JEE Main 29 January 2023 Shift 1 Question Paper

Question 1

Let $\alpha$ and $\beta$ be real numbers. Consider a $3 	imes 3$ matrix $A$ such that $A^2 = 3A + \alpha I$. If $A^4 = 21A + \beta I$, then

Accepted Answer

(A) $\beta = -8$

Answer

(B) $\beta = 8$

Answer

(C) $\alpha = 4$

Answer

(D) $\alpha = 1$

Question 2

Let $x = 2$ be a root of the equation $x^2 + px + q = 0$ and $f(x) = \begin{cases} \dfrac{1 - \cos(x^2 - 4px + q^2 + 8q + 16)}{(x - 2p)^4}, & x 
eq 2p \ 0, & x = 2p \end{cases}$. Then $\lim_{x 	o 2p^+} [f(x)]$, where $[\cdot]$ denotes greatest integer function, is

Accepted Answer

(A) 0

Answer

(B) $-1$

Answer

(C) 2

Answer

(D) 1

Question 3

Let $B$ and $C$ be the two points on the line $y + x = 0$ such that $B$ and $C$ are symmetric with respect to the origin. Suppose $A$ is a point on $y - 2x = 2$ such that $	riangle ABC$ is an equilateral triangle. Then, the area of the $	riangle ABC$ is

Accepted Answer

(D) $\dfrac{8}{\sqrt{3}}$

Answer

(A) $\dfrac{10}{\sqrt{3}}$

Answer

(B) $3\sqrt{3}$

Answer

(C) $2\sqrt{3}$

Question 4

Consider the following system of equations $\alpha x + 2y + z = 1$, $2\alpha x + 3y + z = 1$, $3x + \alpha y + 2z = \beta$ for some $\beta \in \mathbb{R}$. Then which of the following is NOT correct.

Accepted Answer

(D) It has no solution for $\alpha = -1$ and for all $\beta \in \mathbb{R}$

Answer

(A) It has a solution if $\alpha = -1$ and $\beta 
eq 2$

Answer

(B) It has a solution for all $\alpha 
eq -1$ and $\beta = 2$

Answer

(C) It has no solution for $\alpha = 3$ and for all $\beta 
eq 2$

Question 5

Let $y = f(x)$ be the solution of the differential equation $y(x + 1)dx - x^2 dy = 0$, $y(1) = e$. Then $\lim_{x 	o 0^+} f(x)$ is equal to

Accepted Answer

(C) 0

Answer

(A) $\dfrac{1}{e^2}$

Answer

(B) $e^2$

Answer

(D) $\dfrac{1}{e}$

Question 6

The domain of $f(x) = \dfrac{\log_{(x+1)}(x - 2)}{e^{2\log_e x} - (2x + 3)}$, $x \in \mathbb{R}$ is

Accepted Answer

(C) $(2, \infty) - \{3\}$

Answer

(A) $\mathbb{R} - \{3\}$

Answer

(B) $(-1, \infty) - \{3\}$

Answer

(D) $\mathbb{R} - \{-1, 3\}$

Question 7

Fifteen football players of a club-team are given 15 T-shirts with their names written on the backside. If the players pick up the T-shirts randomly, then the probability that at least 3 players pick the correct T-shirt is

Accepted Answer

(B) $\dfrac{1}{6}$

Answer

(A) $\dfrac{5}{24}$

Answer

(C) $\dfrac{5}{36}$

Answer

(D) $\dfrac{2}{15}$

Question 8

Let $[x]$ denote the greatest integer $\leq x$. Consider the function $f(x) = \max\{x^2, 1 + [x]\}$. Then the value of the integral $\int_0^2 f(x)\,dx$ is

Accepted Answer

(A) $\dfrac{5 + 4\sqrt{2}}{3}$

Answer

(B) $\dfrac{4 + 5\sqrt{2}}{3}$

Answer

(C) $\dfrac{1 + 5\sqrt{2}}{3}$

Answer

(D) $\dfrac{8 + 4\sqrt{2}}{3}$

Question 9

For two non-zero complex numbers $z_1$ and $z_2$, if $ ext{Re}(z_1 \bar{z_2}) = 0$ and $ ext{Re}(z_1 + z_2) = 0$, then which of the following are possible? A. $ ext{Im}(z_1) > 0$ and $ ext{Im}(z_2) > 0$ B. $ ext{Im}(z_1) < 0$ and $ ext{Im}(z_2) > 0$ C. $ ext{Im}(z_1) > 0$ and $ ext{Im}(z_2) < 0$ D. $ ext{Im}(z_1) < 0$ and $ ext{Im}(z_2) < 0$. Choose the correct answer from the options given below:

Accepted Answer

(C) B and C

Answer

(A) B and D

Answer

(B) A and B

Answer

(D) A and C

Question 10

If the vectors $\vec{a} = \lambda\hat{i} + \mu\hat{j} + 4\hat{k}$, $\vec{b} = 2\hat{i} + 4\hat{j} - 2\hat{k}$ and $\vec{c} = 2\hat{i} + 3\hat{j} + \hat{k}$ are coplanar and the projection of $\vec{a}$ on the vector $\vec{b}$ is $\sqrt{54}$ units, then the sum of all possible values of $\lambda + \mu$ is equal to

Accepted Answer

(B) 24

Answer

(A) 0

Answer

(C) 6

Answer

(D) 18

Question 11

Let $f(	heta) = 3\left(\sin^4\left(\dfrac{3\pi}{2} - 	hetaight) + \sin^4(3\pi + 	heta)ight) - 2(1 - \sin^2 2	heta)$ and $S = \left\{	heta \in [0, \pi] : f'(	heta) = -\dfrac{\sqrt{3}}{2}ight\}$. If $\beta \in S$, then $f(\beta)$ is equal to

Accepted Answer

(B) $\dfrac{3}{2}$

Answer

(A) $\dfrac{5}{4}$

Answer

(C) $\dfrac{9}{8}$

Answer

(D) $\dfrac{11}{8}$

Question 12

If $p$, $q$ and $r$ three propositions, then which of the following combination of truth values of $p$, $q$ and $r$ makes the logical expression $\{(p \vee q) \wedge ((\sim p) \vee r)\} 	o ((\sim q) \vee r)$ false?

Accepted Answer

(C) p = F, q = T, r = F

Answer

(A) p = T, q = T, r = F

Answer

(B) p = T, q = F, r = T

Answer

(D) p = T, q = F, r = F

Question 13

Let $\Delta$ be the area of the region $\{(x, y) \in \mathbb{R}^2 : x^2 + y^2 \leq 21, y^2 \leq 4x, x \geq 1\}$. Then $\dfrac{1}{2}\left(\Delta - 21\sin^{-1}\dfrac{2}{\sqrt{7}}\right)$ is equal to

Accepted Answer

(B) $\sqrt{3} - \dfrac{4}{3}$

Answer

(A) $2\sqrt{3} - \dfrac{2}{3}$

Answer

(C) $\sqrt{3} - \dfrac{2}{3}$

Answer

(D) $2\sqrt{3} - \dfrac{1}{3}$

Question 14

A light ray emits from the origin making an angle $30^\circ$ with the positive x-axis. After getting reflected by the line $x + y = 1$, if this ray intersects x-axis at $Q$, then the abscissa of $Q$ is

Accepted Answer

(B) $\dfrac{2}{3+\sqrt{3}}$

Answer

(A) $\dfrac{\sqrt{3}}{2(\sqrt{3}+1)}$

Answer

(C) $\dfrac{2}{\sqrt{3}-1}$

Answer

(D) $\dfrac{2}{3-\sqrt{3}}$

Question 15

Let $A = \{(x, y) \in \mathbb{R}^2 : y \geq 0, 2x \leq y \leq \sqrt{4 - (x - 1)^2}\}$ and $B = \{(x, y) \in \mathbb{R} 	imes \mathbb{R} : 0 \leq y \leq \min\{2x, \sqrt{4 - (x - 1)^2}\}\}$. Then the ratio of the area of $A$ to the area of $B$ is

Accepted Answer

(C) $\dfrac{\pi - 1}{\pi + 1}$

Answer

(A) $\dfrac{\pi + 1}{\pi - 1}$

Answer

(B) $\dfrac{\pi}{\pi - 1}$

Answer

(D) $\dfrac{\pi}{\pi + 1}$

Question 16

Let $\lambda 
eq 0$ be a real number. Let $\alpha, \beta$ be the roots of the equation $14x^2 - 31x + 3\lambda = 0$ and $\alpha, \gamma$ be the roots of the equation $35x^2 - 53x + 4\lambda = 0$. Then $\dfrac{3\alpha}{\beta}$ and $\dfrac{4\alpha}{\gamma}$ are the roots of the equation

Accepted Answer

(A) $49x^2 - 245x + 250 = 0$

Answer

(B) $7x^2 + 245x - 250 = 0$

Answer

(C) $7x^2 - 245x + 250 = 0$

Answer

(D) $49x^2 + 245x + 250 = 0$

Question 17

Let the tangents at the points $A(4, -11)$ and $B(8, -5)$ on the circle $x^2 + y^2 - 3x + 10y - 15 = 0$, intersect at the point $C$. Then the radius of the circle, whose centre is $C$ and the line joining $A$ and $B$ is its tangent, is equal to

Accepted Answer

(D) $\dfrac{2\sqrt{13}}{3}$

Answer

(A) $2\sqrt{13}$

Answer

(B) $\sqrt{13}$

Answer

(C) $\dfrac{3\sqrt{3}}{4}$

Question 18

Let $f(x) = x + \dfrac{a}{\pi^2 - 4}\sin x + \dfrac{b}{\pi^2 - 4}\cos x$, $x \in \mathbb{R}$ be a function which satisfies $f(x) = x + \int_0^{\pi/2} \sin(x + y) f(y)\,dy$. Then $(a + b)$ is equal to

Accepted Answer

(B) $-2\pi(\pi + 2)$

Answer

(A) $-2\pi(\pi - 2)$

Answer

(C) $-\pi(\pi - 2)$

Answer

(D) $-\pi(\pi + 2)$

Question 19

Let $f: \mathbb{R} 	o \mathbb{R}$ be a function such that $f(x) = \dfrac{x^2 + 2x + 1}{x^2 + 1}$. Then

Accepted Answer

(A) $f(x)$ is one-one in $[1, \infty)$ but not in $(-\infty, \infty)$

Answer

(B) $f(x)$ is one-one in $(-\infty, \infty)$

Answer

(C) $f(x)$ is many-one in $(-\infty, -1)$

Answer

(D) $f(x)$ is many-one in $(1, \infty)$

Question 20

Three rotten apples are mixed accidently with seven good apples and four apples are drawn one by one without replacement. Let the random variable $X$ denote the number of rotten apples. If $\mu$ and $\sigma^2$ represent mean and variance of $X$, respectively, then $10(\mu^2 + \sigma^2)$ is equal to

Accepted Answer

(D) 20

Answer

(A) 250

Answer

(B) 25

Answer

(C) 30

Question 21

Let the co-ordinates of one vertex of $	riangle ABC$ be $A(0, 2, \alpha)$ and the other two vertices lie on the line $\dfrac{x + \alpha}{5} = \dfrac{y - 1}{2} = \dfrac{z + 4}{3}$. For $\alpha \in \mathbb{Z}$, if the area of $	riangle ABC$ is 21 sq. units and the line segment $BC$ has length $2\sqrt{21}$ units, then $\alpha^2$ is equal to ________ .

Accepted Answer

9

Question 22

Let $f: \mathbb{R} 	o \mathbb{R}$ be a differentiable function that satisfies the relation $f(x + y) = f(x) + f(y) - 1$, $\forall x, y \in \mathbb{R}$. If $f'(0) = 2$, then $|f(-2)|$ is equal to ________ .

Accepted Answer

3

Question 23

Suppose $f$ is a function satisfying $f(x + y) = f(x) + f(y)$ for all $x, y \in \mathbb{N}$ and $f(1) = \dfrac{1}{5}$. If $\sum_{n=1}^{m} \dfrac{f(n)}{n(n + 1)(n + 2)} = \dfrac{1}{12}$, then $m$ is equal to ________ .

Accepted Answer

10

Question 24

Let the coefficients of three consecutive terms in the binomial expansion of $(1 + 2x)^n$ be in the ratio $2 : 5 : 8$. Then the coefficient of the term, which is in the middle of these three terms, is ________ .

Accepted Answer

1120

Question 25

Let $a_1, a_2, a_3, \ldots$ be a GP of increasing positive numbers. If the product of fourth and sixth terms is 9 and the sum of fifth and seventh terms is 24, then $a_1 a_9 + a_2 a_4 a_9 + a_5 + a_7$ is equal to ________ .

Accepted Answer

60

Question 26

Let the equation of the plane $P$ containing the line $x + 10 = \dfrac{8 - y}{2} = z$ be $ax + by + 3z = 2(a + b)$ and the distance of the plane $P$ from the point $(1, 27, 7)$ be $c$. Then $a^2 + b^2 + c^2$ is equal to ________ .

Accepted Answer

355

Question 27

If the co-efficient of $x^9$ in $\left(\alpha x^3 + \dfrac{1}{\beta x}\right)^{11}$ and the co-efficient of $x^{-9}$ in $\left(\alpha x - \dfrac{1}{\beta x^3}\right)^{11}$ are equal, then $(\alpha\beta)^2$ is equal to ________ .

Accepted Answer

1

Question 28

Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be three non-zero non-coplanar vectors. Let the position vectors of four points $A$, $B$, $C$ and $D$ be $\vec{a} - \vec{b} + \vec{c}$, $\lambda\vec{a} - 3\vec{b} + 4\vec{c}$, $-\vec{a} + 2\vec{b} - 3\vec{c}$ and $2\vec{a} - 4\vec{b} + 6\vec{c}$ respectively. If $\vec{AB}$, $\vec{AC}$ and $\vec{AD}$ are coplanar, then $\lambda$ is equal to ________ .

Accepted Answer

2

Question 29

Five digit numbers are formed using the digits 1, 2, 3, 5, 7 with repetitions and are written in descending order with serial numbers. For example, the number 77777 has serial number 1. Then the serial number of 35337 is ________ .

Accepted Answer

1436

Question 30

If all the six digit numbers $x_1 x_2 x_3 x_4 x_5 x_6$ with $0 < x_1 < x_2 < x_3 < x_4 < x_5 < x_6$ are arranged in the increasing order, then the sum of the digits in the 72th number is ________ .

Accepted Answer

32

Question 31

Find the mutual inductance in the arrangement, when a small circular loop of wire of radius $R$ is placed inside a large square loop of wire of side $L$ ($L \gg R$). The loops are coplanar and their centers coincide:

Accepted Answer

(D) $M=\dfrac{2\sqrt{2}\,\mu_0 R^2}{L}$

Answer

(A) $M=\dfrac{\sqrt{2}\,\mu_0 R^2}{L}$

Answer

(B) $M=\dfrac{2\sqrt{2}\,\mu_0 R}{L^2}$

Answer

(C) $M=\dfrac{\sqrt{2}\,\mu_0 R}{L^2}$

Question 32

The threshold wavelength for photoelectric emission from a material is 5500 Å. Photoelectrons will be emitted, when this material is illuminated with monochromatic radiation from a: (A) 75 W infra-red lamp (B) 10 W infra-red lamp (C) 75 W ultra-violet lamp (D) 10 W ultra-violet lamp. Choose the correct answer from the options given below:

Accepted Answer

(D) C and D only

Answer

(A) B and C only

Answer

(B) A and D only

Answer

(C) C only

Question 33

Match List I with List II and choose the correct answer from the options given below:

Accepted Answer

(C) A-III, B-II, C-IV, D-I

Answer

(A) A-II, B-III, C-I, D-IV

Answer

(B) A-II, B-III, C-IV, D-I

Answer

(D) A-III, B-II, C-I, D-IV

Question 34

In a cuboid of dimension $2L 	imes 2L 	imes L$, a charge $q$ is placed at the center of the surface '$S$' having area $4L^2$. The flux through the opposite surface to '$S$' is given by:

Accepted Answer

(B) $\dfrac{q}{6\varepsilon_0}$

Answer

(A) $\dfrac{q}{12\varepsilon_0}$

Answer

(C) $\dfrac{q}{3\varepsilon_0}$

Answer

(D) $\dfrac{q}{2\varepsilon_0}$

Question 35

A person observes two moving trains, 'A' reaching the station and 'B' leaving the station with equal speed of 30 m/s. If both trains emit sounds with frequency 300 Hz (speed of sound = 330 m/s), the approximate difference of frequencies heard by the person will be:

Accepted Answer

(A) 55 Hz

Answer

(B) 80 Hz

Answer

(C) 33 Hz

Answer

(D) 10 Hz

Question 36

A block of mass $m$ slides down the plane inclined at angle $30^\circ$ with an acceleration $\dfrac{g}{4}$. The value of coefficient of kinetic friction will be:

Accepted Answer

(A) $\dfrac{1}{2\sqrt{3}}$

Answer

(B) $\dfrac{\sqrt{3}}{2}$

Answer

(C) $\dfrac{2\sqrt{3}+1}{2}$

Answer

(D) $\dfrac{2\sqrt{3}-1}{2}$

Question 37

A bicycle tyre is filled with air having pressure of 270 kPa at 27 °C. The approximate pressure of the air in the tyre when the temperature increases to 36 °C is:

Accepted Answer

(D) 278 kPa

Answer

(A) 270 kPa

Answer

(B) 262 kPa

Answer

(C) 360 kPa

Question 38

A single current carrying loop of wire carrying current $I$ flowing in anticlockwise direction seen from $+z$ direction and lying in the $xy$ plane is shown in figure. The plot of $\hat{j}$ component of magnetic field ($B_y$) at a distance '$a$' (less than radius of the coil) and on the $yz$ plane vs $z$ coordinate looks like:

Accepted Answer

(A) (graph 1)

Answer

(B) (graph 2)

Answer

(C) (graph 3)

Answer

(D) (graph 4)

Question 39

Surface tension of a soap bubble is $2.0 	imes 10^{-2}$ N m$^{-1}$. Work done to increase the radius of soap bubble from 3.5 cm to 7 cm will be: (Take $\pi = \dfrac{22}{7}$)

Accepted Answer

(D) $18.48 	imes 10^{-4}$ J

Answer

(A) $9.24 	imes 10^{-4}$ J

Answer

(B) $5.76 	imes 10^{-4}$ J

Answer

(C) $0.72 	imes 10^{-4}$ J

Question 40

Given below are two statements: One is labelled as Assertion A and the other is labelled as Reason R.
Assertion A: If $dQ$ and $dW$ represent the heat supplied to the system and the work done on the system respectively, then according to the first law of thermodynamics $dQ = dU - dW$.
Reason R: First law of thermodynamics is based on law of conservation of energy.
In the light of the above statements, choose the correct answer from the options given below:

Accepted Answer

(A) Both A and R are correct and R is the correct explanation of A

Answer

(B) A is not correct but R is correct

Answer

(C) A is correct but R is not correct

Answer

(D) Both A and R are correct but R is not the correct explanation of A

Question 41

If a radioactive element having half-life of 30 min is undergoing beta decay, the fraction of radioactive element remains undecayed after 90 min will be:

Accepted Answer

(A) $\dfrac{1}{8}$

Answer

(B) $\dfrac{1}{2}$

Answer

(C) $\dfrac{1}{4}$

Answer

(D) $\dfrac{1}{16}$

Question 42

Two particles of equal mass '$m$' move in a circle of radius '$r$' under the action of their mutual gravitational attraction. The speed of each particle will be:

Accepted Answer

(B) $\sqrt{\dfrac{Gm}{4r}}$

Answer

(A) $\sqrt{\dfrac{4Gm}{r}}$

Answer

(C) $\sqrt{\dfrac{Gm}{r}}$

Answer

(D) $\sqrt{\dfrac{Gm}{2r}}$

Question 43

If the height of transmitting and receiving antennas are 80 m each, the maximum line of sight distance will be: (Given: Earth's radius = $6.4 	imes 10^6$ m)

Accepted Answer

(D) 64 km

Answer

(A) 28 km

Answer

(B) 36 km

Answer

(C) 32 km

Question 44

A car is moving on a horizontal curved road with radius 50 m. The approximate maximum speed of the car will be, if friction between tyres and road is 0.34. [take $g = 10$ ms$^{-2}$]

Accepted Answer

(B) 13 ms$^{-1}$

Answer

(A) 17 ms$^{-1}$

Answer

(C) 22.4 ms$^{-1}$

Answer

(D) 3.4 ms$^{-1}$

Question 45

Ratio of thermal energy released in two resistors $R$ and $3R$ connected in parallel in an electric circuit is:

Accepted Answer

(D) 3 : 1

Answer

(A) 1 : 27

Answer

(B) 1 : 1

Answer

(C) 1 : 3

Question 46

A stone is projected at angle $30^\circ$ to the horizontal. The ratio of kinetic energy of the stone at point of projection to its kinetic energy at the highest point of flight will be:

Accepted Answer

(D) 4 : 3

Answer

(A) 1 : 2

Answer

(B) 1 : 4

Answer

(C) 4 : 1

Question 47

Which of the following are true? A. Speed of light in vacuum is dependent on the direction of propagation. B. Speed of light in a medium is independent of the wavelength of light. C. The speed of light is independent of the motion of the source. D. The speed of light in a medium is independent of intensity. Choose the correct answer from the options given below:

Accepted Answer

(A) C and D only

Answer

(B) B and C only

Answer

(C) A and C only

Answer

(D) B and D only

Question 48

In a Young's double slit experiment, two slits are illuminated with a light of wavelength 800 nm. The line joining $A_1P$ is perpendicular to $A_1A_2$ as shown in the figure. If the first minimum is detected at $P$, the value of slits separation '$a$' will be: (The distance of screen from slits $D = 5$ cm)

Accepted Answer

(D) 0.2 mm

Answer

(A) 0.5 mm

Answer

(B) 0.1 mm

Answer

(C) 0.4 mm

Question 49

Which one of the following statements is not correct in the case of light emitting diodes? A. It is a heavily doped p-n junction. B. It emits light only when it is forward biased. C. It emits light only when it is reverse biased. D. The energy of the light emitted is equal to or slightly less than the energy gap of the semiconductor used. Choose the correct answer from the options given below:

Accepted Answer

(C) C

Answer

(A) A

Answer

(B) C and D

Answer

(D) B

Question 50

The magnitude of magnetic induction at mid point $O$ due to the current arrangement as shown in the figure will be:

Accepted Answer

(A) $\dfrac{\mu_0 I}{\pi a}$

Answer

(B) $\dfrac{\mu_0 I}{2\pi a}$

Answer

(C) $0$

Answer

(D) $\dfrac{\mu_0 I}{4\pi a}$

Question 51

As shown in the figure, three identical polaroids $P_1$, $P_2$ and $P_3$ are placed one after another. The pass axis of $P_2$ and $P_3$ are inclined at angle of $60^\circ$ and $90^\circ$ with respect to the axis of $P_1$. The source $S$ has an intensity of $256\ \dfrac{W}{m^2}$. The intensity of light at point $O$ is ________ $\dfrac{W}{m^2}$.

Accepted Answer

24

Question 52

A 0.4 kg mass takes 8 s to reach the ground when dropped from a certain height '$P$' above the surface of earth. The loss of potential energy in the last second of fall is ________ J. (Take $g = 10$ m/s$^2$)

Accepted Answer

300

Question 53

Two simple harmonic waves having equal amplitudes of 8 cm and equal frequency of 10 Hz are moving along the same direction. The resultant amplitude is also 8 cm. The phase difference between the individual waves is ________ degree.

Accepted Answer

120

Question 54

A tennis ball is dropped on to the floor from a height of 9.8 m. It rebounds to a height 5.0 m. The ball comes in contact with the floor for 0.2 s. The average acceleration during contact is ________ ms$^{-2}$. (Given $g = 10$ ms$^{-2}$)

Accepted Answer

120

Question 55

A certain elastic conducting material is stretched into a circular loop. It is placed with its plane perpendicular to a uniform magnetic field $B = 0.8$ T. When released the radius of the loop starts shrinking at a constant rate of 2 cm s$^{-1}$. The induced emf in the loop at an instant when the radius of the loop is 10 cm will be ________ mV.

Accepted Answer

10

Question 56

A solid sphere of mass 2 kg is making pure rolling on a horizontal surface with kinetic energy 2240 J. The velocity of centre of mass of the sphere will be ________ ms$^{-1}$.

Accepted Answer

40

Question 57

A body cools from $60^\circ$C to $40^\circ$C in 6 minutes. If the temperature of surroundings is $10^\circ$C, then after the next 6 minutes its temperature will be ________ $^\circ$C.

Accepted Answer

28

Question 58

In a metre bridge experiment the balance point is obtained when the gaps are closed by $2\,\Omega$ and $3\,\Omega$. A shunt of $X$ is added to the $3\,\Omega$ resistor to shift the balancing point by 22.5 cm. The value of $X$ is ________ $\Omega$.

Accepted Answer

2

Question 59

A point charge $q_1 = 4q_0$ is placed at the origin. Another point charge $q_2 = -q_0$ is placed at $x = 12$ cm. The charge of a proton is $q_0$. The proton is placed on the x-axis so that the electrostatic force on the proton is zero. In this situation, the position of the proton from the origin is ________ cm.

Accepted Answer

24

Question 60

A radioactive element $_{92}^{242}X$ emits two $\alpha$-particles, one electron and two positrons. The product nucleus is represented by $_{P}^{234}Y$. The value of $P$ is ________ .

Accepted Answer

87

Question 61

"A" obtained by Ostwald's method involving air oxidation of NH$_3$, upon further air oxidation produces "B". "B" on hydration forms an oxoacid of Nitrogen along with evolution of "A". The oxoacid also produces "A" and gives positive brown ring test. Identify A and B, respectively.

Accepted Answer

(D) NO, NO$_2$

Answer

(A) N$_2$O$_3$, NO$_2$

Answer

(B) NO$_2$, N$_2$O$_4$

Answer

(C) NO$_2$, N$_2$O$_5$

Question 62

Correct statement about smog is:

Accepted Answer

(D) Photochemical smog has high concentration of oxidizing agents

Answer

(A) Classical smog also has high concentration of oxidizing agents

Answer

(B) Both NO$_2$ and SO$_2$ are present in classical smog

Answer

(C) NO$_2$ is present in classical smog

Question 63

The standard electrode potential ($M^{3+}/M^{2+}$) for V, Cr, Mn and Co are $-0.26$ V, $-0.41$ V, $+1.57$ V and $+1.97$ V, respectively. The metal ions which can liberate H$_2$ from a dilute acid are:

Accepted Answer

(C) V$^{2+}$ and Cr$^{2+}$

Answer

(A) Mn$^{2+}$ and Co$^{2+}$

Answer

(B) Cr$^{2+}$ and Co$^{2+}$

Answer

(D) V$^{2+}$ and Mn$^{2+}$

Question 64

The shortest wavelength of hydrogen atom in Lyman series is $\lambda$. The longest wavelength in Balmer series of He$^+$ is:

Accepted Answer

(B) $\dfrac{9\lambda}{5}$

Answer

(A) $\dfrac{36\lambda}{5}$

Answer

(C) $\dfrac{5}{9\lambda}$

Answer

(D) $\dfrac{5\lambda}{9}$

List I (Physical Quantity)	List II (Dimensional Formula)
A.Pressure gradient	I. $[M^0 L^2 T^{-2}]$
B.Energy density	II. $[M^1 L^{-1} T^{-2}]$
C.Electric Field	III. $[M^1 L^{-2} T^{-2}]$
D.Latent heat	IV. $[M^1 L^1 T^{-3} A^{-1}]$

List I (Antimicrobials)	List II (Names)
A.Narrow Spectrum Antibiotic	I.Furacin
B.Antiseptic	II.Sulphur dioxide
C.Disinfectants	III.Penicillin G
D.Broad spectrum antibiotic	IV.Chloramphenicol

List I (Reaction)	List II (Reagents)
A.Hoffmann Degradation	I.Conc. KOH, $\Delta$
B.Clemmensen reduction	II.CHCl $_3$ , NaOH/H $_3$ O $^+$
C.Cannizzaro reaction	III.Br $_2$ , NaOH
D.Reimer-Tiemann Reaction	IV.Zn-Hg/HCl

JEE Main 29 January 2023 Shift 1 Question Paper with Answers

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