Q.No:1 JEST-2012
If \(\lfloor x \rfloor\) denotes the greatest integer not exceeding \(x\), then \(\int_{0}^{\infty} \lfloor x\rfloor e^{-x} dx\) is
(a)
\(\frac{1}{e-1}\)
(b)
\(1\)
(c)
\(\frac{e-1}{e}\)
(d)
\(\frac{e}{e^2-1}\)
Check Answer
Option a
Q.No:2 JEST-2013
A particle of mass \(m\) is thrown upward with velocity \(v\) and there is retarding air resistance proportional to the square of the velocity with proportonality constant \(k\). If the particle attains a maximum height after time \(t\), and \(g\) is the gravitational acceleration, what is the velocity \(v\)?
(a)
\(\sqrt{\frac{k}{g}}\tan{\left(\sqrt{\frac{g}{k}}t\right)}\)
(b)
\(\sqrt{gk}\tan{\left(\sqrt{\frac{g}{k}}t\right)}\)
(c)
\(\sqrt{\frac{g}{k}}\tan{\left(\sqrt{gk}t\right)}\)
(d)
\(\sqrt{gk}\tan{\left(\sqrt{gk}t\right)}\)
Check Answer
Option c
Q.No:3 JEST-2013
The operator
\[
\left(\frac{d}{dx}-x\right)\left(\frac{d}{dx}+x\right)
\]
is equivalent to
(a)
\(\frac{d^2}{dx^2}-x^2\)
(b)
\(\frac{d^2}{dx^2}-x^2+1\)
(c)
\(\frac{d^2}{dx^2}-x\frac{d}{dx}x^2+1\)
(d)
\(\frac{d^2}{dx^2}-2x\frac{d}{dx}-x^2\)
Check Answer
Option b
Q.No:4 JEST-2013
Consider the differential equation
\[
\frac{dG(x)}{dx}+kG(x)=\delta(x),
\]
where \(k\) is a constant. Which of the following statements is true?
(a)
Both \(G(x)\) and \(G'(x)\) are continuous at \(x=0\).
(b)
\(G(x)\) is continuous at \(x=0\) but \(G'(x)\) is not.
(c)
\(G(x)\) is discontinuous at \(x=0\).
(d)
The continuity properties of \(G(x)\) and \(G'(x)\) at \(x=0\) depend on the value of \(k\).
Check Answer
Option c
Q.No:5 JEST-2014
What are the solutions to \(f''(x)-2f'(x)+f(x)=0\)?
(a)
\(c_1 e^x/x\)
(b)
\(c_1 x+c_2/x\)
(c)
\(c_1 xe^x+c_2\)
(d)
\(c_1 e^x+c_2 xe^x\)
Check Answer
Option d
Q.No:6 JEST-2015
What is the maximum number of extrema of the function \(f(x)=P_k(x)e^{-\left(\frac{x^4}{4}+\frac{x^2}{2}\right)}\) where \(x\in (-\infty, \infty)\) and \(P_k(x)\) is an arbitrary polynomial of degree \(k\)?
(a)
\(k+2\)
(b)
\(k+6\)
(c)
\(k+3\)
(d)
\(k\)
Check Answer
Option c
Q.No:7 JEST-2015
The Bernoulli polynomials \(B_n(s)\) are defined by, \(\frac{xe^{xs}}{e^x-1}=\sum B_n(s)\frac{x^n}{n!}\). Which one of the following relations is true?
(a)
\(k\)\(\frac{xe^{x(1-s)}}{e^x-1}=\sum B_n(s)\frac{x^n}{(n+1)!}\)
(b)
\(\frac{xe^{x(1-s)}}{e^x-1}=\sum B_n(s)(-1)^n \frac{x^n}{(n+1)!}\)
(c)
\(\frac{xe^{x(1-s)}}{e^x-1}=\sum B_n(-s)(-1)^n \frac{x^n}{n!}\)
(d)
\(\frac{xe^{x(1-s)}}{e^x-1}=\sum B_n(s)(-1)^n \frac{x^n}{n!}\)
Check Answer
Option d
Q.No:8 JEST-2015
Consider the differential equation \(G'(x)+kG(x)=\delta(x)\); where \(k\) is a constant. Which following statements is true?
(a)
Both \(G(x)\) and \(G'(x)\) are continuous at \(x=0\).
(b)
\(G(x)\) is continuous at \(x=0\) but \(G'(x)\) is not.
(c)
\(G(x)\) is discontinuous at \(x=0\).
(d)
The continuity properties of \(G(x)\) and \(G'(x)\) at \(x=0\) depends on the value of \(k\).
Check Answer
Option c
Q.No:9 JEST-2016
The sum of the infinite series \(1-1/3+1/5-1/7+...\) is:
(A)
\(2\pi\)
(B)
\(\pi\)
(C)
\(\pi/2\)
(D)
\(\pi/4\)
Check Answer
Option D
Q.No:10 JEST-2016
The half-life of a radioactive nuclear source is \(9\) days. The fraction of nuclei which are left undecayed after \(3\) days is:
(A)
\(7/8\)
(B)
\(1/3\)
(C)
\(5/6\)
(D)
\(1/2^{1/3}\)
Check Answer
Option D
Q.No:11 JEST-2018
For which of the following conditions does the integral \(\int_{0}^{1} P_m(x) P_n(x)dx\) vanish for \(m\neq n\), where \(P_m(x)\) and \(P_n(x)\) are the Legendre polynomials of order \(m\) and \(n\) respectively?
(A)
all \(m\), \(m\neq n\)
(B)
\(m-n\) is an odd integer
(C)
\(m-n\) is a nonzero even integer
(D)
\(n=m\pm 1\)
Check Answer
Option C
Q.No:12 JEST-2018
If \(y(x)\) satisfies
\[
\frac{dy}{dx}=y[1+(\log{y})^2]
\]
and \(y(0)=1\) for \(x\geq 0\), then \(y(\pi/2)\) is
(A)
\(0\)
(B)
\(1\)
(C)
\(\pi/2\)
(D)
infinity
Check Answer
Option D
Q.No:13 JEST-2019
Consider a function \(f(x)=P_k(x)e^{-(x^4+2x^2)}\) in the domain \(x\in (-\infty, \infty)\), where \(P_k\) is any polynomial of degree \(k\). What is the maximum possible number of extrema of the function?
(A)
\(k+3\)
(B)
\(k-3\)
(C)
\(k+2\)
(D)
\(k+1\)
Check Answer
Option D
Q.No:14 JEST-2019
The Euler polynomials are defined by
\[
\frac{2e^{xs}}{e^x+1}=\sum_{n=0}^{\infty} E_n(s)\frac{x^n}{n!}.
\]
What is the value of \(E_5(2)+E_5(3)\)?
Check Answer
Ans 64
Q.No:15 JEST-2020
The solution of the differential equation \(y''-2y'-3y=e^{2t}\) is given as \(C_1 e^{-t}+C_2 e^{2t}+C_3 e^{3t}\). The values of the coefficients \(C_1, C_2\) and \(C_3\) are:
(A)
\(C_1, C_2\) and \(C_3\) are arbitrary
(B)
\(C_1, C_3\) are arbitrary and \(C_2=-1/3\)
(C)
\(C_2, C_3\) are arbitrary and \(C_1=-1/3\)
(D)
\(C_1, C_2\) are arbitrary and \(C_3=-1/3\)
Check Answer
Option B
Q.No:16 JEST-2020
A particle moving in two dimensions satisfies the equations of motion
\[
\begin{array}{lll}
\dot{x}(t) &=& x(t)+y(t), \\
\dot{y}(t) &=& x(t)-y(t),
\end{array}
\]
with \(\dot{x}(0)=0\). What is the ratio of \(x(\infty)/y(\infty)\)?
(A)
\(1-1/\sqrt{2}\)
(B)
\(1+1/\sqrt{2}\)
(C)
\(\sqrt{2}-1\)
(D)
\(\sqrt{2}+1\)
Check Answer
Option D
Q.No:17 JEST-2020
Some bacteria are added to a bucket at time 10 am. The number of bacteria doubles every minute and reaches a number \(16\times 10^{15}\) at 10:18 am. How many seconds after 10 am were there \(25\times 10^{13}\) bacteria?
Check Answer
Ans 720
Q.No:18 JEST-2022
If three real variables \(x, y\) and \(z\) evolve with time \(t\) following
\[
\frac{dx}{dt}=x(y-z), \frac{dy}{dt}=y(z-x), \frac{dz}{dt}=z(x-y),
\]
then which of the following quantities remains invariant in time?
(a)
\(\frac{1}{xy}+\frac{1}{yz}+\frac{1}{zx}\)
(b)
\(x^2+y^2+z^2\)
(c)
\(xy+yz+zx\)
(d)
\(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\)
Check Answer
Option a
Q.No:19 JEST-2023
Solve the differential equation,
\[\frac{dy}{dx}=xy+xy^2\]
If \(y(x=\sqrt{2})=\frac{e}{2-e}\) where \(e\) is the base of natural logarithms, compute \(y(x = 0)\).
(a) \(-1\)
(b) \(1\)
(c) \(e\)
(d) \(0\)
Check Answer
Option b
Q.No:20 JEST-2023
If a power series
\[y=\sum_{j=0} ^\infty a_j x^j\]
analysis is carried out of the following differential equation
\[\frac{d^y}{dx^2}+\frac{1}{x^2} \frac{dy}{dx}-\frac{4}{x^2} y =0,\]
which of the following recurrence relations results?
(a) \(a_{j+1}=a_j \frac{4-j(j+1)}{j+1},\hspace{1mm} j=0,1,2,...\)
(b) \(a_{j+2}=a_j \frac{4-j(j-1)}{j+1},\hspace{1mm} j=0,1,2,...\)
(c) \(a_{j+2}=a_j \frac{4-j(j+1)}{j+1},\hspace{1mm} j=0,1,2,...\)
(d) \(a_{j+1}=a_j \frac{4-j(j-1)}{j+1},\hspace{1mm} j=0,1,2,...\)
Check Answer
Option d
Q.No:21 JEST-2024
A polynomial \( C_n(x) \) of degree \( n \) defined on the domain \( x \in [-1, 1] \) satisfies the differential equation
\[ (1 - x^2) \frac{d^2C_n}{dx^2} - x \frac{dC_n}{dx} + n^2C_n = 0. \]
The polynomials satisfy the orthogonality relation
\[ \int_{-1}^{1} \sigma(x) C_n(x) C_m(x) dx = 0 \]
for \( n \neq m \). What is \( \sigma(x) \)?
(a) \( (1 - x^2)^{-1/2} \)
(b) \( (1 - x^2) \)
(c) \( 1 \)
(d) \( \exp(-x^2) \)
Check Answer
Option a
Q.No:1 TIFR-2013
The differential equation
\[
\frac{d^2 y}{dx^2}-2\frac{dy}{dx}+y=0
\]
has the complete solution, in terms of arbitrary constants \(A\) and \(B\),
(a)
\(A\exp{x}+Bx\exp{x}\)
(b)
\(A\exp{x}+Bx\exp{(-x)}\)
(c)
\(A\exp{x}+B\exp{(-x)}\)
(d)
\(x\{A\exp{x}+B\exp{(-x)}\}\)
Check Answer
Option a
Q.No:2 TIFR-2014
A body of mass \(m\) falls from rest at a height \(h\) under gravity (acceleration due to gravity \(g\)) through a dense medium which provides a resistive force \(F=-k\nu^2\), where \(k\) is a constant and \(\nu\) is the speed. It will hit the ground with a kinetic energy
(a)
\(\frac{m^2 g}{2k} \exp{\left(-\frac{2kh}{m}\right)}\)
(b)
\(\frac{m^2 g}{2k} \tanh{\frac{2kh}{m}}\)
(c)
\(\frac{m^2 g}{2k} \left\{1+\exp{\left(-\frac{2kh}{m}\right)}\right\}\)
(d)
\(\frac{m^2 g}{2k} \left\{1-\exp{\left(-\frac{2kh}{m}\right)}\right\}\)
Check Answer
Option d
Q.No:3 TIFR-2015
Consider the differential equation
\[
\frac{d^2 y}{dx^2}=-4\left(y+\frac{dy}{dx}\right)
\]
with the boundary condition that \(y(x)=0\) at \(x=1/5\). When plotted as a function of \(x\), for \(x\geq 0\), we can say \(\underline{with \hspace{2mm} certainty}\) that the value of \(y\)
(a)
oscillates from positive to negative with amplitude decreasing to zero
(b)
has an extremum in the range \(0<x<1\)
(c)
first increases, then decreases to zero
(d)
first decreases, then increases to zero
Check Answer
Option b
Q.No:4 TIFR-2015
The generating function for a set of polynomials in \(x\) is given by
\[
f(x, t)=(1-2xt+t^2)^{-1}
\]
The third polynomial (order \(x^2\)) in this set is
(a)
\(2x^2+1\)
(b)
\(2x^2-x\)
(c)
\(4x^2+1\)
(d)
\(4x^2-1\)
Check Answer
Option d
Q.No:5 TIFR-2016
The function \(y(x)\) satisfies the differential equation
\[
x\frac{dy}{dx}=y(\ln{y}-\ln{x}+1)
\]
with the initial condition \(y(1)=3\). What will be the value of \(y(3)\)?
Check Answer
Ans 81
Q.No:6 TIFR-2017
Write down \(x(t)\), where \(x(t)\) is the solution of the following differential equation
\[
\left(\frac{d}{dt}+2\right)\left(\frac{d}{dt}+1\right)x=1,
\]
with the boundary conditions
\[
\left.\frac{dx}{dt}\right|_{t=0}=0, x(t)|_{t=0}=-\frac{1}{2}
\]
Check Answer
Ans \(\exp{(-2t)}-2\exp{(-t)}+1/2\)
Q.No:7 TIFR-2018
If \(y(x)\) satisfies the differential equation
\[
y''-4y'+4y=0
\]
with boundary conditions \(y(0)=1\) and \(y'(0)=0\), then \(y\left(-\frac{1}{2}\right)=\)
(a)
\(\frac{2}{e}\)
(b)
\(\frac{1}{2}\left(e+\frac{1}{e}\right)\)
(c)
\(\frac{1}{e}\)
(d)
\(-\frac{e}{2}\)
Check Answer
Option a
Q.No:8 TIFR-2019
A set of polynomials of order are given by the formula
\[
P_n(x)=(-1)^n\exp{\left(\frac{x^2}{2}\right)}\frac{d^n}{dx^n}\exp{\left(-\frac{x^2}{2}\right)}
\]
The polynomial \(P_7(x)\) of order \(n=7\) is
(a)
\(x^7-21x^5+105x^4+35x^3-105x\)
(b)
\(x^6-21x^5+105x^4-105x^3+21x^2+x\)
(c)
\(x^7-21x^5+105x^3-105x+21\)
(d)
\(x^7-21x^5+105x^3-105x\)
Check Answer
Option d
Q.No:9 TIFR-2019
On a compact stellar object the gravity is so strong that a body falling from rest will soon acquire a velocity comparable with that of light. If the force on this body is \(F=mg\) where \(m\) is the relativistic mass and \(g\) is a constant, the velocity of this falling body will vary with time as
(a)
\(\nu=\frac{c}{1-\frac{2c}{gt}}\)
(b)
\(\nu=c\tanh{\frac{gt}{c}}\)
(c)
\(\nu=\frac{2c}{\pi}\tan{\frac{gt}{c}}\)
(d)
\(\nu=c\left\{1-\exp{\left(-\frac{gt}{c}\right)}\right\}\)
Check Answer
Option b
Q.No:10 TIFR-2020
The limit
\[
\lim_{x\to \infty} x\log{\frac{x+1}{x-1}}
\]
evaluates to
(a)
\(2\)
(b)
\(0\)
(c)
\(\infty\)
(d)
\(1\)
Check Answer
Option a
Q.No:11 TIFR-2020
Consider the improper differential
\[
ds=(1+y^2)dx+xydy
\]
An integrating factor for this is
(a)
\(-x\)
(b)
\(1+x^2\)
(c)
\(xy\)
(d)
\(-1+y^2\)
Check Answer
Option a
Q.No:12 TIFR-2020
The solution of the differential equation
\[
\frac{dy}{dx}=1+\frac{y}{x}-\frac{y^2}{x^2}
\]
for \(x>0\) with the boundary condition \(y=0\) at \(x=1\), is given by \(y(x)=\)
(a)
\(\frac{x(x^2-1)}{x^2+1}\)
(b)
\(\frac{x(x-1)}{x+1}\)
(c)
\(\frac{x-1}{x+1}\)
(d)
\(\frac{x^2-1}{x^2+1}\)
Check Answer
Option a
Q.No:13 TIFR-2021
A differentiable function \(f(x)\) obeys
\[
x\int_0^x \frac{f(y)}{y^2} dy=f(x)
\]
If \(f(1)=1\), it follows that \(f(2)=\)
(a)
\(4\)
(b)
\(3/4\)
(c)
\(1\)
(d)
\(6\)
Check Answer
Option a
Q.No:14 TIFR-2021
If \(y(x)\) satisfies the following differential equation
\[
x\frac{dy}{dx}=\cot{y}-\csc{y}\cos{x}
\]
and we have
\[
\lim_{x\to 0}y(x)=0
\]
then \(y(\pi/2)=\)
(a)
\(-\cos^{-1}{(2/\pi-2)}\)
(b)
\(\sin^{-1}{(2/\pi)}\)
(c)
\(\pi/2\)
(d)
\(0\)
Check Answer
Option a
Q.No:15 TIFR-2024
Consider a universe that always expands with a scale factor \( a \) that increases with time as \( a(t) = Ct^{2/3} \) where \( C \) is a constant. Its expansion rate at time \( t \) is defined by the Hubble parameter
\[ H(t) = \frac{1}{a(t)} \frac{da(t)}{dt} \]
The current value of \( H(t) \) in the universe is given by \( H_0 = 975 \) km s\(^{-1}\) \(Mpc^{-1}\) where 1 Mpc = \( 3.1 \times 10^{22} \) m. What is the approximate age of this universe?
(a) \( 10^9 \) years
(b) \( 10^7 \) years
(c) \( 10^{11} \) years
(d) \( 10^{13} \) years
Check Answer
Option a
Q.No:16 TIFR-2024
Consider the following differential equations:
\[
\frac{dx}{dt} = ay(t), \quad \frac{dy}{dt} = a
\]
where \( a \) is a positive constant. The solutions to these equations define a family of curves in the \(x, y\) plane. What are these curves?
(a) Parabolas
(b) Circles
(c) Hyperbolas
(d) Ellipses
Check Answer
Option a
Q.No:17 TIFR-2024
Consider an object falling in air. In addition to gravity, it experiences an air resistance force, \( R \), given by \( R = bv \), where \( v \) is the speed and \( b \) is a constant. If the object is dropped from rest (\( v = 0 \) at \( t = 0 \)), the distance traversed by the object at \( t = m/b \) is:
(a) \( \frac{m^2g}{b^2} \left( \frac{1}{e} \right) \)
(b) \( \frac{m^2g}{b^2} \left( 1 - \frac{1}{e} \right) \)
(c) \( \frac{m^2g}{b^2} (e - 1) \)
(d) \( \frac{m^2g}{b^2} \left( 2 - \frac{1}{e} \right) \)
Check Answer
Option a
Q.No:18 TIFR-2025
The general solution of the differential equation
\[
\frac{d^3 y}{dx^3} + k^3 y = 0 \quad (k>0)
\]
is given by:
a) \( C_1 e^{-kx}
+ C_2 e^{kx/2}\cos\!\left(\frac{\sqrt{3}kx}{2}\right)
+ C_3 e^{kx/2}\sin\!\left(\frac{\sqrt{3}kx}{2}\right) \)
b) \( C_1 e^{-kx}
+ C_2 e^{-kx/2}\cos\!\left(\frac{\sqrt{3}kx}{2}\right)
+ C_3 e^{-kx/2}\sin\!\left(\frac{\sqrt{3}kx}{2}\right) \)
c) \( C_1 e^{-kx}
+ C_2 e^{kx/2}\cos\!\left(\frac{\sqrt{3}kx}{2}\right)
+ C_3 e^{kx/2}\sin\!\left(\frac{kx}{2}\right) \)
d) \( C_1 e^{-kx}
+ C_2 e^{kx/2}\cos\!\left(\frac{kx}{2}\right)
+ C_3 e^{kx/2}\sin\!\left(\frac{\sqrt{3}kx}{2}\right) \)