JEE Main 6 April 2024 Shift 2 question paper with solutions

Question 1

Maths · Properties of Triangles · Single correct

Let ABC be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle ABC and the same process is repeated infinitely many times. If P is the sum of perimeters and Q is the sum of areas of all the triangles formed in this process, then :

1. $P^2 = 6\sqrt{3}Q$
2. $P^2 = 36\sqrt{3}Q$
3. $P = 36\sqrt{3}Q^2$
4. $P^2 = 72\sqrt{3}Q$

Answer: (b)

Question 2

Maths · Relations and Functions · Single correct

Let A = $\{1, 2, 3, 4, 5\}$. Let R be a relation on A defined by $xRy$ if and only if $4x \leq 5y$. Let $m$ be the number of elements in R and $n$ be the minimum number of elements from $A \times A$ that are required to be added to R to make it a symmetric relation. Then $m + n$ is equal to:

1. 25
2. 24
3. 26
4. 23

Answer: (a)

Question 3

Maths · Probability · Single correct

If three letters can be posted to any one of the 5 different addresses, then the probability that the three letters are posted to exactly two addresses is:

1. $\frac{18}{25}$
2. $\frac{12}{25}$
3. $\frac{6}{25}$
4. $\frac{4}{25}$

Answer: (b)

Question 4

Maths · Differential Equations · Single correct

Suppose the solution of the differential equation $\frac{dy}{dx} = \frac{(2+\alpha)x - \beta y + 2}{\beta x - 2\alpha y - (\beta \gamma - 4\alpha)}$ represents a circle passing through origin. Then the radius of this circle is:

1. 2
2. $\sqrt{17}$
3. $\frac{1}{2}$
4. $\frac{\sqrt{17}}{2}$

Answer: (d)

Question 5

Maths · Conic Sections · Single correct

If the locus of the point, whose distances from the point $(2,1)$ and $(1,3)$ are in the ratio $5:4$, is $$ax^2 + by^2 + cxy + dx + ey + 170 = 0,$$ then the value of $a^2 + 2b + 3c + 4d + e$ is equal to:

1. 37
2. 437
3. -27
4. 5

Answer: (a)

Question 6

Maths · Sequences and Series · Single correct

$\lim_{n \to \infty}$ $\frac{(1^2-1)(n-1)+(2^2-2)(n-2)+\cdots+((n-1)^2-(n-1)) \cdot 1}{(1^3+2^3+\cdots+n^3)-(1^2+2^2+\cdots+n^2)}$ is equal to :

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

Answer: (b)

Question 7

Maths · Binomial Theorem · Single correct

Let $0 \leq r \leq n$. If $\binom{n+1}{r+1} : \binom{n}{r} : \binom{n-1}{r-1} = 55 : 35 : 21$, then $2n + 5r$ is equal to:

1. 50
2. 62
3. 55
4. 60

Answer: (a)

Question 8

Maths · Sequences and Series · Single correct

A software company sets up $m$ number of computer systems to finish an assignment in 17 days. If 4 computer systems crashed on the start of the second day, 4 more computer systems crashed on the start of the third day and so on, then it took 8 more days to finish the assignment. The value of $m$ is equal to:

1. 150
2. 180
3. 160
4. 125

Answer: (a)

Question 9

Maths · Complex Numbers and Quadratic Equations · Single correct

If $z_1, z_2$ are two distinct complex number such that $$\left| \frac{z_1 - 2z_2}{\frac{1}{2} - z_1 z_2} \right| = 2,$$ then

1. $z_1$ lies on a circle of radius $\frac{1}{2}$ and $z_2$ lies on a circle of radius $1$.
2. both $z_1$ and $z_2$ lie on the same circle.
3. either $z_1$ lies on a circle of radius $\frac{1}{2}$ or $z_2$ lies on a circle of radius $1$.
4. either $z_1$ lies on a circle of radius $1$ or $z_2$ lies on a circle of radius $\frac{1}{2}$.

Answer: (d)

Question 10

Maths · Applications of Derivatives · Single correct

If the function $f(x) = \left(\frac{1}{x}\right)^{2x}$; $x > 0$ attains the maximum value at $x = \frac{1}{e}$ then:

1. $e^{\pi} < \pi^{e}$
2. $e^{\pi} > \pi^{e}$
3. $(2e)^{\pi} > \pi^{(2e)}$
4. $e^{2\pi} < (2\pi)^{e}$

Answer: (b)

Question 11

Maths · Vector Algebra · Single correct

Let $$\vec{a}=6\hat{i}+\hat{j}-\hat{k}\quad\text{and}\quad\vec{b}=\hat{i}+\hat{j}.$$ If $\vec{c}$ is a vector such that $$|\vec{c}|\ge6,\quad \vec{a}\cdot\vec{c}=6|\vec{c}|,\quad |\vec{c}-\vec{a}|=2\sqrt{2},$$ and the angle between $(\vec{a}\times\vec{b})$ and $\vec{c}$ is $60^\circ$, then $$|(\vec{a}\times\vec{b})\times\vec{c}|$$ is equal to:

1. $\frac{9}{2}$(6 - $\sqrt{6}$)
2. $\frac{3}{2}$$\sqrt{6}$
3. $\frac{9}{2}$(6 + $\sqrt{6}$)
4. $\frac{3}{2}$$\sqrt{3}$

Answer: (c)

Question 12

Maths · Permutations and Combinations · Single correct

If all the words with or without meaning made using all the letters of the word "NAGPUR" are arranged as in a dictionary, then the word at $315^{th}$ position in this arrangement is :

1. NRAGUP
2. NRAPUG
3. NRAPGU
4. NRAGPU

Answer: (c)

Question 13

Maths · Continuity and Differentiability · Single correct

Suppose for a differentiable function $h$, $h(0) = 0$, $h(1) = 1$ and $h'(0) = h'(1) = 2$. If $g(x) = h(e^x) e^{h(x)}$, then $g'(0)$ is equal to:

1. 5
2. 4
3. 8
4. 3

Answer: (b)

Question 14

Maths · Three Dimensional Geometry · Single correct

Let $\mathrm{P}(\alpha, \beta, \gamma)$ be the image of the point $\mathrm{Q}(3, -3, 1)$ in the line $\frac{x-0}{1} = \frac{y-3}{1} = \frac{z-1}{-1}$ and $\mathrm{R}$ be the point $(2, 5, -1)$. If the area of the triangle $\mathrm{PQR}$ is $\lambda$ and $\lambda^2 = 14K$, then $K$ is equal to:

1. 36
2. 81
3. 72
4. 18

Answer: (b)

Question 15

Maths · Properties of Triangles · Single correct

If P(6, 1) be the orthocentre of the triangle whose vertices are A(5, -2), B(8, 3) and C(h, k), then the point C lies on the circle:

1. $x^2 + y^2 - 61 = 0$
2. $x^2 + y^2 - 52 = 0$
3. $x^2 + y^2 - 65 = 0$
4. $x^2 + y^2 - 74 = 0$

Answer: (c)

Question 16

Maths · Relations and Functions · Single correct

Let $f(x) = \frac{1}{7 - \sin 5x}$ be a function defined on $\mathbb{R}$. Then the range of the function $f(x)$ is equal to ;

1. $\left[ \frac{1}{7}, \frac{1}{6} \right]$
2. $\left[ \frac{1}{8}, \frac{1}{5} \right]$
3. $\left[ \frac{1}{7}, \frac{1}{5} \right]$
4. $\left[ \frac{1}{8}, \frac{1}{6} \right]$

Answer: (d)

Question 17

Maths · Vector Algebra · Single correct

Let $\vec{a}$ = 2$\hat{i}$ + $\hat{j}$ - $\hat{k}$, $\vec{b}$ = (($\vec{a}$ $\times$ ($\hat{i}$ + $\hat{j}$)) $\times$ $\hat{i}$) $\times$ $\hat{i}$ . Then the square of the projection of $\vec{a}$ on $\vec{b}$ is :

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

Answer: (c)

Question 18

Maths · Applications of Integrals · Single correct

If the area of the region \[ \left\{ (x,y): \frac{a}{x^2}\le y\le\frac{1}{x},\; 1\le x\le2,\; 0<a<1 \right\} \] is \[ (\log_e2)-\frac{1}{7}, \] then the value of $7a-3$ is equal to:

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

Answer: (c)

Question 19

Maths · Trigonometric Functions · Single correct

If $\int \dfrac{1}{a^2\sin^2 x + b^2\cos^2 x}\, dx = \dfrac{1}{12}\tan^{-1}(3\tan x) + \text{constant}$, then the maximum value of $a\sin x + b\cos x$, is :

1. $\sqrt{40}$
2. $\sqrt{41}$
3. $\sqrt{39}$
4. $\sqrt{42}$

Answer: (a)

Question 20

Maths · Matrices · Single correct

If $A$ is a square matrix of order 3 such that $\det(A) = 3$ and $\det(\mathrm{adj}(-4 \mathrm{adj}(-3 \mathrm{adj}(3 \mathrm{adj}((2 \, A)^{-1})))) = 2^m 3^n$, then $m + 2n$ is equal to:

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

Answer: (d)

Question 21

Maths · Continuity and Differentiability · Numerical

Let [t] denote the greatest integer less than or equal to t. Let $f : [0, \infty) \to \mathbb{R}$ be a function defined by $$f(x) = \left[ \frac{x}{2} + 3 \right] - \left[ \sqrt{x} \right].$$ Let $S$ be the set of all points in the interval $[0, 8]$ at which $f$ is not continuous. Then $$\sum_{a \in S} a$$ is equal to .

Answer: 17

Question 22

Maths · Conic Sections · Numerical

The length of the latus rectum and directrices of a hyperbola with eccentricity $e$ are 9 and $x = \pm \frac{4}{\sqrt{13}}$, respectively. Let the line $y - \sqrt{3}x + \sqrt{3} = 0$ touch this hyperbola at $(x_0, y_0)$. If $m$ is the product of the focal distances of the point $(x_0, y_0)$, then $4e^2 + m$ is equal to

Answer: 61

Question 23

Maths · Sequences and Series · Numerical

If $S(x) = (1 + x) + 2(1 + x)^2 + 3(1 + x)^3 + \cdots + 60(1 + x)^{60}, x \neq 0$, and $(60)^2 S(60) = a(b)^b + b$, where $a, b \in \mathbb{N}$, then $(a + b)$ equal to

Answer: 3660

Question 24

Maths · Integrals · Numerical

Let [t] denote the largest integer less than or equal to t. If $$\int_0^3 \left( \left[ x^2 \right] + \left[ \frac{x^2}{2} \right] \right) \, dx = a + b\sqrt{2} - \sqrt{3} - \sqrt{5} + c\sqrt{6} - \sqrt{7},$$ where $a, b, c \in \mathbb{Z}$, then $a + b + c$ is equal to

Answer: 23

Question 25

Maths · Probability · Numerical

From a lot of 12 items containing 3 defectives, a sample of 5 items is drawn at random. Let the random variable X denote the number of defective items in the sample. Let items in the sample be drawn one by one without replacement. If variance of X is $\frac{m}{n}$, where $\mathrm{gcd}(m, n) = 1$, then $n - m$ is equal to

Answer: 71

Question 26

Maths · Properties of Triangles · Numerical

In a triangle ABC, BC = 7, AC = 8, AB = $\alpha$ $\in$ $\mathbb{N}$ and $\cos$ A = $\frac{2}{3}$. If 49 $\cos$(3C) + 42 = $\frac{m}{n}$, where gcd(m, n) = 1, then m + n is equal to

Answer: 39

Question 27

Maths · Three Dimensional Geometry · Numerical

If the shortest distance between the lines $\frac{x-\lambda}{3} = \frac{y-2}{-1} = \frac{z-1}{1}$ and $\frac{x+2}{-3} = \frac{y+5}{2} = \frac{z-4}{4}$ is $\frac{44}{\sqrt{30}}$, then the largest possible value of $|\lambda|$ is equal to _______.

Answer: 43

Question 28

Maths · Complex Numbers and Quadratic Equations · Numerical

Let $\alpha$, $\beta$ be roots of $x^2 + \sqrt{2}x - 8 = 0$. If $U_n = \alpha^n + \beta^n$, then $\frac{U_{10} + \sqrt{2}U_9}{2U_8}$ is equal to

Answer: 4

Question 29

Maths · Determinants · Numerical

If the system of equations $$2x + 7y + \lambda z = 3$$ $$3x + 2y + 5z = 4$$ $$x + \mu y + 32z = -1$$ has infinitely many solutions, then $(\lambda - \mu)$ is equal to

Answer: 38

Question 30

Maths · Differential Equations · Numerical

If the solution $y(x)$ of the given differential equation $(e^y + 1) \cos x \, dx + e^y \sin x \, dy = 0$ passes through the point $\left( \frac{\pi}{2}, 0 \right)$, then the value of $e^{y\left( \frac{\pi}{6} \right)}$ is equal to

Answer: 3

Question 31

Physics · Atoms · Single correct

The longest wavelength associated with Paschen series is : (Given $R_H = 1.097 \times 10^7$ SI unit)

1. 3.646 $\times$ $10^{-6}$m
2. 1.876 $\times$ $10^{-6}$m
3. 2.973 $\times$ $10^{-6}$m
4. 1.094 $\times$ $10^{-6}$m

Answer: (b)

Question 32

Physics · Thermodynamics · Single correct

A total of 48 J heat is given to one mole of helium kept in a cylinder. The temperature of helium increases by $2^\circ \mathrm{C}$. The work done by the gas is: Given, $R = 8.3 \, \mathrm{J} \, \mathrm{K}^{-1} \, \mathrm{mol}^{-1}$.

1. 24.9 $\mathrm{J}$
2. 72.9 $\mathrm{J}$
3. 48 $\mathrm{J}$
4. 23.1 $\mathrm{J}$

Answer: (d)

Question 33

Physics · Ray Optics and Optical Instruments · Single correct

In finding out refractive index of glass slab the following observations were made through travelling microscope 50 vernier scale division = 49MSD; 20 divisions on main scale in each cm For mark on paper MSR = 8.45 $\mathrm{cm}$, VC = 26 For mark on paper seen through slab MSR = 7.12 $\mathrm{cm}$, VC = 41 For powder particle on the top surface of the glass slab MSR = 4.05 $\mathrm{cm}$, VC = 1 (MSR = Main Scale Reading, VC = Vernier Coincidence) Refractive index of the glass slab is :

1. 1.52
2. 1.35
3. 1.42
4. 1.24

Answer: (c)

Question 34

Physics · Electromagnetic Waves · Single correct

In the given electromagnetic wave $E_y = 600 \sin(\omega t - kx) \mathrm{Vm}^{-1}$, intensity of the associated light beam is (in $\mathrm{W/m}^2$): (Given $\epsilon_0 = 9 \times 10^{-12} \mathrm{C}^2 \mathrm{N}^{-1} \mathrm{m}^{-2}$)

1. 243
2. 729
3. 972
4. 486

Answer: (d)

Question 35

Physics · Gravitation · Single correct

Assuming the earth to be a sphere of uniform mass density, a body weighed 300 $\mathrm{N}$ on the surface of earth. How much it would weigh at R/4 depth under surface of earth ?

1. 75 $\mathrm{N}$
2. 300 $\mathrm{N}$
3. 375 $\mathrm{N}$
4. 225 $\mathrm{N}$

Answer: (d)

Question 36

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

The acceptor level of a p-type semiconductor is $6\,\mathrm{eV}$. The maximum wavelength of light which can create a hole would be : Given $\mathrm{hc} = 1242\,\mathrm{eVnm}$.

1. 414 nm
2. 103.5 nm
3. 207 nm
4. 407 nm

Answer: (c)

Question 37

Physics · Laws of Motion · Single correct

A car of 800 kg is taking turn on a banked road of radius 300 m and angle of banking $30^\circ$. If coefficient of static friction is 0.2 then the maximum speed with which car can negotiate the turn safely: $\left( g = 10 \, \mathrm{m/s^2}, \sqrt{3} = 1.73 \right)$

1. 264 $\mathrm{m/s}$
2. 51.4 $\mathrm{m/s}$
3. 70.4 $\mathrm{m/s}$
4. 102.8 $\mathrm{m/s}$

Answer: (b)

Question 38

Physics · Electric Charges and Fields · Single correct

Two identical conducting spheres $P$ and $S$ with charge $Q$ on each, repel each other with a force $16 \, \mathrm{N}$. A third identical uncharged conducting sphere $R$ is successively brought in contact with the two spheres. The new force of repulsion between $P$ and $S$ is:

1. 1 N
2. 6 N
3. 12 N
4. 4 N

Answer: (b)

Question 39

Physics · Electromagnetic Induction · Single correct

In a coil, the current changes from $-2 \, \mathrm{A}$ to $+2 \, \mathrm{A}$ in $0.2 \, \mathrm{s}$ and induces an emf of $0.1 \, \mathrm{V}$. The self inductance of the coil is:

1. 4$\mathrm{mH}$
2. 1$\mathrm{mH}$
3. 5$\mathrm{mH}$
4. 2.5$\mathrm{mH}$

Answer: (c)

Question 40

Physics · Ray Optics and Optical Instruments · Single correct

For the thin convex lens, the radii of curvature are at 15 cm and 30 cm respectively. The focal length the lens is 20 cm. The refractive index of the material is :

1. 1.2
2. 1.8
3. 1.5
4. 1.4

Answer: (c)

Question 41

Physics · Kinetic Theory · Single correct

Energy of 10 non rigid diatomic molecules at temperature $T$ is :

1. $70 \, K_B T$
2. $35 \, K_B T$
3. $\frac{7}{2} RT$
4. $35 RT$

Answer: (b)

Question 42

Physics · Laws of Motion · Single correct

A body of weight 200 $\mathrm{N}$ is suspended from a tree branch through a chain of mass 10 $\mathrm{kg}$. The branch pulls the chain by a force equal to (if $g = 10 \, \mathrm{m/s^2}$):

1. 100 $\mathrm{N}$
2. 200 $\mathrm{N}$
3. 300 $\mathrm{N}$
4. 150 $\mathrm{N}$

Answer: (c)

Question 43

Physics · Dual Nature of Radiation and Matter · Single correct

When UV light of wavelength 300 nm is incident on the metal surface having work function 2.13 eV, electron emission takes place. The stopping potential is: (Given $hc = 1240 \, \mathrm{eV} \, \mathrm{nm}$ )

1. 1.5 V
2. 4.1 V
3. 2 V
4. 4 V

Answer: (c)

Question 44

Physics · Current Electricity · Single correct

The number of electrons flowing per second in the filament of a 110 W bulb operating at 220 V is : ( Given $e = 1.6 \times 10^{-19} \mathrm{C}$)

1. $6.25 \times 10^{17}$
2. $1.25 \times 10^{19}$
3. $6.25 \times 10^{18}$
4. $31.25 \times 10^{17}$

Answer: (d)

Question 45

Physics · System of Particles and Rotational Motion · Single correct

When kinetic energy of a body becomes 36 times of its original value, the percentage increase in the momentum of the body will be:

1. 6$\%$
2. 600$\%$
3. 60$\%$
4. 500$\%$

Answer: (d)

Question 46

Physics · Mechanical Properties of Fluids · Single correct

Pressure inside a soap bubble is greater than the pressure outside by an amount: (given: $R =$ Radius of bubble $S =$ Surface tension of bubble)

1. $\frac{2S}{R}$
2. $\frac{4R}{S}$
3. $\frac{S}{R}$
4. $\frac{4S}{R}$

Answer: (d)

Question 47

Physics · Moving Charges and Magnetism · Single correct

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

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

Answer: (d)

Question 48

Physics · Physical World, Units and Measurements · Single correct

In a vernier calliper, when both jaws touch each other, zero of the vernier scale shifts towards left and its $4^{th}$ division coincides exactly with a certain division on main scale. If 50 vernier scale divisions equal to 49 main scale divisions and zero error in the instrument is $0.04 \, \mathrm{mm}$ then how many main scale divisions are there in 1 cm?

1. 10
2. 5
3. 20
4. 40

Answer: (c)

Question 49

Physics · Physical World, Units and Measurements · Single correct

Given below are two statements: Statement (I): Dimensions of specific heat is $[L^2 \, T^{-2} \, K^{-1}]$. Statement (II): Dimensions of gas constant is $[M \, L^2 \, T^{-1} \, K^{-1}]$. 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. Both statement (I) and statement (II) are incorrect
4. Statement (I) is incorrect but statement (II) is correct

Answer: (b)

Question 50

Physics · Motion in a Straight Line · Single correct

A body projected vertically upwards with a certain speed from the top of a tower reaches the ground in $t_1$. If it is projected vertically downwards from the same point with the same speed, it reaches the ground in $t_2$. Time required to reach the ground, if it is dropped from the top of the tower, is:

1. $\sqrt{t_1 t_2}$
2. $\sqrt{t_1 - t_2}$
3. $\sqrt{\frac{t_1}{t_2}}$
4. $\sqrt{t_1 + t_2}$

Answer: (a)

Question 51

Physics · Atoms · Fill in the blank

In Franck-Hertz experiment, the first dip in the current-voltage graph for hydrogen is observed at 10.2 $\mathrm{V}$. The wavelength of light emitted by hydrogen atom when excited to the first excitation level is ________ nm. (Given $hc = 1245 \, \mathrm{eVnm}$, $e = 1.6 \times 10^{-19} \, \mathrm{C}$).

Answer: 122

Question 52

Physics · Alternating Current · Numerical

For a given series LCR circuit it is found that maximum current is drawn when value of variable capacitance is $25\,\mathrm{nF}$. If resistance of $200\,\Omega$ and $100\,\mathrm{mH}$ inductor is being used in the given circuit. The frequency of ac source is ______ $\times 10^3 \, \mathrm{Hz}$. (given $\pi^2 = 10$)

Answer: 10

Question 53

Physics · Motion in a Straight Line · Numerical

A particle moves in a straight line so that its displacement $x$ at any time $t$ is given by $x^2 = 1 + t^2$. Its acceleration at any time $t$ is $x^{-n}$ where $n = \ldots$

Answer: 3

Question 54

Physics · System of Particles and Rotational Motion · Numerical

Three balls of masses 2 kg, 4 kg and 6 kg respectively are arranged at centre of the edges of an equilateral triangle of side 2 m. The moment of inertia of the system about an axis through the centroid and perpendicular to the plane of triangle, will be _____ $kgm^2$.

Answer: 4

Question 55

Physics · Moving Charges and Magnetism · Numerical

A coil having 100 turns, area of $5 \times 10^{-3} \, \mathrm{m}^2$, carrying current of $1 \, \mathrm{mA}$ is placed in uniform magnetic field of $0.20 \, \mathrm{T}$ such a way that plane of coil is perpendicular to the magnetic field. The work done in turning the coil through $90^\circ$ is _______ $\mu \mathrm{J}$.

Answer: 100

Question 56

Physics · Current Electricity · Numerical

In the given figure an ammeter A consists of a 240 $\Omega$ coil connected in parallel to a 10 $\Omega$ shunt. The reading of the ammeter is ____ mA

Answer: 160

Question 57

Physics · Mechanical Properties of Solids · Numerical

A wire of cross sectional area $A$, modulus of elasticity $2 \times 10^{11} \, \mathrm{Nm}^{-2}$ and length $2 \, \mathrm{m}$ is stretched between two vertical rigid supports. When a mass of $2 \, \mathrm{kg}$ is suspended at the middle it sags lower from its original position making angle $\theta = \frac{1}{100}$ radian on the points of support. The value of $A$ is $\times 10^{-4} \, \mathrm{m}^2$ (consider $x \ll L$).(given: g= 10 m/s^2)

Answer: 1

Question 58

Physics · Wave Optics · Numerical

Two coherent monochromatic light beams of intensities $I$ and $4I$ are superimposed. The difference between maximum and minimum possible intensities in the resulting beam is $xI$. The value of $x$ is ___.

Answer: 8

Question 59

Physics · Waves · Numerical

Two open organ pipes of lengths 60 cm and 90 cm resonate at $6^{th}$ and $5^{th}$ harmonics respectively. The difference of frequencies for the given modes is _______ Hz. (Velocity of sound in air = $333 \, \mathrm{m/s}$)

Answer: 740

Question 60

Physics · Electrostatic Potential and Capacitance · Numerical

A capacitor of $10 \, \mu \mathrm{F}$ capacitance whose plates are separated by $10 \, \mathrm{mm}$ through air and each plate has area $4 \, \mathrm{cm}^2$ is now filled equally with two dielectric media of $K_1 = 2$, $K_2 = 3$ respectively as shown in figure. If new force between the plates is $8 \, \mathrm{N}$. The supply voltage is ________ $\times 10^{-4} \, \mathrm{V}$.

Answer: 93

Question 61

Chemistry · Haloalkanes and Haloarenes · Single correct

The correct arrangement for decreasing order of electrophilic substitution for above compounds is :

1. (III) > (I) > (II) > (IV)
2. (IV) > (I) > (II) > (III)
3. (III) > (IV) > (II) > (I)
4. (II) > (IV) > (III) > (I)

Answer: (a)

Question 62

Chemistry · Solutions · Single correct

Molality ( m ) of 3M aqueous solution of NaCl is : (Given : Density of solution = 1.25 $\mathrm{g \, mL^{-1}}$, Molar mass in gmol^{-1} : Na = 23, Cl = 35.5)

1. 1.9 m
2. 3.85 m
3. 2.79 m
4. 2.90 m

Answer: (c)

Question 63

Chemistry · Biomolecules · Single correct

The incorrect statements regarding enzymes are : (A) Enzymes are biocatalysts. (B) Enzymes are non-specific and can catalyse different kinds of reactions. $(C)$ Most Enzymes are globular proteins. (D) Enzyme - oxidase catalyses the hydrolysis of maltose into glucose.

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

Answer: (b)

Question 64

Chemistry · Aldehydes, Ketones and Carboxylic Acids · Single correct

Consider the above chemical reaction. Product "A" is:

Answer: (a)

Question 65

Chemistry · Analytical Chemistry · Single correct

During the detection of acidic radical present in a salt, a student gets a pale yellow precipitate soluble with difficulty in $\mathrm{NH_4OH}$ solution when sodium carbonate extract was first acidified with dil. $\mathrm{HNO_3}$ and then $\mathrm{AgNO_3}$ solution was added. This indicates presence of:

1. $\mathrm{Br^-}$
2. $\mathrm{I^-}$
3. $\mathrm{Cl^-}$
4. $\mathrm{CO_3^{2-}}$

Answer: (a)

Question 66

Chemistry · Electrochemistry · Single correct

How can an electrochemical cell be converted into an electrolytic cell?

1. Applying an external opposite potential lower than $E^0$ cell.
2. Reversing the flow of ions in salt bridge.
3. Applying an external opposite potential greater than $E^0$ cell.
4. Exchanging the electrodes at anode and cathode.

Answer: (c)

Question 67

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

Arrange the following elements in the increasing order of number of unpaired electrons in it. (A) Sc (B) Cr ( C ) V (D) Ti (E) Mn Choose the correct answer from the options given below :

1. $< (D) < (C) < (B) < (E)$
2. ( C ) $< (E) < (B) < (A) < (D)$
3. $< (D) < (C) < (E) < (B)$
4. $< (C) < (D) < (E) < (A)$

Answer: (c)

Question 68

Chemistry · The s-Block Elements · Single correct

Match List-I with List-II. \begin{tabular}{|c|c|c|c|} \hline \multicolumn{2}{|c|}{\textbf{List-I}} & \multicolumn{2}{c|}{\textbf{List-II}} \\ \multicolumn{2}{|c|}{\textbf{Reaction}} & \multicolumn{2}{c|}{\textbf{Type of redox reaction}} \\ \hline (A) & Li & (I) & 589.2 \\ \hline (B) & Na & (II) & 455.5 \\ \hline (C) & Rb & (III) & 670.8 \\ \hline (D) & Cs & (IV) & 780.0 \\ \hline \end{tabular} Choose the correct answer from the options given below:

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

Answer: (d)

Question 69

Chemistry · Aldehydes, Ketones and Carboxylic Acids · Single correct

The major products formed: A and B respectively are:

Answer: (a)

Question 70

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

The incorrect statement regarding the geometrical isomers of 2-butene is :

1. cis-2-butene and trans-2-butene are not interconvertible at room temperature.
2. cis-2-butene and trans-2-butene are stereoisomers.
3. cis-2-butene has less dipole moment than trans-2-butene.
4. trans-2-butene is more stable than cis-2-butene.

Answer: (c)

Question 71

Chemistry · Co-ordination Compounds · Single correct

Given below are two statements : Statement I : $\mathrm{PF}_5$ and $\mathrm{BrF}_5$ both exhibit $sp^3d$ hybridisation. Statement II : Both $\mathrm{SF}_6$ and $[\mathrm{Co(NH}_3)_6]^{3+}$ exhibit $sp^3d^2$ hybridisation. In the light of the above statements, choose the correct answer from the options 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 true
4. Both Statement I and Statement II are false

Answer: (d)

Question 72

Chemistry · The p-Block Elements (Group-13 and 14) · Single correct

The number of ions from the following that are expected to behave as oxidising agent is : $\mathrm{Sn}^{4+}, \mathrm{Sn}^{2+}, \mathrm{Pb}^{2+}, \mathrm{Tl}^{3+}, \mathrm{Pb}^{4+}, \mathrm{Tl}^{+}$

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

Answer: (b)

Question 73

Chemistry · Amines · Single correct

Identify the product (A) in the following reaction.

Answer: (d)

Question 74

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

The correct statement among the following, for a "chromatography" purification method is :

1. Organic compounds run faster than solvent in the thin layer chromatographic plate.
2. $R_f$ is an integral value.
3. $R_f$ of a polar compound is smaller than that of a non-polar compound.
4. Non-polar compounds are retained at top and polar compounds come down in column chromatography.

Answer: (c)

Question 75

Chemistry · The p-Block Elements (Group-13 and 14) · Single correct

Evaluate the following statements related to group 14 elements for their correctness. (A) Covalent radius decreases down the group from C to Pb in a regular manner. (B) Electronegativity decreases from C to Pb down the group gradually. (C) Maximum covalence of C is 4 whereas other elements can expand their covalence due to presence of d orbitals. (D) Heavier elements do not form $p\pi - p\pi$ bonds. (E) Carbon can exhibit negative oxidation states. Choose the correct answer from the options given below :

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

Answer: (b)

Question 76

Chemistry · Redox Reactions · Single correct

Match List - I with List - II. \begin{tabular}{|l|l|} \hline \textbf{List - I} & \textbf{List - II} \\ Reaction & Type of redox reaction \\ \hline (A) $\mathrm{N_2(g) + O_2(g)}$ & (I) Decomposition \\ $\mathrm{\rightarrow 2NO_{(g)}}$ & \\ \hline (B) $\mathrm{2Pb(NO_3)_2(s)}$ & (II) Displacement \\ $\mathrm{\rightarrow 2PbO_{(s)} + 4NO_2(g) + O_2(g)}$ & \\ \hline (C) $\mathrm{2Na_{(s)} + 2H_2O_{(l)}}$ & (III) Disproportionation \\ $\mathrm{\rightarrow 2NaOH_{(aq.)} + H_2(g)}$ & \\ \hline (D) $\mathrm{2NO_2(g) + 2OH^-_{(aq.)}}$ & (IV) Combination \\ $\mathrm{\rightarrow NO_2^-{}_{(aq.)} + NO_3^-{}_{(aq.)} + H_2O_{(l)}}$ & \\ \hline \end{tabular} Choose the correct answer from the options given below:

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

Answer: (c)

Question 77

Chemistry · Alcohols, Phenols and Ethers · Single correct

Consider the given reaction, identify the major product $P$. $$CH_3 - COOH \xrightarrow{(i) LiAlH_4 \ (ii) PCC \ (iii) HCN/\overline{OH} \ (iv) H_2O/\overline{OH}, \Delta} ''P''$$

2. CH_3 - CH_2 - CH_2 - OH

Answer: (c)

Question 78

Chemistry · Co-ordination Compounds · Single correct

The correct IUPAC name of $[PtBr_2(PMe_3)_2]$ is:

1. dibromodi(trimethylphosphine)platinum(II)
2. bis(trimethylphosphine)dibromoplatinum(II)
3. dibromobis(trimethylphosphine)platinum(II)
4. bis[bromo(trimethylphosphine)]platinum(II)

Answer: (c)

Question 79

Chemistry · Redox Reactions · Single correct

Match List - I with List - II. \begin{tabular}{|c|l|c|l|} \hline \multicolumn{2}{|c|}{List - I (Reaction)} & \multicolumn{2}{c|}{List - II (Type of redox reaction)} \\ \hline (A) & $\mathrm{TiCl_4}$ & (I) & $e^2,\ t_2^0$ \\ \hline (B) & $[\mathrm{FeO_4}]^{2-}$ & (II) & $e^4,\ t_2^3$ \\ \hline (C) & $[\mathrm{FeCl_4}]^{-}$ & (III) & $e^0,\ t_2^0$ \\ \hline (D) & $[\mathrm{CoCl_4}]^{2-}$ & (IV) & $e^2,\ t_2^3$ \\ \hline \end{tabular} Choose the correct answer from the options given below:

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

Answer: (c)

Question 80

Chemistry · Equilibrium · Single correct

The ratio $\frac{K_P}{K_C}$ for the reaction : CO(g) + ½ O₂(g) ⇌ CO₂(g) is :

1. $\frac{1}{\sqrt{RT}}$
2. $(RT)^{1/2}$
3. $RT$
4. $1$

Answer: (a)

Question 81

Chemistry · Amines · Numerical

An amine (X) is prepared by ammonolysis of benzyl chloride. On adding p-toluenesulphonyl chloride to it the solution remains clear. Molar mass of the amine (X) formed is ______ gmol$^{-1}$. (Given molar mass in gmol$^{-1}$ C : 12, H : 1, O : 16, N : 14)

Answer: 287

Question 82

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

Consider the following reactions \[ \mathrm{NiS + HNO_3 + HCl \rightarrow A + NO + S + H_2O} \] The number of protons that do not involve in hydrogen bonding in the product $B$ is \(\underline{\hspace{1cm}}\).

Question 83

Chemistry · Solutions · Numerical

When $x \times 10^{-2}$ $\mathrm{mL}$ $\mathrm{methanol}$ ($\mathrm{molar \ mass}$ = 32 $\mathrm{g}$; $\mathrm{density}$ = 0.792 $\mathrm{g/cm^3}$) $\mathrm{is \ added \ to}$ 100 $\mathrm{mL \ water}$ ($\mathrm{density}$ = 1 $\mathrm{g/cm^3}$), $\mathrm{the \ following \ diagram \ is \ obtained.}$ $x =$ ($\mathrm{nearest \ integer}$). [$\mathrm{Given : \ Molal \ freezing \ point \ depression \ constant \ of \ water \ at \ 273.15 \ K \ is \ 1.86 \ K \ kg \ mol}^{-1}$]

Answer: 543

Question 84

Chemistry · The p-Block Elements (Group 15, 16, 17 & 18) · Fill in the blank

The ratio of number of oxygen atoms to bromine atoms in the product Q is ___ $\times$ 10^{-1}.

Answer: 15

Question 85

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

Number of carbocations from the following that are not stabilized by hyperconjugation is _______

Answer: 5

Question 86

Chemistry · Thermodynamics · Fill in the blank

For the reaction at 298 K, \[ 2\,\mathrm{A}+\mathrm{B}\rightarrow \mathrm{C} \] \[ \Delta H = 400\,\mathrm{kJ\,mol^{-1}} \] and \[ \Delta S = 0.2\,\mathrm{kJ\,mol^{-1}\,K^{-1}}. \] The reaction will become spontaneous above \[ \underline{\hspace{2cm}}\ \mathrm{K}. \]

Answer: 2000

Question 87

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

Total number of species from the following with central atom utilising $sp^2$ hybrid orbitals for bonding is _____. NH$_3$, SO$_2$, SiO$_2$, BeCl$_2$, C$_2$H$_2$, C$_2$H$_4$, BCl$_3$, HCHO, C$_6$H$_6$, BF$_3$, C$_2$H$_4$Cl$_2$

Answer: 6

Question 88

Chemistry · Chemical Kinetics and Nuclear Chemistry · Numerical

Consider the two different first order reactions given below $$\mathrm{A + B \rightarrow C \ (Reaction \ 1)}$$ $$\mathrm{P \rightarrow Q \ (Reaction \ 2)}$$ The ratio of the half life of Reaction 1 : Reaction 2 is 5 : 2. If $t_1$ and $t_2$ represent the time taken to complete $$\frac{2}{3}^{\mathrm{rd}}$$ and $$\frac{4}{5}^{\mathrm{th}}$$ of Reaction 1 and Reaction 2, respectively, then the value of the ratio $t_1 : t_2$ is ______ $\times 10^{-1}$ (nearest integer). [Given : $\log_{10}(3) = 0.477$ and $\log_{10}(5) = 0.699$]

Answer: 17

Question 89

Chemistry · Structure of Atom · Numerical

For hydrogen atom, energy of an electron in first excited state is $-3.4 \, \mathrm{eV}$, K. E. of the same electron of hydrogen atom is $x \, \mathrm{eV}$. Value of $x$ is ____ $\times$ $10^{-1}$ \, $\mathrm{eV}$. (Nearest integer)

Answer: 34

Question 90

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

Among $\mathrm{VO}_2^+$, $\mathrm{MnO}_4^-$ and $\mathrm{Cr}_2\mathrm{O}_7^{2-}$, the spin-only magnetic moment value of the species with least oxidising ability is ______ BM (Nearest integer). (Given atomic member $\mathrm{V} = 23$, $\mathrm{Mn} = 25$, $\mathrm{Cr} = 24$)

Answer: 0