Functions of Single Variable - Calculus - Engineering Sciences Previous Year Questions

1

2007 · Engineering Sciences · Calculus · Functions of Single Variable

Engineering Sciences (XE) 2007

Which of the following statements are true in a C program?
P: A local variable is used only within the block where it is defined, and its sub-blocks
Q: Global variables are declared outside the scope of all blocks
R: Extern variables are used by linkers for sharing between other compilation units
S: By default, all global variables are extern variables

A

P and Q

B

P, Q and R

C

P, Q and S

D

P, Q, R and S

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2

2007 · Engineering Sciences · Calculus · Functions of Single Variable

Engineering Sciences (XE) 2007

Let \( L = \lim_{x \to \pi/2} \frac{\sin^2 2x}{(x - \pi/2)^2} \). Then \( L \) is equal to

A

-4

B

0

C

2

D

4

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3

2007 · Engineering Sciences · Calculus · Functions of Single Variable

Engineering Sciences (XE) 2007

Let \( f : \Re \to \Re \) be a twice differentiable real valued function such that \( f(1/n) = 1 \) for \( n = 1, 2, 3, ... \). Then

A

\(f'(0) = 0 \)

B

\(f'(0) = 1 \)

C

\(0 < f'(0) < 1 \)

D

\(f'(0) > 1 \)

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4

2007 · Engineering Sciences · Calculus · Functions of Single Variable

Engineering Sciences (XE) 2007

Let \( f(x) = \int_{0}^{x^2} \sin \sqrt{t} dt \) for \( x \geq 0 \). Then \( f'(\pi/2) \) is equal to

A

\(0 \)

B

\(\pi \)

C

\(1 \)

D

\(\pi/2 \)

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5

2008 · Engineering Sciences · Calculus · Functions of Single Variable

Engineering Sciences (XE) 2008

The hypotenuse \( c \) and a side \( b \) of a right triangle are found by measurement to be 13 cm and 5 cm respectively. The possible error in the measurement of the hypotenuse is 0.2 cm and that in the side \( b \) is 0.1 cm. The maximum possible error (in cm) in the calculation of the third side is

A

0.26

B

0.22

C

0.17

D

0.10

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6

2008 · Engineering Sciences · Calculus · Functions of Single Variable

Engineering Sciences (XE) 2008

The root of the derivative of f(x) in the same interval is

A

1.500

B

1.200

C

1.578

D

1.495

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7

2008 · Engineering Sciences · Calculus · Functions of Single Variable

Engineering Sciences (XE) 2008

The value of
\[ \left( \int_{0}^{\pi/2} (\sin \theta)^{-1/4} d\theta \right) \times \left( \int_{0}^{\pi/2} (\sin \theta)^{-3/4} d\theta \right) \] is

A

\(\frac{2\pi}{3}(\sqrt{2} + 1)\)

B

\(\frac{2\pi}{3}(\sqrt{2} - 1)\)

C

\(\frac{\pi}{2} \sqrt{3}\)

D

\(\frac{\pi}{2} \sqrt{3}\)

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8

2008 · Engineering Sciences · Calculus · Functions of Single Variable

Engineering Sciences (XE) 2008

A C function named func is defined as follows:
int func(int a, int b) {
if ( (a == 1) || (b == 0) || (a == b) ) return 1;
return func(a-1,b) + func(a-1,b-1);
}
What is the result of func(4,2) ?

A

12

B

6

C

3

D

1

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9

2009 · Engineering Sciences · Calculus · Functions of Single Variable

Engineering Sciences (XE) 2009

Let \(f(x)\) be continuous and satisfy \(m \leq f(x) \leq M\) in \(1 \leq x \leq 10\). Then, \(\mu = \frac{\int_{1}^{10} (f(x)x^{2})dx}{\int_{1}^{10} (x^{2})dx}\) satisfies

A

\(\mu \leq 333 m\)

B

\(333 \mu \geq M\)

C

\(m \leq \mu \leq M\)

D

\(m \leq \mu \leq \frac{M}{333}\)

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10

2010 · Engineering Sciences · Calculus · Functions of Single Variable

Engineering Sciences (XE) 2010

Which of the following options is the closest in meaning to the word below:
Circuitous

A

cyclic

B

indirect

C

confusing

D

crooked

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11

2010 · Engineering Sciences · Calculus · Functions of Single Variable

Engineering Sciences (XE) 2010

The integral \( \int_{1}^{\infty} x^{-6} dx \) is

A

an improper integral converging to \(-6 \).

B

an improper integral converging to 0.

C

not an improper integral but has value \(-6 \).

D

a divergent improper integral.

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12

2010 · Engineering Sciences · Calculus · Functions of Single Variable

Engineering Sciences (XE) 2010

Let \(f(x) = \begin{cases} \frac{\sin x}{x} & \text{if } x \neq 0 \\ 1 & \text{if } x=0. \end{cases}\) Then

A

\(f\) is not continuous at \(x=0\).

B

\(f\) is continuous at \(x=0\) but not differentiable at \(x=0\).

C

\(f\) is differentiable at \(x=0\) and \(f'(0)=0\).

D

\(f\) is differentiable at \(x=0\) and \(f'(0)=1\).

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13

2011 · Engineering Sciences · Calculus · Functions of Single Variable

Engineering Sciences (XE) 2011

The function \(f(x)\) defined by \(f(x) = \begin{cases} 3 - x^2, & x \leq 1, \\ 3 - x, & 1 < x \leq 2, \\ x - 1, & x > 2, \end{cases}\) has

A

a local maxima at \(x = 3\) and a local minima at \(x = 0\)

B

a local maxima at \(x = 0\) and no local minima

C

a local maxima at \(x = 0\) and a local minima at \(x = 2\)

D

no local maxima and a local minima at \(x = 1\)

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14

2011 · Engineering Sciences · Calculus · Functions of Single Variable

Engineering Sciences (XE) 2011

Given that f(y) = | y | / y, and q is any non-zero real number, the value of | f(q) - f(-q) | is

A

0

B

-1

C

1

D

2

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15

2011 · Engineering Sciences · Calculus · Functions of Single Variable

Engineering Sciences (XE) 2011

The integral $\int_{-5\pi/2}^{5\pi/2} f(x)dx$, where $f(x) = e^{\pi x^2} \sin^5 x + 4\cos x$, equals

A

4

B

8

C

$\frac{5\pi}{2}$

D

$-\frac{5\pi}{2}$

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16

2012 · Engineering Sciences · Calculus · Functions of Single Variable

Engineering Sciences (XE) 2012

If \((1.001)^{1259} = 3.52\) and \((1.001)^{2062} = 7.85\), then \((1.001)^{3321} =\)

A

2.23

B

4.33

C

11.37

D

27.64

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17

2012 · Engineering Sciences · Calculus · Functions of Single Variable

Engineering Sciences (XE) 2012

If \(f(x) = x\sin(x)\) and \(g(x) = |x|\sin(x)\), then

A

\(g(x) = |f(x)|\)

B

\(g(x)\) is an even function

C

The x-coordinates corresponding to the various local maxima are identical for both \(f(x)\) and \(g(x)\)

D

\(g(x)\) is differentiable at \(x = 0\)

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18

2013 · Engineering Sciences · Calculus · Functions of Single Variable

Engineering Sciences (XE) 2013

Velocity of an object fired directly in upward direction is given by \( V = 80 - 32 t \), where \( t \) (time) is in seconds. When will the velocity be between 32 m/sec and 64 m/sec?

A

\((1, 3/2) \)

B

\((1/2, 1) \)

C

\((1/2, 3/2) \)

D

\((1, 3) \)

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19

2013 · Engineering Sciences · Calculus · Functions of Single Variable

Engineering Sciences (XE) 2013

If \( |1 - 2X + 9| = 3 \) then the possible value of \( |1 - X| - X^2 \) would be:

A

30

B

-30

C

-42

D

42

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20

2013 · Engineering Sciences · Calculus · Functions of Single Variable

Engineering Sciences (XE) 2013

The value of the integral \(\int_0^1 \frac{dt}{\sqrt{(-\log_e t)}}\) is

A

\(\frac{\sqrt{\pi}}{2}\)

B

\(\sqrt{\pi}\)

C

\(-\sqrt{\pi}\)

D

\(-\frac{\sqrt{\pi}}{2}\)

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