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SECTION – I (Single Correct Choice Type)
This Section contains 8 multiple choice questions. Each question has four choices A), B), C) and D) out of which ONLY ONE is correct.
Q. Incandescent bulbs are designed by keeping in mind that the resistance of their filament increases with the
increase in temperature. If at room temperature, 100 W, 60 W and 40 W bulbs have filament resistances
R_{100}, R_{60} and R_{40}, respectively, the relation between these resistances is
To Verify Ohm’s law, a student is provided with a test resistor R_{T}, a high resistance R_{1}, a small resistance
R_{2}, two identical galvanometers G_{1} and G_{2}, and a variable voltage source V. The correct circuit to carry out
the experiment is
G_{1} is acting as voltmeter and G_{2} is acting as ammeter.
An AC voltage source of variable angular frequency ω and fixed amplitude V_{0} is connected in series with a
capacitance C and an electric bulb of resistance R (inductance zero). When ω is increased
A thin flexible wire of length L is connected to two adjacent fixed points and carries a current I in the
clockwise direction, as shown in the figure. When the system is put in a uniform magnetic field of strength
B going into the plane of the paper, the wire takes the shape of a circle. The tension in the wire is
A block of mass m is on an inclined plane of angle θ. The coefficient of friction between the block and the plane is μ and tan θ > μ. The block is held stationary by applying a force P parallel to the plane. The direction of force pointing up the plane is taken to be positive. As P is varied from P_{1} = mg(sinθ − μ cosθ) to
P_{2} = mg(sinθ + μ cosθ), the frictional force f versus P graph will look like
Initially the frictional force is upwards as P increases frictional force decreases.
A thin uniform annular disc (see figure) of mass M has outer radius 4R and inner radius 3R. The work required to take a unit mass from point P on its axis to infinity is
Consider a thin square sheet of side L and thickness t, made of a material of resistivity ρ. The resistance
between two opposite faces, shown by the shaded areas in the figure is
SECTION – II (Multiple Correct Choice Type)
This section contains 5 multiple choice questions. Each question has four choices A), B), C) and D) out of which ONE OR MORE may be correct. Partially correct answers will be not be awarded any marks.
Q. A point mass of 1 kg collides elastically with a stationary point mass of 5 kg. After their collision, the 1 kg
mass reverses its direction and moves with a speed of 2 ms^{−1}. Which of the following statement(s) is (are)
correct for the system of these two masses?
One mole of an ideal gas in initial state A undergoes a cyclic process ABCA, as shown in the figure. Its pressure at A is P_{0}. Choose the correct option(s) from the following
Process AB is isothermal process
A student uses a simple pendulum of exactly 1m length to determine g, the acceleration due to gravity. He
uses a stop watch with the least count of 1 sec for this and records 40 seconds for 20 oscillations. For this
observation, which of the following statement(s) is (are) true?
A few electric field lines for a system of two charges Q_{1} and Q_{2} fixed at two different points on the xaxis are shown in the figure. These lines suggest that
No. of electric field lines of forces emerging from Q_{1} are larger than terminating at Q_{2}
A ray OP of monochromatic light is incident on the face AB of prism ABCD near vertex B at an incident angle of 60° (see figure). If the refractive index of the material of the prism is , which of the following is (are) correct?
SECTION –III (Paragraph Type)
This section contains 2 paragraphs. Based upon the first paragraph 2 multiple choice questions and based upon the second paragraph 3 multiple choice questions have to be answered. Each of these questions has four choices A), B), C) and D) out of WHICH ONLY ONE CORRECT.
Paragraph for Questions 14 to 15
Q. Electrical resistance of certain materials, known as superconductors, changes abruptly from a nonzero value to zero as their temperature is lowered below a critical temperature T_{c}(0). An interesting property of superconductors is that their critical temperature becomes smaller than T_{c}(0) if they are placed in a magnetic field, i.e., the critical temperature T_{c}(B) is a function of the magnetic field strength B. The dependence of T_{c}(B) on B is shown in the figure.
Q. In the graphs below, the resistance R of a superconductor is shown as a function of its temperature T for
two different magnetic fields B_{1 }(solid line) and B_{2} (dashed line). If B_{2} is larger than B_{1} which of the following graphs shows the correct variation of R with T in these fields?
Larger the magnetic field smaller the critical temperature.
Paragraph for Questions 14 to 15
Q. Electrical resistance of certain materials, known as superconductors, changes abruptly from a nonzero value to zero as their temperature is lowered below a critical temperature T_{c}(0). An interesting property of superconductors is that their critical temperature becomes smaller than T_{c}(0) if they are placed in a magnetic field, i.e., the critical temperature T_{c}(B) is a function of the magnetic field strength B. The dependence of T_{c}(B) on B is shown in the figure.
Q. A superconductor has T_{c} (0) = 100 K. When a magnetic field of 7.5 Tesla is applied, its T_{c} decreases to
75 K. For this material one can definitely say that when
Paragraph for Questions 16 to 18
When a particle of mass m moves on the xaxis in a potential of the form V(x) = kx^{2} it performs simple harmonic motion. The corresponding time period is proportional to , as can be seen easily using dimensional analysis. However, the motion of a particle can be periodic even when its potential energy increases on both sides of x = 0 in a way different from kx^{2} and its total energy is such that the particle does not escape to infinity. Consider a particle of mass m moving on the xaxis. Its potential energy is V(x) = αx^{4} (α > 0) for x near the origin and becomes a constant equal to V_{0} for x ≥ X_{0} (see figure).
Q. If the total energy of the particle is E, it will perform periodic motion only if
Energy must be less than V_{0}
Paragraph for Questions 16 to 18
When a particle of mass m moves on the xaxis in a potential of the form V(x) = kx^{2} it performs simple harmonic motion. The corresponding time period is proportional to , as can be seen easily using dimensional analysis. However, the motion of a particle can be periodic even when its potential energy increases on both sides of x = 0 in a way different from kx^{2} and its total energy is such that the particle does not escape to infinity. Consider a particle of mass m moving on the xaxis. Its potential energy is V(x) = αx^{4} (α > 0) for x near the origin and becomes a constant equal to V_{0} for x ≥ X_{0} (see figure).
Q. For periodic motion of small amplitude A, the time period T of this particle is proportional to
Paragraph for Questions 16 to 18
When a particle of mass m moves on the xaxis in a potential of the form V(x) = kx^{2} it performs simple harmonic motion. The corresponding time period is proportional to , as can be seen easily using dimensional analysis. However, the motion of a particle can be periodic even when its potential energy increases on both sides of x = 0 in a way different from kx^{2} and its total energy is such that the particle does not escape to infinity. Consider a particle of mass m moving on the xaxis. Its potential energy is V(x) = αx^{4} (α > 0) for x near the origin and becomes a constant equal to V_{0} for x ≥ X_{0} (see figure).
Q. The acceleration of this particle for x > X_{0} is
As potential energy is constant for x > X_{0}, the force on the particle is zero hence acceleration is zero.
SECTION –IV (Integer Type)
This section contains TEN questions. The answer to each question is a single digit integer ranging from 0 to 9.
Enter only the numerical value in the space provided below.
Q. Gravitational acceleration on the surface of a planet is where g is the gravitational acceleration on
the surface of the earth. The average mass density of the planet is times that of the earth. If the escape
speed on the surface of the earth is taken to be 11 kms^{1}, the escape speed on the surface of the planet in
kms^{1} will be
A piece of ice (heat capacity = 2100 J kg^{1} °C^{1} and latent heat = 3.36 ×105J kg^{1}) of mass m grams is at
5°C at atmospheric pressure. It is given 420 J of heat so that the ice starts melting. Finally when the
icewater mixture is in equilibrium, it is found that 1 gm of ice has melted. Assuming there is no other heat
exchange in the process, the value of m is
420 = (m × 2100 × 5 + 1 × 3.36 × 105 )×10^{−3}
where m is in gm.
A stationary source is emitting sound at a fixed frequency f_{0}, which is reflected by two cars approaching the
source. The difference between the frequencies of sound reflected from the cars is 1.2% of f_{0}. What is the
difference in the speeds of the cars (in km per hour) to the nearest integer ? The cars are moving at constant
speeds much smaller than the speed of sound which is 330 ms^{1}.
The focal length of a thin biconvex lens is 20 cm. When an object is moved from a distance of 25 cm in
front of it to 50 cm, the magnification of its image changes from m25 to m50. The ratio
An α particle and a proton are accelerated from rest by a potential difference of 100 V. After this, their de
Broglie wavelengths are λ_{α} and λ_{p} respectively. The ratio , to the nearest integer, is
When two identical batteries of internal resistance 1Ω each are connected in series across a resistor R, the
rate of heat produced in R is J_{1}. When the same batteries are connected in parallel across R, the rate is J_{2}.
If J_{1} = 2.25 J_{2} then the value of R in Ω is
Two spherical bodies A (radius 6 cm ) and B(radius 18 cm ) are at temperature T_{1} and T_{2}, respectively.
The maximum intensity in the emission spectrum of A is at 500 nm and in that of B is at 1500 nm.
Considering them to be black bodies, what will be the ratio of the rate of total energy radiated by A to that
of B?
When two progressive waves y_{1} = 4 sin(2x  6t) and are superimposed, the
amplitude of the resultant wave is
Two waves have phase difference π/2.
A 0.1 kg mass is suspended from a wire of negligible mass. The length of the wire is 1m and its crosssectional area is 4.9 × 10^{7} m^{2}. If the mass is pulled a little in the vertically downward direction and
released, it performs simple harmonic motion of angular frequency 140 rad s^{−1}. If the Young’s modulus of
the material of the wire is n × 10^{9} Nm^{2}, the value of n is
A binary star consists of two stars A (mass 2.2M_{s}) and B (mass 11M_{s}), where M_{s} is the mass of the sun.
They are separated by distance d and are rotating about their centre of mass, which is stationary. The ratio
of the total angular momentum of the binary star to the angular momentum of star B about the centre of
mass is
SECTION – I (Single Correct Choice Type)
This Section contains 8 multiple choice questions. Each question has four choices A), B), C) and D) out of which ONLY ONE is correct.
Q. The synthesis of 3 – octyne is achieved by adding a bromoalkane into a mixture of sodium amide and an
alkyne. The bromoalkane and alkyne respectively are
The correct statement about the following disaccharide is
Plots showing the variation of the rate constant (k) with temperature (T) are given below. The plot that
follows Arrhenius equation is
The species which by definition has ZERO standard molar enthalpy of formation at 298 K is
Cl_{2} is gas at 298 K while Br_{2} is a liquid.
The bond energy (in kcal mol^{–1} ) of a C – C single bond is approximately
The correct structure of ethylenediaminetetraacetic acid (EDTA) is
Cl^{–} is replaced by 2NO^{−} in ionization sphere.
SECTION – II (Multiple Correct Choice Type)
This section contains 5 multiple choice questions. Each question has four choices A), B), C) and D) out of which ONE OR MORE may be correct.
Q. In the Newman projection for 2,2dimethylbutane
X and Y can respectively be
Aqueous solutions of HNO_{3}, KOH, CH_{3}COOH, and CH_{3}COONa of identical concentrations are provided.
The pair (s) of solutions which form a buffer upon mixing is(are)
In option (C), if HNO_{3} is present in limiting amount then this mixture will be a buffer. And the mixture given in option (D), contains a weak acid (CH_{3}COOH) and its salt with strong base NaOH, i.e. CH_{3}COONa.
In the reaction the intermediate(s) is(are)
Phenoxide ion is para* and ortho directing. (* preferably)
The reagent(s) used for softening the temporary hardness of water is(are)
Among the following, the intensive property is (properties are)
Resistance and heat capacity are mass dependent properties, hence extensive.
SECTIONIII (Paragraph Type)
This section contains 2 paragraphs. Based upon the first paragraph 2 multiple choice questions and based upon the second paragraph 3 multiple choice questions have to be answered. Each of these questions has four choices A), B), C) and D) out of WHICH ONLY ONE CORRECT.
Paragraph for Question Nos. 42 to 43
The concentration of potassium ions inside a biological cell is at least twenty times higher than the outside.
The resulting potential difference across the cell is important in several processes such as transmission of nerve impulses and maintaining the ion balance. A simple model for such a concentration cell involving a metal M is:
For the above electrolytic cell the magnitude of the cell potential E_{cell}= 70 mV.
Q. For the above cell
Paragraph for Question Nos. 42 to 43
The concentration of potassium ions inside a biological cell is at least twenty times higher than the outside.
The resulting potential difference across the cell is important in several processes such as transmission of nerve impulses and maintaining the ion balance. A simple model for such a concentration cell involving a metal M is:
For the above electrolytic cell the magnitude of the cell potential E_{cell}= 70 mV.
Q. If the 0.05 molar solution of M^{+} is replaced by 0.0025 molar M^{+} solution, then the magnitude of the cell
potential would be
Paragraph for Question Nos. 44 to 46
Copper is the most noble of the first row transition metals and occurs in small deposits in several countries. Ores of copper include chalcanthite (CuSO_{4} ⋅5H_{2}O) , atacamite (Cu_{2}Cl(OH)_{3}) , cuprite (Cu_{2}O) , copper glance (Cu_{2}S) and malachite (Cu_{2}(OH)_{2}CO_{3}). However, 80% of the world copper production comes from the ore of chalcopyrite (CuFeS_{2}) . The extraction of copper from chalcopyrite involves partial roasting, removal of iron and selfreduction.
Q. Partial roasting of chalcopyrite produces
Paragraph for Question Nos. 44 to 46
Copper is the most noble of the first row transition metals and occurs in small deposits in several countries. Ores of copper include chalcanthite (CuSO_{4} ⋅5H_{2}O) , atacamite (Cu_{2}Cl(OH)_{3}) , cuprite (Cu_{2}O) , copper glance (Cu_{2}S) and malachite (Cu_{2}(OH)_{2}CO_{3}). However, 80% of the world copper production comes from the ore of chalcopyrite (CuFeS_{2}) . The extraction of copper from chalcopyrite involves partial roasting, removal of iron and selfreduction.
Q. Iron is removed from chalcopyrite as
Paragraph for Question Nos. 44 to 46
Copper is the most noble of the first row transition metals and occurs in small deposits in several countries. Ores of copper include chalcanthite (CuSO_{4} ⋅5H_{2}O) , atacamite (Cu_{2}Cl(OH)_{3}) , cuprite (Cu_{2}O) , copper glance (Cu_{2}S) and malachite (Cu_{2}(OH)_{2}CO_{3}). However, 80% of the world copper production comes from the ore of chalcopyrite (CuFeS_{2}) . The extraction of copper from chalcopyrite involves partial roasting, removal of iron and selfreduction.
Q. In selfreduction, the reducing species is
SECTIONIV (Integer Type)
This section contains TEN questions. The answer to each question is a single digit integer ranging from 0 to 9.
Enter only the numerical value in the space provided below.
Q. A student performs a titration with different burettes and finds titre values of 25.2 mL, 25.25 mL and 25.0
mL. The number of significant figures in the average titre value is
The concentration of R in the reaction R → P was measured as a function of time and the following data is
obtained:
The order of the reaction is
The number of neutrons emitted when undergoes controlled nuclear fission to is
The total number of basic groups in the following form of lysine is
The total number of cyclic isomers possible for a hydrocarbon with the molecular formula C_{4}H_{6} is
In C_{4}H_{6}, possible cyclic isomers are
In the scheme given below, the total number of intra molecular aldol condensation products formed from ‘Y’ is
Amongst the following, the total number of compound soluble in aqueous NaOH is
These four are soluble in aqueous NaOH.
Amongst the following, the total number of compounds whose aqueous solution turns red litmus paper blue
is
KCN K_{2}SO_{4} (NH_{4})_{2}C_{2}O_{4} NaCl Zn(NO_{3})_{2} FeCl_{3} K_{2}CO_{3} NH_{4}NO_{3} LiCN
KCN, K_{2}CO_{3}, LiCN are basic in nature and their aqueous solution turns red litmus paper blue.
Based on VSEPR theory, the number of 90 degree F−Br−F angles in BrF_{5} is
The value of n in the molecular formula Be_{n}Al_{2}Si_{6}O_{18} is
Be_{3}Al_{2}Si_{6}O_{18} (Beryl)
(according to charge balance in a molecule)
2n + 6 + 24 − 36 = 0
n = 3
SECTION – I (Single Correct Choice Type)
This Section contains 8 multiple choice questions. Each question has four choices A), B), C) and D) out of which ONLY ONE is correct.
Q. Let ω be a complex cube root of unity with ω ≠ 1. A fair die is thrown three times. If r_{1}, r_{2} and r_{3} are the
numbers obtained on the die, then the probability that ω^{r1} + ω^{r2} + ω^{r3 }= 0 is
Let P, Q, R and S be the points on the plane with position vectors respectively. The quadrilateral PQRS must be a
Hence, PQRS is a parallelogram but not rhombus or rectangle.
The number of 3 × 3 matrices A whose entries are either 0 or 1 and for which the system has
exactly two distinct solutions, is
Three planes cannot intersect at two distinct points.
Let p and q be real numbers such that p ≠ 0, p^{3} ≠ q and p^{3} ≠ − q. If α and β are nonzero complex numbers
satisfying α + β = − p and α^{3} + β^{3} = q, then a quadratic equation having as its roots is
Let f, g and h be realvalued functions defined on the interval [0, 1] by f(x) = ,
g(x) = If a, b and c denote, respectively, the absolute maximum of f, g and h on [0, 1], then
If the angles A, B and C of a triangle are in an arithmetic progression and if a, b and c denote the lengths of
the sides opposite to A, B and C respectively, then the value of the expression is
Equation of the plane containing the straight line and perpendicular to the plane containing the
straight lines is
SECTION − II (Multiple Correct Choice Type)
This section contains 5 multiple choice questions. Each question has four choices A), B), C) and D) out of which ONE OR MORE may be correct.
Q. Let z_{1} and z_{2} be two distinct complex numbers and let z = (1 − t) z_{1} + tz_{2} for some real number t with 0 < t < 1. If Arg (w) denotes the principal argument of a nonzero complex number w, then
Let ABC be a triangle such that ∠ACB = and let a, b and c denote the lengths of the sides opposite to A,
B and C respectively. The value(s) of x for which a = x^{2} + x + 1, b = x^{2} − 1 and c = 2x + 1 is (are)
Let A and B be two distinct points on the parabola y^{2} = 4x. If the axis of the parabola touches a circle of radius r having AB as its diameter, then the slope of the line joining A and B can be
Let f be a realvalued function defined on the interval Then which of the following statement(s) is (are) true?
SECTION − III (Paragraph Type)
This section contains 2 paragraphs. Based upon the first paragraph 2 multiple choice questions and based upon the second paragraph 3 multiple choice questions have to be answered. Each of these questions has four choices A), B), C) and D) out of WHICH ONLY ONE CORRECT.
Paragraph for Questions 70 to 71
The circle x^{2} + y^{2} − 8x = 0 and hyperbola intersect at the points A and B.
Q. Equation of a common tangent with positive slope to the circle as well as to the hyperbola is
Paragraph for Questions 70 to 71
The circle x^{2} + y^{2} − 8x = 0 and hyperbola intersect at the points A and B.
Q. Equation of the circle with AB as its diameter is
Paragraph for Questions 72 to 74
Let p be an odd prime number and T_{p} be the following set of 2 × 2 matrices :
Q. The number of A in T_{p} such that A is either symmetric or skewsymmetric or both, and det(A) divisible by
p is
Paragraph for Questions 72 to 74
Let p be an odd prime number and T_{p} be the following set of 2 × 2 matrices :
Q. The number of A in Tp such that the trace of A is not divisible by p but det (A) is divisible by p is
[Note: The trace of a matrix is the sum of its diagonal entries.]
Paragraph for Questions 72 to 74
Let p be an odd prime number and T_{p} be the following set of 2 × 2 matrices :
Q. The number of A in Tp such that det (A) is not divisible by p is
SECTION − IV (Integer Type)
This section contains TEN questions. The answer to each question is a single digit integer ranging from 0 to 9.
Enter only the numerical value in the space provided below.
Q. Let S_{k}, k = 1, 2, ….. , 100, denote the sum of the infinite geometric series whose first term is and the common ratio is . Then the value of is
The number of all possible values of θ, where 0 < θ < π, for which the system of equations
have a solution (x_{0}, y_{0}, z_{0}) with y_{0}z_{0} ≠ 0, is
Let f be a realvalued differentiable function on R (the set of all real numbers) such that f(1) = 1. If the y intercept of the tangent at any point P(x, y) on the curve y = f(x) is equal to the cube of the abscissa of P,
then the value of f(−3) is equal to
The number of values of θ in the interval , such that for n = 0, ±1, ±2 and tanθ = cot 5θ as well as sin 2θ = cos 4θ is
If are vectors in space given by then the value of is
The line 2x + y = 1 is tangent to the hyperbola . If this line passes through the point of intersection of the nearest directrix and the xaxis, then the eccentricity of the hyperbola is
If the distance between the plane Ax − 2y + z = d and the plane containing the lines
and , then d is
For any real number x, let x denote the largest integer less than or equal to x. Let f be a real valued
function defined on the interval [−10, 10] by
Then the value of
Let ω be the complex number . Then the number of distinct complex numbers z satisfying is equal to
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