question: A thin uniform rod, pivoted at O, is rotating in the horizontal plane with constant angular speed ω, as shown in the figure. At time t = 0, a small insect starts from O and moves with constant speed v with respect to the rod towards the other end. It reaches the end of the rod at t = T and stops. The angular speed of the system remains ω throughout. The magnitude of the torque (|τ|) on the system about O, as a function of time is best represented by which plot?

response_choices: A, B, C, D

answer: B

image_available: yes

question_subject: physics


question: Three very large plates of same area are kept parallel and close to each other. They are considered as ideal black surfaces and have very high thermal conductivity. The first and third plates are maintained at temperatures 2T and 3T respectively. The temperature of the middle (i.e. second) plate under steady state condition is

answer: C

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question_subject: physics


question: Consider a thin spherical shell of radius R with its centre at the origin, carrying uniform positive surface charge density. The variation of the magnitude of the electric field | E(r)| and the electric potential V(r) with the distance r from the centre, is best represented by which graph ?

response_choices: (A) |E(r)| and V(r) graphs (B) |E(r)| and V(r) graphs (C) |E(r)| and V(r) graphs (D) |E(r)| and V(r) graphs

answer: D

image_available: yes

question_subject: physics


question: In the determination of Young's modulus (Y = 4ML/πd^3) by using Searle's method, a wire of length L = 2 m and diameter d = 0.5 mm is used. For a load M = 2.5 kg, an extension l = 0.25 mm in the length of the wire is observed. Quantities d and l are measured using a screw gauge and a micrometer, respectively. They have the same pitch of 0.5 mm. The number of divisions on their circular scale is 100. The contributions to the maximum probable error of the Y measurement

response_choices: (A) due to the errors in the measurements of d and l are the same. (B) due to the error in the measurement of d is twice that due to the error in the measurement of l. (C) due to the error in the measurement of l is twice that due to the error in the measurement of d. (D) due to the error in the measurement of d is four times that due to the error in the measurement of l.

answer: A

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question_subject: physics


question: A small block is connected to one end of a massless spring of un-stretched length 4.9 m. The other end of the spring (see the figure) is fixed. The system lies on a horizontal frictionless surface. The block is stretched by 0.2 m and released from rest at t = 0. It then executes simple harmonic motion with angular frequency ω = π/2 rad/s. Simultaneously at t = 0, a small pebble is projected with speed v from point P at an angle of 45° as shown in the figure. Point P is at a horizontal distance of 10 m from O. If the pebble hits the block at t = 1 s, the value of v is (take g = 10 m/s²)

answer: A

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question_subject: physics


question: Young's double slit experiment is carried out by using green, red and blue light, one color at a time. The fringe widths recorded are βG, βR and βB, respectively. Then,

response_choices: (A) βG > βB > βR, (B) βB > βG > βR, (C) βR > βB > βG, (D) βR > βG > βB

answer: D

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question_subject: physics


question: A small mass m is attached to a massless string whose other end is fixed at P as shown in the figure. The mass is undergoing circular motion in the x-y plane with centre at O and constant angular speed ω. If the angular momentum of the system, calculated about O and P are denoted by Lo and Lp respectively, then

response_choices: A) Lo and Lp do not vary with time. B) Lo varies with time while Lp remains constant. C) Lo remains constant while Lp varies with time. D) Lo and Lp both vary with time.

answer: C

image_available: yes

question_subject: physics


question: A mixture of 2 moles of helium gas (atomic mass = 4 amu) and 1 mole of argon gas (atomic mass = 40 amu) is kept at 300 K in a container. The ratio of the r m s speeds (v_rms(helium))/(v_rms(argon)) is

answer: D

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question_subject: physics


question: Two large vertical and parallel metal plates having a separation of 1 cm are connected to a DC voltage source of potential difference X. A proton is released at rest midway between the two plates. It is found to move at 45° to the vertical JUST after release. Then X is nearly

answer: C

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question_subject: physics


question: A bi-convex lens is formed with two thin plano-convex lenses as shown in the figure. Refractive index n of the first lens is 1.5 and that of the second lens is 1.2. Both the curved surfaces are of the same radius of curvature R = 14 cm. For this bi-convex lens, for an object distance of 40 cm, the image distance will be

answer: B

image_available: yes

question_subject: physics


question: A cubical region of side a has its centre at the origin. It encloses three fixed point charges, −q at (0, −a/4,0), +3q at (0,0,0) and −q at (0,+a/4,0). Choose the correct option(s).

response_choices: (A) The net electric flux crossing the plane x = +a/2 is equal to the net electric flux crossing the plane x = − a/2. (B) The net electric flux crossing the plane y = +a/2 is more than the net electric flux crossing the plane y = − a/2. (C) The net electric flux crossing the entire region is q/ε0. (D) The net electric flux crossing the plane z = +a/2 is equal to the net electric flux crossing the plane x = +a/2.

answer: ACD

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question_subject: physics


question: For the resistance network shown in the figure, choose the correct option(s).

response_choices: (A) The current through P Q is zero. (B) I1 = 3 A. (C) The potential at S is less than that at Q. (D) I2 = 2 A.

answer: ABCD

image_available: yes

question_subject: physics


question: A small block of mass of 0.1 kg lies on a fixed inclined plane P Q which makes an angle θ with the horizontal. A horizontal force of 1 N acts on the block through its center of mass as shown in the figure. The block remains stationary if (take g = 10 m/s²)

response_choices: (A) θ = 45° (B) θ > 45° and a frictional force acts on the block towards P. (C) θ > 45° and a frictional force acts on the block towards Q. (D) θ < 45° and a frictional force acts on the block towards Q.

answer: AC

image_available: yes

question_subject: physics


question: Consider the motion of a positive point charge in a region where there are simultaneous uniform electric and magnetic fields E = E0 j and B = B0 i. At time t = 0, this charge has velocity v in the x-y plane, making an angle θ with the x-axis. Which of the following option(s) is(are) correct for time t > 0?

response_choices: (A) If θ = 0°, the charge moves in a circular path in the x-z plane. (B) If θ = 0°, the charge undergoes helical motion with constant pitch along the y-axis. (C) If θ = 10°, the charge undergoes helical motion with its pitch increasing with time, along the y-axis. (D) If θ = 90°, the charge undergoes linear but accelerated motion along the y-axis.

answer: CD

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question_subject: physics


question: A person blows into open-end of a long pipe. As a result, a high-pressure pulse of air travels down the pipe. When this pulse reaches the other end of the pipe,

response_choices: (A) a high-pressure pulse starts traveling up the pipe, if the other end of the pipe is open. (B) a low-pressure pulse starts traveling up the pipe, if the other end of the pipe is open. (C) a low-pressure pulse starts traveling up the pipe, if the other end of the pipe is closed. (D) a high-pressure pulse starts traveling up the pipe, if the other end of the pipe is closed.

answer: BD

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question_subject: physics


question: An infinitely long solid cylinder of radius R has a uniform volume charge density ρ. It has a spherical cavity of radius R/2 with its centre on the axis of the cylinder, as shown in the figure. The magnitude of the electric field at the point P, which is at a distance 2R from the axis of the cylinder, is given by the expression 23ρR/16πε₀. The value of k is

answer: 6

image_available: yes

question_subject: physics


question: A cylindrical cavity of diameter a exists inside a cylinder of diameter 2a as shown in the figure. Both the cylinder and the cavity are infinitely long. A uniform current density J flows along the length. If the magnitude of the magnetic field at the point P is given by N/12 μ₀aJ, then the value of N is

answer: 5

image_available: yes

question_subject: physics


question: A lamina is made by removing a small disc of diameter 2R from a bigger disc of uniform mass density and radius 2R, as shown in the figure. The moment of inertia of this lamina about axes passing through O and P is Io and Ip, respectively. Both these axes are perpendicular to the plane of the lamina. The ratio Ip/Io to the nearest integer is

response_choices: No response choices provided

answer: 3

image_available: yes

question_subject: physics


question: A circular wire loop of radius R is placed in the x-y plane centered at the origin O. A square loop of side a (a << R) having two turns is placed with its center at z = √3 R along the axis of the circular wire loop, as shown in figure. The plane of the square loop makes an angle of 45° with respect to the z-axis. If the mutual inductance between the loops is given by μ0a^2/2√2 R, then the value of p is

response_choices: No response choices provided

answer: 7

image_available: yes

question_subject: physics


question: A proton is fired from very far away towards a nucleus with charge Q = 120 e, where e is the electronic charge. It makes a closest approach of 10 fm to the nucleus. The de Broglie wavelength (in units of fm) of the proton at its start is: (take the proton mass, mp = (5/3) x 10^-27 kg; h/e = 4.2 x 10^15 J.s/C; 4πε0 = 9 x 10^9 m/F; 1 fm = 10^-15 m)

response_choices: No response choices provided

answer: 7

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question_subject: physics


question: In allene (C3H4), the type(s) of hybridisation of the carbon atoms is (are)

response_choices: (A) sp and sp³ (B) sp and sp² (C) only sp² (D) sp² and sp³

answer: B

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question_subject: chemistry


question: For one mole of a van der Waals gas when b = 0 and T = 300 K, the PV vs. 1/V plot is shown below. The value of the van der Waals constant a (atm.liter² mol⁻²) is

answer: C

image_available: yes

question_subject: chemistry


question: The number of optically active products obtained from the complete ozonolysis of the given compound is

answer: A

image_available: yes

question_subject: chemistry


question: A compound M¥Xq has cubic close packing (ccp) arrangement of X. Its unit cell structure is shown below. The empirical formula of the compound is

response_choices: (A) MX, (B) MX₂, (C) M₂X, (D) M₆X₁₄

answer: B

image_available: yes

question_subject: chemistry


question: The number of aldol reaction(s) that occurs in the given transformation is

answer: C

image_available: yes

question_subject: chemistry


question: The colour of light absorbed by an aqueous solution of CuSO4 is

response_choices: (A) orange-red, (B) blue-green, (C) yellow, (D) violet

answer: A

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question_subject: chemistry


question: The carboxyl functional group (–COOH) is present in

response_choices: (A) picric acid, (B) barbituric acid, (C) ascorbic acid, (D) aspirin

answer: D

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question_subject: chemistry


question: The kinetic energy of an electron in the second Bohr orbit of a hydrogen atom is [a0 is Bohr radius]

answer: C

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question_subject: chemistry


question: Which ordering of compounds is according to the decreasing order of the oxidation state of nitrogen?

response_choices: (A) HNO3, NO, NH4Cl, N2, (B) HNO3, NO, N2, NH4Cl, (C) HNO3, NH4Cl, NO, N2, (D) NO, HNO3, NH4Cl, N2

answer: B

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question_subject: chemistry


question: As per IUPAC nomenclature, the name of the complex [Co(H2O)4(NH3)2]Cl3 is

response_choices: (A) Tetraaquadiaminecobalt (III) chloride, (B) Tetraaquadimminecobalt (III) chloride, (C) Diaminetetraaquacobalt (III) chloride, (D) Diamminetraaquacobalt (III) chloride

answer: D

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question_subject: chemistry


question: Identify the binary mixture(s) that can be separated into individual compounds, by differential extraction, as shown in the given scheme.

response_choices: (A) C6H5OH and C6H5COOH, (B) C6H5COOH and C6H5CH3OH, (C) C6H5CH2OH and C6H5OH, (D) C6H5CH2OH and C6H5CH2COOH

answer: BD

image_available: yes

question_subject: chemistry


question: Choose the correct reason(s) for the stability of the lyophobic colloidal particles. (A) Preferential adsorption of ions on their surface from the solution, (B) Preferential adsorption of solvent on their surface from the solution, (C) Attraction between different particles having opposite charges on their surface, (D) Potential difference between the fixed layer and the diffused layer of opposite charges around the colloidal particles

answer: AD

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question_subject: chemistry


question: Which of the following molecules, in pure form, is (are) unstable at room temperature?

response_choices: (A) [Cyclohexane ring], (B) [Benzene ring], (C) [Cycloheptatriene oxide], (D) [Benzene oxide]

answer: BC

image_available: yes

question_subject: chemistry


question: Which of the following hydrogen halides react(s) with AgNO3(aq) to give a precipitate that dissolves in Na2S2O3(aq) ?

response_choices: (A) HCl (B) HF (C) HBr (D) HI

answer: ACD

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question_subject: chemistry


question: For an ideal gas, consider only P-V work in going from an initial state X to the final state Z. The final state Z can be reached by either of the two paths shown in the figure. Which of the following choice(s) is (are) correct ? [Take ∆S as change in entropy and w as work done]

response_choices: (A) ∆Sx→z = ∆Sx→y + ∆Sy→z (B) wx→z = wx→y + wy→z (C) wx→y→z = wx→y (D) ∆Sx→y→z = ∆Sx→y

answer: AC

image_available: yes

question_subject: physics


question: The substituents R1 and R2 for nine peptides are listed in the table given below. How many of these peptides are positively charged at pH = 7.0 ?

answer: 4

image_available: yes

question_subject: chemistry


question: The periodic table consists of 18 groups. An isotope of copper, on bombardment with protons, undergoes a nuclear reaction yielding element X as shown below. To which group, element X belongs in the periodic table ?

answer: 8

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question_subject: chemistry


question: When the following aldohexose exists in its D-configuration, the total number of stereoisomers in its pyranose form is CHO CH2 CHOH CHOH CHOH CH2OH

response_choices: No response choices provided

answer: 8

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question_subject: chemistry


question: 29.2% (w/w) HCl stock solution has a density of 1.25 g mL-1. The molecular weight of HCl is 36.5 g mol-1. The volume (mL) of stock solution required to prepare a 200 mL solution of 0.4 M HCl is

response_choices: No response choices provided

answer: 8

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question_subject: chemistry


question: An organic compound undergoes first-order decomposition. The time taken for its decomposition to 1/8 and 1/10 of its initial concentration are t1/8 and t1/10 respectively. What is the value of [t1/8] / [t1/10] x 10? (take loge2 = 0.3)

response_choices: No response choices provided

answer: 9

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question_subject: chemistry


question: The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that each person gets at least one ball is

answer: B

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question_subject: mathematics


question: Let f(x) = { (x^2 cos (π/x)), x ≠ 0, x ∈ R, 0, x = 0 } then f is

response_choices: (A) differentiable both at x = 0 and at x = 2, (B) differentiable at x = 0 but not differentiable at x = 2, (C) not differentiable at x = 0 but differentiable at x = 2, (D) differentiable neither at x = 0 nor at x = 2

answer: B

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question_subject: mathematics


question: The function f : [0, 3] → [1, 29], defined by f(x) = 2x^3 − 15x^2 + 36x + 1, is

response_choices: (A) one-one and onto, (B) onto but not one-one, (C) one-one but not onto, (D) neither one-one nor onto

answer: B

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question_subject: mathematics


question: If lim (x→∞) (x^2 + x + 1 / x + 1) − ax − b = 4 , then

answer: B

image_available: yes

question_subject: mathematics


question: Let z be a complex number such that the imaginary part of z is nonzero and a = z² + z + 1 is real. Then a cannot take the value

answer: D

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question_subject: mathematics


question: The ellipse E₁ : x²/9 + y²/4 = 1 is inscribed in a rectangle R whose sides are parallel to the coordinate axes. Another ellipse E₂ passing through the point (0, 4) circumscribes the rectangle R. The eccentricity of the ellipse E₂ is

answer: C

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question_subject: mathematics


question: Let P = [aij] be a 3x3 matrix and let Q = [bij], where bij = 2^(i+j-2)*aij for 1 ≤ i, j ≤ 3. If the determinant of P is 2, then the determinant of the matrix Q is

answer: D

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question_subject: mathematics


question: The integral ∫ (sec² x / (sec x + tan x)½) dx equals (for some arbitrary constant K)

answer: C

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question_subject: mathematics


question: The point P is the intersection of the straight line joining the points Q(2,3.5) and R(1, -1, 4) with the plane 5x - 4y - z = 1. If S is the foot of the perpendicular drawn from the point T(2, 1, 4) to QR, then the length of the line segment PS is

answer: A

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question_subject: mathematics


question: The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line 4x - 5y = 20 to the circle x² + y² = 9 is

answer: A

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question_subject: mathematics


question: Let θ, φ∈ [0, 2π] be such that 2 cos θ (1− sin φ) = sin^2 θ (tan (θ/2) + cot (θ/2)) cos φ − 1, tan ( 2π− θ ) > 0 and −1 < sin θ < − √3/2. Then φ cannot satisfy

answer: ACD

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question_subject: mathematics


question: Let S be the area of the region enclosed by y = e^(x^2), y = 0, x = 0, and x = 1. Then

answer: ABD

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question_subject: mathematics


question: A ship is fitted with three engines E1, E2 and E3. The engines function independently of each other with respective probabilities 1/2, 1/4 and 1/4. For the ship to be operational at least two of its engines must function. Let X denote the event that the ship is operational and let X1, X2 and X3 denote respectively the events that the engines E1, E2 and E3 are functioning. Which of the following is (are) true ?

response_choices: (A) P [ X1^c | X ] = 3/16, (B) P (Exactly two engines of the ship are functioning | X) = 7/8, (C) P [ X | X2 ] = 5/16, (D) P [ X | X1 ] = 7/16

answer: BD

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question_subject: mathematics


question: Tangents are drawn to the hyperbola x^2/9 - y^2/4 = 1, parallel to the straight line 2x - y = 1. The points of contact of the tangents on the hyperbola are

answer: AB

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question_subject: mathematics


question: If y (x) satisfies the differential equation y' - y tan x = 2x sec x and y (0) = 0 , then

answer: AD

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question_subject: mathematics


question: Let f : IR → IR be defined as f(x) = [x] + [x^2 - 1]. The total number of points at which f attains either a local maximum or a local minimum is

answer: 5

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question_subject: mathematics


question: The value of 6 + log₂ ( (1/√2) / (4 - (1/√2)) / (4 - (1/√2)) / (4 - (1/√2)) ... ) is

answer: 4

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question_subject: mathematics


question: Let p(x) be a real polynomial of least degree which has a local maximum at x = 1 and a local minimum at x = 3. If p(1) = 6 and p(3) = 2, then p'(0) is

answer: 9

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question_subject: mathematics


question: If ā, b̄ and c̄ are unit vectors satisfying | ā - b̄ |^2 + | b̄ - c̄ |^2 + | c̄ - ā |^2 = 9, then | 2ā + 5b̄ + 5c̄ | is

answer: 3

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question_subject: mathematics


question: Let S be the focus of the parabola y^2 = 8x and let PQ be the common chord of the circle x^2 +y^2 - 2x - 4y = 0 and the given parabola. The area of the triangle PQS is

answer: 4

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question_subject: mathematics