Sequences and Series - Calculus - Engineering Sciences Previous Year Questions

1

2007 · Engineering Sciences · Calculus · Sequences and Series

Engineering Sciences (XE) 2007

The minimum number of terms required in the series expansion of \( e^x \) to evaluate at x = 1 correct up to 3 places of decimals is

A

8

B

7

C

6

D

5

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2

2007 · Engineering Sciences · Calculus · Sequences and Series

Engineering Sciences (XE) 2007

The Fourier series of $f$ in $[-\pi,\pi]$ is

A

$\frac{\pi^2}{3}+4\sum_{n=1}^{\infty}\frac{\cos nx}{n^2}$

B

$\frac{\pi^2}{3}+\sum_{n=1}^{\infty}\frac{(-1)^n \cos nx}{n^2}$

C

$\frac{\pi^2}{3}+4\sum_{n=1}^{\infty}\frac{(-1)^n \cos nx}{n^2}$

D

$\frac{\pi^2}{3}+\sum_{n=1}^{\infty}\frac{\cos nx}{n^2}$

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3

2007 · Engineering Sciences · Calculus · Sequences and Series

Engineering Sciences (XE) 2007

The sum of the absolute values of the Fourier coefficients of $f$ is

A

$\pi^2/6$

B

$\pi^2/3$

C

$2\pi^2/3$

D

$\pi^2$

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4

2008 · Engineering Sciences · Calculus · Sequences and Series

Engineering Sciences (XE) 2008

The radius of convergence of the real power series
\[ \sum_{m=0}^{\infty} \frac{(m!)^3}{(2m+1)!} x^m \] is

A

4

B

3

C

2

D

1

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5

2008 · Engineering Sciences · Calculus · Sequences and Series

Engineering Sciences (XE) 2008

The function e^x is expanded about the point x=0 in a Taylor polynomial P_n(x) of degree n. The value of n necessary to approximate e^x to an accuracy of 10^{-2} in [0,0.5] is

A

5

B

6

C

7

D

8

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6

2009 · Engineering Sciences · Calculus · Sequences and Series

Engineering Sciences (XE) 2009

The infinite series \(\sum_{n=1}^{\infty} \frac{(-1)^{n} x^{2}}{(1+x^{2})^{n}}\) is

A

Divergent for all \(x\)

B

Convergent only for \(x \geq 1\)

C

Convergent for all \(x\)

D

Divergent only for \(-1 \leq x \leq 1\)

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7

2011 · Engineering Sciences · Calculus · Sequences and Series

Engineering Sciences (XE) 2011

The sum of n terms of the series 4+44+444+.... is

A

(4/81) [10^{n+1} - 9n - 1]

B

(4/81) [10^{n-1} - 9n - 1]

C

(4/81) [10^{n+1} - 9n - 10]

D

(4/81) [10^n - 9n - 10]

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8

2012 · Engineering Sciences · Calculus · Sequences and Series

Engineering Sciences (XE) 2012

Taylor series of \( f(x) = \frac{1}{1-x} \) about \( x=0 \) is given as \( TS|_f = 1 + x + x^2 + x^3 + \ldots \). This series can be used to evaluate \( f(x) \) for

A

\(|x| \neq 1 \)

B

\(x < -1 \)

C

\(|x| < 1 \)

D

\(-1 \le x < 1 \)

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9

2013 · Engineering Sciences · Calculus · Sequences and Series

Engineering Sciences (XE) 2013

The Fourier series of the periodic function \(f(x) = |x|, \; -1 < x < 1, \; f(x+2) = f(x), \; x \in \mathbb{R}\) is given by \(\frac{1}{2} - \sum_{n=1}^{\infty} \frac{4 \cos(2n-1)\pi x}{(2n-1)^2 \pi^2}\). Using the above, the sum of the infinite series \(1 + \frac{1}{3^2} + \frac{1}{5^2} + \ldots\) is

A

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

B

\(\frac{3\pi^2}{8}\)

C

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

D

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

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10

2015 · Engineering Sciences · Calculus · Sequences and Series

Engineering Sciences (XE) 2015

The radius of convergence of the following power series is \[\sum_{n=0}^{\infty} \frac{(x-3)^n}{3^n n!}\]

A

zero

B

1

C

3

D

\infty

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11

2017 · Engineering Sciences · Calculus · Sequences and Series

Engineering Sciences (XE) 2017

Let
\(a_k = 2^{-k} k^4 \sin k\) and \(b_k = 2^{-k^2} k^2 \sin^2 k\)
for \(k = 1, 2, ...\). Then

A

\(\sum_{k=1}^{\infty} a_k\) converges but \(\sum_{k=1}^{\infty} b_k\) does NOT converge

B

\(\sum_{k=1}^{\infty} a_k\) does NOT converge but \(\sum_{k=1}^{\infty} b_k\) converges

C

both \(\sum_{k=1}^{\infty} a_k\) and \(\sum_{k=1}^{\infty} b_k\) converge

D

neither \(\sum_{k=1}^{\infty} a_k\) nor \(\sum_{k=1}^{\infty} b_k\) converges

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12

2019 · Engineering Sciences · Calculus · Sequences and Series

Engineering Sciences (XE) 2019

For the series \(\sum_{n=1}^{\infty} \frac{(x+1)^n}{n 2^n}, -\infty < x < \infty\), which of the following statements is NOT correct?

A

The series converges at \(x = -3\)

B

The series converges at \(x = -1\)

C

The series converges at \(x = 0\)

D

The series converges at \(x = 1\)

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13

2020 · Engineering Sciences · Calculus · Sequences and Series

Engineering Sciences (XE) 2020

Select the word that fits the analogy:
Cover : Uncover :: Associate : _____

A

Unassociate

B

Inassociate

C

Misassociate

D

Dissociate

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14

2020 · Engineering Sciences · Calculus · Sequences and Series

Engineering Sciences (XE) 2020

Hit by floods, the kharif (summer sown) crops in various parts of the country have been affected. Officials believe that the loss in production of the kharif crops can be recovered in the output of the rabi (winter sown) crops so that the country can achieve its food-grain production target of 291 million tons in the crop year 2019-20 (July-June). They are hopeful that good rains in July-August will help the soil retain moisture for a longer period, helping winter sown crops such as wheat and pulses during the November-February period.
Which of the following statements can be inferred from the given passage?

A

Officials declared that the food-grain production target will be met due to good rains.

B

Officials want the food-grain production target to be met by the November-February period.

C

Officials feel that the food-grain production target cannot be met due to floods.

D

Officials hope that the food-grain production target will be met due to a good rabi produce.

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15

2020 · Engineering Sciences · Calculus · Sequences and Series

Engineering Sciences (XE) 2020

The difference between the sum of the first \(2n\) natural numbers and the sum of the first \(n\) odd natural numbers is _____.

A

\(n^2 - n\)

B

\(n^2 + n\)

C

\(2n^2 - n\)

D

\(2n^2 + n\)

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16

2020 · Engineering Sciences · Calculus · Sequences and Series

Engineering Sciences (XE) 2020

A hydroelectric power plant takes in 30 m³/s of water through its turbine and discharges it to the atmosphere with V₂ = 2 m/s. The total head loss in the turbine and penstock system is 20 m. (Assume turbulent flow with kinetic energy correction factor as 1.1. Density of water is 1000 kg/m³ and acceleration due to gravity, g is 10 m/s²).
The net head available to the turbine for power generation is _______________ m.
(rounded off to one decimal place).

A

converges for all \(z\).

B

converges for \(|z| \le 1\) and diverges for \(|z| > 1\).

C

converges for \(z = 0\) and diverges for any \(z \neq 0\).

D

converges for \(|z| < 1\) and diverges for \(|z| \ge 1\).

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17

2020 · Engineering Sciences · Calculus · Sequences and Series

Engineering Sciences (XE) 2020

Let \( f(t) \) be a real-valued differentiable function on \( (-1,1) \) such that \( f(0) = 0 \) and \( |\frac{df}{dt}| < 1 \) for \( 0 < t < 1 \). Then the series \( \sum_{n=0}^{\infty} f(0.5)^n \)

A

converges but not absolutely.

B

is unbounded.

C

converges absolutely.

D

is bounded but does not converge.

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18

2024 · Engineering Sciences · Calculus · Sequences and Series

Engineering Sciences (XE) 2024

In the sequence 6, 9, 14, \( x \), 30, 41, a possible value of \( x \) is

A

25

B

21

C

18

D

20

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19

2024 · Engineering Sciences · Calculus · Sequences and Series

Engineering Sciences (XE) 2024

Let
\( f(x) = \begin{cases} \pi + x, & -\pi \le x < 0, \\ 0, & 0 \le x < \pi, \end{cases} \)
with \( f(x + 2\pi) = f(x) \). If \( F(x) \) represents the Fourier series of \( f(x) \), then the value of \( F\left(-\frac{\pi}{2}\right) + F(0) \) is

A

0

B

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

C

\(\pi \)

D

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

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20

2025 · Engineering Sciences · Calculus · Sequences and Series

Engineering Sciences (XE) 2025

Consider the infinite series
\((P): \sum_{n=2}^{\infty} \frac{1}{(n \log n)^{1/n}}\)
\((Q): \sum_{n=1}^{\infty} \frac{n^n}{(2n)!}\)
Then which one of the following statements is correct ?

A

Series \((P)\) and \((Q)\) both converge

B

Series \((P)\) converges and series \((Q)\) diverges

C

Series \((P)\) and \((Q)\) both diverge

D

Series \((P)\) diverges and series \((Q)\) converges

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