Differential Equations and Fourier Analysis - Mathematical Physics - Physics Previous Year Questions

1

2007 · Physics · Mathematical Physics · Differential Equations and Fourier Analysis

Physics (PH) 2007

Inverse Laplace transform of \(\frac{s+1}{s^2 - 4}\) is

A

\(\cos 2x + \frac{1}{2} \sin 2x\)

B

\(\cos x + \frac{1}{2} \sin x\)

C

\(\cosh x + \frac{1}{2} \sinh x\)

D

\(\cosh 2x + \frac{1}{2} \sinh 2x\)

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2

2007 · Physics · Mathematical Physics · Differential Equations and Fourier Analysis

Physics (PH) 2007

The points, where the series solution of the Legendre differential equation \((1-x^2)\frac{d^2y}{dx^2} - 2x\frac{dy}{dx} + \frac{3}{2}(\frac{3}{2}+1)y = 0\) will diverge, are located at

A

0 and 1

B

0 and -1

C

-1 and 1

D

\(\frac{3}{2}\) and \(\frac{5}{2}\)

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3

2007 · Physics · Mathematical Physics · Differential Equations and Fourier Analysis

Physics (PH) 2007

Solution of the differential equation \(x\frac{dy}{dx} + y = x^4\), with the boundary condition that \(y = 1\), at \(x = 1\), is

A

\(y = 5x^4 - 4\)

B

\(y = \frac{x^4}{5} + \frac{4x}{5}\)

C

\(y = \frac{4x^4}{5} + \frac{1}{5x}\)

D

\(y = \frac{x^4}{5} + \frac{4}{5x}\)

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4

2007 · Physics · Mathematical Physics · Differential Equations and Fourier Analysis

Physics (PH) 2007

The points, where the series solution of the Legendre differential equation \((1-x^2)\frac{d^2 y}{dx^2} - 2x\frac{dy}{dx} + \frac{3}{2}(\frac{3}{2}+1)y = 0\) will diverge, are located at

A

0 and 1

B

0 and -1

C

-1 and 1

D

\(\frac{3}{2}\) and \(\frac{5}{2}\)

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5

2007 · Physics · Mathematical Physics · Differential Equations and Fourier Analysis

Physics (PH) 2007

Solution of the differential equation \(x\frac{dy}{dx} + y = x^4\), with the boundary condition that \(y=1\), at \(x=1\), is

A

\(y = 5x^4 - 4\)

B

\(y = \frac{x^4}{5} + \frac{4x}{5}\)

C

\(y = \frac{4x^4}{5} + \frac{1}{5x}\)

D

\(y = \frac{x^4}{5} + \frac{4}{5x}\)

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6

2008 · Physics · Mathematical Physics · Differential Equations and Fourier Analysis

Physics (PH) 2008

A finite wave train, of an unspecified nature, propagates along the positive x axis with a constant speed v and without any change of shape. The differential equation among the four listed below, whose solution it must be, is

A

$\left( \frac{\partial^2}{\partial x^2} - \frac{1}{v^2} \frac{\partial^2}{\partial t^2} \right) \psi(x,t) = 0$

B

$\left( \nabla^2 - \frac{1}{v^2} \frac{\partial^2}{\partial t^2} \right) \psi(\vec{r},t) = 0$

C

$\left( -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} - i\hbar \frac{\partial}{\partial t} \right) \psi(x,t) = 0$

D

$\left( \nabla^2 + \frac{\partial}{\partial t} \right) \psi(\vec{r},t) = 0$

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7

2008 · Physics · Mathematical Physics · Differential Equations and Fourier Analysis

Physics (PH) 2008

If the Fourier transform \(F[\delta(x-a)] = \exp(-i2\pi v a)\), then \(F^{-1}(\cos 2\pi a v)\) will correspond to

A

\(\delta(x-a) - \delta(x+a)\)

B

a constant

C

\(\frac{1}{2}[\delta(x-a) + i\delta(x+a)]\)

D

\(\frac{1}{2}[\delta(x-a) + \delta(x+a)]\)

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8

2008 · Physics · Mathematical Physics · Differential Equations and Fourier Analysis

Physics (PH) 2008

Consider the Bessel equation \(\left(\nu = 0\right)\), \(\frac{d^2 y}{d z^2} + \frac{1}{z} \frac{d y}{d z} + y = 0\). Which one of the following statements is correct?

A

Equation has regular singular points at \(z = 0\) and \(z = \infty\).

B

Equation has 2 linearly independent solutions that are entire.

C

Equation has an entire solution and a second linearly independent solution singular at \(z = 0\).

D

Limit \(z \to \infty\), taken along \(x\) axis, exists for both the linearly independent solutions.

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9

2009 · Physics · Mathematical Physics · Differential Equations and Fourier Analysis

Physics (PH) 2009

Which one of the following curves gives the solution of the differential equation \( k_1 \frac{dx}{dt} + k_2 x = k_3 \), where \( k_1, k_2 \), and \( k_3 \) are positive constants with initial conditions \( x = 0 \) at \( t = 0 \) ?

A

Graph showing x increasing and approaching a horizontal asymptote.

B

Graph showing x increasing and approaching a vertical asymptote.

C

Graph showing x increasing to a maximum then decreasing to zero.

D

Graph showing x decreasing then increasing with a vertical asymptote.

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10

2010 · Physics · Mathematical Physics · Differential Equations and Fourier Analysis

Physics (PH) 2010

If \( f(x) = \begin{cases} 0 & \text{for } x < 3 \\ x-3 & \text{for } x \geq 3 \end{cases} \), then the Laplace transform of \( f(x) \) is

A

\(s^{-2} e^{3s} \)

B

\(s^{2} e^{-3s} \)

C

\(s^{-2} \)

D

\(s^{-2} e^{-3s} \)

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11

2010 · Physics · Mathematical Physics · Differential Equations and Fourier Analysis

Physics (PH) 2010

The solution of the differential equation for \( y(t) \): \( \frac{d^2 y}{dt^2} - y = 2 \cosh(t) \), subject to the initial conditions \( y(0) = 0 \) and \( \left. \frac{dy}{dt} \right|_{t=0} = 0 \), is

A

\(\frac{1}{2} \cosh(t) + t \sinh(t) \)

B

\(- \sinh(t) + t \cosh(t) \)

C

\(t \cosh(t) \)

D

\(t \sinh(t) \)

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12

2010 · Physics · Mathematical Physics · Differential Equations and Fourier Analysis

Physics (PH) 2010

Given the recurrence relation for the Legendre polynomials
\( (2n+1) x P_n(x) = (n+1) P_{n+1}(x) + n P_{n-1}(x) \),
which of the following integrals has a non zero value?

A

\(\int_{-1}^{1} x^2 P_0(x) P_{n+1}(x) \, dx \)

B

\(\int_{-1}^{1} x P_n(x) P_{n+2}(x) \, dx \)

C

\(\int_{-1}^{1} x [P_n(x)]^2 \, dx \)

D

\(\int_{-1}^{1} x^2 P_n(x) P_{n+2}(x) \, dx \)

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13

2010 · Physics · Mathematical Physics · Differential Equations and Fourier Analysis

Physics (PH) 2010

Hari (H), Gita (G), Irfan (I) and Saira (S) are siblings (i.e. brothers and sisters). All were born on 1st January. The age difference between any two successive siblings (that is born one after another) is less than 3 years. Given the following facts:
i. Hari's age + Gita's age > Irfan's age + Saira's age.
ii. The age difference between Gita and Saira is 1 year. However, Gita is not the oldest and Saira is not the youngest.
iii. There are no twins.
In what order were they born (oldest first)?

A

HSIG

B

SGHI

C

IGSH

D

IHSG

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14

2010 · Physics · Mathematical Physics · Differential Equations and Fourier Analysis

Physics (PH) 2010

Given digits 2, 2, 3, 3, 3, 4, 4, 4, 4 how many distinct 4 digit numbers greater than 3000 can be formed?

A

50

B

51

C

52

D

54

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15

2011 · Physics · Mathematical Physics · Differential Equations and Fourier Analysis

Physics (PH) 2011

The solutions to the differential equation \( \frac{dy}{dx} = -\frac{x}{y+1} \) are a family of

A

circles with different radii

B

circles with different centres

C

straight lines with different slopes

D

straight lines with different intercepts on the y-axis

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16

2011 · Physics · Mathematical Physics · Differential Equations and Fourier Analysis

Physics (PH) 2011

Choose the most appropriate word(s) from the options given below to complete the following sentence.
I contemplated __________ Singapore for my vacation but decided against it.

A

to visit

B

having to visit

C

visiting

D

for a visit

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17

2011 · Physics · Mathematical Physics · Differential Equations and Fourier Analysis

Physics (PH) 2011

If \(\log (P) = (1/2)\log (Q) = (1/3) \log (R)\), then which of the following options is TRUE?

A

\(P^2 = Q^3 R^2\)

B

\(Q^2 = PR\)

C

\(Q^2 = R^3 P\)

D

\(R = P^2 Q^2\)

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18

2011 · Physics · Mathematical Physics · Differential Equations and Fourier Analysis

Physics (PH) 2011

A transporter receives the same number of orders each day. Currently, he has some pending orders (backlog) to be shipped. If he uses 7 trucks, then at the end of the 4th day he can clear all the orders. Alternatively, if he uses only 3 trucks, then all the orders are cleared at the end of the 10th day. What is the minimum number of trucks required so that there will be no pending order at the end of the 5th day?

A

4

B

5

C

6

D

7

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19

2011 · Physics · Mathematical Physics · Differential Equations and Fourier Analysis

Physics (PH) 2011

The variable cost (V) of manufacturing a product varies according to the equation \(V = 4q\), where q is the quantity produced. The fixed cost (F) of production of same product reduces with q according to the equation \(F = 100/q\). How many units should be produced to minimize the total cost (V+F)?

A

5

B

4

C

7

D

6

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20

2011 · Physics · Mathematical Physics · Differential Equations and Fourier Analysis

Physics (PH) 2011

Choose the most appropriate word(s) from the options given below to complete the following sentence.
I contemplated ______________ Singapore for my vacation but decided against it.

A

to visit

B

having to visit

C

visiting

D

for a visit

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