Calculus and Vector Analysis - Engineering Mathematics - Naval Architecture & Marine Engineering Previous Year Questions

1

2022 · Naval Architecture & Marine Engineering · Engineering Mathematics · Calculus and Vector Analysis

Naval Architecture & Marine Engineering (NM) 2022

If \(\frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos nx\) is the Fourier cosine series of the function \(f(x) = \sin x, \quad 0 < x < \pi,\) then which of the following are TRUE?

A

\(a_0 + a_1 = 4/\pi\)

B

\(a_0 = 4/\pi\)

C

\(a_0 + a_1 = 2/\pi\)

D

\(a_1 = 2/\pi\)

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2

2022 · Naval Architecture & Marine Engineering · Engineering Mathematics · Calculus and Vector Analysis

Naval Architecture & Marine Engineering (NM) 2022

Let \( f(x, y, z) = xy + yz + xz \). If a point \( (0, 0, \lambda) \) lies on the tangent plane to the surface \( f(x, y, z) = 3 \) at the point \( (1, 1, 1) \), then the value of \( \lambda \) is ________.

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3

2022 · Naval Architecture & Marine Engineering · Engineering Mathematics · Calculus and Vector Analysis

Naval Architecture & Marine Engineering (NM) 2022

The value of the line integral \(\oint_C -3y\,dx + 3x\,dy + z\,dz\) along the circle \(C: x^2 + y^2 = 1, z = 1\) oriented in the clockwise sense as seen from the origin, is

A

2π

B

4π

C

6π

D

8π

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4

2023 · Naval Architecture & Marine Engineering · Engineering Mathematics · Calculus and Vector Analysis

Naval Architecture & Marine Engineering (NM) 2023

References : ________ : : Guidelines : Implement
(By word meaning)

A

Sight

B

Site

C

Cite

D

Plagiarise

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5

2023 · Naval Architecture & Marine Engineering · Engineering Mathematics · Calculus and Vector Analysis

Naval Architecture & Marine Engineering (NM) 2023

In the given figure, PQRS is a parallelogram with PS = 7 cm, PT = 4 cm and PV = 5 cm. What is the length of RS in cm? (The diagram is representative.)

A

\(\frac{20}{7}\)

B

\(\frac{28}{5}\)

C

\(\frac{9}{2}\)

D

\(\frac{35}{4}\)

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6

2023 · Naval Architecture & Marine Engineering · Engineering Mathematics · Calculus and Vector Analysis

Naval Architecture & Marine Engineering (NM) 2023

A line of symmetry is defined as a line that divides a figure into two parts in a way such that each part is a mirror image of the other part about that line.
The given figure consists of 16 unit squares arranged as shown. In addition to the three black squares, what is the minimum number of squares that must be coloured black, such that both PQ and MN form lines of symmetry? (The figure is representative)

A

3

B

4

C

5

D

6

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7

2023 · Naval Architecture & Marine Engineering · Engineering Mathematics · Calculus and Vector Analysis

Naval Architecture & Marine Engineering (NM) 2023

The coefficient of \( x^4 \) in the polynomial \( (x - 1)^3 (x - 2)^3 \) is equal to _____.

A

33

B

-3

C

30

D

21

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8

2023 · Naval Architecture & Marine Engineering · Engineering Mathematics · Calculus and Vector Analysis

Naval Architecture & Marine Engineering (NM) 2023

Consider the function \( z = \tan^{-1}\left(\frac{x}{y}\right) \), where \( x = u \sin v \) and \( y = u \cos v \). The partial derivative, \( \frac{\partial z}{\partial v} \) is

A

0

B

1

C

2

D

3

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9

2023 · Naval Architecture & Marine Engineering · Engineering Mathematics · Calculus and Vector Analysis

Naval Architecture & Marine Engineering (NM) 2023

Consider the function \( z = x^3 - 2x^2 y + xy^2 + 1 \). The directional derivative of \( z \) at the point \( (1, 2) \) along the direction \( 3\hat{i} + 4\hat{j} \) is

A

0

B

-1

C

1

D

-2

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10

2023 · Naval Architecture & Marine Engineering · Engineering Mathematics · Calculus and Vector Analysis

Naval Architecture & Marine Engineering (NM) 2023

In dimensional analysis, according to Buckingham’s \(\pi\)-theorem, if \(n\) is the total number of variables and \(m\) is the number of independent dimensions, then the maximum number of independent dimensionless \(\pi\)-groups will be

A

\(m - n\)

B

\(mn\)

C

\(m + n\)

D

\(n - m\)

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11

2023 · Naval Architecture & Marine Engineering · Engineering Mathematics · Calculus and Vector Analysis

Naval Architecture & Marine Engineering (NM) 2023

The value of the surface integral \(\iint (x^2 dy dz + y^2 dz dx + z^2 dx dy)\) over the surface of the cube given by \(0 \le x \le 2, 0 \le y \le 2, 0 \le z \le 2\), is

A

12

B

24

C

36

D

48

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12

2024 · Naval Architecture & Marine Engineering · Engineering Mathematics · Calculus and Vector Analysis

Naval Architecture & Marine Engineering (NM) 2024

The tangent plane to the surface \( x^2 + y^2 + z = 9 \) at the point (1, 2, 4) is

A

\(2x + 4y + z = 14 \)

B

\(4x + 2y + z = 12 \)

C

\(x + 4y + 2z = 17 \)

D

\(4x + y + 2z = 14 \)

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13

2024 · Naval Architecture & Marine Engineering · Engineering Mathematics · Calculus and Vector Analysis

Naval Architecture & Marine Engineering (NM) 2024

The value of the line integral \(\oint x^2 \, dx + 2x \, dy\) along the ellipse \(4x^2 + y^2 = 4\) oriented in the counterclockwise sense is

A

\(\pi\)

B

\(2\pi\)

C

\(4\pi\)

D

\(8\pi\)

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14

2024 · Naval Architecture & Marine Engineering · Engineering Mathematics · Calculus and Vector Analysis

Naval Architecture & Marine Engineering (NM) 2024

The arc length of the one arch of the cycloid given by \( x = t - \sin t \) and \( y = 1 - \cos t \) is ________.

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15

2024 · Naval Architecture & Marine Engineering · Engineering Mathematics · Calculus and Vector Analysis

Naval Architecture & Marine Engineering (NM) 2024

Consider the function \(f(x, y) = x^4 + y^4 - 4xy + 1\). Which of the following is/are correct?

A

The minimum value of \(f\) occurs at \((0, 0)\)

B

The point \((0, 0)\) is a point of inflection

C

\(f\) has three critical points

D

The minimum value of \(f\) is \(-1\)

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16

2024 · Naval Architecture & Marine Engineering · Engineering Mathematics · Calculus and Vector Analysis

Naval Architecture & Marine Engineering (NM) 2024

Consider the \(2\pi\) – periodic function defined by
\(f(x) = \begin{cases} -1 & \text{if } -\pi < x \le 0 \\ 1 & \text{if } 0 < x < \pi \end{cases}\)
Which of the following is/are correct about its Fourier series expansion, \(\frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos nx + b_n \sin nx\)?

A

\(a_n = \frac{1}{n} \ \forall n = 1, 2, ...\)

B

\(a_0 = 0\)

C

\(b_n = \frac{4}{n\pi}\) if n is odd

D

\(b_n = -\frac{4}{n\pi}\) if n is even

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17

2024 · Naval Architecture & Marine Engineering · Engineering Mathematics · Calculus and Vector Analysis

Naval Architecture & Marine Engineering (NM) 2024

The value of the triple integral \(\iiint (xy^2 + yz^3) \, dx \, dy \, dz\) over the region given by \(-1 \le x \le 1\), \(3 \le y \le 4\), \(0 \le z \le 2\), is ______.

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18

2025 · Naval Architecture & Marine Engineering · Engineering Mathematics · Calculus and Vector Analysis

Naval Architecture & Marine Engineering (NM) 2025

The value of \(\lim_{t\to 1} \frac{\ln t}{t^2-1}\) is ______.

A

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

B

0

C

1

D

\(\infty\)

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19

2025 · Naval Architecture & Marine Engineering · Engineering Mathematics · Calculus and Vector Analysis

Naval Architecture & Marine Engineering (NM) 2025

The Fourier series expansion of \(f(x) = \sin^2 x\) in the interval \((-\pi, \pi)\) is ______.

A

\(f(x) = \frac{1}{2} - \frac{1}{2}\cos 2x\)

B

\(f(x) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1} \cos nx}{n^2}\)

C

\(f(x) = 2 \sum_{n=1}^{\infty} \frac{\sin 2nx}{n}\)

D

\(f(x) = \frac{1}{2} - \frac{4}{\pi^2} \sum_{n=1}^{\infty} \frac{\cos(2n-1)\pi x}{(2n-1)^2}\)

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20

2025 · Naval Architecture & Marine Engineering · Engineering Mathematics · Calculus and Vector Analysis

Naval Architecture & Marine Engineering (NM) 2025

The value of \( \int_1^3 dy \int_2^5 x^2 y \, dx \) is __________ (answer in integer).

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