No.
of Printed Pages : 12 BMTC–131
BACHELOR OF SCIENCE/BACHELOR
OF ARTS [B. SC.(G)/B. A.(G)/BSCM]
Term-End Examination
December, 2024
BMTC-131 : CALCULUS
Time : 3 Hours Maximum Marks : 100
Note : (i) Question No. 1 is compulsory.
(ii) Attempt any eight questions from
Question No. 2 to 10.
(iii) Use of calculator is not allowed.
1. Which of the following statements are true and
which are false ? Give a short proof or a
counter-example, in support of your answer : 20
(i) If { }
Z, A =x ∈ Q | x ≥ 3 ,U =
Q, then
AC ∪ ZC =Q − {1,2,3,...} .
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(ii) The value of k for which the function f
defined by :
x 2 − 2, if x < 1
f ( x) =
kx − 2, if x ≥ 1
is continuous at x = 1 is 0.
(iii) The only solution of the equation :
x2 2x
=log x log − log
2 1 + x
1 − x
1
is x = .
2
(iv) The maximum possible domain of a
4 − x2
function f given by f ( x) = is
x2 + x
R − {0,1} .
4 9
(v) The curve 2
− 1 has no asymptotes
=
x y2
parallel to axes.
(vi)
d
dx
(
sin x ≠
d
dx
) ( )
sin x .
16 x 2 − 25
(vii) lim = 10 .
x→−
5 4x + 5
4
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(viii) The function f, defined by f ( x=
) | x − 2| is
differentiable at x = 2 .
1
(ix) − ,0,1 ∈ Q × Z × N .
2
(x) If f ( x ) = x3 and ) 2x + 1,
g ( x= then
f o g (1) = g o f (1) .
2. (a) Find all the roots α,β and γ of the cubic
equation 3 x3 + 11 x 2 + 12 x + 4 =
0 such that
the roots are in H.P. 5
(b) Prove that the function f : R →]1, ∞[
x ) 32 x + 1 is invertible and
defined as f (=
find a formula for the inverse. 5
3. (a) If y = sin x and n is any positive integer,
then find yn2 + yn2+1 . 5
(b) Find the nth Taylor’s polynomial for
1
about x = 1 . 5
x
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, when y = x( x ) .
dy x
4. (a) Find 5
dx
(b) Evaluate : 5
2
∫ −1 (| x|+| x − 1|) dx
5. (a) Test the continuity and differentiability of
the function : 5
x 2 − 1, x ≥ 1
f ( x) =
1 − x, x < 1
at x = 1 .
(b) Draw the rough sketch of the curve
=r 2 3 r sin θ . 5
6. (a) Find the fourth roots of the complex
number 1+i and show the solution
geometrically. 5
(b) A bee follows the trajectory x = t − 3sin t
and y= 4 − 3cos t , where t ≥ 0 . It lands on
a wall at time t = 10 . A what time was the
bee flying horizontally ? 5
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7. Trace the curve y 2 ( a + x )= x 2 ( a − x ) , a > 0,
stating all the properties you use to trace it. 10
8. (a) Using the ε-δ definition of limit, find δ
such that lim (4 x + 5) =
13 ; ε =0.04 . 5
x →2
(b) Verify Lagrange’s mean value theorem for
the function f, defined by f ( x ) = x3 on the
interval [ −1,1] . 5
9. (a) Find the area of the region bounded by the
curves y = x 2 and y = x3 . 5
(b) Find all possible relative extreme values of
the function f defined by
f ( x ) = x3 − 6 x 2 + 9 x − 4 by using first
derivative test. 5
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10. (a) Find the length of the curve given by the
equations
= x 2 cos t + 3 and
= y 2sin t + 4 ;
0 ≤ t ≤ 2π . 5
d2y
(b) If x= a ( θ − sin θ), y= a (1 − cos θ) , find
dx 2
at θ = π . 5
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BMTC–131
foKku Lukrd@dyk Lukrd
[ch- ,l&lh- (th-)@ ch- ,- (th-)@
ch- ,l- lh- ,e-]
l=kkar ijh{kk
fnlEcj] 2024
ch-,e-Vh-lh--131 % dyu
le; % 3 ?k.Vs vf/dre vad % 100
uksV % (i) iz'u la- 1 djuk vfuok;Z gSA
(ii) iz'u la- 2 ls 10 rd dksbZ vkB iz'u
dhft,A
(iii) dSYdqysVj dk iz;ksx djus dh vuqefr ugha
gSA
1- fuEufyf[kr esa ls dkSu&ls dFku lR; vkSj dkSu&ls
dFku vlR; gSa\ vius mÙkj ds i{k esa laf{kIr miifÙk
;k izfr&mnkgj.k nhft, % 20
(i) ;fn { }
Z, A =x ∈ Q | x ≥ 3 ,U =
Q gS]a rks
AC ∪ ZC =Q − {1,2,3,...} gksxkA
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(ii) k dk og eku] ftlds fy,
x 2 − 2, ;fn x < 1
f ( x) =
kx − 2, ;fn x ≥ 1
}kjk ifjHkkf"kr iQyu f ,x =1 ij larr~ gS] 0 gSA
x2 2x
(iii) lehdj.k
= log x log − log dk
1 − x 2
1 + x
1
dsoy gy x= gSA
2
4 − x2
(iv) f ( x ) = }kjk ifjHkkf"kr iQyu f dk
x2 + x
vf/dre laHkkfor izkar R − {0,1} gSA
4 9
(v) oØ − 1 dh
= v{kksa ds lekukarj dksbZ
x2 y2
vuarLi'khZ ugha gSA
(vi)
d
dx
(
sin x ≠
d
dx
) ( sin x )
16 x 2 − 25
(vii) lim = 10
x→−
5 4x + 5
4
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(viii) f ( x=
) | x − 2| }kjk ifjHkkf"kr iQyu f ,x = 2
ij vodyuh; gSA
1
(ix) − ,0,1 ∈ Q × Z × N
2
(x) ;fn f ( x ) = x3 vkSj g ( x=
) 2x + 1 gSa] rks
f o g (1) = g o f (1) A
2- (d) ?ku cgqin 3 x3 + 11 x 2 + 12 x + 4 =
0 ds lHkh
ewy α, β vkSj γ] tks fd H.P. esa gSa]
fudkfy,A 5
([k) x ) 32 x + 1
f (= }kjk ifjHkkf"kr iQyu
f : R →]1, ∞[ O;qRØe.kh; gSa vkSj O;qRØe dk
lw=k fudkfy,A 5
3- (d) ;fn y = sin x vkSj n ,d /ukRed iw.kk±d
gS] rks yn2 + yn2+1 Kkr dhft,A 5
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1
([k) ds fy, x =1 ds lkis{k noha Vsyj cgqin
x
fudkfy,A 5
y = x( x
x) dy
4- (d) ds fy, fudkfy,A 5
dx
2
([k) ∫ −1 (| x |+| x − 1|) dx dk eku fudkfy,A 5
5- (d) iQyu 5
x 2 − 1, x ≥ 1
f ( x) =
1 − x, x < 1
dh x =1 ij lrrrk vkSj vodyuh;rk dk
ijh{k.k dhft,A
([k) oØ=r2 3 r sin θ dk jiQ LdSp [khafp,A 5
6- (d) lfEeJ la[;k 1+i dk prqFkZ ewy Kkr dhft,
vkSj bls T;kfefr :i esa Hkh n'kkZb,A 5
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([k) ,d eD[kh iz{ksioØ x = t − 3sin t vkSj
y= 4 − 3cos t ] tgk¡ t >0] esa mM+rh gSA ;g
t = 10 le; ij nhokj ij vkrh gSA eD[kh
fdrus le; ij {kSfrt :i ls mM+ jgh Fkh \ 5
7- oØ y 2 ( a + x )= x 2 ( a − x ) ] a>0 dk vkjs[k.k
dhft, vkSj ,slk djus esa iz;ksx fd, x;s xq.k/eks± dks
fyf[k,A 10
8- (d) lhek dh ε-δ ifjHkk"kk dk iz;ksx djds δ dk
og eku Kkr dhft, ftlds fy,
lim (4 x + 5) =
13 ] ε =0.04 gSA 5
x →2
([k) f ( x ) = x3 }kjk ifjHkkf"kr iQyu f ds fy,
varjky [ −1,1] ij ySxzkat ekè;eku izes;
lR;kfir dhft,A 5
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9- (d) oØksa y = x2 vkSj y = x3 ls f?kjs izns'k dk
{ks=kiQy Kkr dhft,A 5
([k) izFke vodyt ijh{k.k dk iz;ksx djds
f ( x ) = x3 − 6 x 2 + 9 x − 4 }kjk ifjHkkf"kr iQyu
f ds lHkh laHko lkis{k b"Vre eku Kkr
dhft,A 5
10- (d) lehdj.kks
= a x 2 cos t + 3 vkS=
j y 2sin t + 4 _
0 ≤ t ≤ 2π ] }kjk fn, x, oØ dh yackbZ Kkr
dhft,A 5
([k) ;fn x= a ( θ − sin θ), y= a (1 − cos θ) gS] rks
d2y
θ=π ij Kkr dhft,A 5
dx 2
×××××××
B–1414/BMTC–131
