Q. 1 Carry one mark each.

Q.1 “When she fell down the *, she received many* but little help.”

The words that best fill the blanks in the above sentence are

(A) stairs, stares

(B) stairs, stairs

(C) stares, stairs

(D) stares, stares

Ans:- A

Q.2 “In spite of being warned repeatedly, he failed to correct his ** _** behaviour.”

The word that best fills the blank in the above sentence is

(A) rational

(B) reasonable

(C) errant

(D) good

Ans:- C

Q.3 For 0 ≤ ???? ≤2????, sin???? and cos???? are both decreasing functions in the interval ** __**.

(A) $latex \left(0,\frac{\mathrm\pi}2\right)$

(B) $latex \left(\frac{\mathrm\pi}2,\mathrm\pi\right)$

(C) $latex \left(\mathrm\pi,\frac{3\mathrm\pi}2\right)$

(D) $latex \left(\mathrm\pi,\frac{3\mathrm\pi}2\right)$

Ans:- B

Q.4 The area of an equilateral triangle is √3. What is the perimeter of the triangle?

(A) 2

(B) 4

(C) 6

(D) 8

Ans:- C

Q.5 Arrange the following three-dimensional objects in the descending order of their volumes:

(i) A cuboid with dimensions 10 cm, 8 cm and 6 cm

(ii) A cube of side 8 cm

(iii) A cylinder with base radius 7 cm and height 7 cm

(iv) A sphere of radius 7 cm

(A) (i), (ii), (iii), (iv)

(B) (ii), (i), (iv), (iii)

(C) (iii), (ii), (i), (iv)

(D) (iv), (iii), (ii), (i)

Ans:- D

Q.6 An automobile travels from city A to city B and returns to city A by the same route. The speed of the vehicle during the onward and return journeys were constant at 60 km/h and 90 km/h, respectively. What is the average speed in km/h for the entire journey?

(A) 72

(B) 73

(C) 74

(D) 75

Ans:- A

Q.7 A set of 4 parallel lines intersect with another set of 5 parallel lines. How many parallelograms are formed?

(A) 20

(B) 48

(C) 60

(D) 72

Ans:- C

Q.8 To pass a test, a candidate needs to answer at least 2 out of 3 questions correctly. A total of 6,30,000 candidates appeared for the test. Question A was correctly answered by 3,30,000 candidates. Question B was answered correctly by 2,50,000 candidates. Question C was answered correctly by 2,60,000 candidates. Both questions A and B were answered correctly by 1,00,000 candidates. Both questions B and C were answered correctly by 90,000 candidates. Both questions A and C were answered correctly by 80,000 candidates. If the number of students answering all questions correctly is the same as the number answering none, how many candidates failed to clear the test?

(A) 30,000

(B) 2,70,000

(C) 3,90,000

(D) 4,20,000

Ans:- D

Q.9 If x2 + x – 1 = 0 what is the value of x4 + $latex \frac1{x^4}$ ?

(A) 1

(B) 5

(C) 7

(D) 9

Ans:- C

Q.10 In a detailed study of annual crow births in India, it was found that there was relatively no growth during the period 2002 to 2004 and a sudden spike from 2004 to 2005. In another unrelated study, it was found that the revenue from cracker sales in India which remained fairly flat from 2002 to 2004, saw a sudden spike in 2005 before declining again in 2006. The solid line in the graph below refers to annual sale of crackers and the dashed line refers to the annual crow births in India. Choose the most appropriate inference from the above data.

(A) There is a strong correlation between crow birth and cracker sales.

(B) Cracker usage increases crow birth rate.

(C) If cracker sale declines, crow birth will decline.

(D) Increased birth rate of crows will cause an increase in the sale of crackers.

Ans:- A

Q. 1 – Q. 25 carry one mark each.

Q.1 The eigenvalues of a Hermitian matrix are all

(A) real

(B) imaginary

(C) of modulus one

(D) real and positive

Ans:-

Q.2 Which one of the following represents the 3p radial wave function of hydrogen atom? (a0 is the Bohr radius)

Q.3 Given the following table,

which one of the following correctly matches the experiments from Group I to their inferences in Group II?

(A) P-2, Q-3, R-4, S-1

(B) P-1, Q-3, R-2, S-4

(C) P-3, Q-4, R-2, S-1

(D) P-2, Q-1, R-4, S-3

Q.4 In spherical polar coordinates $latex \left(r,\theta,\phi\right)$ the unit vector $latex \left(\widehat\theta\right)$ at $latex \left(10,\mathrm\pi/4,\mathrm\pi/2\right)$ is

(A) $latex \widehat k$

(B) $latex \frac1{\sqrt2}\left(\widehat j+\widehat k\right)$

(C) $latex \frac1{\sqrt2}\left(\widehat -j+\widehat k\right)$

(D) $latex \frac1{\sqrt2}\left(\widehat j-\widehat k\right)$

Q.5 The scale factors corresponding to the covariant metric tensor gij in spherical polar coordinates are

(A) 1, r2 , r2 sin2θ

(B) 1, r2,sin2θ

(C) 1,1,1

(D) 1,r,rsinθ

Q.6 In the context of small oscillations, which one of the following does NOT apply to the normal coordinates?

(A) Each normal coordinate has an eigen-frequency associated with it

(B) The normal coordinates are orthogonal to one another

(C) The normal coordinates are all independent

(D) The potential energy of the system is a sum of squares of the normal coordinates with constant coefficients

Q.7 For the given unit cells of a two dimensional square lattice, which option lists all the primitive cells?

For the given unit cells of a two dimensional square lattice, which option lists all the primitive cells?

(A) $latex \boxed1\;and\;\boxed2$

(B) $latex \boxed1\;,\;\boxed2\;and\;\boxed3$

(C) $latex \boxed1\;,\;\boxed2\;and\;\boxed3;\boxed4$

(D) $latex \boxed1\;,\;\boxed2\;\boxed3;\boxed4;and\boxed5$

Q.8 Among electric field $latex \left(\overset\rightharpoonup E\right)$ ,magnetic field $latex \left(\overset\rightharpoonup B\right)$ angular momentum $latex \left(\overset\rightharpoonup L\right)$ and vector potential $latex \left(\overset\rightharpoonup A\right)$, which is/are odd under parity (space inversion) operation?

(A) $latex \left(\overset\rightharpoonup E\right)$ only

(B) $latex \left(\overset\rightharpoonup E\right)$ & $latex \left(\overset\rightharpoonup A\right)$ only

(C) $latex \left(\overset\rightharpoonup E\right)$ & $latex \left(\overset\rightharpoonup B\right)$ only

(D) $latex \left(\overset\rightharpoonup B\right)$ & $latex \left(\overset\rightharpoonup L\right)$ only

Q.9 The expression for the second overtone frequency in the vibrational absorption spectra of a diatomic molecule in terms of the harmonic frequency we and anharmonicity constant xe is

(A) 2????????(1−????????)

(B) 2????????(1−3????????)

(C) 3????????(1−2????????)

(D) 3????????(1−4????????)

Ans:- (D)

Q.10 Match the physical effects and order of magnitude of their energy scales given below, where $latex \alpha=\frac{e^2}{4\mathrm\pi\in_0\mathrm{hc}}$ is fine structure constant; ???????? and ???????? are electron and proton mass,respectively.

(A) P-3, Q-1, R-2, S-4

(B) P-2, Q-3, R-1, S-4

(C) P-4, Q-2, R-1, S-3

(D) P-2, Q-4, R-1, S-3

Ans:- (C)

Q.11 The logic expression $latex \overline ABC\;+\overline A\overline BC\;+AB\overline C\;\;+\;A\overline{BC}$ can be simplified to

(A) ???? XOR ????

(B) $latex A\;AND\;\overline C$

(C) 0

(D) 1

Ans:- (A)

Q.12 At low temperatures (????), the specific heat of common metals is described by (with ???? and ???? as constants)

(A) ???? ???? + ???? ????3

(B) ???? ????3

(C) exp(−????/????)

(D) ???? ???? + ???? ????5

Ans:- (A)

Q.13 In a 2-to-1 multiplexer as shown below, the output X = A0 if C = 0, and X = A1 if C = 1.

Which one of the following is the correct implementation of this multiplexer?

(A)

(B)

(C)

(D)

Ans:- (A)

Q.14 The elementary particle Ξ0 is placed in the baryon decuplet, shown below, at

(A) P

(B) Q

(C) R

(D) S

Ans:- (C)

Q.15 The intrinsic/permanent electric dipole moment in the ground state of hydrogen atom is (????0 is the Bohr radius)

(A) −3????????0

(B) zero

(C) ????????0

(D) 3????????0

Ans:- (B)

Q.16 The high temperature magnetic susceptibility of solids having ions with magnetic moments can be described by $latex x\;\alpha\frac1{T+\theta}$ with ???? as absolute temperature and ???? as constant. The three behaviors i.e. paramagnetic, ferromagnetic and anti-ferromagnetic are described,respectively, by

(A) ????<0,????>0,????=0

(B) ????>0,????<0,????=0 (C) ????=0,????<0,????>0

(D) ????=0,????>0,????<0

Ans:- (C)

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