Functions of Two Variables - Calculus - Engineering Sciences Previous Year Questions

1

2007 · Engineering Sciences · Calculus · Functions of Two Variables

Engineering Sciences (XE) 2007

The maximum value of the function \( 2x + 3y + 4z \) on the ellipsoid \( 2x^2 + 3y^2 + 4z^2 = 1 \) is

A

\(2 \)

B

\(3 \)

C

\(6 \)

D

\(9 \)

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2

2007 · Engineering Sciences · Calculus · Functions of Two Variables

Engineering Sciences (XE) 2007

The volume of the prism whose base is the triangle in the \( xy \)-plane bounded by the \( x \)-axis and the lines \( y=x \) and \( x=2 \) and whose top lies in the plane \( z=5-x-y \) is

A

2

B

4

C

6

D

10

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3

2008 · Engineering Sciences · Calculus · Functions of Two Variables

Engineering Sciences (XE) 2008

\(\lim_{(x,y) \to (0,0)} \frac{x^4 + xy}{x^3 - y^3}\) is

A

0

B

1

C

-1

D

does not exist

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4

2008 · Engineering Sciences · Calculus · Functions of Two Variables

Engineering Sciences (XE) 2008

The Karnaugh Map for F is given by (X being don't care)

A

AB \ CD	00	01	11	10
00	X	X	X	1
01	X	X	X	X
11	1	X	X	1
10	1	1	1	1

B

AB \ CD	00	01	11	10
00	1	1	1	X
01	1	1	1	1
11	X	1	1	X
10	X	X	X	X

C

AB \ CD	00	01	11	10
00	1	X	1	X
01	X	1	X	1
11	1	X	X	X
10	X	1	X	1

D

AB \ CD	00	01	11	10
00	1	1	X	X
01	X	X	X	X
11	X	1	1	1
10	X	1	1	X

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5

2010 · Engineering Sciences · Calculus · Functions of Two Variables

Engineering Sciences (XE) 2010

Let \(u(x,y) = \tan\{xy(x+y)\}\). Then

A

\(x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = \frac{1}{3} xy(x+y) \sec^2\{xy(x+y)\}\)

B

\(x \frac{\partial u}{\partial x} - y \frac{\partial u}{\partial y} = \frac{1}{3} xy(x+y) \sec^2\{xy(x+y)\}\)

C

\(x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = 3 xy(x+y) \sec^2\{xy(x+y)\}\)

D

\(x \frac{\partial u}{\partial x} - y \frac{\partial u}{\partial y} = 3 xy(x+y) \sec^2\{xy(x+y)\}\)

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6

2012 · Engineering Sciences · Calculus · Functions of Two Variables

Engineering Sciences (XE) 2012

Let \( f(u,v) = u \ln(v) \) and \( F(x,y) = f(u(x,y), v(x,y)) \), where \( u = x/y \) and \( v = x-y \). Then \( \partial F / \partial y \) is

A

\(-\frac{x}{y^2} \ln(x-y) + \frac{x}{y(x-y)} \)

B

\(-\frac{x}{y^2} \ln v - \frac{u}{v} \)

C

\(\frac{x}{y^2} \ln v - \frac{u}{v} \)

D

\(\frac{x}{y^2} \ln v + \frac{x}{yv} \)

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7

2013 · Engineering Sciences · Calculus · Functions of Two Variables

Engineering Sciences (XE) 2013

The integral \(\int_{0}^{1} \int_{x^{2}}^{x}\left(\frac{x}{y}\right) e^{-x^{2} / y} d y d x\) equals

A

\(\frac{e-2}{e}\)

B

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

C

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

D

\(\frac{e-2}{2 e}\)

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8

2014 · Engineering Sciences · Calculus · Functions of Two Variables

Engineering Sciences (XE) 2014

The perimeter of a rectangle having the largest area that can be inscribed in the ellipse $\frac{x^2}{8} + \frac{y^2}{32} = 1$, is ______.

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9

2017 · Engineering Sciences · Calculus · Functions of Two Variables

Engineering Sciences (XE) 2017

If \[ \int_{0}^{\alpha} \int_{x}^{\pi/\alpha} \frac{\sin y}{y} \, dy \, dx = \frac{1}{2} \] for some \( \alpha \geq 1 \), then the value of \( \alpha \) is ______________.

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10

2019 · Engineering Sciences · Calculus · Functions of Two Variables

Engineering Sciences (XE) 2019

The number of critical points of the function \( f(x, y) = x^3 + 3xy^2 - 15x - 12y \) at which there is neither maximum nor minimum is ______.

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11

2020 · Engineering Sciences · Calculus · Functions of Two Variables

Engineering Sciences (XE) 2020

The number of points at which the function \( f(x, y) = \frac{x^3}{2} + \frac{y^4}{4} - \frac{y^2}{2} \) has local minima is ____.

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12

2021 · Engineering Sciences · Calculus · Functions of Two Variables

Engineering Sciences (XE) 2021

Let \( f(x, y) = x^4 + y^4 - 2x^2 + 4xy - 2y^2 + \alpha \) be a real valued function. Then, which one of the following statements is TRUE for all \( \alpha \) ?

A

(0, 0) is not a stationary point of \(f \)

B

\(f \) has a local maxima at (0, 0)

C

\(f \) has a local minima at (0, 0)

D

\(f \) has a saddle point at (0, 0)

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13

2022 · Engineering Sciences · Calculus · Functions of Two Variables

Engineering Sciences (XE) 2022

Let \( f: \mathbb{R}^2 \to \mathbb{R} \) be given by \( f(x,y) = 4xy - 2x^2 - y^4 + 1 \). The number of critical points where \( f \) has local maximum is equal to ______.

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14

2023 · Engineering Sciences · Calculus · Functions of Two Variables

Engineering Sciences (XE) 2023

Let f: ℝ^2 → ℝ be a function defined by f(x, y) = { xy / (|x| + y) , y ≠ -|x| ; 0 , otherwise. } Then which one of the following statement is TRUE?

A

f is NOT continuous at (0,0).

B

∂f/∂x (0,0) = 0, and ∂f/∂y (0,0) = 1.

C

∂f/∂x (0,0) = 1, and ∂f/∂y (0,0) = 0.

D

∂f/∂x (0,0) = 1, and ∂f/∂y (0,0) = 1.

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15

2024 · Engineering Sciences · Calculus · Functions of Two Variables

Engineering Sciences (XE) 2024

If \(\int_{0}^{\alpha} \int_{\sqrt{x}/\alpha}^{1} e^{y^3} \, dy \, dx = e - 1\), \(\alpha > 0\), then the value (in integer) of \(\alpha\) is ______

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16

2025 · Engineering Sciences · Calculus · Functions of Two Variables

Engineering Sciences (XE) 2025

Consider the function
\(f(x,y) = x^2y + 2xy^2 - 2x^2y^2\).
Then which one of the following statements is correct ?

A

\(\left(\frac{3}{2}, 0\right)\) is a point of local maxima of \(f\).

B

\(\left(0, \frac{3}{4}\right)\) is a point of local minima of \(f\).

C

\(\left(\frac{3}{2}, \frac{3}{4}\right)\) is a point of local maxima of \(f\).

D

\(\left(\frac{3}{2}, \frac{3}{4}\right)\) is a saddle point of \(f\).

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17

2026 · Engineering Sciences · Calculus · Functions of Two Variables

Engineering Sciences (XE) 2026

Circles \(C_1\), \(C_2\), and \(C_3\), with centers \(O_1\), \(O_2\), and \(O_3\), and radii \(r_1\), \(r_2\), and \(r_3\), respectively, touch each other as shown in the following figure. Given \(r_1 = 2\) cm, \(r_2 = 1\) cm and the angle \(\angle O_1 O_3 O_2\) is \(90^\circ\), \(r_3 = \) ______ cm.

A

\(\frac{1}{2}(-3 + \sqrt{17})\)

B

\(\frac{1}{2}(3 + \sqrt{17})\)

C

\(\frac{1}{2}(-2 + \sqrt{17})\)

D

\(\frac{1}{2}(-3 + 2\sqrt{17})\)

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18

2026 · Engineering Sciences · Calculus · Functions of Two Variables

Engineering Sciences (XE) 2026

Let \( L \) be the lamina of the form \( x^2 + 4y^2 \le 64, 0 \le y \le 4 \), with density \( \rho(x,y) = |x|y \). Then, the mass of \( L \) (in integer) is ______

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