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Consider a branch of the hyperbola x² - 2y² - 2√2x - 4√2y - 6 = 0 with vertex at the point A. Let B be one of the end points of its latus rectum. If C is the focus of the hyperbola nearest to the point A, then the area of the triangle ABC is

(A) 1 - 2/√3, (B) √(3/2) - 1, (C) 1 + 2/√3, (D) 3/√2 + 1

B

no

mathematics

The area of the region between the curves y = √(1+sinx/cosx) and y = √(1-sinx/cosx) bounded by the lines x = 0 and x = π/4 is

(A) ∫₀^(π/2√1+t²)dt/(1+t²√1-t²), (B) ∫₀^(π/2√1-t²)4t/(1+t²√1-t²) dt, (C) ∫₀^(π/4√1+t²)4t/(1+t²√1-t²) dt, (D) ∫₀^(π/1√1-t²)t/(1+t²√1-t²) dt

B

yes

mathematics

Consider three points P = (- sin(β - α), - cos β), Q = (cos(β - α), sin β) and R = (cos(β - α + θ), sin(β - θ)), where 0 < α , β , θ < π/4. Then,

(A) P lies on the line segment RQ (B) Q lies on the line segment PR (C) R lies on the line segment QP (D) P, Q, R are non-collinear

D

no

mathematics

An experiment has 10 equally likely outcomes. Let A and B be two non-empty events of the experiment. If A consists of 4 outcomes, the number of outcomes that B must have so that A and B are independent, is

(A) 2, 4 or 8 (B) 3, 6 or 9 (C) 4 or 8 (D) 5 or 10

D

no

mathematics

Let two non-collinear unit vectors â and b̂ form an acute angle. A point P moves so that at any time t the position vector OP⃗ (where O is the origin) is given by â cost + b̂ sin t. When P is farthest from origin O , let M be the length of OP⃗ and û be the unit vector along OP⃗ . Then,

(A) û = (â + b̂) / |â + b̂| and M = (1 + â . b̂)^(1/2) (B) û = (â - b̂) / |â - b̂| and M = (1 + â . b̂)^(1/2) (C) û = (â + b̂) / |â + b̂| and M = (1 + 2â . b̂)^(1/2) (D) û = (â - b̂) / |â - b̂| and M = (1 + 2â . b̂)^(1/2)

A

no

mathematics

Let I = ∫(e^x / e^4x + e^2x + 1) dx, J = ∫(e^-x / e^-4x + e^-2x + 1) dx. Then, for an arbitrary constant C, the value of J - I equals

(A) (1/2) log[((e^4x - e^2x + 1) / (e^4x + e^2x + 1))] + C (B) (1/2) log[((e^2x + e^x + 1) / (e^2x - e^x + 1))] + C (C) (1/2) log[((e^2x - e^x + 1) / (e^2x + e^x + 1))] + C (D) (1/2) log[((e^4x + e^2x + 1) / (e^4x - e^2x + 1))] + C

C

no

mathematics

Let g(x) = log f(x) where f(x) is a twice differentiable positive function on (0, ∞) such that f(x + 1) = x f(x). Then, for N = 1, 2, 3,..., g'(N + 1/2) - g'(1/2) =

(A) -4{1 + 1/9 + 1/25 + ... + 1/(2N - 1)^2}, (B) 4{1 + 1/9 + 1/25 + ... + 1/(2N - 1)^2}, (C) -4{1 + 1/9 + 1/25 + ... + 1/(2N + 1)^2}, (D) 4{1 + 1/9 + 1/25 + ... + 1/(2N + 1)^2}

(A)

no

mathematics

Suppose four distinct positive numbers a1, a2, a3, a4 are in G.P. Let b1 = a1, b2 = b1 + a2, b3 = b2 + a3 and b4 = b3 + a4. STATEMENT-1 : The numbers b1, b2, b3, b4 are neither in A.P. nor in G.P. and STATEMENT-2 : The numbers b1, b2, b3, b4 are in H.P.

(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1, (B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1, (C) STATEMENT-1 is True, STATEMENT-2 is False, (D) STATEMENT-1 is False, STATEMENT-2 is True

(C)

no

mathematics

Let a, b, c, p, q be real numbers. Suppose α, β are the roots of the equation x^2 + 2px + q = 0 and α, 1/β are the roots of the equation ax^2 + 2bx + c = 0, where β^2 ∈ {-1, 0, 1}. STATEMENT-1 : (p^2 - q)(b^2 - ac) ≥ 0 and STATEMENT-2 : b ≠ pa or c ≠ qa

(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1, (B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1, (C) STATEMENT-1 is True, STATEMENT-2 is False, (D) STATEMENT-1 is False, STATEMENT-2 is True

(B)

no

mathematics

Consider L1 : 2x + 3y + p - 3 = 0, L2 : 2x + 3y + p + 3 = 0, where p is a real number, and C : x^2 + y^2 + 6x - 10y + 30 = 0. STATEMENT-1 : If line L1 is a chord of circle C, then line L2 is not always a diameter of circle C. and STATEMENT-2 : If line L1 is a diameter of circle C, then line L2 is not a chord of circle C.

(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1, (B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1, (C) STATEMENT-1 is True, STATEMENT-2 is False, (D) STATEMENT-1 is False, STATEMENT-2 is True

C

no

mathematics

Let a solution y = y(x) of the differential equation x√(x^2 - 1) dy - y√(y^2 - 1) dx = 0 satisfy y(2) = 2/√3. STATEMENT-1 : y(x) = sec(sec^-1 x - π/6) and STATEMENT-2 : y(x) is given by 1/(2√3 - √(1 - x^2))

(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1, (B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1, (C) STATEMENT-1 is True, STATEMENT-2 is False, (D) STATEMENT-1 is False, STATEMENT-2 is True

C

no

mathematics

Which of the following is true?

(A) (2 + a)^2 f''(1) + (2 - a)^2 f''(-1) = 0 (B) (2 - a)^2 f''(1) - (2 + a)^2 f''(-1) = 0 (C) f'(1) f'(-1) = (2 - a)^2 (D) f'(1) f'(-1) = -(2 + a)^2

A

no

mathematics

Which of the following is true?

(A) f(x) is decreasing on (-1, 1) and has a local minimum at x = 1 (B) f(x) is increasing on (-1, 1) and has a local maximum at x = 1 (C) f(x) is increasing on (-1, 1) but has neither a local maximum nor a local minimum at x = 1 (D) f(x) is decreasing on (-1, 1) but has neither a local maximum nor a local minimum at x = 1

A

no

mathematics

Let g(x) = ∫(0 to x) f'(t) / (1 + t^2) dt. Which of the following is true?

(A) g'(x) is positive on (- ∞, 0) and negative on (0, ∞) (B) g'(x) is negative on (- ∞, 0) and positive on (0, ∞) (C) g'(x) changes sign on both (- ∞, 0) and (0, ∞) (D) g'(x) does not change sign on (- ∞, ∞)

B

no

mathematics

The unit vector perpendicular to both L1 and L2 is

A) -i + 7j + 7k/√99, B) -i - 7j + 5k/5√3, C) -i + 7j + 5k/5√3, D) 7i - 7j - k/√99

B

yes

mathematics

The shortest distance between L1 and L2 is

A) 0, B) 17/√3, C) 41/5√3, D) 17/5√3

D

yes

mathematics

The distance of the point (1, 1, 1) from the plane passing through the point (-1, -2, -1) and whose normal is perpendicular to both the lines L1 and L2 is

A) 2/√75, B) 7/√75, C) 13/√75, D) 23/√75

C

yes

mathematics

Consider the lines given by L1 : x + 3y - 5 = 0 L2 : 3x - ky - 1 = 0 L3 : 5x + 2y - 12 = 0 Match the Statements / Expressions in Column I with the Statements / Expressions in Column II and indicate your answer by darkening the appropriate bubbles in the 4 x 4 matrix given in the ORS.

(A) L1, L2, L3 are concurrent, if (B) One of L1, L2, L3 is parallel to at least one of the other two, if (C) L1, L2, L3 form a triangle, if (D) L1, L2, L3 do not form a triangle, if (p) k = -9 (q) k = - 6/5 (r) k = 5/6 (s) k = 5

p:O, q:O, r:O, s:*, A:O, B:*, C:O, D:*

yes

mathematics

Match the Statements / Expressions in Column I with the Statements / Expressions in Column II and indicate your answer by darkening the appropriate bubbles in the 4 x 4 matrix given in the ORS.

(A) The minimum value of x^2 + 2x + 4/x + 2 is (B) Let A and B be 3 x 3 matrices of real numbers, where A is symmetric, B is skew-symmetric, and (A + B) (A - B) = (A - B) (A + B). If (AB)' = (-1)^k AB, where (AB)' is the transpose of the matrix AB, then the possible values of k are (C) Let a = log₃ 2, log₂ 2. An integer k satisfying 1 < 2^(-k + 3 - a) < 2, must be less than (D) If sin θ = cos φ, then the possible values of 1/π (θ + φ - π/2) are (p) 0 (q) 1 (r) 2 (s) 3

p:O, q:*, r:*, s:O, A:O, B:O, C:*, D:*

yes

mathematics

Consider all possible permutations of the letters of the word ENDEANOEL. Match the Statements / Expressions in Column I with the Statements / Expressions in Column II and indicate your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in the ORS.

Column I: (A) The number of permutations containing the word ENDEA is, (B) The number of permutations in which the letter E occurs in the first and the last positions is, (C) The number of permutations in which none of the letters D, L, N occurs in the last five positions is, (D) The number of permutations in which the letters A, E, O occur only in odd positions is Column II: (p) 5!, (q) 2 × 5!, (r) 7 × 5!, (s) 21 × 5!

A: p, B: s, C: r, D: q

no

mathematics

Consider a system of three charges q, q and 2q/3 placed at points A, B and C, respectively, as shown in the figure. Take O to be the centre of the circle of radius R and angle CAB = 60° Figure: (A) The electric field at point O is q/8πε₀R² directed along the negative x-axis (B) The potential energy of the system is zero (C) The magnitude of the force between the charges at C and B is q²/54πε₀R² (D) The potential at point O is q/12πε₀R

(A), (B), (C), (D)

(C)

yes

physics

A radioactive sample S1 having an activity of 5 μCi has twice the number of nuclei as another sample S2 which has an activity of 10 μCi. The half lives of S1 and S2 can be

(A) 20 years and 5 years, respectively (B) 20 years and 10 years, respectively (C) 10 years each (D) 5 years each

A

no

physics

A transverse sinusoidal wave moves along a string in the positive x-direction at a speed of 10 cm/s. The wavelength of the wave is 0.5 m and its amplitude is 10 cm. At a particular time t, the snap-shot of the wave is shown in figure. The velocity of point P when its displacement is 5 cm is

(A) √(3π÷50) m/s (B) -√(3π÷50) m/s (C) √(3π÷50) m/s (D) -√(3π÷50) m/s

A

yes

physics

A block (B) is attached to two unstretched springs S1 and S2 with spring constants k and 4 k, respectively (see figure 1). The other ends are attached to identical supports M1 and M2 not attached to the walls. The springs and supports have negligible mass. There is no friction anywhere. The block B is displaced towards wall 1 by a small distance x (figure II) and released. The block returns and moves a maximum distance y towards wall 2. Displacements x and y are measured with respect to the equilibrium position of the block B. The ratio y/x is

A) 4, B) 2, C) 1/2, D) 1/4

C

yes

physics

A bob of mass M is suspended by a massless string of length L. The horizontal velocity V at position A is just sufficient to make it reach the point B. The angle θ at which the speed of the bob is half of that at A, satisfies

A) θ = π/4, B) π/4 < θ < π/2, C) π/2 < θ < 3π/4, D) 3π/4 < θ < π

D

yes

physics

A glass tube of uniform internal radius (r) has a valve separating the two identical ends. Initially, the valve is in a tightly closed position. End 1 has a hemispherical soap bubble of radius r. End 2 has sub-hemispherical soap bubble as shown in figure. Just after opening the valve,

(A) air from end 1 flows towards end 2. No change in the volume of the soap bubbles (B) air from end 1 flows towards end 2. Volume of the soap bubble at end 1 decreases (C) no change occurs (D) air from end 2 flows towards end 1. Volume of the soap bubble at end 1 increases

B

yes

physics

A vibrating string of certain length ℓ under a tension T resonates with a mode corresponding to the first overtone (third harmonic) of an air column of length 75 cm inside a tube closed at one end. The string also generates 4 beats per second when excited along with a tuning fork of frequency n. Now when the tension of the string is slightly increased the number of beats reduces to 2 per second. Assuming the velocity of sound in air to be 340 m/s, the frequency n of the tuning fork in Hz is

(A) 344 (B) 336 (C) 117.3 (D) 109.3

A

no

physics

A parallel plate capacitor C with plates of unit area and separation d is filled with a liquid of dielectric constant K = 2. The level of liquid is d/3 initially. Suppose the liquid level decreases at a constant speed V, the time constant as a function of time t is

(A) 6εoR / 5d+3Vt, (B) (15d+9Vt)εoR / 2d^2-3dVt-9V^2t^2, (C) 6εoR / 5d-3Vt, (D) (15d-9Vt)εoR / 2d^2+3dVt-9V^2t^2

A

yes

physics

A light beam is traveling from Region I to Region IV (Refer Figure). The refractive index in Regions I, II, III and IV are no, no/2, no/6 and no/8, respectively. The angle of incidence θ for which the beam just misses entering Region IV is

(A) sin^-1(3/4), (B) sin^-1(1/5), (C) sin^-1(1/4), (D) sin^-1(1/3)

B

yes

physics

STATEMENT-1 For an observer looking out through the window of a fast moving train, the nearby objects appear to move in the opposite direction to the train, while the distant objects appear to be stationary. and STATEMENT-2 If the observer and the object are moving at velocities V1 and V2 respectively with reference to a laboratory frame, the velocity of the object with respect to the observer is V2 - V1.

(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1 (B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1 (C) STATEMENT-1 is True, STATEMENT-2 is False (D) STATEMENT-1 is False, STATEMENT-2 is True

B

no

physics

STATEMENT-1 It is easier to pull a heavy object than to push it on a level ground. and STATEMENT-2 The magnitude of frictional force depends on the nature of the two surfaces in contact.

(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1 (B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1 (C) STATEMENT-1 is True, STATEMENT-2 is False (D) STATEMENT-1 is False, STATEMENT-2 is True

B

no

physics

STATEMENT-1 For practical purposes, the earth is used as a reference at zero potential in electrical circuits. and STATEMENT-2 The electrical potential of a sphere of radius R with charge Q uniformly distributed on the surface is given by Q/4πε0R.

(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1 (B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1 (C) STATEMENT-1 is True, STATEMENT-2 is False (D) STATEMENT-1 is False, STATEMENT-2 is True

A

no

physics

STATEMENT-1 The sensitivity of a moving coil galvanometer is increased by placing a suitable magnetic material as a core inside the coil. and STATEMENT-2 Soft iron has a high magnetic permeability and cannot be easily magnetized or demagnetized.

(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1 (B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1 (C) STATEMENT-1 is True, STATEMENT-2 is False (D) STATEMENT-1 is False, STATEMENT-2 is True

C

no

physics

The electric field at r = R is

(A) independent of a, (B) directly proportional to a, (C) directly proportional to a^2, (D) inversely proportional to a

A

yes

physics

For a = 0, the value of d (maximum value of ρ as shown in the figure) is

(A) 3Ze/(4πR^3), (B) 3Ze/(πR^3), (C) 4Ze/(3πR^3), (D) Ze/(3πR^3)

B

yes

physics

The electric field within the nucleus is generally observed to be linearly dependent on r. This implies

(A) a = 0, (B) a = R/2, (C) a = R, (D) a = 2R/3

C

yes

physics

The net external force acting on the disk when its centre of mass is at displacement x with respect to its equilibrium position is

(A) -kx, (B) -2kx, (C) -2kx/3, (D) 4kx/3

C

yes

physics

The centre of mass of the disk undergoes simple harmonic motion with angular frequency ω equal to

(A) √(k/M), (B) √(2k/M), (C) √(2k/3M), (D) √(4k/3M)

D

yes

physics

The maximum value of V0 for which the disk will roll without slipping is

(A) μg√(k/M), (B) μg√(2k/M), (C) μg√(3M/k), (D) μg√(5M/2k)

C

yes

physics

Column I gives a list of possible set of parameters measured in some experiments. The variations of the parameters in the form of graphs are shown in Column II. Match the set of parameters given in Column I with the graphs given in Column II. Indicate your answer by darkening the appropriate bubbles of the 4x4 matrix given in the ORS.

A) Potential energy of a simple pendulum (y axis) as a function of displacement (x axis), B) Displacement (y axis) as a function of time (x axis) for a one dimensional motion at zero or constant acceleration when the body is moving along the positive x-direction, C) Range of a projectile (y axis) as a function of its velocity (x axis) when projected at a fixed angle, D) The square of the time period (y axis) of a simple pendulum as a function of its length (x axis)

B, C, D

yes

physics

An optical component and an object S placed along its optic axis are given in Column I. The distance between the object and the component can be varied. The properties of images are given in Column II. Match all the properties of images from Column II with the appropriate components given in Column I. Indicate your answer by darkening the appropriate bubbles of the 4 x 4 matrix given in the ORS.

A, B, C, D. Matrix response with options p, q, r, s

p, q, r, s

yes

physics

Column I contains a list of processes involving expansion of an ideal gas. Match this with Column II describing the thermodynamic change during this process. Indicate your answer by darkening the appropriate bubbles of the 4 x 4 matrix given in the ORS.

A) ... B) ... C) ... D) ...

A: q, B: r and q, C: p and s, D: s

yes

physics

The correct stability order for the following species is

(A) (II) > (IV) > (I) > (III), (B) (I) > (II) > (III) > (IV), (C) (II) > (I) > (IV) > (III), (D) (I) > (III) > (II) > (IV)

D

yes

chemistry

Cellulose upon acetylation with excess acetic anhydride/H₂SO₄ (catalytic) gives cellulose triacetate whose structure is

Options shown as molecular structure diagrams (A), (B), (C), and (D)

A

yes

chemistry

In the following reaction sequence, the correct structures of E, F and G are

(A) E = Ph* CH3, F = Ph* O Na, G = CH3; (B) E = Ph* CH3, F = Ph* ONa, G = CH3; (C) E = Ph* CH3, F = Ph* O Na, G = CH3; (D) E = Ph* CH3, F = Ph* ONa, G = CH3I

C

yes

chemistry

Among the following, the coloured compound is

(A) CuCl; (B) K3 [Cu(CN)4]; (C) CuF2; (D) [Cu(CH3CN)4]BF4

C

no

chemistry

Both [Ni(CO)4] and [Ni(CN)4]2- are diamagnetic. The hybridisations of nickel in these complexes, respectively, are

(A) sp3, sp3; (B) sp3, dsp2; (C) dsp2, sp3; (D) dsp2, dsp2

B

no

chemistry

The IUPAC name of [Ni(NH3)4] [NiCl4] is

(A) Tetrachloronickel (II) - tetraammineniickel (II), (B) Tetraammineniickel (II) - tetrachloronickel (II), (C) Tetraammineniickel (II) - tetrachloronickelate (II), (D) Tetrachloronickel (II) - tetraammineniickelate (I)

C

no

chemistry

Electrolysis of dilute aqueous NaCl solution was carried out by passing 10 milli ampere current. The time required to liberate 0.01 mol of H2 gas at the cathode is (1 Faraday = 96500 C mol^-1)

(A) 9.65 × 10^4 sec, (B) 19.3 × 10^4 sec, (C) 28.95 × 10^4 sec, (D) 38.6 × 10^4 sec

B

no

chemistry

Among the following, the surfactant that will form micelles in aqueous solution at the lowest molar concentration at ambient conditions is

(A) CH3(CH2)15 N^+(CH3)3 Br^-, (B) CH3(CH2)11 OSO3^- Na^+, (C) CH3(CH2)9 COO^- Na^+, (D) CH3(CH2)11 N^+(CH3)3 Br^-

A

no

chemistry

Solubility product constants (Ksp) of salts of types MX, MX2 and M3X at temperature 'T' are 4.0 x 10^-8, 3.2 x 10^-14 and 2.7 x 10^-16, respectively. Solubilities (mol dm^-3) of the salts at temperature 'T' are in the order

(A) MX > MX2 > M3X, (B) M3X > MX2 > MX, (C) MX2 > M3X > MX, (D) MX > M3X > MX2

D

no

chemistry

STATEMENT-1 : Aniline on reaction with NaNO2/HCl at 0 °C followed by coupling with β-napththol gives a dark blue coloured precipitate. and STATEMENT-2 : The colour of the compound formed in the reaction of aniline with NaNO2/HCl at 0 °C followed by coupling with β-napththol is due to the extended conjugation.

(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1 (B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1 (C) STATEMENT-1 is True, STATEMENT-2 is False (D) STATEMENT-1 is False, STATEMENT-2 is True

D

no

chemistry

STATEMENT-1 : [Fe(H2O)6NO]SO4 is paramagnetic. and STATEMENT-2 : The Fe in [Fe(H2O)6 NO]SO4 has three unpaired electrons.

(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1 (B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1 (C) STATEMENT-1 is True, STATEMENT-2 is False (D) STATEMENT-1 is False, STATEMENT-2 is True

A

no

chemistry

STATEMENT-1 : The geometrical isomers of the complex [M(NH3)4Cl2] are optically inactive. and STATEMENT-2 : Both geometrical isomers of the complex [M(NH3)4Cl2] possess axis of symmetry.

(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1 (B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1 (C) STATEMENT-1 is True, STATEMENT-2 is False (D) STATEMENT-1 is False, STATEMENT-2 is True

B

no

chemistry

STATEMENT-1 : There is a natural asymmetry between converting work to heat and converting heat to work. and STATEMENT-2 : No process is possible in which the sole result is the absorption of heat from a reservoir and its complete conversion into work.

(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1, (B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1, (C) STATEMENT-1 is True, STATEMENT-2 is False, (D) STATEMENT-1 is False, STATEMENT-2 is True

(A)

no

physics

Compound H is formed by the reaction of

(A) Ph CH3 + PhMgBr, (B) Ph CH3 + PhCH2 MgBr, (C) Ph H + PhCH2MgBr, (D) Ph H + Ph MgBr

(B)

yes

chemistry

The structure of compound I is

(A) Ph CH3, (B) H3C Ph, (C) Ph CH3, (D) H3C CH3

A

yes

chemistry

The structures of compounds J, K and L, respectively, are

(A) PhCOCH3, PhCH2COCH3 and PhCH2COO-K+, (B) PhCHO, PhCH2CHO and PhCOO-K+, (C) PhCOCH3, PhCH2CHO and CH3COO-K+, (D) PhCHO, PhCOCH3 and PhCOO-K+

D

no

chemistry

The number of atoms in this HCP unit cell is

(A) 4, (B) 6, (C) 12, (D) 17

B

no

chemistry

The volume of this HCP unit cell is

(A) 24√2 r^3, (B) 16√2 r^3, (C) 12√2 r^3, (D) (64)/(3√3) r^3

A

no

chemistry

The empty space in this HCP unit cell is

(A) 74%, (B) 47.6%, (C) 32%, (D) 26%

D

no

chemistry

Match the compounds in Column I with their characteristic test(s)/reaction(s) given in Column II. Indicate your answer by darkening the appropriate bubbles of the 4 x 4 matrix given in the ORS.

A, B, C, D - Matrix Response

B: C, D; C: A, B, C; D: A, B, D OR A: B, C; B: A, B; C: A, B, C; D: A

yes

chemistry

Match the conversions in Column I with the type(s) of reaction(s) given in Column II. Indicate your answer by darkening the appropriate bubbles of the 4 x 4 matrix given in the ORS

(A) PbS -> PbO, (B) CaCO3 -> CaO, (C) ZnS -> Zn, (D) Cu2S -> Cu, (p) roasting, (q) calcination, (r) carbon reduction, (s) self reduction

A:p, B:q, C:r,s, D:s

yes

chemistry

Match the entries in Column I with the correctly related quantum number(s) in Column II. Indicate your answer by darkening the appropriate bubbles of the 4 x 4 matrix given in the ORS

(A) Orbital angular momentum of the electron in a hydrogen-like atomic orbital, (B) A hydrogen-like one-electron wave function obeying Pauli principle, (C) Shape, size and orientation of hydrogen-like atomic orbitals, (D) Probability density of electron at the nucleus in hydrogen-like atom, (p) Principal quantum number, (q) Azimuthal quantum number, (r) Magnetic quantum number, (s) Electron spin quantum number

A:q,r, B:p,q,r,s, C:p,q,r, D:p,s

yes

physics

Given that the abundances of isotopes 54Fe, 56Fe and 57Fe are 5%, 90% and 5%, respectively, the atomic mass of Fe is

(A) 55.85 (B) 55.95 (C) 55.75 (D) 56.05

B

no

chemistry

The term that corrects for the attractive forces present in a real gas in the van der Waals equation is

(A) nb (B) an^2/V^2 (C) an^2/V (D) - nb

B

no

chemistry

Among the electrolytes Na2SO4, CaCl2, Al2(SO4)3 and NH4Cl, the most effective coagulating agent for Sb2S3 sol is

(A) Na2SO4 (B) CaCl2 (C) Al2(SO4)3 (D) NH4Cl

C

no

chemistry

The Henry's law constant for the solubility of N2 gas in water at 298 K is 1.0 x 10^4 atm. The mole fraction of N2 in air is 0.8. The number of moles of N2 from air dissolved in 10 moles of water at 298 K and 5 atm pressure is

(A) 4.0 x 10^4 (B) 4.0 x 10^5 (C) 5.0 x 10^4 (D) 4.0 x 10^6

A

no

chemistry

The reaction of P4 with X leads selectively to P4O6. The X is

(A) Dry O2 (B) A mixture of O2 and N2 (C) Moist O2 (D) O2 in the presence of aqueous NaOH

B

no

chemistry

The correct acidity order of the following is

(A) (III) > (IV) > (II) > (I) (B) (IV) > (III) > (I) > (II) (C) (III) > (II) > (IV) > (I) (D) (II) > (III) > (IV) > (I)

A

yes

chemistry

Among cellulose, poly(vinyl chloride), nylon and natural rubber, the polymer in which the intermolecular force of attraction is weakest is

(A) Nylon (B) Poly(vinyl chloride) (C) Cellulose (D) Natural Rubber

D

no

chemistry

The IUPAC name of the following compound is

(A) 4-Bromo-3-cyanophenol (B) 2-Bromo-5-hydroxybenzonitrile (C) 2-Cyano-4-hydroxybromobenzene (D) 6-Bromo-3-hydroxybenzonitrile

B

yes

chemistry

The correct statement(s) regarding defects in solids is(are)

(A) Frenkel defect is usually favoured by a very small difference in the sizes of cation and anion (B) Frenkel defect is a dislocation defect (C) Trapping of an electron in the lattice leads to the formation of F-center (D) Schottky defects have no effect on the physical properties of solids

B C

no

chemistry

The compound(s) that exhibit(s) geometrical isomerism is(are)

(A) [Pt(en)Cl2] (B) [Pt(en)2]Cl2 (C) [Pt(en)2Cl2]Cl2 (D) [Pt(NH3)2Cl2]

C D

no

chemistry

The compound(s) formed upon combustion of sodium metal in excess air is(are)

(A) Na2O2 (B) Na2O (C) NaO2 (D) NaOH

A OR B D

no

chemistry

The correct statement(s) about the compound H3C(HO)HC-CH-CH-CH(OH)CH3 (X) is(are)

(A) The total number of stereoisomers possible for X is 6 (B) The total number of diastereomers possible for X is 3 (C) If the stereochemistry about the double bond in X is trans, the number of enantiomers possible for X is 4 (D) If the stereochemistry about the double bond in X is cis, the number of enantiomers possible for X is 2

A D

no

chemistry

The compound X is

(A) Na3NO3 (B) NaCl (C) Na2SO4 (D) Na2S

D

no

chemistry

The compound Y is

(A) MgCl2 (B) FeCl2 (C) FeCl3 (D) ZnCl2

C

no

chemistry

The compound Z is

(A) Mg2[Fe(CN)6] (B) Fe[Fe(CN)6] (C) Fe3[Fe(CN)6]4 (D) K2Zn3[Fe(CN)6]4

B

no

chemistry

The structure of the carbonyl compound P is

(A) (image A) (B) (image B) (C) (image C) (D) (image D)

B

yes

chemistry

The structures of the products Q and R, respectively, are

(A) (image A) (B) (image B) (C) (image C) (D) (image D)

A

yes

chemistry

The structure of the product S is

(A) (B) (C) (D)

B

yes

chemistry

Match each of the diatomic molecules in Column I with its property/properties in Column II.

Column I: (A) B₂ (B) N₂ (C) O₂⁻ (D) O₂ Column II: (p) Paramagnetic (q) Undergoes oxidation (r) Undergoes reduction (s) Bond order > 2 (t) Mixing of 's' and 'p' orbitals

p q r s t, p q r s t, p q r s t, p q r s t

no

chemistry

Match each of the compounds in Column I with its characteristic reaction(s) in Column II.

A) CH₃CH₂CH₂CN, (B) CH₃CH₂OCOCH₃, (C) CH₃ – CH = CH – CH₂OH, (D) CH₃CH₂CH₂CH₂NH₂

A) p, q, r, s, t, B) p, q, r, s, t, C) p, q, r, s, t, D) p, q, r, s, t

no

chemistry

Let P(3, 2, 6) be a point in space and Q be a point on the line r = (j - j + 2k) + μ(-3i + j + 5k). Then the value of μ for which the vector PQ is parallel to the plane x - 4y + 3z = 1 is

(A) 1/4, (B) -1/4, (C) 1/8, (D) -1/8

A

no

mathematics

Tangents drawn from the point P(1, 8) to the circle x² + y² - 6x - 4y - 11 = 0 touch the circle at the points A and B. The equation of the circumcircle of the triangle PAB is

(A) x² + y² + 4x - 6y + 19 = 0, (B) x² + y² - 4x - 10y +19 = 0, (C) x² + y² - 2x + 6y - 29 = 0, (D) x² + y² - 6x - 4y +19 = 0

B

no

mathematics

Let f be a non-negative function defined on the interval [0, 1]. If ∫[0, 1] (f(t))^2 dt = ∫[0, 1] f(t) dt, 0 ≤ x ≤ 1, and f(0) = 0, then

(A) f(1/2) < 1/2 and f(1/3) > 1/3, (B) f(1/2) > 1/2 and f(1/3) < 1/3, (C) f(1/2) < 1/2 and f(1/3) < 1/3, (D) f(1/2) > 1/2 and f(1/3) > 1/3

C

no

mathematics

Let z = x + iy be a complex number where x and y are integers. Then the area of the rectangle whose vertices are the roots of the equation z^2 + z^3 = 350 is

(A) 48, (B) 32, (C) 40, (D) 80

A

no

mathematics

The line passing through the extremity A of the major axis and extremity B of the minor axis of the ellipse x^2 + 9y^2 = 9 meets its auxiliary circle at the point M. Then the area of the triangle with vertices at A, M and the origin O is

(A) 31/10, (B) 29/10, (C) 21/10, (D) 27/10

D

no

mathematics

If a, b, c and d are unit vectors such that |a × b| |c × d| = 1 and a.c = 1/2, then

(A) a, b, c are non-coplanar, (B) b, c, d are non-coplanar, (C) b, d are non-parallel, (D) a, d are parallel and b, c are parallel

C

no

mathematics

Let z = cos θ + i sin θ. Then the value of Σ[n=1 to 18] Im(z^2n+1) at θ = 2° is

(A) 1/sin 2°, (B) 1/3 sin 2°, (C) 1/2 sin 2°, (D) 1/4 sin 2°

D

no

mathematics

The number of seven digit integers, with sum of the digits equal to 10 and formed by using the digits 1, 2 and 3 only, is

(A) 55 (B) 66 (C) 77 (D) 88

C

no

mathematics

Area of the region bounded by the curve y = e^x and lines x = 0 and y = e is

(A) e - 1 (B) \int_{1}^{e} (e + 1 - y) dy (C) e - \int_{e}^{1} e^x dx (D) \int_{1}^{e} \ln y dy

B C D

yes

mathematics

Let L = \lim_{x \to 0} \frac{a - \sqrt{a^2 - x^2} - \frac{x^2}{4}}{x^4}, a > 0. If L is finite, then

(A) a = 2 (B) a = 1 (C) L = \frac{1}{64} (D) L = \frac{1}{32}

A C

no

mathematics

In a triangle ABC with fixed base BC, the vertex A moves such that cos B + cos C = 4 sin^2 \frac{A}{2}. If a, b and c denote the lengths of the sides of the triangle opposite to the angles A, B and C, respectively, then

(A) b + c = 4a (B) b + c = 2a (C) locus of point A is an ellipse (D) locus of point A is a pair of straight lines

B C

no

mathematics

If (sin^4 x)/2 + (cos^4 x)/3 = 1/5, then

(A) tan^2 x = 2/3, (B) (sin^8 x)/(8) + (cos^8 x)/(27) = 1/125, (C) tan^2 x = 1/3, (D) (sin^8 x)/(8) + (cos^8 x)/(27) = 2/125

A, B

no

mathematics

Let M be the set of all 3×3 symmetric matrices all of whose entries are either 0 or 1. Five of these entries are 1 and four of them are 0. The number of matrices in M is

(A) 12, (B) 6, (C) 9, (D) 3

A

no

mathematics

The number of matrices A in M for which the system of linear equations [x 1 A] [y 0 z 0] has a unique solution, is

(A) less than 4, (B) at least 4 but less than 7, (C) at least 7 but less than 10, (D) at least 10

C

yes

mathematics

The number of matrices A in M for which the system of linear equations [x 1 A] [y 0 z 0] is inconsistent, is

(A) 0, (B) more than 2, (C) 2, (D) 1

B

yes

mathematics

The probability that X ≥3 equals

(A) 25/216 (B) 25/36 (C) 5/36 (D) 125/216

A

no

mathematics