RvZxq wek¦we`¨vj‡qi cÖkœcÎ - 2022 1 2 K¨vjKzjvm-I

a
NUH-2022 (U) †Kvb k‡Z©  a
f ( x) dx  0 n‡e? [What is the condition for
K-wefvM a
 a
f ( x) dx  0 ?] [Ch-10A: Quiz-14]
1| (K) y  f ( x) Øviv Kx †evSvq? [What is meant by y  f ( x) ?]
(V) r  a Kx wb‡`©k K‡i? Bnvi wPÎ AvuK| [What is denoted by r  a ?
[Ch-1: Quiz-1] Draw its graph.]
1
(L) f ( x)  dvsk‡bi †Wv‡gb I †iÄ KZ? [What is the domain L-wefvM
x 1
2| hw` f ( x)  x  16 Ges g ( x)  x nq Z‡e ( g  f ) ( x) wbY©q Ki|
2
1
and range of the function f ( x)  ?] AZtci ( g  f ) ( x) Gi †Wv‡gb wbY©q Ki| [If f ( x)  x 2  16 and
x 1
(M) †Kvb e¨ewa‡Z f ( x)  ln x dvskbwU Aš—ixKiY‡hvM¨? [Which is the g ( x)  x then find ( g  f ) ( x ) . Hence find domain of
( g  f ) ( x ) .]
interval where the functions f ( x)  ln x is differentiable?]
x3  1
dy dy 3| wjwg‡Ui (   ) msÁv e¨envi K‡i †`LvI †h, lim  3 | [Using
x 1 x  1
(N) R¨vwgwZKfv‡e Øviv Kx †evSvq? [What is is geometrically?]
dx dx x3  1
[Ch-3A: Quiz-3] the (   ) definition of limit to show that, lim  3 .]
x 1 x  1

(O) x Gi mv‡c‡¶ y  e ax Gi n -Zg Aš—iR KZ? [What is the n -th [Ch-2A: Prob-2(ii)]
2t 2t dy
derivative of y  e ax with respect to x ?] 4| tan y  Ges sin x  n‡j, wbY©q Ki| [If
1 t 2
1 t 2
dx
(P) y  f ( x) eµ‡iLvi Dci¯’ ( x, y ) we›`y‡Z Awfj‡¤^i mgxKiY †jL| 2t 2t dy
tan y  and sin x  then find .]
[Write down the equation of normal at ( x, y ) of the curve 1 t 2
1 t 2
dx
[Ch-3A: Exercise-D(2)]
y  f ( x ) .] [Ch-7A: Quiz-9]
5| y  x  2 x  1 eµ‡iLvi (1, 4) we›`y‡Z ¯úk©K Ges Awfj‡¤^i mgxKiY
2

(Q) y  f ( x) Ges y  g ( x) eµ‡iLvØq j¤^ nIqvi kZ© Kx? [What is the wbY©q Ki| [Find the equations of tangent and normal of the curve
condition of perpendicularity of two curves y  f ( x) and y  x 2  2 x  1 at (1, 4) .] [Ch-7A: Prob-1(ii)]
y  g ( x) ?] 1 1
 tan x  x  tan x  x
6| lim   Gi gvb wbY©q Ki| [Find the value of lim   .]
x 0  x  x 0  x 
1 1
(R) lim e Gi gvb KZ? [What is the value of lim e ?]
x x
x  0 x  0 [Ch-8: Exercise-23]
7|  Gi jNy KiY m~ Î cÖ w Zôv Ki|
n
(S) GKwU e¨ewa‡Z µgea©gvb dvskb ej‡Z Kx eyS? [What do you mean sin x dx [Find the reduction formula

 sin
n
by increasing function in a interval?] [Ch-6A: Quiz-5] for x dx .] [Ch-10C: Art-10C.2(vii)]
 1 
(T) e  ln x   dx Gi gvb KZ? [What is the value of
x

 x 8| gvb wbY©q Ki [Find the value of]:  0

4
tan x sec4 x dx

 1 1
e
x
 ln x   dx ?] 9| †`LvI †h [Show that],      [Ch-12: Abywm×vš—-1]
 x 2
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 RvZxq wek¦we`¨vj‡qi cÖkœcÎ - 2022 3 4 K¨vjKzjvm-I
M-wefvM  p 1  q 1
    

10| x  we›`y‡Z dvskbwUi Awew”QbœZv I Aš—ixKiY‡hvM¨Zv Av‡jvPbv Ki| 15| †`LvI †h [Show that],  02 sin p  cos q  d   2   2 
2  pq2
2  
  2 
[Discuss the continuity and differentiability at x  of the
2 [Ch-12: Th-18]
function] 16| Bw›UMÖvj K¨vjKzjv‡mi †gŠwjK Dccv`¨ eY©bv I cÖgvY Ki| [State and prove
 the fundamental theorem of integral calculus.][Ch-10A: Art-10A.5]

 1 ; hLb [when] x  0
17| x3  y 3  3axy †iLvi GKwU duv‡mi †¶Îdj wbY©q Ki| [Find the area of
 
f  x    1  sin x ; hLb [when] 0  x 
2 a loop of the curve x3  y 3  3axy .] [Ch-14B: Prob-2(ii)]

  
2

2   x   ; hLb [when] x 
  2 2
----------
[Ch-2C: Prob-4(i)]
11| †iv‡ji Dccv`¨ eY©bvmn cÖgvY Ki| [State and prove Rolle’s theorem.]
[Ch-4: Art-4.2]

12| hw` y  A ( x  x 2  1)m  B ( x  x 2  1) m nq Z‡e †`LvI †h [If

y  A ( x  x 2  1)m  B ( x  x 2  1) m then show that],

(1  x 2 ) yn  2  (2n  1) xyn 1  (n2  m2 ) yn  0

[Ch-3C: Exercise-5]
13| (0, 9) e¨ewa‡Z f ( x)  x3  18 x 2  96 x dvskbwUi m‡e©v”P I me©wbgœ gvb
wbY©q Ki| [Find the maximum and minimum values of the function
f ( x)  x3  18 x 2  96 x in the interval (0, 9) .] [Ch-6B: Prob-2(iv)]
14| gvb wbY©q Ki [Evaluate]:
x3
(i)  x 1
dx [Ch-9B: Prob-1(i)]


dx
(ii)  0
2

1  cot x
[Ch-10B: Prob-1(v)]

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