Q.No:1 JAM-2015
Consider the equation \(\frac{dy}{dx}=\frac{y^2}{x}\) with the boundary condition \(y(1)=1\). Out of the following, the
range of \(x\) in which \(y\) is real and finite, is
(A)
\(-\infty \leq x\leq -3\)
(B)
\(-3 \leq x\leq 0\)
(C)
\(0 \leq x\leq 3\)
(D)
\(3 \leq x\leq \infty\)
Check Answer
Option D
Q.No:2 JAM-2017
Consider two particles moving along the x - axis. In terms of their coordinates \(x_1\) and \(x_2\) , their velocities are given as \(\frac{dx_1}{dt}=x_2-x_1\) and \(\frac{dx_2}{dt}=x_1-x_2\), respectively. When they start moving from their initial locations of \(x_1(0)=1\) and \(x_2(0)=-1\), the time dependence of
both \(x_1\) and \(x_2\) contains a term of the form \(e^{at}\) , where \(a\) is a constant. The value of \(a\) (an integer) is ____________.
Check Answer
Ans 2
Q.No:3 JAM-2017
Consider the differential equation \(y^{\prime \prime}+2y'+y=0\). If \(y(0)=0\) and \(y'(0)=1\), then the value of \(y(2)\) is ______________.
(Specify your answer to two digits after the decimal point)
Check Answer
Ans 0.25-0.29
Q.No:4 JAM-2018
Which one of the following curves correctly represents (schematically) the solution for the equation \(\frac{df}{dx}+2f=3 ; f(0)=0\) ?
Check Answer
Option B
Q.No:5 JAM-2019
The function \(f(x)=\frac{8x}{x^2+9}\) is continuous everywhere except at
(A)
\(x=0\)
(B)
\(x=\pm 9\)
(C)
\(x=\pm 9i\)
(D)
\(x=\pm 3i\)
Check Answer
Option D
Q.No:6 JAM-2020
Which one of the following functions has a discontinuity in the second derivative at \(x = 0\), where \(x\)
is a real variable?
(A)
\(f(x)=|x|^3\)
(B)
\(f(x)=x|x|\)
(C)
\(f(x)=cos |x|\)
(D)
\(f(x)=|x|^2\)
Check Answer
Option B
Q.No:7 JAM-2020
If a function \(f(x)\) is described by the initial-value problem, \(\frac{d^2y}{dx^2}+5\frac{dy}{dx}+6y=0\), with initial
conditions \(y(0)=2\) and \((\frac{dy}{dx})_{x=0}=0\), then the value of \(y\) at \(x=1\) is ________________.
(Round off to 2 decimal places)
Check Answer
Ans 0.60-0.62
Q.No:8 JAM-2021
The solution \(y(x)\) of the differential equation
\(y\frac{dy}{dx}+3x=0 , y(1)=0\),is described by
(A)
an ellipse
(B)
a circle
(C)
a parabola
(D)
a straight line
Check Answer
Option A
Q.No:9 JAM-2021
At \(t=0\), \(N_0\) number of a radioactive nuclei \(A\) start decaying into \(B\) with a decay constant \(\lambda_a\) The daughter nuclei \(B\) decay into nuclei \(C\) with a decay constant \(\lambda_b\). Then, the number of nuclei \(B\) at small time \(t\) (to the leading order) is
(A)
\(\lambda_a N_0 t\)
(B)
\((\lambda_a -\lambda_b) N_0 t\)
(C)
\((\lambda_a +\lambda_b) N_0 t\)
(D)
\(\lambda_b N_0 t\)
Check Answer
Option A
Q.No:10 JAM-2021
Consider the following differential equation that describes the oscillations of a physical system:
\[\alpha(\frac{d^2y}{dt^2})+\beta(\frac{dy}{dt})+\gamma y=0\]
If \(\alpha\) and \(\beta\) are held fixed, and \(\gamma\) is increased, then,
(A)
the frequency of oscillations increases
(B)
the oscillations decay faster
(C)
the frequency of oscillations decreases
(D)
the oscillations decay slower
Check Answer
Option A
Q.No:11 JAM-2022
In a dilute gas, the number of molecules with free path length \(\geq x\) is given by \(N(x)=N_0 e^{-x/\lambda}\), where \(N_0\) is the total number of molecules and \(\lambda \) is the mean free path. The fraction of molecules with free path lengths between \(\lambda\) and \(2\lambda\) is
(A)
\(\frac{1}{e}\)
(B)
\(\frac{e}{e-1}\)
(C)
\(\frac{e^2}{e-1}\)
(D)
\(\frac{e-1}{e^2}\)
Check Answer
Option D
Q.No:12 JAM-2022
A radioactive nucleus has a decay constant \(\lambda\) and its radioactive daughter nucleus has a decay constant \(10\lambda\). At time \(t=0, N_0\) is the number of parent nuclei and there are no daughter nuclei present. \(N_1(t)\) and \(N_2(t)\) are the number of parent
and daughter nuclei present at time \(t\), The ratio \(N_2(t)/N_1(t)\) is
(A)
\(\frac{1}{9}[1-e^{-9\lambda t}]\)
(B)
\(\frac{1}{10}[1-e^{-10\lambda t}]\)
(C)
\([1-e^{-10\lambda t}]\)
(D)
\([1-e^{-9\lambda t}]\)
Check Answer
Option A
Q.No:13 JAM-2022
Consider the second order ordinary differential equation, \(y^{\prime \prime} +4y'+5y = 0\). If \(y(0)=0\) and \(y'(0)=1\), then the value of \(y(\pi /2)\) is _____________ (Round off to 3 decimal places).
Check Answer
Ans 0.041-0.045
Q.No:14 JAM-2024
In the Taylor expansion of the function
\[
F(x) = e^{x}\sin x,
\]
around \(x = 0\), the coefficient of \(x^5\) is ______.
(Rounded off to three decimal places)
Check Answer
Ans -0.034 to -0.032
Q.No:15 JAM-2025
Consider radioactive decays A → B with half-life \((T_{1/2})_A\) and B → C with
half-life \((T_{1/2})_B\).
At any time t, the number of nuclides of B is given by
\[
(N_B)_t
=
\frac{\lambda_A}{\lambda_B - \lambda_A}
(N_A)_0
\left(e^{-\lambda_A t} - e^{-\lambda_B t}\right),
\]
where \((N_A)_0\) is the number of nuclides of A at t = 0.
The decay constants of A and B are \(\lambda_A\) and \(\lambda_B\),
respectively.
If \((T_{1/2})_B < (T_{1/2})_A\), then the ratio
\[
\frac{(N_B)_t}{(N_A)_t}
\]
at time \(t \gg (T_{1/2})_A\) is\(\_\_\_\_\)[\((N_A)_t\) is the number of nuclides of A at a time t]
A) \(
\frac{\lambda_A}{\lambda_B - \lambda_A}
\)
B) \(
\frac{\lambda_B}{\lambda_A}
\)
C) \(
\frac{\lambda_A}{\lambda_B}
\)
D) \(
\frac{\lambda_B}{\lambda_B - \lambda_A}
\)
Check Answer
Option A
Q.No:16 JAM-2025
Given a function
\(
f(x,y) = \frac{x}{a}e^{y} + \frac{y}{b}e^{x},
\)
where \(x = at\) and \(y = bt\) (\(a\) and \(b\) are non-zero constants),
the value of
\(
\frac{df}{dt}
\)
at \(t = 0\) is
A) -1
B) 0
C) 1
D) 2