Calculus - Mathematics Foundation - Environmental Science & Engineering Previous Year Questions

1

2024 · Environmental Science & Engineering · Mathematics Foundation · Calculus

Environmental Science & Engineering (ES) 2024

\(f(x) = x^3 - 4.5x^2 - 12x\) has a local maximum at \(x = \) ______ (an integer value) in the range \(x = -2\) to \(+2\).

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2

2024 · Environmental Science & Engineering · Mathematics Foundation · Calculus

Environmental Science & Engineering (ES) 2024

Consider the equation \(\frac{dy}{dx} - x^2 + e^x = 0\), with \(y = 1\) at \(x = 0\). The value of \(y\) at \(x = 1\) is ______ (rounded off to 2 decimal places). Take the value of \(e\) (base of natural logarithm) as 2.7.

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3

2024 · Environmental Science & Engineering · Mathematics Foundation · Calculus

Environmental Science & Engineering (ES) 2024

Which one is the solution y(x) for the following ordinary differential equation and the specified boundary conditions?

\( \frac{d^2y}{dx^2} - 3\frac{dy}{dx} + 2y = 2e^{-x}; \; y(0)=2; \; \left(\frac{dy}{dx}\right)_{x=0} = 1 \)

A

\(y(x)=\frac{1}{3}e^{-x} - 2e^{x} - \frac{1}{3}e^{2x} \)

B

\(y(x)=\frac{1}{3}e^{x} + 2e^{x} - \frac{1}{3}e^{2x} \)

C

\(y(x)=\frac{1}{3}e^{-x} + 2e^{-x} - \frac{1}{3}e^{2x} \)

D

\(y(x)=\frac{1}{3}e^{-x} + 2e^{x} - \frac{1}{3}e^{2x} \)

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4

2024 · Environmental Science & Engineering · Mathematics Foundation · Calculus

Environmental Science & Engineering (ES) 2024

Consider the function \( f(x) = \ln(\sin(x)) \).

Expand \( f(x+h) \) using Taylor's series. In this context, the correct statement(s) is/are

A

Second term in the Taylor's series i.e., the term which includes \(h \) is: \(h.\ln(\sin(x)) \)

B

First term is \(\ln(\sin(x)) \)

C

Third term in the Taylor's series i.e., the term which includes \(h^2 \) is: \(\frac{-h^2}{2(\sin(x))^2} \)

D

Third term in the Taylor's series i.e., the term which includes \(h^2 \) is: \(\frac{2h^2}{(\sin(x))^2} \)

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5

2024 · Environmental Science & Engineering · Mathematics Foundation · Calculus

Environmental Science & Engineering (ES) 2024

Consider the equation for a curve, \( y = f(x) = x^2 + x \).
The area enclosed by the curve, the \( x \)-axis (\( y = 0 \) line); the vertical lines passing through \( x = 1 \) and \( x = 2 \) is __________ (rounded off to 2 decimal places)

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6

2025 · Environmental Science & Engineering · Mathematics Foundation · Calculus

Environmental Science & Engineering (ES) 2025

Assuming \(s > |a|\); Laplace transform of \(f(x) = \cosh ax\) is

A

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

B

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

C

\(\frac{s}{s^2-a^2}\)

D

\(\frac{a}{s^2-a^2}\)

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7

2025 · Environmental Science & Engineering · Mathematics Foundation · Calculus

Environmental Science & Engineering (ES) 2025

For the ordinary differential equation \(\frac{d^2y}{dx^2} + 4y = 0\), the general solution is

A

\(y = c_1 \cos 2x + c_2 \sin 2x\)

B

\(y = c_1 \cosh 2x + c_2 \sinh 2x\)

C

\(y = c_1 e^{2x} + c_2 e^{-2x}\)

D

\(y = c_1 e^{2x} \cos 2x + c_2 e^{-2x} \sin 2x\)

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8

2025 · Environmental Science & Engineering · Mathematics Foundation · Calculus

Environmental Science & Engineering (ES) 2025

Consider the following two series
P: \(\sum_{n=1}^{\infty} \frac{1}{n}\)
Q: \(\sum_{n=1}^{\infty} \frac{1}{n^2}\)
Choose the correct option from the following

A

P is convergent series; Q is divergent series

B

P is divergent series; Q is convergent series

C

Both P and Q are convergent series

D

Both P and Q are divergent series

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9

2025 · Environmental Science & Engineering · Mathematics Foundation · Calculus

Environmental Science & Engineering (ES) 2025

\[ \lim_{x\to 0} \left( \frac{\ln(1+x)}{2\sin x} \right) \] is __________ (rounded off to two decimal places).

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10

2025 · Environmental Science & Engineering · Mathematics Foundation · Calculus

Environmental Science & Engineering (ES) 2025

The value of \( \int_{0}^{\infty} \frac{\sin 4x}{\pi x} dx \) is _________. (rounded off to two decimal places)

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11

2025 · Environmental Science & Engineering · Mathematics Foundation · Calculus

Environmental Science & Engineering (ES) 2025

A tank has inflow, outflow and stirring mechanism. Initially, the tank holds 500 L of a brine solution of concentration 200 g/L. At \( t = 0 \), an inflow of another brine solution of concentration 100 g/L starts entering the tank at the rate of 15 L/minute. At the same time the outflow of thoroughly stirred mixture also takes place at the same rate so that the volume of brine in the tank remains constant. The brine concentration \( C \) (g/L) in the tank at any time \( t \) (minute) can be expressed by the following differential equation \[ \frac{dC}{dt} + 0.03 C = 3 \] The brine concentration in the tank at \( t = 1.5 \) hour is ________ g/L. (rounded off to two decimal places)

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12

2021 · Environmental Science & Engineering · Mathematics Foundation · Calculus

Environmental Science & Engineering (ES) 2021

A tangent is drawn on the curve of the function y = x^2 at the point (x, y) = (3, 9). The slope of the tangent is ______.

A

3

B

6

C

9

D

12

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13

2021 · Environmental Science & Engineering · Mathematics Foundation · Calculus

Environmental Science & Engineering (ES) 2021

\[\lim_{x \to 0} \frac{x^2}{\sin x} = ______\]

A

0

B

1

C

2

D

-1

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14

2021 · Environmental Science & Engineering · Mathematics Foundation · Calculus

Environmental Science & Engineering (ES) 2021

The ordinary differential equation \[ \frac{dy}{dx} = x^2 y \] has y as the dependent variable and x as the independent variable. Which of the following classification(s) is/are applicable to the equation?

A

Linear

B

Non-linear

C

First order

D

Second order

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15

2021 · Environmental Science & Engineering · Mathematics Foundation · Calculus

Environmental Science & Engineering (ES) 2021

Consider the following equation: \[ x^3 - 10x^2 + 31x - 30 = 0 \] Which of the following is/are the root(s) of the above equation?

A

1

B

2

C

3

D

4

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16

2021 · Environmental Science & Engineering · Mathematics Foundation · Calculus

Environmental Science & Engineering (ES) 2021

If \( f(x) + 3f\left(g(x)\right) = x - 2 \), where \( g(x) = \frac{3x+1}{x-3} \), then the value of the ratio \( \frac{f(5)}{f(8)} \) is ______________ (answer in integer).

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17

2021 · Environmental Science & Engineering · Mathematics Foundation · Calculus

Environmental Science & Engineering (ES) 2021

Consider a function \( y = f(x) \) which satisfies the following equation: \( \frac{d^2 y}{dx^2} - \frac{dy}{dx} = 0 \)
As \( x \to -\infty \), \( y = 1 \), and at \( x = 0 \), \( y = 2 \).
The value of \( \frac{dy}{dx} \) at \( x = 0 \) is ______________ (answer in integer).

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18

2022 · Environmental Science & Engineering · Mathematics Foundation · Calculus

Environmental Science & Engineering (ES) 2022

Assuming s > 0; Laplace transform for f(x) = sin ax is

A

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

B

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

C

\(\frac{a}{s^2 - a^2}\)

D

\(\frac{s}{s^2 - a^2}\)

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19

2022 · Environmental Science & Engineering · Mathematics Foundation · Calculus

Environmental Science & Engineering (ES) 2022

\[ \lim_{x \to 0} \frac{\sqrt[4]{1+x}-1}{x} \] is __________ (rounded off to one decimal place)

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20

2022 · Environmental Science & Engineering · Mathematics Foundation · Calculus

Environmental Science & Engineering (ES) 2022

Given, \( y = f(x) \); \( \frac{d^2y}{dx^2} + 4y = 0; y(0) = 0; \frac{dy}{dx}(0) = 1 \). The problem is a/an

A

initial value problem having solution \(y = x \)

B

boundary value problem having solution \(y = x \)

C

initial value problem having solution \(y = \frac{1}{2} \sin 2x \)

D

boundary value problem having solution \(y = \frac{1}{2} \sin 2x \)

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