p + (c − a)log q + (a − b)log r Gi gvb wbY©q
ab −1
= =ab = (Wvbcÿ) Ki‡Z n‡e|
a1
a a cÖ`Ë ivwk = (b − c)log p + (c − a)log q + (a − b) log r
(a )
 b = a = (†`Lv‡bv n‡jv)
b b
−1
= (b − c)log (xya−1) + (c − a)log (xyb−1)
+ (a − b) (xyc−1)
L. †`Iqv Av‡Q, p = xya − 1, q = xyb − 1, r = xyc − 1
= (b − c)log x + (b − c)logy + (c − a)logx
a−1
p q r
evgcÿ = (b + a)log q + (c + b)log r + (a + c)log p + (c−a)logyb−1 + (a − b)logx + (a − b)logc−1
p q r = (b − c)logx + (b − c) (a − 1) logy + (c − a)logx + (c − a)
= (a + b)log
q
+ (b + c)log + (c + a)log
r q (b − 1) logy + (a − 1) logx + (a − b) (c − 1) logy
xya−1 xyb−1 xyc−1 = (b − c + c − a + a − b)logx + {(b − c) (a − 1)
= (a + b)log b−1 + (b + c)log c−1 + (c + a)log a−1
xy xy xy + (c − a) (b − 1) + (a − b)(c − 1)} logy
ya−1 yb−1 yc−1 = 0  logx + {(b − c) (a − 1)
= (a + b)log b−1 + (b + c)log c−1 + (c + a)log a−1 + (c − a) (b − 1) + (a − b)(c − 1)} logy
y y y
= 0 + {(ab − ca − b + c) + (bc − ab − c + a)
= (a + b)log ya − 1 − b + 1 + (b + c)log yb − 1 − c + 1
+ (ca − bc − a + b)}logy
+ (c + a)log yc − 1 − a + 1
− − = (ab − ca − b + c + bc − ab − c + a + ca − bc − a + b)logy
= (a + b)log y a b + (b + c) log y b c + (c + a) log yc − a
= 0  log y = 0
= (a + b) (a − b)log y + (b + c)(b − c)log y + (c + a) (c − a)log y
= (a2 − b2)log y + (b2 − c2)log y + (c2 − a2)log y
wb‡Y©q gvb 0
= (a2 − b2 + b2 − c2 + c2 − a2)log y
= 0  log y = 0 = (Wvbcÿ)

A byk xj b g ~j K K v‡R i A v‡j v‡K m„R b kxj cÖk œ I mg vavb

2 2
loga logb logc
œ 2  hw` b − c = c − a = a − b nq, Z‡eÑ
cÖk- ev, logca + ab + b = k (a3 − b3) ................... (iii)
mgxKiY (i), (ii) I (iii) †hvM K‡i cvB,
K. AbycvZ¸‡jvi gvb k a‡i, logaa Gi gvb wbY©q Ki| 2 2 2 2 2 2 2
loga b + bc + c + logb c + ca + a + logc a + ab + b = k (b3 − c3) +
L. aa.bb.cc Gi gvb wbY©q Ki| 4 k (c3 − a3) + k(a3 − b3)
2 2 2 2 2 2
2 2 2 2 2 2
M. cÖgvY Ki †h, ab + bc + c  bc + ca + a  ca + ab + b = aabbcc. 4 ev, log (a b + bc + c  b c + ca + a  c a + ab + b ) = 0
2 2 2 2 2 2

 2bs cª‡kœi mgvavb  ev, log (a b + bc + c  b c + ca + a . c a + ab + b = log1

2 2 2 2 2 2
ev, a b + bc + c . b c + ca + a  c a + ab + b = 1
loga logb logc
 a b + bc + c  b c + ca + a  c a + ab + b = aa.bb.cc [ÔLÕ n‡Z] (cÖgvwYZ)
2 2 2 2 2 2
K. awi, b − c = c − a = a − b = k
 loga = k (b − c) œ 3  hw` x = 1 + logabc, y = 1 + logbca Ges z = 1 + logcab
cÖk-
ev, a log a = ka (b − c); [Dfq cÿ‡K a Øviv ¸Y K‡i] nq, Z‡eÑ
 logaa = ka (b − c) .............. (i) 1

L. GLb, log b = k (c − a) K. †`LvI †h, a = (abc)x 2

ev, b log a = kb (c − a); [Dfq cÿ‡K b Øviv ¸Y K‡i] L. cÖgvY Ki †h, xyz = xy + yz + zx 4
ev, logbb = kb (c − a) .............. (ii) M. †`LvI †h, ax − 3. by − 3. cz − 3 = 1 4
Ges log c = k (a − b)  3bs cª‡kœi mgvavb 
ev, c log c = kc (a − b) .............. (iii) K. †`Iqv Av‡Q, x = 1 + logabc
GLb, (i), (ii) I (iii) †hvM K‡i cvB, ev, x = logaa + logabc
logaa + logbb + logcc = k (ab − ac + bc − ab + ac − bc)
ev, log (aabbcc) = k 0 = 0 ev, x = logaabc
 aabbcc = 1 (Ans.) ev, ax = abc
1
M. ÔKÕ †_‡K cvB, loga = k (b − c)
ev, (b2 + bc + c2) loga = k (b − c) (b2 + bc + ca) a = (abc) x
(†`Lv‡bv n‡jv)
2 2
1
ev, log ab + bc + c = k (b3 − c3) ....................... (i) L. ÔKÕ n‡Z cvB, a = (abc) ............. (i)
x
ÔLÕ †_‡K cvB, logb = k(c − a) 1
ev, (c2 + ca + a2) logb = k(c − a) (c2 + ca + a2) Abyiƒcfv‡e, b = (abc) y............ (ii)
2 2
ev, logbc + ca + a = k(c3 − a3) .................... (ii) 1

Ges, logc = k (a − b) Ges c = (abc) z.............. (iii)

ev, (a2 + ab + b2) log c = k 9a − b) (a2 + ab + b2) (i), (ii) I (iii) ¸Y K‡i cvB,
 beg-`kg †kÖwY : D”PZi MwYZ  388
1 1 1
Y
abc = (abc) x.(abc) y .(abc)z
2, 25
1 1 1
+ +
†¯ ‹j : X A‡ÿ eivei ÿ z`ªZg 5 eM© Ni = 1 GKK
ev, (abc)1 = (abc)x y z Y A‡ÿ eivei ÿ z`ªZg 2 eM© Ni = 1 GKK

1 1 1
ev, 1 = x + y + z
yz + zx + xy
ev, xyz
=1
 xyz = zy + yz + zx (cÖgvwYZ)
M. †`Iqv Av‡Q, x = 1 + logabc
ev, x − 1 = logabc
ev, ax − 1 = bc ................ (i)
Avevi, y = 1 + logbca
ev, y − 1 = logbca
ev, by − 1 = ca ............... (ii)
Abyiƒcfv‡e, cz − 1 = ab .......... (iii) 1, 5
(i), (ii) I (iii) ¸Y K‡i cvB,
ax −1. by − 1. cz − 1 = bc. ca. ab
ev, ax − 1. by − 1. cz − 1 = a2.b2.c2 (−2 251 ) −1, 51 (0, 1)
ax − 1 by − 1 cz − 1 −1 −2 −3 O
ev, a2 . b2 . c2 = 1 X
Y
1 2 3 X

ev, ax − 1 − 2. by − 1 − 2. cz − 1 − 2 = 1 M. †jLwPÎ †_‡K †`Lv hvq †h, hLb x = 0

 ax − 3. by − 3. cz − 3 = 1 (†`Lv‡bv n‡jv) ZLb y = 5 = 1 Kv‡RB †jLwU (0, 1) we›`yMvgx|
œ 4  wb‡Pi QKwU jÿ Ki :
cÖk- Avevi x G‡i FYvZ¥K gv‡bi Rb¨ y Gi gvb µgvš^‡q k~‡b¨i
x −2 −1 0 1 2 LyeB KvQvKvwQ †cuŠQvq wKš‘ 0 nq bv A_©vr x → − , y → 0+.
y 1 1 1 5 25 x Gi †h‡Kv‡bv abvZ¥K gv‡bi Rb¨ dvskbwUi gvb Amx‡gi
25 5 KvQvKvwQ A_©vr x → , y → .
Avevi, dvskbwU (x) = ax AvKv‡ii †hLv‡b a > 0 Ges a  0|
K. QKwU †Kvb dvskb Øviv eY©bv Kiv hvq| 2 myZivs y = 5x GKwU m~PKxq dvskb|
L. dvskbwUi †jLwPÎ A¼b Ki| 4 myZivs dvskbwUi †Wv‡gb mKj ev¯Íe msL¨vi †mU A_©vr (−, )
M. dvskbwUi cÖK…wZ eY©bv Ki Ges †Wv‡gb I †iÄ Ges dvskbwUi †iÄ mKj abvZ¥K ev¯Íe msL¨vi †mU A_©vr (0, )|
wbY©q Ki| 4
œ 5  y = 2−x GKwU dvskb †hLv‡b −3  x  3
cÖk-
 4bs cª‡kœi mgvavb  K. cÖ`Ë mxgvi g‡a¨ dvskbwUi K‡qKwU gv‡bi ZvwjKv
K. QKwU‡Z ewY©Z (x, y) µg‡Rv‡oi gvb¸‡jv y = 5x dvskb Øviv cÖ¯‘Z Ki| 2
eY©bv Kiv hvq, †hLv‡b x-ev¯Íe msL¨v| L. dvskbwUi †jLwPÎ A¼b Ki| 4
L. QK KvM‡R myweavgZ x-Aÿ eivei XOX Ges y-Aÿ eivei M. dvsk‡bi †Wv‡gb I †iÄ wbY©q Ki Ges wecixZ
YOY AuvwK| x-Aÿ eivei 5 eM© Ni = 1 GKK Ges y-Aÿ dvskbwUI wbY©q Ki| 4
eivei 2 eM© Ni = 1 GKK we‡ePbv K‡i (x, y) we›`y¸‡jv QK  5bs cª‡kœi mgvavb 
KvM‡R ¯’vcb Kwi| we›`y¸‡jv mvejxjfv‡e eµ‡iLvq hy³ K‡i
dvskbwUi †jL cvIqv hvq| hv wb‡¤œ †`Lv‡bv n‡jv : K. awi, y = (x) = 2−x
x Gi −3 †_‡K 3 Gi g‡a¨ K‡qKwU gvb wb‡q mswkøó y Gi
gvb wb‡Pi Q‡K †`Lv‡bv n‡jvÑ
x −3 −2 −1 0 1 2 3
y 8 4 2 1 0.5 0.25 0.125
L. QK KvM‡Ri myweavgZ x-Aÿ XOX Ges YOY AuvwK| x-
Aÿ eivei ÿz`ªZg 5 eM© Ni = 1 GKK Ges y-Aÿ eivei
ÿz`ªZg 5 eM© Ni = 1 GKK a‡i (x, y) we›`y¸‡jv cvZb Kwi|
we›`y¸‡jv‡K mvejxjfv‡e eµ‡iLvq hy³ K‡i y = (x) Gi †jLv
cvIqv hvq| hv wb‡¤œ †`Lv‡bv n‡jvÑ
 beg-`kg †kÖwY : D”PZi MwYZ  389
−3, 8 Y y 1 0.5 −1 −4 AmsÁvwqZ 8 5 35 3 275
ÿ z`Z
ª g 5 eM© Ni = 1 GKK
L. ÔKÕ Gi cÖvß we›`y¸‡jv QK KvM‡R myweavgZ x-Aÿ XOX
Ges y-Aÿ YOY AuvwK| x-Aÿ eivei ÿz`ªZg 5 eM© Ni = 1
GKK Ges y-Aÿ eivei ÿz`ªZg 2 eM© Ni = 1 GKK a‡i
(x, y) we›`y¸‡jv cvZb Kwi| we›`y¸‡jv‡K mvejxjfv‡e
eµ‡iLvq hy³ K‡i y = (x) Gi †jL cvIqv hvq|
(−2, 4) Y
xAÿ eivei ¶z`ªZg 5 eM© = 1 GKK
Aÿ eivei ¶z`ªZg 2 eM© = 1 GKK
(15, 8)
(−1, 2)
(2, 5)
(3, 35)
0, 1 1,05 (2, 25 (4, 3)
(3, 125 )
−2, 1 (5, 275)
X −3 −2 −1 O 1 2 X −1, 0.5
3 )
Y
X O X
(0, −1)
M. †jLwPÎ †_‡K †`Lv hvq †h, x Gi abvZ¥K gvb e„w×i Rb¨
dvskwUi gvb µgk: k~‡b¨i KvQvKvwQ †cuŠQvq wKš‘ k~b¨ nq (0.5, −4)
1 1
bv| x = 0 n‡j dvskbwUi gvb, y = 2−0 = 20 = 1 = 1 Kv‡RB
dvskbwU (0, 1) we›`yMvgx| Avevi, x Gi D”PZi FYvZ¥K Y
gv‡bi Rb¨ dvskbwUi gvb e„w× cvq| myZivs cÖ`Ë mxgvi g‡a¨
dvskbwU x = 1 Gi Rb¨ AmsÁvwqZ
dvskwUi †Wv‡gb = [−3, 3] Ges dvskbwUi †iÄ = 8 8 [1 ] †Wv‡gb D = R − {1}
wecixZ dvskb wbY©q : M. awi, y = (x) = x − 1
2x + 1
y =  (x) = 2−x
2x + 1
GLb, y = 2−x GLb, y = x − 1
ev, log2y = −x ev, y(x − 1) = 2x + 1
ev, x = − log2y ev, yx − 2x + y + 1
ev, x = log2y−1 ev, x(y − 2) = y + 1
1 y+1
 x = log2 x=
y y−2
1
wecixZ dvskb, f−1 : y → x hLb x = log2 y wecixZ dvskb −1 : y → x †hLv‡b, x = y − 2
y+1

1
ev, f−1 : y → log2 y ev, −1 : y → y − 2
y+1

y Gi ¯’‡j x ¯’vcb K‡i cvB, x+1

1 y Gi ¯’‡j x ¯’vcb K‡i cvB, −1 : x → x − 2
f−1 : x → log2
x x+1
1  −1 (x) = ;x2
 f−1 (x) = log2 x−2
x
5+x
2x + 1 œ 7  y = ln 5 − x GKwU jMvwi`g dvskb|
cÖk-
œ 6  y = x − 1 GKwU dvskb|
cÖk-
K. dvskbwU †h k‡Z©i Rb¨ AmsÁvwqZ †mme kZ© wbY©q
K. cÖ`Ë dvsk‡bi †jLwPÎ A¼‡bi R‡b¨ x I y Gi Ki| 2
gv‡bi ZvwjKv cÖ¯‘Z Ki| 2 L. dvskbwUi †Wv‡gb wbY©q Ki| 4
L. dvskbwUi †jLwPÎ A¼b Ki Ges †Wv‡gb wbY©q Ki| 4 M. dvskbwUi †iÄ wbY©q Ges wecixZ dvsk‡bi †Wv‡gb
M. dvskbwUi wecixZ dvskb wbY©q Ki| 4 I †iÄ †ei Ki| 4
 6bs cª‡kœi mgvavb   7bs cª‡kœi mgvavb 
2x + 1 K. x = 5 Gi Rb¨ dvskbwU AmsÁvwqZ| Avevi, jMvwi`g dvskb
K. awi, y = (x) = x − 1
5+x
cÖ`Ë dvskb (x) Gi †jLwPÎ A¼‡bi Rb¨ x Ges y Gi FYvZ¥K gv‡bi Rb¨I AmsÁvwqZ| ZvB 5 − x < 0 dvskbwU
gvb¸‡jvi ZvwjKv cÖ¯‘Z Kwi| AmsÁvwqZ|
x −2 −1 0 0.5 1 1.5 2 3 4 5
 beg-`kg †kÖwY : D”PZi MwYZ  390
5+x y 014 036 1 271 74 2008 546
L. awi, y = (x) = ln 5 − x
L. GLb, ÔKÕ G cÖvß we›`y¸‡jv QK KvM‡R myweavgZ X-Aÿ
†h‡nZz jMvwi`g dvskb ïaygvÎ abvZ¥K ev¯Íe msL¨vi Rb¨ XOX Ges Y-Aÿ YOY AuvwK| X-Aÿ eivei ÿz`ªZg 5 eM©
msÁvwqZ nq| Ni = 1 GKK Ges Y-Aÿ eivei ÿz`ªZg 1 eM© Ni = 1 GKK
5+x
 > 0 hw` (i) 5 + x > 0 Ges 5 − x > 0 nq a‡i (x, y) we›`y¸‡jv cvZb Kwi| we›`y¸‡jv‡K mvejxj
5−x
eµ‡iLvi hy³ K‡i y = (x) Gi †jL cvIqv hvq|
A_ev, (ii) 5 + x < 0 Ges 5 − x < 0 nq|
(i) bs n‡Z cvB, x > −5 Ges − x > − 5
hv wb‡¤œ †`Lv‡bv n‡jvÑ
(−4, 546)
ev, x > −5 Ges x < 5 Y

 †Wv‡gb = {x : −5 < x} Ges {x : x < 5} X Aÿ eivei ÿ z`Z

ª g 5 eM© Ni = 1 GKK
= (−5, )  (− , 5) = (−5, 5) Y Aÿ eivei ÿ z`Z
ª g 1 eM© Ni = 1 GKK
(ii) bs n‡Z cvB, x < −5 Ges −x < −5
ev, x < − 5 Ges x > 5
 †Wv‡gb = {x : x < −5}  (x : x > 5} = 
 cÖ`Ë dvsk‡bi †Wv‡gb,
D = (i) I (ii) G cÖvß †Wv‡g‡bi ms‡hvM = (−5, 5)   = (−5, 5)
5+x
M. awi, y = (x) = ln 5 − x
5+x
ev, ey = 5 − x (−3, 2008)
ev, 5 + x = 5ey − xey
ev, x(1 + ey) = 5(ey − 1)
5(ey − 1)
x = ey + 1 (−2, 74)
y Gi mKj ev¯Íe gv‡bi Rb¨ x Gi gvb ev¯Íe nq| (0, 1) (1, 36)
(−1,271) (2, 14)
 cÖ`Ë dvsk‡bi †iÄ R = R
−4 −3 −1 O 1
5(ey − 1) X
Y
X
wecixZ dvskb −1 : y → x †hLv‡b, x = ey + 1
5(ey − 1)
M. x Gi mKj ev¯Íe gv‡bi Rb¨ cÖ`Ë dvskb (x) msÁvwqZ|
ev, −1 : y → ey + 1  dvskbwUi †Wv‡gb D = R
y Gi cwie‡Z© x ewm‡q cvB, Ges x hLb +  Gi KvQvKvwQ nq ZLb (x) Gi gvb k~‡b¨i
5(e − 1)
x KvQvKvwQ nq Ges x Gi gvb n«v‡mi mv‡_ mv‡_ (x) Gi gvb
−1 : x → x
e +1 Amx‡gi w`‡K e„w× cvq|
5(ex − 1)
  (x) = x
−1  cÖ`Ë dvsk‡bi †iÄ R = (0, )
e +1
myZivs, wecixZ dvsk‡bi †Wv‡gb n‡e dvskbwU †iÄ Ges †iÄ ÔKÕ n‡Z cvB, y = e−x
n‡e dvskbwUi †Wv‡gb| ev, logey = −x
−1 ev, x = −logey
 D−1 = R Ges R = (−5, 5) (Ans.)
ev, x = logey−1
1
œ 8  (x) = e−x GKwU dvskb|
cÖk- ev, x = loge y
K. cÖ`Ë dvsk‡bi †jLwPÎ A¼‡bi Rb¨ GKwU mviwY wecixZ dvskb −1 : y → x †hLv‡b, x = loge y
1
ˆZwi Ki| 2
1
L. dvskbwUi †jLwPÎ A¼b Ki| 4 ev, −1 : y → loge y
M. dvskbwUi †Wv‡gb I †iÄ wbY©q Ki Ges wecixZ y Gi cwie‡Z© x ewm‡q cvB,
dvskb wbY©q Ki| 4 1
−1 : x → loge
 8bs cª‡kœi mgvavb  x
1
K. awi, y = (x) = e−x  −1(x) = loge
x
x Gi K‡qKwU gvb wb‡q mswkøó y Gi gvb wb‡gœi Q‡K
†`Lv‡bv n‡jv-
x 2 1 0 −1 −2 −3 −4
 beg-`kg †kÖwY : D”PZi MwYZ  391

AwZwi³ m„R bkxj cÖkœ I mgvavb

log p log q log r  †Wv‡gb D =
œ 9  y −kz = z −kx = x −ky
cÖk-
†iÄ : y = 1 − 2−x
K. cÖgvY Ki †h, pqr = 1 2 ev, 2−x = 1 − y
L. py + z.qz + x.rx + y = 1 4 ev, − x = log2(1 − y)
2 2 2 2 2 2
M. py + yz + z .qz + zx + x .rx + xy + y = 1 4
 9bs cª‡kœi mgvavb  ev, x = log2(1 − y)−1

logkp logkq logkr  x = log  1 

K. awi, y − z = z − x = x − y = c 1 − y
2

 logkp = c(y − z) ..........(i) ïaygvÎ abvZ¥K ev¯Íe msL¨vi Rb¨ jMvwi`g msÁvwqZ nq|
logkq = c(z − x) ........... (ii) 1
 >0 hw` 1 − y > 0 nq|
1−y
logkr = c(x − y) ........... (iii)
ev, 1 > y
(i), (ii) I (iii) bs †hvM K‡i cvB,
logkp + logkq + logkr = c(y − z + z − x + x − y) y<1
ev, logkpqr = c.0 = 0 = logk1  †iÄ R = (− , 1)
 pqr = 1 (cÖgvwYZ) L. †jLwPÎ A¼b : cÖ`Ë dvskb, y = 1 − 2−x
L. mgxKiY (i), (ii) I (iii) †K h_vµ‡g (y + z), (z + x) I (x + y) cÖ`Ë dvsk‡bi †jLwPÎ A¼‡bi Rb¨ x I y Gi gv‡bi GKwU
Øviv ¸Y Kivi ci †hvM K‡i cvB, ZvwjKv cÖ¯‘Z Kwi :
(y + z)logkp + (z + x)logkq + (x + y)logkr = x −3 −2 −1 0 1 2
c {(y + z)(y − z) + (z + x)(z − x) + (x + y)(x − y)} y −7 −3 −1 0 0.5 0.75
ev, logkp(y + z) + logkq(z + x) + logkr(x + y) =
c{y2 − z2 + z2 − x2 + x2 − y2}
QK KvM‡R gvb¸‡jv ¯’vcb Ki‡j wbgœiƒc †jLwPÎ cvIqv hvq|
ev, logk(py + z.qz + x.rx + y) = c.0 = 0 = logk1 x-Aÿ : cÖwZ 4 eM© = 1 GKK a‡i
 py + z.qz + x.rx + y = 1 (cÖgvwYZ) y-Aÿ : cÖwZ 4 eM© = 1 GKK a‡i

M. mgxKiY (i), (ii) I (iii) †K h_vµ‡g (y2 + yz + z2), (z2 + zx + x2)

I (x2 + xy + y2) Øviv ¸Y Kivi ci †hvM K‡i cvB,
(y2 + yz + z2)logkp + (z2 + zx + x2)logkq +
(x2 + xy + y2)logkr = c {(y −z)(y2 + yz + z2) +
(z − x)(z2 + zx + x2) + (x − y)(x2 + xy + y2)}
2 + yz + z2) 2 + zx + x2) 2 + xy + y2)
ev, logkp(y + logkq(z + logkr(x
3
= c{y − z3 + z3 − x3 + x3 − y3}
2 2 2 2 2 2
ev, log k(py + yz + z .qz + zx + x .rz + xy + y ) = c.0 = 0 = logk1
2 + yz + z2 2 + zx + x2 z2 + xy + y2
 py .q z .r = 1 (cÖgvwYZ)
œ 10  †`Iqv Av‡Q, y = 1 − 2 − x
cÖk- ˆewkó¨ :
1. †iLvwU g~jwe›`yMvgx|
K. cÖ`Ë dvsk‡bi †Wv‡gb I †iÄ wbY©q Ki| 2
2. dvskbwUi †Wv‡gb D =
L. dvskbwUi †jLwPÎ A¼b Ki Ges Gi ˆewk󨸇jv 3. dvskbwUi †iÄ R = (−,1)
†jL| 4 M. wecixZ dvskb wbY©q :
M. dvskbwUi wecixZ dvskb wbY©q K‡i Zv GK-GK GLv‡b, y = 1 − 2−x = (x) (awi)
wKbv Zv wba©viY Ki| 4 ev, 2−x = 1 − y
ev, − x = log2(1 − y)
 10bs cª‡kœi mgvavb 
ev, x = log21 − y
1
K. GLv‡b, y = 1 − 2−x −
y = (x) n‡j  1(y) = x
x Gi mKj ev¯Íe gv‡bi Rb¨ dvskbwU msÁvwqZ|
 beg-`kg †kÖwY : D”PZi MwYZ  392
logkc = p(a − b) .......... (iii)
  1(y) = log2 
− 1
1 − y (i)  a + (ii)  b + (iii)  c K‡i cvB,
ev,  −1(x) = log21 − x
1 a logka + b logkb + c logkc = p{a(b − c) + b(c − a) + c(a − b)}
ev, logkaa + logkbb + logkcc = p (ab − ca + bc − ab + ca − bc)
wb‡Y©q wecixZ dvskb  −1(x) = log21 − x
1
a
ev, logk(a .bb.cc) = p.0 = 0 = logk1
x1 , x2  aa.bb.cc = 1 (Ans.)
  (x1) = log2 
−1 1
œ 12  a, b, c > 0 Ges a, b, c  1
cÖk-
1 − x1
K. loga(abc) = x n‡j, a = KZ? 2
Ges  −1(x2) = log21 − x 
1
 2 L. †`LvI †h,
1
+
1
+
1
=1 4
loga(abc) logb(abc) logc(abc)
GLb,  −1 (x1) =  −1(x2)
M. hw` p = loga(bc), q = logb(ca) Ges r = logc(ab)
ev, log21 − x  = log21 − x 
1 1
 
1  
2
1 1 1
nq Z‡e †`LvI †h, 1 + p + 1 + q + 1 + r = 1 4
1 1
ev, 1 − x = 1 − x  12bs cª‡kœi mgvavb 
1 2
ev, 1 − x1 = 1 − x2 K. †`Iqv Av‡Q, loga(abc) = x
ev, − x1 = −x2 x
ev, a = abc
 x1 = x2 x
a
†h‡nZz  −1(x1) =  −1(x2) Gi Rb¨ x1 = x2 nq| ev, a = bc
  1(x) = log2 
− 1
1 − x dvskbwU GKwU GK-GK dvskb| ev, ax − 1 = bc
1 2 1
logka −
œ 11  a, b, c  ; †hLv‡b b = (1 + 3 + 3 ) Ges
cÖk- 3 3
=  a = (bc) 1
x
b−c
logkb logkc 1
=
c−a a−b L. ÔKÕ n‡Z cvB, a = (abc)x
b awi, logb(abc) = y Ges logc(abc) = z
K. †`LvI †h, logalogaloga(a ) = b aa
2 1 1
L. †`LvI †h, b3 − 3b2 − 6b − 4 = 0 4 ev, b = (abc)y Ges c = (abc) z
a b c
M. a .b .c Gi gvb †ei Ki| 4 1 1 1
 11bs cª‡kœi mgvavb   abc = (abc) .(abc) .(abc)z
x y
1 1 1
ab b + +
K. evgcÿ = logalogalogaaa = logalogaaa logaa ev, (abc) = (abc)x y z
1

ab b
1 1 1
= logalogaa .1 = logaa logaa ev, x + y + z = 1
= logaab.1 = blogaa = b.1 = b = Wvbcÿ
1 1 1
ab  log (abc) + log (abc) + log (abc) = 1 (†`Lv‡bv n‡jv)
A_©vr logalogaloga(aa ) = b (†`Lv‡bv n‡jv) a b c

1 2 M. †`Iqv Av‡Q, p = loga(bc), q = logb(ca) Ges r = logc(ab)

3 3
L. †`Iqv Av‡Q, b = 1 + 3 + 3  1 + p = logaa + loga(bc) = loga(abc)
1 2 1 + q = logbb + logb(ca) = logb(abc)
3 3
ev, b − 1 = 3 + 3 .................. (i) 1 + r = logcc + logc(ab) = logc(abc)
 
1 2 3
Avevi, ÔLÕ n‡Z cvB,
ev, (b − 1) = 3 + 3 
3 3 3
[Nb K‡i] 1 1 1
+ + =1
 3  3  3 3
1 3 2 3 1 2 1 2 loga(abc) logb(abc) logc(abc)
ev, b − 1− 3b + 3b = 3  + 3  + 3.3 .3 3 + 3 
3 3
3 2
1 1 1
3 2 2 1  1 + p + 1 + q + 1 + r = 1 (†`Lv‡bv n‡jv)
ev, b −1−3b + 3b = 3 + 3 + 3.3 .(b −1) [(i) †_‡K]
3 2
ev, b − 3b + 3b − 1 = 12 + 9b − 9 2x + 1
3 2
ev, b − 3b + 3b − 1 − 12 − 9b + 9 = 0 œ 13  (x) = x −1
cÖk-
3 2
 b − 3b − 6b − 4 = 0 (†`Lv‡bv n‡jv) K. cÖ`Ë dvskbwUi †Wv‡gb I †iÄ wbY©q Ki| 2
logka logkb logkc L. dvskbwUi wecixZ dvskb wbY©q Ki Ges wecixZ
M. b − c = c − a = a − b = p (awi) dvsk‡bi †Wv‡gb I †iÄ wbY©q Ki| 4
5+x
Zvn‡j, logka = p(b − c) .......... (i) M. hw` y = ln 5 − x nq, Z‡e dvskbwUi †Wv‡gb I
logkb = p(c − a) .......... (ii)
 beg-`kg †kÖwY : D”PZi MwYZ  393
†iÄ wbY©q Ki| 4 ev, −5 < x Ges x < 5
 13bs cª‡kœi mgvavb   †Wv‡gb = {x : −5 < x}  {x : x < 5}
2x + 1 = {−5, }  {, 5}
K. cÖ`Ë dvskb, (x) = x − 1 −G = {−5, 5}
x − 1 = 0 ev, x = 1 emv‡j dvskbwU AmsÁvwqZ nq| (ii) n‡Z x < −5 Ges 5 < x

 †Wv‡gb  = − {1} ev, x < −5 Ges x > 5

2x + 1  †Wv‡gb = {x : x < −5} {x : x < 5}
Avevi awi, y = x − 1 =
ev, 2x + 1 = xy − y  cÖ`Ë dvsk‡bi †Wv‡gb D = (i) I (ii) †ÿ‡Î cÖvß †Wv‡g‡bi ms‡hvM
ev, 2x − xy = −1 − y = {−5, 5}  
ev, x(2 − y) = −(1 + y) = {−5, 5}
−(y + 1) 5+x
†iÄ : y = ln 5 − x
ev, x = −(y − 2)
y+1 5+x
x= ............... (i) ev, ey = 5 − x
y−2
(i)-G y = 2 emv‡j x Gi gvb AmsÁvwqZ nq| ev, xey + x = 5ey − 5
 †iÄ  = − {2} 5 (ey − 1)
x=
†Wv‡gb  = − {1}, †iÄ  = − {2} (Ans.) ey + 1
yGi mKj ev¯Íe gv‡bi Rb¨ x Gi gvb ev¯Íe nq|
L. msÁvbymv‡i, ( −1(x)) =x
 cÖ`Ë dvsk‡bi †iÄ R =
(y) = x .......... (i) [†h‡nZz −1(x) = y]
logk1 + x
†`Iqv Av‡Q, (x) = x − 1
2x + 1 œ 14 
cÖk- logkx
=2

2y + 1
ev, (y) = y − 1 K. cÖgvY Ki †h, x2 − x − 1 = 0 2
1+ 5
2y + 1 L. †`LvI †h, x = 4
ev, x = y − 1 2
1+ 5
ev, 2y + 1 = xy − x M. x = 2
Ges log Gi wfwË 2 a‡i DcwiD³
ev, 2y − xy = −1− x mgxKi‡Yi mZ¨Zv hvPvB Ki| 4
ev, y (2 − x) = −(1 + x)  14 bs cª‡kœi mgvavb 
−(x + 1)
ev, y = −(x − 2) logk(1 + x)
K. †`Iqv Av‡Q, logkx
=2
x+1
ev, y = x − 2 ev, logk(1 + x) = 2logkx
 −1(x) =
x+1 ev, logk(1 + x) = logkx2
x−2
ev, 1 + x = x2
x − 2 = 0 ev, x = 2 emv‡j dvskbwU AmsÁvwqZ nq|  x2 − x − 1 = 0 (cÖgvwYZ)
−1
 †Wv‡gb  = − {2} (Ans.)
x+1 L. ÔKÕ †_‡K cvB, x2 − x − 1 = 0
Avevi awi, y = x − 2 1 2

ev, xy − 2y = x + 1
ev, (x)2 − 2.x.2 + 2 (1) − 14 − 1 = 0
ev, xy − x = 2y + 1 1 2
5  52
ev, x(y − 1) = 2y + 1
( )
ev, x − 2 =
4 2
=

1 5
x=
2y + 1 ev, x − 2 =  2
y−1
y = 1 emv‡j x Gi gvb AmsÁvwqZ nq| 1
nq, x − 2 = 2
5 1
A_ev, x − 2 = − 2
5
−1
 †iÄ  = − {1} (Ans.)
5 1 5 1
M. †h‡nZz jMvwi`g ïaygvÎ abvZ¥K ev¯Íe msL¨vi Rb¨ msÁvwqZ ev, x = 2 + 2 ev, x = − 2 + 2
5+x
 >0 1+ 5 − 5 + 1 −( 5 − 1)
5−x x=  x=− =
2 2 2
hw` (i) 5 + x > 0Ges 5 − x >0 nq −( 5 − 1)
A_ev, (ii) 5 + x<0 Ges 5 − x<0 nq| GLv‡b, x = 2
MÖnY‡hvM¨ bq|
n‡Z x > −5 Ges 5 > x KviY x Gi FYvZ¥K gv‡bi Rb¨ logx Gi gvb msÁvwqZ bq|
 beg-`kg †kÖwY : D”PZi MwYZ  394

x=
1+ 5
2
(†`Lv‡bv n‡jv)  xlogk (ba) = log a (†`Lv‡bv n‡jv)
k

logk(1 + x)
M. †`Iqv Av‡Q, log x = 2 œ 16  x = 1 + logabc, y = 1 + logbca Ges z = 1 + logcab
cÖk-
k 1
cÖkœg‡Z, k = 2 [ wfwË = 2] K. †`LvI †h, a = (abc)x 2
logk(1 + x) log2(1 + x) L. cÖgvY Ki †h, xyz = xy + yz + zx 4
evgcÿ =
logkx
=
log2x
M. †`LvI †h, ax − 3.by − 3.cz − 3 = 1 4
log210  log10 (1 + x) log(1 + x)
=
log210  log10x
=
logx
 16bs cª‡kœi mgvavb 
K. †`Iqv Av‡Q x = 1 + logabc
log 1 +
1 + 5
 2  log 2.618 ev, x = logaa + logabc
= = = 2.000006
ev, x = logaabc
log 
1 + 5 log 1.618
 2  ev, ax = abc
= 2 Wvbcÿ (†`Lv‡bv n‡jv) 1
 a = (abc)x (†`Lv‡bv n‡jv)
œ 15  a3 − x b5x = a5 + xb3x
cÖk-
K. hw` x = 0 nq Z‡e cÖgvY Ki 2logka = 0 2 1
x
L. †`LvI †h, (1 + x)logka = xlogkb 4 L. ÔKÕ n‡Z cvB, a = (abc) .............. (i)
1
M. †`LvI †h, xlogk a = logka (b) 4 Abyiƒcfv‡e, b = (abc)y .............. (ii)
 15bs cª‡kœi mgvavb  1

K. †`Iqv Av‡Q, a3 − x b5x = a5 + x b3x Ges c = (abc)z .............. (iii)

ev, a3 − 0b5.0 = a5 + 0b3.0 [ x = 0] (i), (ii) I (iii) ¸Y K‡i cvB,
1 1 1
ev, a3b0 = a5b0
ev, a3 = a5 abc = (abc) .(abc) .(abc)z
x y
1 1 1
a5 + +
ev, =1 ev, (abc)1 = (abc)x y z
a3
1 1 1
ev, a2 = 1 ev, 1 = x + y + z
ev, logka2 = logk1 xy + yz + zx
ev, 1 =
 2logka = 0 (cÖgvwYZ) xyz
L. †`Iqv Av‡Q, a3 − x b5x = a5 + x b3x  xyz = xy + yz + zx (cÖgvwYZ)
5x 5+x
ev,
b
=
a M. †`Iqv Av‡Q, x = 1 + logabc
b3x a3 − x ev, x − 1 = logabc
ev, b2x = a2 + 2x
ev, ax − 1 = bc ................ (i)
ev, (bx)2 = (a1 + x)2
Avevi, y = 1 + logbca
ev, bx = a1 + x
x ev, y − 1 = logbca
ev, logkb = logka1 + x −1
 by = ca ................ (ii)
ev, xlogkb = (1 + x) logka
Abyiƒcfv‡e, cz − 1 = ab ...... (iii)
 (1 + x)logka = xlogkb (†`Lv‡bv n‡jv)
(i), (ii) I (iii) ¸Y K‡i cvB,
−1 −1 z−1
M. ÔLÕ bs †_‡K, b2x = a2 + 2x ax .by .c = bc.ab.ca
ev, b2x = a2x.a2 ev, a .by −1.cz −1 = a2b2c2
x −1

b2x x−1 y−1 z−1

ev, a2x = a2 a b c
ev, a2 . b2 . c2 = 1
b 2x − − −
 ax 3.by 3.cz 3 = 1 (†`Lv‡bv n‡jv)
( ) =a
ev, a 2

x
b 2x 2
ev, log (a) = log a
k k
2 œ 17  y = 2 GKwU m~PK dvskb Ges −3  x  3
cÖk-
b K. cÖ`Ë dvsk‡bi †jLwPÎ A¼‡bi Rb¨ x I y Gi
ev, 2x log (a) = 2 log a
k k gv‡bi ZvwjKv cÖ¯‘Z Ki| 2
L. dvskbwUi †jLwPÎ A¼b Ki| 4
M. dvskbwUi †Wv‡gb I †iÄ wbY©q Ki| 4
 beg-`kg †kÖwY : D”PZi MwYZ  395
 17bs cª‡kœi mgvavb  x
x M. †`Iqv Av‡Q, y = 22
x
K. awi, y = (x) = 22
x Gi K‡qKwU wbw`©ó gv‡bi Rb¨ y-Gi Avmbœ Abym½x gvb awi, y = (x) = 22
wbY©q Kwi Ges Q‡K wjwL : x Gi †h‡Kv‡bv ev¯Íe gv‡bi Rb¨ y = (x) Gi gvb msÁvwqZ
x −3 −2 −1 0 1 2 3 nq|
y myZivs dvskbwUi †Wv‡gb D =
0.35 0.5 0.70 1 1.41 2 2.82
GLb,
L. ÔKÕ Gi cÖvß we›`y¸‡jv QK KvM‡R myweavg‡Zv x Aÿ XOX (x) = y
Ges y-Aÿ YOY AuvwK| x-Aÿ eivei 4 ¶z`ªZg eM© = 1 ev, −1(y) = x .......... (i)
GKK Ges y Aÿ eivei ¶z`ªZg 10 eM© Ni = 1 GKK a‡i (x, x
y) we›`y¸‡jv cvZb Kwi| we›`y¸‡jv‡K mnRfv‡e eµ‡iLvq hy³ Ges y = 22
K‡i y = (x) Gi †jL cvIqv hvq| x
ev, log2y = 2
hv wb‡P †`Lv‡bv n‡jv Ñ
ev, x = 2log2y .......... (ii)
ïaygvÎ abvZ¥K ev¯Íe msL¨vi jMvwi`g msÁvwqZ nq|
myZivs y-Gi abvZ¥K ev¯Íe gv‡bi Rb¨ x-Gi ev¯Íe gvb Av‡Q|
 dvskbwUi †iÄ R = {xR : x > 0}

wbe©vwPZ m„Rbkxj cÖkœ I mgvavb

œ 18  p2 + q2 = 9pq
cÖk- ev, p2 − 2pq + q2 = 9pq − 2pq
K. †`LvI †h, log(p2 + q2) = 2 log3 + logp + logq. 2 ev, (p − q)2 = 7pq
ev, log(p − q)2 = log7pq [Dfq w`‡K log wb‡q]
L. †`LvI †h, log(p4 + q4) = log 79 + 2 (logp + ev, 2log (p − q) = log7 + logpq
logq) 4 ev, 2log (p − q) = log7 + logp + logq (cÖgvwYZ)
M. cÖgvY Ki †h, 2log(p − q) = log7 + logp + logq 4 logkp logkq logkr
 18bs cª‡kœi mgvavb  œ 19  y − z = z − x = x − y
cÖk-
K. †`Iqv Av‡Q, p2 + q2 = 9pq K. cÖgvY Ki †h, pqr = 1 2
mgxKi‡Yi Dfq cv‡k¦© log wb‡q cvB, L. py + z. qz + x. yx + y = 1 4
2 2 2 2 2 2
log (p2 + q2) = log 9pq M. py + yz + z  qz + zx + x  rx + xy + y = 1 4
= log9 + logp + logq
= log32 + logp + logq  19bs cª‡kœi mgvavb 
 log(p2 + q2) = 2log3 + logp + logq (†`Lv‡bv n‡jv) logkp logkq logkr
L. †`Iqv Av‡Q, p2 + q2 = 9pq K. awi, y − z = z − x = x − y = T
ev, (p2 + q2)2 = (9pq)2 [eM© K‡i]  logkp = T(y − z) ................. (i)
ev, p4 + q4 + 2p2q2 = 81p2q2 logkq = T (z − x) ................. (ii)
ev, p4 + q4 = 79p2q2
logkr = T (x − y) .................. (iii)
ev, log (p4 + q4) = log(79 p2q2) [Dfq w`‡K log wb‡q]
= log 79 + log (pq)2 mgxKiY (i), (ii) I (iii) †hvM K‡i cvB,
= log79 + 2log(pq) logkp + logkq + logkr = T (y − z + z − x + x − y)
 log (p4 + q4) = log79 + 2(logp + logq) (†`Lv‡bv n‡jv)
ev, logk (pqr) = T  0
M. †`Iqv Av‡Q, p2 + q2 = 9pq
 beg-`kg †kÖwY : D”PZi MwYZ  396
ev, logk (pqr) = 0 =
1
+
1
+
1
ev, logk (pqr) = logk1 1 + xa + x−b 1 + xb + x−c 1 + xc + x−a
1 1 1
 pqr = 1 (cÖgvwYZ) = b + +
x + x−c + 1 xc + x−a + 1 xa + x−b + 1
L. ÔKÕ Ask n‡Z cÖvß, logkp = T(y − z) 1 1 1
= + c +
ev, p = kT(y − z) b
1 x + x−a + 1 xa + x−b + 1
x + c+1
x
ev, py + z = kT(y − z) (y + z)
2 2
xc 1 1
 py + z = kT(y − z ) ......... (i) = + + [a + b + c = 0
1 + x c + xb + c 1 + xc + xb + c a 1
x + b+1
Abyiƒcfv‡e, q + x = kT(z − x ) ................(ii)
2 2
z x
Ges rx + y = kT(x
2
− y2) ................. (iii)  b + c = − a]
2 2 2 2 2 2 xc 1 xb
 py + z.qz + x.rx + y
= kT(y − z + z − x + x − y ) = + +
1 + xc + xb + c 1 + xc + xb + c xa+b + xb + 1
= kT.O = k0 = 1 xc 1 xb
 p .q .r
y + z z + x x + y = 1 (cÖgvwYZ) = + +
1 + xc + bb + c 1 + xc + bb + c x−c + xb + 1
M. ÔKÕ Ask n‡Z cvB, logkp = T(y − z) xc 1 xb
= c b + c + c b + c +
1+x +b 1+x +b 1
ev, p = kT(y − z) [j‡Mi msÁv n‡Z] xc
+ xb + 1
2 2 2 2
ev, py + yz +z = kT(y − z) (y + yz + z ) xc 1 xb.xc
= + +
 py
2 + yz + z2
= kT(y
3 − z3)
.............. (i) 1 + x c + bb + c 1 + x c + bb + c 1 + x c + bb + c
2 2 3 3 xc + 1 + x b + c 1 + x c + xb + c
Abyiƒcfv‡e, qz + zx + x = kT(z − x ) ............ (ii) = =
1 + x c + xb + c 1 + x c + xb + c
=1
2 2 3 3
Ges rx + xy + y = kT(z − x ) ..................... (iii) 1 1 1
 + + = 1 (†`Lv‡bv n‡jv)
mgxKiY (i), (ii) I (iii) ¸Y K‡i cvB, 1 + p + q−1 1 + q + r−1 1 + r + p−1
œ 21  f(x) = log (1 + x) − 2log(x)
2 + yz + z2 2 +zx + x2 2 + xy + y2 3 3 3 3 + x3 − y3)
py .qz . rx = kT(y − z + z − x cÖk-
= kT.0 = k0 = 1 K. †`LvI †h, logaxm = mlogax 2
2 + yz + z2 2 +zx + x2 2 + xy + y2
 py .qz . rx = 1 (cÖgvwYZ)
1+ 5
L. f(x) = 0 n‡j, †`LvI †h, x = 2 4
œ 20  p = xa, q = xb, r = xc Ges a + b + c = 0
cÖk-
K. (pqr)2 Gi gvb †ei Ki| 2 M. D Ges R wbY©q Ki| 4
a2 + ab + b2 b2 + bc + c2  21bs cª‡kœi mgvavb 
L. †`LvI †h, q−1   −1  
p q r
r  q−1 K. awi, log ax = p
c2 + ca + a2 ev, x = ap
=1 4 ev, xm = amp
M. cÖgvY Ki †h, ev, logaxm = logaamp
1 1 1 ev, logaxm = mp  logaa
+ −
1 + p + q−1 1 + q + r 1 + r + p−1
=1 4
ev, logaxm = mp
 20bs cª‡kœi mgvavb   logaxm = mlogax [†`Lv‡bv n‡jv]
K. †`Iqv Av‡Q, p = xa, q = xb, r = xc Ges a + b + c = 0 L. †`Iqv Av‡Q, (x) = log(1 + x) − 2log(x)
 (pqr)2 = (xa.xb.xc)2 = (xa + b + c)2 = (x0)2 = (1)2 = 1 = log (1 + x) − logx2
 (pqr)2 = 1 (Ans.) 1+x
a2 + ab + b2 b2 + bc + c2 c2 + ca + a2
= log 2
x
L. evgcÿ = q−1   −1   −1
p q r
r  q  GLb (x) = 0 n‡j,
a2 + ab + b2 b2 + bc + c2 c2 + ca + a2
= (xx )
a
b  (xx )
b
c 
c

(xx )
a
ev, log (1 x+ x) = 0 = log1
2

1+x
2 2
= (xa − b)a + ab + b  (xb − c)b + bc + c  (xc − a)c
2 2 2 + ca + a2
ev, x2 = 1
3 3 3 3 3 3
= x(a − b )  x(b − c )  x(c − a ) ev, x2 = 1 + x
3 3 3 3 3 3
= xa − b + b − c + c − a
ev, x2 − x − 1 = 0
= x0 = 1 = Wvbcÿ
1 1 1
a2 + ab + b2 b2 + bc + c2 c2 + ca + a2 ev, x2 − 2x.2 + 4 − 1 − 4 = 0
 q−1   −1   −1
p q r
r  q  =1 (†`Lv‡bv n‡jv) 2

( 1) = 54
ev, x − 2
1 1 1
M. 1 + p + q−1 + 1 + q + r−1 + 1 + r + p−1 1 5
ev, x − 2 = 2 [FYvZ¥K gvb eR©b K‡i]
 beg-`kg †kÖwY : D”PZi MwYZ  397
5 1 1+ 5 Y
x=
2 2
+ =
2
(†`Lv‡bv n‡jv) (35, 468)

M. (x) = log(1 + x) − 2 log(x)

log (1 + x) dvskbwU 1 + x > 0 ev, x > −1 Gi Rb¨
msÁvwqZ|
Avevi, logx dvskbwU x > 0 Gi Rb¨ msÁvwqZ
 f(x) = log (1 + x) − 2log(x) dvskbwU x > 0 Gi Rb¨
msÁvwqZ
(3, 27)
Df = {x  R : x  0} (Ans.)
1+x
 (x) = log 2 Gi †iÄ R = (0, ) (Ans.)
x
log (1 + y)
œ 22  †`Iqv Av‡Q, y = 3x Ges
cÖk- logy
=2
(25, 156)
K. y = Gi †Wv‡gb Ges †iÄ wbY©q Ki|
3x 2
L. y = 3x Gi †jLwPÎ A¼b Ki| 4 (2, 9)
M. wØZxq mgxKiY †_‡K †`LvI †h, y Gi †Kej GKwU (05, 173)
gvb mgxKiYwU‡K wm× K‡i| 4 (0, 1) (15, 519)
(1, 3)
 22bs cª‡kœi mgvavb  0 0.5 1 1.5 2 2.5 3 35 X

K. †`Iqv Av‡Q, y = 3x M. †`Iqv Av‡Q,

log (1 + y)
=2
log y
x -Gi †h‡Kv‡bv ev¯Íe gv‡bi Rb¨ y ev¯Íe n‡e|
ev, log (1 + y) = 2 log y
myZivs †Wv‡gb = R (Ans.)
ev, log (1 + y) = log y2
Avevi, y = 3x
ev, 1 + y = y2
ev, logy = log3x ev, y2 − y − 1= 0
ev, logy = x log3 ev, 4y2 − 4y + 1 − 5 = 0
ev, (2y − 1)2 = 5
log y
 x = log3 ev, 2y − 1 =  5
1 5
GLv‡b y- Gi gvb AFYvZ¥K n‡jB †Kej x Gi ev¯Íe gvb y=
2
cvIqv hv‡e| wKš‘ y FYvZ¥K
n‡j logy AmsÁvwqZ nq|
 y Gi gvb FYvZ¥K n‡Z cv‡i bv|
 †iÄ = {x : x  R Ges x > 0} (Ans.)
1+ 5
L. awi, x = (x) = 3x myZivs y = 2
0 †_‡K 3.5 Gi g‡a¨ x Gi K‡qKwU gvb wb‡q mswkøó y Gi  y Gi †Kej GKwU gvb mgxKiY‡K wm× K‡i (†`Lv‡bv n‡jv)
gvb wb‡¤œi Q‡K †`Lv‡bv n‡jvÑ œ 23  f(x) = − 5−x + 1, x  R n‡j,
cÖk-
x 0 0 .5 1 1 .5 2 2 .5 3 3 .5 3a 1
y 1 1.73 3 5.19 9 15.6 27 46.8 K. †`LvI 3b = 3b − a hLb a, b  N , a < b 2
GLb, QK KvM‡R myweavgZ X Aÿ YOY Ges Y Aÿ AuvwK|
X-Aÿ eivei ÿz`ªZg 10 eM© Ni = 1 GKK Ges Y-Aÿ eivei
L. f(x) Gi wecixZ dvskb‡K log b Gi gva¨‡g (a )
ÿz`ªZg 1 eM©Ni = 1 GKK a‡i (x, y) we›`y¸‡jv ¯’vcb Kwi| cÖKvk Ki| 4
we›`y¸‡jv‡K mnRfv‡e eµ‡iLvq hy³ K‡i y = (x) = 3x Gi M. †jLwP‡Îi gva¨‡g dvskbwUi †iÄ wbY©q Ki| 4
†jL cvIqv hvq| hv wb‡gœ †`Lv‡bv n‡jv:  23bs cª‡kœi mgvavb 
3a 1
K. †`Lv‡Z n‡e, 3b = 3b − a
3a 1 1
evgcÿ = 3b = 3b.3−a = 3b − a
= Wvbcÿ (†`Lv‡bv n‡jv)
L. †`Iqv Av‡Q, f(x) = −5−x + 1
ev, y = f(x) = −5−x + 1
 beg-`kg †kÖwY : D”PZi MwYZ  398
ev, 5−x = 1 − y M. cÖ`Ë dvskb, f(x) = −5−x + 1
ev, log5−x = log(1 − y) [Dfq c‡ÿ log wb‡q] awi, y = f(x) = −5−x + 1
ev, − xlog5 = log(1 − y) x Gi K‡qKwU gv‡bi Rb¨ y Gi cÖwZiƒcx gvb wb‡Pi Q‡K
log(1 − y) †`Iqv n‡jv :
ev, − 1 = log5
x −1 0 1 2 3
log(1 − y)
ev, x = − log5 y −4 0 0.8 0.96 0992
log(1 − y) Y
 f−1(y) = − (2, 0.96)
log5 (1, 0.8)
 y †K x Øviv cÖwZ¯’vcb K‡i, (3, 0.992)
log(1 − x) (0, 0)
f−1(x) = − (Ans.)
log5
x x

†¯ ‹j : X-Aÿ eivei ¶z`ªZg

5 eM© Ni = 1 GKK
Y-Aÿ eivei ¶z`Z ª g 5 eM©
Ni = 1 GKK
(−1, −4)

Y
†jLwPÎ n‡Z †`Lv hvq †h, x Gi gvb hZ e„w× cvq, y Gi
gvb ZZB 1 Gi KvQvKvwQ †cŠQvq wKš‘ 1 nq bv| A_©vr x →
, y →  ZLb y → 1| x Gi gvb hZB FYvZ¥K w`‡K e„w×
cvq, y Gi gvb ZZB n«vm †c‡Z _v‡K Ges µgvš^‡q − 
w`‡K avweZ nq| A_©vr x → − , y → − 
†Wv‡gb Dr = (−,); †iÄ Rr = (− , 1) (Ans.)

m„R bkxj cÖkœe¨vsK DËimn

œ 24  wb‡Pi mgxKiY¸‡jv jÿ Ki :
cÖk- œ 27  log4 x = a Ges log2y = b
cÖk-
(i) x2 − 5x + 6 = 0
(ii) 5x + 52−x = 26 K. x Ges y Gi gvb wbY©q Ki| 2
logk(3 + x) x
(iii)
(logkx) L. xy Ges y
†K 2
Gi kw³iƒ‡c cÖKvk Ki| 4
K. (i) bs mgxKi‡Yi wbðvqK †ei Ki| 2 x
L. (ii) bs mgxKiYwUi mgvavb Ki| 4 M. hw` xy = 128 Ges y = 4 nq, Z‡e a Ges b Gi gvb wbY©q Ki| 4
1 + 13
M. (iii) bs mgxKiY Øviv cÖgvY Ki †h, x= 2 4 x 9 5
DËi : K. 22a, 2b; L. xy = 22a + b, y = 22a-b; M. 4 , 2
DËi : K. 1; L. 0, 2
ab logk (ab) bc logk(bc) ca logk (ca) œ 28  y = logex GKwU jMvwi`wgK dvskb|
cÖk-
œ 25 
cÖk- a+b
=
b+c
=
c+a
=m
K. x I y Gi gv‡bi GKwU †Uwej ˆZwi Ki| 2
K. logk (ab) Ges logk (bc) Gi gvb KZ? 2
L. cÖgvY Ki †h, cc = km 4 L. dvskbwUi †jLwPÎ Au vK| 4
M. cÖgvY Ki †h, aa = bb = cc 4 M. †`LvI †h, dvskbwUi wecixZ dvskb = ex| GB dvskbwUi
DËi : K. m(aab+ b), m(bbc+ c) †Wv‡gb I †iÄ wbY©q Ki| 4
log a log b log c
œ 26  ax = b, by = c Ges cz = a.
cÖk- œ 29  b −kc = c −ka = a −kb
cÖk-
K. cÖ_g k‡Z© a = 3 I b = 81 n‡j, x Gi gvb KZ n‡e? 2
L. cÖ`Ë k‡Z©i mvnv‡h¨ xyz Gi gvb wbY©q Ki| 4 K. abc Gi gvb KZ? 2
1 1 1 L. cÖgvY Ki †h, aa.bb.cc = 1 4
M. xa = yb = cc Ges ÔLÕ bs n‡Z cÖvß gv‡bi Rb¨ cÖgvY Ki M. cÖgvY Ki †h, a(b + c).b(c + a).c(a + b) = 1 4
a + b + c = 0| 4
DËi : K. 4; L. 1 DËi : K. 1
 beg-`kg †kÖwY : D”PZi MwYZ  399

Aa¨vq mgwš^Z m„R bkxj cÖkœ I mgvavb

xa xb xc = (xyp − 1)q − r.(xyq − 1)r − p.(xyr − 1)p − q ................. (i)
œ 30  P = xb , Q = xc Ges R = xa.
cÖk-
= xq − r.(yp − 1)q − r.xr − p.(yq − 1)r −p.(xp − q)(yr − 1)p − q
K. Q = 1 n‡j, †`LvI †h, b = c. 2 = xq − r + r − p + p − q.ypq − q − rp + r.yqr − r − pq + p.yrp − p − qr + q
− − −− − −
L. †`LvI †h, = x0.ypq q rp + r + qr r pq + p + rp p qr + q
Pa + b − c . Qb + c − a. Rc + a − b = 1. 4 = x .y = 1.1 = 1 = Wvbcÿ
0 0

M. cÖgvY Ki †h,  xa − r br − pCp − q = 1 (†`Lv‡bv n‡jv)

(a2 + ab + b2) logkP + (b2 + bc + c2) logkQ + (c2
M. (q − r) loga + (r − p) logb + (p − q) logc
+ ca + a2) logkR = 0. 4 − − −
= (q − r) logxyp 1 + (r − p)logxyq 1 + (p − q)logxyr 1
 30bs cª‡kœi mgvavb  − − − −
= log(xyp 1)q r + log(xyq 1)r p + log(xyr 1)p q
− −

p−1 q−r q −1 r − p r −1 p − q
xb = log{(xy ) .(xy ) .(xy ) }
K. †`Iqv Av‡Q, Q = xc = xb–c = log1 [(i) Gi mvnv‡h¨]
hw` Q = 1 nq, = 0 (Ans.)
1 = xb–c œ 32  x = logay †hLv‡b a > 0, a 1
cÖk-
ev, x = xb–c x
ev, 0 = b – c  1 x − y x − y
2 2

 x x + y 
 b = c (†`Lv‡bv n‡jv) K. 2   Gi gvb KZ? 2
L. †`Iqv Av‡Q, pa+b–c . Qb+c–a . Rc+a–b 1
−
1
a a+b–c b b+c–a c c+a–b
L. y = 23 + 2 3 n‡j, †`LvI †h, 2y3 − 6y − 5 = 0 4
= (xx ) (xx ) (xx )
b c a
log10(1 + x)
= (xa–b)a+b–c . (xb–c)b+c–a . (xc–a)c+a–b M. x Gi †Kvb gv‡bi Rb¨ log10x
=2 n‡e? 4
2 2 2 2 2 2
= xa +ab–ac–ab–b +bc . xb +bc–ab–bc–c +ac . xc +ac–bc–ac–a +ab
=x a2–ac–b2+bc
.x b 2–ab–c2+ac
.x c2–bc–a2+ab  32bs cª‡kœi mgvavb 
= x = 1 x x
 1 x − y x − y  1 (x − y) (x + y)x − y
2 2
 pa+b–c . Qb+c–a . Rc+a–b =1 (†`Lv‡bv n‡jv)  
 x x+y   (x + y) 
M. (a2 + ab + b2) logkP + (b2 + bc + c2) logkQ + (c2 + ca + a2) logkR K. 2   = 2x 
xa xb (x − y)x
= (a2 + ab + b2) logk b + (b2 + bc + c2) logk c +  1 x−y  1x
= 2x = 2x = 21 = 2 (Ans.)
x x
2 2
xc 1 1
(c + ca + a ) logk a −
x 3 3
L. y=2 +2 ........................ (i)
= (a2 + ab + b2) logkxa–b + (b2 + bc + c2) logkxb–c +
(c2 + ca + a2) logkxc–a
 − 33
1 1

= 2 +2 
3
= (a – b) (a2 + ab + b2) logkx + (b2 + bc + c2) (b – c) logkx ev, y 3
[Nb K‡i]
+ (c2 + ca + a2) (c – a) logkx
 33  − 33  − 3
1 1 1 1
= (a3 – b3) logkx + (b3 – c3) logkx + (c3 – a3) logkx 1
−
1
ev, y3 = 2  + 2  + 3.23.2 32 + 2 
3
= (a3 – b3 + b3 – c3 + c3 – a3) logkx
= 0.logkx
=0 ev, y3 = 2 + 2−1 + 3.20.y [(i) †_‡K]
 (a2 + ab + b2) logkP + (b2 + bc + c2) logkQ + (c2 + ca + 1
a2) logkR = 0 (cÖgvwYZ) ev, y3 = 2 + 2 + 3y
œ 31  a = xyp − 1, b = xyq − 1 Ges C = xyr − 1
cÖk- ev, 2y3 = 4 + 1 + 6y
K. aq−r Gi mij gvb wbY©q Ki| 2  2y3 − 6y − 5 = 0 (†`Lv‡bv n‡jv)
L. †`LvI †h, aq − rbr − pcp − q = 1 4
log10(1 + x)
M. mij Ki : (q − r) loga + (r − p) logb + (p − q) M. =2
log10x
logc 4
 31bs cª‡kœi mgvavb  ev, 2log10x = log10(1 + x)

K. †`Iqv Av‡Q, a = xyp − 1 ev, log10x2 = log10(1 + x)

 aq−r = (xyp − 1)q−r = xq − r.ypq − q − pr + r (Ans.) ev, x2 = 1 + x
L. evgcÿ = a q−r r−p p−q
b c
ev, x2 − x − 1 = 0
 beg-`kg †kÖwY : D”PZi MwYZ  400
−(−1)  (−1)2 − 4.1.(−1) 1  1 + 4 (x − x2 − 1)2 (x − x2 − 1)2
ev, x = 2.1
=
2
= logk = logk
x2 − x2 + 1
x2 − ( x2 − 1)2
1 5 = logk (x − x2 − 1)2 = 2logk (x − x2 − 1)
=
2
= Wvbcÿ (†`Lv‡bv n‡jv)
wKš‘ FYvZ¥K gvb MÖnY‡hvM¨ bq, KviY log10x > 0
œ 34  (x) = ln(x − 4)
cÖk-
1 5 K. dvskbwUi wecixZ dvskb †ei Ki| 2
x= 2 (Ans.)
L. (x) Gi †Wv‡gb I †iÄ †ei Ki| 4
œ 33  a  0, Ges m, n  Z Ges FYvZ¥K c~Y© mvswL¨K m~P‡Ki
cÖk- M. (x) dvskbwUi †jLwPÎ A¼b Ki| 4
n
Rb¨ (am) = amn m~ÎwU mZ¨|  34bs cª‡kœi mgvavb 
m n
K. †`LvI †h, (a ) = amn, †hLv‡b m < 0 Ges n < 0 2 K. †`Iqv Av‡Q, (x) = ln (x − 4)
b c a awi, y = (x) = ln (x − 4)
bc ca ab  y = (x) Ges y = ln (x − 4)
xc xa x b
L. c

a

b
Gi gvb ev, x = −1(y) ev, ey = x − 4 ....... (i)
xb xc xa
 x = ey + 4 ........ (ii)
I (ii) †_‡K  (y) = e + 4
(i) −1 y
wbY©q Ki| 4
 −1(x) = ex + 4
x− x2−1
M. cÖgvY Ki †h, logk = 2logk(x − x2−1) 4 L. †h‡nZz jMvwi`g ïaygvÎ abvZ¥K ev¯Íe msL¨vi Rb¨ msÁvwqZ nq|
x+ x2−1 x−4>0
 33bs cª‡kœi mgvavb  ev, x > 4
ev, {x : x > 4}
K. †`Iqv Av‡Q, m < 0 Ges n < 0 = (4, )
awi, m = − q Ges n = − r, †hLv‡b, q, r x>4
msL¨v‡iLv :
G‡ÿ‡Î evgcÿ = (am)n = (a−q)−r O
1 1 1  cÖ`Ë dvsk‡bi †Wv‡gb = (4, )
= −q r = =
(a )  1 r 1 Avevi ÔKÕ n‡Z cvB, x = ey + 4 hv y Gi Rb¨ x ev, nq|
aq aqr  cÖ`Ë dvsk‡bi †iÄ = .
= aqr = a(−q)(−r) = amn = Wvbcÿ M. cÖ`Ë dvskb, y = (x) = ln(x − 4)
 (am)n = amn (†`Lv‡bv n‡jv) dvskbwUi †jLwPÎ A¼‡bi Rb¨ x I y Gi gvb¸‡jvi ZvwjKv
ˆZwi Kwi :
b c a
bc ca ab x 4 4.5 5 6 7 8 10
xc xa xb
L. cÖ`Ë ivwk = c

a

b
y − −0.693 0 0.693 1.09 1.39 1.79

xb xc xa

 b 1  c 1  a 1
xcbc xaca xbab
=  
 c 1  a 1  b 1
x 
b bc x  xaab
c ca

1 1 1
2 2 2
x c xa x b
=   = 1 (Ans.)
1 1 1
2 2 2
xb xc xa

x− x2 − 1
M. evgcÿ = logk g‡b Kwi, QK KvM‡Ri XOX eivei x-Aÿ, YOY eivei y
x+ x2 − 1
Aÿ Ges O g~jwe›`y| x-A‡ÿ cÖwZ ¶z`ªZg 2 eM© = 1 GKK
(x − x2 − 1)(x − x2 − 1) Ges y A‡ÿ cÖwZ ¶z`ªZg 10 eM© = 1 GKK a‡i Q‡K cÖvß
= logk
(x + x − 1)(x −
2
x2 − 1) (x, y) we›`y¸‡jv QK KvM‡R ¯’vcb Kwi Ges mvejxjfv‡e hy³
K‡i cÖ`Ë dvsk‡bi †jLwPÎ A¼b Kwi|
 beg-`kg †kÖwY : D”PZi MwYZ  401
b b+c c c+a a a+b
ev, x2 − 12x + 32 = 0
œ 35  A = xc
cÖk- (x )  (xx ) a  (xx )b
ev, x(x − 8) − 4(x − 8) = 0
2 2  (x − 4) (x − 8) = 0
B = a2 − 33 − 33 + 2 Ges a  0  x = 4 A_ev 8
P = loga(bc), q = logb(ca), r = logc(ab) n‡j, wb‡Y©q mgvavb, x = 4 A_ev 8
K. †`LvI †h, A = 1 2 3 3
L. †`Iqv Av‡Q, a2 − b2 = c3 Ges x = a + b + a − b
L. B = 0 n‡j †`LvI †h, 3a3 + 9a = 8 4
evgcÿ = x3 − 3cx − 2a
1 1 1
M. cÖgvY Ki †h, p + 1 + q + 1 + r + 1 = 1 4 3
= ( 3
a+b+ ) + 3.c ( a + b + a − b) − 2a
3 3
a−b
3

 35bs cª‡kœi mgvavb 

K. Abykxjbx-9.1 Gi c„ôv-184, D`vniY-12 `ªóe¨|
[ x = a + b + (a − b)] 3 3

= ( a + b) + 3. a + b. a − b( a + b + a − b)
3 3 3
2 2 3 3 3
−
L. †`Iqv Av‡Q, B = a2 − 33 − 3 3 + 2 Ges B = 0
+ ( a − b) −3.c( a + b + a − b) − 2a
3 3 3
2 2 3
−
A_©vr a2 + 2 + 33 −3 3 =0
3 2
2
−
2 = a + b + 3. a − b2. x + a − b − 3cx − 2a
ev, a2 +2= 33 +3 3
3 3
1 2 1 2 = 2a + 3 − 3cx − 2a = 3cx − 3cx = 0 = Wvbcÿ
c .x
( ) + (3 ) − 2
ev, a2 = 33
−
3  x3 − 3cx − 2a = 0 (†`Lv‡bv n‡jv)
1 2 1 2 1 1 1 1
ev, a = (3 ) + (3 ) − 23
− −
[ −
] 3
3 3 3 3 3 33. 3 3 = 3 = 1
(ba) + (ba) = 1
2 2 2
2
M. cÖgvY Ki‡Z n‡e, a+
1 1 3
ev, a = (3 ) − b
2 3−3 3
3 3 2

(ba) + (ba)
1 1
− evgcÿ =
ev, a = 33 − 3 3 [Dfqc‡ÿ eM©g~j Ges
 a  0 abvZ¥K gvb wb‡q]
a3 3 b2 1 3 b2  a = b3
= a3. 3 +
b3  a2 = b3 
1 1 3 = +
ev, a3 ( −
33 − 3 3 ) [Dfqcÿ‡K Nb K‡i]
b3 a2 b
1 3 1 3 1 1 1 1 a a 3 1 1
ev, a3 = ( ) − (3 ) −
33
−
3
−
3.33.3 3 (3 −
3−3 3 ) =
a
+
b
= a+
3
= Wvbcÿ
[ (a − b)3 = a3 − b3 − 3ab (a − b)] b
ev, a3 =3− 3−1 − 3.30.a a 3 3
b 2 1
1 1 1 1 1 1  ()b
+ ()= a
a+ (cÖgvwYZ)
− − − 3
[ 33 .3 3 = 33 3 = 3 Ges 33 −3 3 = a] b
1 loge(1 + x)
ev, a3 = 3 − 3 − 3a œ 37 
cÖk- logex
=2 GKwU jMvwi`wgK mgxKiY|
8 K. cÖ`Ë mgxKiYwU‡K x PjK msewjZ GKwU exRMvwYwZK
ev, a3 + 3a = 3
wØNvZ mgxKi‡Yi Av`k©iƒ‡c cÖKvk Ki| 2
(†`Lv‡bv n‡jv)
 3a3 + 9a = 8
L. ÔK' n‡Z cÖvß wØNvZ mgxKibwUi g~‡ji cÖK…wZ
M. Abykxjbx- 9.2 c„ôv-192, D`vniY-10 bs `ªóe¨|
wbY©q Ki Ges †jLwP‡Îi mvnv‡h¨ mgvavb Ki| 4
3 3 − x 5x
3
œ 36  hw` a > 0 Ges x = a + b + a − b Ges a = b3 nq
cÖk- M. hw` a b = a b nq Z‡e †`LvI †h,
5 + x 3x

Z‡e,
K. mgvavb Ki : log10 [98 + − 12x + 36] = 2 x2 2
x loge (ba) = log a e 4

L. hw` a2 − b2 = c3 Zvn‡j †`LvI †h, x3 − 3cx − 2a = 0 4  37bs cª‡kœi mgvavb 

loge(1 + x)
a 3 3
b 2 1 K. †`Iqv Av‡Q, =2
M. cÖgvY Ki : () b
+ () a
= a+
3
b
4
ev,
logex
2logex = logc( 1 + x) [Avo ¸bb K‡i]
 36bs cª‡kœi mgvavb  ev, logex2 = loge (1 + x)
ev, x2 = 1 + x
K. log10[98 + x2 − 12x + 36] = 2
 x2 − x − 1 = 0
ev, [98 + x2 − 12x + 36] = 102 [ logax = b n‡j x = ab]
BnvB wb‡Y©q wØNvZ mgxKi‡Yi Av`k©iƒc|
ev, 98 + x2 − 12x + 36 = 100 L. ÔKÕ n‡Z cÖvß mgxKiY,
ev, x2 − 12x + 36 = 2 x2 − x − 1 = 0 †hLv‡b, a = 1, b = −1 Ges c = −1|
ev, x2 − 12x + 36 = 4 [eM© K‡i]
GLv‡b wbðvqK = b2 − 4ac = (−1) − {4.1.(−1)}
 beg-`kg †kÖwY : D”PZi MwYZ  402
= 1 + 4 = 5 > 0 wKš‘
c~Y©eM© bq| k n
 mgxKiYwUi g~jØq ev¯Íe, Amgvb I Ag~j`|
(ii) A = x + 2
x ( )
GKwU wØc`x ivwk Ges D³ ivwki we¯Í…wZ‡Z

†jLwP‡Îi mvnv‡h¨ mgvavb wbY©q : PZz_© c` x gy³ we‡ePbv Kiv n‡jv|

K. cÖgvY Ki †h, m(n − 2) + n(m − 2) = 0 2
awi, y = x2 − x − 1.................... (i)
L. DÏxc‡Ki we¯Í…wZ †_‡K n Gi gvb wbY©q Ki| 4
(i) bs mgxKi‡Y x Gi wewfbœ gv‡bi Rb¨ y Gi gvb wb‡Pi
M. x Gi mnM 144 n‡j, †`LvI †h, k =  2
3 4
Q‡K wbY©q Kwi|
 38bs cª‡kœi mgvavb 
x −3 (1 2 3 4 5
y 11 1 1 5 11 19 K. †`Iqv Av‡Q, am  an = (am)n
GLv‡b, †j‡Li K‡qKwU we›`y n‡jvÑ ev, am + n = amn
 m + n = mn
(−3, 11), (−1, 1), 2, 1), (3, 5), (4, 11) I (6, 29)
evgcÿ = m(n − 2) + n(m − 2)
GLb, QK KvM‡Ri XOX eivei X- Aÿ, YOY eivei Y- = mn − 2m + mn − 2n
Aÿ Ges O g~jwe›`y| = 2mn − 2 (m + n)
Dfq A‡ÿ ÿz`ªZg e‡M©i cÖwZ evûi ˆ`N©¨‡K GKK a‡i = 2mn − 2mn [ m + n = mn]
= 0 = Wvbcÿ
we›`y¸‡jv ¯’vcb Kwi Ges †hvM Kwi|
 evgcÿ = Wvbcÿ
Y
A_©vr, m(n − 2) + n(m − 2) = 0 (cÖgvwYZ)
L. wØc`x Dccv`¨ e¨envi K‡i cvB,
(5, 19)

(x + xk ) = x + C x (xk ) + C x (xk ) + C
n 2
n n n−1 n n−2 n
2 1 2 2 2 3

(xk ) + ...................
3
n−3
x 2
(−3, 11) (4, 11)
k n k2 k3
= xn + nxn − 1  2 + C2 xn − 2  4 + nC3 xn − 3  6 + .......
x x x
(3, 5) = xn + nxn − 3 k + nC2 xn − 6 k2 + nC3 xn − 9 k3 + ............
we¯Í„wZwUi 4_© c` nC3 xn − 9 k3
(−1, 1) (2, 1) ivwkwU x gy³ e‡j
X X xn − 9 = x0
Y ev, n − 9 = 0
 n = 9 (Ans.)
Aw¼Z †jLwU X- Aÿ‡K x = 16 Ges M. ÔLÕ Ask n‡Z cÖvß, n = 9, we¯Í„wZwU‡Z ewm‡q cvB,
x = −06 we›`y‡Z †Q` K‡i‡Q| k 9
wb‡Y©q mgvavb : x = −06, 16 ( )
x + 2 = x9 + 9c1x9 − 3 k + 9c2x9 − 6 k2 + 9c3x9 − 9 k3 +
x
.............
M. †`Iqv Av‡Q, a3 − x b5x = a5 + x b3x = x9 + 9c1x6 k + 9c2x3 k2 + 9c3 k3 + ..................
b5x a5 + x cÖkœg‡Z, c2k2 = 144
9
ev, b3x = a3 − x [Dfqcÿ‡K a3 −x. b3x Øviv fvM K‡i]
98
ev, b5x − 3x = a5 + x − 3 + x ev, 12 k2 = 144
ev, b2x = a2 + 2x 72
ev, 2 k2 = 144
ev, b2x = a2.a2x
ev, 36 k2 = 144
b2x
ev, a2x = a2 [Dfqcÿ‡K a2x Øviv fvM K‡i] 144
ev, k2 = 36
b2x
ev, logea2x = logea2 [Dfqc‡ÿ loge wb‡q] ev, k2 = 4
2x  k =  2 (†`Lv‡bv n‡jv)
(b) = log a
ev, loge a e
2
x3 + 2x2 + 1
œ 39  (x) = x2 − 2x − 3 Ges g(y) = 22y − 3.2y + 2 + 32.
cÖk-
b
ev, 2x log (a ) = 2log a

b
e e
( 1)
K.  − 3 wbY©q Ki| 2
x log (a ) = log a (†`Lv‡bv n‡jv)
e e L. g(y) = 0 n‡j y Gi gvb wbY©q Ki| 4
M. (x) †K AvswkK fMœvs‡k cÖKvk Ki| 4
cÖk-œ 38  wb‡Pi Z_¨¸‡jv j¶ Ki Ges cÖkœ¸‡jvi DËi `vI:  39 bs cª‡kœi mgvavb 
(i) am.an = (am)n GKwU m~PKxq mgxKiY| x3 + 2x2 + 1
K. †`Iqv Av‡Q, (x) = x2 − 2x − 3
 beg-`kg †kÖwY : D”PZi MwYZ  403
1 1 3 2 x(x2 − 2x − 3) + 4x2 + 3x + 1
1
 (− ) =
(− ) + 2 (− ) + 1
3 3
=
2
x2 − 2x − 3
3 2 4x + 3x + 1
(−13) − 2 (−13) − 3 =x+ 2
x − 2x − 3
1 2 −1 + 6 + 27 4(x2 − 2x − 3) + 11x + 13
− + +1 =x+
x2 − 2x − 3
27 9 27
= =
1 2 1 + 6 − 27 11x + 13
+ −3 =x+4+ 2
9 3 9 x − 2x − 3
32 11x + 13
=x+4+
27 32 9 (x + 1)(x − 3)
= = 
−20 27 −20 11x + 13
9
GLv‡b, (x + 1)(x − 3) GKwU cÖK…Z fMœvsk|
8 11x + 13 A B
=−
15
(Ans.) awi, (x + 1)(x − 3)  (x + 1) + (x − 3) ............(i)
L. †`Iqv Av‡Q, (i) bs mgxKi‡Yi Dfqcÿ‡K (x + 1)(x − 3) Øviv ¸Y K‡i cvB,
g(y) = 22y − 32y + 2 + 32 11x + 13  A(x − 3) + B(x + 1) .......................(ii)
GLb, g(y) = 0 (ii) bs mgxKi‡Y x = 3 ewm‡q cvB,
ev, 22y − 32y + 2 + 32 = 0 33 + 13 = 4B
ev, 22y − 32y. 22 + 32 = 0 ev, 4B = 46
ev, 22y − 32y 4 + 32 = 0 23
B=
2
ev, (2y)2 − 122y + 32 = 0
Avevi, (ii) bs mgxKiY x = − 1 ewm‡q cvB,
ev, x2 − 12x + 32 = 0 [2y = x a‡i]
−11 + 13 = − 4A
ev, x2 − 8x − 4x + 32 = 0 ev, − 4A = 2
ev, x(x − 8) − 4(x − 8) = 0 1
A=−
ev, (x − 8)(x − 4) = 0 2
nq, x − 8 = 0 A_ev, x − 4 = 0 A I B Gi gvb (i) bs mgxKi‡Y ewm‡q cvB,
ev, x = 8 ev, x = 4 11x + 13
=
23
−
1
(x + 1)(x − 3) 2(x − 3) 2(x + 1)
ev, 2 = 2
y 3 ev, 2y = 22
wb‡Y©q AvswkK fMœvsk,
y=3 y=2
23 1
 y Gi gvb 2, 3 (Ans.) f(x) = x + 4 + − (Ans.)
2(x − 3) 2(x + 1)
x3 + 2x2 + 1
M. †`Iqv Av‡Q, f(x) = x2 − 2x − 3