Calculus - Engineering Mathematics - Aerospace Engineering Previous Year Questions

1

2007 · Aerospace Engineering · Engineering Mathematics · Calculus

Aerospace Engineering (AE) 2007

An athlete starts running with a speed \(V_0\). Subsequently, his speed decreases by an amount that is proportional to the distance that he has already covered. The distance covered will be

A

Linear in time

B

Quadratic in time

C

Exponential in time

D

Logarithmic in time

Open complete paper

2

2007 · Aerospace Engineering · Engineering Mathematics · Calculus

Aerospace Engineering (AE) 2007

At a stationary point of a multi-variable function, which of the following is true?

A

Curl of the function becomes unity

B

Gradient of the function vanishes

C

Divergence of the function vanishes

D

Gradient of the function is maximum

Open complete paper

3

2008 · Aerospace Engineering · Engineering Mathematics · Calculus

Aerospace Engineering (AE) 2008

The function defined by
\( f(x) = \begin{cases} \sin x, & x < 0 \\ 0, & x = 0 \\ 3x^3, & x > 0 \end{cases} \)
is

A

neither continuous nor differentiable at \(x = 0 \)

B

continuous and differentiable at \(x = 0 \)

C

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

D

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

Open complete paper

4

2008 · Aerospace Engineering · Engineering Mathematics · Calculus

Aerospace Engineering (AE) 2008

The function \( f(x, y, z) = \frac{1}{2} x^2 y^3 z^2 \) satisfies

A

\(\text{grad } f = 0 \)

B

\(\text{div}(\text{grad } f) = 0 \)

C

\(\text{curl}(\text{grad } f) = 0 \)

D

\(\text{grad}(\text{div}(\text{grad } f)) = 0 \)

Open complete paper

5

2008 · Aerospace Engineering · Engineering Mathematics · Calculus

Aerospace Engineering (AE) 2008

The value of the line integral \( \frac{1}{2\pi} \oint (x \, dy - y \, dx) \) taken anticlockwise along a circle of unit radius is

A

0.5

B

1

C

2

D

\(\pi \)

Open complete paper

6

2009 · Aerospace Engineering · Engineering Mathematics · Calculus

Aerospace Engineering (AE) 2009

In the interval \(1 \le x \le 2\), the function \(f(x) = e^{\pi x} + \sin \pi x\) is

A

maximum at \(x = 1\)

B

maximum at \(x = 2\)

C

maximum at \(x = 1.5\)

D

monotonically decreasing

Open complete paper

7

2009 · Aerospace Engineering · Engineering Mathematics · Calculus

Aerospace Engineering (AE) 2009

After 10 seconds, the change in specific internal energy of the air is

A

21,525 J/kg

B

-21,525 J/kg

C

30,000 J/kg

D

-8,475 J/kg

Open complete paper

8

2009 · Aerospace Engineering · Engineering Mathematics · Calculus

Aerospace Engineering (AE) 2009

Given that the specific heats of air at constant pressure and volume are \( c_p = 1004.5 \) J/kg·K and \( c_v = 717.5 \) J/kg·K respectively, the corresponding change in the temperature of the air is

A

21.4 K

B

-21.4 K

C

30 K

D

-30 K

Open complete paper

9

2010 · Aerospace Engineering · Engineering Mathematics · Calculus

Aerospace Engineering (AE) 2010

The definite integral \(\int_{-1}^{1} \frac{dx}{x^2}\)

A

does not exist

B

is equal to 2

C

is equal to 0

D

is equal to –2

Open complete paper

10

2010 · Aerospace Engineering · Engineering Mathematics · Calculus

Aerospace Engineering (AE) 2010

In a certain region a hill is described by the shape \(z(x, y) = \frac{1}{50} x^2 + y^2 - xy - 3y\), where the axes \(x\) and \(y\) are in the horizontal plane and axis \(z\) points vertically upward. If \(\hat{i}\), \(\hat{j}\) and \(\hat{k}\) are unit vectors along \(x\), \(y\) and \(z\), respectively, then at the point \(x = 5, y = 10\) the unit vector in the direction of the steepest slope of the hill will be:

A

\(\hat{i}\)

B

\(\hat{j}\)

C

\(\hat{k}\)

D

\(\hat{i} + \hat{j} + \hat{k}\)

Open complete paper

11

2010 · Aerospace Engineering · Engineering Mathematics · Calculus

Aerospace Engineering (AE) 2010

The function \(f(x, y) = x^2 + y^2 - xy - 3y\) has an extremum at the point

A

(1,2)

B

(3,0)

C

(2,2)

D

(1,1)

Open complete paper

12

2010 · Aerospace Engineering · Engineering Mathematics · Calculus

Aerospace Engineering (AE) 2010

If \(e\) is the base of the natural logarithms then the equation of the tangent from the origin to the curve \(y = e^x\) is

A

\(y = x\)

B

\(y = \pi x\)

C

\(y = \frac{x}{e}\)

D

\(y = ex\)

Open complete paper

13

2011 · Aerospace Engineering · Engineering Mathematics · Calculus

Aerospace Engineering (AE) 2011

The function \(f(x_1, x_2, x_3) = x_1^2 + x_2^2 + x_3^2 - 2x_1 - 4x_2 - 6x_3 + 14\) has its minimum value at

A

\((1, 2, 3)\)

B

\((0, 0, 0)\)

C

\((3, 2, 1)\)

D

\((1, 1, 3)\)

Open complete paper

14

2011 · Aerospace Engineering · Engineering Mathematics · Calculus

Aerospace Engineering (AE) 2011

Consider the function \(f(x_1, x_2) = x_1^2 + 2x_2^2 + e^{-x_1 - x_2}\). The vector pointing in the direction of maximum increase of the function at the point \((1, -1)\) is

A

\(\begin{pmatrix} 2 \\ -5 \end{pmatrix}\)

B

\(\begin{pmatrix} 1 \\ -5 \end{pmatrix}\)

C

\(\begin{pmatrix} -0.73 \\ -6.73 \end{pmatrix}\)

D

\(\begin{pmatrix} 2 \\ -4 \end{pmatrix}\)

Open complete paper

15

2012 · Aerospace Engineering · Engineering Mathematics · Calculus

Aerospace Engineering (AE) 2012

The \(n^{th}\) derivative of the function \(y = \frac{1}{x+3}\) is

A

\(\frac{(-1)^n n!}{(x+3)^{n+1}}\)

B

\(\frac{(-1)^{n+1} n!}{(x+3)^{n+1}}\)

C

\(\frac{(-1)^n (n+1)!}{(x+3)^n}\)

D

\(\frac{(-1)^n n!}{(x+3)^n}\)

Open complete paper

16

2012 · Aerospace Engineering · Engineering Mathematics · Calculus

Aerospace Engineering (AE) 2012

The volume of a solid generated by rotating the region between semi-circle \(y = 1 - \sqrt{1-x^2}\) and straight line \(y = 1\), about \(x\) axis, is

A

\(\pi^2 - \frac{4}{3}\pi\)

B

\(4\pi^2 - \frac{1}{3}\pi\)

C

\(\pi^2 - \frac{3}{4}\pi\)

D

\(\frac{3}{4}\pi^2 - \pi\)

Open complete paper

17

2012 · Aerospace Engineering · Engineering Mathematics · Calculus

Aerospace Engineering (AE) 2012

Two points \((4, p)\) and \((0, q)\) lie on a straight line having a slope of 3/4. The value of \((p - q)\) is

A

-3

B

0

C

3

D

4

Open complete paper

18

2014 · Aerospace Engineering · Engineering Mathematics · Calculus

Aerospace Engineering (AE) 2014

If \( y = 5x^2 + 3 \), then the tangent at \( x = 0 \), \( y = 3 \)

A

passes through \(x = 0, y = 0 \)

B

has a slope of +1

C

is parallel to the x-axis

D

has a slope of −1

Open complete paper

19

2014 · Aerospace Engineering · Engineering Mathematics · Calculus

Aerospace Engineering (AE) 2014

The series \( s = \sum_{m=1}^{\infty} \frac{m^2}{3^m} (x-2)^m \) converges for all \( x \) with \( |x-2| \le R \) given by

A

\(R = 0 \)

B

\(R = 3 \)

C

\(R = \infty \)

D

\(R = 1/3 \)

Open complete paper

20

2014 · Aerospace Engineering · Engineering Mathematics · Calculus

Aerospace Engineering (AE) 2014

The function given by \( f(x) = \begin{cases} \sin(1/x), & x \neq 0 \\ 0, & x = 0 \end{cases} \) is

A

Unbounded everywhere

B

Bounded and continuous everywhere

C

Bounded but not continuous at \(x = 0 \)

D

Continuous and differentiable everywhere

Open complete paper