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fo/u fopkjr Hkh# tu] ugha vkjEHks dke]
foifr ns[k NksM+s rqjra e/;e eu dj ';keA
iq#"k flag ladYi dj] lgrs foifr vusd]
^cuk^ u NksM+s /;s; dks] j?kqcj jk[ks VsdAA
jfpr% ekuo /keZ iz.ksrk
ln~xq# Jh j.kNksM+nklth egkjkt

Basics

A la[;k fudk; (NUMBER SYSTEM) :

(i) izkd`r la[;k,W(a natural Numbers) : -
Lka[;k,Wa 1] 2] 3] 4]---------- vkfn izkd`r la[;k,Wa dgykrh gSA
Ikzkd`r la[;k,Wa ds leqPp; dks N ls iznf’kZr djrs gSA
N dks I+ ;k Z+ ls Hkh iznf’kZr fd;k tk ldrk gSA
N={1, 2, 3, 4,….}

(ii) iw.kZ la[;k,Wa (Whole Numbers) : -

izkd`r la[;kvksa esa ;fn 'kwU; dks feyk fn;k tk, rks ;s lHkh la[;k,Wa feydj
iw.kZ la[;k,Wa dgykrh gSA
iw.kZ la[;kvksa ds leqPp; dks W ls iznf’kZr djrs gSA
W={0, 1, 2,….}

(iii) iw.kkZd (Integers) : -

Lak[;k,Wa ---- -3, -2, -1, 0, 1, 2, 3, ….. iw.kkZd dgykrh gSAa buds
leqPp; dks I ;k Z ls iznf’kZr djrs gSA
I (;k Z)={.. -3, -2, -1, 0, 1, 2, 3, …..}
(a) _.kkRed iw.kkZd a ks ds leqPp; dks I- ls iznf’kZr djrs gSA
-
I ={…., -3, -2, -1}
(b) v_.kkRed iw.kkZd a ks ds leqPp; dks W ls iznf’kZr djrs gSA
(c) v/kukRed iw.kkZd
a ksa ds leqPp; esa {…., -3, -2, -1, 0} vkrs gSAa
uksV % 'kwU; u rks /kukRed iw.kkZd
a gS vkSj u gh _.kkRed iw.kZd
a A
(iv) le iw.kkZd (Even Inteters) : -
,sls iw.kkZd
a tks 2 ls HkkT; gksrs gs] le iw.kkZd dgykrs gSAa
tSls 0, ± 2, ± 4,……

(v) fo"ke iw.kkZda (Odd Integers) : -

,sls iw.kkZd
a tks 2 ls vHkkT; gksr]s fo"ke iw.kkZd
a dgykrs gSAa
tSls ± 1, ±3, ± 5, ±7,…

(vi) vHkkT; la[;k,Wa (Prime Numbers) : -

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,sls izkd`r la[;k,Wa tks dsoy Loa; vFkok 1 ls gh HkkT; gks] vHkkT; la[;k,Wa
dgykrh gSa
mnkgj.k 2] 3] 5] 7] 11] 13] 17]19] 23] 29] 31 ----------------
(vii) la;Dq r la[;k,Wa (Composite Number) : -
,slh izkd`r la[;k,Wa ¼1 dks NksMd
+ j ½ tks vHkkT; ugha gS] la;Dq r la[;k,Wa
dgykrh gSAa

uksV %
(a) ‘1’ u rks vHkkT; la[;k gS u gh la;Dq r la[;k gSA
(b) ‘2’ gh dsoy ,slh le la[;k gS tks vHkkT; gSA
(c) ‘4’ gh lcls NksVh la;Dq r la[;k gSA
(viii) lg&vHkkT; la[;k,Wa (Co-prime Numbers) : -
nks izkd`r la[;k,Wa ¼;g t:jh ugha gS fd og vHkkT; gh gksa½ lg&vHkkT; dgkykrh gS] ;fn mudk e-l-i- bdkbZ gksA
mnkgj.k % ¼4] 9½] ¼3] 4½] ¼3] 10½] ¼3] 8½] ¼5] 6½] ¼7] 8½ vkfnA
uksV % (a) nks fHkUu &fHkUu vHkkT; la[;k,Wa ges’k gh lg&vHkkT; gksrh gS ysfdu bldk foifjr lR; ugha gSA
(b) nks dzekxr izkd`r la[;k,Wa ges’kk lg&vHkkT; gksrh gSA

(ix) ;qxr vHkkT; la[;k,Wa (Twin Prime) :-

;fn nks vHkkT; la[;kvksa ds e/; vUrj nks gS rks os ;qxy vHkkT; la[;k,Wa dgykrh gSA
mnkgj.k % ¼3] 5½] ¼5] 7½] ¼11] 13½] ¼17] 19½] ¼29] 31½

(x) ifjes; la[;k,Wa (Rational Numbers) : -

,slh la[;k,Wa ftUgs p/q :Ik esa fy[k ldrs gS] tgkWa p ,oa q iw.kkZd gS ,oa q≠0, ifjes; la[;k,Wa dgykrh gS ,oa buds
leqPp; dks Q ls iznf’kZr djrs gSA
p
vr% Q = { : p, q ∈ Ι and q ≠ 0}
q
1 22
mnkgj.k % ,2,0,−5, ,2.5,0.3333...... vkfnA
2 7
uksV % (i) izR;sd iw.kkZd
a la[;k ifjes; la[;k gksr h gS D;ksfa d bUgs p/1 ds :Ik esa fy[kk tk ldrk gSA
(ii) ifjes; lla[;kvksa dk n’keyo Hkkx ;k rks ifjfer gksrk gS ;k mldh iqujko`fŸk gksr h gSA

(xi) vifjes; la[;k,Wa (Irrational Numbers) : -

,slh okLrfod la[;k,Wa ftUgsa p/q :Ik esa ugha fy[kk tk lds] vifjes; la[;k dgykrh gSA muds leqPp; dks Qc ;k Q }kjk
iznf’kZr djrs gSA
mnkgj.k % 2,1 + 3 , e, π vkfnA

(xii) okLrfod la[;k,Wa (Real Numbers) : -

lHkh ifjes; rFkk vifjes; la[;k,W]a okLrfod la[;k,Wa dgykrh gSAa buds leqPp; dks R ls iznf’kZr djrs gSA
vr% R=Q∪Qc
izR;sd okLrfod la[;k ds laxr] la[;k js[kk ij ,d fcUnq gksr k gSA bl la[;k js[kk dks okLrfod js[kk Hkh dgrs gSA nwljs
'kCnksa es]a okLrfod js[kk dk izR;sd fcUnq ,d okLrfod la[;k dks n’kkZr k gSA

bl izdkjj ifjHkkf"kr lHkh okLrfod la[;k,W d

a ez xq.k/keZ dk ikyu djrh gS vFkkZr~ a vkSj b nks fHkUu&fHkUu okLrfod
la[;k,Wa gks] rks a<b ;k a>b.

uksV %& (a) lHkh iw.kkZd ifjes; la[;k gksrh gS ijUrq bldk foifjr lR; ugha gSA

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(b) vifjes; la[;k dk _.kkRed Hkh vifjes; la[;k gksrh gSA
(c) ,d ifjes; vkSj ,d vifjes; la[;k dk ;ksx ;k O;odyu lnSo ,d vifjes; la[;k gksrh gSA
mnkgj.k % 2 + 3 ,3 − 5

(d) ,d v’kwU; ifjes; la[;k vkSj ,d vifjes; la[;k dk xq.kuQy ;k HkkxQy lnSo ,d vifjes; la[;k
gksrh gSA

(e) ;fn a∈Q ,oa b∈Q gks] rks xq.kk ab ,d ifjes; la[;k gksxk dsoy ;fn a=0

(f) nks vifjes; la[;kvksa dk ;ksx] O;odyu] xq.kuQy vkSj HkkxQy ,d ifjes; ;k vifjes; la[;k gks
ldrh gSA
(xiii) lfEeJ la[;k,Wa (Complex Numbers) : -

os lHkh la[;kvksa tks a+ib ds :Ik esa fu:fir dh tk ldrh gS] lfEeJ la[;k,Wa dgykrh gSAa tgkWa a vkSj b okLrfod
la[;k,Wa ,oa i = − 1 gSA lfEeJ la[;k dks lkekU;r% z ls iznf’kZr fd;k tkrk gS vkSj lfEeJ la[;kvksa ds leqPp; dks C
ls fu:fir djrs gSA
uksV % N ⊂ W ⊂ Ι ⊂ Q ⊂ R ⊂ C .
(a) la;Xq eh lfEeJ la[;k
;k % ;fn z=a+ib, tgkWa a,b∈R, ,d lfEeJ la[;k gks] rks z dh la;Xq eh lfEeJ la[;k dks
z ls iznf’kZr djrs gSa rFkk z = a − ib

B. dqN egŸoiw.kz lw= %

2 2 2 2
(i) (a+b) =a +2ab+b =(a-b) +4ab
2 2 2 2
(ii) (a-b) =a -2ab+b =(a+b) -4ab
2 2
(iii) a -b =(a+b)(a-b)
3 3 3
(iv) (a+b) =a +b +3ab(a+b)
3 3 3
(v) (a-b) =a -b -3ab(a-b)
3 3 3 2 2
(vi) a +b =(a+b) -3ab(a+b)=(a+b)(a +b -ab)
3 3 3 2 2
(vii) a -b =(a-b) +3ab(a-b)=(a-b)(a +b +ab)
2 2 2 2
(viii) (a+b+c) =a +b +c +2ab+2bc+2ca
 1 1 1
2
=a +b +c +2abc 
2 2
+ + 
a b c
2 2 2 2 2 2
(ix) a +b +c -ab-bc-ca= [(a-b) +(b-c) +(c-a) ]
2 2 2 2 2 2
(x) a +b +c -3abc=(a+b+c)( a +b +c -ab-bc-ca)
2 2 2
=(a+b+c) [(a-b) +(b-c) +(c-a) ]
4 4 2 2
(xi) a -b =(a+b)(a-b)(a +b )
4 2 2 2 2 2 2
(xii) a +a +1=(a +1) -a =(1+a+a ) (1-a+a )

C. Hktdrk ds fu;r (Divisibility Test) : -

(i) dksbZ la[;k 2 ls HkkT; dgykrh gS ;fn vkSj dsoy ;fn mldk bdkbZ dk vad 2 ls foHkkftr gksA

(ii) dksbZ la[;k 3 ls HkkT; dgykrh gS ;fn vkSj dsoy ;fn mlds lHkh vadks dk ;ksx 3 ls foHkkftr gksA

(iii) dksbZ la[;k 4 ls HkkT; dgykrh gS ;fn vkSj dsoy ;fn mlds vfUre nks vad 4 ;s foHkkftr gksA
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(iv) dksbZ la[;k 5 ls HkkT; dgykrh gS ;fn vkSj dsoy ;fn mldk bdkbZ dk vad 0 ;k 5 gksA

(v) dksbZ la[;k 6 ls HkkT; dgykrh gS ;fn vkSj dsoy ;fn og la[;k 2 vkSj 3 nksuksa ls HkkT; gksA

(vi) dksbZ la[;k 8 ls HkkT; dgykrh gS ;fn vkSj dsoy ;fn mlds vfUre 3 vad 8 ls foHkkftr gksA

(vii) dksbZ la[;k 9 ls HkkT; dgykrh g ;fn vkSj dsoy ;fn mlds lHkh vadks dk ;ksx 9 ls foHkkftr gksA
(viii) dksbZ la[;k 10 ls HkkT; dgykrh gS ;fn vkSj dsoy ;fn mldk bdkbZ dk vad 0 gksA

(ix) dksbZ la[;k 11 ls HkkT; dgykrh gS ;fn vkSj dsoy ;fn le LFkkuksa ij vkus okys vad ks ds ;ksx vkSj fo"ke LFkkuksa
ij vkus okys vadks ds ;ksx dk vUrj 11 dk xq.kt gksA
mnkgj.kr% 1298] 1221] 123321] 12344321] 1234554321] 123456654321
D. ?kkrkad (Indices) : -
;fn a dksbZ v’kwU; okLrfod ;k dkYifud la[;k gks vkSj m ?kukRed iw.kkZd gS] rks am=a.a.a. ……a(m ckj)
;gkWa ‘a’ dks vk/kkj vkSj ‘m’ dks ?kkr dgrs gSA
(1) ?kkrkad fu;r (Law of indices) :
(i) a0=1 , (a≠0)

1
(ii) a −m = , (a≠0)
am
m+n m n
(iii) a =a .a ] tgkWa m ,oa n okLrfod la[;k,Wa gSA

am
(iv) a m −n = ] tgkWa m ,oa n okLrfod la[;k,Wa gS] (a≠0)
an
m n mn
(v) (a ) =a

q
(vi) ap / q = ap

E. (1) vuqikr (Ratio)

A
(i) ;fn A vkSj B leku izdkj dh nks jkf’k;kWa gS rks mudk vuqikr A : B gksrk gS] ftls ls Hkh iznf’kZr fd;k tkrk
B
gSA
a ma na
(ii) ,d vuqikr dks dbZ rjhdksa ls iznf’kZr fd;k tk ldrk gS tsl = = = ...... m,n,…. v’kwU; la[;k,Wa
b mb nb
gSAa
(iii) nks ;k nks ls vf/kd vuqikrksa dh rqyuk djus ds fy, muds gj dks leku cuk;k tkrk gSA

(2) lekuqikr (Proportion) : -

a c
Tkc nks vuqikr a : b vkSj c : d leku gks] rks pkjksa jkf’k;kWa a,b,c,d lekuqikrh dgykrh gSA ;fn = gks] rks a:b=c:d
b d
;k a:b::c:d
(i) a vkSj d dks ckg~; in dgrs gSa rFkk b vkSj c e/; in dgykrs gSAa
(ii) lekuqikr dk egŸoiw.kZ xq.k %& ckg~; inksa dk xq.kuQy = e/; inksa dk xq.kuQy
(iii) ;fn a : b=c : d gks] rks
b : a=d : c ¼izfryksekuqikr½

(iv) ;fn a : b =c : d gks] rks

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a : c =b : d ¼,dkUrjkuqikr½

(v) ;fn a : b =c : d gks] rks

a+b c +d
= ¼;ksxkuqikr½
b d

(vi) ;fn a : b =c : d gks] rks

a−b c −d
= ¼vUrjkuqikr ½
b d

(vii) ;fn a : b =c : d gks] rks

a+b c +d
= ¼;ksxkUrjkuqikr½
a−b c −d

F. cgqin (Polynomial) : -

;fn ,d O;atd f(x)=a0xn+a1xn-1+a2xn-2+…..+an-1x+an }kjk ifjHkkf"kr gS] tgkWa n v_.kkRed iw.kkZd

a rFkk
a0,a1,a2,…….a n okLrfod la[;k,Wa ,oa a 0≠0 gks] rc f(x), n ?kkr dk cgqin dgykrk gSA
(i) 'ks"kQy ize;
s (Remainder Theorem)
Ekkuk fd P(x) dksbZ ,d ;k ,d ls vf/kd ?kkr dk cgqin gS vkSj a dksbZ okLrfod la[;k gSA ;fn P(x) dks (x-a) ls
foHkkftr fd;k tk;s rks 'ks"kQy P(a) ds cjkcj gksr k gSA
(ii) xq.ku[k.M ize;
s (Factor Theorem)
Ekkuk P(x) dksbZ ,d ;k ,d ls vf/kd ?kkr dk cgqin gS vkSj a dksbZ okLrfod la[;k bl izdkj gS fd P(a)=0, rks (x-a),
P(x) dk ,d xq.ku[k.M gksrk gSA foykser% ;fn (x-a), P(x) dk ,d xq.ku[k.M gS rks P(a)=0

G. vUrjky (Intervals) :
vUrjky ewyr% R ds mileqPp; gksrs gS vkSj lkekU;r;k budk mi;ksx vlfedkvksa dks gy djus ;k izkUr Kkr djus esa
fd;k tkrk gSA ;fn a vkSj b nks okLrfod la[;k,Wa bl izdkj gS fd a<b gS rks ge rhu izdkj ds vUrjky fuEukuqlkj
ifjHkkf"kr dj ldrs gS%
iz;Dq r izr hd
(i) [kqyk ¼foo`Ÿk½ vUrjky : (a,b)={x:a<x<b} () ;k ] [
(ii) cUn ¼laoŸ` k½ vUrjky :[a,b]={x:a≤x≤b} []
(iii) v)Z&[kqyk ;k v)Z&cUn vUrjky :(a,b]={x:a<x≤b} ( ] ;k ] ]
;k [a,b)={x:a≤x<b} [ ) ;k [ [
vuUr vUrjky fuEukuqlkj ifjHkkf"kr fd;s tkrs gS %

(i) (a,∞)={x:x>a}

(ii) [a,∞)={x:x≥a}

(iii) (-∞,b)={x:x<b}

(iv) (-∞,b]={x:x≤b}

(v) (-∞,∞)={x:x∈R}
uksV % (a) x ds dqN fo’ks"k ekuksa ds fy, ge {} fpUg dk mi;ksx djrs gSA mnkgj.kFkZ% ;fn x=1, 2 gks] rks bls x∈{1,2} }kjk
fy[kk tkrk gSA
(b) ;fn x dk dksbz eku ugha gks] rks djrs gSa fd x∈φ ¼’kwU; leqPp;½

H. Qyuksa ds fofHkUu izdkj (Various Types of Functions) :

(i) ifjes; Qyu (Rational Function) :

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g( x )
y = f (x) = ] :Ik dk Qyu ifjes; gksrk gs] tgkWa g(x) ,oa h(x) cgqin Qyu gSA
h( x )
(ii) fujsi{s k eku Qyu@ekikad Qyu (Absolute Value Function / Modulus Function) :
x if x≥0
Ekikad Qyu dk izrhd f(x)=x gS rFkk bls y = x =  }kjk ifjHkkf"kr djrs gSA
− x if x<0

y=x

(iii) egŸke iw.kkZd

a Qyu ;k Lh<+h Qyu (Greatest Integer Function or Step Up Function) :
Qyu y=f(x)=[x] egŸke iw.kkZda Qyu dgykrk gs] tgkWa [x] ml egŸke iw.kkZd
a ds cjkcj gksr k gS tks ;k rks x ds
cjkcj gS ;k mlls NksV k gSA
mnkgj.k % [0.8]=0, [1.5]=1, [7.8]=7, [-1.2]= -2 vkfnA
(iv) pj?kkrar h; Qyu (Exponential Function) :
Qyu f(x)=ax=ex in a(a>0,a≠1,x∈R) pj?kkrakdh; Qyu dgykrk gS] pj?kkrkadh; Qyuksa ds vkjs[k fuEu izdkj
ds gks ldrs gS &
LFkfr -ΙΙ LFkfr -ΙΙΙ
a>1 ds fy, 0<a<1 ds fy,

Ι. Yk?kqx.kd
(i) la[;kvksa dk y?kqx.kd %
fdlh la[;k N dk vk/kkj a ij y?kqx.kd] ml ?kkrkad dks fu:fir djrk gS] ftldks a ij yxkus ls la[;k N izkIr gksr h
gS] bl la[;k dks logaN ls iznf’kZr djrs gSA

⇔ax=N, a>0, a≠
LogaN=x⇔ ≠1 & N>0

;fn a=10 gks] rks log10b dh ctk; logb fy[krs gSAa

;fn a=e gks] rks logeb dh ctk;
nb fy[krs gS] ;gkWa e usfi;j vk/kkj gS]a ftldk la[;kRed eku 2.7182 gksrk gSA

;kn j[ksa %
log102≈0.3010 ; log103≈0.4771

n2≈0.693 ;
n10≈2.303

(ii) izkUr :

La[;k LogaN ds vfLrRo ,oa vf}rh;rk dks izfrcU/kksa a>0, a≠1 ,oa N>0 dh lgk;rk ls Kkr fd;k tk ldrk gSA
Yk?kqx.kd dk vk/kkj ‘a’, bdkbZ ds cjkcj ugha gksuk pkfg, vU;kFk bdkbZ ds vykok vU; la[;kvksa ds y?kqx.kd ugha gksxas
rFkk izR;sd la[;k bdkbZ dk y?kqx.kd gksxhA
(iii) vk/kkjHkwr y?kqx.kdh; loZlfedk %

alogaN=N,a>0, a≠1 ,oa N>0

J. y?kqx.kd ds eq[; xq.k/kZe %

Ekkukfd M vkSj N LosPN /kukRed la[;k,Wa a>0, a≠1,b>0,b≠1 rFkk α,β dksbZ okLrfod la[;k,Wga ]S rks &
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(i) loga(M.N)=logaM+logaN; O;kid :Ik esa loga(x1 x2 …..xn)=logax1+ logax2+…….+ logaxn

(ii) loga(M/N)=logaM-logaN

(iii) logaMα =α.logaM

1
(iv) logaβ M = loga M
β

loga M
(v) logb M = ¼vk/kkj ifjorZu ize;
s ½
loga b
uksV %

Loga1=0
logaa=1
1

Log1/aa=-1
logb a =
loga b


x
a =e
x
na

alogc b = blogc a
K. y?kqx.krh; Qyuksa ds xzkQ (y=logax) :
LFkfr-ΙΙ LFkfr -ΙΙΙ
a>1ds fy,fy, 0<a<1 ds fy,

uksV % (i) ;fn la[;k vkSj vk/kkj bdkbZ ds ,d gh vksj fLFkr gks] rks y?kqx.kd dk eku /kukRed gksrk gSA
(ii) ;fn la[;k vkSj vk/kkj bdkbZ ds foijhr vksj fLFkr gks] rks y?kqx.kd dk eku _.kkRed gksr k gSA

L. y?kqx.kdh; lehdj.k
lehdj.k

Lkehdj.k Logax=logay laHko gS ;fn vkSj dsoy ;fn x=y vFkkZr~

⇔x=y
logax=logay⇔

lnSo nh xbZ lehdj.k dh oS/krk Kkr djuh pkfg, vFkkZr x>0, y>0, a>0, a≠1 gksus pkfg,A

Exercise – 1 Objective Questions

Hkkx-A : dsoy ,d lgh fodYi

1. ;fn A o B nks ifjes; la[;k,Wa gS rFkk AB, A+B,A-B ifjes; la[;k,Wa gks] rks A/B
(A) lnso ifjes; (B) dHkh Hkh ifjes; ugha
(C) ifjes; tc B≠0 (D) ifjes; tc A≠0

2. lHkh vifjes; la[;kvksa dks la[;k js[kk ij O;Dr fd;k tk ldrk gSA ;g dFku gS &
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(A) lnSo lR; (B) vlR;
(C) dqN ifjfLFkfr;ksa esa lR; (D) buesa ls dksbZ ugha

3. ,d ifjes; la[;k ‘x’ rFkk ,d vifjes; la[;k ‘y’ dk xq.kk &

(A) lnSo ifjes; gksrk gSA (B) ifjes; gksrk gS] tcfd y=π u gksA
(C) lnSo vifjes; gksrk gSA (D) vifjes; gksrk gS] tcfd x=0 u gksA

4. ;fn x,y ifjes; la[;k,Wa bl izdkj gS fd ( x + y ) + ( x − 2y ) 2 = 2x − y + ( x − y − 1) 6 ] rks

(A) x=1, y=1 (B) x=2, y=1

(C) x=5,y=1 (D) x ,oa y ds vuUr eku gks ldrs gSA

1 1 1 1
5. Js.kh + + + ...... + dk ;ksxQy cjkcj gS &
(1× 2) (2 × 3) (3 × 4) (100 × 101)
(A) 99/100 (B) 1/100 (C) 100/101 (D) 101/102

1 1
6. ;fn x + = 2 gks] rks x 2 + 2 dk eku gS &
x x
(A) 0 (B) 1 (C) 2 (D) 3

7. lehdj.k (x-1)2+(x-2)2+(x-3)2=0 ds okLrfod gyksa dh la[;k gS &

(A) 0 (B) 1 (C) 2 (D) 3

8. ;fn a,b,c okLrfod gks] rks a(a-b)+b(b-c)+c(c-a)=0 gksxk dsoy ;fn &
(A) a+b+c=0 (B) a=b=c
(C) a=b or b=c or c=a (D) a-b-c=0

(a − b)3 + (b − c )3 + (c − a)3
9. ;fn a,b,c fHkUu&fHkUu o okLrfod la[;k,Wa gks] rks dk eku gS &
(a − b)(b − c )(c − a)
(A) 1 (B) abc (C) 2 (D) 3

10. ;fn x-a O;atd x3-a2x+x+2 dk ,d xq.ku[k.M gks] rks a dk eku gS &
(A) 0 (B) 2 (C) -2 (D) 1

11. ;fn P(x)=kx3+3x2-3 ,oa Q(x)=2x3-5x+k dks (x-4) ls foHkkftr djus ij leku 'ks"kQy cprk gS] rks k dk eku gS &
(A) 2 (B) 1 (C) 0 (D) -1

12. ;fn 2x3-5x2+x+2=(x-2)(ax2-bx-1) gks] rks a o b ds eku dze’k% gS &

(A) 2,1 (B) 2,-1 (C) 1,2 (D) -1,1/2

13. lehdj.k 4x+3+3x-4=12 dk gy gS &

7 3 5 2 11 13 3 7
(A) x = − , (B) x = − , (C) x = − , (D) x = − ,
3 7 2 5 7 7 7 5

14. lehdj.k x2-3x+2=0 ds okLrfod gyksa dh la[;k gS &

(A) 1 (B) 2 (C) 3 (D) 4

15. [e]-[-π] dk eku gS] tgkWa [.] egŸke iw.kkZd

a Qyu dks iznf’kZr djrk gSA
(A) 5 (B) 6 (C) 7 (D) 8

1 1 1
16. + + dk eku gS &
log bc
abc log ca
abc log ab
abc
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(A) 1/2 (B) 1 (C) 2 (D) 4

17. log215.log1/6 2.log1/6 ds cjkcj ;k blls NksV egŸke iw.kkZd

a gS &
(A) 4 (B) 3 (C) 2 (D) 1

1
18. ;fn log x log18 ( 2 + 8 ) = gks] rks 1000 x dk eku gS &
3

(A) 8 (B) 1/8 (C) 1/125 (D) 125

log a 2
+1)3
2 21 / 4
− 3log27 ( a − 2a
19. vuqikr ds ljyhdj.k ls izkIr gksrk gS &
7 4 log49 a
− a −1

(A) a2-a-1 (B) a2+a-1 (C) a2-a+1 (D) a2+a+1

1 1 1
20. + + dk eku gS &
1 + logb a + logb c 1 + logc a + logc b 1 + loga b + loga c

1
(A) abc (B) (C) 0 (D) 1
abc

21. ;fn 3 2 l0g3 x − 2x − 3 = 0 gks] rks x ds fdrus eku lehdj.k dks lUrq"V djrs gS &
(A) 'kwU; (B) 1 (C) 2 (D) 2 ls vf/kd

22. lehdj.k log10 ( − x ) = log10 x 2 ds okLrfod gyksa dh la[;k gS &

(A) 'kwU; (B) dsoy ,d (C) dsoy nks (D) 4

3 x 2 −10 x + 3
23. lehdj.k x − 3 = 1 ds okLrfod gyksa dh la[;k gS &
(A) dsoy pkj (B) dsoy rhu (C) dsoy nks (D) dsoy ,d

24. log27 dk eku gS &

(A) ,d iw.kkZd
a (B) ,d ifjes; la[;k (C) ,d vifjes; l[a;k (D) ,d vHkkT; la[;k

25. 0.75 dk vk/kkj 16 ij izfry?kqx.kd dk eku gS &

(A) 4 (B) 6 (C) 8 (D) 12

Hkkx -(B) : ,d ;k ,d ls vf/kd lgh fodYi

y
26. ;fn x o y okLrfod la[;k,Wa gS ,oa = x gks] rks y dk eku ugh gks ldrk &
x
(A) -1 (B) 0 (C) 1 (D) 2

log3 135 log3 5

27. ;fn N = − gks] rks N gS &
log15 3 log405 3
(A) ,d izkd`r la[;k (B) ,d vHkkT; la[;k (C) ,d ifjes; la[;k (D) ,d iw.kkZd
a

2
28. lehdj.k fudk; log3x+log3y=2+log32 ,oa log27 ( x + y ) = ds gyksa dk leqPp; gS &
3
(A) (6,3) (B) (3,6) (C) (6,12) (D) (12,6)

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29. lehdj.k log x 2 16 + log2 x 64 = 3 j[krh gS &
(A) ,d vifjes; gy (B) dksbZ vHkkT; gy ugha (C) nks okLrfod gy (D) ,d iw.kkZd
a gy
 
 (log3 x ) − 2 log3 x + 5 
2 9

30. lehdj.k x  
= 3 3 j[krh gS &
(A) Bhd rhu okLrfod gy (B) de ls de ,d okLrfod gy
(C) ,d ,d vifjes; gy (D) lfEeJ ewy

Exercise – 2 Subjective Questions

1. fuEu dks fHkUukRed :Ik esa fyf[k, (p/q, tgkWa p,q∈Ι rFkk q≠0)
(i) 2.35 (ii) 1.1 4 (iii) 3.379 (iv) 12
2. fuEu esa ls dkSu cM+k gS &

7 6 9 6
(i) , (ii) 13 − 12 , 14 − 13 (iii) ,
8 7 11 − 2 3 3

3. fl) dhft, fd nks fHkUu fo"ke izkd`r la[;kvksa ds oxksZ dk vUrj ges’kk 8 dk xq.kt gksrk gSA

4. gj esa mifLFkr vifjes; inksa dks foyqIr dhft,A

2 −1 1
(i) (ii)
2 +1 1+ 2 + 3

5. fuEu ds xq.ku[k.M dhft,

1
(i) (x-y)3 - y3 (ii) a 3 − +4
a3
3 2 3
(iii) x - 6x +11x – 6 (iv) x - 9x - 10
2 2 2
(v) a (b-c) + b (c-a) + c (a-b)

6. xq.ku[k.M dhft, &

4 8 4
(i) 1+x +x (ii) x +4

a c e 2a 4b 4 + 3a 2c 2 − 5e 4 f
7. ;fn = = gks] rks dk eku a o b ds inksa esa Kkr dhft,A
b d f 2b 6 + 3b 2d2 − 5f 5

8. fuEu Qyuksa ds vkjs[k [khafp, &

(i) y=4x+5 (ii) y=2x-3

9. ;fn ;g Kkr gS fd a1 + a2 + a3 + ……+an=0 rks la[;kvksa a1,a2,…..,an ds ckjs esa D;k dgk tk ldrk gS
\
10. fuEufyf[kr lehdj.kksa dks gy dhft, &
(i) x+2=3 (ii) x-2x+5=0

(iii) xx=4 (iv) x-1-2=1

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(v) x 2-x+4=2x2-3x+1

(vi) x-3+2x+1=4

(vii) x-1-2=x-3

11. lehdj.k fudk; x+2+y=5,x-y=1 dks gy dhft,A

12. 7log3 5 + 3log5 7 − 5log3 7 − 7log5 3 dk eku Kkr dhft,A

13. ;fn 4A+9B=10C, tgkWa A=log164, B=log39 & C=logx83 gks] rks x dk eku Kkr dhft,A

14. ;fn logba. logca+logab. logcb+logac. logbc=3 (tgkWa a, b, c fHkUu&fHkUu /kukRed okLrfod la[;k,a ≠1gS) , gks rks] abc
dk eku Kkr dhft, A

15. ;fn a=log1218 ,oa b=log2454 gks] rks iznf’kZr dhft, fd ab+5(a-b) dk eku Kkr dhft,A

log a log b log c

16. ;fn = = gks] rks iznf’kZr dhft, fd a a.bb.cc=1.
b−c c −a a−b

17. fuEu esa dkSu cM+k gS &

(a) log23 ;k log1/25 (b) log711 ;k log85

fuEu (18-27) dks x ds fy, gy dhft, %

2
18. log10(x -12x+36)=2 19. log4log3log2x=0

 1 
20. log3  log9 x + + 9 x  = 2x 21. 2log4(4-x)=4-log2(-2-x)
 2 

log x + 5
22. 2
log10 x + log10 x 2 = log10
2
2 −1 23. x 3 = 10 5 +log x

5
24. log52 x + log5 x   = 1 25. log4(log2x)+log2(log4x)=2
x
26. log 3
x − 2 − log3 x − 2 = 2 27. 5 x.x 8 x −1 500

28. ;fn log102=0.3010 ,oa log103=0.4771 gks] rks Kkr dhft, &
15
(a) 6 esa vdksa dh la[;k
-100
(b) 3 esa n’keyo ds Bhd ckn vkus okys 'kwU;ksa dh la[;k

29. lehdj.kksa log100x+y=1/2,log10y-log10x=log1004 dks x ,oa y ds fy, gy dhft,A

30. lehdj.k x-1A=(x-1)7 tgkWa A=log3x2-2logx9 ] dks lUrq"V djus okys x ds eku Kkr dhft,A
31. x ds os lHkh okLrfod eku Kkr dhft, tks lehdj.k 2 log2 log2 x + log1/ 2 log2 (2 2x ) = 1 dks lUrq"V djrs
gSA

32. lehdj.k log3/4log8(x2+7)+log1/2log1/4(x2+7)-1=-2 gks gy dhft,A

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Answers
EXERCISE – 1

1. C 2. A 3. D 4. B 5. C 6. C 7. A

8. B 9. D 10. C 11. B 12. A 13. C 14. D a4

7.
15. B 16. B 17. C 18. D 19. D 20. D 21. B b4

22. C 23. B 24. C 25. C 26. AB 27. ABCD

28. AB 29. ABCD 30. ABCD

EXERCISE – 2 8. (i) (ii)

47 103 1673
1. (i) (ii) (iii) (iv) not possible
20 90 495 9. a1=a2=a3=………=an=0

7 9
2. (i) (ii) 13 − 12 (iii)
10.
8 11 − 2
(i) x=±1 (ii) x=5
2+ 2 − 6
4. (i) 2 −1 (ii)
4 (iii) x=2 (iv) x=-2, 0, 2, 4
2 2
5. (i) (x-2y) (x +y -xy) (v) x=-3, 3 (vi) x=-1

 1  2 1 1  (vii) x∈[1, ∞)
(ii)  a − + 1 a + 2 − a + + 2 
 a  a a  11. x=2,y=1 12. 0 13. x=10 14. abc=1
15. 1 17. (a) log23 (b) log711
(iii) (x-1) (x-2) (x-3) 18. x=16 or x=-4 19. 8 20. {1/3}
1 1 -5 3
(iv) (x+2) (x2-2x-5) 21. {-4} 22. , 23. {10 , 10 }
20 5
(v) –(a-b) (b-c) (c-a)  1
24. 1,5,  25. x=16 26. 1/9,9
4 2 2 2  25 
6. (i) (x -x +1) (x +x+1) (x -x+1)
2 2 27. x=3 28. (a) 12 (b) 47
(ii) (x -2x+2) (x +2x+2) 29. x=10/3,y=20/3 & x=-10,y=20
30. x=2 or 81 31. x=8 32. x=3 or -3

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