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QUESTION: 1

A current carrying wire heats a metal rod. The wire provides a constant power (P) to the rod. The metal rod is enclosed in an insulated container. It is observed that the temperature (T) in the metal rod changes with time (t) as:

T(t) = T_{0} (1 + βt^{1/4})

where β is a constant with appropriate dimension while T_{0} is a constant with dimension of temperature. The heat capacity of the metal is:

Solution:

QUESTION: 2

A thin spherical insulating shell of radius R carries a uniformly distributed charge such that the potential at its surface is V_{0}. A hole with a small area α4πR^{2} (α<<1) is made on the shell without affecting the rest of the shell. Which one of the following statements is correct?

Solution:

Let charge on the sphere initially be Q.

∴

and charge removed = αQ

QUESTION: 3

Consider a spherical gaseous cloud of mass density ρ(r) in free space where r is the radial distance from its center. The gaseous cloud is made of particles of equal mass m moving in circular orbits about the common center with the same kinetic energy K. The force acting on the particles is their mutual gravitational force. If ρ(r) is constant in time, the particle number density n(r) = ρ(r)/m is :

[G is universal gravitational constant]

Solution:

Let total mass included in a sphere of radius r be M.

For a particle of mass m,

QUESTION: 4

In a radioactive sample, nuclei either decay into stable nuclei with decay constant 4.5 × 10^{–10} per year or into stable nuclei with decay constant 0.5 × 10^{–10} per year. Given that in this sample all the stable nuclei are produced by the nuclei only. In time t × 10^{9} years, if the ratio of the sum of stable nuclei to the radioactive nuclei is 99, the value of t will be: [Given ln 10 = 2.3]

Solution:

Parallel radioactive decay

*Multiple options can be correct

QUESTION: 5

One mole of a monoatomic ideal gas goes through a thermodynamic cycle, as shown in the volume versus temperature (V - T) diagram. The correct statement(s) is/are:

[R is the gas constant]

Solution:

From graph

*Multiple options can be correct

QUESTION: 6

In the circuit shown, initially, there is no charge on capacitors and keys S_{1} and S_{2} are open. The values of the capacitors are C_{1} = 10 μF, C_{2} = 30 μF and C_{3} = C_{4} = 80 μF.

Which of the statement(s) is/are correct?

Solution:

(1) at t = 0, capacitor C_{1} acts as a battery of 4V, C_{4} & C_{3 }of 1/2 V each, C_{2} is shorted Circuit is

(2) and (4)

At steady state,

When capacitor is fully charged it behave as open circuit and current through it zero.

Hence, Charge on each capacitor is same.

Now,

(3) At t = 0, S_{1} is closed, capacitor act as short circuit.

*Multiple options can be correct

QUESTION: 7

A thin convex lens is made of two materials with refractive indices n_{1} and n_{2}, as shown in figure. The radius of curvature of the left and right spherical surfaces are equal. f is the focal length of the lens when n_{1} = n_{2} = n. The focal length is f + Δf when n_{1} = n and n_{2} = n + Δn. Assuming Δn << (n–1) and 1 < n < 2, the correct statement(s) is/are:

Solution:

when n_{1} = n_{2} = n

...(1)

2^{nd} case:

...(2)

(1) Relation between is independent of R so (1) is correct.

(2) 2n – 2 < n because n < 2

*Multiple options can be correct

QUESTION: 8

Let us consider a system of units in which mass and angular momentum are dimensionless. If length has dimension of L, which of the following statement (s) is/are correct?

Solution:

Mass = M^{0}L^{0}T^{0}

L^{2} = T^{1} ...(1)

= M^{0}L^{2}L^{–4} (In new system from equation (1))

= L^{-2}

*Multiple options can be correct

QUESTION: 9

A cylindrical capillary tube of 0.2 mm radius is made by joining two capillaries T1 and T2 of different materials having water contact angles of 0° and 60°, respectively. The capillary tube is dipped vertically in water in two different configurations, case I and II as shown in figure. Which of the following option(s) is(are) correct?

(Surface tension of water = 0.075 N/m, density of water = 1000 kg/m^{3}, take g = 10 m/s^{2})

Solution:

⇒ h_{1} = 75 mm (in T1) [If we assume entire tube of T1]

⇒ mm (in T2) [If we assume entire tube of T2]

Option (1): Since contact angles are different so correction in the height of water column raised in the tube will be different in both the cases, so option (1) is correct

Option (2): If joint is 5 cm is above water surface, then let's say water crosses the joint by height h, then:

⇒ h = –ve, not possible, so liquid will not cross the interface, but angle of contact at the interface will change, to balance the pressure,

So option (2) is wrong.

Option (3): If interface is 8 cm above water then water will not even reach the interface, and water will rise till 7.5 cm only in T1, so option (3) is right.

Option (4): If interface is 5 cm above the water in vessel, then water in capillary will not even reach the interface. Water will reach only till 3.75 cm, so option (4) is right.

*Multiple options can be correct

QUESTION: 10

Two identical moving coil galvanometer have 10 Ω resistance and full scale deflection at 2 µA current. One of them is converted into a voltmeter of 100 mV full scale reading and the other into an Ammeter of 1 mA full scale current using appropriate resistors. These are then used to measure the voltage and current in the Ohm's law experiment with R = 1000 Ω resistor by using an ideal cell. Which of the following statement(s) is/are correct?

Solution:

*Multiple options can be correct

QUESTION: 11

A charged shell of radius R carries a total charge Q. Given φ as the flux of electric field through a closed cylindrical surface of height h, radius r and with its center same as that of the shell. Here, center of the cylinder is a point on the axis of the cylinder which is equidistant from its top and bottom surfaces. Which of the following option(s) is/are correct ? [∈_{0} is the permittivity of free space]

Solution:

For option (1), cylinder encloses the shell, thus option is correct

For option (2),

cylinder perfectly enclosed by shell,

thus φ = 0, so option is correct.

For option (3)

For option (4) :

Flux enclosed by cylinder

*Multiple options can be correct

QUESTION: 12

A conducting wire of parabolic shape, initially y = x^{2}, is moving with velocity in a non-uniform magnetic field as shown in figure. If V_{0}, B_{0}, L and β are positive constants and Δφ is the potential difference developed between the ends of the wire, then the correct statement(s) is/are:

Solution:

y = x^{2}

Δφ will be same if the wire is repalced by the straight wire of length √2L and y = x

∵ range of y remains same

∴ option 1 is correct.

*Answer can only contain numeric values

QUESTION: 13

A block of weight 100 N is suspended by copper and steel wires of same cross sectional area 0.5 cm^{2} and, length √3 m and 1 m, respectively. Their other ends are fixed on a ceiling as shown in figure. The angles subtended by copper and steel wires with ceiling are 30° and 60°, respectively. If elongation in copper wire is (Δl_{C}) and elongation in steel wire is (Δl_{s}), then the ratio is _____.

[Young's modulus for copper and steel are 1 × 10^{11} N/m^{2 }and 2 × 10^{11} N/m^{2} respectively]

Solution:

Let T_{S} = tension in steel wire

T_{C} = Tension in copper wire

in x direction

in y direction

Solving equation (i) & (ii)

We know

On solving above equation

*Answer can only contain numeric values

QUESTION: 14

A planar structure of length L and width W is made of two different optical media of refractive indices n_{1} = 1.5 and n_{2} = 1.44 as shown in figure. If L >> W, a ray entering from end AB will emerge from end CD only if the total internal reflection condition is met inside the structure. For L = 9.6 m, if the incident angle θ is varied, the maximum time taken by a ray to exit the plane CD is t × 10^{–9} s, where t is ____. [Speed of light c = 3 × 10^{8} m/s]

Solution:

For maximum time the ray of light must undergo TIR at all surfaces at minimum angle i.e. θ_{C}

For TIR n_{1}sinθ_{C} = n_{2}

where L = length of tube, D = length of path of light

Time taken by light

*Answer can only contain numeric values

QUESTION: 15

A particle is moved along a path AB-BC-CD-DE-EF-FA, as shown in figure, in presence of a force where x and y are in meter and α = –1 N/m^{–1}. The work done on the particle by this force will be ____ Joule.

Solution:

Similarly,

W_{BC} = 1J

W_{CD} = 0.25J

W_{DE} = 0.5 J

W_{EF} = W_{FA} = 0 J

∴ New work in cycle = 0.75 J

*Answer can only contain numeric values

QUESTION: 16

A parallel plate capacitor of capacitance C has spacing d between two plates having area A. The region between the plates is filled with N dielectric layers, parallel to its plates, each with thickness δ = d/N. The dielectric constant of the m^{th} layer is For a very large N (> 10^{3}), the capacitance C is The value of α will be _____.

[∈_{0} is the permittivity of free space]

Solution:

*Answer can only contain numeric values

QUESTION: 17

A train S1, moving with a uniform velocity of 108 km/h, approaches another train S2 standing on a platform.

An observer O moves with a uniform velocity of 36 km/h towards S2, as shown in figure. Both the trains are blowing whistles of same frequency 120 Hz. When O is 600 m away from S2 and distance between S1 and S2 is 800 m, the number of beats heard by O is ____.

[Speed of the sound = 330 m/s]

Solution:

Frequency observed by O from S_{2}

frequency observed by O from S_{1}

beat frequency = 131.76 – 123.63 = 8.128 ≈ 8.12 to 8.13 Hz

*Answer can only contain numeric values

QUESTION: 18

A liquid at 30°C is poured very slowly into a Calorimeter that is at temperature of 110°C. The boiling temperature of the liquid is 80°C. It is found that the first 5 gm of the liquid completely evaporates. After pouring another 80 gm of the liquid the equilibrium temperature is found to be 50°C. The ratio of the Latent heat of the liquid to its specific heat will be ______ °C.

[Neglect the heat exchange with surrounding]

Solution:

Let m = mass of calorimeter,

x = specific heat of calorimeter

s = specific heat of liquid

L = latent heat of liquid

First 5 g of liquid at 30° is poured to calorimeter at 110°C

∴ m × x × (110 – 80) = 5 × s × (80 × 30) + 5 L

⇒ mx × 30 = 250 s + 5 L ... (i)

Now, 80 g of liquid at 30° is poured into calorimeter at 80°C, the equilibrium temperature reaches to 50°C.

QUESTION: 19

Molar conductivity (∧_{m}) of aqueous solution of sodium stearate, which behaves as a strong electrolyte, is recorded at varying concentrations (c) of sodium stearate. Which one of the following plots provides the correct representation of micelle formation in the solution?

(Critical micelle concentration (CMC) is marked with an arrow in the figures.)

Solution:

QUESTION: 20

The correct order of acid strength of the following carboxylic acids is -

Solution:

I > II > III > IV

QUESTION: 21

Calamine, malachite, magnetite and cryolite, respectively are

Solution:

So correct answer is option(2)

QUESTION: 22

The green colour produced in the borax bead test of a chromium(III) salt is due to -

Solution:

*Multiple options can be correct

QUESTION: 23

Fusion of MnO_{2} with KOH in presence of O_{2} produces a salt W. Alkaline solution of W upon electrolytic oxidation yields another salt X. The manganese containing ions present in W and X, respectively, are Y and Z. Correct statement(s) is (are)

Solution:

∵ In acidic solution; Y undergoes disproportionation reaction

*Multiple options can be correct

QUESTION: 24

Which of the following statement(s) is (are) correct regarding the root mean square speed (U_{rms}) and average translational kinetic energy (ε_{av}) of a molecule in a gas at equilibrium?

Solution:

*Multiple options can be correct

QUESTION: 25

In the decay sequence:

x_{1}, x_{2}, x_{3} and x_{4} are particles/ radiation emitted by the respective isotopes. The correct option(s) is/are

Solution:

U and Z are isotopes

*Multiple options can be correct

QUESTION: 26

Which of the following statement(s) is(are) true?

Solution:

(2) TRUE: Six member hemiacetal on anomeric carbon gives α-D glucose & β-D glucose.

(3) TRUE:

(4) TRUE: Monosaccharide cannot be hydrolysed to give polyhydroxy aldehydes and ketones

*Multiple options can be correct

QUESTION: 27

A tin chloride Q undergoes the following reactions (not balanced)

X is a monoanion having pyramidal geometry. Both Y and Z are neutral compounds. Choose the correct option(s).

Solution:

*Multiple options can be correct

QUESTION: 28

Choose the correct option(s) for the following set of reactions

Solution:

*Multiple options can be correct

QUESTION: 29

Each of the following options contains a set of four molecules. Identify the option(s) where all four molecules possess permanent dipole moment at room temperature.

Solution:

Polar molecule

Non-polar molecule

So correct answer is option (2) and (4)

*Multiple options can be correct

QUESTION: 30

Choose the reaction(s) from the following options, for which the standard enthalpy of reaction is equal to the standard enthalpy of formation.

Solution:

Enthalpy of formation is defined as enthalpy change for formation of 1 mole of substance from its elements, present in their natural most stable form.

*Answer can only contain numeric values

QUESTION: 31

For the following reaction, the equilibrium constant K_{c} at 298 K is 1.6 × 10^{17}.

When equal volumes of 0.06 M Fe^{2+}(aq) and 0.2 M S^{2–}(aq) solutions are mixed, the equilibrium concentration of Fe^{2+}(aq) is found to be Y × 10^{–17} M. The value of Y is ––––––––

Solution:

y = 8.93

*Answer can only contain numeric values

QUESTION: 32

Among B_{2}H_{6}, B_{3}N_{3}H_{6}, N_{2}O, N_{2}O_{4}, H_{2}S_{2}O_{3} and H_{2}S_{2}O_{8}, the total number of molecules containing covalent bond between two atoms of the same kind is ––––––––

Solution:

So correct answer is 4

*Answer can only contain numeric values

QUESTION: 33

Consider the kinetic data given in the following table for the reaction A + B + C → Product.

The rate of the reaction for [A] = 0.15 mol dm^{–3}, [B] = 0.25 mol dm^{–3} and [C] = 0.15 mol dm^{–3} is found to be Y × 10^{–5} mol dm^{–3} s^{–1}. The value of Y is ––––––––

Solution:

r = K[A]^{n1} [B]^{n2} [C]^{n3}

From table

n_{1} = 1

n_{2} = 0

n_{3} = 1

r = K[A] [C]

From Exp-1

*Answer can only contain numeric values

QUESTION: 34

On dissolving 0.5 g of a non-volatile non-ionic solute to 39 g of benzene, its vapor pressure decreases from 650 mm Hg to 640 mm Hg. The depression of freezing point of benzene (in K) upon addition of the solute is _____

(Given data: Molar mass and the molal freezing point depression constant of benzene are 78 g mol^{–1} and 5.12 K kg mol^{–1}, respectively)

Solution:

QUESTION: 35

Scheme 1 and 2 describe the conversion of P to Q and R to S, respectively. Scheme 3 describes the synthesis of T from Q and S. The total number of Br atoms in a molecule of T is ________

Scheme 1:

Scheme 2:

Scheme 3:

Solution:

Scheme 1:

Scheme 2:

Scheme 3:

QUESTION: 36

At 143 K. the reaction of XeF_{4} with O_{2}F_{2} produces a xenon compound Y. The total number of lone pair(s) of electrons present on the whole molecule of Y is ______

Solution:

Y has 3 lone pair of electron in each fluorine and one lone pair of electron in xenon.

Hence total lone pairs of electrons is 19.

QUESTION: 37

Let

where α = α(β) and β = β(θ) are real number, and I is the 2 × 2 identity matrix. If

α* is the minimum of the set {α(θ) : θ ∈ [0, 2π)} and

β* is the minimum of the set {β(θ) : θ ∈ [0, 2π),

then the value of α* + β* is

Solution:

Given M = αI + βM^{–1}

⇒ M^{2} – αM – βI = O

By putting values of M and M^{2}, we get

⇒

QUESTION: 38

A line y = mx + 1 intersects the circle (x – 3)^{2} + (y + 2)^{2} = 25 at the points P and Q. If the midpoint of the line segment PQ has x-coordinate -3/5, then which one of the following options is correct?

Solution:

⇒ m^{2} – 5m + 6 = 0

⇒ m = 2, 3

QUESTION: 39

Let S be the set of all complex numbers z satisfying |z - 2 + i| ≥ √5. If the complex number z_{0} is such that is the maximum of the set then the principal argument of

Solution:

= arg(–ki) ; k > 0 (as Rez_{0} < 2 & Imz_{0} > 0)

= - π/2

QUESTION: 40

The area of the region {(x, y): xy ≤ 8, 1 ≤ y ≤ x^{2}} is

Solution:

For intersection,8/y = √y ⇒ y = 4

Hence, required area =

*Multiple options can be correct

QUESTION: 41

There are three bags B_{1}, B_{2} and B_{3}. The bag B_{1} contains 5 red and 5 green balls, B_{2} contains 3 red and 5 green balls, and B_{3} contains 5 red and 3 green balls, Bags B_{1}, B_{2} and B_{3} have probabilities respectively of being chosen. A bag is selected at random and a ball is chosen at random from the bag. Then which of the following options is/are correct?

Solution:

*Multiple options can be correct

QUESTION: 42

Define the collections {E_{1}, E_{2}, E_{3}, .....} of ellipses and {R_{1}, R_{2}, R_{3}, .....} of rectangles as follows:

R_{1}: rectangle of largest area, with sides parallel to the axes, inscribed in E_{1} ;

E_{n}: ellipse of largest area inscribed in

R_{n}: rectangle of largest area, with sides parallel to the axes, inscribed in E_{n}, n > 1.

Then which of the following options is/are correct?

Solution:

Area of R_{1} = 3sin2θ ; for this to be maximum

Hence for subsequent areas of rectangles R_{n} to be maximum the coordinates will be in GP with common ratio

Eccentricity of all the ellipses will be same

Distance of a focus from the centre in

Length of latus rectum of

*Multiple options can be correct

QUESTION: 43

Let where a and b are real numbers. Which of the following options is/are correct?

Solution:

(adjM)_{11} = 2 – 3b = –1 ⇒ b = 1

Also, (adjM)_{22} = –3a = –6 ⇒ a = 2

⇒ (α, β, γ) = (1, –1, 1)

*Multiple options can be correct

QUESTION: 44

Let ƒ: be given by

Then which of the following options is/are correct?

Solution:

For x ≥ 3, ƒ(x) is again continuous and

⇒

Hence, range of ƒ(x) is

Hence ƒ' has a local maximum at x = 1 and ƒ' is NOT differentiable at x = 1.

*Multiple options can be correct

QUESTION: 45

Let α and β be the roots of x^{2} – x – 1 = 0, with α > β. For all positive integers n, define

Then which of the following options is/are correct?

Solution:

α, β are roots of x^{2} – x –1

*Multiple options can be correct

QUESTION: 46

Let denote a curve y = y(x) which is in the first quadrant and let the point (1, 0) lie on it. Let the tangent to at a point P intersect the y-axis at Y_{P}. If PY_{P} has length 1 for each point P on T, then which of the following options is/are correct?

Solution:

Y – y = y'(X – x)

So,Y_{P} = (0, y – xy')

So,

[dy/dx can not be positive i.e. ƒ(x) can not be increasing in first quadrant, for x ∈ (0, 1)]

Hence,

*Multiple options can be correct

QUESTION: 47

In a non-right-angled triangle ΔPQR, let p, q, r denote the lengths of the sides opposite to the angles at P, Q, R respectively. The median from R meets the side PQ at S, the perpendicular from P meets the side QR at E, and RS and PE intersect at O. If p = √3, q = 1, and the radius of the circumcircle of the ΔPQR equals 1, then which of the following options is/are correct?

Solution:

*Multiple options can be correct

QUESTION: 48

Let L_{1} and L_{2} denotes the lines

respectively. If L_{3} is a line which is perpendicular to both L_{1} and L_{2} and cuts both of them, then which of the following options describe(s) L_{3}?

Solution:

Points on L_{1} and L_{2} are respectively A(1 – λ, 2λ, 2λ) and B(2µ, –µ, 2µ)

and vector along their shortest distance

*Answer can only contain numeric values

QUESTION: 49

If

then 27I^{2} equals _____

Solution:

⇒

*Answer can only contain numeric values

QUESTION: 50

Let the point B be the reflection of the point A(2, 3) with respect to the line 8x – 6y – 23 = 0. Let and be circles of radii 2 and 1 with centres A and B respectively. Let T be a common tangent to the circles and such that both the circles are on the same side of T. If C is the point of intersection of T and the line passing through A and B, then the length of the line segment AC is _____

Solution:

Distance of point A from given line = 5/2

⇒ AC = 2 x 5 = 10

*Answer can only contain numeric values

QUESTION: 51

Let AP (a; d) denote the set of all the terms of an infinite arithmetic progression with first term a and common difference d > 0. If AP(1; 3) ∩ AP(2; 5) ∩ AP(3; 7) = AP(a; d) then a + d equals ___

Solution:

We equate the general terms of three respective

*Answer can only contain numeric values

QUESTION: 52

Let S be the sample space of all 3 × 3 matrices with entries from the set {0, 1}. Let the events E_{1} and E_{2 }be given by

E_{1 }= {A ∈ S : det A = 0} and

E_{2} = {A ∈ S : sum of entries of A is 7}.

If a matrix is chosen at random from S, then the conditional probability P(E_{1}|E_{2}) equals ____

Solution:

n(E_{2}) =^{ 9}C_{2} (as exactly two cyphers are there)

Now, det A = 0, when two cyphers are in the same column or same row

Hence,

*Answer can only contain numeric values

QUESTION: 53

Three lines are given by

Let the lines cut the plane x + y + z = 1 at the points A, B and C respectively. If the area of the triangle ABC is Δ then the value of (6Δ)^{2} equals ___

Solution:

*Answer can only contain numeric values

QUESTION: 54

Let ω ≠ 1 be a cube root of unity. Then the minimum of the set { |a + bω + cω^{2}|^{2} : a, b, c distinct non-zero integers} equals __

Solution:

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