JEE Main 30 January 2024 Shift 2 question paper with solutions

Question 1

Maths · Determinants · Single correct

Consider the system of linear equations $$x + y + z = 5, \; x + 2y + \lambda^2 z = 9$$ $$x + 3y + \lambda z = \mu, \; where \; \lambda, \mu \in \mathbb{R}.$$ Then, which of the following statement is NOT correct?

1. System has infinite number of solution if $\lambda = 1$ and $\mu = 13$
2. System is inconsistent if $\lambda = 1$ and $\mu \neq 13$
3. System is consistent if $\lambda \neq 1$ and $\mu = 13$
4. System has unique solution if $\lambda \neq 1$ and $\mu \neq 13$

Answer: (d)

Question 2

Maths · Trigonometric Functions · Single correct

For $\alpha, \beta \in \left(0, \frac{\pi}{2}\right)$, let $3 \sin(\alpha + \beta) = 2 \sin(\alpha - \beta)$ and a real number $k$ be such that $\tan \alpha = k \tan \beta$. Then the value of $k$ is equal to:

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

Answer: (b)

Question 3

Maths · Conic Sections · Single correct

Let $A(\alpha, 0)$ and $B(0, \beta)$ be the points on the line $5x + 7y = 50$. Let the point $P$ divide the line segment $AB$ internally in the ratio $7 : 3$. Let $3x - 25 = 0$ be a directrix of the ellipse $E : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ and the corresponding focus be $S$. If from $S$, the perpendicular on the x-axis passes through $P$, then the length of the latus rectum of $E$ is equal to

1. $\frac{25}{3}$
2. $\frac{32}{9}$
3. $\frac{25}{9}$
4. $\frac{32}{5}$

Answer: (d)

Question 4

Maths · Vector Algebra · Single correct

Let $\vec{a}$ = $\hat{i}$ + $\alpha$ $\hat{j}$ + $\beta$ $\hat{k}$, $\alpha$, $\beta$ $\in$ $\mathbb{R}$. Let a vector $\vec{b}$ be such that the angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{4}$ and |$\vec{b}|^2$ = 6, If $\vec{a}$ $\cdot$ $\vec{b}$ = 3$\sqrt{2}$, then the value of ($\alpha^2$ + $\beta^2$) $|\vec{a}\times\vec{b}|^2$ is equal to

1. 90
2. 75
3. 95
4. 85

Answer: (a)

Question 5

Maths · Applications of Derivatives · Single correct

Let $f(x) = (x + 3)^2(x - 2)^3$, $x \in [-4, 4]$. If $M$ and $m$ are the maximum and minimum values of $f$, respectively in $[-4, 4]$, then the value of $M - m$ is:

1. 600
2. 392
3. 608
4. 108

Answer: (c)

Question 6

Maths · Sequences and Series · Single correct

Let $a$ and $b$ be two distinct positive real numbers. Let $11^{th}$ term of a GP, whose first term is $a$ and third term is $b$, is equal to $p^{th}$ term of another GP, whose first term is $a$ and fifth term is $b$. Then $p$ is equal to

1. 20
2. 25
3. 21
4. 24

Answer: (c)

Question 7

Maths · Conic Sections · Single correct

If $x^2 - y^2 + 2hxy + 2gx + 2fy + c = 0$ is the locus of a point, which moves such that it is always equidistant from the lines $x + 2y + 7 = 0$ and $2x - y + 8 = 0$, then the value of $g + c + h - f$ equals

1. 14
2. 6
3. 8
4. 29

Answer: (a)

Question 8

Maths · Vector Algebra · Single correct

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{b}| = 1$ and $|\vec{b} \times \vec{a}| = 2$. Then $|(\vec{b} \times \vec{a}) - \vec{b}|^2$ is equal to

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

Answer: (b)

Question 9

Maths · Applications of Integrals · Single correct

Let $y = f(x)$ be a thrice differentiable function in $(-5, 5)$. Let the tangents to the curve $y = f(x)$ at $(1, f(1))$ and $(3, f(3))$ make angles $\frac{\pi}{6}$ and $\frac{\pi}{4}$, respectively with positive $x$-axis. If $$27 \int_{1}^{3} \left( (f'(t))^2 + 1 \right) f''(t) dt = \alpha + \beta \sqrt{3}$$ where $\alpha$, $\beta$ are integers, then the value of $\alpha + \beta$ equals

1. -14
2. 26
3. -16
4. 36

Answer: (b)

Question 10

Maths · Conic Sections · Single correct

Let P be a point on the hyperbola H : $\frac{x^2}{9} - \frac{y^2}{4} = 1$, in the first quadrant such that the area of triangle formed by P and the two foci of H is $2\sqrt{13}$. Then, the square of the distance of P from the origin is

1. 18
2. 26
3. 22
4. 20

Answer: (c)

Question 11

Maths · Probability · Single correct

Bag A contains 3 white, 7 red balls and bag B contains 3 white, 2 red balls. One bag is selected at random and a ball is drawn from it. The probability of drawing the ball from the bag A, if the ball drawn in white, is :

1. $\frac{1}{4}$
2. $\frac{1}{9}$
3. $\frac{1}{3}$
4. $\frac{3}{10}$

Answer: (c)

Question 12

Maths · Integrals · Single correct

Let $f : \mathbb{R} \to \mathbb{R}$ be defined $f(x) = ae^{2x} + be^x + cx$. If $f(0) = -1$, $f'(\log_e 2) = 21$ and $$\int_0^{\log_4 4} (f(x) - cx) \, dx = \frac{39}{2}$$, then the value of $|a + b + c|$ equals:

1. 16
2. 10
3. 12
4. 8

Answer: (d)

Question 13

Maths · Three Dimensional Geometry · Single correct

Let $L_1 : \vec{r} = (\hat{i} - \hat{j} + 2\hat{k}) + \lambda (\hat{i} - \hat{j} + 2\hat{k}), \lambda \in \mathbb{R}$, $L_2 : \vec{r} = (\hat{j} - \hat{k}) + \mu (3\hat{i} + \hat{j} + p\hat{k}), \mu \in \mathbb{R}$ and $L_3 : \vec{r} = \delta (\ell \hat{i} + m \hat{j} + n \hat{k}) \delta \in \mathbb{R}$ Be three lines such that $L_1$ is perpendicular to $L_2$ and $L_3$ is perpendicular to both $L_1$ and $L_2$. Then the point which lies on $L_3$ is

1. $(-1, 7, 4)$
2. $(-1, -7, 4)$
3. $(1, 7, -4)$
4. $(1, -7, 4)$

Answer: (a)

Question 14

Maths · Continuity and Differentiability · Single correct

Let $a$ and $b$ be real constants such that the function $f$ defined by $f(x) = \begin{cases} x^2 + 3x + a, & x \leq 1 \\ bx + 2, & x > 1 \end{cases}$ be differentiable on $\mathbb{R}$. Then, the value of $\int_{-2}^{2} f(x) \, dx$ equals

1. $\($ $\frac{15}{6}$ $\)$
2. $\($ $\frac{19}{6}$ $\)$
3. 21
4. 17

Answer: (d)

Question 15

Maths · Continuity and Differentiability · Single correct

Let $f : \mathbb{R} - \{0\} \to \mathbb{R}$ be a function satisfying $f\left(\frac{x}{y}\right) = \frac{f(x)}{f(y)}$ for all $x, y$, $f(y) \neq 0$. If $f'(1) = 2024$, then

1. $x f'(x) - 2024 f(x) = 0$
2. $x f'(x) + 2024 f(x) = 0$
3. $x f'(x) + f(x) = 2024$
4. $x f'(x) - 2023 f(x) = 0$

Answer: (a)

Question 16

Maths · Complex Numbers and Quadratic Equations · Single correct

If $z$ is a complex number, then the number of common roots of the equation $z^{1985} + z^{100} + 1 = 0$ and $z^3 + 2z^2 + 2z + 1 = 0$, is equal to:

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

Answer: (b)

Question 17

Maths · Binomial Theorem · Single correct

Suppose $2 - p$, $p$, $2 - \alpha$, $\alpha$ are the coefficient of four consecutive terms in the expansion of $(1 + x)^n$. Then the value of $p^2 - \alpha^2 + 6\alpha + 2p$ equals

1. 4
2. 10
3. 8
4. Data Inconsistent

Answer: (a)

Question 18

Maths · Inverse Trigonometric Functions · Single correct

If the domain of the function $f(x) = \log_e \left( \frac{2x+3}{4x^2+x-3} \right) + \cos^{-1} \left( \frac{2x-1}{x+2} \right)$ is $(\alpha, \beta]$, then the value of $5\beta - 4\alpha$ is equal to

1. 10
2. 12
3. 11
4. 9

Answer: (b)

Question 19

Maths · Integrals · Single correct

Let $f : \mathbb{R} \to \mathbb{R}$ be a function defined $f(x) = \frac{x}{(1+x^4)^{1/4}}$ and $g(x) = f(f(f(f(x))))$ then $18 \int_0^{\sqrt{2\sqrt{5}}} x^2 g(x) \, dx$

1. 33
2. 36
3. 42
4. 39

Answer: (d)

Question 20

Maths · Matrices · Single correct

Let $R = \begin{pmatrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{pmatrix}$ be a non-zero $3 \times 3$ matrix, where $x \sin \theta = y \sin \left( \theta + \frac{2\pi}{3} \right) = z \sin\left(\theta+\frac{4\pi}{3}\right) \neq 0, \qquad \theta\in(0,2\pi)$. For a square matrix $M$, let trace $(M)$ denote the sum of all the diagonal entries of $M$. Then, among the statements: (I) Trace $(R) = 0$ (II) If trace $(adj(adj(R))) = 0$, then $R$ has exactly one non-zero entry.

1. Both (I) and (II) are true
2. Neither (I) nor (II) is true
3. Only (II) is true
4. Only (I) is true

Answer: (b)

Question 21

Maths · Applications of Integrals · Numerical

Let $Y = Y(X)$ be a curve lying in the first quadrant such that the area enclosed by the line $$Y - y = Y'(x)(X - x)$$ and the co-ordinate axes, where $(x, y)$ is any point on the curve, is always $$\frac{-y^2}{2Y'(x)} + 1, \; Y'(x) \neq 0.$$ If $Y(1) = 1$, then $12Y(2)$ equals .

Answer: 20

Question 22

Maths · Three Dimensional Geometry · Numerical

Let a line passing through the point $(-1, 2, 3)$ intersect the lines $L_1 : \frac{x-1}{3} = \frac{y-2}{2} = \frac{z+1}{-2}$ at $M(\alpha, \beta, \gamma)$ and $L_2 : \frac{x+2}{-3} = \frac{y-2}{-2} = \frac{z-1}{4}$ at $N(a, b, c)$. Then the value of $\frac{(\alpha+\beta+\gamma)^2}{(a+b+c)^2}$ equals .

Answer: 196

Question 23

Maths · Conic Sections · Numerical

Consider two circles $C_1 : x^2 + y^2 = 25$ and $C_2 : (x - \alpha)^2 + y^2 = 16$, where $\alpha \in (5, 9)$. Let the angle between the two radii (one to each circle) drawn from one of the intersection points of $C_1$ and $C_2$ be $\sin^{-1}\left(\frac{\sqrt{63}}{8}\right)$. If the length of common chord of $C_1$ and $C_2$ is $\beta$, then the value of $(\alpha \beta)^2$ equals

Answer: 1575

Question 24

Maths · Binomial Theorem · Numerical

Let $\alpha=\sum_{k=0}^{n}\left(\frac{\left({}^{n}C_{k}\right)^2}{k+1}\right)$ and $\beta=\sum_{k=0}^{n-1}\left(\frac{{}^{n}C_{k}\,{}^{n}C_{k+1}}{k+2}\right)$ If $5\alpha=6\beta$, then $n$ equals

Answer: 10

Question 25

Maths · Sequences and Series · Numerical

Let $S_n$ be the sum to n-terms of an arithmetic progression 3, 7, 11, $\ldots$ $\ldots$ . If $40 < \left( \frac{6}{n(n+1)} \sum_{k=1}^{n} s_k \right) < 42$, then $n$ equals ________.

Answer: 9

Question 26

Maths · Permutations and Combinations · Numerical

In an examination of Mathematics paper, there are 20 questions of equal marks and the question paper is divided into three sections: A, B and C. A student is required to attempt total 15 questions taking at least 4 questions from each section. If section A has 8 questions, section B has 6 questions and section C has 6 questions, then the total number of ways a student can select 15 questions is

Answer: 11376

Question 27

Maths · Relations and Functions · Numerical

The number of symmetric relations defined on the set {$1, 2, 3, 4$\} which are not reflexive is $\ldots$

Answer: 960

Question 28

Maths · Complex Numbers and Quadratic Equations · Numerical

The number of real solutions of the equation $x \left( x^2 + 3|x| + 5|x - 1| + 6|x - 2| \right) = 0$ is_____.

Answer: 1

Question 29

Maths · Applications of Integrals · Numerical

The area of the region enclosed by the parabola $(y - 2)^2 = x - 1$, the line $x - 2y + 4 = 0$ and the positive coordinate axes is ________.

Answer: 5

Question 30

Maths · Statistics · Fill in the blank

The variance $\sigma^2$ of the data \begin{tabular}{|c|c|c|c|c|c|c|c|} \hline $x_i$ & $0$ & $1$ & $5$ & $6$ & $10$ & $12$ & $17$ \\ \hline $f_i$ & $3$ & $2$ & $3$ & $2$ & $6$ & $3$ & $3$ \\ \hline \end{tabular} Is ___

Answer: 29

Question 31

Physics · Experimental Physics · Single correct

If 50 Vernier divisions are equal to 49 main scale divisions of a travelling microscope and one smallest reading of main scale is $0.5 \, \mathrm{mm}$, the Vernier constant of travelling microscope is:

1. $0.1 \, \mathrm{mm}$
2. $0.1 \, \mathrm{cm}$
3. $0.01 \, \mathrm{cm}$
4. $0.01 \, \mathrm{mm}$

Answer: (d)

Question 32

Physics · Laws of Motion · Single correct

A block of mass 1 kg is pushed up a surface inclined to horizontal at an angle of $60^\circ$ by a force of 10 $\,$ $\mathrm{N}$ parallel to the inclined surface as shown in figure. When the block is pushed up by 10 m along inclined surface, the work done against frictional force is: [g = 10 $\,$ $\mathrm{m/s^2}$]

1. 5 $\sqrt{3}$ $\,$ $\mathrm{J}$
2. 5 $\,$ $\mathrm{J}$
3. 5 $\times$ 10^3 $\,$ $\mathrm{J}$
4. 10 $\,$ $\mathrm{J}$

Answer: (b)

Question 33

Physics · Dual Nature of Radiation and Matter · Single correct

For the photoelectric effect, the maximum kinetic energy $(E_k)$ of the photoelectrons is plotted against the frequency $(\nu)$ of the incident photons as shown in figure. The slope of the graph gives

1. Ratio of Planck's constant to electric charge
2. Work function of the metal
3. Charge of electron
4. Planck's constant

Answer: (d)

Question 34

Physics · Thermal Properties of Matter · Single correct

A block of ice at $-10^\circ \mathrm{C}$ is slowly heated and converted to steam at $100^\circ \mathrm{C}$. Which of the following curves represent the phenomenon qualitatively:

Answer: (d)

Question 35

Physics · Nuclei · Single correct

In a nuclear fission reaction of an isotope of mass M, three similar daughter nuclei of same mass are formed. The speed of a daughter nuclei in terms of mass defect $\Delta M$ will be:

1. $\sqrt{\frac{2c \Delta M}{M}}$
2. $\frac{\Delta M c^2}{3}$
3. $c \sqrt{\frac{2 \Delta M}{M}}$
4. $c \sqrt{\frac{3 \Delta M}{M}}$

Answer: (c)

Question 36

Physics · Thermodynamics · Single correct

Choose the correct statement for processes A $\&$ B shown in figure.

1. $PV^\gamma = k$ for process B and $PV = k$ for process A.
2. $PV = k$ for process B and A.
3. $\frac{T^\gamma}{P^{\gamma-1}} = k$ for process A and $PV = k$ for process B.
4. None of these

Answer: (a)

Question 37

Physics · Atoms · Single correct

An electron revolving in $n^{th}$ Bohr orbit has magnetic moment $\mu_n$. If $\mu_n \alpha n^x$, the value of $x$ is:

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

Answer: (b)

Question 38

Physics · Alternating Current · Single correct

An alternating voltage $V(t) = 220 \sin 100 \pi t$ volt is applied to a purely resistive load of $50 \Omega$. The time taken for the current to rise from half of the peak value to the peak value is:

1. $5\,\mathrm{ms}$
2. $3.3\,\mathrm{ms}$
3. $7.2\,\mathrm{ms}$
4. $2.2\,\mathrm{ms}$

Answer: (b)

Question 39

Physics · Laws of Motion · Single correct

A block of mass $m$ is placed on a surface having vertical cross section given by $y = x^2/4$. If coefficient of friction is $0.5$, the maximum height above the ground at which block can be placed without slipping is:

1. 1/4 m
2. 1/2 m
3. 1/6 m
4. 1/3 m

Answer: (a)

Question 40

Physics · System of Particles and Rotational Motion · Single correct

If the total energy transferred to a surface in time $t$ is $6.48 \times 10^5 \, \mathrm{J}$, then the magnitude of the total momentum delivered to this surface for complete absorption will be:

1. $2.46 \times 10^{-3} \, \mathrm{kg \, m/s}$
2. $2.16 \times 10^{-3} \, \mathrm{kg \, m/s}$
3. $1.58 \times 10^{-3} \, \mathrm{kg \, m/s}$
4. $4.32 \times 10^{-3} \, \mathrm{kg \, m/s}$

Answer: (b)

Question 41

Physics · Wave Optics · Single correct

A beam of unpolarised light of intensity $I_0$ is passed through a polaroid A and then through another polaroid B which is oriented so that its principal plane makes an angle of $45^\circ$ relative to that of A. The intensity of emergent light is :

1. $I_0/4$
2. $I_0$
3. $I_0/2$
4. $I_0/8$

Answer: (a)

Question 42

Physics · Gravitation · Single correct

Escape velocity of a body from earth is $11.2 \, \mathrm{km/s}$. If the radius of a planet be one-third the radius of earth and mass be one-sixth that of earth, the escape velocity from the plate is:

1. $11.2 \, \mathrm{km/s}$
2. $8.4 \, \mathrm{km/s}$
3. $4.2 \, \mathrm{km/s}$
4. $7.9 \, \mathrm{km/s}$

Answer: (d)

Question 43

Physics · Electric Charges and Fields · Single correct

A particle of charge ' -q ' and mass ' m ' moves in a circle of radius ' r ' around an infinitely long line charge of linear density ' +$\lambda$ '. Then time period will be given as: (Consider k as Coulomb's constant)

1. $T^2 = \frac{4\pi^2 \, m}{2k\lambda q} \, r^3$
2. $T = 2\pi r \sqrt{\frac{m}{2k\lambda q}}$
3. $T = \frac{1}{2\pi} \sqrt{\frac{m}{2k\lambda q}}$
4. $T = \frac{1}{2\pi} \sqrt{\frac{2k\lambda q}{m}}$

Answer: (b)

Question 44

Physics · Physical World, Units and Measurements · Single correct

If mass is written as $m = k \, c^P \, G^{-1/2} \, h^{1/2}$ then the value of $P$ will be: (Constants have their usual meaning with $k$ a dimensionless constant)

1. 1/2
2. 1/3
3. 2
4. -1/3

Answer: (a)

Question 45

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

In the given circuit, the voltage across load resistance ($R_L$) is:

1. 8.75 V
2. 9.00 V
3. 8.50 V
4. 14.00 V

Answer: (a)

Question 46

Physics · Kinetic Theory · Single correct

If three moles of monoatomic gas $\left( \gamma = \frac{5}{3} \right)$ is mixed with two moles of a diatomic gas $\left( \gamma = \frac{7}{5} \right)$, the value of adiabatic exponent $\gamma$ for the mixture is:

1. 1.75
2. 1.40
3. 1.52
4. 1.35

Answer: (c)

Question 47

Physics · Laws of Motion · Single correct

Three blocks A, B and C are pulled on a horizontal smooth surface by a force of 80 N as shown in figure The tensions $T_1$ and $T_2$ in the string are respectively:

1. 40 N, 64 N
2. 60 N, 80 N
3. 88 N, 96 N
4. 80 N, 100 N

Answer: (a)

Question 48

Physics · Current Electricity · Single correct

When a potential difference $V$ is applied across a wire of resistance $R$, it dissipates energy at a rate $W$. If the wire is cut into two halves and these halves are connected mutually parallel across the same supply, the same supply, the energy dissipation rate will become:

1. 1/4 $W$
2. 1/2 $W$
3. 2 $W$
4. 4 $W$

Answer: (d)

Question 49

Physics · Electric Charges and Fields · Single correct

Match List-I with List-II. \begin{tabular}{|c|l|c|l|} \hline & \textbf{List-I} & & \textbf{List-II} \\ \hline A. & Gauss's law of magnetism & I. & $\displaystyle \oint \vec{E}\cdot d\vec{a}=\frac{1}{\varepsilon_0}\int \rho\,dV$ \\ \hline B. & Faraday's law of electromagnetic induction & II. & $\displaystyle \oint \vec{B}\cdot d\vec{a}=0$ \\ \hline C. & Ampere's law & III. & $\displaystyle \oint \vec{E}\cdot d\vec{l}=-\frac{d}{dt}\int \vec{B}\cdot d\vec{a}$ \\ \hline D. & Gauss's law of electrostatics & IV. & $\displaystyle \oint \vec{B}\cdot d\vec{l}=-\mu_0 I$ \\ \hline \end{tabular} Choose the correct answer from the options given below:

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

Answer: (d)

Question 50

Physics · Motion in a Plane · Single correct

Projectiles A and B are thrown at angles of $45^\circ$ and $60^\circ$ with vertical respectively from top of a $400 \, \mathrm{m}$ high tower. If their ranges and times of flight are same, the ratio of their speeds of projection $v_A : v_B$ is :

1. $1 : \sqrt{3}$
2. $\sqrt{2} : 1$
3. $1 : 2$
4. $1 : 1$

Answer: (d)

Question 51

Physics · Alternating Current · Numerical

A power transmission line feeds input power at $2.3 \, \mathrm{kV}$ to a step down transformer with its primary winding having 3000 turns. The output power is delivered at $230 \, \mathrm{V}$ by the transformer. The current in the primary of the transformer is $5 \, \mathrm{A}$ and its efficiency is $90\%$. The winding of transformer is made of copper. The output current of transformer is_____ $\mathrm{A}$.

Answer: 45

Question 52

Physics · Mechanical Properties of Fluids · Numerical

A big drop is formed by coalescing 1000 small identical drops of water. If $E_1$ be the total surface energy of 1000 small drops of water and $E_2$ be the surface energy of single big drop of water, the $E_1 : E_2$ is $x : 1$ where $x =$

Answer: 10

Question 53

Physics · System of Particles and Rotational Motion · Numerical

Two discs of moment of inertia $I_1 = 4 \, \mathrm{kg} \, \mathrm{m}^2$ and $I_2 = 2 \, \mathrm{kg} \, \mathrm{m}^2$ about their central axes $\&$ normal to their planes, rotating with angular speeds $10 \, \mathrm{rad/s}$ $\&$ $4 \, \mathrm{rad/s}$ respectively are brought into contact face to face with their axe of rotation coincident. The loss in kinetic energy of the system in the process is_____ J.

Answer: 24

Question 54

Physics · Ray Optics and Optical Instruments · Numerical

In an experiment to measure the focal length (f) of a convex lens, the magnitude of object distance (x) and the image distance (y) are measured with reference to the focal point of the lens. The $y-x$ plot is shown in figure. The focal length of the lens is_____cm.

Answer: 20

Question 55

Physics · Mathematics in Physics · Numerical

A vector has magnitude same as that of $\vec{A} = 3\hat{j} + 4\hat{j}$ and is parallel to $\vec{B} = 4\hat{i} + 3\hat{j}$. The $x$ and $y$ components of this vector in first quadrant are $x$ and 3 respectively where $x =$

Answer: 4

Question 56

Physics · Moving Charges and Magnetism · Numerical

The current of 5 A flows in a square loop of sides 1 m is placed in air. The magnetic field at the centre of the loop is $X \sqrt{2} \times 10^{-7} \, \mathrm{T}$. The value of $X$ is_____

Answer: 40

Question 57

Physics · Electric Charges and Fields · Numerical

Two identical charged spheres are suspended by string of equal lengths. The string make an angle of $37^\circ$ with each other. When suspended in a liquid of density $0.7 \, \mathrm{g/cm^3}$, the angle remains same. If density of material of the sphere is $1.4 \, \mathrm{g/cm^3}$, the dielectric constant of the liquid is_____ ( $tan 37^\circ$ = $\frac{3}{4}$ ).

Answer: 2

Question 58

Physics · Oscillations · Numerical

A simple pendulum is placed at a place where its distance from the earth's surface is equal to the radius of the earth. If the length of the string is 4 m, then the time period of small oscillations will be _____ s. [ take $g = \pi^2 \, \mathrm{ms^{-2}}$]

Answer: 8

Question 59

Physics · Waves · Numerical

A point source is emitting sound waves of Power $16 \times 10^{-8} \, \mathrm{W}$ at the origin. The ratio of intensity (magnitude only) at two points located at a distances of $2 \, \mathrm{m}$ and $4 \, \mathrm{m}$ from the origin respectively will be _____ $\mathrm{Wm}^{-2}$.

Answer: 4

Question 60

Physics · Current Electricity · Fill in the blank

Two resistances of $100\,\Omega$ and $200\,\Omega$ are connected in series with a battery of $4\,\mathrm{V}$ and negligible internal resistance. A voltmeter is used to measure voltage across $100\,\Omega$ resistance, which gives reading as $1\,\mathrm{V}$. The resistance of voltmeter must be ___ $\Omega$.

Answer: 200

Question 61

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

Which among the following purification methods is based on the principle of "Solubility" in two different solvents?

1. Column Chromatography
2. Sublimation
3. Distillation
4. Differential Extraction

Answer: (d)

Question 62

Chemistry · Alcohols, Phenols and Ethers · Single correct

Salicylaldehyde is synthesized from phenol, when reacted with

1. CO_2, NaOH
2. CCl_4, NaOH
3. HCCl_3, NaOH
4. $\mathrm{CHCl_3,\ NaOH}$

Answer: (d)

Question 63

Chemistry · Haloalkanes and Haloarenes · Single correct

Given below are two statements:\ Statement-I: High concentration of strong nucleophilic reagent with secondary alkyl halides which do not have bulky substituents will follow $S_N2$ mechanism.\ Statement-II: A secondary alkyl halide when treated with a large excess of ethanol follows $S_N1$ mechanism.\ In the light of the above statements, choose the most appropriate from the questions given below:

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

Answer: (d)

Question 64

Chemistry · Aldehydes, Ketones and Carboxylic Acids · Single correct

m-chlorobenzaldehyde on treatment with 50$\%$ KOH solution yields

Answer: (b)

Question 65

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

Given below are two statements: One is labelled as Assertion A and the other is labelled as Reason R.\ Assertion A: $\mathrm{H}_2\mathrm{Te}$ is more acidic than $\mathrm{H}_2\mathrm{S}$.\ Reason R: Bond dissociation enthalpy of $\mathrm{H}_2\mathrm{Te}$ is lower than $\mathrm{H}_2\mathrm{S}$.\ In the light of the above statements. Choose the most appropriate from the options given below.

1. Both A and R are true but R is NOT the correct explanation of A.
2. Both A and R are true and R is the correct explanation of A.
3. A is false but R is true.
4. A is true but R is false.

Answer: (b)

Question 66

Chemistry · Hydrocarbons · Single correct

Product A and B formed in the following set of reactions are:

Answer: (b)

Question 67

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

IUPAC name of following compound is

1. 2-Aminopentanenitrile
2. 2-Aminobutanenitrile
3. 3-Aminobutanenitrile
4. 3-Aminopropanenitrile

Answer: (c)

Question 68

Chemistry · Amines · Single correct

The products A and B formed in the following reaction scheme are respectively

Answer: (c)

Question 69

Chemistry · Co-ordination Compounds · Single correct

The molecule/ion with square pyramidal shape is:

1. $[Ni(CN)_4]^{2-}$
2. $PCl_5$
3. $BrF_5$
4. $PF_5$

Answer: (c)

Question 70

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

The orange colour of $\mathrm{K_2Cr_2O_7}$ and purple colour of $\mathrm{KMnO_4}$ is due to

1. Charge transfer transition in both.
2. $d \rightarrow d$ transition in $\mathrm{KMnO_4}$ and charge transfer transitions in $\mathrm{K_2Cr_2O_7}$.
3. $d \rightarrow d$ transition in $\mathrm{K_2Cr_2O_7}$ and charge transfer transitions in $\mathrm{KMnO_4}$.
4. $d \rightarrow d$ transition in both.

Answer: (a)

Question 71

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

Alkaline oxidative fusion of $\mathrm{MnO_2}$ gives "A" which on electrolytic oxidation in alkaline solution produces B. A and B respectively are:

1. $\mathrm{Mn_2O_7}$ and $\mathrm{MnO_4^-}$
2. $\mathrm{MnO_4^{2-}}$ and $\mathrm{MnO_4^-}$
3. $\mathrm{Mn_2O_3}$ and $\mathrm{MnO_4^{2-}}$
4. $\mathrm{MnO_4^{2-}}$ and $\mathrm{Mn_2O_7}$

Answer: (b)

Question 72

Chemistry · Solutions · Single correct

If a substance ' A ' dissolves in solution of a mixture of ' B ' and ' C ' with their respective number of moles as $n_A, n_B$ and $n_C$, mole fraction of C in the solution is:

1. $\frac{n_C}{n_A \times n_B \times n_C}$
2. $\frac{n_C}{n_A + n_B + n_C}$
3. $\frac{n_C}{n_A - n_B - n_C}$
4. $\frac{n_B}{n_A + n_B}$

Answer: (b)

Question 73

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

Given below are two statements:\ Statement-I: Along the period, the chemical reactivity of the element gradually increases from group 1 to group 18.\ Statement-II: The nature of oxides formed by group 1 element is basic while that of group 17 elements is acidic.\ In the light above statements, choose the most appropriate from the questions given below:

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

Answer: (c)

Question 74

Chemistry · Co-ordination Compounds · Single correct

The coordination geometry around the manganese in decacarbonyl dimanganese(0)

1. Octahedral
2. Trigonal bipyramidal
3. Square pyramidal
4. Square planar

Answer: (a)

Question 75

Chemistry · Chemical Bonding and Molecular Structure · Single correct

Given below are two statements:\ Statement-I: Since fluorine is more electronegative than nitrogen, the net dipole moment of $\mathrm{NF}_3$ is greater than $\mathrm{NH}_3$.\ Statement-II: In $\mathrm{NH}_3$, the orbital dipole due to lone pair and the dipole moment of N-H bonds are in opposite direction, but in $\mathrm{NF}_3$ the orbital dipole due to lone pair and dipole moments of N-F bonds are in same direction.\ In the light of the above statements. Choose the most appropriate from the options given below.

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

Answer: (b)

Question 76

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

The correct stability order of carbocations is

1. \quad (CH_3)_3C^{+}>CH_3-CH_2^{+}>(CH_3)_2CH^{+}>CH_3^{+}
2. \quad CH_3^{+}>(CH_3)_2CH^{+}>CH_3-CH_2^{+}>(CH_3)_3C^{+}
3. \quad (CH_3)_3C^{+}>(CH_3)_2CH^{+}>CH_3-CH_2^{+}>CH_3^{+}
4. \quad CH_3^{+}>CH_3-CH_2^{+}>CH_3-CH(CH_3)^{+}>(CH_3)_3C^{+}

Answer: (c)

Question 77

Chemistry · Solutions · Single correct

The solution from the following with highest depression in freezing point/lowest freezing point is

1. 180 g of acetic acid dissolved in water
2. 180 g of acetic acid dissolved in benzene
3. 180 g of benzoic acid dissolved in benzene
4. 180 g of glucose dissolved in water

Answer: (a)

Question 78

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

A and B formed in the following reactions are: $$\mathrm{CrO_2Cl_2 + 4NaOH \rightarrow A + 2NaCl + 2H_2O}$$ $$\mathrm{A + 2HCl + 2H_2O_2 \rightarrow B + 3H_2O}$$

1. $A = \mathrm{Na_2CrO_4}$, $B = \mathrm{CrO_5}$
2. $A = \mathrm{Na_2Cr_2O_4}$, $B = \mathrm{CrO_4}$
3. $A = \mathrm{Na_2Cr_2O_7}$, $B = \mathrm{CrO_3}$
4. $A = \mathrm{Na_2Cr_2O_7}$, $B = \mathrm{CrO_5}$

Answer: (a)

Question 79

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

Choose the correct statements about the hydrides of group 15 elements. A. The stability of the hydrides decreases in the order $\mathrm{NH}_3 > \mathrm{PH}_3 > \mathrm{AsH}_3 > \mathrm{SbH}_3 > \mathrm{BiH}_3$ B. The reducing ability of the hydrides increases in the order $\mathrm{NH}_3 < \mathrm{PH}_3 < \mathrm{AsH}_3 < \mathrm{SbH}_3 < \mathrm{BiH}_3$ C. Among the hydrides, $\mathrm{NH}_3$ is strong reducing agent while $\mathrm{BiH}_3$ is mild reducing agent. D. The basicity of the hydrides increases in the order $\mathrm{NH}_3 < \mathrm{PH}_3 < \mathrm{AsH}_3 < \mathrm{SbH}_3 < \mathrm{BiH}_3$\ Choose the most appropriate from the option given below:

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

Answer: (c)

Question 80

Chemistry · Electrochemistry · Single correct

Reduction potential of ions are given below: $\begin{array}{ccc} \mathrm{ClO_4^-} & \mathrm{IO_4^-} & \mathrm{BrO_4^-} \\ E^\circ = 1.19\,\mathrm{V} & E^\circ = 1.65\,\mathrm{V} & E^\circ = 1.74\,\mathrm{V} \end{array}$ $ The correct order of their oxidising power is:

1. $\mathrm{ClO_4^- > IO_4^- > BrO_4^-}$
2. $\mathrm{BrO_4^- > IO_4^- > ClO_4^-}$
3. $\mathrm{BrO_4^- > ClO_4^- > IO_4^-}$
4. $\mathrm{IO_4^- > BrO_4^- > ClO_4^-}$

Answer: (b)

Question 81

Chemistry · Co-ordination Compounds · Numerical

Number of complexes which show optical isomerism among the following is $\underline{\phantom{123456789}}$ cis $-[\mathrm{Cr(ox)_2Cl_2}]^{3-}$, $[\mathrm{Co(en)_3}]^{3+}$,\ cis $-[\mathrm{Pt(en)_2Cl_2}]^{2+}$, cis $-[\mathrm{Co(en)_2Cl_2}]^{+}$,\ trans $-[\mathrm{Pt(en)_2Cl_2}]^{2+}$, trans $-[\mathrm{Cr(ox)_2Cl_2}]^{3-}$

Answer: 4

Question 82

Chemistry · Chemical Kinetics and Nuclear Chemistry · Numerical

$\mathrm{NO_2}$ required for a reaction is produced by decomposition of $\mathrm{N_2O_5}$ in $\mathrm{CCl_4}$ as by equation $$2\,\mathrm{N_2O_{5(g)}} \rightarrow 4\,\mathrm{NO_{2(g)}} + \mathrm{O_{2(g)}}$$ The initial concentration of $\mathrm{N_2O_5}$ is $3\,\mathrm{mol\,L^{-1}}$ and it is $2.75\,\mathrm{mol\,L^{-1}}$ after $30$ minutes. The rate of formation of $\mathrm{NO_2}$ is $x \times 10^{-3}\,\mathrm{mol\,L^{-1}\,min^{-1}}$, value of $x$ is ______

Answer: 17

Question 83

Chemistry · Thermodynamics · Numerical

Two reactions are given below: $$2\mathrm{Fe}_{(s)} + \frac{3}{2}\mathrm{O}_{2(g)} \rightarrow \mathrm{Fe}_2\mathrm{O}_{3(s)}, \Delta H^\circ = -822 \, \mathrm{kJ/mol}$$ $$\mathrm{C}_{(s)} + \frac{1}{2}\mathrm{O}_{2(g)} \rightarrow \mathrm{CO}_{(g)}, \Delta H^\circ = -110 \, \mathrm{kJ/mol}$$ Then enthalpy change for following reaction $$3\mathrm{C}_{(s)} + \mathrm{Fe}_2\mathrm{O}_{3(s)} \rightarrow 2\mathrm{Fe}_{(s)} + 3\mathrm{CO}_{(g)}$$

Answer: 492

Question 84

Chemistry · Biomolecules · Single correct

The total number of correct statements, regarding the nucleic acids is

1. RNA is regarded as the reserve of genetic information.
2. DNA molecule self-duplicates during cell division
3. DNA synthesizes proteins in the cell.
4. The message for the synthesis of particular proteins is present in DNA
5. Identical DNA strands are transferred to daughter cells.

Answer: (c)

Question 85

Chemistry · Equilibrium · Numerical

The pH of an aqueous solution containing 1M benzoic acid ($pK_a = 4.20$) and 1M sodium benzoate is 4.5. The volume of benzoic acid solution in 300 mL of this buffer solution is _____ mL.

Answer: 100

Question 86

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

Number of geometrical isomers possible for the given structure is/are _____

Answer: 4

Question 87

Chemistry · Redox Reactions · Numerical

Total number of species from the following which can undergo disproportionation reaction . $\mathrm{H_2O_2, ClO_3^-, P_4, Cl_2, Ag,}$$Cu^{+1}$, $F_2,$ $NO_2,$ $K^+$

Answer: 6

Question 88

Chemistry · Analytical Chemistry · Numerical

Number of metal ions characterized by flame test among the following is . $\mathrm{Sr}^{2+}, \mathrm{Ba}^{2+}, \mathrm{Ca}^{2+}, \mathrm{Cu}^{2+}, \mathrm{Zn}^{2+}, \mathrm{Co}^{2+}, \mathrm{Fe}^{2+}$

Answer: 4

Question 89

Chemistry · Haloalkanes and Haloarenes · Numerical

2-chlorobutane + $\mathrm{Cl}_2 \rightarrow \mathrm{C}_4\mathrm{H}_8\mathrm{Cl}_2$ (isomers) Total number of optically active isomers shown by $\mathrm{C}_4\mathrm{H}_8\mathrm{Cl}_2$, obtained in the above reaction is _______.

Answer: 6

Question 90

Chemistry · Structure of Atom · Numerical

Number of spectral lines obtained in $\mathrm{He}^+$ spectra, when an electron makes transition from fifth excited state to first excited state will be

Answer: 10