JEE Main 28 July 2022 Shift 1 question paper with solutions

Question 1

Maths · Differential Equations · Single correct

Let the solution curve of the differential equation $$x \, dy = \left( \sqrt{x^2 + y^2} + y \right) \, dx, \; x > 0,$$ intersect the line $$x = 1$$ at $$y = 0$$ and the line $$x = 2$$ at $$y = \alpha$$. Then the value of $$\alpha$$ is:

1. $\frac{1}{2}$
2. $\frac{3}{2}$
3. $-\frac{3}{2}$
4. $\frac{5}{2}$

Answer: (b)

Question 2

Maths · Inverse Trigonometric Functions · Single correct

Considering only the principal values of the inverse trigonometric functions, the domain of the function $f(x) = \cos^{-1}\left(\frac{x^2 - 4x + 2}{x^2 + 3}\right)$ is:

1. $(-\\infty, \\frac{1}{4}]$
2. $[-\\frac{1}{4}, \\infty)$
3. $( -\\frac{1}{3}, \\infty)$
4. $(-\\infty, \\frac{1}{3}]$

Answer: (b)

Question 3

Maths · Vector Algebra · Single correct

Let the vectors $\vec{a} = (1+t)\hat{i} + (1-t)\hat{j} + \hat{k}$, $\vec{b} = (1-t)\hat{i} + (1+t)\hat{j} + 2\hat{k}$ and $\vec{c} = t\hat{i} - t\hat{j} + \hat{k}$, $t \in \mathbb{R}$ be such that for $\alpha, \beta, \gamma \in \mathbb{R}$, $\alpha \vec{a} + \beta \vec{b} + \gamma \vec{c} = \vec{0}$ $\Rightarrow \alpha = \beta = \gamma = 0$. Then, the set of all values of $t$ is:

1. a non-empty finite set
2. equal to $\mathbb{N}$
3. equal to $\mathbb{R} - \{0\}$
4. equal to $\mathbb{R}$

Answer: (c)

Question 4

Maths · Inverse Trigonometric Functions · Single correct

Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the equation $\cos^{-1}(x) - 2\sin^{-1}(x) = \cos^{-1}(2x)$ is equal to:

1. 0
2. 1
3. $\frac{1}{2}$
4. $-\frac{1}{2}$

Question 5

Maths · Mathematical Reasoning · Single correct

Let the operations $*$, $\odot \in \{\land, \lor\}$. If $(p * q) \odot (p \odot \sim q)$ is a tautology, then the ordered pair $(*, \odot)$ is:

1. $(\lor, \land)$
2. $(\lor, \lor)$
3. $(\land, \land)$
4. $(\land, \lor)$

Question 6

Maths · Vector Algebra · Single correct

Let a vector $\vec{a}$ has a magnitude 9. Let a vector $\vec{b}$ be such that for every $(x,y) \in \mathbb{R} \times \mathbb{R} - \{(0,0)\}$, the vector $(x\vec{a} + y\vec{b})$ is perpendicular to the vector $(6y\vec{a} - 18x\vec{b})$. Then the value of $|\vec{a} \times \vec{b}|$ is equal to:

1. $9\sqrt{3}$
2. $27\sqrt{3}$
3. 9
4. 81

Answer: (b)

Question 7

Maths · Properties of Triangles · Single correct

For $t \in (0, 2\pi)$, if $ABC$ is an equilateral triangle with vertices $A(\sin t, -\cos t)$, $B(\cos t, \sin t)$ and $C(a, b)$ such that its orthocentre lies on a circle with centre $\left(1, \frac{1}{3}\right)$, then $(a^2 - b^2)$ is equal to:

1. $\frac{8}{3}$
2. 8
3. $\frac{77}{9}$
4. $\frac{80}{9}$

Answer: (b)

Question 8

Maths · Relations and Functions · Single correct

For $\alpha \in \mathbb{N}$, consider a relation $R$ on $\mathbb{N}$ given by $R = \{(x,y) : 3x + \alpha y$ is a multiple of $7\}$. The relation $R$ is an equivalence relation if and only if:

1. $\alpha = 14$
2. $\alpha$ is a multiple of $4$
3. $4$ is the remainder when $\alpha$ is divided by $10$
4. $4$ is the remainder when $\alpha$ is divided by $7$

Answer: (d)

Question 9

Maths · Probability · Single correct

Out of 60$\%$ female and 40$\%$ male candidates appearing in an exam, 60$\%$ candidates qualify it. The number of females qualifying the exam is twice the number of males qualifying it. A candidate is randomly chosen from the qualified candidates. The probability, that the chosen candidate is a female, is :

1. $\frac{2}{3}$
2. $\frac{11}{16}$
3. $\frac{23}{32}$
4. $\frac{13}{16}$

Answer: (a)

Question 10

Maths · Differential Equations · Single correct

If $$ y = y(x), \; x \in \left(0, \frac{\pi}{2}\right) \,$$ be the solution curve of the differential equation $$\left(\sin^2 2x\right) \frac{dy}{dx} + \left(8 \sin^2 2x + 2 \sin 4x\right) y = 2e^{-4x} \left(2 \sin 2x + \cos 2x\right), \; with \; y\left(\frac{\pi}{4}\right) = e^{-\pi},$$ then $$\, y\left(\frac{\pi}{6}\right) \,$$ is equal to:

1. $\frac{2}{\sqrt{3}} e^{-2\pi/3}$
2. $\frac{2}{\sqrt{3}} e^{2\pi/3}$
3. $\frac{1}{\sqrt{3}} e^{-2\pi/3}$
4. $\frac{1}{\sqrt{3}} e^{2\pi/3}$

Answer: (a)

Question 11

Maths · Conic Sections · Single correct

If the tangents drawn at the points P and Q on the parabola $y^2 = 2x - 3$ intersect at the point R(0, 1), then the orthocentre of the triangle PQR is:

1. (0, 1)
2. (2, -1)
3. (6, 3)
4. (2, 1)

Answer: (b)

Question 12

Maths · Conic Sections · Single correct

Let C be the centre of the circle $x^2 + y^2 - x + 2y = \frac{11}{4}$ and P be a point on the circle. A line passes through the point C, makes an angle of $\frac{\pi}{4}$ with the line CP and intersects the circle at the points Q and R. Then the area of the triangle PQR (in unit$^2$) is :

1. 2
2. 2$\sqrt{2}$
3. 8\sin\left(\frac{\pi}{8}\right)
4. 8\cos\left(\frac{\pi}{8}\right)

Answer: (b)

Question 13

Maths · Binomial Theorem · Single correct

The remainder when $7^{2022} + 3^{2022}$ is divided by 5 is:

1. 0
2. 2
3. 3
4. 4

Answer: (c)

Question 14

Maths · Matrices · Single correct

Let the matrix $A = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{bmatrix}$ and the matrix $B_0 = A^{49} + 2A^{98}$. If $B_n = Adj(B_{n-1})$ for all $n \geq 1$, then $\det(B_4)$ is equal to:

1. $3^{28}$
2. $3^{30}$
3. $3^{32}$
4. $3^{36}$

Answer: (c)

Question 15

Maths · Complex Numbers and Quadratic Equations · Single correct

Let $S_1 = \left\{ z_1 \in \mathbb{C} : |z_1 - 3| = \frac{1}{2} \right\}$ and $S_2 = \left\{ z_2 \in \mathbb{C} : |z_2 - |z_2 + 1|| = |z_2 + |z_2 - 1|| \right\}$. Then, for $z_1 \in S_1$ and $z_2 \in S_2$, the least value of $|z_2 - z_1|$ is :

1. 0
2. $\frac{1}{2}$
3. $\frac{3}{2}$
4. $\frac{5}{2}$

Answer: (c)

Question 16

Maths · Three Dimensional Geometry · Single correct

The foot of the perpendicular from a point on the circle $x^2 + y^2 = 1$, $z = 0$ to the plane $2x + 3y + z = 6$ lies on which one of the following curves?

1. $(6x + 5y - 12)^2 + 4(3x + 7y - 8)^2 = 1$, $z = 6 - 2x - 3y$
2. $(5x + 6y - 12)^2 + 4(3x + 5y - 9)^2 = 1$, $z = 6 - 2x - 3y$
3. $(6x + 5y - 14)^2 + 9(3x + 5y - 7)^2 = 1$, $z = 6 - 2x - 3y$
4. $(5x + 6y - 14)^2 + 9(3x + 7y - 8)^2 = 1$, $z = 6 - 2x - 3y$

Answer: (b)

Question 17

Maths · Applications of Derivatives · Single correct

If the minimum value of $f(x) = \frac{5x^2}{2} + \frac{\alpha}{x^5}$, $x > 0$, is 14, then the value of $\alpha$ is equal to:

1. 32
2. 64
3. 128
4. 256

Answer: (c)

Question 18

Maths · Relations and Functions (Advanced) · Single correct

Let $\alpha$, $\beta$ and $\gamma$ be three positive real numbers. Let $f(x) = \alpha x^5 + \beta x^3 + \gamma x$, $x \in \mathbb{R}$ and $g : \mathbb{R} \to \mathbb{R}$ be such that $g(f(x)) = x$ for all $x \in \mathbb{R}$. If $a_1, a_2, a_3, \ldots, a_n$ be in arithmetic progression with mean zero, then the value of $f\left(g\left(\dfrac{1}{n}\sum_{i=1}^{n} f(a_i)\right)\right)$ is equal to:

1. 0
2. 3
3. 9
4. 27

Answer: (a)

Question 19

Maths · Sequences and Series · Single correct

Consider the sequence $a_1, a_2, a_3, \ldots$ such that $$a_1 = 1, \ a_2 = 2 \ and \ a_{n+2} = \frac{2}{a_{n+1}} + a_n \ for \ n = 1, 2, 3, \ldots$$ If $$\left( \frac{a_1 + \frac{1}{a_2}}{a_3} \right) \cdot \left( \frac{a_2 + \frac{1}{a_3}}{a_4} \right) \cdot \left( \frac{a_3 + \frac{1}{a_4}}{a_5} \right) \cdot \ldots \cdot \left( \frac{a_{30} + \frac{1}{a_{31}}}{a_{32}} \right) = 2^\alpha \left( ^{61}C_{31} \right),$$ then $\alpha$ is equal to:

1. $-30$
2. $-31$
3. $-60$
4. $-61$

Answer: (c)

Question 20

Maths · Integrals · Single correct

The minimum value of the twice differentiable function $f(x) = \int_{0}^{x} e^{x-t} f'(t) \, dt - (x^2 - x + 1) e^x$, $x \in \mathbb{R}$, is:

1. -$\frac{2}{\sqrt{e}}$
2. -2$\sqrt{e}$
3. -$\sqrt{e}$
4. $\frac{2}{\sqrt{e}}$

Answer: (a)

Question 21

Maths · Permutations and Combinations · Fill in the blank

Let S be the set of all passwords which are six to eight characters long, where each character is either an alphabet from $\{$A, B, C, D, E$\}$ or a number from {1, 2, 3, 4, 5} with the repetition of characters allowed. If the number of passwords in S whose at least one character is a number from {1, 2, 3, 4, 5} is $\alpha \times 5^6$, then $\alpha$ is equal to _____.

Answer: 7073

Question 22

Maths · Three Dimensional Geometry · Fill in the blank

Let P(-2, -1, 1) and Q $(\frac{56}{17}, \frac{43}{17}, \frac{111}{17})$ be the vertices of the rhombus PRQS. If the direction ratios of the diagonal RS are $\alpha$, -1, $\beta$, where both $\alpha$ and $\beta$ are integers of minimum absolute values, then $\alpha^2$ + $\beta^2$ is equal to ______.

Answer: 450

Question 23

Maths · Applications of Derivatives · Numerical

Let $f : [0, 1] \to \mathbb{R}$ be a twice differentiable function in $(0, 1)$ such that $f(0) = 3$ and $f(1) = 5$. If the line $y = 2x + 3$ intersects the graph of $f$ at only two distinct points in $(0, 1)$, then the least number of points $x \in (0, 1)$, at which $f''(x) = 0$, is ______.

Answer: 2

Question 24

Maths · Integrals · Numerical

If $$\int_{0}^{\sqrt{3}} \frac{15x^3}{\sqrt{1+x^2 + \sqrt{\left(1+x^2\right)^3}}} \, dx = \alpha \sqrt{2} + \beta \sqrt{3}$$, where $\alpha$, $\beta$ are integers, then $\alpha + \beta$ is equal to

Answer: 10

Question 25

Maths · Matrices · Numerical

Let $A = \begin{bmatrix} 1 & -1 \\ 2 & \alpha \end{bmatrix}$ and $B = \begin{bmatrix} \beta & 1 \\ 1 & 0 \end{bmatrix}$, $\alpha, \beta \in \mathbb{R}$. Let $\alpha_1$ be the value of $\alpha$ which satisfies $(A+B)^2 = A^2 + \begin{bmatrix} 2 & 2 \\ 2 & 2 \end{bmatrix}$ and $\alpha_2$ be the value of $\alpha$ which satisfies $(A+B)^2 = B^2$. Then $|\alpha_1 - \alpha_2|$ is equal to _______.

Answer: 2

Question 26

Maths · Sequences and Series · Numerical

For $p, q \in \mathbb{R}$, consider the real valued function $f(x) = (x - p)^2 - q$, $x \in \mathbb{R}$ and $q > 0$. Let $a_1, a_2, a_3$ and $a_4$ be in an arithmetic progression with mean $p$ and positive common difference. If $\left| f(a_i) \right| = 500$ for all $i = 1, 2, 3, 4$, then the absolute difference between the roots of $f(x) = 0$ is

Answer: 50

Question 27

Maths · Conic Sections · Fill in the blank

For the hyperbola H : $x^2-y^2=1$ and the ellipse E : $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, $a>b>0$, let the (1) eccentricity of E be reciprocal of the eccentricity of H, and (2) the line $y=\sqrt{\frac{5}{2}}x+K$ be a common tangent of E and H. Then $4(a^2+b^2)$ is equal to ______.

Answer: 3

Question 28

Maths · Sequences and Series · Numerical

Let $x_1, x_2, x_3, \ldots, x_{20}$ be in geometric progression with $x_1 = 3$ and the common ratio $\frac{1}{2}$. A new data is constructed replacing each $x_i$ by $(x_i - i)^2$. If $\overline{x}$ is the mean of new data, then the greatest integer less than or equal to $\overline{x}$ is

Answer: 142

Question 29

Maths · Limits and Derivatives · Numerical

$\lim_{x\to0} \left( \frac{ (x+2\cos x)^3 +2(x+2\cos x)^2 +3\sin(x+2\cos x) }{ (x+2)^3 +2(x+2)^2 +3\sin(x+2) } \right)^{\frac{100}{x}}$ is equal to ______.

Answer: 1

Question 30

Maths · Complex Numbers and Quadratic Equations · Numerical

The sum of all real values of $x$ for which $$\frac{3x^2 - 9x + 17}{x^2 + 3x + 10} = \frac{5x^2 - 7x + 19}{3x^2 + 5x + 12}$$ is equal to ___.

Answer: 6

Question 31

Physics · Physical World, Units and Measurements · Single correct

The dimensions of $\frac{B^2}{\mu_0}$ will be: (if $\mu_0$: permeability of free space and B: magnetic field)

1. $[M\,L^{2}\,T^{-2}]$
2. $[M\,L\,T^{-2}]$
3. $[M\,L^{-1}\,T^{-2}]$
4. $[M\,L^{2}\,T^{-2}\,A^{-1}]$

Answer: (c)

Question 32

Physics · Motion in a Plane · Single correct

A NCC parade is going at a uniform speed of 9 km/h under a mango tree on which a monkey is sitting at a height of 19.6 m. At any particular instant, the monkey drops a mango. A cadet will receive the mango whose distance from the tree at time of drop is: (Given g = 9.8 m/s²)

1. 5 m
2. 10 m
3. 19.8 m
4. 24.5 m

Answer: (a)

Question 33

Physics · Laws of Motion · Single correct

In two different experiments, an object of mass $5 \, \mathrm{kg}$ moving with a speed of $25 \, \mathrm{ms}^{-1}$ hits two different walls and comes to rest within (i) $3 \, \mathrm{seconds}$, (ii) $5 \, \mathrm{seconds}$, respectively. Choose the correct option out of the following:

1. Impulse and average force acting on the object will be same for both the cases.
2. Impulse will be same for both the cases but the average force will be different.
3. Average force will be same for both the cases but the impulse will be different.
4. Average force and impulse will be different for both the cases.

Answer: (b)

Question 34

Physics · Laws of Motion · Single correct

A balloon has mass of 10 g in air. The air escapes from the balloon at a uniform rate with velocity $4.5 \, \mathrm{cm/s}$. If the balloon shrinks in $5 \, \mathrm{s}$ completely. Then, the average force acting on that balloon will be (in dyne).

1. 3
2. 9
3. 12
4. 18

Answer: (b)

Question 35

Physics · Gravitation · Single correct

If the radius of earth shrinks by 2$\%$ while its mass remains same. The acceleration due to gravity on the earth’s surface will approximately :

1. decrease by 2$\%$
2. decrease by 4$\%$
3. increase by 2$\%$
4. increase by 4$\%$

Answer: (d)

Question 36

Physics · Mechanical Properties of Solids · Single correct

The force required to stretch a wire of cross-section $1 \, \mathrm{cm}^2$ to double its length will be: (Given Yong's modulus of the wire $= 2 \times 10^{11} \, \mathrm{N/m}^2$)

1. $1 \times 10^7 \, \mathrm{N}$
2. $1.5 \times 10^7 \, \mathrm{N}$
3. $2 \times 10^7 \, \mathrm{N}$
4. $2.5 \times 10^7 \, \mathrm{N}$

Answer: (c)

Question 37

Physics · Thermodynamics · Single correct

A Carnot engine has efficiency of 50$\%$. If the temperature of sink is reduced by $40^\circ$$\mathrm{C}$, its efficiency increases by 30$\%$. The temperature of the source will be:

1. $166.7\,\mathrm{K}$
2. $255.1\,\mathrm{K}$
3. $266.7\,\mathrm{K}$
4. $367.7\,\mathrm{K}$

Answer: (c)

Question 38

Physics · Kinetic Theory · Single correct

Given below are two statements : Statement I : The average momentum of a molecule in a sample of an ideal gas depends on temperature. Statement II : The rms speed of oxygen molecules in a gas is $v$. If the temperature is doubled and the oxygen molecules dissociate into oxygen atoms, the rms speed will become $2v$. In the light of the above statements, choose the correct answer from the options given below :

1. Both Statement I and Statement II are true
2. Both Statement I and Statement II are false
3. Statement I is true but Statement II is false
4. Statement I is false but Statement II is true

Answer: (d)

Question 39

Physics · Waves · Single correct

In the wave equation $$y = 0.5 \sin \frac{2\pi}{\lambda} (400\, t - x)\, \mathrm{m}$$ the velocity of the wave will be:

1. 200 $\mathrm{m/s}$
2. 200$\sqrt{2}$ $\mathrm{m/s}$
3. 400 $\mathrm{m/s}$
4. 400$\sqrt{2}$ $\mathrm{m/s}$

Answer: (c)

Question 40

Physics · Electrostatic Potential and Capacitance · Single correct

Two capacitors, each having capacitance $40 \, \mu\mathrm{F}$ are connected in series. The space between one of the capacitors is filled with dielectric material of dielectric constant $K$ such that the equivalence capacitance of the system became $24 \, \mu\mathrm{F}$. The value of $K$ will be:

1. 1.5
2. 2.5
3. 1.2
4. 3

Answer: (a)

Question 41

Physics · Current Electricity · Single correct

A wire of resistance $R_1$ is drawn out so that its length is increased by twice of its original length. The ratio of new resistance to original resistance is:

1. 9 : 1
2. 1 : 9
3. 4 : 1
4. 3 : 1

Answer: (a)

Question 42

Physics · Moving Charges and Magnetism · Multiple correct

The current sensitivity of a galvanometer can be increased by: (A) decreasing the number of turns (B) increasing the magnetic field (C) decreasing the area of the coil (D) decreasing the torsional constant of the spring Choose the most appropriate answer from the options given below:

1. and (C) only
2. and (D) only
3. and (C) only
4. and (D) only

Answer: (d)

Question 43

Physics · Moving Charges and Magnetism · Single correct

As shown in the figure, a metallic rod of linear density $0.45 \, \mathrm{kg \, m^{-1}}$ is lying horizontally on a smooth incline plane which makes an angle of $45^\circ$ with the horizontal. The minimum current flowing in the rod required to keep it stationary, when $0.15 \, \mathrm{T}$ magnetic field is acting on it in the vertical upward direction, will be: $\{$Use $g = 10 \, \mathrm{m/s^2}$$\}$

1. 30 A
2. 15 A
3. 10 A
4. 3 A

Answer: (a)

Question 44

Physics · Alternating Current · Single correct

The equation of current in a purely inductive circuit is $5\sin(49\pi t - 30^\circ)$. If the inductance is $30\,\mathrm{mH}$ then the equation for the voltage across the inductor, will be: $$\left\{\text{Let } \pi = \frac{22}{7}\right\}$$

1. $1.47 \sin(49 \pi t - 30^\circ)$
2. $1.47 \sin(49 \pi t + 60^\circ)$
3. $23.1 \sin(49 \pi t - 30^\circ)$
4. $23.1 \sin(49 \pi t + 60^\circ)$

Answer: (d)

Question 45

Physics · Ray Optics and Optical Instruments · Single correct

As shown in the figure, after passing through the medium 1. The speed of light $v_2$ in medium 2 will be: (Given $c = 3 \times 10^8 \, \mathrm{ms}^{-1}$)

1. $1.0 \times 10^8 \, \mathrm{ms}^{-1}$
2. $0.5 \times 10^8 \, \mathrm{ms}^{-1}$
3. $1.5 \times 10^8 \, \mathrm{ms}^{-1}$
4. $3.0 \times 10^8 \, \mathrm{ms}^{-1}$

Answer: (a)

Question 46

Physics · Ray Optics and Optical Instruments · Single correct

In normal adjustment, for a refracting telescope, the distance between objective and eye piece is 30 $\mathrm{cm}$. The focal length of the objective, when the angular magnification of the telescope is 2, will be:

1. 20 $\mathrm{cm}$
2. 30 $\mathrm{cm}$
3. 10 $\mathrm{cm}$
4. 15 $\mathrm{cm}$

Answer: (a)

Question 47

Physics · Dual Nature of Radiation and Matter · Single correct

The equation $\lambda = \frac{1.227}{x} \, \mathrm{nm}$ can be used to find the de-Brogli wavelength of an electron. In this equation $x$ stands for: Where, m = mass of electron P = momentum of electron K = Kinetic energy of electron V = Accelerating potential in volts for electron

1. $\sqrt{mK}$
2. $\sqrt{P}$
3. $\sqrt{K}$
4. $\sqrt{V}$

Answer: (d)

Question 48

Physics · Nuclei · Single correct

The half life period of a radioactive substance is 60 days. The time taken for $\frac{7}{8}$ of its original mass to disintegrate will be:

1. 120 days
2. 130 days
3. 180 days
4. 20 days

Answer: (c)

Question 49

Physics · Semiconductor Electronics: Materials, Devices and Simple Circuits · Single correct

Identify the solar cell characteristics from the following options:

Answer: (b)

Question 50

Physics · Communication Systems · Single correct

In the case of amplitude modulation to avoid distortion the modulation index ($\mu$) should be:

1. $\mu \leq 1$
2. $\mu \geq 1$
3. $\mu = 2$
4. $\mu = 0$

Answer: (a)

Question 51

Physics · Mathematics in Physics · Numerical

If the projection of $2\hat{i} + 4\hat{j} - 2\hat{k}$ on $\hat{i} + 2\hat{j} + \alpha \hat{k}$ is zero. Then, the value of $\alpha$ will be

Answer: 5

Question 52

Physics · Nuclei · Numerical

A freshly prepared radioactive source of half life 2 hours 30 minutes emits radiation which is 64 times the permissible safe level. The minimum time, after which it would be possible to work safely with source, will be ______ hours.

Answer: 15

Question 53

Physics · Wave Optics · Numerical

In a Young's double slit experiment, a laser light of 560 $\mathrm{nm}$ produces an interference pattern with consecutive bright fringes' separation of 7.2 $\mathrm{mm}$. Now another light is used to produce an interference pattern with consecutive bright fringes' separation of 8.1 $\mathrm{mm}$. The wavelength of second light is $\mathrm{nm}$.

Answer: 630

Question 54

Physics · Alternating Current · Numerical

The frequencies at which the current amplitude in an LCR series circuit becomes $\frac{1}{\sqrt{2}}$ times its maximum value, are $212 \, \mathrm{rad} \, \mathrm{s}^{-1}$ and $232 \, \mathrm{rad} \, \mathrm{s}^{-1}$. The value of resistance in the circuit is $R = 5 \, \Omega$. The self inductance in the circuit is ______ mH.

Answer: 250

Question 55

Physics · Current Electricity · Numerical

As shown in the figure, a potentiometer wire of resistance $20\,\Omega$ and length $300\,\mathrm{cm}$ is connected with resistance box (R.B.) and a standard cell of emf $4\,\mathrm{V}$. For a resistance ‘R’ of resistance box introduced into the circuit, the null point for a cell of $20\,\mathrm{mV}$ is found to be $60\,\mathrm{cm}$. The value of ‘R’ is _________ $\Omega$.

Answer: 780

Question 56

Physics · Electric Charges and Fields · Numerical

Two electric dipoles of dipole moments $1.2 \times 10^{-30} \, \mathrm{cm}$ and $2.4 \times 10^{-30} \, \mathrm{cm}$ are placed in two different uniform electric fields of strengths $5 \times 10^{4} \, \mathrm{NC}^{-1}$ and $15 \times 10^{4} \, \mathrm{NC}^{-1}$ respectively. The ratio of maximum torque experienced by the electric dipoles will be $\frac{1}{x}$. The value of $x$ is _______.

Answer: 6

Question 57

Physics · Waves · Fill in the blank

The frequency of echo will be $\underline{\hspace{1cm}}$ Hz if the train blowing a whistle of frequency $320$ Hz is moving with a velocity of $36$ km/h towards a hill from which an echo is heard by the train driver. Velocity of sound in air is $330$ m/s.

Answer: 340

Question 58

Physics · Mechanical Properties of Fluids · Numerical

The diameter of an air bubble which was initially 2 $\mathrm{mm}$, rises steadily through a solution of density $1750 \, \mathrm{kg \, m^{-3}}$ at the rate of $0.35 \, \mathrm{cm \, s^{-1}}$. The coefficient of viscosity of the solution is ______ poise (in nearest integer). (the density of air is negligible).

Answer: 11

Question 59

Physics · Work, Energy and Power · Numerical

A block of mass 'm' (as shown in figure) moving with kinetic energy $E$ compresses a spring through a distance $25 \, \mathrm{cm}$ when, its speed is halved. The value of spring constant of used spring will be $nE \, \mathrm{Nm}^{-1}$ for n = _____________.

Answer: 24

Question 60

Physics · System of Particles and Rotational Motion · Numerical

Four identical discs each of mass 'M' and diameter 'a' are arranged in a small plane as shown in figure. If the moment of inertia of the system about OO' is $\frac{x}{4}Ma^2$. Then, the value of $x$ will be _____.

Answer: 3

Question 61

Chemistry · Structure of Atom · Single correct

Identify the incorrect statement from the following.

1. A circular path around the nucleus in which an electron moves is proposed as Bohr's orbit.
2. An orbital is the one electron wave function ($\Psi$) in an atom.
3. The existence of Bohr's orbits is supported by hydrogen spectrum.
4. Atomic orbital is characterised by the quantum numbers $n$ and $l$ only.

Answer: (d)

Question 62

Chemistry · Thermodynamics · Single correct

Which of the following relation is not correct?

1. $\Delta H = \Delta U - P \Delta V$
2. $\Delta U = q + W$
3. $\Delta S_{sys} + \Delta S_{surr} \geq 0$
4. $\Delta G = \Delta H - T \Delta S$

Answer: (a)

Question 63

Chemistry · Electrochemistry · Single correct

Match List-I with List-II. Choose the correct answer from the options given below :

1. $(A)(a) - (I), (b) - (II), (c) - (III), (b) - (IV)$
2. $(B)(a) - (II), (b) - (I), (c) - (IV), (d) - (III)$
3. $(C)(a) - (II), (b) - (I), (c) - (IV), (d) - (III)$
4. $(D)(a) - (IV), (b) - (II), (c) - (I), (d) - (IV)$

Answer: (c)

Question 64

Chemistry · The p-Block Elements (Group 15, 16, 17 & 18) · Single correct

Match List-I with List-II. \begin{tabular}{|c|p{7cm}|c|l|} \hline \multicolumn{2}{|c|}{\textbf{List-I}} & \multicolumn{2}{c|}{\textbf{List-II}} \\ \multicolumn{2}{|c|}{\textbf{Reaction}} & \multicolumn{2}{c|}{\textbf{Catalyst}} \\ \hline (A) & $4NH_3(g)+5O_2(g)\rightarrow4NO(g)+6H_2O(g)$ & (I) & NO(g) \\ \hline (B) & $N_2(g)+3H_2(g)\rightarrow2NH_3(g)$ & (II) & H$_2$SO$_4(l)$ \\ \hline (C) & $C_{12}H_{22}O_{11}(aq)+H_2O(l)\rightarrow C_6H_{12}O_6$ (Glucose) $+$ C$_6$H$_{12}$O$_6$ (Fructose) & (III) & Pt(s) \\ \hline (D) & $2SO_2(g)+O_2(g)\rightarrow2SO_3(g)$ & (IV) & Fe(s) \\ \hline \end{tabular} Choose the correct answer from the options given below:

1. (A) - (II), (B) - (III), $(C)$ - (I), (D) - (IV)
2. (A) - (III), (B) - (II), $(C)$ - (I), (D) - (IV)
3. (A) - (III), (B) - (IV), $(C)$ - (II), (D) - (I)
4. (A) - (III), (B) - (II), $(C)$ - (IV), (D) - (I)

Answer: (c)

Question 65

Chemistry · Classification of Elements and Periodicity in Properties · Single correct

In which of the following pairs, electron gain enthalpies of constituent elements are nearly the same or identical? (A) Rb and Cs (B) Na and K $(C)$ Ar and Kr (D) I and At Choose the correct answer from the options given below:

1. $(A)$ and (B) only
2. $(B)$ and $(C)$ only
3. $(A)$ and $(C)$ only
4. $(C)$ and (D) only

Answer: (c)

Question 66

Chemistry · General Principles and Processes of Isolation of Elements · Single correct

Which of the reaction is suitable for concentrating ore by leaching process?

1. $2Cu_2S + 3O_2 \rightarrow 2Cu_2O + 2SO_2$
2. $Fe_3O_4 + CO \rightarrow 3FeO + CO_2$
3. $Al_2O_3 + 2NaOH + 3H_2O \rightarrow 2Na[Al(OH)_4]$
4. $Al_2O_3 + 6Mg \rightarrow 6MgO + 4Al$

Answer: (c)

Question 67

Chemistry · Hydrogen · Single correct

The metal salts formed during softening of hardwater using Clark’s method are:

1. $Ca(OH)_2$ and $Mg(OH)_2$
2. $CaCO_3$ and $Mg(OH)_2$
3. $Ca(OH)_2$ and $MgCO_3$
4. $CaCO_3$ and $MgCO_3$

Answer: (b)

Question 68

Chemistry · The s-Block Elements · Single correct

Which of the following statement is incorrect ?

1. Low solubility of LiF in water is due to its small hydration enthalpy.
2. $\mathrm{KO_2}$ is paramagnetic.
3. Solution of sodium in liquid ammonia is conducting in nature.
4. Sodium metal has higher density than potassium metal

Answer: (a)

Question 69

Chemistry · The p-Block Elements (Group 15, 16, 17 & 18) · Single correct

Match List-I with List-II, match the gas evolved during each reaction. \begin{tabular}{|c|p{5.5cm}|c|c|} \hline \multicolumn{2}{|c|}{List-I} & \multicolumn{2}{c|}{List-II} \\ \hline (A) & $(NH_4)_2Cr_2O_7\xrightarrow{\Delta}$ & (I) & H$_2$ \\ \hline (B) & $KMnO_4+HCl\rightarrow$ & (II) & N$_2$ \\ \hline (C) & $Al+NaOH+H_2O\rightarrow$ & (III) & O$_2$ \\ \hline (D) & $NaNO_3\xrightarrow{\Delta}$ & (IV) & Cl$_2$ \\ \hline \end{tabular} Choose the correct answer from the options given below :

1. (A) - (II), (B) - (III), $C)$ - (I), (D) - (IV)
2. (A)- (III), (B) - (I), $C)$ - (IV), (D) - (II)
3. (A)- (II), (B) - (IV), $C)$ - (I), (D) - (III)
4. (A)- (III), (B) - (IV), $C)$ - (I), (D) - (II)

Answer: (c)

Question 70

Chemistry · The d-and f-Block Elements · Single correct

Which of the following has least tendency to liberate $\mathrm{H}_2$ from mineral acids?

1. Cu
2. Mn
3. Ni
4. Zn

Answer: (a)

Question 71

Chemistry · Environmental Chemistry · Single correct

Given below are two statements : Statement I : In polluted water values of both dissolved oxygen and BOD are very low. Statement II : Eutrophication results in decrease in the amount of dissolved oxygen. In the light of the above statements, choose the most appropriate answer from the options given below :

1. Both Statement I and Statement II are true
2. Both Statement I and Statement II are false
3. Statement I is true but Statement II is false
4. Statement I is false but Statement II is true

Answer: (d)

Question 72

Chemistry · Co-ordination Compounds · Single correct

Match List-I with List-II. Choose the correct answer from the options given below:

1. (A) - (II), (B) - (I), ($C$) - (IV), (D) - (III)
2. (A) - (IV), (B) - (III), ($C$) - (I), (D) - (II)
3. (A) - (III), (B) - (IV), ($C$) - (I), (D) - (II)
4. (A) - (IV), (B) - (III), ($C$) - (II), (D) - (I)

Answer: (c)

Question 73

Chemistry · Hydrocarbons · Single correct

Choose the correct option for the following reactions.

1. 'A' and 'B' are both Markovnikov addition products.
2. 'A' is Markovnikov product and 'B' is anti-Markovnikov product.
3. 'A' and 'B' are both anti-Markovnikov products.
4. 'B' is Markovnikov and 'A' is anti-Markovnikov product.

Answer: (b)

Question 74

Chemistry · Organic Chemistry — Some Basic Principles & Techniques · Single correct

Among the following marked proton of which compound shows lowest $pK_a$ value?

Answer: (c)

Question 75

Chemistry · Alcohols, Phenols and Ethers · Single correct

Identify the major product A and B for the below given reaction sequence.

Answer: (b)

Question 76

Chemistry · Amines · Single correct

Identify the correct statement for the below given transformation.

1. $A - CH_3CH_2CH = CH-CH_3$, $B - CH_3CH_2CH_2CH = CH_2$, Saytzeff products
2. $A - CH_3CH_2CH = CH-CH_3$, $B - CH_3CH_2CH_2CH = CH_2$, Hafmann products
3. $A - CH_3CH_2CH_2CH = CH_2$, $B - CH_3CH_2CH = CHCH_3$, Hofmann products
4. $A - CH_3CH_2CH_2CH = CH_2$, $B - CH_3CH_2CH = CHCH_3$, Saytzeff products

Answer: (c)

Question 77

Chemistry · Polymers · Single correct

Terylene polymer is obtained by condensation of:

1. Ethane-1, 2-diol and Benzene-1, 3 dicarboxylic acid
2. Propane-1, 2-diol and Benzene-1, 4 dicarboxylic acid
3. Ethane-1, 2-diol and Benzene-1, 4 dicarboxylic acid
4. Ethane-1, 2-diol and Benzene-1, 2 dicarboxylic acid

Answer: (c)

Question 78

Chemistry · Biomolecules · Single correct

For the below given cyclic hemiacetal (X), the correct pyranose structure is:

Answer: (d)

Question 79

Chemistry · Chemistry in Everyday Life · Single correct

Statements about Enzyme Inhibitor Drugs are given below : (A) There are Competitive and Non-competitive inhibitor drugs. (B) These can bind at the active sites and allosteric sites. (C) Competitive Drugs are allosteric site blocking drugs. (D) Non-competitive Drugs are active site blocking drugs. Choose the correct answer from the options given below :

1. $(A), (D)$ only
2. $(A), (C)$ only
3. $(A), (B)$ only
4. $(A), (B), (C)$ only

Answer: (c)

Question 80

Chemistry · Chemical Kinetics and Nuclear Chemistry · Single correct

For kinetic study of the reaction of iodide ion with $H_2O_2$ at room temperature: (A) Always use freshly prepared starch solution. (B) Always keep the concentration of sodium thiosulphate solution less than that of KI solution. (C) Record the time immediately after the appearance of blue colour. (D) Record the time immediately before the appearance of blue colour. (E) Always keep the concentration of sodium thiosulphate solution more than that of KI solution. Choose the correct answer from the options given below:

1. , (B), $(C)$ only
2. , (D), (E) only
3. , (E) only
4. , (B), (E) only

Answer: (a)

Question 81

Chemistry · Some Basic Concepts of Chemistry · Numerical

In the given reaction, $$\mathrm{X + Y + 3Z \rightleftharpoons XYZ_3}$$ if one mole of each of X and Y with 0.05 mol of Z gives compound XYZ$_3$. (Given : Atomic masses of X, Y and Z are 10, 20 and 30 amu, respectively). The yield of XYZ$_3$ is _________ g. (Nearest integer)

Answer: 2

Question 82

Chemistry · The Solid State · Numerical

An element M crystallises in a body centred cubic unit cell with a cell edge of 300 pm. The density of the element is $6.0 \, \mathrm{g \, cm^{-3}}$. The number of atoms present in 180 g of the element is _____ $\times \, 10^{23}$. (Nearest integer)

Answer: 22

Question 83

Chemistry · Chemical Bonding and Molecular Structure · Numerical

The number of paramagnetic species among the following is __________. $\mathrm{B_2}$, $\mathrm{Li_2}$, $\mathrm{C_2}$, $\mathrm{C_2^-}$, $\mathrm{O_2^{2-}}$, $\mathrm{O_2}$ and $\mathrm{He_2^+}$

Answer: 4

Question 84

Chemistry · Solutions · Numerical

$150\,\mathrm{g}$ of acetic acid was contaminated with $10.2\,\mathrm{g}$ of ascorbic acid ($\mathrm{C_6H_8O_6}$) to lower down its freezing point by $(x \times 10^{-1})°\mathrm{C}$. The value of $x$ is \_\_\_\_. (Nearest integer) [Given $K_f = 3.9\,\mathrm{K\,kg\,mol^{-1}}$; Molar mass of ascorbic acid $= 176\,\mathrm{g\,mol^{-1}}$]

Answer: 15

Question 85

Chemistry · Equilibrium · Numerical

$K_a$ for butyric acid $(\mathrm{C_3H_7COOH})$ is $2 \times 10^{-5}$. The pH of $0.2 \, \mathrm{M}$ solution of butyric acid is $\_\_\_ \times 10^{-1}$. (Nearest integer) [Given $\log 2 = 0.30$]

Answer: 27

Question 86

Chemistry · Chemical Kinetics and Nuclear Chemistry · Numerical

For the given first order reaction A $\rightarrow$ B the half life of the reaction is 0.3010 min. The ratio of the initial concentration of reactant to the concentration of reactant at time 2.0 min will be equal to . (Nearest integer)

Answer: 100

Question 87

Chemistry · The p-Block Elements (Group 15, 16, 17 & 18) · Numerical

The number of interhalogens from the following having square pyramidal structure is : $ClF_3, IF_7, BrF_5, BrF_3, I_2Cl_6, IF_5, ClF, ClF_5$

Answer: 3

Question 88

Chemistry · The d-and f-Block Elements · Numerical

The disproportionation of $\mathrm{MnO}_4^{2-}$ in acidic medium resulted in the formation of two manganese compounds A and B. If the oxidation state of Mn in B is smaller than that of A, then the spin-only magnetic moment ($\mu$) value of B in BM is ___________. (Nearest integer)

Answer: 4

Question 89

Chemistry · Co-ordination Compounds · Numerical

Total number of relatively more stable isomer(s) possible for octahedral complex $[Cu(en)_2(SCN)_2]$ will be __________.

Answer: 3

Question 90

Chemistry · Some Basic Concepts of Chemistry · Numerical

On complete combustion of $0.492$ g of an organic compound containing C, H and O, $0.7938$ g of $\mathrm{CO_2}$ and $0.4428$ g of $\mathrm{H_2O}$ was produced. The % composition of oxygen in the compound is _____.

Answer: 46