1. Fermi energy of a certain metal M₁ is 5 eV. A second metal M₂ has an electron density which is 6% higher than that of M₁. Assuming that the free electron theory is valid for both the metals, the Fermi energy of M₂ is closest to

(a) 5.6 eV

(b) 5.2 eV

(c) 4.8 eV

(d) 4.4 eV

Ans. (b)

Sol.

2. The following histogram represents the binding energy per particle (B.E./A) in MeV as a function of the mass number A of a nucleus.

A nucleus with mass number A = 180 fissions into two nuclei of equal masses. In the process.

(a) 180 MeV of energy is released

(b) 180 MeV of energy is absorbed.

(c) 360 MeV of energy is released

(d) 360 MeV of energy is absorbed.

Ans. (c)

Sol.

Energy released (Q) = B.E. products – B.E. of reactants.

B.E. of ⁹⁰Y = 6 × 90 = 540 MeV

B.E. of ⁹⁰X = 4 × 180 = 720 MeV

Q = [2 × 540 – 720] MeV = 360 MeV

3. A particle is confined in a one dimensional box with impenetrable walls at x = +a. Its energy eigen-value is 2 eV and the corresponding eigenfunction is as shown below.

The lowest possible energy of the particle is:

(a) 4 eV

(b) 2 eV

(c) 1 eV

(d) 0.5 eV

Ans. (d)

Sol. Energy eigenvalue of the particle in the n^th state is

(width of box is 2a units)

(n = 1, 2, 3, ......)

Eigenfunction of the particle (Given in question) has only one node (except x = + a)

Therefore, the given eigenfunction of particle represent the first excited state of the particle i.e. n = 2

So, ground state (n = 1) energy =

4. Experimental measurements of heat capacity per mole of aluminium at low temperature show that the data can be fitted to the formula, C_V = aT + bT³, where a = 0.00135 J K^–2 mole^–1, b = 2.48 × 10^–5 J K^–4 mole^–1 and T is the temperature in Kelvin. Change in entropy is:

(a)

(b)

(c)

(d)

Ans. (a)

Sol.

5. A uniform and constant magnetic field B coming out of the plane of the paper exists in a rectangular region as shown in the figure. A conducting rod PQ is rotated about O with a uniform angular speed in the plane of the paper. The emf E_PQ induced between P and Q is best represented by the graph.

(a)

(b)

(c)

(d)

Ans. (a)

Sol. We know that when a conducting rod is rotated in uniform magnetic field about an axis which coincides with the direction of field then a motional e.m.f. develops between the ends of the rod which is equal to .

= length of the portion inside the magnetic field. Positive polarity is in the direction .

Therefore, when end P is inside the magnetic field E_PO > 0 and when end Q is inside the magnetic field E_PQ < 0. Therefore, variation of E_PQ with time will be given by option (a).

6. Three polarizers P, Q and R are placed parallel to each other with their planes perpendicular to the z-axis. Q is placed between P and R. Initially the polarizing directions of P and Q are parallel, but that of R is perpendicular to them. In this arrangement when unpolarized light of intensity I₀ is incident on P, the intensity coming out of R is zero. The polarizer Q is now rotated about the z-axis. As a function of angle of rotation, the intensity of light coming out of R is best represented by

(a)

(b)

(c)

(d)

Ans. (b)

Sol.

7. The black body spectrum of an object O₁ is such that its radiant intensity (i.e. intensity per unit wavelength interval) is maximum at a wavelength of 200 nm. Another object O₂ has the maximum radiant intensity at 600 nm. The ratio of power emitted per unit area by O₁ to that of O₂ is

(a)

(b)

(c) 9

(d) 81

Ans. (b)

Sol.

8. In terms of the basic units of mass (M), length (L), time (T) and charge (Q), the dimensions of magnetic permeability of a vacuum (µ₀) are

(a) MLQ

(b) ML²T^–1Q^–

(c) LTQ^–1

(d) LT^–1Q^–1

Ans. (a)

Sol.

9. When two simple harmonic oscillations represented by

are superposed at right angle, the result is an ellipse with its major axis along the y-axis as shown in the figure. The conditions which corresponds to this are

(a)

(b)

(c)

(d)

Ans. (c)

Sol. The lissajous figure is ellipse only when phase difference is and amplitude of y-vibration is twice of amplitude of a x-vibration.

10. A projectile is fired from the origin 'O' at an angle of 45º from the horizontal. At the highest point 'P' of its trajectory the radial and traverse components of its acceleration in terms of the gravitational acceleration 'g' are

(a)

(b)

(c)

(d)

Ans. (d)

Sol.

11. A satellite moves around a planet in a circular orbit at a distance R from its centre. The time period of revolution of the satellite is T. If the same satellite is taken to an orbit of radius 4R around the same planet, the time period would be

(a) 8T

(b) 4T

(c) T/4

(d) T/8

Ans. (a)

Sol. Using Kepler's third law,

12. The speed of an electron, whose de-Broglie wavelength is equal to its compton wavelength, is (c is the speed of light)

(a) c

(b)

(c) c / 2

(d) c / 3

Ans. (a)

Sol. de-Broglie wavelength,

[where, 'm' is the relativistic mass of electron and v is the velocity of particle]

[where 'm₀' is the rest mass of electron]

Compton wavelength

13.

The matrix equation above represents.

(a) A circle of radius units

(b) An ellipse of semi major axis units

(c) An ellipse of semi major axis 5 units

(d) A hyperbola

Ans. (b)

Sol.

It is equation of ellipse with semi-major axis units.

14.

Figures (i) and (ii) represents respectively

(a) NOR, NOR

(b) NOR, NAND

(c) NAND, NAND

(d) OR, NAND

Ans. (c)

Sol. NAND and NAND

15. f(x) is a periodic function of x with a period of 2. In the interval – < x < , f(x) is given by

In the expansion of f(x) as a Fourier series of sine and cosine functions, the coefficients of cos (2x) is:

(a)

(b)

(c) 0

(d)

Ans. (d)

Sol.

16. If the total surface area (including the area of the top and bottom ends) of a cylinder is to be kept fixed (=A), what is its maximum possible volume?

For such cylinders of fixed total area, plot in the axes shown below their volume (V) versus the radius (R) clearly indicating the values of R for which the volume is maximum and zero.

Sol.

17. A horizontal square platform of mass 'm' and side 'a' is free to rotate about a vertical axis passing through its centre O. The platform is stationary and a person of the same mass 'm' as the platform is standing on it at point A. The person now starts walking along the edge from A to B (see figure). The e speed 'v' of the person with respect to the platform is constant. Find the time the person takes to reach B. Also find his distance r(t) from O as a function of time. Further find the angle through which the platform has rotated by the time the person reaches B.

Sol. Since speed with respect to plateform is constant

Therefore, time taken to reach

Let w be angular speed of platform at some intant of time.

18. Two identical parallel plate capacitors are connected across terminals A and B as shown. Each of the capacitors is made of square plates of side l with a distance d between them. A dielectric slab (relative permitivity k) of thickness 'd' is kept between the plates. The slab covers only half of the length of the plates in each of the capacitors as shown. Find the total capacitance of the assembly. The capacitors are charged by a battery and then the battery is disconnnected. If the slab is now displaced slightly by a distance , show that it will perform simple harmonic oscillations.

Sol. C´ = capacitance of each capacitor with dielectric.

Equivalent capacitance of the combination

Now, if dielectric slab is displaced slightly.

19. For the circuit shown below, calculate the output voltage V₀. What would V₀ be if the polarity of 2V battery is reversed at terminal B? (Assume the operational amplifier to be ideal).

Sol. Potential at B, V_B = 2V

Voltage at A and B will be equal due to virtual ground.

Now, let the polarity of 2V battery is reversed at terminal B.

Therefore, potential at A, V_A = –2V

Since, the supply voltage are +12V, the output saturates at +12V

So, output will be –12 volt.

20. A beam of light of wavelength 400 nm and power 1.55 mW is directed at the cathode of a photoelectric cell. (given hc = 1240 eV nm, e = 1.6 × 10^–19 C). If only 10% of the incident photons effectively produce photoelectrons, find the current due to these electrons. If the wavelength of light is now reduced to 200 nm, keeping its power the same, the kinetic energy of the electrons is found to increase by a factor of 5. What are the values of the stopping potentials for the two wavelengths?

Sol. Wavelength of incident light, = 400 nm, P = 1.55 mW

Energy of one photon

Number of photon falling per second,

Number of photoelectrons ejected is 0% of number of photon falling per second.

per second

Current constituted by flow of ejected electrons

1 = Ne = 50 µA

K.E. of ejected electron = Energy of incident photon - work function

When = 400 nm is reduced to = 200 nm, KE is increased by 5 times.

Substracting equation (1) from (2)

21. Two thin lenses L₁ and L₂ of focal length 15 cm and 10 cm respectively, are kept 15 cm apart from each other. Their axes are separated by 0.5 cm as shown in figure (not to scale). If a point object P is placed on the axis of L₁ of its left at a distance of 30 cm, find the x and y coordinates (origin O) of the image formed by the combination,

Sol. For L₁ u₁ = –30 cm, f₁ = 15 cm; since v₁ = 30 cm

For L₂, the image of L₁ will act as a virtual object.

u₂ = 15 m, f₂ = 10 cm since

The object for L₂ lies on the principal axis of L₁, so, height of object h = –0.5 cm, magnification, .

Magnification,

Negative sign means the location of image is below the principal axis of L₂.

x coordinate of image = 21 cm

y coordinate of image = 0.5 – 0.2 = 0.3 cm

So, co-ordinate of the image (21, 0.3)

22. The circuit shown consists of two identical capacitors of capacitance C each and an inductor of inductance L. Initially, both switches are open and capacitor 1 is charged with a charge Q₀ while the second capacitor has no charge. Switches S₁ and S₂ are closed simultaneously at t = 0. The circuit now becomes oscillatory.

(a) Calculate the maximum current in the circuit.

(b) Obtain expressions for the charge on the capacitors 1 and 2 as a function of time.

Sol. Let a be the charge capacitors 2 at time t,

Therefore, charge on capacitor 1 is (Q₀ – a)

Therefore, applying kirchoff's voltage law,

Solution of this equation is , q(t) = C.F. + P.I.

23. A particle travels along the diameter of the earth at a relativistic speed. It crosses the earth in a time 3 × 10^–2 s in its own frame. An observer, located on the earth, measures the same time interval to be 5 × 10^–2 s. Find the speed of the particle with respect to the earth and the diameter of the earth.

Sol. Let the speed of particle with respect to earth surfaces is v

Diameter of earth = velocity of particle × time taken

24. 1 m³ of an ideal gas with g = C_P / C_V = 1.5 is at a pressure of 100 kPa and a temperature of 300 K. Initially the state of the gas at the point a of the PV diagram shown. The gas is taken through a reversible cycle a b c a. The pressure at point 'b' is 200 kPa and the line ba, when extended, passes through the origin.

(a) Calculate the work done by the gas in each of the steps a b, b c and c a.

(b) Calculate the change in entropy of the gas in eah of the three steps above.

Sol. At point a P_a = 100 kPa

T_a = 300 K

At point b P_b = 200 kPa

Since, ab when produced passes through origin

W_bc = 0; W_ca = P_a(V_a – V_c) = –100 × 10³ (–6 × 10^–3 R + 3 × 10^–3 R) = –300 R Joule

S_ab = 2R ln 4 + R ln 2 = 5R ln (2);

25. How many work is done when an object moves from O P Q R O in a force field given by

along the rectangular path shown. Find the answer by evaluating the line integral and also by using the Stoke's theorem.

Sol.

For PQ, y = b, dy = 0,