JEE Main 1 February 2024 Shift 1 question paper with solutions

Question 1

Maths · Probability · Single correct

A bag contains 8 balls, whose colours are either white or black. 4 balls are drawn at random without replacement and it was found that 2 balls are white and other 2 balls are black. The probability that the bag contains equal number of white and black balls is:

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

Answer: (b)

Question 2

Maths · Integrals · Single correct

The value of the integral $$\int_{0}^{\frac{\pi}{4}} \frac{xdx}{\sin^4(2x) + \cos^4(2x)}$$ equals :

1. $\($ $\frac{\sqrt{2}\pi^2}{8}$ $\)$
2. $\($ $\frac{\sqrt{2}\pi^2}{16}$ $\)$
3. $\($ $\frac{\sqrt{2}\pi^2}{32}$ $\)$
4. $\($ $\frac{\sqrt{2}\pi^2}{64}$ $\)$

Answer: (c)

Question 3

Maths · Matrices · Single correct

If $A=\begin{bmatrix} \sqrt{2} & 1\\ -1 & \sqrt{2} \end{bmatrix}$, $B=\begin{bmatrix} 1 & 0\\ 1 & 1 \end{bmatrix}$, $C=ABA^{T}$ and $X=A^{T}C^{2}A$, then $\det X$ is equal to:

1. 243
2. 729
3. 27
4. 891

Answer: (b)

Question 4

Maths · Trigonometric Functions · Single correct

If $\tan A = \frac{1}{\sqrt{x(x^2+x+1)}}$, $\tan B = \frac{\sqrt{x}}{\sqrt{x^2+x+1}}$ and $\tan C = \left(x^{-3} + x^{-2} + x^{-1}\right)^{\frac{1}{2}}$, $0 < A, B, C < \frac{\pi}{2}$, then $A + B$ is equal to :

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

Answer: (a)

Question 5

Maths · Permutations and Combinations · Single correct

If n is the number of ways five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero, then n is equal to:

1. 47
2. 53
3. 51
4. 43

Answer: (c)

Question 6

Maths · Complex Numbers and Quadratic Equations · Single correct

Let $S = \{ z \in \mathbb{C} : |z - 1| = 1 and (\sqrt{2} - 1)(z + \bar{z}) - i(z - \bar{z}) = 2\sqrt{2} \}$. Let $z_1, \ z_2 \in S$ be such that $|z_1| = \max_{z \in S} |z|$ and $|z_2| = \min_{z \in S} |z|$. Then $\left| \sqrt{2} z_1 - z_2 \right|^2$ equals:

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

Answer: (d)

Question 7

Maths · Statistics · Single correct

Let the median and the mean deviation about the median of 7 observation 170, 125, 230, 190, 210, a, b be 170 and $\frac{205}{7}$ respectively. Then the mean deviation about the mean of these 7 observations is :

1. 31
2. 28
3. 30
4. 32

Answer: (c)

Question 8

Maths · Vector Algebra · Single correct

Let $\vec{a}$ = -5$\hat{i}$ + $\hat{j}$ - 3$\hat{k}$, $\vec{b}$ = $\hat{i}$ + 2$\hat{j}$ - 4$\hat{k}$ and $\vec{c}$ = ((($\vec{a}$ $\times$ $\vec{b}$) $\times$ $\hat{i}$) $\times$ $\hat{i}$) $\times$ $\hat{i}$ . Then $\vec{c}$ $\cdot$ (-$\hat{i}$ + $\hat{j}$ + $\hat{k}$) is equal to

1. -12
2. -10
3. -13
4. -15

Answer: (a)

Question 9

Maths · Sequences and Series · Single correct

Let $S = \left\{x \in \mathbb{R} : \left(\sqrt{3} + \sqrt{2}\right)^x + \left(\sqrt{3} - \sqrt{2}\right)^x = 10\right\}$. Then the number of elements in $S$ is:

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

Answer: (c)

Question 10

Maths · Applications of Integrals · Single correct

The area enclosed by the curves $xy + 4y = 16$ and $x + y = 6$ is equal to:

1. 28 - 30 $\log$_2 2
2. 30 - 28 $\log$_2 2
3. 30 - 32 $\log$_2 2
4. 32 - 30 $\log$_2 2

Answer: (c)

Question 11

Maths · Relations and Functions · Single correct

Let f : $\mathbb{R}$ $\to$ $\mathbb{R}$ and g : $\mathbb{R}$ $\to$ $\mathbb{R}$ be defined as $$f(x) = \begin{cases} \log_e x , & x > 0 \\ e^{-x} , & x \leq 0 \end{cases}$$ and $$g(x) = \begin{cases} x , & x \geq 0 \\ e^x , & x < 0 \end{cases}.$$ Then, gof : $\mathbb{R}$ $\to$ $\mathbb{R}$ is :

1. one-one but not onto
2. neither one-one nor onto
3. onto but not one-one
4. both one-one and onto

Answer: (b)

Question 12

Maths · Determinants · Single correct

If the system of equations $$2x + 3y - z = 5$$ $$x + \alpha y + 3z = -4$$ $$3x - y + \beta z = 7$$ has infinitely many solutions, then $13\alpha\beta$ is equal to

1. 1110
2. 1120
3. 1210
4. 1220

Answer: (b)

Question 13

Maths · Conic Sections · Single correct

For $0 < \theta < \pi/2$, if the eccentricity of the hyperbola $x^2 - y^2 \operatorname{cosec}^2\theta = 5$ is $\sqrt{7}$ times eccentricity of the ellipse $x^2 \operatorname{cosec}^2\theta + y^2 = 5$, then the value of $\theta$ is :

1. $\frac{\pi}{6}$
2. $\frac{5\pi}{12}$
3. $\frac{\pi}{3}$
4. $\frac{\pi}{4}$

Answer: (c)

Question 14

Maths · Differential Equations · Single correct

Let y = y(x) be the solution of the differential equation $\frac{dy}{dx}$ = 2x(x+y)^3 - x(x+y) - 1, y(0) = 1. Then, ( $\frac{1}{\sqrt{2}}$ + y( $\frac{1}{\sqrt{2}}$ ) )^2 equals:

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

Answer: (d)

Question 15

Maths · Continuity and Differentiability · Single correct

Let f : $\mathbb{R} \to \mathbb{R}$ be defined as $$f(x) = \begin{cases} \frac{a - b \cos 2x}{x^2} & ; x 1 \end{cases}$$ If $f$ is continuous everywhere in $\mathbb{R}$ and $m$ is the number of points where $f$ is NOT differential then $m + a + b + c$ equals :

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

Answer: (d)

Question 16

Maths · Conic Sections · Single correct

Let $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, $a > b$ be an ellipse, whose eccentricity is $\frac{1}{\sqrt{2}}$ and the length of the latus rectum is $\sqrt{14}$. Then the square of the eccentricity of $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ is:

1. 3
2. 7/2
3. 3/2
4. 5/2

Answer: (c)

Question 17

Maths · Sequences and Series · Single correct

Let 3, a, b, c be in A.P. and 3, a - 1, b + 1, c + 9 be in G.P. Then, the arithmetic mean of a, b and c is :

1. -4
2. -1
3. 13
4. 11

Answer: (d)

Question 18

Maths · Conic Sections · Single correct

Let $C : x^2 + y^2 = 4$ and $C' : x^2 + y^2 - 4\lambda x + 9 = 0$ be two circles. If the set of all values of $\lambda$ so that the circles $C$ and $C'$ intersect at two distinct points, is $\mathbb{R} - [a, b]$, then the point $(8a + 12, 16b - 20)$ lies on the curve :

1. $x^2 + 2y^2 - 5x + 6y = 3$
2. $5x^2 - y = -11$
3. $x^2 - 4y^2 = 7$
4. $6x^2 + y^2 = 42$

Answer: (d)

Question 19

Maths · Applications of Derivatives · Single correct

If $5f(x) + 4f\left(\frac{1}{x}\right) = x^2 - 2, \forall x \neq 0$ and $y = 9x^2 f(x)$, then $y$ is strictly increasing in :

1. $\left(0, \frac{1}{\sqrt{5}}\right) \cup \left(\frac{1}{\sqrt{5}}, \infty\right)$
2. $\left(-\frac{1}{\sqrt{5}}, 0\right) \cup \left(\frac{1}{\sqrt{5}}, \infty\right)$
3. $\left(-\frac{1}{\sqrt{5}}, 0\right) \cup \left(0, \frac{1}{\sqrt{5}}\right)$
4. $(-\infty, \frac{1}{\sqrt{5}}) \cup \left(0, \frac{1}{\sqrt{5}}\right)$

Answer: (b)

Question 20

Maths · Three Dimensional Geometry · Single correct

If the shortest distance between the lines $\frac{x-\lambda}{-2} = \frac{y-2}{1} = \frac{z-1}{1}$ and $\frac{x-\sqrt{3}}{1} = \frac{y-1}{-2} = \frac{z-2}{1}$ is 1, then the sum of all possible values of $\lambda$ is:

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

Answer: (b)

Question 21

Maths · Differential Equations · Numerical

If $x = x(t)$ is the solution of the differential equation $(t + 1) dx = \left(2x + (t + 1)^4\right) dt, x(0) = 2$, then, $x(1)$ equals

Answer: 14

Question 22

Maths · Sets · Numerical

The number of elements in the set $$S = \{(x, y, z) : x, y, z \in \mathbb{Z}, x + 2y + 3z = 42, x, y, z \geq 0\}$$ equals $\ldots$.

Answer: 169

Question 23

Maths · Binomial Theorem · Numerical

If the Coefficient of $x^{30}$ in the expansion of $$\left(1 + \frac{1}{x}\right)^6 (1 + x^2)^7 (1 - x^3)^8$$ ; $x \neq 0$ is $\alpha$, then $|\alpha|$ equals

Answer: 678

Question 24

Maths · Sequences and Series · Numerical

Let 3, 7, 11, 15, $\ldots$, 403 and 2, 5, 8, 11, $\ldots$, 404 be two arithmetic progressions. Then the sum, of the common terms in them, is equal to

Answer: 6699

Question 25

Maths · Limits and Derivatives · Numerical

Let $\{$x$\}$ denote the fractional part of $x$ and $f(x) = \frac{\cos^{-1}(1-\{x\}^2) \sin^{-1}(1-\{x\})}{\{x\}-\{x\}^3}$, $x \neq 0$. If $L$ and $R$ respectively denotes the left hand limit and the right hand limit of $f(x)$ at $x = 0$, then $\frac{32}{\pi^2} \left(L^2 + R^2\right)$ is equal to

Answer: 18

Question 26

Maths · Conic Sections · Numerical

Let the line $L : \sqrt{2}x + y = \alpha$ pass through the point of the intersection $P$ (in the first quadrant) of the circle $x^2 + y^2 = 3$ and the parabola $x^2 = 2y$. Let the line $L$ touch two circles $C_1$ and $C_2$ of equal radius $2\sqrt{3}$. If the centres $Q_1$ and $Q_2$ of the circles $C_1$ and $C_2$ lie on the y-axis, then the square of the area of the triangle $PQ_1Q_2$ is equal to

Answer: 72

Question 27

Maths · Complex Numbers and Quadratic Equations · Numerical

Let $\mathbb{P} = \{ z \in \mathbb{C} : |z + 2 - 3i| \leq 1 \}$ and $\mathbb{Q} = \{ z \in \mathbb{C} : z(1+i) + \bar{z}(1-i) \leq -8 \}$. Let in $\mathbb{P} \cap \mathbb{Q}$, $|z - 3 + 2i|$ be maximum and minimum at $z_1$ and $z_2$ respectively. If $|z_1|^2 + 2|z_2|^2 = \alpha + \beta\sqrt{2}$, where $\alpha$, $\beta$ are integers, then $\alpha + \beta$ equals

Answer: 36

Question 28

Maths · Integrals · Numerical

If $$\int_{-\pi/2}^{\pi/2} \frac{8\sqrt{2} \cos x \, dx}{(1+e^{\sin x})(1+\sin^4 x)} = \alpha \pi + \beta \log_e(3 + 2\sqrt{2})$$, where $\alpha$, $\beta$ are integers, then $\alpha^2 + \beta^2$ equals

Answer: 8

Question 29

Maths · Three Dimensional Geometry · Numerical

Let the line of the shortest distance between the lines $$L_1 : \vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda (\hat{i} - \hat{j} + \hat{k})$$ and $$L_2 : \vec{r} = (4\hat{i} + 5\hat{j} + 6\hat{k}) + \mu (\hat{i} + \hat{j} - \hat{k})$$ intersect $L_1$ and $L_2$ at $P$ and $Q$ respectively. If $(\alpha, \beta, \gamma)$ is the midpoint of the line segment $PQ$, then $$2(\alpha + \beta + \gamma)$$ is equal to

Answer: 21

Question 30

Maths · Relations and Functions · Numerical

Let A = {1, 2, 3, $\ldots$, 20}. Let $R_1$ and $R_2$ two relation on A such that $R_1$ = $\{$(a, b) : b is divisible by a$\}$ $R_2$ = $\{$(a, b) : a is an integral multiple of b$\}$. Then, number of elements in $R_1 - R_2$ is equal to $\ldots$.

Answer: 46

Question 31

Physics · Mechanical Properties of Solids · Single correct

With rise in temperature, the Young's modulus of elasticity

1. changes erratically
2. decreases
3. increases
4. remains unchanged

Answer: (b)

Question 32

Physics · Gravitation · Single correct

If $R$ is the radius of the earth and the acceleration due to gravity on the surface of earth is $g = \pi^2 \, \mathrm{m/s^2}$, then the length of the second's pendulum at a height $h = 2R$ from the surface of earth will be,:

1. $\frac{2}{9} \, \mathrm{m}$
2. $\frac{1}{9} \, \mathrm{m}$
3. $\frac{4}{9} \, \mathrm{m}$
4. $\frac{8}{9} \, \mathrm{m}$

Answer: (b)

Question 33

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

In the given circuit if the power rating of Zener diode is 10 $\mathrm{mW}$, the value of series resistance $R_s$ to regulate the input unregulated supply is:

1. 5k$\Omega$
2. 10$\Omega$
3. 1k$\Omega$
4. None of these

Answer: (d)

Question 34

Physics · Current Electricity · Single correct

The reading in the ideal voltmeter (V) shown in the given circuit diagram is :

1. 5 \, $\mathrm{V}$
2. 10 \, $\mathrm{V}$
3. 0 \, $\mathrm{V}$
4. 3 \, $\mathrm{V}$

Answer: (c)

Question 35

Physics · Electrostatic Potential and Capacitance · Single correct

Two identical capacitors have same capacitance $C$. One of them is charged to the potential $V$ and other to the potential $2 \, V$. The negative ends of both are connected together. When the positive ends are also joined together, the decrease in energy of the combined system is :

1. $\frac{1}{4} CV^2$
2. $2CV^2$
3. $\frac{1}{2} CV^2$
4. $\frac{3}{4} CV^2$

Answer: (a)

Question 36

Physics · Kinetic Theory · Single correct

Two moles a monoatomic gas is mixed with six moles of a diatomic gas. The molar specific heat of the mixture at constant volume is :

1. $\frac{9}{4} R$
2. $\frac{7}{4} R$
3. $\frac{3}{2} R$
4. $\frac{5}{2} R$

Answer: (a)

Question 37

Physics · System of Particles and Rotational Motion · Single correct

A ball of mass 0.5 kg is attached to a string of length 50 cm. The ball is rotated on a horizontal circular path about its vertical axis. The maximum tension that the string can bear is 400 N. The maximum possible value of angular velocity of the ball in rad/s is,;

1. 1600
2. 40
3. 1000
4. 20

Answer: (b)

Question 38

Physics · Alternating Current · Single correct

A parallel plate capacitor has a capacitance $C = 200 \, \mathrm{pF}$. It is connected to $230 \, \mathrm{V}$ ac supply with an angular frequency $300 \, \mathrm{rad/s}$. The rms value of conduction current in the circuit and displacement current in the capacitor respectively are :

1. $1.38 \, \mu \mathrm{A}$ and $1.38 \, \mu \mathrm{A}$
2. $14.3 \, \mu \mathrm{A}$ and $143 \, \mu \mathrm{A}$
3. $13.8 \, \mu \mathrm{A}$ and $138 \, \mu \mathrm{A}$
4. $13.8 \, \mu \mathrm{A}$ and $13.8 \, \mu \mathrm{A}$

Answer: (d)

Question 39

Physics · Thermodynamics · Single correct

The pressure and volume of an ideal gas are related as $PV^{3/2} = K$ (Constant). The work done by gas when the gas is taken from state $A (P_1, V_1, T_1)$ to state $B (P_2, V_2, T_2)$ is :

1. $2 (P_1 V_1 - P_2 V_2)$
2. $2 (P_2 V_2 - P_1 V_1)$
3. $2 (\sqrt{P_1} V_1 - \sqrt{P_2} V_2)$
4. $2 (P_2 \sqrt{V_2} - P_1 \sqrt{V_1})$

Answer: (a)

Question 40

Physics · Current Electricity · Single correct

A galvanometer has a resistance of $50\,\Omega$ and it allows maximum current of $5\,\mathrm{mA}$. It can be converted into voltmeter to measure upto $100\,\mathrm{V}$ by connecting in series a resistor of resistance

1. $5975\,\Omega$
2. $20050\,\Omega$
3. $19950\,\Omega$
4. $19500\,\Omega$

Answer: (c)

Question 41

Physics · Dual Nature of Radiation and Matter · Single correct

The de Broglie wavelengths of a proton and an $\alpha$ particle are $\lambda$ and $2\lambda$ respectively. The ratio of the velocities of proton and $\alpha$ particle will be:

1. 1 : 8
2. 1 : 2
3. 4 : 1
4. 8 : 1

Answer: (d)

Question 42

Physics · Experimental Physics · Single correct

10 divisions on the main scale of a Vernier calliper coincide with 11 divisions on the Vernier scale. If each division on the main scale is of 5 units, the least count of the instrument is:

1. $\frac{1}{2}$
2. $\frac{10}{11}$
3. $\frac{50}{11}$
4. $\frac{5}{11}$

Answer: (d)

Question 43

Physics · Alternating Current · Single correct

In series LCR circuit, the capacitance is changed from C to 4C. To keep the resonance frequency unchanged, the new inductance should be:

1. reduced by $\frac{1}{4} L$
2. increased by $2 L$
3. reduced by $\frac{3}{4} L$
4. increased to $4 L$

Answer: (c)

Question 44

Physics · Mathematics in Physics · Single correct

The radius $(r)$, length $(l)$ and resistance $(R)$ of a metal wire was measured in the laboratory as $r = (0.35 \pm 0.05)\, \mathrm{cm}$ $R = (100 \pm 10)\, \mathrm{ohm}$ $l = (15 \pm 0.2)\, \mathrm{cm}$ The percentage error in resistivity of the material of the wire is:

1. 25.6%
2. 39.9%
3. 37.3%
4. 35.6%

Answer: (b)

Question 45

Physics · Physical World, Units and Measurements · Single correct

The dimensional formula of angular impulse is:

1. [M L^{-2} T^{-1}]
2. [M L^2 T^{-2}]
3. [M L T^{-1}]
4. [M L^2 T^{-1}]

Answer: (d)

Question 46

Physics · System of Particles and Rotational Motion · Single correct

A simple pendulum of length 1 m has a wooden bob of mass 1 kg. It is struck by a bullet of mass $10^{-2} \, \mathrm{kg}$ moving with a speed of $2 \times 10^2 \, \mathrm{ms^{-1}}$. The bullet gets embedded into the bob. The height to which the bob rises before swinging back is. (use $g = 10 \, \mathrm{m/s^2}$)

1. 0.30 m
2. 0.20 m
3. 0.35 m
4. 0.40 m

Answer: (b)

Question 47

Physics · Motion in a Plane · Single correct

A particle moving in a circle of radius $R$ with uniform speed takes time $T$ to complete one revolution. If this particle is projected with the same speed at an angle $\theta$ to the horizontal, the maximum height attained by it is equal to $4R$. The angle of projection $\theta$ is then given by:

1. $\sin^{-1} \left[ \frac{2gT^2}{\pi^2 R} \right]^{\frac{1}{2}}$
2. $\sin^{-1} \left[ \frac{\pi^2 R}{2gT^2} \right]^{\frac{1}{2}}$
3. $\cos^{-1} \left[ \frac{2gT^2}{\pi^2 R} \right]^{\frac{1}{2}}$
4. $\cos^{-1} \left[ \frac{\pi R}{2gT^2} \right]^{\frac{1}{2}}$

Answer: (a)

Question 48

Physics · Laws of Motion · Single correct

Consider a block and trolley system as shown in figure. If the coefficient of kinetic friction between the trolley and the surface is 0.04, the acceleration of the system in $\mathrm{ms^{-2}}$ is: (Consider that the string is massless and unstretchable and the pulley is also massless and frictionless):

1. 3
2. 4
3. 2
4. 1.2

Answer: (c)

Question 49

Physics · Atoms · Single correct

The minimum energy required by a hydrogen atom in ground state to emit radiation in Balmer series is nearly:

1. 1.5 $\mathrm{eV}$
2. 13.6 $\mathrm{eV}$
3. 1.9 $\mathrm{eV}$
4. 12.1 $\mathrm{eV}$

Answer: (d)

Question 50

Physics · Wave Optics · Single correct

A monochromatic light of wavelength $6000\,\mathrm{\AA}$ is incident on the single slit of width $0.01\,\mathrm{mm}$. If the diffraction pattern is formed at the focus of the convex lens of focal length $20\,\mathrm{cm}$, the linear width of the central maximum is:

1. 60 mm
2. 24 mm
3. 120 mm
4. 12 mm

Answer: (b)

Question 51

Physics · Moving Charges and Magnetism · Numerical

A regular polygon of 6 sides is formed by bending a wire of length $4\pi$ meter. If an electric current of $4\pi\sqrt{3}$ A is flowing through the sides of the polygon, the magnetic field at the centre of the polygon would be $x \times 10^{-7}$ T. The value of $x$ is_____

Answer: 72

Question 52

Physics · Electromagnetic Induction · Numerical

A rectangular loop of sides $12\,\mathrm{cm}$ and $5\,\mathrm{cm}$, with its sides parallel to the x-axis and y-axis respectively moves with a velocity of $5\,\mathrm{cm/s}$ in the positive x axis direction, in a space containing a variable magnetic field in the positive z direction. The field has a gradient of $10^{-3}\,\mathrm{T/cm}$ along the negative x direction and it is decreasing with time at the rate of $10^{-3}\,\mathrm{T/s}$. If the resistance of the loop is $6\,\mathrm{m\Omega}$, the power dissipated by the loop as heat is $\underline{\hspace{1cm}}\times10^{-9}\,\mathrm{W}$.

Answer: 216

Question 53

Physics · Ray Optics and Optical Instruments · Fill in the blank

The distance between object and its 3 times magnified virtual image as produced by a convex lens is 20cm. The focal length of the lens used is ____ cm

Answer: 15

Question 54

Physics · Electric Charges and Fields · Numerical

Two identical charged spheres are suspended by strings of equal lengths. The strings make an angle $\theta$ with each other. When suspended in water the angle remains the same. If density of the material of the sphere is $1.5 \, \mathrm{g/cc}$, the dielectric constant of water will be _____ (Take density of water $= 1 \, \mathrm{g/cc}$)

Answer: 3

Question 55

Physics · Nuclei · Numerical

The radius of a nucleus of mass number 64 is 4.8 fermi. Then the mass number of another nucleus having radius of 4 fermi is $\frac{1000}{x}$, where $x$ is ____.

Answer: 27

Question 56

Physics · System of Particles and Rotational Motion · Numerical

The identical spheres each of mass $2M$ are placed at the corners of a right angled triangle with mutually perpendicular sides equal to $4 \, \mathrm{m}$ each. Taking point of intersection of these two sides as origin, the magnitude of position vector of the centre of mass of the system is $\frac{4\sqrt{2}}{x}$, where the value of $x$ is _____

Answer: 3

Question 57

Physics · Waves · Numerical

A tuning fork resonates with a sonometer wire of length $1\,\mathrm{m}$ stretched with a tension of $6\,\mathrm{N}$. When the tension in the wire is changed to $54\,\mathrm{N}$, the same tuning fork produces $12$ beats per second with it. The frequency of the tuning fork is ___ Hz.

Answer: 6

Question 58

Physics · Mechanical Properties of Fluids · Numerical

A plane is in level flight at constant speed and each of its two wings has an area of $40 \, \mathrm{m}^2$. If the speed of the air is $180 \, \mathrm{km/h}$ over the lower wing surface and $252 \, \mathrm{km/h}$ over the upper wing surface, the mass of the plane is _____ kg. (Take air density to be $1 \, \mathrm{kg} \, \mathrm{m}^{-3}$ and $g = 10 \, \mathrm{ms}^{-2}$)

Answer: 9600

Question 59

Physics · Current Electricity · Numerical

The current in a conductor is expressed as $I = 3t^2 + 4t^3$, where $I$ is in Ampere and $t$ is in second. The amount of electric charge that flows through a section of the conductor during $t = 1 \, \mathrm{s}$ to $t = 2 \, \mathrm{s}$ is_____ C.

Answer: 22

Question 60

Physics · Motion in a Straight Line · Numerical

A particle is moving in one dimension (along x axis) under the action of a variable force. It's initial position was 16 m right of origin. The variation of its position (x) with time (t) is given as $x = -3t^3 + 18t^2 + 16t$, where $x$ is in $m$ and $t$ is in $s$. The velocity of the particle when its acceleration becomes zero is_____ m/s.

Answer: 52

Question 61

Chemistry · Biomolecules · Single correct

If one strand of a DNA has the sequence ATGCTTCA, sequence of the bases in complementary strand is:

1. CATTAGCT
2. TACGAAGT
3. GTACTTAC
4. ATGCGACT

Answer: (b)

Question 62

Chemistry · Haloalkanes and Haloarenes · Single correct

Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason $(R)$. Assertion (A): Haloalkanes react with KCN to form alkyl cyanides as a main product while with AgCN form isocyanide as the main product. Reason $(R)$: KCN and AgCN both are highly ionic compounds. In the light of the above statement, choose the most appropriate answer from the options given below:

1. (A) is correct but $(R)$ is not correct
2. Both (A) and $(R)$ are correct but $(R)$ is not the correct explanation of (A)
3. (A) is not correct but $(R)$ is correct
4. Both (A) and $(R)$ are correct and $(R)$ is the correct explanation of (A)

Answer: (a)

Question 63

Chemistry · Redox Reactions · Single correct

In acidic medium, $K_2Cr_2O_7$ shows oxidising action as represented in the half reaction $$Cr_2O_7^{2-} + XH^+ + Ye^- \rightarrow 2A + ZH_2O$ $X, Y, Z$ and $A$ are respectively are:

1. $8, 6, 4$ $and$ $Cr_2O_3$
2. $14, 7, 6$ $and$ $Cr^{3+}$
3. $8, 4, 6$ $and$ $Cr_2O_3$
4. $14, 6, 7$ $and$ $Cr^{3+}$

Answer: (d)

Question 64

Chemistry · Redox Reactions · Single correct

Which of the following reactions are disproportionation reactions? (A) $\mathrm{Cu}^+ \rightarrow \mathrm{Cu}^{2+} + \mathrm{Cu}$ (B) $3\mathrm{MnO}_4^{2-} + 4\mathrm{H}^+ \rightarrow 2\mathrm{MnO}_4^- + \mathrm{MnO}_2 + 2\mathrm{H}_2\mathrm{O}$ (C) $2\mathrm{KMnO}_4 \rightarrow \mathrm{K}_2\mathrm{MnO}_4 + \mathrm{MnO}_2 + \mathrm{O}_2$ (D) $2\mathrm{MnO}_4^- + 3\mathrm{Mn}^{2+} + 2\mathrm{H}_2\mathrm{O} \rightarrow 5\mathrm{MnO}_2 + 4\mathrm{H}^+$ Choose the correct answer from the options given below:

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

Answer: (a)

Question 65

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

In case of isoelectronic species the size of $\mathrm{F}^-$, Ne and $\mathrm{Na}^+$ is affected by:

1. Principal quantum number (n)
2. None of the factors because their size is the same
3. Electron-electron interaction in the outer orbitals
4. Nuclear charge (z)

Answer: (d)

Question 66

Chemistry · Structure of Atom · Single correct

According to the wave-particle duality of matter by de-Broglie, which of the following graph plot presents most appropriate relationship between wavelength of electron ($\lambda$) and momentum of electron (p) ?

Answer: (a)

Question 67

Chemistry · Co-ordination Compounds · Single correct

Given below are two statements: Statement (I): A solution of $[\mathrm{Ni(H_2O)_6}]^{2+}$ is green in colour. Statement (II): A solution of $[\mathrm{Ni(CN)_4}]^{2-}$ is colourless. 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 incorrect
2. Both Statement I and Statement II are correct
3. Statement I is incorrect but Statement II is correct
4. Statement I is correct but Statement II is incorrect

Answer: (b)

Question 68

Chemistry · Chemical Bonding and Molecular Structure · Single correct

Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason $(R)$. Assertion (A) : PH$_3$ has lower boiling point than NH$_3$. Reason $(R)$ : In liquid state NH$_3$ molecules are associated through vander waal's forces, but PH$_3$ molecules are associated through hydrogen bonding. In the light of the above statements, choose the most appropriate answer from the options given below:

1. Both (A) and $(R)$ are correct and $(R)$ is not the correct explanation of (A)
2. is not correct but $(R)$ is correct
3. Both (A) and $(R)$ are correct but $(R)$ is the correct explanation of (A)
4. is correct but $(R)$ is not correct

Answer: (d)

Question 69

Chemistry · Haloalkanes and Haloarenes · Single correct

Identify $A$ and $B$ in the following sequence of reaction

Answer: (b)

Question 70

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

Given below are two statements: Statement (I) : Aminobenzene and aniline are same organic compounds. Statement (II) : Aminobenzene and aniline are different organic compounds. 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 71

Chemistry · Co-ordination Compounds · Single correct

Which of the following complex is homoleptic?

1. $[\mathrm{Ni(CN)_4}]^{2-}$
2. $[\mathrm{Ni(NH_3)_2Cl_2}]$
3. $[\mathrm{Fe(NH_3)_4Cl_2}]^{+}$
4. $[\mathrm{Co(NH_3)_4Cl_2}]^{+}$

Answer: (a)

Question 72

Chemistry · Hydrocarbons · Single correct

Which of the following compound will most easily be attacked by an electrophile?

Answer: (d)

Question 73

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

Ionic reactions with organic compounds proceed through: (A) Homolytic bond cleavage (B) Heterolytic bond cleavage (C) Free radical formation (D) Primary free radical (E) Secondary free radical Choose the correct answer from the options given below:

1. only
2. $(C)$ only
3. only
4. and (E) only

Answer: (c)

Question 74

Chemistry · Chemical Bonding and Molecular Structure · Single correct

Arrange the bonds in order of increasing ionic character in the molecules. $LiF, K_2O, N_2, SO_2$ and $ClF_3.$

1. ClF_3 < N_2 < SO_2 < K_2O < LiF
2. LiF < K_2O < ClF_3 < SO_2 < N_2
3. N_2 < SO_2 < ClF_3 < K_2O < LiF
4. N_2 < ClF_3 < SO_2 < K_2O < LiF

Answer: (c)

Question 75

Chemistry · Solutions · Single correct

We have three aqueous solutions of $\mathrm{NaCl}$ labelled as 'A', 'B' and 'C' with concentration $0.1\,\mathrm{M}$, $0.01\,\mathrm{M}$ $\&$ $0.001\,\mathrm{M}$, respectively. The value of van t' Hoff factor $(i)$ for these solutions will be in the order.

1. $i_A < i_B < i_C$
2. $i_A < i_C < i_B$
3. $i_A = i_B = i_C$
4. $i_A > i_B > i_C$

Answer: (a)

Question 76

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

In Kjeldahl's method for estimation of nitrogen, $CuSO_4$ acts as:

1. Reducing agent
2. Catalytic agent
3. Hydrolysis agent
4. Oxidising agent

Answer: (b)

Question 77

Chemistry · Redox Reactions · Single correct

Given below are two statements: Statement (I): Potassium hydrogen phthalate is a primary standard for standardisation of sodium hydroxide solution. Statement (II): In this titration phenolphthalein can be used as indicator. 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: (a)

Question 78

Chemistry · Aldehydes, Ketones and Carboxylic Acids · Single correct

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

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

Answer: (b)

Question 79

Chemistry · Thermodynamics · Single correct

Choose the correct option for free expansion of an ideal gas under adiabatic condition from the following:

1. $q = 0, \Delta T \neq 0, w = 0$
2. $q = 0, \Delta T < 0, w \neq 0$
3. $q \neq 0, \Delta T = 0, w = 0$
4. $q = 0, \Delta T = 0, w = 0$

Answer: (d)

Question 80

Chemistry · Hydrocarbons · Single correct

Given below are two statements: Statement (I) : The $\mathrm{NH}_2$ group in Aniline is ortho and para directing and a powerful activating group. Statement (II) : Aniline does not undergo FriedelCraft's reaction (alkylation and acylation). 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. Both Statement I and Statement II are incorrect
3. Statement I is incorrect but Statement II is correct
4. Statement I is correct but Statement II is incorrect

Answer: (a)

Question 81

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

Number of optical isomers possible for 2-chlorobutane

Answer: 2

Question 82

Chemistry · Electrochemistry · Fill in the blank

The potential for the given half cell at 298 K is $(-) \ldots \times 10^{-2} \mathrm{V}$. $2\mathrm{H}^+_{(aq)} + 2\mathrm{e}^- \rightarrow \mathrm{H}_2(g)$ $[\mathrm{H}^+] = 1\mathrm{M}, \;P_{\mathrm{H}_2} = 2 \mathrm{atm}$ (Given: $2.303RT/F = 0.06 \mathrm{V}, \log 2 = 0.3$)

Answer: 1

Question 83

Chemistry · Analytical Chemistry · Single correct

The number of white coloured salts among the following is \begin{enumerate} \item[(A)] $SrSO_4$ \item[(B)] $Mg(NH_4)PO_4$ \item[(C)] $BaCrO_4$ \item[(D)] $Mn(OH)_2$ \item[(E)] $PbSO_4$ \item[(F)] $PbCrO_4$ \item[(G)] $AgBr$ \item[(H)] $PbI_2$ \item[(I)] $CaC_2O_4$ \item[(J)] $[Fe(OH)_2(CH_3COO)]$ \end{enumerate}

1. SrSO_4
2. $\mathrm{Mg(NH_4)PO_4}$
3. BaCrO_4
4. Mn(OH)_2

Answer: e

Question 84

Chemistry · Chemical Kinetics and Nuclear Chemistry · Numerical

The ratio of $\frac{^{14}C}{^{12}C}$ in a piece of wood is $\frac{1}{8}$ part that of atmosphere. If half life of $^{14}C$ is 5730 years, the age of wood sample is _____ years.

Answer: 17190

Question 85

Chemistry · Chemical Bonding and Molecular Structure · Numerical

The number of molecules/ion/s having trigonal bipyramidal shape is ........ . $\mathrm{PF}_5$, $\mathrm{BrF}_5$, $\mathrm{PCl}_5$, $[\mathrm{PtCl}_4]^{2-}$, $\mathrm{BF}_3$, $\mathrm{Fe(CO)}_5$

Answer: 3

Question 86

Chemistry · Hydrocarbons · Numerical

Total number of deactivating groups in aromatic electrophilic substitution reaction among the following is

Answer: 2

Question 87

Chemistry · Classification of Elements and Periodicity in Properties · Numerical

Lowest Oxidation number of an atom in a compound $\mathrm{A_2 B}$ is $-2$. The number of an electron in its valence shell is

Answer: 6

Question 88

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

Among the following oxide of p-block elements, number of oxides having amphoteric nature is $\mathrm{Cl_2O_7}$, CO, $\mathrm{PbO_2}$, $\mathrm{N_2O}$, NO, $\mathrm{Al_2O_3}$, $\mathrm{SiO_2}$, $\mathrm{N_2O_5}$, $\mathrm{SnO_2}$

Answer: 3

Question 89

Chemistry · Some Basic Concepts of Chemistry · Numerical

Consider the following reaction: $$3\mathrm{PbCl}_2 + 2(\mathrm{NH}_4)_3\mathrm{PO}_4 \rightarrow \mathrm{Pb}_3(\mathrm{PO}_4)_2 + 6\mathrm{NH}_4\mathrm{Cl}$$ If $72\,\mathrm{mmol}$ of $\mathrm{PbCl}_2$ is mixed with $50\,\mathrm{mmol}$ of $(\mathrm{NH}_4)_3\mathrm{PO}_4$, then amount of $\mathrm{Pb}_3(\mathrm{PO}_4)_2$ formed is ___ mmol. (nearest integer)

Answer: 24

Question 90

Chemistry · Equilibrium · Numerical

$K_a$ for $\mathrm{CH_3COOH}$ is $1.8 \times 10^{-5}$ and $K_b$ for $\mathrm{NH_4OH}$ is $1.8 \times 10^{-5}$. The pH of ammonium acetate solution will be

Answer: 7