Calculus - Engineering Mathematics - Textile Engineering & Fibre Science Previous Year Questions

1

2007 · Textile Engineering & Fibre Science · Engineering Mathematics · Calculus

Textile Engineering & Fibre Science (TF) 2007

A function g(t) is defined as follows $g(t) = \begin{cases} \frac{1}{2\tau}, & \text{when } t_0 - \tau < t < t_0 + \tau \\ 0, & \text{when } t \leq t_0 - \tau \text{ and } t \geq t_0 + \tau \end{cases}$ The Laplace transform of the function g(t) is given by

A

$\frac{1}{2s} [\exp(s\tau) - \exp(-s\tau)] \exp(-st_0)$

B

$\frac{1}{2s} [\exp(s\tau) + \exp(-s\tau)] \exp(-st_0)$

C

$\frac{1}{2s\tau} [\exp(s\tau) - \exp(-s\tau)] \exp(-st_0)$

D

$\frac{1}{2s\tau} [\exp(s\tau) + \exp(-s\tau)] \exp(-st_0)$

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2

2007 · Textile Engineering & Fibre Science · Engineering Mathematics · Calculus

Textile Engineering & Fibre Science (TF) 2007

If $f(x,y,z) = 4(x^2 + y^2) - z^2$, then $\nabla f$ at a point (1,0,2) is given by

A

$8\hat{i} - 4\hat{j}$

B

$8\hat{i} - 4\hat{k}$

C

$8\hat{j} - 4\hat{k}$

D

$8\hat{k} - 4\hat{i}$

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3

2007 · Textile Engineering & Fibre Science · Engineering Mathematics · Calculus

Textile Engineering & Fibre Science (TF) 2007

Given a vector $u(x,y,z) = xy\hat{i} + (z+x)\hat{j} + y\hat{k}$, the points where the $\nabla \times u$ vanishes lie on the plane

A

y = 2

B

y = 1

C

z = 1

D

x = 1

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4

2007 · Textile Engineering & Fibre Science · Engineering Mathematics · Calculus

Textile Engineering & Fibre Science (TF) 2007

If \( y_n = \frac{d^n y}{d x^n} \) and \( y = (x^2 - 1)^n \), then the expression \( (x^2 - 1) y_{n+2} + 2 x y_{n+1} \) is equal to

A

\((n^2 + 1) y_n \)

B

\((n^2 - 1) y_n \)

C

\(n(n - 1) y_n \)

D

\(n(n + 1) y_n \)

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5

2007 · Textile Engineering & Fibre Science · Engineering Mathematics · Calculus

Textile Engineering & Fibre Science (TF) 2007

A function f(x) is defined by
f(x) = \begin{cases} -x, & \text{for } -2 \le x < 0 \\ x, & \text{for } 0 \le x < 2 \end{cases} \quad \text{and } f(x+4)=f(x).
This periodic function f(x) with a period 4 has its Fourier series expansion as
f(x) = \frac{a_0}{2} + \sum_{m=1}^{\infty} a_m \cos \frac{m\pi x}{2}, \quad a_m = \frac{1}{2} \int_{-2}^{2} f(x) \cos \frac{m\pi x}{2} \, dx
The coefficient of the term \cos \frac{5\pi x}{2} in the above expansion is

A

-1

B

\(-\frac{8}{5\pi}\)

C

\(-\frac{8}{(5\pi)^2}\)

D

\(-\frac{8}{(5\pi)^3}\)

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6

2008 · Textile Engineering & Fibre Science · Engineering Mathematics · Calculus

Textile Engineering & Fibre Science (TF) 2008

\(\lim_{x \to 2} \frac{x^2 - 4}{x - 2}\) is equal to

A

-2

B

2

C

4

D

\(\infty\)

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7

2008 · Textile Engineering & Fibre Science · Engineering Mathematics · Calculus

Textile Engineering & Fibre Science (TF) 2008

The function \(f(x) = x^3 - 3x + 3\) defined in the interval \([-2, 2]\) has a minimum at

A

\(x = -1\)

B

\(x = 0\)

C

\(x = 1\)

D

\(x = 2\)

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8

2008 · Textile Engineering & Fibre Science · Engineering Mathematics · Calculus

Textile Engineering & Fibre Science (TF) 2008

The total derivative of a function \( u = f(x, y, z) \) is expressed as \( du = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy + \frac{\partial f}{\partial z} dz \). If \( u = \exp(x^2 + y^2) \sin z \), then the expression for \( du \) is given by

A

\(du = \exp(x^2 + y^2)[2x dx + 2y dy] \cos z + \sin z \, dz \)

B

\(du = \exp(x^2 + y^2)[2x dx + 2y dy] \sin z + \cos z \, dz \)

C

\(du = \exp(x^2 + y^2)[(2x dx + 2y dy) \cos z + \sin z \, dz] \)

D

\(du = \exp(x^2 + y^2)[(2x dx + 2y dy) \sin z + \cos z \, dz] \)

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9

2008 · Textile Engineering & Fibre Science · Engineering Mathematics · Calculus

Textile Engineering & Fibre Science (TF) 2008

A curve in space is represented by a vector \(\vec{r}(t) = x(t) \hat{i} + y(t) \hat{j} + z(t) \hat{k}\). Given a vector function \(\vec{F}(r) = 5z \hat{i} + xy \hat{j} + x^2 z \hat{k}\) and \(\vec{r}(t) = t \hat{i} + t \hat{j} + t \hat{k}\), \(0 \le t \le 1\), the value of the integral \(\int_0^1 [\vec{F}(\vec{r}(t)) \cdot \frac{d\vec{r}}{dt}] dt\) is

A

\(\frac{7}{12}\)

B

\(\frac{17}{12}\)

C

\(\frac{27}{12}\)

D

\(\frac{37}{12}\)

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10

2008 · Textile Engineering & Fibre Science · Engineering Mathematics · Calculus

Textile Engineering & Fibre Science (TF) 2008

The unit normal vector \(\mathbf{n}\) to a surface \(S(x,y,z) = 0\) is defined as \(\mathbf{n} = \frac{\nabla S}{|\nabla S|}\), \(|\nabla S|\) is the modulus of \(\nabla S\). If the equation of the surface is \(S = x^2 + y^2 + z^2 - a^2 = 0\), then unit normal to this surface is given by

A

\(x \hat{i} + y \hat{j} + z \hat{k}\)

B

\(\frac{1}{a} (\hat{i} + \hat{j} + \hat{k})\)

C

\(\frac{x}{a} \hat{i} + \frac{y}{a} \hat{j} + \frac{z}{a} \hat{k}\)

D

\(a(x \hat{i} + y \hat{j} + z \hat{k})\)

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11

2009 · Textile Engineering & Fibre Science · Engineering Mathematics · Calculus

Textile Engineering & Fibre Science (TF) 2009

The coefficient of \(\cos 4x\) in the Fourier Series of the function \(f(x) = \begin{cases} -1, & \text{when } -\pi < x < 0 \\ 1, & \text{when } 0 < x < \pi \end{cases}\) and \(f(x) = f(x + 2\pi)\), for all \(x\) is

A

0

B

1

C

-1

D

\(\pi\)

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12

2009 · Textile Engineering & Fibre Science · Engineering Mathematics · Calculus

Textile Engineering & Fibre Science (TF) 2009

The partial differential equation \(\frac{\partial^2 u}{\partial r^2} + \frac{1}{r}\frac{\partial u}{\partial r} + \frac{1}{r^2}\frac{\partial^2 u}{\partial \theta^2} = 0\) is known as the polar form of

A

Laplace's equation

B

Heat equation

C

Wave equation

D

None of these

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13

2009 · Textile Engineering & Fibre Science · Engineering Mathematics · Calculus

Textile Engineering & Fibre Science (TF) 2009

Radius of the circle passing through \(P(2, 4)\) and the points of intersection with the \(x\) - axis of the tangent and normal drawn at \(P\) to the curve \(y^2 = 8x\) is

A

1

B

2

C

3

D

4

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14

2009 · Textile Engineering & Fibre Science · Engineering Mathematics · Calculus

Textile Engineering & Fibre Science (TF) 2009

If \(S\) is the largest possible set of real numbers \(x\) for which the series \(\sum_{n=1}^{\infty} \frac{x^n}{n}\) is convergent, then \(S\) is

A

(-1, 1)

B

[-1, 1)

C

[-1, 1]

D

(-\infty, \infty)

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15

2009 · Textile Engineering & Fibre Science · Engineering Mathematics · Calculus

Textile Engineering & Fibre Science (TF) 2009

If the Laplace transform of a function \(f(t)\) is \(\frac{1}{s^3 - s}\), then \(f(t)\) is

A

\cosh t - 1

B

\cosh t + 1

C

\sinh t - 1

D

\sinh t + 1

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16

2009 · Textile Engineering & Fibre Science · Engineering Mathematics · Calculus

Textile Engineering & Fibre Science (TF) 2009

The absolute value of the directional derivative of the surface given by \(\frac{x^2}{1} + \frac{y^2}{4} + \frac{z^2}{9} = 3\) at \(P(1, 2, 3)\) in the direction of the line \(OP\), where \(O\) denotes the origin, is

A

\(\frac{3}{\sqrt{14}}\)

B

\(\frac{\sqrt{14}}{3}\)

C

\(\frac{6}{\sqrt{14}}\)

D

\(\frac{\sqrt{14}}{6}\)

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17

2010 · Textile Engineering & Fibre Science · Engineering Mathematics · Calculus

Textile Engineering & Fibre Science (TF) 2010

The divergence of the vector field \((x - y)\hat{i} + (y - x)\hat{j} + (x + y + z)\hat{k}\) is

A

0

B

1

C

2

D

3

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18

2010 · Textile Engineering & Fibre Science · Engineering Mathematics · Calculus

Textile Engineering & Fibre Science (TF) 2010

When \(x \to 4\), \(\lim \frac{x^3 - 64}{\log_e(x - 3)}\) will be equal to

A

24

B

48

C

72

D

96

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19

2010 · Textile Engineering & Fibre Science · Engineering Mathematics · Calculus

Textile Engineering & Fibre Science (TF) 2010

For a circular rod with volume \(16\pi \text{ cm}^3\), the value of radius for which the surface area (including the top and bottom surfaces) will be minimum is

A

1 cm

B

2 cm

C

3 cm

D

4 cm

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20

2010 · Textile Engineering & Fibre Science · Engineering Mathematics · Calculus

Textile Engineering & Fibre Science (TF) 2010

Consider the function \(f(x) = \max(7-x, x+3)\). In which range does \(f\) take its minimum value?

A

\(-6 \leq x < -2\)

B

\(-2 \leq x < 2\)

C

\(2 \leq x < 6\)

D

\(6 \leq x < 10\)

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