JEE Main 4 April 2024 Shift 1 question paper with solutions

Question 1

Maths · Continuity and Differentiability · Single correct

Let $f : \mathbb{R} \to \mathbb{R}$ be a function given by $$f(x) = \begin{cases} \frac{1 - \cos 2x}{x^2}, & x 0 \end{cases}$$ where $\alpha, \beta \in \mathbb{R}$. If $f$ is continuous at $x = 0$, then $\alpha^2 + \beta^2$ is equal to :

1. 3
2. 12
3. 48
4. 6

Answer: (b)

Question 2

Maths · Probability · Single correct

Three urns A, B and C contain 7 red, 5 black; 5 red, 7 black and 6 red, 6 black balls, respectively. One of the urn is selected at random and a ball is drawn from it. If the ball drawn is black, then the probability that it is drawn from urn A is :

1. $\frac{5}{18}$
2. $\frac{5}{16}$
3. $\frac{4}{17}$
4. $\frac{7}{18}$

Answer: (a)

Question 3

Maths · Straight Lines and Pair of Straight Lines · Single correct

The vertices of a triangle are A(-1, 3), B(-2, 2) and C(3, -1). A new triangle is formed by shifting the sides of the triangle by one unit inwards. Then the equation of the side of the new triangle nearest to origin is :

1. $x + y + \left(2 - \sqrt{2}\right) = 0$
2. $-x + y - \left(2 - \sqrt{2}\right) = 0$
3. $x + y - \left(2 - \sqrt{2}\right) = 0$
4. $x - y - \left(2 + \sqrt{2}\right) = 0$

Answer: (c)

Question 4

Maths · Differential Equations · Single correct

If the solution $y = y(x)$ of the differential equation $(x^4 + 2x^3 + 3x^2 + 2x + 2) \, dy - (2x^2 + 2x + 3) \, dx = 0$ satisfies $y(-1) = -\frac{\pi}{4}$, then $y(0)$ is equal to:

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

Answer: (d)

Question 5

Maths · Applications of Derivatives · Single correct

Let the sum of the maximum and the minimum values of the function $f(x) = \frac{2x^2 - 3x + 8}{2x^2 + 3x + 8}$ be $\frac{m}{n}$, where $\gcd(m, n) = 1$. Then $m + n$ is equal to:

1. 195
2. 201
3. 217
4. 182

Answer: (b)

Question 6

Maths · Applications of Integrals · Single correct

One of the points of intersection of the curves $y = 1 + 3x - 2x^2$ and $y = \frac{1}{x}$ is $\left( \frac{1}{2}, 2 \right)$. Let the area of the region enclosed by these curves be $\frac{1}{24} \left( l \sqrt{5} + m \right) - n \log_e \left( 1 + \sqrt{5} \right)$, where $l, m, n \in \mathbb{N}$. Then $l + m + n$ is equal to

1. 29
2. 31
3. 30
4. 32

Answer: (c)

Question 7

Maths · Determinants · Single correct

If the system of equations $$x + (\sqrt{2} \sin \alpha) y + (\sqrt{2} \cos \alpha) z = 0$$ $$x + (\cos \alpha) y + (\sin \alpha) z = 0$$ $$x + (\sin \alpha) y - (\cos \alpha) z = 0$$ has a non-trivial solution, then $\alpha \in \left(0, \frac{\pi}{2}\right)$ is equal to:

1. $\frac{11\pi}{24}$
2. $\frac{5\pi}{24}$
3. $\frac{7\pi}{24}$
4. $\frac{3\pi}{4}$

Answer: (b)

Question 8

Maths · Permutations and Combinations · Single correct

There are 5 points $P_1, P_2, P_3, P_4, P_5$ on the side $AB$, excluding $A$ and $B$, of a triangle $ABC$. Similarly there are 6 points $P_6, P_7, \ldots, P_{11}$ on the side $BC$ and 7 points $P_{12}, P_{13}, \ldots, P_{18}$ on the side $CA$ of the triangle. The number of triangles, that can be formed using the points $P_1, P_2, \ldots, P_{18}$ as vertices, is:

1. 776
2. 796
3. 751
4. 771

Answer: (c)

Question 9

Maths · Integrals · Single correct

Let $f(x) = \begin{cases} -2, & -2 \leq x \leq 0 \\ x - 2, & 0 < x \leq 2 \end{cases}$ and $h(x) = f(|x|) + |f(x)|$. Then $\int_{-2}^{2} h(x) \, dx$ is equal to :

1. 1
2. 6
3. 4
4. 2

Answer: (d)

Question 10

Maths · Binomial Theorem · Single correct

The sum of all rational terms in the expansion of $\left(2^{\frac{1}{5}} + 5^{\frac{1}{3}}\right)^{15}$ is equal to:

1. 3133
2. 931
3. 6131
4. 633

Answer: (a)

Question 11

Maths · Vector Algebra · Single correct

Let a unit vector which makes an angle of $60^\circ$ with $2\hat{i} + 2\hat{j} - \hat{k}$ and angle $45^\circ$ with $\hat{i} - \hat{k}$ be $\vec{C}$. Then $\vec{C} + \left( -\frac{1}{2} \hat{i} + \frac{1}{3\sqrt{2}} \hat{j} - \frac{\sqrt{2}}{3} \hat{k} \right)$ is :

1. $\frac{\sqrt{2}}{3} \hat{i} - \frac{1}{2} \hat{k}$
2. $\left( \frac{1}{\sqrt{3}} + \frac{1}{2} \right) \hat{i} + \left( \frac{1}{\sqrt{3}} - \frac{1}{3\sqrt{2}} \right) \hat{j} + \left( \frac{1}{\sqrt{3}} + \frac{\sqrt{2}}{3} \right) \hat{k}$
3. $\frac{\sqrt{2}}{3} \hat{i} + \frac{1}{3\sqrt{2}} \hat{j} - \frac{1}{2} \hat{k}$
4. $-\frac{\sqrt{2}}{3} \hat{i} + \frac{\sqrt{2}}{3} \hat{j} + \left( \frac{1}{2} + \frac{2\sqrt{2}}{3} \right) \hat{k}$

Answer: (a)

Question 12

Maths · Sequences and Series · Single correct

Let the first three terms $2, p$ and $q$, with $q \neq 2$, of a G.P. be respectively the $7^{th}$, $8^{th}$ and $13^{th}$ terms of an A.P. If the $5^{th}$ term of the G.P. is the $n^{th}$ term of the A.P., then $n$ is equal to:

1. 163
2. 151
3. 177
4. 169

Answer: (a)

Question 13

Maths · Statistics · Single correct

Let $\alpha, \beta \in \mathbb{R}$. Let the mean and the variance of 6 observations $-3, 4, 7, -6, \alpha, \beta$ be 2 and 23, respectively. The mean deviation about the mean of these 6 observations is:

1. $\frac{13}{3}$
2. $\frac{16}{3}$
3. $\frac{11}{3}$
4. $\frac{14}{3}$

Answer: (a)

Question 14

Maths · Complex Numbers and Quadratic Equations · Single correct

If 2 and 6 are the roots of the equation $ax^2 + bx + 1 = 0$, then the quadratic equation, whose roots are $\frac{1}{2a+b}$ and $\frac{1}{6a+b}$, is:

1. $2x^2 + 11x + 12 = 0$
2. $x^2 + 8x + 12 = 0$
3. $4x^2 + 14x + 12 = 0$
4. $x^2 + 10x + 16 = 0$

Answer: (b)

Question 15

Maths · Complex Numbers and Quadratic Equations · Single correct

Let $\alpha$ and $\beta$ be the sum and the product of all the non-zero solutions of the equation $\left(\bar{z}\right)^2 + |z| = 0, z \in \mathbb{C}$. Then $4 \left(\alpha^2 + \beta^2\right)$ is equal to:

1. 6
2. 8
3. 2
4. 4

Answer: (d)

Question 16

Maths · Three Dimensional Geometry · Single correct

Let the point, on the line passing through the points $P(1, -2, 3)$ and $Q(5, -4, 7)$, farther from the origin and at distance of 9 units from the point $P$, be $(\alpha, \beta, \gamma)$. Then $\alpha^2 + \beta^2 + \gamma^2$ is equal to:

1. 165
2. 160
3. 155
4. 150

Answer: (c)

Question 17

Maths · Conic Sections · Single correct

A square is inscribed in the circle $x^2 + y^2 - 10x - 6y + 30 = 0$. One side of this square is parallel to $y = x + 3$. If $(x_i, y_i)$ are the vertices of the square, then $\Sigma (x_i^2 + y_i^2)$ is equal to:

1. 148
2. 152
3. 160
4. 156

Answer: (b)

Question 18

Maths · Inverse Trigonometric Functions · Single correct

If the domain of the function $\sin^{-1}\left(\frac{3x-22}{2x-19}\right) + \log_e\left(\frac{3x^2-8x+5}{x^2-3x-10}\right)$ is $(\alpha, \beta]$, then $3\alpha + 10\beta$ is equal to:

1. 100
2. 95
3. 97
4. 98

Answer: (c)

Question 19

Maths · Continuity and Differentiability · Single correct

Let $f(x) = x^5 + 2e^{x/4}$ for all $x \in \mathbb{R}$. Consider a function $g(x)$ such that $(g \circ f)(x) = x$ for all $x \in \mathbb{R}$. Then the value of $8g'(2)$ is:

1. 2
2. 8
3. 4
4. 16

Answer: (d)

Question 20

Maths · Determinants · Single correct

Let $\alpha \in (0, \infty)$ and $A = \begin{bmatrix} 1 & 2 & \alpha \\ 1 & 0 & 1 \\ 0 & 1 & 2 \end{bmatrix}$. If $\det(adj(2A - A^T) \cdot adj(A - 2A^T)) = 2^8$, then $(\det(A))^2$ is equal to:

1. 36
2. 16
3. 1
4. 49

Answer: (b)

Question 21

Maths · Limits and Derivatives · Numerical

If $\lim_{x \to 1} \frac{(5x+1)^{1/3}-(x+5)^{1/3}}{(2x+3)^{1/2}-(x+4)^{1/2}} = \frac{m\sqrt{5}}{n(2n)^{2/3}}$, where gcd$(m, n) = 1$, then $8m + 12n$ is equal to

Answer: 100

Question 22

Maths · Sets · Numerical

In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both Physics and Chemistry; 50 studied both Chemistry and Mathematics; 40 studied both Mathematics and Physics and 10 studied none of these subjects. Let $m$ and $n$ respectively be the least and the most number of students who studied all the three subjects. Then $m + n$ is equal to

Answer: 45

Question 23

Maths · Differential Equations · Numerical

Let the solution $y = y(x)$ of the differential equation $\frac{dy}{dx} - y = 1 + 4 \sin x$ satisfy $y(\pi) = 1$. Then $y\left(\frac{\pi}{2}\right) + 10$ is equal to

Answer: 7

Question 24

Maths · Three Dimensional Geometry · Numerical

If the shortest distance between the lines $\frac{x+2}{2} = \frac{y+3}{3} = \frac{z-5}{4}$ and $\frac{x-3}{1} = \frac{y-2}{-3} = \frac{z+4}{2}$ is $\frac{38}{3\sqrt{5}}k$, and $\int_0^k [x^2] \, dx = \alpha - \sqrt{\alpha}$, where $[x]$ denotes the greatest integer function, then $6\alpha^3$ is equal to .

Answer: 48

Question 25

Maths · Matrices · Fill in the blank

Let $A$ be a square matrix of order $2$ such that $|A|=2$ and the sum of its diagonal elements is $-3$. If the points $(x,y)$ satisfying \[ A^2+xA+yI=0 \] lie on a hyperbola, whose transverse axis is parallel to the $x$-axis, eccentricity is $e$ and the length of the latus rectum is $\ell$, then $e^4+\ell^4$ is equal to ____

Answer: 233

Question 26

Maths · Binomial Theorem · Numerical

Let $a = 1 + \frac{{^2C_2}}{{3!}} + \frac{{^3C_2}}{{4!}} + \frac{{^4C_2}}{{5!}} + \cdots,$ $b = 1 + \frac{{^1C_0 + ^1C_1}}{{1!}} + \frac{{^2C_0 + ^2C_1 + ^2C_2}}{{2!}} + \frac{{^3C_0 + ^3C_1 + ^3C_2 + ^3C_3}}{{3!}} + \cdots$ Then $\frac{{2b}}{{a^2}}$ is equal to

Answer: 8

Question 27

Maths · Matrices · Numerical

Let $A$ be a $3 \times 3$ matrix of non-negative real elements such that $A \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = 3 \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$. Then the maximum value of $\det(A)$ is .

Answer: 27

Question 28

Maths · Conic Sections · Numerical

Let the length of the focal chord PQ of the parabola $y^2 = 12x$ be 15 units. If the distance of PQ from the origin is $p$, then $10p^2$ is equal to

Answer: 72

Question 29

Maths · Vector Algebra · Numerical

Let ABC be a triangle of area $15\sqrt{2}$ and the vectors $\overrightarrow{AB} = \hat{i} + 2\hat{j} - 7\hat{k}$, $\overrightarrow{BC} = a\hat{i} + b\hat{j} + c\hat{k}$ and $\overrightarrow{AC} = 6\hat{i} + d\hat{j} - 2\hat{k}$, $d > 0$. Then the square of the length of the largest side of the triangle ABC is

Answer: 54

Question 30

Maths · Integrals · Fill in the blank

If $\displaystyle\int_0^{\frac{\pi}{4}} \frac{\sin^2 x}{1+\sin x\cos x}\,dx = \frac{1}{a}\log_e\left(\frac{a}{3}\right) + \frac{\pi}{b\sqrt{3}}$, where $a, b \in \mathbb{N}$, then $a+b$ is equal to _____

Answer: 8

Question 31

Physics · Moving Charges and Magnetism · Single correct

An electron is projected with uniform velocity along the axis inside a current carrying long solenoid. Then :

1. the electron will continue to move with uniform velocity along the axis of the solenoid.
2. the electron will be accelerated along the axis.
3. the electron path will be circular about the axis.
4. the electron will experience a force at 45^$\circ$ to the axis and execute a helical path.

Answer: (a)

Question 32

Physics · Electromagnetic Waves · Single correct

The electric field in an electromagnetic wave is given by $\vec{E} = \hat{i} 40 \cos \omega (t - z/c) \mathrm{NC}^{-1}$. The magnetic field induction of this wave is (in SI unit):

1. $\vec{B} = \hat{k} \frac{40}{c} \cos \omega (t - z/c)$
2. $\vec{B} = \hat{j} 40 \cos \omega (t - z/c)$
3. $\vec{B} = \hat{i} \frac{40}{c} \cos \omega (t - z/c)$
4. $\vec{B} = \hat{j} \frac{40}{c} \cos \omega (t - z/c)$

Answer: (d)

Question 33

Physics · Nuclei · Single correct

Which of the following nuclear fragments corresponding to nuclear fission between neutron $\left( ^1_0 \mathrm{n} \right)$ and uranium isotope $\left( ^{235}_{92} \mathrm{U} \right)$ is correct:

1. $^{144}_{56} \mathrm{Ba} + ^{89}_{36} \mathrm{Kr} + 4 \, ^{1}_{0} \mathrm{n}$
2. $^{144}_{56} \mathrm{Ba} + ^{89}_{36} \mathrm{Kr} + 3 \, ^{1}_{0} \mathrm{n}$
3. $^{140}_{56} \mathrm{Xe} + ^{94}_{38} \mathrm{Sr} + 3 \, ^{1}_{0} \mathrm{n}$
4. $^{153}_{51} \mathrm{Sb} + ^{99}_{41} \mathrm{Nb} + 3 \, ^{1}_{0} \mathrm{n}$

Answer: (b)

Question 34

Physics · Mathematics in Physics · Single correct

In an experiment to measure focal length $(f)$ of convex lens, the least counts of the measuring scales for the position of object $(u)$ and for the position of image $(v)$ are $\Delta u$ and $\Delta v$, respectively. The error in the measurement of the focal length of the convex lens will be:

1. $2f \left[ \frac{\Delta u}{u} + \frac{\Delta v}{v} \right]$
2. $\frac{\Delta u}{u} + \frac{\Delta v}{v}$
3. $f^2 \left[ \frac{\Delta u}{u^2} + \frac{\Delta v}{v^2} \right]$
4. $f \left[ \frac{\Delta u}{u} + \frac{\Delta v}{v} \right]$

Answer: (c)

Question 35

Physics · Mechanical Properties of Fluids · Single correct

Given below are two statements: Statement I : When speed of liquid is zero everywhere, pressure difference at any two points depends on equation $P_1 - P_2 = \rho g \left( h_2 - h_1 \right)$. Statement II : In ventury tube shown $2gh = v_1^2 - v_2^2$ 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 correct.
2. Statement I is correct but Statement II is incorrect.
3. Statement I is incorrect but Statement II is correct.
4. Both Statement I and Statement II are incorrect.

Answer: (b)

Question 36

Physics · Thermal Properties of Matter · Single correct

The resistances of the platinum wire of a platinum resistance thermometer at the ice point and steam point are $8\,\Omega$ and $10\,\Omega$ respectively. After inserting in a hot bath of temperature $400^\circ\mathrm{C}$, the resistance of platinum wire is:

1. $10\,\Omega$
2. $8\,\Omega$
3. $16\,\Omega$
4. $2\,\Omega$

Answer: (c)

Question 37

Physics · Gravitation · Single correct

A metal wire of uniform mass density having length $L$ and mass $M$ is bent to form a semicircular arc and a particle of mass $m$ is placed at the centre of the arc. The gravitational force on the particle by the wire is :

1. $\frac{GmM\pi^2}{L^2}$
2. $\frac{GMm\pi}{2L^2}$
3. 0
4. $\frac{2GmM\pi}{L^2}$

Answer: (d)

Question 38

Physics · Thermal Properties of Matter · Single correct

On celcius scale the temperature of body increases by $40^\circ \mathrm{C}$. The increase in temperature on Fahrenheit scale is:

1. $68^\circ \mathrm{F}$
2. $75^\circ \mathrm{F}$
3. $72^\circ \mathrm{F}$
4. $70^\circ \mathrm{F}$

Answer: (c)

Question 39

Physics · Ray Optics and Optical Instruments · Single correct

An effective power of a combination of 5 identical convex lenses which are kept in contact along the principal axis is 25D. Focal length of each of the convex lens is :

1. 20 $\mathrm{\ cm}$
2. 50 $\mathrm{\ cm}$
3. 500 $\mathrm{\ cm}$
4. 25 $\mathrm{\ cm}$

Answer: (a)

Question 40

Physics · Dual Nature of Radiation and Matter · Single correct

Which figure shows the correct variation of applied potential difference (V) with photoelectric current (I) at two different intensities of light ($I_1 < I_2$) of same wavelengths :

Answer: (b)

Question 41

Physics · Laws of Motion · Single correct

A wooden block, initially at rest on the ground, is pushed by a force which increases linearly with time $t$. Which of the following curve best describes acceleration of the block with time:

Answer: (d)

Question 42

Physics · Work, Energy and Power · Single correct

If a rubber ball falls from a height $h$ and rebounds upto the height of $h/2$. The percentage loss of total energy of the initial system as well as velocity ball before it strikes the ground, respectively, are :

1. 50$\%$, $\sqrt{2gh}$
2. 50$\%$, $\sqrt{gh}$
3. 40$\%$, $\sqrt{2gh}$
4. 50$\%$, $\sqrt{\frac{gh}{2}}$

Answer: (a)

Question 43

Physics · Physical World, Units and Measurements · Single correct

The equation of stationary wave is : $$y = 2a \sin\left(\frac{2\pi nt}{\lambda}\right) \cos\left(\frac{2\pi x}{\lambda}\right).$$ Which of the following is NOT correct :

1. The dimensions of $n/\lambda$ is $[T]$
2. The dimensions of $n$ is $[LT^{-1}]$
3. The dimensions of $x$ is $[L]$
4. The dimensions of $nt$ is $[L]$

Answer: (a)

Question 44

Physics · Motion in a Straight Line · Single correct

A body travels $102.5\,\mathrm{m}$ in $n^{\text{th}}$ second and $115.0\,\mathrm{m}$ in $(n+2)^{\text{th}}$ second. The acceleration is:

1. $9\,\mathrm{m/s^2}$
2. $6.25\,\mathrm{m/s^2}$
3. $12.5\,\mathrm{m/s^2}$
4. $5\,\mathrm{m/s^2}$

Answer: (b)

Question 45

Physics · Current Electricity · Single correct

To measure the internal resistance of a battery, potentiometer is used. For $R = 10\Omega$, the balance point is observed at $l = 500 \, \mathrm{cm}$ and for $R = 1\Omega$ the balance point is observed at $l = 400 \, \mathrm{cm}$. The internal resistance of the battery is approximately :

1. 0.2$\Omega$
2. 0.3$\Omega$
3. 0.4$\Omega$
4. 0.1$\Omega$

Answer: (b)

Question 46

Physics · Electric Charges and Fields · Single correct

An infinitely long positively charged straight thread has a linear charge density $\lambda \mathrm{Cm}^{-1}$. An electron revolves along a circular path having axis along the length of the wire. The graph that correctly represents the variation of the kinetic energy of electron as a function of radius of circular path from the wire is:

Answer: (c)

Question 47

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

The value of net resistance of the network as shown in the given figure is :

1. 6$\Omega$
2. ($\frac{5}{2}$)$\Omega$
3. ($\frac{15}{4}$)$\Omega$
4. ($\frac{30}{11}$)$\Omega$

Answer: (a)

Question 48

Physics · Thermodynamics · Single correct

P-T diagram of an ideal gas having three different densities $\rho_1, \rho_2, \rho_3$ (in three different cases) is shown in the figure. Which of the following is correct:

1. $\rho_1 > \rho_2$
2. $\rho_2 < \rho_3$
3. $\rho_1 = \rho_2 = \rho_3$
4. $\rho_1 < \rho_2$

Answer: (a)

Question 49

Physics · Motion in a Plane · Single correct

The co-ordinates of a particle moving in $x - y$ plane are given by : $x = 2 + 4t, y = 3t + 8t^2$. The motion of the particle is :

1. uniformly accelerated having motion along a parabolic path.
2. uniform motion along a straight line.
3. uniformly accelerated having motion along a straight line.
4. non-uniformly accelerated.

Answer: (a)

Question 50

Physics · Alternating Current · Single correct

In an ac circuit, the instantaneous current is zero, when the instantaneous voltage is maximum. In this case, the source may be connected to: \begin{enumerate} \item[(A)] pure inductor. \item[(B)] pure capacitor. \item[(C)] pure resistor. \item[(D)] combination of an inductor and capacitor. \end{enumerate} Choose the correct answer from the options given below:

1. A, B and C only
2. A and B only
3. B, C and D only
4. A, B and D only

Answer: (d)

Question 51

Physics · Electric Charges and Fields · Numerical

An infinite plane sheet of charge having uniform surface charge density $+\sigma_s \, \mathrm{C/m^2}$ is placed on $x-y$ plane. Another infinitely long line charge having uniform linear charge density $+\lambda_e \, \mathrm{C/m}$ is placed at $z = 4 \, \mathrm{m}$ plane and parallel to $y$-axis. If the magnitude values $|\sigma_s| = 2 \, |\lambda_e|$ then at point $(0, 0, 2)$, the ratio of magnitudes of electric field values due to sheet charge to that of line charge is $\pi \sqrt{n} : 1$. The value of $n$ is _______.

Answer: 16

Question 52

Physics · Atoms · Numerical

A hydrogen atom changes its state from $n = 3$ to $n = 2$. Due to recoil, the percentage change in the wavelength of emitted light is approximately $1 \times 10^{-n}$. The value of $n$ is_____. [Given $\mathrm{Rhc}$ = 13.6 $\mathrm{eV}$, $\mathrm{hc}$ = 1242 $\mathrm{eVnm}$, $\mathrm{h}$ = 6.6 $\times$ $10^{-34}$ $\mathrm{Js}$ mass of the hydrogen atom = 1.6 $\times$ $10^{-27}$ $\mathrm{kg}$ ]

Answer: 7

Question 53

Physics · Moving Charges and Magnetism · Numerical

The magnetic field existing in a region is given by $\vec{B} = 0.2(1 + 2x)\hat{k}$ T. A square loop of edge 50 cm carrying 0.5 A current is placed in $x-y$ plane with its edges parallel to the $x-y$ axes, as shown in figure. The magnitude of the net magnetic force experienced by the loop is_____ mN.

Answer: 50

Question 54

Physics · Alternating Current · Numerical

A alternating current at any instant is given by $i = [6 + \sqrt{56} \sin(100 \pi t + \pi/3)] \, \mathrm{A}$. The $rms$ value of the current is ________ A.

Answer: 8

Question 55

Physics · Current Electricity · Numerical

Twelve wires each having resistance $2\,\Omega$ are joined to form a cube. A battery of $6\,\mathrm{V}$ emf is joined across point $a$ and $c$. The voltage difference between $e$ and $f$ is _____ $\mathrm{V}$.

Answer: 1

Question 56

Physics · Mechanical Properties of Fluids · Numerical

A soap bubble is blown to a diameter of 7 cm. 36960 $\mathrm{erg}$ of work is done in blowing it further. If surface tension of soap solution is 40 $\mathrm{dyne/cm}$ then the new radius is ________ cm. Take ( $\pi$ = $\frac{22}{7}$ ) .

Answer: 7

Question 57

Physics · Wave Optics · Numerical

Two wavelengths $\lambda_1$ and $\lambda_2$ are used in Young's double slit experiment. $\lambda_1 = 450 \, \mathrm{nm}$ and $\lambda_2 = 650 \, \mathrm{nm}$. The minimum order of fringe produced by $\lambda_2$ which overlaps with the fringe produced by $\lambda_1$ is $n$. The value of $n$ is _____.

Answer: 9

Question 58

Physics · Mechanical Properties of Solids · Numerical

An elastic spring under tension of 3 N has a length $a$. Its length is $b$ under tension 2 N. For its length $(3a - 2b)$, the value of tension will be ______N.

Answer: 5

Question 59

Physics · Mathematics in Physics · Numerical

Two forces $\vec{F}_1$ and $\vec{F}_2$ are acting on a body. One force has magnitude thrice that of the other force and the resultant of the two forces is equal to the force of larger magnitude. The angle between $\vec{F}_1$ and $\vec{F}_2$ is $\cos^{-1}\left(\frac{1}{n}\right)$. The value of $|n|$ is .

Answer: 6

Question 60

Physics · System of Particles and Rotational Motion · Numerical

A solid sphere and a hollow cylinder roll up without slipping on same inclined plane with same initial speed $v$. The sphere and the cylinder reaches upto maximum heights $h_1$ and $h_2$, respectively, above the initial level. The ratio $h_1 : h_2$ is $\frac{n}{10}$. The value of $n$ is _____.

Answer: 7

Question 61

Chemistry · Electrochemistry · Single correct

What pressure (bar) of $\mathrm{H}_2$ would be required to make emf of hydrogen electrode zero in pure water at $25^\circ \mathrm{C}$?

1. $10^{-7}$
2. $0.5$
3. $1$
4. $10^{-14}$

Answer: (d)

Question 62

Chemistry · Co-ordination Compounds · Single correct

The correct sequence of ligands in the order of decreasing field strength is :

1. $\mathrm{NCS}^{-} > \mathrm{EDTA}^{4-} > \mathrm{CN}^{-} > \mathrm{CO}$
2. $\mathrm{CO} > \mathrm{H_2O} > \mathrm{F}^{-} > \mathrm{S}^{2-}$
3. $\mathrm{S}^{2-} > \mathrm{-OH} > \mathrm{EDTA}^{4-} > \mathrm{CO}$
4. $\mathrm{-OH} > \mathrm{F}^{-} > \mathrm{NH_3} > \mathrm{CN}^{-}$

Answer: (b)

Question 63

Chemistry · Co-ordination Compounds · Single correct

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

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

Answer: (a)

Question 64

Chemistry · Equilibrium · Single correct

What will be the decreasing order of basic strength of the following conjugate bases? $\mathrm{OH^-},\ \mathrm{RO^-},\ \mathrm{CH_3COO^-},\ \mathrm{Cl^-}$

1. \quad $\mathrm{RO^-} > \mathrm{OH^-} > \mathrm{CH_3COO^-} > \mathrm{Cl^-}$
2. \quad $\mathrm{Cl^-} > \mathrm{RO^-} > \mathrm{OH^-} > \mathrm{CH_3COO^-}$
3. \quad $\mathrm{OH^-} > \mathrm{RO^-} > \mathrm{CH_3COO^-} > \mathrm{Cl^-}$
4. \quad $\mathrm{Cl^-} > \mathrm{OH^-} > \mathrm{RO^-} > \mathrm{CH_3COO^-}$

Answer: (a)

Question 65

Chemistry · Analytical Chemistry · Single correct

In the precipitation of the iron group (III) in qualitative analysis, ammonium chloride is added before adding ammonium hydroxide to :

1. increase concentration of $\mathrm{Cl}^-$ ions
2. increase concentration of $\mathrm{NH}_4^+$ ions
3. prevent interference by phosphate ions
4. decrease concentration of $\mathrm{OH}^-$ ions

Answer: (d)

Question 66

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

Identify (B) and $(C)$ and how are (A) and $(C)$ related?

Answer: (c)

Question 67

Chemistry · Electrochemistry · Single correct

One of the commonly used electrode is calomel electrode. Under which of the following categories, calomel electrode comes?

1. Oxidation - Reduction electrodes
2. Metal ion - Metal electrodes
3. Gas - Ion electrodes
4. Metal - Insoluble Salt - Anion electrodes

Answer: (d)

Question 68

Chemistry · Co-ordination Compounds · Single correct

Number of complexes from the following with even number of unpaired " d " electrons is $$[\mathrm{V(H_2O)_6}]^{3+}, \ [\mathrm{Cr(H_2O)_6}]^{2+}, \ [\mathrm{Fe(H_2O)_6}]^{3+}, \ [\mathrm{Ni(H_2O)_6}]^{3+}, \ [\mathrm{Cu(H_2O)_6}]^{2+}$$ [Given atomic numbers : $V = 23$, $Cr = 24$, $Fe = 26$, $Ni = 28$, $Cu = 29$]

1. 2
2. 1
3. 4
4. 5

Answer: (a)

Question 69

Chemistry · Chemical Bonding and Molecular Structure · Single correct

Which one of the following molecules has maximum dipole moment?

1. $\mathrm{NF}_3$
2. $\mathrm{CH}_4$
3. $\mathrm{PF}_5$
4. $\mathrm{NH}_3$

Answer: (d)

Question 70

Chemistry · Chemical Bonding and Molecular Structure · Single correct

Number of molecules/ions from the following in which the central atom is involved in $sp^3$ hybridization is _____ $\mathrm{NO_3^-}$, $\mathrm{BCl_3}$, $\mathrm{ClO_2^-}$, $\mathrm{ClO_3}$

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

Answer: (c)

Question 71

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

Which among the following is incorrect statement?

1. Electromeric effect dominates over inductive effect
2. The electromeric effect is, temporary effect
3. Hydrogen ion ($H^+$) shows negative electromeric effect
4. The organic compound shows electromeric effect in the presence of the reagent only.

Answer: (c)

Question 72

Chemistry · Aldehydes, Ketones and Carboxylic Acids · Single correct

Given below are two statements: Statements I: Acidity of $\alpha$-hydrogens of aldehydes and ketones is responsible for Aldol reaction. Statement II: Reaction between benzaldehyde and ethanal will NOT give Cross - Aldol product. In the light of the above statements, choose the most appropriate answer from the options given below:

1. Statement I is correct but Statement II is incorrect
2. Both Statement I and Statement II are correct
3. Both Statement I and Statement II are incorrect
4. Statement I is incorrect but Statement II is correct

Answer: (a)

Question 73

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

Which of the following nitrogen containing compound does not give Lassaigne's test?

1. Urea
2. Phenyl hydrazine
3. Glycene
4. Hydrazine

Answer: (d)

Question 74

Chemistry · Biomolecules · Single correct

Which of the following is the correct structure of L-Glucose?

Answer: (a)

Question 75

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

The element which shows only one oxidation state other than its elemental form is :

1. Cobalt
2. Titanium
3. Nickel
4. Scandium

Answer: (d)

Question 76

Chemistry · Aldehydes, Ketones and Carboxylic Acids · Single correct

Identify the product in the following reaction:

Answer: (a)

Question 77

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

Number of elements from the following that CANNOT form compounds with valencies which match with their respective group valencies is _____. B, C, N, S, O, F, P, Al, Si

1. 7
2. 3
3. 5
4. 6

Answer: (b)

Question 78

Chemistry · Solutions · Single correct

The Molarity (M) of an aqueous solution containing 5.85 g of NaCl in 500 mL water is : (Given : Molar Mass Na : 23 and Cl : 35.5 $\mathrm{g/mol}$ )

1. 2
2. 20
3. 4
4. 0.2

Answer: (d)

Question 79

Chemistry · Haloalkanes and Haloarenes · Single correct

Identify the correct set of reagents or reaction conditions 'X' and 'Y' in the following set of transformation

1. X = dil.aq. NaOH, 20°C, Y = Br₂/CHCl₃
2. X = conc.alc. NaOH, 80°C, Y = Br₂/CHCl₃
3. X = dil.aq. NaOH, 20°C, Y = HBr /acetic acid
4. X = conc.alc. NaOH, 80°C, Y = HBr/ acetic acid

Answer: (d)

Question 80

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

The correct order of first ionization enthalpy values of the following elements is : (A) O (B) N (C) Be (D) F (E) B Choose the correct answer from the options given below :

1. E < C < A < B < D
2. C < E < A < B < D
3. B < D < C < E < A
4. A < B < D < C < E

Answer: (a)

Question 81

Chemistry · Thermodynamics · Numerical

The enthalpy of formation of ethane ($C_2H_6$) from ethylene by addition of hydrogen where the bond-energies of $\mathrm{C-H}$, $\mathrm{C-C}$, $\mathrm{C=C}$, $\mathrm{H-H}$ are $414 \, \mathrm{kJ}$, $347 \, \mathrm{kJ}$, $615 \, \mathrm{kJ}$ and $435 \, \mathrm{kJ}$ respectively is kJ

Answer: 125

Question 82

Chemistry · Aldehydes, Ketones and Carboxylic Acids · Numerical

The number of the correct reaction(s) among the following is

Answer: (a)

Question 83

Chemistry · Amines · Numerical

X $\mathrm{g}$ of ethylamine is subjected to reaction with $\mathrm{NaNO_2/HCl}$ followed by water; evolved dinitrogen gas which occupied 2.24 $\mathrm{L}$ volume at STP. X is __ $\times 10^{-1}$ $\mathrm{g}$.

Answer: 45

Question 84

Chemistry · Structure of Atom · Numerical

The de-Broglie's wavelength of an electron in the 4th orbit is _______$\pi a_0 \cdot (a_0 = Bohr's radius)$

Answer: 8

Question 85

Chemistry · Redox Reactions · Numerical

Only 2 mL of $KMnO_4$ solution of unknown molarity is required to reach the end point of a titration of 20 $\mathrm{\, mL}$ of oxalic acid (2M) in acidic medium. The molarity of $KMnO_4$ solution should be____ $\mathrm{M.}$

Answer: 8

Question 86

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

Consider the following reaction $$\mathrm{MnO_2 + KOH + O_2 \rightarrow A + H_2O}$$. Product ' A ' in neutral or acidic medium disproportionate to give products ' B ' and ' C ' along with water. The sum of spin-only magnetic moment values of B and C is ______$\mathrm{BM}$ . (nearest integer) (Given atomic number of Mn is 25)

Answer: 4

Question 87

Chemistry · Chemical Kinetics and Nuclear Chemistry · Numerical

Consider the following transformation involving first order elementary reaction in each step at constant temperature as shown below. \[ A+B \xrightleftharpoons[\text{Step 3}]{\text{Step 1}} C \xrightarrow{\text{Step 2}} P \] Some details of the above reactions are listed below. \[ \begin{array}{ccc} \text{Step} & \text{Rate constant }(\mathrm{sec}^{-1}) & \text{Activation energy }(\mathrm{kJ\,mol}^{-1})\\ 1 & k_1 & 300\\ 2 & k_2 & 200\\ 3 & k_3 & E_{a3} \end{array} \] If the overall rate constant of the above transformation $(k)$ is given as \[ k=\frac{k_1k_2}{k_3} \] and the overall activation energy $(E_a)$ is $400\,\mathrm{kJ\,mol^{-1}}$, then the value of $E_{a3}$ is \[ \underline{\hspace{2cm}}\,\mathrm{kJ\,mol^{-1}} \] (nearest integer).

Answer: 100

Question 88

Chemistry · Solutions · Numerical

2.5 $\mathrm{g}$ of a non-volatile, non-electrolyte is dissolved in 100 $\mathrm{g}$ of water at $25^\circ$ $\mathrm{C}$. The solution showed a boiling point elevation by $2^\circ$ $\mathrm{C}$. Assuming the solute concentration is negligible with respect to the solvent concentration, the vapor pressure of the resulting aqueous solution is _____ mm of Hg (nearest integer) [Given : Molal boiling point elevation constant of water ($K_b$) = 0.52 $\mathrm{K}$ $\cdot$ $\mathrm{kgmol}^{-1}$, 1 atm pressure = 760 mm of Hg, molar mass of water = 18 $\mathrm{g}$ $\cdot$ $\mathrm{mol}^{-1}$]

Answer: 707

Question 89

Chemistry · Organic Chemistry — Some Basic Principles & Techniques · Numerical

The number of different chain isomers for $\mathrm{C_7H_{16}}$ is

Answer: 9

Question 90

Chemistry · Chemical Bonding and Molecular Structure · Numerical

Number of molecules/species from the following having one unpaired electron is ____$\mathrm{O_2}$, $\mathrm{O_2^{-1}}$, $\mathrm{NO}$, $\mathrm{CN^{-1}}$, $\mathrm{O_2^{2-}}$

Answer: 2