Multivariable Differentiation and Optimization - Calculus - Mathematics Previous Year Questions

1

2008 · Mathematics · Calculus · Multivariable Differentiation and Optimization

Mathematics (MA) 2008

For \((x, y) \in \mathbb{R}^2\), let \( f(x, y) = \begin{cases} \frac{2xy}{x^2 + y^2} & \text{if } (x, y) \neq (0, 0), \\ 0 & \text{if } (x, y) = (0, 0). \end{cases} \) Then

A

\(f_x\) and \(f_y\) exist at \((0,0)\), and \(f\) is continuous at \((0,0)\)

B

\(f_x\) and \(f_y\) exist at \((0,0)\), and \(f\) is discontinuous at \((0,0)\)

C

\(f_x\) and \(f_y\) do not exist at \((0,0)\), and \(f\) is continuous at \((0,0)\)

D

\(f_x\) and \(f_y\) do not exist at \((0,0)\), and \(f\) is discontinuous at \((0,0)\)

Open complete paper

2

2008 · Mathematics · Calculus · Multivariable Differentiation and Optimization

Mathematics (MA) 2008

Let \(q_1, q_2, \cdots, q_n\) be the generalized coordinates and \(\dot{q}_1, \dot{q}_2, \cdots, \dot{q}_n\) be the generalized velocities in a conservative force field. If under a transformation \(\varphi\), the new coordinate system has the generalized coordinates \(Q_1, Q_2, \cdots, Q_n\) and velocities \(\dot{Q}_1, \dot{Q}_2, \cdots, \dot{Q}_n\). Then the equation \(\frac{\partial L}{\partial q_i} = \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right)\) takes the form

A

\(\frac{\partial L}{\partial Q_k} = \varphi \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{Q}_k}\right)\)

B

\(\varphi \frac{\partial L}{\partial Q_k} = \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{Q}_k}\right)\)

C

\(\frac{\partial L}{\partial Q_k} = -\varphi \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{Q}_k}\right)\)

D

\(\frac{\partial L}{\partial Q_k} = \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{Q}_k}\right)\)

Open complete paper

3

2008 · Mathematics · Calculus · Multivariable Differentiation and Optimization

Mathematics (MA) 2008

Let \(E = \{(x, y) \in \mathbb{R}^2 : |x| \le 1, |y| \le 1\}\). Define \(f: E \to \mathbb{R}\) by \(f(x, y) = \frac{x + y}{1 + x^2 + y^2}\). Then the range of \(f\) is a

A

connected open set

B

connected closed set

C

bounded open set

D

closed and unbounded set

Open complete paper

4

2009 · Mathematics · Calculus · Multivariable Differentiation and Optimization

Mathematics (MA) 2009

The value of \( f \) at a local minimum in the rectangular region \( R=\left\{(x, y) \in \mathbb{R}^{2}:|x|<\frac{3}{2},|y|<\frac{3}{2}\right\} \) is

A

\(-2 \)

B

\(-3 \)

C

\(-\frac{7}{8} \)

D

\(0 \)

Open complete paper

5

2010 · Mathematics · Calculus · Multivariable Differentiation and Optimization

Mathematics (MA) 2010

Let \( f(x,y) = \begin{cases} \frac{xy}{(x^2+y^2)^{3/2}} [1-\cos(x^2+y^2)], & (x,y) \neq (0,0) \\ k, & (x,y) = (0,0) \end{cases} \) Then the value of \( k \) for which \( f(x,y) \) is continuous at \( (0,0) \) is

A

0

B

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

C

1

D

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

Open complete paper

6

2011 · Mathematics · Calculus · Multivariable Differentiation and Optimization

Mathematics (MA) 2011

If \( x, y \) and \( z \) are positive real numbers, then the minimum value of \( x^2 + 8y^2 + 27z^2 \) where \( \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 1 \) is

A

108

B

216

C

405

D

1048

Open complete paper

7

2011 · Mathematics · Calculus · Multivariable Differentiation and Optimization

Mathematics (MA) 2011

The fuel consumed by a motorcycle during a journey while traveling at various speeds is indicated in the graph below.
The distances covered during four laps of the journey are listed in the table below

Lap	Distance (kilometres)	Average speed (kilometres per hour)
P	15	15
Q	75	45
R	40	75
S	10	10

From the given data, we can conclude that the fuel consumed per kilometre was least during the lap

A

P

B

Q

C

R

D

S

Open complete paper

8

2012 · Mathematics · Calculus · Multivariable Differentiation and Optimization

Mathematics (MA) 2012

The maximum value of the function \( f(x,y,z) = xyz \) subject to the constraint
\( xy + yz + zx - a = 0, a > 0 \) is

A

\(a^{3/2} \)

B

\((a/3)^{3/2} \)

C

\((3/a)^{3/2} \)

D

\((3a/2)^{3/2} \)

Open complete paper

9

2015 · Mathematics · Calculus · Multivariable Differentiation and Optimization

Mathematics (MA) 2015

Let \(D = \{(x,y) \in \mathbb{R}^2: 1 \le x \le 1000, 1 \le y \le 1000\}\). Define
\(f(x,y) = \frac{x y}{2} + \frac{500}{x} + \frac{500}{y}\).
Then the minimum value of \(f\) on \(D\) is equal to ______

Check answer

Open complete paper

10

2016 · Mathematics · Calculus · Multivariable Differentiation and Optimization

Mathematics (MA) 2016

Maximum {x + y : (x,y) ∈ B̄(0,1)} is equal to __________

Check answer

Open complete paper

11

2016 · Mathematics · Calculus · Multivariable Differentiation and Optimization

Mathematics (MA) 2016

For any \((x,y) \in \mathbb{R}^2 \setminus \overline{B(0,1)}\), let
\[ f(x,y) = \text{distance}\left((x,y), \overline{B(0,1)}\right) \\ = \inf\left\{\sqrt{(x-x_1)^2 + (y-y_1)^2} : (x_1,y_1) \in \overline{B(0,1)}\right\}. \]
Then, \(\|\nabla f(3,4)\|\| is equal to __________

Check answer

Open complete paper

12

2017 · Mathematics · Calculus · Multivariable Differentiation and Optimization

Mathematics (MA) 2017

If \(u(x, y) = 1 + x + y + f(xy)\), where \(f : \mathbb{R}^2 \to \mathbb{R}\) is a differentiable function, then \(u\) satisfies

A

\(x \frac{\partial u}{\partial x} - y \frac{\partial u}{\partial y} = x^2 - y^2\)

B

\(x \frac{\partial u}{\partial x} - y \frac{\partial u}{\partial y} = 0\)

C

\(x \frac{\partial u}{\partial x} - y \frac{\partial u}{\partial y} = x - y\)

D

\(y \frac{\partial u}{\partial x} - x \frac{\partial u}{\partial y} = x - y\)

Open complete paper

13

2017 · Mathematics · Calculus · Multivariable Differentiation and Optimization

Mathematics (MA) 2017

Let \(f:\mathbb{R}^2\to\mathbb{R}\) be defined by \(f(x,y)=\begin{cases}\sin\left(\frac{y^2}{x}\right)\sqrt{x^2+y^2}, & x\neq 0,\\ 0, & x=0.\end{cases}\) Then, at \((0,0)\),

A

\(f\) is continuous and the directional derivative of \(f\) does NOT exist in some direction

B

\(f\) is NOT continuous and the directional derivatives of \(f\) exist in all directions

C

\(f\) is NOT differentiable and the directional derivatives of \(f\) exist in all directions

D

\(f\) is differentiable

Open complete paper

14

2018 · Mathematics · Calculus · Multivariable Differentiation and Optimization

Mathematics (MA) 2018

Let \(u(x, y, z) = x^2 - 2y + 4z^2\) for \((x, y, z) \in \mathbb{R}^3\). Then the directional derivative of \(u\) in the direction \(\frac{3}{5}\hat{i} - \frac{4}{5}\hat{k}\) at the point \((5, 1, 0)\) is ________.

Check answer

Open complete paper

15

2019 · Mathematics · Calculus · Multivariable Differentiation and Optimization

Mathematics (MA) 2019

Let \(f : \mathbb{R}^2 \to \mathbb{R}\) be defined by \[f(x,y) = x^5 - 2x^3 y - x^2 y + 2y^3.\] ( \(\mathbb{R}\) is the set of all real numbers and \(\mathbb{R}^2 = \{(x,y) : x,y \in \mathbb{R}\}\) ) Which one of the following statements is TRUE?

A

\(f\) has a local maximum at origin

B

\(f\) has a local minimum at origin

C

\(f\) has a saddle point at origin

D

The origin is not a critical point of \(f\)

Open complete paper

16

2019 · Mathematics · Calculus · Multivariable Differentiation and Optimization

Mathematics (MA) 2019

Let \(g : \mathbb{R}^2 \to \mathbb{R}^2\) be a function defined by \(g(x,y) = (e^x \cos y, e^x \sin y)\) and \((a,b) = g \left( 1, \frac{\pi}{3} \right)\). ( \(\mathbb{R}\) is the set of all real numbers and \(\mathbb{R}^2 = \{(x,y) : x,y \in \mathbb{R}\}\) ) Which one of the following statements is TRUE?

A

\(g\) is injective

B

If \(h\) is the continuous inverse of \(g\), defined in some neighbourhood of \((a,b) \in \mathbb{R}^2\), such that \(h(a,b) = \left( 1, \frac{\pi}{3} \right)\), then the Jacobian of \(h\) at \((a,b)\) is \(e^2\)

C

If \(h\) is the continuous inverse of \(g\), defined in some neighbourhood of \((a,b) \in \mathbb{R}^2\), such that \(h(a,b) = \left( 1, \frac{\pi}{3} \right)\), then the Jacobian of \(h\) at \((a,b)\) is \(e^{-2}\)

D

\(g\) is surjective

Open complete paper

17

2019 · Mathematics · Calculus · Multivariable Differentiation and Optimization

Mathematics (MA) 2019

The temperature \(T: \mathbb{R}^3 \setminus \{(0,0,0)\} \to \mathbb{R}\) at any point \(P(x,y,z)\) is inversely proportional to the square of the distance of \(P\) from the origin. If the value of the temperature \(T\) at the point \(R(0,0,1)\) is \(\sqrt{3}\), then the rate of change of \(T\) at the point \(Q(1,1,2)\) in the direction of \(\overrightarrow{QR}\) is equal to ______ (round off to 2 places of decimal). (\(\mathbb{R}\) is the set of all real numbers, \(\mathbb{R}^3 = \{(x,y,z): x,y,z \in \mathbb{R}\}\) and \(\mathbb{R}^3 \setminus \{(0,0,0)\}\) denotes \(\mathbb{R}^3\) excluding the origin)

Check answer

Open complete paper

18

2019 · Mathematics · Calculus · Multivariable Differentiation and Optimization

Mathematics (MA) 2019

Let \(f: \mathbb{R}^2 \to \mathbb{R}\) be defined by \(f(x,y) = \begin{cases} (x^2+y^2)\sin\left(\frac{1}{x^2+y^2}\right), & \text{if } (x,y) \neq (0,0) \\ 0, & \text{if } (x,y) = (0,0). \end{cases}\) Consider the following statements: I. The partial derivatives \(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\) exist at \((0,0)\) but are unbounded in any neighbourhood of \((0,0)\). II. \(f\) is continuous but not differentiable at \((0,0)\). III. \(f\) is not continuous at \((0,0)\). IV. \(f\) is differentiable at \((0,0)\). (\(\mathbb{R}\) is the set of all real numbers and \(\mathbb{R}^2 = \{(x,y): x,y \in \mathbb{R}\}\)) Which of the above statements is/are TRUE? (A) I and II only (B) I and IV only (C) IV only (D) III only

A

I and II only

B

I and IV only

C

IV only

D

III only

Open complete paper

19

2020 · Mathematics · Calculus · Multivariable Differentiation and Optimization

Mathematics (MA) 2020

If \((4,0)\) and \(\left(0, -\frac{1}{2}\right)\) are critical points of the function \( f(x,y) = 5 - (\alpha + \beta)x^2 + \beta y^2 + (\alpha + 1)y^3 + x^3, \) where \(\alpha, \beta \in \mathbb{R}\), then

A

\(\left(4, -\frac{1}{2}\right)\) is a point of local maxima of \(f\)

B

\(\left(4, -\frac{1}{2}\right)\) is a saddle point of \(f\)

C

\(\alpha = 4, \beta = 2\)

D

\(\left(4, -\frac{1}{2}\right)\) is a point of local minima of \(f\)

Open complete paper

20

2021 · Mathematics · Calculus · Multivariable Differentiation and Optimization

Mathematics (MA) 2021

The family of surfaces given by \(u = xy + f(x^2 - y^2)\), where \(f: \mathbb{R} \to \mathbb{R}\) is a differentiable function, satisfies

A

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

B

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

C

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

D

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

Open complete paper