Calculus - Engineering Mathematics - Chemical Engineering Previous Year Questions

1

2007 · Chemical Engineering · Engineering Mathematics · Calculus

Chemical Engineering (CH) 2007

The directional derivative of
\( f = \frac{1}{2} \sqrt{x^2 + y^2} \)
at (1,1) in the direction of \( \vec{b} = \hat{i} - \hat{j} \) is

A

0

B

\(1/\sqrt{2} \)

C

\(\sqrt{2} \)

D

2

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2

2007 · Chemical Engineering · Engineering Mathematics · Calculus

Chemical Engineering (CH) 2007

Evaluate the following integral ( \( n \neq 0 \) )
\( \iint (-xy^n dx + x^{n+1} y dy) \)
within the area of a triangle with vertices (0,0), (1,0) and (1,1) (counter-clockwise)

A

0

B

\(1/(n+1) \)

C

\(1/2 \)

D

\(n/2 \)

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3

2007 · Chemical Engineering · Engineering Mathematics · Calculus

Chemical Engineering (CH) 2007

The Laplace transform of
\( f(t) = \frac{1}{\sqrt{t}} \)
is

A

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

B

\(\frac{1}{\sqrt{s}} \)

C

\(\frac{1}{s^{3/2}} \)

D

does not exist

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4

2008 · Chemical Engineering · Engineering Mathematics · Calculus

Chemical Engineering (CH) 2008

Q.3 The limit of \( \frac{\sin x}{x} \) as \( x \rightarrow \infty \) is

A

-1

B

0

C

1

D

\(\infty \)

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5

2008 · Chemical Engineering · Engineering Mathematics · Calculus

Chemical Engineering (CH) 2008

Q.4 The unit normal vector to the surface of the sphere \( x^2 + y^2 + z^2 = 1 \) at the point \( \left( \frac{1}{\sqrt{2}}, 0, \frac{1}{\sqrt{2}} \right) \) is \( (\hat{i}, \hat{j}, \hat{k} \text{ are unit normal vectors in the cartesian coordinate system}) \)

A

\(\frac{1}{\sqrt{2}} \hat{i} + \frac{1}{\sqrt{2}} \hat{j} \)

B

\(-\frac{1}{\sqrt{2}} \hat{i} + \frac{1}{\sqrt{2}} \hat{k} \)

C

\(\frac{1}{\sqrt{2}} \hat{j} + \frac{1}{\sqrt{2}} \hat{k} \)

D

\(\frac{1}{\sqrt{3}} \hat{i} + \frac{1}{\sqrt{3}} \hat{j} + \frac{1}{\sqrt{3}} \hat{k} \)

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6

2008 · Chemical Engineering · Engineering Mathematics · Calculus

Chemical Engineering (CH) 2008

The Laplace transform of the function \( f(t) = t \sin t \) is

A

\(\frac{2s}{(s^2 + 1)^2} \)

B

\(\frac{1}{s^2(s^2 + 1)} \)

C

\(\frac{1}{s^2} + \frac{1}{(s^2 + 1)} \)

D

\(\frac{1}{(s-1)^2 + 1} \)

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7

2008 · Chemical Engineering · Engineering Mathematics · Calculus

Chemical Engineering (CH) 2008

The value of the surface integral \( \iint_S (x \hat{i} + y \hat{j}) \cdot \hat{n} \, dA \) evaluated over the surface of a cube having sides of length \( a \) is (\( \hat{n} \) is unit normal vector)

A

0

B

\(a^3 \)

C

\(2a^3 \)

D

\(3a^3 \)

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8

2008 · Chemical Engineering · Engineering Mathematics · Calculus

Chemical Engineering (CH) 2008

The first four terms of the Taylor series expansion of \( \cos x \) about the point \( x = 0 \) are

A

\(1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} \)

B

\(1 - x + \frac{x^2}{2!} - \frac{x^3}{3!} \)

C

\(1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} \)

D

\(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} \)

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9

2009 · Chemical Engineering · Engineering Mathematics · Calculus

Chemical Engineering (CH) 2009

The direction of largest increase of the function \( x y^3 - x^2 \) at the point (1,1) is

A

\(3\hat{i} + \hat{j} \)

B

\(\hat{i} + 3\hat{j} \)

C

\(-\hat{i} + 3\hat{j} \)

D

\(-\hat{i} - 3\hat{j} \)

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10

2009 · Chemical Engineering · Engineering Mathematics · Calculus

Chemical Engineering (CH) 2009

The value of the limit \(\lim_{x \to \pi/2} \frac{\cos x}{(x - \pi/2)^3}\) is

A

\(-\infty\)

B

0

C

1

D

\(\infty\)

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11

2009 · Chemical Engineering · Engineering Mathematics · Calculus

Chemical Engineering (CH) 2009

Consider the integral \(\iint (2x\hat{i} - 2y\hat{j} + 5z\hat{k}) \cdot \hat{n} dS\) over the surface of a sphere of radius = 3 with center at the origin, and surface unit normal \(\hat{n}\) pointing away from the origin. Using the Gauss divergence theorem, the value of this integral is

A

\(-180\pi\)

B

0

C

\(90\pi\)

D

\(180\pi\)

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12

2010 · Chemical Engineering · Engineering Mathematics · Calculus

Chemical Engineering (CH) 2010

The Laplace transform of the function shown in the figure below is

A

\(\frac{V e^{(a-b)s}}{s} \)

B

\(\frac{V}{s} (e^{-bs} - e^{-as}) \)

C

\(\frac{V}{s} (e^{-as} - e^{-bs}) \)

D

\(\frac{V}{s^2} (e^{-as} - e^{bs}) \)

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13

2010 · Chemical Engineering · Engineering Mathematics · Calculus

Chemical Engineering (CH) 2010

If \( \vec{u} = y \hat{i} + xy \hat{j} \) and \( \vec{v} = x^2 \hat{i} + xy^2 \hat{j} \), then \( \text{curl}(\vec{u} \times \vec{v}) \) is

A

(2xy^2) \hat{i} - (x + y^2) \hat{j}

B

(xy - x^2) \hat{i} - (y - 3xy) \hat{j}

C

(2x^2 y^2 - 3x^2) \hat{i} - (y^3 - 3xy^2) \hat{k}

D

(3xy^2 - x^3) \hat{i} - (y^3 - 3x^2 y) \hat{j}

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14

2010 · Chemical Engineering · Engineering Mathematics · Calculus

Chemical Engineering (CH) 2010

For a function \( g(x) \), if \( g(0) = 0 \) and \( g'(0) = 2 \), then \[ \lim_{x \to 0} \frac{\int_0^{x+1} \frac{2t}{x} dt}{x} \] is equal to

A

∞

B

2

C

0

D

-∞

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15

2010 · Chemical Engineering · Engineering Mathematics · Calculus

Chemical Engineering (CH) 2010

5 skilled workers can build a wall in 20 days; 8 semi-skilled workers can build a wall in 25 days; 10 unskilled workers can build a wall in 30 days. If a team has 2 skilled, 6 semi-skilled and 5 unskilled workers, how long will it take to build the wall?

A

20 days

B

18 days

C

16 days

D

15 days

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16

2011 · Chemical Engineering · Engineering Mathematics · Calculus

Chemical Engineering (CH) 2011

\(\mathbf{R}\) is a closed planar region as shown by the shaded area in the figure below. Its boundary \(C\) consists of the circles \(C_1\) and \(C_2\).
If \(F_1(x,y), F_2(x,y), \frac{\partial F_1}{\partial y}\) and \(\frac{\partial F_2}{\partial x}\) are all continuous everywhere in \(\mathbf{R}\), Green’s theorem states that \(\iint_R \left( \frac{\partial F_2}{\partial x} - \frac{\partial F_1}{\partial y} \right) dxdy = \oint_C (F_1 dx + F_2 dy)\). Which ONE of the following alternatives CORRECTLY depicts the direction of integration along \(C\)?

A

Image of option A

B

Image of option B

C

Image of option C

D

Image of option D

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17

2011 · Chemical Engineering · Engineering Mathematics · Calculus

Chemical Engineering (CH) 2011

Unit vectors in \(x\) and \(z\) directions are \(\hat{i}\) and \(\hat{k}\) respectively. Which ONE of the following is the directional derivative of the function \(F(x,z) = \ln(x^2+z^2)\) at the point P: (4,0), in the direction of \((\hat{i}-\hat{k})\)?

A

\(\frac{\hat{i}}{2\sqrt{2}}\)

B

\(\hat{i}\)

C

1

D

\(\frac{1}{2\sqrt{2}}\)

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18

2011 · Chemical Engineering · Engineering Mathematics · Calculus

Chemical Engineering (CH) 2011

The value of the improper integral \(\int_{-\infty}^{\infty} \frac{dx}{(1+x^2)}\) is

A

-2\(\pi\)

B

0

C

\(\pi\)

D

2\(\pi\)

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19

2011 · Chemical Engineering · Engineering Mathematics · Calculus

Chemical Engineering (CH) 2011

If Log (P) = (1/2)Log (Q) = (1/3) Log (R), then which of the following options is TRUE?

A

\(P^2 = Q^3 R^2\)

B

\(Q^2 = PR\)

C

\(Q^2 = R^3 P\)

D

\(R = P^2 Q^3\)

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20

2012 · Chemical Engineering · Engineering Mathematics · Calculus

Chemical Engineering (CH) 2012

If \( a \) is a constant, then the value of the integral \( a^2 \int_{0}^{a} x e^{-ax} dx \) is

A

1/a

B

a

C

1

D

0

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