Q. 1  10 carry 3 marks each.
Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE THAN ONE are correct.
Q.
A student is performing an experiment using a resonance column and a tuning fork of frequency
244s^{1 }. He is told that the air in the tube has been replaced by another gas (assume that the column remains filled with the gas). If the minimum height at which resonance occurs is (0.350 0.005)m , the gas in the tube is
(Useful information: The molar masses M in grams are given in the options. Take the values of for each gas as given there.)
Calculations for for gases mentioned in options A, B, C and D, work out to be 0.459 m, 0.363 m 0.340 m & 0.348 m respectively. As . Hence correct option is D.
At time t = 0, terminal A in the circuit shown in the figure is connected to B by a key and an alternating
current I(t) = I0 cos (ωt), with I0 = 1A and ω = 500 rad/s starts flowing in it with the initial direction shown
in the figure. At the key is switched from B to D. Now onwards only A and D are connected. A
total charge Q flows from the battery to charge the capacitor fully. If C = 20 μF, R = 10 ? and the battery
is ideal with emf of 50 V, identify the correct statement(s).
As current leads voltage by π/2 in the given circuit initially, then ac voltage can be represent as
A parallel plate capacitor has a dielectric slab of dielectric constant K between its plates that
covers 1/3 of the area of its plates, as shown in the figure. The total capacitance of the
capacitor is C while that of the portion with dielectric in between is C_{1}. When the capacitor is
charged, the plate area covered by the dielectric gets charge Q_{1} and the rest of the area gets
charge Q_{2}. The electric field in the dielectric is E_{1} and that in the other portion is E_{2}. Choose
the correct option/options, ignoring edge effects.
As E = V/d
E_{1}/E_{2} = 1 (both parts have common potential difference)
Assume C_{0} be the capacitance without dielectric for whole capacitor.
One end of a taut string of length 3m along the x axis is fixed at x = 0. The speed of the waves in the string
is 100 ms^{1}. The other end of the string is vibrating in the y direction so that stationary waves are set up in
the string. The possible waveform(s) of these stationary waves is (are)
Taking y(t) = A f(x) g(t) & Applying the conditions:
1; here x = 3m is antinode & x = 0 is node
2; possible frequencies are odd multiple of fundamental frequency.
A transparent thin film of uniform thickness and refractive index n_{1} = 1.4 is coated on the convex spherical surface of radius R at one end of a long solid glass cylinder of refractive index n_{2} = 1.5, as shown in the figure. Rays of light parallel to the axis of the cylinder traversing through the film from air to glass get focused at distance f1 from the film, while rays of light traversing from glass to air get focused at distance f_{2} from the film. Then
Heater of an electric kettle is made of a wire of length L and diameter d. It takes 4 minutes to raise the
temperature of 0.5 kg water by 40 K. This heater is replaced by a new heater having two wires of the same
material, each of length L and diameter 2d. The way these wires are connected is given in the options.
How much time in minutes will it take to raise the temperature of the same amount of water by 40K?
Two ideal batteries of emf V_{1} and V_{2} and three resistances R_{1}, R_{2} and R_{3} are connected as shown in the figure. The current in resistance R_{2} would be zero if
Let E_{1} (r), E_{2} (r) and E_{3} (r) be the respective electric fields at a distance r from a point charge Q, an
infinitely long wire with constant linear charge density λ, and an infinite plane with uniform surface charge
density σ. If E_{1}(r_{0} ) = E_{2} (r_{0} ) = E_{3} (r_{0} ) at a given distance r0, then
A light source, which emits two wavelengths λ_{1} = 400 nm and λ_{2} = 600 nm, is used in a Young’s double
slit experiment. If recorded fringe widths for λ_{1} and λ_{2} are β_{1} and β_{2} and the number of fringes for them
within a distance y on one side of the central maximum are m_{1} and m_{2}, respectively, then
In the figure, a ladder of mass m is shown leaning against a wall. It is in static equilibrium making an angle
θ with the horizontal floor. The coefficient of friction between the wall and the ladder is μ1 and that
between the floor and the ladder is μ_{2}. The normal reaction of the wall on the ladder is N1 and that of the
floor is N_{2}. If the ladder is about to slip, then
Condition of translational equilibrium
Applying torque equation about corner (left) point on the floor
Q. No. 11  20 carry 3 Marks each
Each question, when worked out will result in one integer from 0 to 9 (both inclusive).
Q.
During Searle’s experiment, zero of the Vernier scale lies between 3.20 x 10^{2} m and 3.25 x 10^{2} m of the
main scale. The 20th division of the Vernier scale exactly coincides with one of the main scale divisions.
When an additional load of 2 kg is applied to the wire, the zero of the Vernier scale still lies between 3.20 x
10^{2} m and 3.25 x10^{2} m of the main scale but now the 45th division of Vernier scale coincides with one of
the main scale divisions. The length of the thin metallic wire is 2 m and its crosssectional area is 8 x 10^{7}
m^{2}. The least count of the Vernier scale is 1.0 x 10^{5} m. The maximum percentage error in the Young’s
modulus of the wire is
change in lengths = 25(LC)
and the maximum permissible error in elongation is one LC
Airplanes A and B are flying with constant velocity in the same vertical plane at angles 30^{0} and 60^{0} with
respect to the horizontal respectively as shown in the figure. The speed of A is 100√3 ms^{1}. At time t = 0
s, an observer in A finds B at a distance of 500 m. This observer sees B moving with a constant velocity
perpendicular to the line of motion of A. If at t = t_{0}, A just escapes being hit by B, t_{0} in seconds is
The relative velocity of B with respect to A is perpendicular to line of motion of A.
A thermodynamic system is taken from an initial state i with internal energy Ui = 100 J to the final state f
along two different paths iaf and ibf, as schematically shown in the figure. The work done by the system
along the paths af, ib and bf are W_{af} = 200 J, W_{ib} = 50 J and W_{bf} = 100 J respectively. The heat supplied to
the system along the path iaf, ib and bf are iaf ib Q , Q and Q_{bf} respectively. If the internal energy of the
system in the state b is U_{b} = 200 J and Q_{iaf} = 500 J, the ratio Q_{bf} / Q_{ib} is
U_{b} = 200 J, U_{i} =100 J
Process iaf
Two parallel wires in the plane of the paper are distance X_{0} apart. A point charge is moving with speed u
between the wires in the same plane at a distance X_{1} from one of the wires. When the wires carry current of
magnitude I in the same direction, the radius of curvature of the path of the point charge is R_{1}. In contrast,
if the currents I in the two wires have directions opposite to each other, the radius of curvature of the path is R_{2}. id X_{0}/X_{1 } =3 the value of R_{1}/ R_{2 }is
To find the distance d over which a signal can be seen clearly in foggy conditions, a railways engineer uses
dimensional analysis and assumes that the distance depends on the mass density ρ of the fog, intensity
(power/area) S of the light from the signal and its frequency f. The engineer finds that d is proportional to
S^{1/n}. The value of n is
A rocket is moving in a gravity free space with a constant acceleration of 2 ms^{–2} along + x direction (see figure). The length of a chamber inside the rocket is 4 m. A ball is thrown from the left end of the chamber in + x direction with a speed of 0.3 ms^{–1} relative to the rocket. At the same time, another ball is thrown in +x direction with a speed of 0.2 ms^{–1} from its right end relative to the rocket. The time in seconds when the two balls hit each other is
Maximum displacement of the left ball from the left wall of the chamber is 2.25 cm, so the right ball has to travel almost the whole length of the chamber (4m) to hit the left ball. So the time taken by the right ball is
1.9 sec (approximately 2 sec)
A galvanometer gives full scale deflection with 0.006 A current. By connecting it to a 4990 resistance, it can be converted into a voltmeter of range 0 – 30 V. If connected to a 2n/249 resistance, it becomes an ammeter of range 0 – 1.5 A. The value of n is
A uniform circular disc of mass 1.5 kg and radius 0.5 m isinitially at rest on a horizontal frictionless surface. Three forces of equal magnitude F = 0.5 N are applied simultaneously along the three sides of an equilateral
triangle XYZ with its vertices on the perimeter of the disc (see figure). One second after applying the forces, the angular speed of the disc in rad s^{–1} is
A horizontal circular platform of radius 0.5 m and mass 0.45 kg is free to rotate about its axis. Two massless spring toyguns, each carrying a steel ball of mass 0.05 kg are attached to the platform at a distance 0.25 m from the centre on its either sides along its diameter (see figure). Each gun simultaneously fires the balls horizontally and perpendicular to the diameter in opposite directions. After leaving the platform, the balls have horizontal speed of 9 ms^{–1} with respect to the ground. The rotational speed of the platform in rad s^{–1} after the balls leave the platform is
Since net torque about centre of rotation is zero, so we can apply conservation of angular momentum of the
system about center of disc
L_{i} = L_{f}
0 = Iω + 2mv (r/2); comparing magnitude
Consider an elliptically shaped rail PQ in the vertical plane with OP = 3 m and OQ = 4 m. A block of mass 1 kg is pulled along the rail from P to Q with a force of 18 N, which is always parallel to line PQ (see the figure given). Assuming no frictional losses, the kinetic energy of the block when it reaches Q is (n×10) Joules. The
value of n is (take acceleration due to gravity = 10 ms^{–2})
Q. No. 21 30 carry 3 mark each.
Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE THAN ONE are correct.
Q. The correct combination of names for isomeric alcohols with molecular formula C_{4}H_{10}O is/are
An ideal gas in a thermally insulated vessel at internal pressure = P_{1}, volume = V_{1} and absolute
temperature = T_{1} expands irreversibly against zero external pressure, as shown in the diagram. The final
internal pressure, volume and absolute temperature of the gas are P_{2}, V_{2} and T_{2}, respectively. For this
expansion,
Since container is thermally insulated. So, q = 0, and it is a case of free expansion therefore W = 0 and
ΔE = 0
So, T_{1} = T_{2}
Also, P_{1}V_{1} = P_{2}V_{2}
Hydrogen bonding plays a central role in the following phenomena:
(A) Ice has cagelike structure in which each water molecule is surrounded by four other water molecules tetrahedrally through hydrogen boding, due to this density of ice is less than water and it floats in
water.
The cation (I) more stabilized through hydrogen boding than cation (II) . So, R–NH_{2} is better base than (R)_{3}N in aqueous solution.
(C) HCOOH is stronger acid than CH_{3}COOH due to inductive effect and not due to hydrogen bonding.
(D) Acetic acid dimerizes in benzene through intermolecular hydrogen bonding.
In a galvanic cell, the salt bridge
For the reaction:
The correct statement(s) in the balanced equation is/are:
The balanced equation is,
The reactivity of compound Z with different halogens under appropriate conditions is given below:
The observed pattern of electrophilic substitution can be explained by
The correct statement(s) for orthoboric acid is/are
(A) H_{3}BO_{3} is a weak monobasic Lewis acid.
(B) Equilibrium (i) is shifted in forward direction by the addition of syndiols like ethylene glycol which forms a stable complex with
(C) It has a planar sheet like structure due to hydrogen bonding.
(D) H_{3}BO_{3} is a weak electrolyte in water.
Upon heating with Cu_{2}S, the reagent(s) that give copper metal is/are
The pair(s) of reagents that yield paramagnetic species is/are
(A) sodium (Na) when dissolved in excess liquid ammonia, forms a blue coloured paramagnetic solution.
(B)
(c)
(D)
In the reaction shown below, the major product(s) formed is/are
Only amines undergo acetylation and not acid amides
Q. No. 31  40 Carry 3 marks each
Each question, when worked out will result in one integer from 0 to 9 (both inclusive).
Q.
Among PbS, CuS, HgS, MnS, Ag_{2}S, NiS, CoS, Bi_{2}S_{3} and SnS_{2}, the total number of BLACK coloured
sulphides is
Black coloured sulphides (PbS,CuS,HgS,Ag_{2}S,NiS,CoS)
* Bi_{2}S_{3} in its crystalline form is dark brown but Bi_{2}S_{3} precipitate obtained is black in colour.
The total number(s) of stable conformers with nonzero dipole moment for the following compound is(are)
Consider the following list of reagents:
Acidified K_{2}Cr_{2}O_{7} , alkaline KMnO_{4} ,CuSO_{4},H_{2}O_{2} ,Cl_{2} ,O_{3},FeCl_{3},HNO_{3} and Na_{2}S_{2}O_{3} .
The total number of reagents that can oxidise aqueous iodide to iodine is
A list of species having the formula XZ4 is given below.
Defining shape on the basis of the location of X and Z atoms, the total number of species having a square
planar shape is
Consider all possible isomeric ketones, including stereoisomers of MW = 100. All these isomers are
independently reacted with NaBH_{4} (NOTE: stereoisomers are also reacted separately). The total number of
ketones that give a racemic product(s) is/are
In an atom, the total number of electrons having quantum numbers n = 4, m_{l} = 1 and m_{s} = –1/2 is
If the value of Avogadro number is 6.023 x 1023 mol^{–1} and the value of Boltzmann constant is
1.380x10^{23} JK^{1} , then the number of significant digits in the calculated value of the universal gas constant
is
MX_{2} dissociates in M^{2+} and X^{} ions in an aqueous solution, with a degree of dissociation (α) of 0.5. The
ratio of the observed depression of freezing point of the aqueous solution to the value of the depression of
freezing point in the absence of ionic dissociation is
The total number of distinct naturally occurring amino acids obtained by complete acidic hydrolysis of the
peptide shown below is
This peptide on complete hydrolysis produced 4 distinct amino acids which are given below:
A compound H_{2}X with molar weight of 80g is dissolved in a solvent having density of 0.4 gml^{–1}. Assuming
no change in volume upon dissolution, the molality of a 3.2 molar solution is
Q. No. 41 to 50 carry 3 marks each.
Each question has four choices (A), (B), (C) and (D) out of which ONE or MORE THAN ONE are correct.
Q.
Let be given by
Hence, an odd function
Let a R and let f : R → R be given by f (x) = x^{5}  5x + a, then
For every pair of continuous functions , g : [0, 1] → R such that max{ f (x) : x [0, 1]} = max{g(x) : x
[0, 1]}, the correct statement(s) is(are)
Let f (x) and g (x) achieve their maximum value at x_{1} and x_{2} respectively
h (x) = f (x)  g (x)
h (x1) = f (x_{1})  g (x_{1}) 0
h (x2) = f (x_{2})  g (x_{2}) 0
h (c) = 0 where c [0, 1] f (c) = g (c).
A circle S passes through the point (0, 1) and is orthogonal to the circles (x  1)^{2} + y^{2} = 16 and x^{2} + y^{2} = 1.
Then
Let be three vectors each of magnitude √2 and the angle between each pair of them is π/3. If is a nonzero vector perpendicular to is a nonzero vector perpendicular to and
From a point P(λ, λ, λ), perpendiculars PQ and PR are drawn respectively on the lines y = x, z = 1 and y = x, z = 1. If P is such that QPR is a right angle, then the possible value(s) of λ is(are)
Let M be a 2 x 2 symmetric matrix with integer entries. Then M is invertible if
Let M and N be two 3 x 3 matrices such that MN = NM. Further, if M N^{2} and M^{2} = N^{4}, then
Let be a continuous function and let g : R → R be defined as
Then
Hence g(x) is not differentiable at x = a
Similarly LHD at x = b is greater than 1
g(x) is not differentiable at x = b
Let be given by f(x) = (log(sec x + tan x))^{3}. Then
Each question, when worked out will result in one integer from 0 to 9 (both inclusive).
Q.
Let n_{1} < n_{2} < n_{3} < n_{4} < n_{5} be positive integers such that n_{1} + n_{2} + n_{3} + n_{4} + n_{5} = 20. Then the number of such distinct arrangements (n_{1}, n_{2}, n_{3}, n_{4}, n_{5}) is _________
When n_{5} takes value from 10 to 6 the carry forward moves from 0 to 4 which can be arranged in
Let n 2 be an integer. Take n distinct points on a circle and join each pair of points by a line segment.
Colour the line segment joining every pair of adjacent points by blue and the rest by red. If the number of
red and blue line segments are equal, then the value of n is _________
Let f : R → R and g : R → R be respectively given by by
Then number of points at which h(x) is not differentiable is __________
Let a, b, c be positive integers such that b/a is an integer. If a, b, c are in geometric progression and the
arithmetic mean of a, b, c is b + 2, then the value of
Let be three noncoplanar unit vectors such that the angle between every pair of them is π/3 . if where p, q and r are scalars, then the value of is ___________________
The slope of the tangent to the curve at the point (1, 3) is ______
The value of
The largest value of the nonnegative integer a for which
Let be defined by f (x) = cos^{1}(cos x). The number of points x [0, 4π] satisfying the
equation
For a point P in the plane, let d_{1}(P) and d_{2}(P) be the distances of the point P from the lines x  y = 0 and x +
y = 0 respectively. The area of the region R consisting of all points P lying in the first quadrant of the plane
and satisfying is ___________________
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