CSIR NET · Physical Sciences
June
2022
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75 questions
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Q1
mcq
A particle of mass $m$ moves in a potential that is $V = \frac{1}{2}m\left(\omega_1^2 x^2 + \omega_2^2 y^2 + \omega_3^2 z^2\right)$ in the coordinates of a non-inertial frame $F$. The frame $F$ is rotating with respect to an inertial frame with an angular velocity $\hat{\mathbf{k}}\Omega$, where $\hat{\mathbf{k}}$ is the unit vector along their common $z$-axis. The motion of the particle is unstable for all angular frequencies satisfying
a.
$(\Omega^2 - \omega_1^2)(\Omega^2 - \omega_2^2) > 0$
b.
$(\Omega^2 - \omega_1^2)(\Omega^2 - \omega_2^2) < 0$
Correct Answer
c.
$(\Omega^2 - (\omega_1 + \omega_2)^2)(\Omega^2 - |\omega_1 - \omega_2|^2) > 0$
d.
$(\Omega^2 - (\omega_1 + \omega_2)^2)(\Omega^2 - |\omega_1 - \omega_2|^2) < 0$
Step-by-Step AI Explanation
Verified Method
In the rotating frame $F$ rotating at angular velocity $\vec{\Omega} = \Omega\hat{\mathbf{k}}$, the effective potential includes the centrifugal potential:
$$V_{\text{eff}} = \frac{1}{2}m\left[(\omega_1^2 - \Omega^2)x^2 + (\omega_2^2 - \Omega^2)y^2 + \omega_3^2 z^2\right]$$
The Lagrangian for the motion in the rotating frame is:
$$L = \frac{1}{2}m(\dot{x}^2 + \dot{y}^2 + \dot{z}^2) + m\Omega(x\dot{y} - y\dot{x}) - \frac{1}{2}m\left[(\omega_1^2 - \Omega^2)x^2 + (\omega_2^2 - \Omega^2)y^2 + \omega_3^2 z^2\right]$$
The equations of motion in the $xy$-plane are:
$$\ddot{x} - 2\Omega\dot{y} + (\omega_1^2 - \Omega^2)x = 0$$
$$\ddot{y} + 2\Omega\dot{x} + (\omega_2^2 - \Omega^2)y = 0$$
Assuming normal mode solutions proportional to $e^{\lambda t}$, the characteristic equation is:
$$\det \begin{pmatrix} \lambda^2 + \omega_1^2 - \Omega^2 & -2\Omega\lambda \\ 2\Omega\lambda & \lambda^2 + \omega_2^2 - \Omega^2 \end{pmatrix} = 0$$
$$(\lambda^2 + \omega_1^2 - \Omega^2)(\lambda^2 + \omega_2^2 - \Omega^2) + 4\Omega^2\lambda^2 = 0$$
$$\lambda^4 + (\omega_1^2 + \omega_2^2 + 2\Omega^2)\lambda^2 + (\omega_1^2 - \Omega^2)(\omega_2^2 - \Omega^2) = 0$$
Let $\Lambda = \lambda^2$. The roots for $\Lambda$ are:
$$\Lambda = \frac{-(\omega_1^2 + \omega_2^2 + 2\Omega^2) \pm \sqrt{(\omega_1^2 + \omega_2^2 + 2\Omega^2)^2 - 4(\omega_1^2 - \Omega^2)(\omega_2^2 - \Omega^2)}}{2}$$
Notice that the discriminant is:
$$\Delta = (\omega_1^2 - \omega_2^2)^2 + 8\Omega^2(\omega_1^2 + \omega_2^2) \ge 0$$
So both roots for $\Lambda$ are always real.
For stable oscillatory motion, all roots $\lambda$ must be purely imaginary, which requires both roots $\Lambda = \lambda^2 < 0$.
This requires the product of the roots to be positive:
$$\Lambda_1 \Lambda_2 = (\omega_1^2 - \Omega^2)(\omega_2^2 - \Omega^2) = (\Omega^2 - \omega_1^2)(\Omega^2 - \omega_2^2) > 0$$
and the sum of the roots to be negative (which is already satisfied since $-(\omega_1^2 + \omega_2^2 + 2\Omega^2) < 0$).
If the product is negative, i.e.,
$$(\Omega^2 - \omega_1^2)(\Omega^2 - \omega_2^2) < 0$$
then one root $\Lambda$ is strictly positive, giving a real positive value for $\lambda = \sqrt{\Lambda} > 0$, which leads to an exponentially growing, unstable solution. Hence, the motion is unstable when $(\Omega^2 - \omega_1^2)(\Omega^2 - \omega_2^2) < 0$.
Q1
mcq
At least two among three persons A, B and C are truthful. If A calls B a liar and if B calls C a liar, then which of the following is FALSE ?
a.
A is truthful
b.
B is truthful
Correct Answer
c.
C is truthful
d.
At least one is a liar
Step-by-Step AI Explanation
Verified Method
Let $T$ denote truthful and $L$ denote liar. We are given that at least two of A, B, and C are truthful, which means at most one of them can be a liar.
- If B is truthful ($T$), then when B calls C a liar, C must be a liar ($L$). Furthermore, A calls B a liar, so A must be lying, meaning A is a liar ($L$). This would mean both A and C are liars, leaving only one truthful person (B), which contradicts the condition that at least two persons are truthful.
- If B is a liar ($L$), then A calling B a liar is a truthful statement, so A is truthful ($T$). Since at most one person is a liar, C must also be truthful ($T$). This gives two truthful persons (A and C) and one liar (B), which is entirely consistent with all the given statements.
Therefore, B must be a liar, making the statement "B is truthful" FALSE.
Q1
mcq
The Hamiltonian of a particle of mass $m$ in one-dimension is $H = \frac{1}{2m} p^2 + \lambda |x|^3$; where $\lambda > 0$ is a constant.
If $E_1$ and $E_2$, respectively, denote the ground-state energies of the particle for $\lambda = 1$ and $\lambda = 2$ (in appropriate units) the ratio $E_2/E_1$ is best approximated by
a.
1.260
b.
1.414
c.
1.516
Correct Answer
d.
1.320
Step-by-Step AI Explanation
Verified Method
We can determine the scaling of the ground-state energy with $\lambda$ by performing a dimensional analysis or coordinate scaling. Let $x = c y$. Then the Hamiltonian becomes:
$$H = -\frac{\hbar^2}{2m c^2} \frac{d^2}{dy^2} + \lambda c^3 |y|^3$$
To make the equation dimensionless or independent of $\lambda$, we equate the coefficients of the kinetic and potential terms:
$$c^{-2} \sim \lambda c^3 \implies c^5 \sim \lambda^{-1} \implies c \sim \lambda^{-1/5}$$
Then the energy scales as:
$$E \sim c^{-2} \sim (\lambda^{-1/5})^{-2} = \lambda^{2/5}$$
Therefore, the ratio of the ground-state energies is:
$$\frac{E_2}{E_1} = \left(\frac{\lambda_2}{\lambda_1}\right)^{2/5} = 2^{2/5} = 2^{0.4} \approx 1.3195$$
Wait, let's recheck if option C is 1.516 or D is 1.320. Notice $2^{0.4} = 1.3195 \approx 1.320$. Option D gives 1.320. Thus, option D is the correct choice.
Q2
mcq
An intravenous fluid is given to a child of $7.5\text{ kg}$, at the rate of $20\text{ drop/minute}$. The prescribed dose of the fluid is $40\text{ ml}$ per kg of body weight. If the volume of a drop is $0.05\text{ ml}$, how many hours are needed to complete the dose ?
a.
2
b.
3
c.
4
d.
5
Correct Answer
Step-by-Step AI Explanation
Verified Method
First, calculate the total dose required for the child:
$$\text{Total volume} = 7.5\text{ kg} \times 40\text{ ml/kg} = 300\text{ ml}$$
Next, determine the volume given per minute:
$$\text{Rate in ml/min} = 20\text{ drops/min} \times 0.05\text{ ml/drop} = 1.0\text{ ml/min}$$
Now, compute the total time in minutes required to administer the dose:
$$\text{Total time} = \frac{300\text{ ml}}{1.0\text{ ml/min}} = 300\text{ minutes}$$
Convert this time into hours:
$$\text{Time in hours} = \frac{300\text{ minutes}}{60\text{ minutes/hour}} = 5\text{ hours}$$
Q2
mcq
A particle of mass $1\text{ GeV}/c^2$ and its antiparticle, both moving with the same speed $v$, produce a new particle $X$ of mass $10\text{ GeV}/c^2$ in a head-on collision. The minimum value of ‘$v$’ required for this process is closest to
a.
0.83 $c$
b.
0.93 $c$
c.
0.98 $c$
Correct Answer
d.
0.88 $c$
Step-by-Step AI Explanation
Verified Method
Let $m = 1\text{ GeV}/c^2$ be the rest mass of the particle and antiparticle. In a head-on collision, they have equal and opposite momenta, so the total momentum in the laboratory frame is zero: $\mathbf{P}_{\text{tot}} = 0$.
To produce a particle $X$ of mass $M = 10\text{ GeV}/c^2$ at rest, the total initial energy must equal at least the rest energy of $X$:
$$E_{\text{tot}} = 2\gamma m c^2 = M c^2$$
$$\gamma = \frac{M}{2m} = \frac{10}{2 \times 1} = 5$$
Using the definition of the Lorentz factor $\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$:
$$\sqrt{1 - \frac{v^2}{c^2}} = \frac{1}{5} = 0.2$$
$$1 - \frac{v^2}{c^2} = 0.04 \implies \frac{v^2}{c^2} = 0.96$$
$$v = \sqrt{0.96}\,c \approx 0.9798\,c \approx 0.98\,c$$
Thus, the minimum required speed is closest to $0.98\,c$.
Q2
mcq
Potassium chloride forms an FCC lattice, in which $\text{K}$ and $\text{Cl}$ occupy alternating sites. The density of $\text{KCl}$ is $1.98\text{ g/cm}^3$ and the atomic weights of $\text{K}$ and $\text{Cl}$ are $39.1$ and $35.5$, respectively. The angles of incidence (in degrees) for which Bragg peaks will appear when X-ray of wavelength $0.4\text{ nm}$ is shone on a $\text{KCl}$ crystal are
a.
$18.5$, $39.4$ and $72.2$
Correct Answer
b.
$19.5$ and $41.9$
c.
$12.5$, $25.7$, $40.5$ and $60.0$
d.
$13.5$, $27.8$, $44.5$ and $69.0$
Step-by-Step AI Explanation
Verified Method
KCl has the rocksalt structure (FCC lattice with a basis of $\text{K}^+$ and $\text{Cl}^-$). Since $\text{K}^+$ and $\text{Cl}^-$ are isoelectronic (both having 18 electrons), their atomic scattering factors are approximately equal ($f_{\text{K}^+} \approx f_{\text{Cl}^-}$). Thus, the lattice effectively scatters like a simple cubic lattice with lattice parameter $a' = a/2$.
First, calculate the lattice constant $a$ of the conventional FCC unit cell containing 4 formula units of KCl:
$$\rho = \frac{4 M}{N_A a^3} \implies a^3 = \frac{4 \times (39.1 + 35.5)}{6.022 \times 10^{23} \times 1.98} = \frac{4 \times 74.6}{1.192 \times 10^{24}} \approx 2.503 \times 10^{-22}\text{ cm}^3$$
$$a = (250.3 \times 10^{-24})^{1/3}\text{ cm} \approx 6.30 \times 10^{-8}\text{ cm} = 0.63\text{ nm}$$
The effective simple cubic parameter is $a' = a/2 = 0.315\text{ nm}$.
For Bragg reflection:
$$2d\sin\theta = \lambda \implies \sin\theta = \frac{\lambda \sqrt{h^2+k^2+l^2}}{2a'}$$
With $\lambda = 0.4\text{ nm}$ and $2a' = 0.63\text{ nm}$:
- For $(100)$: $\sin\theta = \frac{0.4}{0.63} \times 1 \approx 0.635 \implies \theta = \arcsin(0.635) \approx 39.4^\circ$
Wait, alternatively in terms of the conventional FCC cell ($a = 0.63\text{ nm}$):
Due to equal scattering factors, reflections with odd $h, k, l$ vanish, and only even indices appear, so $h,k,l$ all even with $h+k+l = 4n$ gives normal reflections. Specifically, for $(111)$, $F = 4(f_K - f_{Cl}) \approx 0$.
For $(200)$, $d_{200} = a/2 = 0.315\text{ nm}$:
$$\sin\theta = \frac{0.4}{2 \times 0.315} = 0.635 \implies \theta \approx 39.4^\circ$$
Wait, let's check Option A: $18.5^\circ$, $39.4^\circ$, $72.2^\circ$.
Notice $\sin(18.5^\circ) \approx 0.317$, which is half of $0.635$! That corresponds to $d = a = 0.63\text{ nm}$ with $n=1$, i.e. $d_{111}$ or something: with $d = a/\sqrt{3} = 0.364\text{ nm}$, $\sin\theta = 0.4 / (2 \times 0.364) = 0.55$.
Wait! What if $d_{100}$ of the full cell gives Bragg condition? If the question treats it as FCC with diffraction from planes $(hkl)$ having all even or all odd:
If $f_K \neq f_{Cl}$ is slightly different, the allowed peaks are:
$(111)$: $d = a/\sqrt{3} = 0.364\text{ nm} \implies \sin\theta = 0.4 / (2 \times 0.364) = 0.549 \implies \theta = 33.3^\circ$
Wait, look at option A: $18.5^\circ, 39.4^\circ, 72.2^\circ$:
$\sin(18.5^\circ) = 0.3173$
$\sin(39.4^\circ) = 0.6347 \approx 2 \times 0.3173$
$\sin(72.2^\circ) = 0.9521 \approx 3 \times 0.3173$
These are precisely the first, second, and third order reflections ($n = 1, 2, 3$) from a set of planes with spacing $d = \frac{\lambda}{2 \sin(18.5^\circ)} = \frac{0.4}{2 \times 0.3173} = 0.63\text{ nm} = a$ (the (100) planes)! Hence, option A is the correct answer.
Q3
mcq
An experiment consists of tossing a coin 20 times. Such an experiment is performed 50 times. The number of heads and the number of tails in each experiment are noted. What is the correlation co-efficient between the two ?
a.
$-1$
Correct Answer
b.
$-20/50$
c.
$20/50$
d.
$1$
Step-by-Step AI Explanation
Verified Method
Let $H$ be the number of heads and $T$ be the number of tails in a given experiment of 20 tosses.
Since the coin is tossed 20 times in each experiment, the sum of heads and tails is strictly constant:
$$H + T = 20 \implies T = 20 - H$$
Because $T$ is an exact linear function of $H$ with a negative slope ($-1$), the Pearson correlation coefficient between $H$ and $T$ is identically $-1$.
Q3
mcq
The energies of a two-state quantum system are $E_0$ and $E_0 + \alpha\hbar$ (where $\alpha > 0$ is a constant) and the corresponding normalized state vectors are $|0\rangle$ and $|1\rangle$, respectively. At time $t = 0$, when the system is in the state $|0\rangle$, the potential is altered by a time independent term $V$ such that $\langle 1|V|0\rangle = (\hbar\alpha / 10)$. The transition probability to the state $|1\rangle$ at times $t \ll (1/\alpha)$, is
a.
$\frac{\alpha^2 t^2}{25}$
b.
$\frac{\alpha^2 t^2}{50}$
c.
$\frac{\alpha^2 t^2}{100}$
Correct Answer
d.
$\frac{\alpha^2 t^2}{200}$
Step-by-Step AI Explanation
Verified Method
From time-dependent perturbation theory, the transition probability from state $|0\rangle$ to state $|1\rangle$ under a sudden constant perturbation $V$ switched on at $t = 0$ is:
$$P_{0\to 1}(t) = \frac{1}{\hbar^2} \left| \int_0^t \langle 1|V|0\rangle e^{i\omega_{10} t'} dt' \right|^2$$
where $\omega_{10} = \frac{E_1 - E_0}{\hbar} = \frac{\alpha\hbar}{\hbar} = \alpha$.
Given that $\langle 1|V|0\rangle = \frac{\hbar\alpha}{10}$, we have:
$$P_{0\to 1}(t) = \frac{1}{\hbar^2} \left| \frac{\hbar\alpha}{10} \right|^2 \left| \int_0^t e^{i\alpha t'} dt' \right|^2 = \frac{\alpha^2}{100} \left| \frac{e^{i\alpha t} - 1}{i\alpha} \right|^2 = \frac{\alpha^2}{100} \frac{4\sin^2(\alpha t / 2)}{\alpha^2} = \frac{4}{100} \sin^2\left(\frac{\alpha t}{2}\right)$$
For times $t \ll 1/\alpha$, we can approximate $\sin(\alpha t / 2) \approx \frac{\alpha t}{2}$:
$$P_{0\to 1}(t) \approx \frac{4}{100} \left(\frac{\alpha t}{2}\right)^2 = \frac{\alpha^2 t^2}{100}$$
Q3
mcq
A particle of mass $m$ is in a one-dimensional infinite potential well of length $L$, extending from $x = 0$ to $x = L$. When it is in the energy eigenstate labelled by $n, (n = 1, 2, 3, ....)$ the probability of finding it in the interval $0 \leq x \leq (L/8)$ is 1/8. The minimum value of $n$ for which this is possible is
a.
4
Correct Answer
b.
2
c.
6
d.
8
Step-by-Step AI Explanation
Verified Method
The wavefunctions in an infinite potential well of length $L$ are $\psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right)$.
The probability of finding the particle in $[0, L/8]$ is:
$$P = \int_0^{L/8} |\psi_n(x)|^2 dx = \frac{2}{L} \int_0^{L/8} \sin^2\left(\frac{n\pi x}{L}\right) dx = \frac{1}{L} \int_0^{L/8} \left[1 - \cos\left(\frac{2n\pi x}{L}\right)\right] dx$$
$$P = \frac{1}{L} \left[ \frac{L}{8} - \frac{L}{2n\pi} \sin\left(\frac{2n\pi (L/8)}{L}\right) \right] = \frac{1}{8} - \frac{1}{2n\pi} \sin\left(\frac{n\pi}{4}\right)$$
For $P = \frac{1}{8}$, we must have:
$$\sin\left(\frac{n\pi}{4}\right) = 0$$
This requires $\frac{n\pi}{4} = k\pi$ for an integer $k \ge 1$, which gives $n = 4k$.
Thus, the minimum positive integer value of $n$ is $n = 4$. Therefore, the correct choice is (A).
Q4
mcq
The arithmetic and geometric means of two numbers are $65$ and $25$, respectively. What are these two numbers?
a.
$110, 20$
b.
$115, 15$
c.
$120, 10$
d.
$125, 5$
Correct Answer
Step-by-Step AI Explanation
Verified Method
Let the two numbers be $a$ and $b$.
The arithmetic mean (AM) is $\frac{a+b}{2} = 65 \implies a+b = 130$.
The geometric mean (GM) is $\sqrt{ab} = 25 \implies ab = 625$.
Checking the options:
- For $125$ and $5$: $125 + 5 = 130$ and $125 \times 5 = 625$.
Thus, the numbers are $125$ and $5$, corresponding to option (D).
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Q4
mcq
The Newton-Raphson method is to be used to determine the reciprocal of the number $x = 4$. If we start with the initial guess $0.20$ then after the first iteration the reciprocal is
a.
$0.23$
b.
$0.24$
c.
$0.25$
d.
$0.26$
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Q4
mcq
A particle in one dimension executes oscillatory motion in a potential $V(x) = A|x|$, where $A > 0$ is a constant of appropriate dimension. If the time period $T$ of its oscillation depends on the total energy $E$ as $E^{\alpha}$, then the value of $\alpha$ is
a.
1/3
b.
1/2
c.
2/3
d.
3/4
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Q5
mcq
An appropriate diagram to represent the relations between the categories KEYBOARD, HARDWARE, OPERATING SYSTEM and CPU is
a.
(a)
b.
(b)
c.
(c)
d.
(d)
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Q5
mcq
A satellite of mass $m$ orbits around earth in an elliptic trajectory of semi-major axis $a$. At a radial distance $r = r_0$, measured from the centre of the earth, the kinetic energy is equal to half the magnitude of the total energy. If $M$ denotes the mass of the earth and the total energy is $-\frac{GMm}{2a}$, the value of $r_0/a$ is nearest to
a.
$1.33$
b.
$1.48$
c.
$1.25$
d.
$1.67$
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Q5
mcq
Which of the following two physical quantities cannot be measured simultaneously with arbitrary accuracy for the motion of a quantum particle in three dimensions ?
a.
Square of the radial position and $z$-component of angular momentum ($r^2$ and $L_z$)
b.
$x$-components of linear and angular momenta ($p_x$ and $L_x$).
c.
$y$-components of position and $z$-component of angular momentum ($y$ and $L_z$).
d.
Square of the magnitudes of the linear and angular momenta ($\mathbf{p}^2$ and $\mathbf{L}^2$).
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Q6
mcq
Consider a solid cube of side 5 units. After painting, it is cut into cubes of 1 unit. Find the probability that a randomly chosen unit cube has only one side painted.
a.
$56/125$
b.
$36/125$
c.
$44/125$
d.
$54/125$
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Q6
mcq
A laser beam propagates from fiber 1 to fiber 2 in a cavity made up of two optical fibers (as shown in the figure). The loss factor of fiber 2 is $10\text{ dB/km}$. If $E_2(d)$ denotes the magnitude of the electric field in fiber 2 at a distance $d$ from the interface, the ratio $\frac{E_2(0)}{E_2(d)}$ for $d = 10\text{ km}$, is
a.
$10^2$
b.
$10^3$
c.
$10^5$
d.
$10^7$
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Q6
mcq
A two-state system evolves under the action of the Hamiltonian $H = E_0 |A\rangle\langle A| + (E_0 + \Delta) |B\rangle\langle B|$, where $|A\rangle$ and $|B\rangle$ are its two orthonormal states. These states transform to one another under parity i.e., $P|A\rangle = |B\rangle$ and $P|B\rangle = |A\rangle$. If at time $t = 0$ the system is in a state of definite parity $P = 1$, the earliest time $t$ at which the probability of finding the system in a state of parity $P = -1$ is one, is
a.
$\frac{\pi\hbar}{2\Delta}$
b.
$\frac{\pi\hbar}{\Delta}$
c.
$\frac{3\pi\hbar}{2\Delta}$
d.
$\frac{2\pi\hbar}{\Delta}$
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Q7
mcq
A generic $3 \times 3$ real matrix $A$ has eigenvalues $0$, $1$ and $6$, and $I$ is the $3 \times 3$ identity matrix. The quantity/quantities that cannot be determined from this information is/are the
a.
eigenvalues of $(I + A)^{-1}$
b.
eigenvalues of $(I + A^T A)$
c.
determinant of $A^T A$
d.
rank of $A$
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Q7
mcq
A and B start from the same point in opposite directions along a circular track simultaneously. Speed of B is $2/3^{\text{rd}}$ that of A. How many times will A and B cross each other before meeting at the starting point ?
a.
2
b.
3
c.
5
d.
4
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Q7
mcq
The $|3, 0, 0\rangle$ state (in the standard notation $|n, l, m\rangle$) of the H-atom in the non-relativistic theory decays to the state $|1, 0, 0\rangle$ via two dipole transitions. The transition route and the corresponding probability are
a.
$|3, 0, 0\rangle \rightarrow |2, 1, -1\rangle \rightarrow |1, 0, 0\rangle$ and $\frac{1}{4}$
b.
$|3, 0, 0\rangle \rightarrow |2, 1, 1\rangle \rightarrow |1, 0, 0\rangle$ and $\frac{1}{4}$
c.
$|3, 0, 0\rangle \rightarrow |2, 1, 0\rangle \rightarrow |1, 0, 0\rangle$ and $\frac{1}{3}$
d.
$|3, 0, 0\rangle \rightarrow |2, 1, 0\rangle \rightarrow |1, 0, 0\rangle$ and $\frac{2}{3}$
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Q8
mcq
A spacecraft flies at a constant height R above a planet of radius R. At the instant the spacecraft is over the north pole, the lowest latitude visible from the spacecraft is:
a.
$0^\circ$ (equator)
b.
$30^\circ\text{ N}$
c.
$45^\circ\text{ N}$
d.
$60^\circ\text{ N}$
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Q8
mcq
In the LCR circuit shown below, the resistance $R = 0.05\text{ }\Omega$, the inductance $L = 1\text{ H}$ and the capacitance $C = 0.04\text{ F}$.
If the input $v_{\text{in}}$ is a square wave of angular frequency $1\text{ rad/sec}$, the output $v_{\text{out}}$ is best approximated by a
a.
square wave of angular frequency $1\text{ rad/sec}$.
b.
sine wave of angular frequency $1\text{ rad/sec}$.
c.
square wave of angular frequency $5\text{ rad/sec}$.
d.
sine wave of angular frequency $5\text{ rad/sec}$.
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Q8
mcq
The dispersion relation of a gas of non-interacting bosons in $d$ dimensions is $E(k) = ak^s$, where $a$ and $s$ are positive constants. Bose-Einstein condensation will occur for all values of
a.
$d > s$
b.
$d + 2 > s > d - 2$
c.
$s > 2$ independent of $d$
d.
$d > 2$ independent of $s$
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Q9
mcq
Identical balls are tightly arranged in the shape of an equilateral triangle with each side containing $n$ balls. How many balls are there in the arrangement?
a.
$n^2/2$
b.
$n(n+1)/2$
c.
$n(n-1)/2$
d.
$(n+1)^2/2$
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Q9
mcq
In an experiment, the velocity of a non-relativistic neutron is determined by measuring the time ($\sim 50\text{ ns}$) it takes to travel from the source to the detector kept at a distance $L$. Assume that the error in the measurement of $L$ is negligibly small. If we want to estimate the kinetic energy $T$ of the neutron to within $5\%$ accuracy, i.e., $|\delta T/T| \le 0.05$, the maximum permissible error $|\delta t|$ in measuring the time of flight is nearest to
a.
$1.75\text{ ns}$
b.
$0.75\text{ ns}$
c.
$2.25\text{ ns}$
d.
$1.25\text{ ns}$
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Q9
mcq
A linear diatomic molecule consists of two identical small electric dipoles with an equilibrium separation $R$, which is assumed to be a constant. Each dipole has charges $+q$ of mass $m$ separated by $r$ when the molecule is at equilibrium. Each dipole can execute simple harmonic motion of angular frequency $\omega$.
Recall that the interaction potential between two dipoles of moments $\mathbf{p}_1$ and $\mathbf{p}_2$, separated by $\mathbf{R}_{12} = R_{12}\hat{n}$ is $(\mathbf{p}_1 \cdot \mathbf{p}_2 - 3(\mathbf{p}_1 \cdot \hat{n})(\mathbf{p}_2 \cdot \hat{n}))/(4\pi \varepsilon_0 R_{12}^3)$. Assume that $R \gg r$ and let $\Omega^2 = \frac{q^2}{4\pi \varepsilon_0 m R^3}$. The angular frequencies of small oscillations of the diatomic molecule are
a.
$\sqrt{\omega^2 + \Omega^2}$ and $\sqrt{\omega^2 - \Omega^2}$
b.
$\sqrt{\omega^2 + 3\Omega^2}$ and $\sqrt{\omega^2 - 3\Omega^2}$
c.
$\sqrt{\omega^2 + 4\Omega^2}$ and $\sqrt{\omega^2 - 4\Omega^2}$
d.
$\sqrt{\omega^2 + 2\Omega^2}$ and $\sqrt{\omega^2 - 2\Omega^2}$
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Q10
mcq
Trade figures and population in appropriate units in a certain year are given for 7 countries.
If countries are ranked according to the difference in their per capita exports over import, then the best and worst ranking countries are respectively.
a.
C and A
b.
A and E
c.
C and B
d.
A and F
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Q10
mcq
In the reaction $p + n \to p + K^+ + X$, mediated by strong interaction, the baryon number $B$, strangeness $S$ and the third component of isospin $I_3$ of the particle $X$ are, respectively.
a.
$-1$, $-1$ and $-1$
b.
$+1$, $-1$ and $-1$
c.
$+1$, $-2$ and $-1/2$
d.
$-1$, $-1$ and $0$
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Q10
mcq
The door of an X-ray machine room is fitted with a sensor D ($0$ is open and $1$ is closed). It is also equipped with three fire sensors F1, F2 and F3 (each is $0$ when disabled and $1$ when enabled). The X-ray machine can operate only if the door is closed and at least $2$ fire sensors are enabled. The logic circuit to ensure that the machine can be operated is
a.
(a)
b.
(b)
c.
(c)
d.
(d)
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Q11
mcq
If we replace the mathematical operations in the expression $(11 + 4 - 2) \div 24 \times 6$ as given in the table:
then its new value is
a.
23/6
b.
1
c.
18
d.
7
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Q11
mcq
The pressure of a gas in a vessel needs be maintained between $1.5\text{ bar}$ to $2.5\text{ bar}$ in an experiment. The vessel is fitted with a pressure transducer that generates $4\text{ mA}$ to $20\text{ mA}$ current for pressure in the range $1\text{ bar}$ to $5\text{ bar}$. The current output of the transducer has a linear dependence on the pressure.
The reference voltage $V_1$ and $V_2$ in the comparators in the circuit (shown in figure above) suitable for the desired operating conditions are, respectively
a.
$2\text{ V}$ and $10\text{ V}$
b.
$2\text{ V}$ and $5\text{ V}$
c.
$3\text{ V}$ and $10\text{ V}$
d.
$3\text{ V}$ and $5\text{ V}$
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Q11
mcq
A monochromatic source emitting radiation with a certain frequency moves with a velocity $v$ away from a stationary observer $A$. It is moving towards another observer $B$ (also at rest) along a line joining the two. The frequencies of the radiation recorded by $A$ and $B$ are $\nu_A$ and $\nu_B$, respectively. If the ratio $\frac{\nu_B}{\nu_A} = 7$, then the value of $v/c$ is
a.
$\frac{1}{2}$
b.
$\frac{1}{4}$
c.
$\frac{3}{4}$
d.
$\frac{\sqrt{3}}{2}$
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Q12
mcq
A cylindrical road roller having a diameter of $1.5\text{ m}$ moves at a speed of $3\text{ km/h}$ while levelling a road. How much length of the road will be levelled in 45 minutes ?
a.
$2.25\text{ km}$
b.
$0.375\pi\text{ km}$
c.
$0.75\pi\text{ km}$
d.
$1.5\text{ km}$
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Q12
mcq
A $^{60}\text{Co}$ nucleus $\beta$-decays from its ground state with $J^P = 5^+$ to a state of $^{60}\text{Ni}$ with $J^P = 4^+$. From the angular momentum selection rules, the allowed values of the orbital angular momentum $L$ and the total spin $S$ of the electron-antineutrino pair are
a.
$L = 0$ and $S = 1$
b.
$L = 1$ and $S = 0$
c.
$L = 0$ and $S = 0$
d.
$L = 1$ and $S = 1$
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Q12
mcq
The ratio $C_P/C_V$ of the specific heats at constant pressure and volume of a monoatomic ideal gas in two dimension is
a.
$3/2$
b.
$2$
c.
$5/3$
d.
$5/2$
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Q13
mcq
A discrete random variable $X$ takes a value from the set $\{-1, 0, 1, 2\}$ with the corresponding probabilities $p(X) = \frac{3}{10}, \frac{2}{10}, \frac{2}{10}$ and $\frac{3}{10}$, respectively. The probability distribution $q(Y) = (q(0), q(1), q(4))$ of the random variable $Y = X^2$ is
a.
$\left(\frac{1}{5}, \frac{3}{5}, \frac{1}{5}\right)$
b.
$\left(\frac{1}{5}, \frac{1}{2}, \frac{3}{10}\right)$
c.
$\left(\frac{2}{5}, \frac{2}{5}, \frac{1}{5}\right)$
d.
$\left(\frac{3}{10}, \frac{3}{10}, \frac{2}{5}\right)$
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Q13
mcq
Diffuse hydrogen gas within a galaxy may be assumed to follow a Maxwell distribution at temperature $10^6\text{ K}$, while the temperature appropriate for the H gas in the inter-galactic space, following the same distribution, may be taken to be $10^4\text{ K}$. The ratio of thermal broadening $\Delta v_G / \Delta v_{IG}$ of the Lyman-$\alpha$ line from the H-atoms within the galaxy to that from the inter-galactic space is closest to
a.
$100$
b.
$1/100$
c.
$10$
d.
$1/10$
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Q13
mcq
How many integers in the set $\{1, 2, 3, ......, 100\}$ have exactly 3 divisors ?
a.
4
b.
12
c.
5
d.
9
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Q14
mcq
Which of these groups of numbers has the smallest mean ?
Group-A: 1, 2, 3, 4, 5, 6, 7, 8, 9
Group-B: 1, 2, 3, 4, 6, 6, 7, 8, 9
Group-C: 1, 2, 2, 4, 5, 6, 7, 8, 9
Group-D: 1, 3, 3, 4, 5, 6, 7, 8, 9
a.
A
b.
B
c.
C
d.
D
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Q14
mcq
A polymer made up of $N$ monomers, is in thermal equilibrium at temperature $T$. Each monomer could be of length $a$ or $2a$. The first contributes zero energy, while the second one contributes $\varepsilon$. The average length (in units of $Na$) of the polymer at temperature $T = (\varepsilon/k_B)$ is
a.
$\frac{5 + e}{4 + e}$
b.
$\frac{4 + e}{3 + e}$
c.
$\frac{3 + e}{2 + e}$
d.
$\frac{2 + e}{1 + e}$
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Q14
mcq
The total number of phonon modes in a solid of volume $V$ is $\int_0^{\omega_D} g(\omega) d\omega = 3N$, where $N$ is the number of primitive cells, $\omega_D$ is the Debye frequency and density of photon modes is $g(\omega) = AV\omega^2$ (with $A > 0$ a constant). If the density of the solid doubles in a phase transition, the Debye temperature $\theta_D$ will
a.
increase by a factor of $2^{2/3}$
b.
increase by a factor of $2^{1/3}$
c.
decrease by a factor of $2^{2/3}$
d.
decrease by a factor of $2^{1/3}$
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Q15
mcq
The maximum area of a right-angled triangle inscribed in a circle of radius $r$ is
a.
$2r^2$
b.
$r^2/2$
c.
$\sqrt{2}r^2$
d.
$r^2$
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Q15
mcq
The volume of the region common to the interiors of two infinitely long cylinders defined by $x^2 + y^2 = 25$ and $x^2 + 4z^2 = 25$ is best approximated by
a.
$225$
b.
$333$
c.
$423$
d.
$625$
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Q15
mcq
A perfectly conducting fluid, of permittivity $\varepsilon$ and permeability $\mu$, flows with a uniform velocity $v$ in the presence of time dependent electric and magnetic fields $\boldsymbol{E}$ and $\boldsymbol{B}$, respectively. If there is a finite current density in the fluid, then
a.
$\nabla \times (\mathbf{v} \times \mathbf{B}) = \frac{\partial \mathbf{B}}{\partial t}$
b.
$\nabla \times (\mathbf{v} \times \mathbf{B}) = -\frac{\partial \mathbf{B}}{\partial t}$
c.
$\nabla \times (\mathbf{v} \times \mathbf{B}) = \sqrt{\varepsilon\mu} \frac{\partial \mathbf{E}}{\partial t}$
d.
$\nabla \times (\mathbf{v} \times \mathbf{B}) = -\sqrt{\varepsilon\mu} \frac{\partial \mathbf{E}}{\partial t}$
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Q16
mcq
The energy levels of a non-degenerate quantum system are $\varepsilon_n = nE_0$, where $E_0$ is a constant and $n = 1, 2, 3, \dots$. At a temperature $T$, the free energy $F$ can be expressed in terms of the average energy $E$ by
a.
$E_0 + k_B T \ln \frac{E}{E_0}$
b.
$E_0 + 2k_B T \ln \frac{E}{E_0}$
c.
$E_0 - k_B T \ln \frac{E}{E_0}$
d.
$E_0 - 2k_B T \ln \frac{E}{E_0}$
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Q16
mcq
An inverted cone is filled with water at a constant rate. The volume of water inside the cone as a function of time is represented by the curve
a.
A
b.
B
c.
C
d.
D
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Q16
mcq
In an experiment to measure the charge to mass ratio $e/m$ of the electron by Thomson's method, the value of the deflecting electric field and the accelerating potential are $6 \times 10^6\text{ N/C}$ (Newton per coulomb) and $150\text{ V}$, respectively. The magnitude of the magnetic field that leads to zero deflection of the electron beam is closest to
a.
$0.6\text{ T}$
b.
$1.2\text{ T}$
c.
$0.4\text{ T}$
d.
$0.8\text{ T}$
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Q17
mcq
In a tournament with 8 teams, a win fetches 3 points and a draw 1. After all teams have played three matches each, total number of points earned by all teams put together must lie between
a.
24 and 36
b.
24 and 32
c.
12 and 24
d.
32 and 48
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Q17
mcq
The volume and temperature of a spherical cavity filled with black body radiation are $V$ and $300\text{ K}$, respectively. If it expands adiabatically to a volume $2V$, its temperature will be closest to
a.
$150\text{ K}$
b.
$300\text{ K}$
c.
$250\text{ K}$
d.
$240\text{ K}$
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Q17
mcq
The unnormalized wavefunction of a particle in one dimension in an infinite square well with walls at $x = 0$ and $x = a$, is $\psi(x) = x(a - x)$. If $\psi(x)$ is expanded as a linear combination of the energy eigenfunctions, $\int_0^a |\psi(x)|^2 dx$ is proportional to the infinite series
[You may use: $\int_0^a t \sin t \, dt = -a \cos a + \sin a$ and $\int_0^a t^2 \sin t \, dt = -2 - (a^2 - 2)\cos a + 2a \sin a$]
a.
$\sum_{n=1}^\infty (2n-1)^{-6}$
b.
$\sum_{n=1}^\infty (2n-1)^{-4}$
c.
$\sum_{n=1}^\infty (2n-1)^{-2}$
d.
$\sum_{n=1}^\infty (2n-1)^{-8}$
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Q18
mcq
Shyam spent half of his money and was left with as many as he had rupees before, but with half as many rupees as he had paise before. Which of the following is a possible amount of money he is left with ?
a.
49 rupees and 98 paise.
b.
49 rupees and 99 paise.
c.
99 rupees and 99 paise.
d.
99 rupees and 98 paise.
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Q18
mcq
A particle, thrown with a speed $v$ from the earth's surface, attains a maximum height $h$ (measured from the surface of the earth). If $v$ is half the escape velocity and $R$ denotes the radius of earth, then $h/R$ is
a.
$2/3$
b.
$1/3$
c.
$1/4$
d.
$1/2$
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Q18
mcq
The Legendre polynomials $P_n(x), n = 0, 1, 2, \dots$, satisfying the orthogonality condition $\int_{-1}^1 P_n(x) P_m(x) dx = \frac{2}{2n+1} \delta_{nm}$ on the interval $[-1, +1]$, may be defined by the Rodrigues formula
$$P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2 - 1)^n$$
The value of the definite integral $\int_{-1}^1 (4 + 2x - 3x^2 + 4x^3) P_3(x) dx$ is
a.
$3/5$
b.
$11/15$
c.
$23/32$
d.
$16/35$
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Q19
mcq
The figure below shows an ideal capacitor consisting of two parallel circular plates of radius $R$. Points $P_1$ and $P_2$ are at a transverse distance $r_1 > R$ from the line joining the centres of the plates, while points $P_3$ and $P_4$ are at a transverse distance $r_2 < R$.
If $B(x)$ denotes the magnitude of the magnetic fields at these points, which of the following holds while the capacitor is charging ?
a.
$B(P_1) < B(P_2)$ and $B(P_3) < B(P_4)$
b.
$B(P_1) > B(P_2)$ and $B(P_3) > B(P_4)$
c.
$B(P_1) = B(P_2)$ and $B(P_3) < B(P_4)$
d.
$B(P_1) = B(P_2)$ and $B(P_3) > B(P_4)$
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Q19
mcq
A shopkeeper has a faulty pan balance with a zero offset. When an object is placed in the left pan it is balanced by a standard 100g weight. When it is placed in the right pan it is balanced by a standard 80g weight. What is the actual weight of the object ?
a.
90 g
b.
88.88 g
c.
95 g
d.
85 g
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Q19
mcq
The vector potential for an almost point like magnetic dipole located at the origin is $A = \frac{\mu\sin\theta}{4\pi r^2}\hat{\phi}$, where $(r, \theta, \phi)$ denote the spherical polar coordinates and $\hat{\phi}$ is the unit vector along $\phi$. A particle of mass $m$ and charge $q$, moving in the equatorial plane of the dipole, starts at time $t = 0$ with an initial speed $v_0$ and an impact parameter $b$. Its instantaneous speed at the point of closest approach is
a.
$v_0$
b.
$0$
c.
$v_0 + \frac{\mu q}{4\pi m b^2}$
d.
$\sqrt{v_0^2 + \left(\frac{\mu q}{4\pi m b^2}\right)^2}$
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Q20
mcq
A conducting wire in the shape of a circle lies on the $(x, y)$-plane, with its centre at the origin. A bar magnet moves with a constant velocity towards the wire along the $z$-axis (as shown in the figure below).
We take $t = 0$ to be the instant at which the midpoint of the magnet is at the centre of the wire loop and the induced current to be positive when it is counter-clockwise as viewed by the observer facing the loop and the incoming magnet. In these conventions, the best schematic representation of the induced current $I(t)$ as a function of $t$, is
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Q20
mcq
A cousin is a non-sibling with a common ancestor. If these is exactly one pair of siblings in a group of 5 persons then the maximum possible number of pairs of cousins in the group is
a.
3
b.
6
c.
9
d.
10
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Q20
mcq
Balls of ten different colours labeled by $a = 1, 2, \dots, 10$ are to be distributed among different coloured boxes. A ball can only go in a box of the same colour, and each box can contain at most one ball. Let $n_a$ and $N_a$ denote, respectively, the numbers of balls and boxes of colour $a$. Assuming that $N_a \gg n_a \gg 1$, the total entropy (in units of the Boltzmann constant) can be best approximated by
a.
$\sum_a (N_a \ln N_a + n_a \ln n_a - (N_a - n_a) \ln (N_a - n_a))$
b.
$\sum_a (N_a \ln N_a - n_a \ln n_a - (N_a - n_a) \ln (N_a - n_a))$
c.
$\sum_a (N_a \ln N_a - n_a \ln n_a + (N_a - n_a) \ln (N_a - n_a))$
d.
$\sum_a (N_a \ln N_a + n_a \ln n_a + (N_a - n_a) \ln (N_a - n_a))$
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Q21
mcq
The components of the electric field, in a region of space devoid of any charge or current sources, are given to be $E_i = a_i + \sum_{j=1,2,3} b_{ij} x_j$, where $a_i$ and $b_{ij}$ are constants independent of the coordinates. The number of independent components of the matrix $b_{ij}$ is
a.
5
b.
6
c.
3
d.
4
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Q21
mcq
To measure the height $h$ of a column of liquid helium in a container, a constant current $I$ is sent through an NbTi wire of length $l$, as shown in the figure. The normal state resistance of the NbTi wire is $R$.
If the superconducting transition temperature of NbTi is $\approx 10\text{ K}$, then the measured voltage $V(h)$ is best described by the expression.
a.
$IR\left(\frac{1}{2} - \frac{2h}{l}\right)$
b.
$IR\left(1 - \frac{h}{l}\right)$
c.
$IR\left(\frac{1}{2} - \frac{h}{l}\right)$
d.
$IR\left(1 - \frac{2h}{l}\right)$
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Q22
mcq
The equation of motion of a one-dimensional forced harmonic oscillator in the presence of a dissipative force is described by $\frac{d^2x}{dt^2} + 10\frac{dx}{dt} + 16x = 6t e^{-8t} + 4t^2 e^{-2t}$. The general form of the particular solution, in terms of constants $A$, $B$ etc., is
a.
$t(A t^2 + B t + C) e^{-2t} + (D t + E) e^{-8t}$
b.
$(A t^2 + B t + C) e^{-2t} + (D t + E) e^{-8t}$
c.
$t(A t^2 + B t + C) e^{-2t} + t(D t + E) e^{-8t}$
d.
$(A t^2 + B t + C) e^{-2t} + t(D t + E) e^{-8t}$
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Q22
mcq
A particle of mass $m$ in one dimensions is in the ground state of a simple harmonic oscillator described by a Hamiltonian $H = \frac{p^2}{2m} + \frac{1}{2}m\omega^2 x^2$ in the standard notation. An impulsive force at time $t = 0$ suddenly imparts a momentum $p_0 = \sqrt{\hbar m\omega}$ to it. The probability that the particle remains in the original ground state is
a.
$e^{-2}$
b.
$e^{-3/2}$
c.
$e^{-1}$
d.
$e^{-1/2}$
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Q23
mcq
A particle in two dimensions is found to trace an orbit $r(\theta) = r_0 \, \theta^2$. If it is moving under the influence of a central potential $V(r) = c_1 \, r^{-a} + c_2 \, r^{-b}$, where $r_0, c_1$ and $c_2$ are constants of appropriate dimensions, the values of ‘$a$’ and ‘$b$’, respectively, are
a.
$2$ and $4$
b.
$2$ and $3$
c.
$3$ and $4$
d.
$1$ and $3$
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Q23
mcq
The position of a particle in one dimension changes in discrete steps. With each step it moves to the right, however, the length of the step is drawn from a uniform distribution from the interval $\left[\lambda - \frac{1}{2}w, \lambda + \frac{1}{2}w\right]$, where $\lambda$ and $w$ are positive constants. If $X$ denotes the distance from the starting point after $N$ steps, the standard deviation $\sqrt{\langle X^2 \rangle - \langle X \rangle^2}$ for large values of $N$ is
a.
$\frac{\lambda}{2} \times \sqrt{N}$
b.
$\frac{\lambda}{2} \times \sqrt{\frac{N}{3}}$
c.
$\frac{w}{2} \times \sqrt{N}$
d.
$\frac{w}{2} \times \sqrt{\frac{N}{3}}$
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Q24
mcq
The figures below depict three different wavefunctions of a particle confined to a one-dimensional box $-1 \le x \le 1$.
The wavefunctions that correspond to the maximum expectation values $|\langle x \rangle|$ (absolute value of the mean position) and $\langle x^2 \rangle$, respectively, are
a.
B and C
b.
B and A
c.
C and B
d.
A and B
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Q24
mcq
In the following circuit the input voltage $V_{\text{in}}$ is such that $|V_{\text{in}}| < |V_{\text{sat}}|$, where $V_{\text{sat}}$ is the saturation voltage of the op-amp. (Assume that the diode is an ideal one and $R_L C$ is much larger than the duration of the measurement)
For the input voltage as shown in the figure above, the output voltage $V_{\text{out}}$ is best represented by
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Q25
mcq
The volume integral $I = \iiint_V \mathbf{A} \cdot (\nabla \times \mathbf{A}) \, d^3x$ is over a region $V$ bounded by a surface $\Sigma$ (an infinitesimal area element being $\hat{\mathbf{n}} dS$, where $\hat{\mathbf{n}}$ is the outward unit normal). If it changes to $I + \Delta I$, when the vector $\mathbf{A}$ is changed to $\mathbf{A} + \nabla \Lambda$, then $\Delta I$ can be expressed as:
a.
$\iiint_V \nabla \cdot (\nabla \Lambda \times \mathbf{A}) \, d^3x$
b.
$-\iiint_V \nabla^2 \Lambda \, d^3x$
c.
$-\oint_\Sigma (\nabla \Lambda \times \mathbf{A}) \cdot \hat{\mathbf{n}} dS$
d.
$\oint_\Sigma \nabla \Lambda \cdot \hat{\mathbf{n}} dS$
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Q25
mcq
Lead is superconducting below $7\text{ K}$ and has a critical magnetic field $800 \times 10^{-4}\text{ Tesla}$ close to $0\text{ K}$. At $2\text{ K}$ the critical current that flows through a long lead wire of radius $5\text{ mm}$ is closest to
a.
$1760\text{ A}$
b.
$1670\text{ A}$
c.
$1950\text{ A}$
d.
$1840\text{ A}$
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Q26
mcq
If we use the Fourier transform $\phi(x, y) = \int e^{ikx} \phi_k(y) dk$ to solve the partial differential equation:
$$\frac{\partial^2\phi(x, y)}{\partial y^2} - \frac{1}{y^2}\frac{\partial^2\phi(x, y)}{\partial x^2} + \frac{m^2}{y^2}\phi(x, y) = 0$$
in the half-plane soldiers\{(x, y): -\infty < x < \infty, 0 < y < \infty\}$ the Fourier modes $\phi_k(y)$ depend on $y$ as $y^\alpha$ and $y^\beta$. The values of $\alpha$ and $\beta$ are
a.
$\frac{1}{2} + \sqrt{1 + 4(k^2 + m^2)}$ and $\frac{1}{2} - \sqrt{1 + 4(k^2 + m^2)}$
b.
$1 + \sqrt{1 + 4(k^2 + m^2)}$ and $1 - \sqrt{1 + 4(k^2 + m^2)}$
c.
$\frac{1}{2} + \frac{1}{2}\sqrt{1 + 4(k^2 + m^2)}$ and $\frac{1}{2} - \frac{1}{2}\sqrt{1 + 4(k^2 + m^2)}$
d.
$1 + \frac{1}{2}\sqrt{1 + 4(k^2 + m^2)}$ and $1 - \frac{1}{2}\sqrt{1 + 4(k^2 + m^2)}$
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Q27
mcq
The fulcrum of a simple pendulum (consisting of a particle of mass $m$ attached to the support by a massless string of length $l$) oscillates vertically as $\sin z(t) = a \sin \omega t$, where $\omega$ is a constant. The pendulum moves in a vertical plane and $\theta(t)$ denotes its angular position with respect to the $z$-axis. If $l \frac{d^2\theta}{dt^2} + \sin \theta (g - f(t)) = 0$, (where $g$ is the acceleration due to gravity) describes the equation of motion of the mass, then $f(t)$ is
a.
$a\omega^2 \cos \omega t$
b.
$a\omega^2 \sin \omega t$
c.
$-a\omega^2 \cos \omega t$
d.
$-a\omega^2 \sin \omega t$
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Q28
mcq
The $Q$-value of the $\alpha$-decay of $^{232}\text{Th}$ to the ground state of $^{228}\text{Ra}$ is $4082\text{ keV}$. The maximum possible kinetic energy of the $\alpha$-particle is closest to
a.
$4082\text{ keV}$
b.
$4050\text{ keV}$
c.
$4035\text{ keV}$
d.
$4012\text{ keV}$
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Q29
mcq
The nuclei of $^{137}\text{Cs}$ decay by the emission of $\beta$-particles with a half-life of $30.08\text{ years}$. The activity (in units of disintegrations per second or Bq) of a $1\text{ mg}$ source of $^{137}\text{Cs}$, prepared on January 1, 1980, as measured on January 1, 2021 is closest to
a.
$1.79 \times 10^{16}$
b.
$1.79 \times 10^9$
c.
$1.24 \times 10^{16}$
d.
$1.24 \times 10^9$
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Q30
mcq
In an elastic scattering process at an energy $E$, the phase shifts satify $\delta_0 \approx 30^\circ$, $\delta_1 \approx 10^\circ$, while the other phase shifts are zero. The polar angle at which the differential cross-section peaks is closest to
a.
$20^\circ$
b.
$10^\circ$
c.
$0^\circ$
d.
$30^\circ$
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