If \({\rm{y}} = {\sec ^{ - 1}}\left( {\frac{{{\rm{x}} + 1}}{{{\rm{x}} - 1}}} \right) + {\sin ^{ - 1}}\left( {\frac{{{\rm{x}} - 1}}{{{\rm{x}} + 1}}} \right)\), then \(\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\) is equal to

\(\frac{{{\rm{x}} - 1}}{{{\rm{x}} + 1}}\)

\(\frac{{{\rm{x}} + 1}}{{{\rm{x}} - 1}}\)

Let \({\rm{f}}\left( {{\rm{x}},{\rm{y}}} \right) = \frac{{{\rm{a}}{{\rm{x}}^2} + {\rm{b}}{{\rm{y}}^2}}}{{{\rm{xy}}}}\), where a and b are constants. If \(\frac{{\partial {\rm{f}}}}{{\partial {\rm{x}}}} = \frac{{\partial {\rm{f}}}}{{\partial {\rm{y}}}}{\rm{\;}}\) at x = 1 and y = 2, then the relation between a and b is

\({\rm{a}} = \frac{{\rm{b}}}{4}\)

\({\rm{a}} = \frac{{\rm{b}}}{2}\)

If \(f\left( x \right) = \frac{{{x^3}}}{3} - \frac{{5{x^2}}}{2} + 6x + 7\) increases in the interval T and decreases in the interval S, then which one of the following is correct?

T = (-∞, 2) ∪ (3, ∞) and S = (2, 3)

T = (2, 3) and S = (-∞, 2) ∪ (3, ∞)

Find the differentiation of \({\cos ^{ - 1}}x\)

\(\frac{1}{{\sqrt {1 - {x^2}} }}\)

\(- \frac{1}{{\sqrt {1 - {x^2}} }}\)

Change the order of integration in\(\mathop{\int }_{0}^{a}\mathop{\int }_{y}^{a}\frac{x}{{{x}^{2}}+{{y}^{2}}}dx~dy\)

\(\mathop{\int }_{0}^{a}\mathop{\int }_{x}^{a}\frac{x}{{{x}^{2}}+{{y}^{2}}}dy~dx\)

\(\mathop{\int }_{0}^{a}\mathop{\int }_{0}^{x}\frac{x}{{{x}^{2}}+{{y}^{2}}}dy~dx\)

\(\mathop{\int }_{0}^{a}\mathop{\int }_{x}^{a}\frac{x}{{{x}^{2}}+{{y}^{2}}}dy~dx\)

\(\mathop{\int }_{x}^{a}\mathop{\int }_{0}^{y}\frac{x}{{{x}^{2}}+{{y}^{2}}}dy~dx\)

\(\mathop{\int }_{x}^{a}\mathop{\int }_{y}^{a}\frac{x}{{{x}^{2}}+{{y}^{2}}}dy~dx\)

If for non-zero x, if \(af\left( x \right) + bf\left( {\frac{1}{x}} \right) = \frac{1}{x} - 25\) where a ≠ b then \(\mathop \smallint \limits_1^2 f\left( x \right)dx\) is

\(\frac{1}{{{a^2} - {b^2}}}\left[ {a\left( {2\;ln\;2 - 25} \right) - \frac{{47b}}{2}} \right]\)

\(\frac{1}{{{a^2} - {b^2}}}\left[ {a\left( {ln\;2 - 25} \right) + \frac{{47b}}{2}} \right]\)

\(\frac{1}{{{a^2} - {b^2}}}\left[ {a\left( {2\;ln\;2 - 25} \right) - \frac{{47b}}{2}} \right]\)

\(\frac{1}{{{a^2} - {b^2}}}\left[ {a\left( {2\;ln\;2 - 25} \right) + \frac{{47b}}{2}} \right]\)

\(\frac{1}{{{a^2} - {b^2}}}\left[ {a\left( {ln\;2 - 25} \right) - \frac{{47b}}{2}} \right]\)

Let f: [0, 2] → R be a twice differentiable function such that f''(x) > 0, for all x ∈ (0, 2). If ϕ(x) = f(x) + f(2 – x), then ϕ is:

Increasing on (0, 1) and decreasing on (1, 2)

Decreasing on (0, 1) and increasing on (1, 2)

If \(V = \frac{{10}}{{{r^2}}}sin\theta .cos\varphi \), the electric flux density at \(\left( {2,\frac{\pi }{2},0} \right)\) is

\(10.2~\overrightarrow {{a_r}} ~pC/{m^2}\)

\(32.1~\overrightarrow {{a_r}} ~pC/{m^2}\)

\(22.1~\overrightarrow {{a_r}}~ pC/{m^2}\)

\(10.2~\overrightarrow {{a_r}} ~pC/{m^2}\)

\(5.8~\overrightarrow {{a_r}} ~pC/{m^2}\)

\(\int {\frac{{dx}}{{(\sin x + 4)(\sin x - 1)}} = \frac{A}{{\tan \frac{x}{2} - 1}} + B{{\tan }^{ - 1}}} \left( {f(x)} \right) + c\), then

\(A = \frac{2}{5}\), \(B = -\frac{2}{{\sqrt {15} }}\), \(f(x) = \frac{{4\tan \left( {\frac{x}{2}} \right) + 1}}{{\sqrt {5} }}\)

\(A = \frac{1}{5}\), \(B = -\frac{2}{{5\sqrt {15} }}\), \(f(x) = \frac{{4\tan x + 3}}{{\sqrt {15} }}\)

\(A = \frac{-1}{5}\), \(B = \frac{1}{{\sqrt {15} }}\), \(f(x) = \frac{{4\tan \left( {\frac{x}{2}} \right) + 1}}{{\sqrt {15} }}\)

\(A = \frac{2}{5}\), \(B = -\frac{2}{{5\sqrt {5} }}\), \(f(x) = \frac{{4\tan x + 1}}{{\sqrt {5} }}\)

\(A = \frac{2}{5}\), \(B = -\frac{2}{{\sqrt {15} }}\), \(f(x) = \frac{{4\tan \left( {\frac{x}{2}} \right) + 1}}{{\sqrt {5} }}\)

If \(I_1 = \int^{\frac {\pi} 2}_0 \sin^2 xdx\) and \(I_2 = \int^{\frac \pi 2}_{0} \frac {\sin^2 x}{1 + 3^x}dx,\) then:

\(I_2 - I_1 = \frac \pi 2 - \log 3\)

\(I_1 + I_2 = \frac \pi 2 + \log 3\)

\(\displaystyle\int \left\lbrace \dfrac{(\log x - 1)}{ 1 + (\log x)^2}\right\rbrace^2dx\) is equal to

\(\rm \dfrac{\log x}{(\log x)^2 + 1}+C\)

\(\rm \dfrac{x}{(\log x)^2 +1}+C\)

\(\rm \dfrac{\log x}{(\log x)^2 + 1}+C\)

For two non-zero vectors A̅ and B̅, if (A̅ + B̅) is perpendicular to (A̅ - B̅), then

The magnitude of A̅ is twice the magnitude of B̅

The magnitude of A̅ is half the magnitude of B̅

A̅ and B̅ cannot be orthogonal

The magnitudes of A̅ and B̅ are equal

If a function is continuous at a point

The limit of the function may not exist at the point

The function must be derivable at the point

The limit of the function at the point tends to infinity

The limit must exist at the point and the value of limit should be same as the value of the function at that point

If 0 < a < b, then \(\mathop \smallint \nolimits_{\rm{a}}^{\rm{b}} \frac{{\left| {\rm{x}} \right|}}{{\rm{x}}}{\rm{dx}}\) is equal to

\(\frac{{\left| {\rm{b}} \right|}}{{\left| {\rm{a}} \right|}}\)

For each t > 0, if \({\rm{\Phi }}\left( {x,t} \right) = \frac{1}{{\sqrt {4\pi t} }}{e^{ - {x^2}/4t}}\) solves the heat equation, then for t > 0

\(\mathop \smallint \limits_{ - \infty }^\infty {\rm{\Phi }}\left( {x,t} \right)dt = \infty\)

\(\mathop \smallint \limits_{ - \infty }^\infty {\rm{\Phi }}\left( {x,t} \right)dt = 0\)

\(\mathop \smallint \limits_{ - \infty }^\infty {\rm{\Phi }}\left( {x,t} \right)dt = 1\)

\(\mathop \smallint \limits_{ - \infty }^\infty {\rm{\Phi }}\left( {x,t} \right)dt = \infty\)

395 + Mcqs in Calculus in UPSEE Page 1 McqOptions

1.	If \({\rm{y}} = {\sec ^{ - 1}}\left( {\frac{{{\rm{x}} + 1}}{{{\rm{x}} - 1}}} \right) + {\sin ^{ - 1}}\left( {\frac{{{\rm{x}} - 1}}{{{\rm{x}} + 1}}} \right)\), then \(\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\) is equal to
A.	0
B.	1
C.	\(\frac{{{\rm{x}} - 1}}{{{\rm{x}} + 1}}\)
D.	\(\frac{{{\rm{x}} + 1}}{{{\rm{x}} - 1}}\)
Answer» B. 1

Discussion

2.	Let \({\rm{f}}\left( {{\rm{x}},{\rm{y}}} \right) = \frac{{{\rm{a}}{{\rm{x}}^2} + {\rm{b}}{{\rm{y}}^2}}}{{{\rm{xy}}}}\), where a and b are constants. If \(\frac{{\partial {\rm{f}}}}{{\partial {\rm{x}}}} = \frac{{\partial {\rm{f}}}}{{\partial {\rm{y}}}}{\rm{\;}}\) at x = 1 and y = 2, then the relation between a and b is
A.	\({\rm{a}} = \frac{{\rm{b}}}{4}\)
B.	\({\rm{a}} = \frac{{\rm{b}}}{2}\)
C.	a = 2b
D.	a = 4b
Answer» E.

Discussion

3.	If \(f\left( x \right) = \frac{{{x^3}}}{3} - \frac{{5{x^2}}}{2} + 6x + 7\) increases in the interval T and decreases in the interval S, then which one of the following is correct?
A.	T = (-∞, 2) ∪ (3, ∞) and S = (2, 3)
B.	T = ϕ and S = (-∞, ∞)
C.	T = (-∞, ∞) and S = ϕ
D.	T = (2, 3) and S = (-∞, 2) ∪ (3, ∞)
Answer» B. T = ϕ and S = (-∞, ∞)

Discussion

4.	Find the differentiation of \({\cos ^{ - 1}}x\)
A.	\(\frac{1}{{\sqrt {1 - {x^2}} }}\)
B.	\(- \frac{1}{{\sqrt {1 - {x^2}} }}\)
C.	\(\frac{1}{{1 + {x^2}}}\)
D.	\(1 + {x^2}\)
Answer» C. \(\frac{1}{{1 + {x^2}}}\)

Discussion

5.	If v = yz î + 3zx ĵ + z k̂, then curl v is
A.	-3xî + yĵ + 2zk̂
B.	3xî – yĵ + 2zk̂
C.	-3xî – yĵ – 2xk̂
D.	3xî + yĵ – 2zk̂
Answer» B. 3xî – yĵ + 2zk̂

Discussion

6.	Change the order of integration in\(\mathop{\int }_{0}^{a}\mathop{\int }_{y}^{a}\frac{x}{{{x}^{2}}+{{y}^{2}}}dx~dy\)
A.	\(\mathop{\int }_{0}^{a}\mathop{\int }_{0}^{x}\frac{x}{{{x}^{2}}+{{y}^{2}}}dy~dx\)
B.	\(\mathop{\int }_{0}^{a}\mathop{\int }_{x}^{a}\frac{x}{{{x}^{2}}+{{y}^{2}}}dy~dx\)
C.	\(\mathop{\int }_{x}^{a}\mathop{\int }_{0}^{y}\frac{x}{{{x}^{2}}+{{y}^{2}}}dy~dx\)
D.	\(\mathop{\int }_{x}^{a}\mathop{\int }_{y}^{a}\frac{x}{{{x}^{2}}+{{y}^{2}}}dy~dx\)
Answer» B. \(\mathop{\int }_{0}^{a}\mathop{\int }_{x}^{a}\frac{x}{{{x}^{2}}+{{y}^{2}}}dy~dx\)

Discussion

7.	Each of four particles move along an x-axis. Their coordinates (in meters) as functions of time (in seconds) are given by1) particle 1: x (t) = 3.5 – 2.7 t32) particle 2: x (t) = 3.5 + 2.7 t33) particle 3: x (t) = 3.5 – 2.7 t24) particle 4: x (t) = 3.5 – 3.4t - 2.7 t2Which of these particles have constant acceleration?
A.	All four
B.	Only (1) and (2)
C.	Only (2) and (3)
D.	Only (3) and (4)
Answer» E.

Discussion

8.	Differentiate {-log (log x), x > 1} with respect to x
A.	-1 / (x log x)
B.	1 / (log x)
C.	1 / x
D.	x log x
Answer» B. 1 / (log x)

Discussion

9.	A function f(x) = 1 - x2 + x3 is defined in the closed interval [-1, 1]. The value of x, in the open interval (-1, 1) for which the mean value theorem is satisfied, is
A.	-1/2
B.	-1/3
C.	1/3
D.	1/2
Answer» C. 1/3

Discussion

10.	If \(u = \log \left( {\frac{{{x^2} + {y^2}}}{{x + y}}} \right)\) then \(x\frac{{\partial u}}{{\partial x}} + y\frac{{\partial u}}{{\partial x}}\) is equal to –
A.	0
B.	1
C.	u
D.	eu
Answer» C. u

Discussion

11.	\(\mathop {\lim }\limits_{x \to 0} \frac{{\cos x - 1}}{{\sin x - x}}\) is equal to
A.	Undefined
B.	∞
C.	1
D.	0
Answer» C. 1

Discussion

12.	If \(\int \frac{dx}{{{x}^{3}}{{\left( 1+{{x}^{6}} \right)}^{2/3}}}=x\cdot f\left( x \right)\cdot {{\left( 1+{{x}^{6}} \right)}^{1/3}}+C\), where C is a constant of integration, then the function f(x) is equal to:
A.	\(\frac{3}{{{x}^{2}}}\)
B.	\(-\frac{1}{6{{x}^{3}}}\)
C.	\(-\frac{1}{2{{x}^{2}}}\)
D.	\(-\frac{1}{2{{x}^{3}}}\)
Answer» E.

Discussion

13.	A 2 m ladder leans against a vertical wall. If the top of the ladder begins to slide down the wall at the rate 25 cm/sec, then the rate (in cm/sec) at which the bottom of the ladder slides away from the wall on the horizontal ground when the top of the ladder is 1 m above the ground is:
A.	25√3
B.	\(\frac{25}{\sqrt{3}}\)
C.	25/3
D.	25
Answer» C. 25/3

Discussion

14.	If \({\rm{y}} = {\cot ^{ - 1}}\left[ {\frac{{\sqrt {1 + \sin {\rm{x}}} + \sqrt {1 - \sin {\rm{x}}} }}{{\sqrt {1 + \sin {\rm{x}}} - \sqrt {1 - \sin {\rm{x}}} }}} \right],\) where \(0 < {\rm{x}} < \frac{{\rm{\pi }}}{2}\), then \(\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\) is equal to
A.	\(\frac{1}{2}\)
B.	2
C.	sin x + cos x
D.	sin x – cos x
Answer» B. 2

Discussion

15.	Consider the shaded triangle region P shown in the figureWhat is \(\iint_P xydx dy \ ?\)
A.	\(\frac{1}{6}\)
B.	\(\frac{2}{9}\)
C.	\(\frac{1}{16}\)
D.	1
Answer» B. \(\frac{2}{9}\)

Discussion

16.	First derivative of cot(x) with respect to x is
A.	tan(x)
B.	-tan(x
C.	-cosec2(x)
D.	-cosec(x)cot(x
Answer» D. -cosec(x)cot(x

Discussion

17.	If for non-zero x, if \(af\left( x \right) + bf\left( {\frac{1}{x}} \right) = \frac{1}{x} - 25\) where a ≠ b then \(\mathop \smallint \limits_1^2 f\left( x \right)dx\) is
A.	\(\frac{1}{{{a^2} - {b^2}}}\left[ {a\left( {ln\;2 - 25} \right) + \frac{{47b}}{2}} \right]\)
B.	\(\frac{1}{{{a^2} - {b^2}}}\left[ {a\left( {2\;ln\;2 - 25} \right) - \frac{{47b}}{2}} \right]\)
C.	\(\frac{1}{{{a^2} - {b^2}}}\left[ {a\left( {2\;ln\;2 - 25} \right) + \frac{{47b}}{2}} \right]\)
D.	\(\frac{1}{{{a^2} - {b^2}}}\left[ {a\left( {ln\;2 - 25} \right) - \frac{{47b}}{2}} \right]\)
Answer» B. \(\frac{1}{{{a^2} - {b^2}}}\left[ {a\left( {2\;ln\;2 - 25} \right) - \frac{{47b}}{2}} \right]\)

Discussion

18.	Evaluate: \(\smallint {e^{\tan x}}{\sec ^2}x\;dx\)
A.	\({- \ e^{\tan x^2}} + C\)
B.	\({- \ e^{\tan x}} + C\)
C.	\({e^{\tan x^2}} + C\)
D.	\({e^{\tan x}} + C\)
Answer» E.

Discussion

19.	If f : R → R f is a differentiable function and \(\begin{array}{*{20}{c}}{{\rm{}}f\left( 2 \right) = 6,{\rm{\;then\;}}\mathop {lim}\limits_{x \to 2} \mathop \smallint \nolimits_6^{f\left( x \right)}\frac{{2tdt}}{{\left( {x - 2} \right)}}}\end{array}\) is
A.	24f' (2)
B.	2f' (2)
C.	0
D.	12f' (2)
Answer» E.

Discussion

20.	Let f: [0, 2] → R be a twice differentiable function such that f''(x) > 0, for all x ∈ (0, 2). If ϕ(x) = f(x) + f(2 – x), then ϕ is:
A.	Increasing on (0, 1) and decreasing on (1, 2)
B.	Decreasing on (0, 2)
C.	Decreasing on (0, 1) and increasing on (1, 2)
D.	Increasing on (0, 2)
Answer» D. Increasing on (0, 2)

Discussion

21.	If \(V = \frac{{10}}{{{r^2}}}sin\theta .cos\varphi \), the electric flux density at \(\left( {2,\frac{\pi }{2},0} \right)\) is
A.	\(32.1~\overrightarrow {{a_r}} ~pC/{m^2}\)
B.	\(22.1~\overrightarrow {{a_r}}~ pC/{m^2}\)
C.	\(10.2~\overrightarrow {{a_r}} ~pC/{m^2}\)
D.	\(5.8~\overrightarrow {{a_r}} ~pC/{m^2}\)
Answer» C. \(10.2~\overrightarrow {{a_r}} ~pC/{m^2}\)

Discussion

22.	Find the angle between two vectors a and b with magnitudes √3 and 2 respectively having a.b = √6
A.	45°
B.	90°
C.	180°
D.	30°
Answer» B. 90°

Discussion

23.	\(\int {\frac{{dx}}{{(\sin x + 4)(\sin x - 1)}} = \frac{A}{{\tan \frac{x}{2} - 1}} + B{{\tan }^{ - 1}}} \left( {f(x)} \right) + c\), then
A.	\(A = \frac{1}{5}\), \(B = -\frac{2}{{5\sqrt {15} }}\), \(f(x) = \frac{{4\tan x + 3}}{{\sqrt {15} }}\)
B.	\(A = \frac{-1}{5}\), \(B = \frac{1}{{\sqrt {15} }}\), \(f(x) = \frac{{4\tan \left( {\frac{x}{2}} \right) + 1}}{{\sqrt {15} }}\)
C.	\(A = \frac{2}{5}\), \(B = -\frac{2}{{5\sqrt {5} }}\), \(f(x) = \frac{{4\tan x + 1}}{{\sqrt {5} }}\)
D.	\(A = \frac{2}{5}\), \(B = -\frac{2}{{\sqrt {15} }}\), \(f(x) = \frac{{4\tan \left( {\frac{x}{2}} \right) + 1}}{{\sqrt {5} }}\)
Answer» D. \(A = \frac{2}{5}\), \(B = -\frac{2}{{\sqrt {15} }}\), \(f(x) = \frac{{4\tan \left( {\frac{x}{2}} \right) + 1}}{{\sqrt {5} }}\)

Discussion

24.	In which one of the following intervals is the function increasing?
A.	(-2, 3)
B.	(3, 4)
C.	(-3, -2)
D.	(-4, -3)
Answer» B. (3, 4)

Discussion

25.	In which one of the following intervals is the function decreasing?
A.	(-2, 3)
B.	(3, 4)
C.	(4, 6)
D.	(6, 9)
Answer» C. (4, 6)

Discussion

26.	If \(f\left( x \right)=\int \frac{5{{x}^{8}}+7{{x}^{6}}}{{{\left( {{x}^{2}}+1+2{{x}^{7}} \right)}^{2}}}dx\), (x ≥ 0), and f(0) = 0, then the value of f(1) is:
A.	\(-\frac{1}{2}\)
B.	\(-\frac{1}{4}\)
C.	\(\frac{1}{2}\)
D.	\(\frac{1}{4}\)
Answer» E.

Discussion

27.	If x is real, then the minimum value of \(\frac {x^2 - x + 1}{x^2 + x + 1}\) is
A.	\(\frac 1 2\)
B.	2
C.	3
D.	\(\frac 1 3\)
Answer» E.

Discussion

28.	If \(I_1 = \int^{\frac {\pi} 2}_0 \sin^2 xdx\) and \(I_2 = \int^{\frac \pi 2}_{0} \frac {\sin^2 x}{1 + 3^x}dx,\) then:
A.	\(I_2 - I_1 = \frac \pi 2 - \log 3\)
B.	\(I_1 + I_2 = \frac \pi 2 + \log 3\)
C.	I1 + I2 = 0
D.	I2 - I1 = 0
Answer» E.

Discussion

29.	\(\displaystyle\int \left\lbrace \dfrac{(\log x - 1)}{ 1 + (\log x)^2}\right\rbrace^2dx\) is equal to
A.	\(\rm \dfrac{xe^x}{1+x^2}+C\)
B.	\(\rm \dfrac{x}{(\log x)^2 +1}+C\)
C.	\(\rm \dfrac{\log x}{(\log x)^2 + 1}+C\)
D.	\(\rm \dfrac{x}{x^2 + 1}+C\)
Answer» C. \(\rm \dfrac{\log x}{(\log x)^2 + 1}+C\)

Discussion

30.	For two non-zero vectors A̅ and B̅, if (A̅ + B̅) is perpendicular to (A̅ - B̅), then
A.	The magnitude of A̅ is twice the magnitude of B̅
B.	The magnitude of A̅ is half the magnitude of B̅
C.	A̅ and B̅ cannot be orthogonal
D.	The magnitudes of A̅ and B̅ are equal
Answer» E.

Discussion

31.	If y = log10 x + logx 10 + logx x + log10 10 then what is \({\left( {\frac{{dy}}{{dx}}} \right)_{x = 10}}\) equal to?
A.	10
B.	2
C.	1
D.	0
Answer» E.

Discussion

32.	\(\mathop {\lim }\limits_{x \to \infty } \left( {\frac{{x + \sin x}}{x}} \right)\) equals to
A.	-∞
B.	0
C.	1
D.	∞
Answer» D. ∞

Discussion

33.	Let f be a differentiable function defined for all x ∈ R such that f(x3) = x5 for all x ∈ R, x ≠ 0. Then the value of \(\dfrac{df}{dx} (8)\) is:
A.	5/3
B.	None of these
C.	20
D.	20/3
Answer» E.

Discussion

34.	Consider the following:1. f(2) = f(1) – f(0)2. f”(2) – 2f’(1) = 12Which of the above is/are correct?
A.	1 only
B.	2 only
C.	Both 1 and 2
D.	Neither 1 nor 2
Answer» D. Neither 1 nor 2

Discussion

35.	A man of height 6 feet walks at uniform rate of 3kmph away from the lamp post of 15 feet height. The rate at which the length of his shadow changes is
A.	2 kmph, decreases
B.	2 kmph increases
C.	4.5 kmph, increases
D.	4.5 kmph decreases
Answer» C. 4.5 kmph, increases

Discussion

36.	If y = cos2 x2, find \(\frac {dy}{dx}\)
A.	4x2 sin x2 cos x2
B.	-4x cos x2 sin x2
C.	2x sin x2 cos x2
D.	-2x cos x2 sin x2
Answer» C. 2x sin x2 cos x2

Discussion

37.	Evaluate\(\int\limits_0^1 {\frac{{\ln (x + 1)}}{{{x^2} + 1}}} ~dx\)
A.	\(\pi \ln \sqrt 2\)
B.	\(\frac{\pi }{8}\ln \sqrt 2\)
C.	\(2\pi \ln \sqrt 2\)
D.	\( \ln \sqrt 2\)
Answer» C. \(2\pi \ln \sqrt 2\)

Discussion

38.	If a function is continuous at a point
A.	The limit of the function may not exist at the point
B.	The function must be derivable at the point
C.	The limit of the function at the point tends to infinity
D.	The limit must exist at the point and the value of limit should be same as the value of the function at that point
Answer» E.

Discussion

39.	If 0 < a < b, then \(\mathop \smallint \nolimits_{\rm{a}}^{\rm{b}} \frac{{\left\| {\rm{x}} \right\|}}{{\rm{x}}}{\rm{dx}}\) is equal to
A.	\|b\| - \|a\|
B.	\|a\|- \|b\|
C.	\(\frac{{\left\| {\rm{b}} \right\|}}{{\left\| {\rm{a}} \right\|}}\)
D.	0
Answer» B. \|a\|- \|b\|

Discussion

40.	Consider the following statements:Fourier series of any periodic function X(t) can be obtained if1. \(\mathop \smallint \limits_0^1 \left\| {x\left( t \right)} \right\|dt < \infty \)2. Finite number of discontinous exist within finite time interval t.Which of the above statements is/are correct ?
A.	1 only
B.	2 only
C.	Both 1 and 2
D.	Neither 1 nor 2
Answer» C. Both 1 and 2

Discussion

41.	For each t > 0, if \({\rm{\Phi }}\left( {x,t} \right) = \frac{1}{{\sqrt {4\pi t} }}{e^{ - {x^2}/4t}}\) solves the heat equation, then for t > 0
A.	\(\mathop \smallint \limits_{ - \infty }^\infty {\rm{\Phi }}\left( {x,t} \right)dt = 0\)
B.	\(\mathop \smallint \limits_{ - \infty }^\infty {\rm{\Phi }}\left( {x,t} \right)dt = 1\)
C.	\(\mathop \smallint \limits_{ - \infty }^\infty {\rm{\Phi }}\left( {x,t} \right)dt = \infty\)
D.	not defined
Answer» C. \(\mathop \smallint \limits_{ - \infty }^\infty {\rm{\Phi }}\left( {x,t} \right)dt = \infty\)

Discussion

42.	Given a vector \(\vec u = \frac{1}{3}\left( { - {y^3}̂ i + {x^3}̂ j + {z^3}̂ k} \right)\)and n̂ as the unit normal vector to the surface of the hemisphere (x2 + y2 + z2 = 1; z ≥ 0), the value of integral \(\smallint \left( {\;\nabla \times u} \right) \bullet \hat n\;dS\) evaluated on the curved surface of the hemisphere S is
A.	– π/2
B.	π/3
C.	π/2
D.	π
Answer» D. π

Discussion

43.	If \(f\left( x \right) = \frac{{ax + b}}{{cx + d}}\) and f(f(x)) = x then
A.	d = a
B.	d = - a
C.	a = b = c = d = 1
D.	a = c = 1
Answer» C. a = b = c = d = 1

Discussion

44.	Find the equation of tangent and normal for f(x) = x2 – 4x + 10 at x = 5.
A.	6y + x = 95 and 6x – y = 15
B.	x + 6y = 95 and 6x – y = 15
C.	x - 6y = - 85 and 6x = y = 15
D.	x - 6y = 85 and 6x = y = -15
Answer» B. x + 6y = 95 and 6x – y = 15

Discussion

45.	If \(f\left( x \right) = \frac{{{{\cos }^2}x}}{{1 + {{\sin }^2}x}}\), then the value \(f\left( {\frac{\pi }{4}} \right) - 3f'\left( {\frac{\pi }{4}} \right)\) is
A.	0
B.	1
C.	3
D.	4
Answer» D. 4

Discussion

46.	Equation of the line normal to function \(f\left( X \right) = {\left( {X - 8} \right)^{\frac{2}{3}}} + 1\) at P(0,5) is
A.	y = 3x - 5
B.	y = 3x + 5
C.	3y = x + 15
D.	3y = x - 15
Answer» C. 3y = x + 15

Discussion

47.	If 7x3 + 3y3 + 4x2 + 6x = 100, then (dy/dx)(2, 4) is
A.	\(-\frac{55}{72}\)
B.	\(-\frac{53}{72}\)
C.	\(-\frac{59}{72}\)
D.	\(-\frac{61}{72}\)
Answer» C. \(-\frac{59}{72}\)

Discussion

48.	If the centre of a circle is (-6, 8) and it passes through the origin, then equation to its tangent at the origin is
A.	2y = x
B.	4y = 3x
C.	3y = 4x
D.	3x + 4y = 0
Answer» C. 3y = 4x

Discussion

49.	\(\mathop \smallint \limits_{ - 1}^2 x\left\| x \right\|\;dx\) is equal to
A.	0
B.	2/3
C.	5/3
D.	7/3
Answer» E.

Discussion

50.	Consider the hemi-spherical tank of radius 13 m as shown in the figure (not drawn to scale). What is the volume of water (in m3) when the depth of water at the centre of the tank is 6 m?
A.	78 π
B.	468 π
C.	156 π
D.	396 π
Answer» E.

Discussion

Explore topic-wise MCQs in UPSEE.

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