JEE-MAINS-2021-Feb 24-SHIFT-1

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Q 1.
In a Young’s double slit experiment, the width of the one of the slit is three times the other slit. The amplitude of the light coming from a slit is proportional to the slit-width. Find the ratio of the maximum to the minimum intensity in the interference pattern.

Q 2.
Each side of a box made of metal sheet in cubic shape is ‘a’ at room temperature ‘T’, the coefficient of linear expansion of the metal sheet is \(‘\alpha’\). The metal sheet is heated uniformly, by a small temperature \(\Delta T\), so that its new temperature is \(T + \Delta T\). Calculate the increase in the volume of the metal box.

Q 3.
The focal length f is related to the radius of curvature r of the spherical convex mirror by

Q 4.
Two equal capacitors are first connected in series and then in parallel. The ratio of the equivalent capacities in the two cases will be

Q 5.
If Y, K and\( \eta\) are the values of Young’s modulus, bulk modulus and modulus of rigidity of any material respectively. Choose the correct relation for these parameters.

Q 6.
A current through a wire depends on time as\(i=\alpha_0t+\beta t^2\) where \(\alpha _0 = 20 A/s\) and \(\beta = 8 As^{–2.}\) Find the charge crossed through a section of the wire in 15 s

Q 7.
If an emitter current is changed by 4 mA, the collector current changes by 3.5 mA. The value of \(\beta\) will be :

Q 8.
The work done by a gas molecule in an isolated system is given by, \(W=\alpha \beta^2e^{\frac{-x^2}{\alpha kT}}\)where x is the displacement, k is the Boltzmann constant and T is the temperature. \(\alpha\)and \(\beta\) are constants. Then the dimensions of \(\beta\) will be :

Q 9.

n mole of a perfect gas undergoes a cyclic process ABCA (see figure) consisting of the following processes

\(A\rightarrow B\) : Isothermal expansion at temperature T so that the volume is doubled from \(V_1\) to\( V_2 = 2V_1\) and pressure changes from \(P_1 \)to\( P_2\)

\(B \to C \): Isobaric compression at pressure\( P_2\) to initial volume \(V_1\).

\(C \to A\) : Isochoric change leading to change of pressure from\( P_2\) to\( P_1.\)

Total workdone in the complete cycle ABCA is :

Q 10.

If the velocity-time graph has the shape AMB, what would be the shape of the corresponding acceleration-time graph?

Q 11.

In the given figure, the energy levels of hydrogen atom have been shown along with some transitions marked A, B, C, D and E.

The transitions A, B and C respectively represent :

Q 12.
Consider two satellites \(S_1\) and \(S_2 \) with periods of revolution 1 hr. and 8 hr. respectively revolving around a planet in circular orbits. The ratio of angular velocity of satellite \( S_1\) to the angular velocity of satellite \( S_2\) is :

Q 13.

Moment of inertia (M. I.) of four bodies, having same mass and radius, are reported as;

\(I _1\) = M.I. of thin circular ring about its diameter,

\(I_2\) = M.I. of circular disc about an axis perpendicular to disc and going through the centre

\(I _3\) = M.I. of solid cylinder about its axis and

\(I _4 \)= M.I. of solid sphere about its diameter.

Then :

Q 14.

A cube of side ‘a’ has point charges +Q located at each of its vertices except at the origin where the charge is –Q. The electric field at the centre of cube is :

Q 15.

In the given figure, a mass M is attached to a horizontal spring which is fixed on one side to a rigid support. The spring constant of the spring is k. The mass oscillates on a frictionless surface with time period T and amplitude A. When the mass is in equilibrium position, as shown in the figure, another mass m is gently fixed upon it. The new amplitude of oscillation will be :.

Q 16.

A cell\( E_1\) of emf 6 V and internal resistance 2\( \Omega\) is connected with another cell\( E_2\) of emf 4 V and internal resistance 8 \(\Omega\) (as shown in the figure). The potential difference across points X and Y is :

Q 17.

Given below are two statements:

Statement I : Two photons having equal linear momenta have equal wavelengths

Statement II : If the wavelength of photon is decreased, then the momentum and energy of a photon will also decrease

In the light of the above statements, choose the correct answer from the options given below

Q 18.
Two stars of masses m and 2m at a distance d rotate about their common centre of mass in free space. The period of revolution is :

Q 19.

Match List I with List II

List I                                       List II

(a) Isothermal                        (i) Pressure constant

(b) Isochoric                           (ii) Temperature constant

(c) Adiabatic                           (iii) Volume constant

(d) Isobaric                             (iv) Heat content is constant

Choose the correct answer from the options given below :

Q 20.
Four identical particles of equal masses 1 kg made to move along the circumference of a circle of radius 1 m under the action of their own mutual gravitational attraction. The speed of each particle will be :

Q 21.
An electromagnetic wave of frequency 5 GHz, is travelling in a medium whose relative electric permittivity and relative magnetic permeability both are 2. Its velocity in this medium is _________ \(× 10^7 m/s.\)
Q 22.

In connection with the circuit drawn below, the value of current flowing through 2\( k\Omega\) resistor is _________ ×\( 10^{–4 }\)A.

Q 23.
An inclined plane is bent in such a way that the vertical cross-section is given by \(y=\frac{x^2}{4}\) where y is in vertical and x in horizontal direction. If the upper surface of this curved plane is rough with coefficient of friction\( \mu\) = 0.5, the maximum height in cm at which a stationary block will not slip downward is _______ cm.
Q 24.
A ball with a speed of 9 m/s collides with another identical ball at rest. After the collision, the direction of each ball makes an angle of 30° with the original direction. The ratio of velocities of the balls after collision is x : y, where x is _________.
Q 25.
An unpolarized light beam is incident on the polarizer of a polarization experiment and the intensity of light beam emerging from the analyzer is measured as 100 Lumens. Now, if the analyzer is rotated around the horizontal axis (direction of light) by 30° in clockwise direction, the intensity of emerging light will be _______ Lumens.
Q 26.
A hydraulic press can lift 100 kg when a mass ‘m’ is placed on the smaller piston. It can lift _______ kg when the diameter of the larger piston is increased by 4 times and that of the smaller piston is decreased by 4 times keeping the same mass ‘m’ on the smaller piston.
Q 27.
The coefficient of static friction between a wooden block of mass 0.5 kg and a vertical rough wall is 0.2. The magnitude of horizontal force that should be applied on the block to keep it adhere to the wall will be _______ N. \([g = 10 ms^{–2}]\)
Q 28.
An audio signal \(\upsilon _m = 20 sin 2\pi(1500t)\) amplitude modulates a carrier \(\upsilon _c = 80 sin 2\pi(100,000t)\). The value of percent modulation is __________.
Q 29.

A resonance circuit having inductance and resistance \(2 × 10^{–4}\) H and \(6.28 \Omega\) respectively oscillates at 10 MHz frequency. The value of quality factor of this resonator is _____.

\([\pi = 3.14]\)

Q 30.
A common transistor radio set requires 12 V (D.C.) for its operation. The D.C. source is constructed by using a transformer and a rectifier circuit, which are operated at 220 V (A.C.) on standard domestic A.C. supply. The number of turns of secondary coil are 24, then the number of turns of primary are ________.
Q 31.

Which of the following are isostructural pairs?

\(SO^{2-}_4 and \ CrO^{2-}_ 4 \)

\(SiCl_4 \ and \ TiCl_4\)

\(NH_3 \ and \ NO_ 3^-\)

\(BCl_3 \ and\ BrCl_3 \)

Q 32.
Out of the following, which type of interaction is responsible for the stabilisation of \(\alpha\)-helix structure of proteins?

Q 33.
The major components in “Gun Metal” are:

Q 34.

In Freundlich adsorption isotherm, slope of AB line is:

Q 35.
Which of the following ore is concentrated using group 1 cyanide salt?

Q 36.

‘A’ and ‘B’ in the following reactions are :

Q 37.
Consider the elements Mg, AI, S, P and Si, the correct increasing order of their first ionization enthalpy is :

Q 38.
The electrode potential of \(M^2+ / M \)of 3d-series elements shows positive value for :

Q 39.

\( (A) HOCl + H_2O_2 \to H_3O^+ + CI^– + O_2\)

\((B) I_2 + H_2O_2 + 2OH^– \to2I^– + 2H_2O + O_2\)

Choose the correct option.

Q 40.

In the following reaction the reason why metanitro product also formed is :

Q 41.
\(Al_2O_3\) was leached with alkali to get X. The solution of X on passing of gas Y, forms Z. X, Y and Z respectively are:

Q 42.

Given below are two statements:

Statement-I : Colourless cupric metaborate is reduced to cuprous metaborate in a luminous flame.

Statement-II : Cuprous metaborate is obtained by heating boric anhydride and copper sulphate in a non-luminous flame.

In the light of the above statements, choose the most appropriate answer from the options given below.

Q 43.
Which of the following compound gives pink colour on reaction with phthalic anhydride in conc. \(H_2SO_4\) followed by treatment with NaOH?

Q 44.
What is the major product formed by HI on reaction with

Q 45.

What is the final product (major) ‘A’ in the given reaction?


Q 46.

Identify Products A and B. 

Q 47.
The gas released during anaerobic degradation of vegetation may lead to :

Q 48.

Match List I with List II.

List I                                List II

(Monomer unit)               (Polymer)

(a) Caprolactum              (i) Natural rubber

(b) 2-Chloro-1,                (ii) Buna-N


(c) Isoprene                     (iii) Nylon 6

(d) Acrylonitrile                (iv) Neoprene

Choose the correct answer from the options given below:

Q 49.

Which of the following reagent is used for the following reaction?

Q 50.
The product formed in the first step of the reaction of \(\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;Br\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,\;\;|\\ CH_3 – CH_2 – CH – CH_2 – CH_ – CH_3 \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,\;|\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;Br \) with excess \(Mg/Et_2O(Et = C_2H_5)\) is:

Q 51.

A proton and a\( Li^{3+}\) nucleus are accelerated by the same potential.If \(\lambda_{Li}\)Li and \(\lambda_{p}\) denote the de Broglie wavelengths of\( Li^{3+}\) and proton respectively, then the value of \(\frac {\lambda_{Li}}{\lambda_{p}}\) is x × \(10^{–1}\)

The value of x is _____. (Rounded off to the nearest integer)

[Mass of \( Li^{3+}\) = 8.3 mass of proton]

Q 52.
Gaseous cyclobutene isomerizes to butadiene in a first order process which has a ‘k’ value of 3.3 × \(10^{–4 }\)\(s^{–1}\) at 153°C. The time in minutes it takes for the isomerization to proceed 40% to completion at this temperature is __________. (Rounded off to the nearest integer)
Q 53.
4.5 g of compound A (MW = 90) was used to make 250 mL of its aqueous solution. The molarity of the solution in M is x × \(10^{–1}\). The value of x is _____. (Rounded off to the nearest integer)
Q 54.

When 9.45 g of\( ClCH_2COOH \)is added to 500 mL of water, its freezing point drops by 0.5°C. The dissociation constant of \(ClCH_2COOH\) is x × \(10^{–3}\). The value of x is _______. (Rounded off to the nearest integer)

\([K _f(H_2O)=1.86K\ kg\ mol^{-1}]\)

Q 55.

The coordination number of an atom in a bodycentered cubic structure is ______.

[Assume that the lattice is made up of atoms.]

Q 56.

At 1990 K and 1 atm pressure, there are equal number of \(Cl_2\) molecules and Cl atoms in the reaction mixture. The value of \(K_p\) for the reaction\(Cl _2(g) \rightleftharpoons 2Cl(g)\)under the above conditions is x × \(10^{–1}\). The value of x is____.

(Rounded off to the nearest integer)

Q 57.

For the reaction \(A_{(g)}\to B_{(g)}\) the value of the equilibrium constant at 300 K and 1 atm is equal to 100.0 The value of\( \Delta _r G\) for the reaction at 300 K and 1 atm in J mol–1 is –xR, where x is________ (Rounded off to the nearest integer)

[R = 8.31 J \(mol^{–1}K^{–1}\) and In 10 = 2.3)

Q 58.

The reaction of sulphur in alkaline medium is given below:

\(s_{8(s)}+a \ OH^{-}(aq)\to b\ S^2_{(aq)}+c S_2O^{2-}_{3(aq)}+d \ H_2O_{(I)}\)

The value of ‘a’ is ______. (Integer answer)

Q 59.

Number of amphoteric compounds among the following is____

\((1) BeO        \\    (2) BaO \\(3) Be(OH)_2\\ (4) Sr(OH)_2\)

Q 60.

The stepwise formation of \([Cu(NH_3)_4]^{2+}\) is given below:

The value of stability constants \(K_1, K_2, K_3\;and \;K_4\; are \;10^4, 1.58 × 10^3, 5 × 10^2 \;and \;10^2\) respectively. The overall equilibrium constants for dissociation of \([Cu(NH_3)_4]^{2+} \) is x \(× 10^{–12}\). The value of x is ____. (Rounded off to the nearest integer)

Q 61.
If \(\int \frac {cosx -sinx}{\sqrt 8- sin2x}dx= asin^{-1}(\frac{sinx +cosx}{b})+c,\) where c is a constant of integration, then the ordered pair (a, b) is equal to :

Q 62.
The area (in sq. units) of the part of the circle \(x^ 2 + y^2 = 36\), which is outside the parabola \(y ^2 = 9x, \)is :

Q 63.
If \(e^{(cos^2 x+cos^4x+cos^6x+.....\infty)}log_e 2\) satisfies the equation \(t^2 – 9t + 8 = 0\), then the value of  \(\frac{2sinx}{sinx +\sqrt3 cosx} \left(0  is:

Q 64.
The population P = P(t) at time ‘t’ of a certain species follows the differential equation \(\frac{dp}{dt}=0.5P-450.\)If\(P(0) = 850\)then the time at which population becomes zero is :

Q 65.
The statement among the following that is a tautology is :

Q 66.
Let p and q be two positive numbers such that \(p + q = 2\) and \(p^4 + q^4 = 272\). Then p and q are roots of the equation :

Q 67.

The system of linear equations

3x – 2y – kz = 10

2x – 4y – 2z = 6

x + 2y – z = 5m

is inconsistent if :

Q 68.
A scientific committee is to be formed from 6 Indians and 8 foreigners, which includes at least 2 Indians and double the number of foreigners as Indians. Then the number of ways, the committee can be formed, is :

Q 69.
The equation of the plane passing through the point (1, 2, –3) and perpendicular to the planes 3x + y – 2z = 5 and 2x – 5y – z = 7, is:

Q 70.
If the tangent to the curve \( y = x^3\) at the point \(P(t, t^3 )\) meets the curve again at Q, then the ordinate of the point which divides PQ internally in the ratio 1 : 2 is :

Q 71.
Let \(f : R \to R \) be defined as \(f(x) = 2x – 1\) and \(g : R – {1} \to R\) be defined as \(g(x)=\frac {x-{\frac 1 2}}{x-1}\) Then the composition function f(g(x)) is :

Q 72.
An ordinary dice is rolled for a certain number of times. If the probability of getting an odd number 2 times is equal to the probability of getting an even number 3 times, then the probability of getting an odd number for odd number of times is :

Q 73.
Two vertical poles are 150 m apart and the height of one is three times that of the other. If from the middle point of the line joining their feet, an observer finds the angles of elevation of their tops to be complementary, then the height of the shorter pole (in meters) is :

Q 74.
If f : R \(\to\) R is a function defined by\( f(x) = [x – 1] cos (\frac{2x-1}{2})\pi\) , where \([\cdot]\) denotes the greatest integer function, then f is :

Q 75.
The distance of the point (1, 1, 9) from the point of intersection of the line \(\frac{x-3}1= \frac{y-4}2=\frac {z-5}2\) and the plane x + y + z = 17 is:

Q 76.
The locus of the mid-point of the line segment joining the focus of the parabola \(y^2 = 4ax\) to a moving point of the parabola, is another parabola whose directrix is

Q 77.
A man is walking on a straight line. The arithmetic mean of the reciprocals of the intercepts of this line on the coordinate axes is \(\frac14\) . Three stones A, B and C are placed at the points (1, 1), (2, 2) and (4, 4) respectively. Then which of these stones is/are on the path of the man?

Q 78.

The function


Q 79.

The value of \(-^{15}c_1+2.^{15}c_2-3.^{15}c_3+.....-15^{15}c_{15} +{14}c_1+^{14}c_3+^{14}c_5+....^{14}c_{11} \ is\)

Q 80.
\(\lim\limits_ {x\to0}\frac{\int\limits_0^{x^2}(sin\sqrt t)dt}{x^3} \;is\;equal\;to\)

Q 81.

Let   A = {n\(\in\)N : n is a 3-digit number}

        B = {9k + 2 : k\(\in\)N}

and  C = {9k + l : k\(\in\)N} for some l (0 < l < 9)

If the sum of all the elements of the set \(A \cap (B \cup C)\) is 274 × 400, then l is equal to _______.

Q 82.
If \(\int\limits_{-a}^a(|x|+|x-2|)dx=22,\{a>2)\) and [x] denotes the greatest integer\( < x\), then \(\int\limits_{-a}^{a}(x+[x])dx\) is equal to __________.
Q 83.
Let \(P=\begin{bmatrix}{3} & {-1} &{-2} \\{2} & {0} &{\alpha} \\{3} & {-5} &{0} \end{bmatrix}\) where \(\alpha \in\)R. Suppose \(Q = [q_{ij}]\) is a matrix satisfying \(PQ = kI_3\) for some non-zero \(k\in R\). If \(q_{23}=-\frac k 8\) and \(|Q|= \frac {k^2} {2}\) then \(\alpha ^ 2 + k^2 \)is equal to _____________
Q 84.
If one of the diameters of the circle \(x ^2 + y^2 – 2x – 6y + 6 = 0\) is a chord of another circle ‘C’, whose center is at (2, 1), then its radius is ________.
Q 85.
Let\( B_i (i = 1, 2, 3)\) be three independent events in a sample space. The probability that only \(B_1\) occurs is \(\alpha\), only\( B_2\) occurs is \(\beta\) and only \(B_3\) occurs is \(\gamma\). Let p be the probability that none of the events\( B_i\) occurs and these 4 probabilities satisfy the equations\( (\alpha – 2\beta) p = \alpha \beta\ and (\beta– 3\gamma) p = 2\beta \gamma\) (All the probabilities are assumed to lie in the interval (0, 1)). Then \(\frac {P(B_1 )} {P(B_3 )}\) is equal to __________.
Q 86.
If the least and the largest real values of \(\alpha\), for which the equation \(z + \alpha|z – 1| + 2i = 0 \) and \((z\in c \ and \ i=\sqrt-1)\) has a solution, are p and q respectively; then \(4(p^2+q^2)\) is equal to ______.
Q 87.
The minimum value of \(\alpha\) for which the equation \(\frac{4}{ sinx }+\frac{1}{1-sinx}=\alpha\) has at least one solution in\((0,\frac{\pi}{2})\) is _______.
Q 88.
Let M be any 3 × 3 matrix with entries from the set (0, 1, 2). The maximum number of such matrices, for which the sum of diagonal elements of \(M^TM\) is seven is _______.
Q 89.
\(\lim\limits_{n \to \infty} tan \left\{\displaystyle\sum_{r=1}^{n} tan^{-1} \left(\frac{1}{1+r+r^2}\right)\right\} \) is equal to ______.
Q 90.
Let three vectors \(\vec a, \vec b\;and\; \vec c\) be such that \(\vec c\) is coplanar with  \(\vec a \; and \; \vec b,\; \vec a.\vec c=7\;and\;\vec b\) is perpendicular to \(\vec c\) , where \(\vec a=- \hat i+\hat j+\hat k\;and\;\vec b=\hat 2i+\hat k,\) then the value of \(2|\vec a+\vec b+\vec c|^2\) is _____.
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