JEE Main 25 July 2021 Shift 1 question paper with solutions

Question 1

Maths · Heights and Distances · Single correct

A spherical gas balloon of radius 16 meter subtends an angle $60^\circ$ at the eye of the observer $A$ while the angle of elevation of its center from the eye of $A$ is $75^\circ$. Then the height (in meter) of the top most point of the balloon from the level of the observer's eye is:

1. 8(2 + 2$\sqrt{3}$ + $\sqrt{2}$)
2. 8($\sqrt{6}$ + $\sqrt{2}$ + 2)
3. 8($\sqrt{2}$ + 2 + $\sqrt{3}$)
4. 8($\sqrt{6}$ - $\sqrt{2}$ + 2)

Answer: (b)

Question 2

Maths · Applications of Derivatives · Single correct

Let \[ f(x)=3\sin^4x+10\sin^3x+6\sin^2x-3,\qquad x\in\left[-\frac{\pi}{6},\frac{\pi}{2}\right]. \] Then, $f$ is:

1. increasing in $\left(-\frac{\pi}{6}, \frac{\pi}{2}\right)$
2. decreasing in $\left(0, \frac{\pi}{2}\right)$
3. increasing in $\left(-\frac{\pi}{6}, 0\right)$
4. decreasing in $\left(-\frac{\pi}{6}, 0\right)$

Answer: (d)

Question 3

Maths · Sequences and Series · Single correct

Let $S_n$ be the sum of the first $n$ terms of an arithmetic progression. If $S_{3n} = 3 \ S_{2n}$, then the value of $\frac{S_{4n}}{S_{2n}}$ is :

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

Answer: (a)

Question 4

Maths · Conic Sections · Single correct

The locus of the centroid of the triangle formed by any point P on the hyperbola $16x^2 - 9y^2 + 32x + 36y - 164 = 0$, and its foci is:

1. $16x^2 - 9y^2 + 32x + 36y - 36 = 0$
2. $9x^2 - 16y^2 + 36x + 32y - 144 = 0$
3. $16x^2 - 9y^2 + 32x + 36y - 144 = 0$
4. $9x^2 - 16y^2 + 36x + 32y - 36 = 0$

Answer: (a)

Question 5

Maths · Vector Algebra · Single correct

Let the vectors $$(2 + a + b)\hat{i} + (a + 2b + c)\hat{j} - (b + c)\hat{k}, (1 + b)\hat{i} + 2\hat{b} - b\hat{k}$$ and $$(2 + b)\hat{i} + 2\hat{b}\hat{j} + (1 - b)\hat{k} a, b, c, \in \mathbb{R}$$ be co-planar. Then which of the following is true?

1. $2b = a + c$
2. $3c = a + b$
3. $a = b + 2c$
4. $2a = b + c$

Answer: (a)

Question 6

Maths · Continuity and Differentiability · Single correct

Let $f : \mathbb{R} \to \mathbb{R}$ be defined as $$f(x) = \begin{cases} \frac{\lambda |x^2 - 5x + 6|}{\mu (5x - x^2 - 6)}, & x 2 \\ \mu, & x = 2 \end{cases}$$ where $[x]$ is the greatest integer less than or equal to $x$. If $f$ is continuous at $x = 2$, then $\lambda + \mu$ is equal to

1. $e(-e + 1)$
2. $e(e - 2)$
3. $2e - 1$

Answer: (a)

Question 7

Maths · Integrals · Single correct

The value of the definite integral $$\int_{x/24}^{5x/24} \frac{dx}{1 + \sqrt[3]{\tan 2x}}$$ is

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

Answer: (c)

Question 8

Maths · Binomial Theorem · Single correct

If b is very small as compared to the value of a, so that the cube and other higher powers of $\frac{b}{a}$ can be neglected in the identity $\frac{1}{a-b} + \frac{1}{a-2b} + \frac{1}{a-3b} + \ldots + \frac{1}{a-nb} = \alpha n + \beta n^2 + \gamma n^3$, then the value of $\gamma$ is:

1. $\frac{a^2+b}{3a^3}$
2. $\frac{a+b}{3a^2}$
3. $\frac{b^2}{3a^3}$
4. $\frac{a+b^2}{3a^3}$

Answer: (c)

Question 9

Maths · Differential Equations · Single correct

Let $y = y(x)$ be the solution of the differential equation $\frac{dy}{dx} = 1 + xe^{y-x}$, $-\sqrt{2} < x < \sqrt{2}, y(0) = 0$ then, the minimum value of $y(x), x \in (-\sqrt{2}, \sqrt{2})$ is equal to:

1. $(2 - \sqrt{3}) - \log_e 2$
2. $(2 + \sqrt{3}) + \log_e 2$
3. $(1 + \sqrt{3}) - \log_e(\sqrt{3} - 1)$
4. $(1 - \sqrt{3}) - \log_e(\sqrt{3} - 1)$

Answer: (d)

Question 10

Maths · Mathematical Reasoning · Single correct

The Boolean expression $(p \Rightarrow q) \land (q \Rightarrow \sim p)$ is equivalent to :

1. $\sim$ q
2. q
3. p
4. p

Answer: (d)

Question 11

Maths · Applications of Integrals · Single correct

The area (in sq. units) of the region, given by the set $\{(x, y) \in \mathbb{R} \times \mathbb{R} \mid x \geq 0, 2x^2 \leq y \leq 4 - 2x\}$ is :

1. $\frac{8}{3}$
2. $\frac{17}{3}$
3. $\frac{13}{3}$
4. $\frac{7}{3}$

Answer: (d)

Question 12

Maths · Trigonometric Functions · Single correct

The sum of all values of $x$ in $[0, 2\pi]$, for which $$\sin x + \sin 2x + \sin 3x + \sin 4x = 0,$$ is equal to:

1. $8\pi$
2. $11\pi$
3. $12\pi$
4. $9\pi$

Answer: (d)

Question 13

Maths · Relations and Functions (Advanced) · Single correct

Let $g : \mathbb{N} \to \mathbb{N}$ be defined as $$g(3n + 1) = 3n + 2$$ $$g(3n + 2) = 3n + 3$$ $g(3n + 3) = 3n + 1$, for all n $\geq 0$ Then which of the following statements is true ?

1. There exists an onto function $f : \mathbb{N} \to \mathbb{N}$ such that fog = f
2. There exists a one-one function $f : \mathbb{N} \to \mathbb{N}$ such that fog = f
3. gogog = g
4. There exists a function $f : \mathbb{N} \to \mathbb{N}$ such that gof = f

Answer: (a)

Question 14

Maths · Integrals · Single correct

Let $f : [0, \infty) \to [0, \infty)$ be defined as $$f(x) = \int_0^x [y] \, dy$$ where $[x]$ is the greatest integer less than or equal to $x$. Which of the following is true?

1. $f$ is continuous at every point in $[0, \infty)$ and differentiable except at the integer points.
2. $f$ is both continuous and differentiable except at the integer points in $[0, \infty)$.
3. $f$ is continuous everywhere except at the integer points in $[0, \infty)$.
4. $f$ is differentiable at every point in $[0, \infty)$. Official

Answer: (a)

Question 15

Maths · Determinants · Single correct

The values of $a$ and $b$, for which the system of equations $2x + 3y + 6z = 8$ $x + 2y + az = 5$ $3x + 5y + 9z = b$ has no solution, are:

1. $a = 3, b \neq 13$
2. $a \neq 3, b \neq 13$
3. $a \neq 3, b = 3$
4. $a = 3, b = 13$

Answer: (a)

Question 16

Maths · Probability · Single correct

Let 9 distinct balls be distributed among 4 boxes, $B_1$, $B_2$, $B_3$ and $B_4$. If the probability than $B_3$ contains exactly 3 balls is $k \left( \frac{3}{4} \right)^9$ then $k$ lies in the set:

1. $\{ x \in \mathbb{R} : |x - 3| < 1 \}$
2. $\{ x \in \mathbb{R} : |x - 2| \leq 1 \}$
3. $\{ x \in \mathbb{R} : |x - 1| < 1 \}$
4. $\{ x \in \mathbb{R} : |x - 5| \leq 1 \}$

Answer: (a)

Question 17

Maths · Conic Sections · Single correct

Let a parabola $P$ be such that its vertex and focus lie on the positive $x$-axis at a distance 2 and 4 units from the origin, respectively. If tangents are drawn from $O(0, 0)$ to the parabola $P$ which meet $P$ at $S$ and $R$, then the area (in sq. units) of $\triangle SOR$ is equal to

1. $16\sqrt{2}$
2. 16
3. 32
4. $8\sqrt{2}$

Answer: (b)

Question 18

Maths · Applications of Derivatives · Single correct

The number of real roots of the equation $e^{6x} - e^{4x} - 2e^{3x} - 12e^{2x} + e^x + 1 = 0$ is:

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

Answer: (a)

Question 19

Maths · Conic Sections · Single correct

Let an ellipse $E : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a^2 > b^2$, passes through $\left( \frac{\sqrt{3}}{2}, 1 \right)$ and has eccentricity $\frac{1}{\sqrt{3}}$. If a circle, centered at focus $F(\alpha, 0), \alpha > 0$, of $E$ and radius $\frac{2}{\sqrt{3}}$, intersects $E$ at two points $P$ and $Q$, then $PQ^2$ is equal to:

1. $\frac{8}{3}$
2. $\frac{4}{3}$
3. $\frac{16}{3}$
4. 3

Answer: (c)

Question 20

Maths · Three Dimensional Geometry · Single correct

Let the foot of perpendicular from a point P(1, 2, -1) to the straight line L : $\frac{x}{1} = \frac{y}{0} = \frac{z}{-1}$ be N. Let a line be drawn from P parallel to the plane $x + y + 2z = 0$ which meets L at point Q. If $\alpha$ is the acute angle between the lines PN and PQ, then $\cos \alpha$ is equal to

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

Answer: (c)

Question 21

Maths · Differential Equations · Numerical

Let $y = y(x)$ be solution of the following differential equation $$e^y \frac{dy}{dx} - 2e^y \sin x + \sin x \cos^2 x = 0, \ y\left(\frac{\pi}{2}\right) = 0$$ If $y(0) = \log_c \left(\alpha + \beta e^{-2}\right)$, then $4(\alpha + \beta)$ is equal to

Answer: 4

Question 22

Maths · Sequences and Series · Numerical

If the value of $$\left(1 + \frac{2}{3} + \frac{6}{3^2} + \frac{10}{3^3} + \ldots upto \infty \right) \log_{0.25} \left(\frac{1}{3} + \frac{1}{3^2} + \frac{1}{3^3} + \ldots upto \infty \right)$$ is $l$, then $l^2$ is equal to

Answer: 3

Question 23

Maths · Statistics · Numerical

Consider the following frequency distribution \begin{tabular}{|l|l|l|l|l|l|} \hline class : & 10 - 20 & 20 - 30 & 30 - 40 & 40 - 50 & 50 - 60 \\ \hline Frequency : & $\alpha$ & 110 & 54 & 30 & $\beta$ \\ \hline \end{tabular} If the sum of all frequencies is 584 and median is 45, then $|\alpha - \beta|$ is equal to

Answer: 164

Question 24

Maths · Vector Algebra · Fill in the blank

Let $\vec{p}=2\hat{i}+3\hat{j}+\hat{k}$ and $\vec{q}=\hat{i}+2\hat{j}+\hat{k}$ be two vectors. If a vector $\vec{r}=\alpha\hat{i}+\beta\hat{j}+\gamma\hat{k}$ is perpendicular to each of the vectors $(\vec{p}+\vec{q})$ and $(\vec{p}-\vec{q})$, and $|\vec{r}|=\sqrt{3}$, then $|\alpha|+|\beta|+|\gamma|$ is equal to ________.

Answer: 3

Question 25

Maths · Binomial Theorem · Numerical

The ratio of the coefficient of the middle term in the expansion of $(1 + x)^{20}$ and the sum of the coefficients of two middle terms in expansion of $(1 - x)^{19}$ is

Answer: 1

Question 26

Maths · Determinants · Numerical

Let $M = \left\{ A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} : a, b, c, d \in \{ \pm 3, \pm 2, \pm 1, 0 \} \right\}$ Define $f : M \rightarrow \mathbb{Z}$, as $f(A) = \det(A)$, for all $A \in M$ where $\mathbb{Z}$ is set of all integers. Then the number of $A \in M$ such that $f(A) = 15$ is equal to

Answer: 16

Question 27

Maths · Permutations and Combinations · Numerical

There are 5 students in class 10, 6 students in class 11 and 8 students in class 12. If the number of ways, in which 10 students can be selected from them so as to include at least 2 students from each class and at most 5 students from the total 11 students of class 10 and 11 is 100k, then k is equal to

Answer: 238

Question 28

Maths · Complex Numbers and Quadratic Equations · Numerical

If $\alpha, \beta$ are roots of the equation $x^2 - 5(\sqrt{2})x + 10 = 0$, $\alpha > \beta$ and $P_n = \alpha^n - \beta^n$ for each positive integer $n$, then the value of $$\left( \frac{P_{17}P_{20} + 5\sqrt{2}P_{17}P_{19}}{P_{18}P_{19} + 5\sqrt{2}P_{18}^2} \right)$$ is equal to

Answer: 1

Question 29

Maths · Binomial Theorem · Numerical

The term independent of $x$ in the expansion of $$\left( \frac{x+1}{x^{2/3} - x^{3} + 1} - \frac{x-1}{x-x^{1/2}} \right)^{10}$$, where $x \neq 0, 1$ is equal to

Answer: 210

Question 30

Maths · Matrices · Numerical

Let $$ S = \left\{ n \in \mathbb{N} \left( \begin{array}{cc} 0 & i \\ 1 & 0 \end{array} \right)^n \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) = \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) \forall a, b, c, d \in \mathbb{R} \right\} $$ where $i = \sqrt{-1}$. Then the number of 2-digit numbers in the set $S$ is

Answer: 11

Question 31

Physics · Kinetic Theory · Single correct

For a gas $C_P - C_V = R$ in a state P and $C_P - C_V = 1.10R$ in a state Q, $T_P$ and $T_Q$ are the temperatures in two different states P and Q respectively. Then

1. $T_P = T_Q$
2. $T_P < T_Q$
3. $T_P = 0.9 \, T_Q$
4. $T_P > T_Q$

Answer: (d)

Question 32

Physics · System of Particles and Rotational Motion · Single correct

Given below are two statements: one is labelled as Assertion $\textbf{A}$ and the other is labelled as Reason $\textbf{R}$. Assertion $\textbf{A}$ : Moment of inertia of a circular disc of mass 'M' and radius 'R' about X, Y axes (passing through its plane) and Z-axis which is perpendicular to its plane were found to be $I_x$, $I_y$ and $I_z$ respectively. The respective radii of gyration about all the three axes will be the same. Reason $\textbf{R}$ : A rigid body making rotational motion has fixed mass and shape. In the light of the above statements, choose the most appropriate answer from the options given below:

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

Answer: (b)

Question 33

Physics · Dual Nature of Radiation and Matter · Single correct

What should be the order of arrangement of de-Broglie wavelength of electron ($\lambda_e$), an $\alpha$-particle ($\lambda_\alpha$) and proton ($\lambda_p$) given that all have the same kinetic energy?

1. $\lambda_e = \lambda_p = \lambda_\alpha$
2. $\lambda_e < \lambda_p < \lambda_\alpha$
3. $\lambda_e > \lambda_p > \lambda_\alpha$
4. $\lambda_e = \lambda_p > \lambda_\alpha$

Answer: (c)

Question 34

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

Identify the logic operation carried out.

1. OR
2. AND
3. NOR
4. NAND

Answer: (b)

Question 35

Physics · Dual Nature of Radiation and Matter · Single correct

A particle of mass 4M at rest disintegrates into two particles of mass M and 3M respectively having non zero velocities. The ratio of de-Broglie wavelength of particle of mass M to that of mass 3M will be :

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

Answer: (d)

Question 36

Physics · Nuclei · Single correct

Some nuclei of a radioactive material are undergoing radioactive decay. The time gap between the instances when a quarter of the nuclei have decayed and when half of the nuclei have decayed is given as: (where $\lambda$ is the decay constant)

1. $\frac{1}{2} \frac{\ln 2}{\lambda}$
2. $\frac{\ln 2}{\lambda}$
3. $\frac{2 \ln 2}{\lambda}$
4. $\frac{\ln d}{2}$

Answer: (d)

Question 37

Physics · Mathematics in Physics · Single correct

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

1. $(a) \to (iv), (b) \to (i), (c) \to (iii), (d) \to (ii)$
2. $(a) \to (iv), (b) \to (iii), (c) \to (i), (d) \to (ii)$
3. $(a) \to (iii), (b) \to (ii), (c) \to (iv), (d) \to (i)$
4. $(a) \to (i), (b) \to (iv), (c) \to (ii), (d) \to (iii)$

Answer: (b)

Question 38

Physics · Electrostatic Potential and Capacitance · Single correct

A parallel plate capacitor with plate area 'A' and distance of separation 'd' is filled with a dielectric. What is the capacity of the capacitor when permittivity of the dielectric varies as: $$\varepsilon(x) = \varepsilon_0 + kx, for \left(0 < x \leq \frac{d}{2}\right)$$ $$\varepsilon(x) = \varepsilon_0 + k(d-x), for \left(\frac{d}{2} \leq x \leq d\right)$$

1. $\left( \varepsilon_0 + \frac{kd}{2} \right)^{2/kA}$
2. $\frac{kA}{2 \ln \left( \frac{2\varepsilon_0 + kd}{2\varepsilon_0} \right)}$
3. 0
4. $\frac{kA}{2} \ln \left( \frac{2\varepsilon_0}{2\varepsilon_0 - kd} \right)$

Answer: (b)

Question 39

Physics · Thermodynamics · Single correct

A monoatomic ideal gas, initially at temperature $T_1$ is enclosed in a cylinder fitted with a frictionless piston. The gas is allowed to expand adiabatically to a temperature $T_2$ by releasing the piston suddenly. If $l_1$ and $l_2$ are the lengths of the gas column, before and after the expansion respectively, then the value of $\frac{T_1}{T_2}$ will be

1. $\left( \frac{l_1}{l_2} \right)^{\frac{2}{3}}$
2. $\left( \frac{l_2}{l_1} \right)^{\frac{2}{3}}$
3. $\frac{l_2}{l_1}$
4. $\frac{l_1}{l_2}$

Answer: (b)

Question 40

Physics · Ray Optics and Optical Instruments · Single correct

A ray of laser of a wavelength 630 $\mathrm{nm}$ is incident at an angle of 30^$\circ$ at the diamond-air interface. It is going from diamond to air. The refractive index of diamond is 2.42 and that of air is 1. Choose the correct option.

1. angle of refraction is 24.41^$\circ$
2. angle of refraction is 30^$\circ$
3. refraction is not possible
4. angle of refraction is 53.4^$\circ$

Answer: (c)

Question 41

Physics · Mechanical Properties of Solids · Single correct

Two wires of same length and radius are joined end to end and loaded. The Young's moduli of the materials of the two wires are $Y_1$ and $Y_2$. The combination behaves as a single wire then its Young's modulus is :

1. $Y = \frac{2Y_1 Y_2}{3(Y_1 + Y_2)}$
2. $Y = \frac{2Y_1 Y_2}{Y_1 + Y_2}$
3. $Y = \frac{Y_1 Y_2}{2(Y_1 + Y_2)}$
4. $Y = \frac{Y_1 Y_2}{Y_1 + Y_2}$

Answer: (b)

Question 42

Physics · Nuclei · Single correct

The half-life of $^{198}\mathrm{Au}$ is 3 days. If atomic weight of $^{198}\mathrm{Au}$ is 198 g/mol then the activity of 2mg of $^{198}\mathrm{Au}$ is [in disintegration/second] :

1. $2.67 \times 10^{12}$
2. $6.06 \times 10^{18}$
3. $32.36 \times 10^{12}$
4. $16.18 \times 10^{12}$

Answer: (d)

Question 43

Physics · Laws of Motion · Single correct

Two billiard balls of equal mass 30 g strike a rigid wall with same speed of 108 $\mathrm{kmph}$ (as shown) but at different angles. If the balls get reflected with the same speed then the ratio of the magnitude of impulses imparted to ball 'a' and ball 'b' by the wall along 'X' direction is :

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

Answer: (b)

Question 44

Physics · Wave Optics · Single correct

In the Young's double slit experiment, the distance between the slits varies in time as $d(t) = d_0 + a_0 \sin \omega t$ ; where $d_0$, $\omega$ and $a_0$ are constants. The difference between the largest fringe width and the smallest fringe width obtained over time is given as :

1. $\frac{2 \lambda D (d_0)}{(d_0^2 - a_0^2)}$
2. $\frac{2 \lambda D a_0}{(d_0^2 - a_0^2)}$
3. $\frac{\lambda D}{d_0^2} a_0$
4. $\frac{\lambda D}{d_0 + a_0}$

Answer: (b)

Question 45

Physics · Thermal Properties of Matter · Single correct

Two different metal bodies A and B of equal mass are heated at a uniform rate under similar conditions. The variation of temperature of the bodies is graphically represented as shown in the figure. The ratio of specific heat capacities is :

1. $\frac{8}{3}$
2. $\frac{3}{8}$
3. $\frac{3}{4}$
4. $\frac{4}{3}$

Answer: (b)

Question 46

Physics · Electromagnetic Waves · Single correct

A linearly polarized electromagnetic wave in vacuum is $$\mathbf{E} = 3.1 \cos \left[(1.8)z - \left(5.4 \times 10^6\right)t \right] \hat{i} \mathrm{N/C}$$ is incident normally on a perfectly reflecting wall at $z = a$. Choose the correct option

1. The wavelength is 5.4 m
2. The frequency of electromagnetic wave is $54 \times 10^4$ Hz
3. The transmitted wave will be $$3.1 \cos \left[(1.8)z - \left(5.4 \times 10^6\right)t \right] \hat{i} \mathrm{N/C}$$
4. The reflected wave will be $1 \cos \left[(1.8)z + \left(5.4 \times 10^6\right)t \right] \hat{i} \mathrm{N/C}$

Answer: (d)

Question 47

Physics · Current Electricity · Single correct

In the given figure, there is a circuit of potentiometer of length AB = 10 $\mathrm{m}$. The resistance per unit length is 0.1 $\Omega$ per cm. Across AB, battery of emf E and internal resistance 'r' is connected. The maximum value of emf measured by this potentiometer is :

1. 5 $\mathrm{V}$
2. 2.25 $\mathrm{V}$
3. 6 $\mathrm{V}$
4. 2.75 $\mathrm{V}$

Answer: (a)

Question 48

Physics · Communication Systems · Single correct

In amplitude modulation, the message signal $V_m(t) = 10 \sin(2\pi \times 10^5 t)$ volts and Carrier signal $V_c(t) = 20 \sin(2\pi \times 10^7 t)$ volts The modulated signal now contains the message signal with lower side band and upper side band frequency, therefore the bandwidth of modulated signal is $\alpha \mathrm{kHz}$. The value of $\alpha$ is :

1. 200$\mathrm{kHz}$
2. 50$\mathrm{kHz}$
3. 100$\mathrm{kHz}$
4. 0

Answer: (a)

Question 49

Physics · Motion in a Straight Line · Single correct

Water droplets are falling from an open tap at a constant rate. The spacing between a droplet observed at the 4$^{\text{th}}$ second after its fall and the next droplet is $34.3\,\mathrm{m}$. At what rate are the droplets falling from the tap? (Take $g = 9.8\,\mathrm{m\,s^{-2}}$)

1. 3 drops /2 seconds
2. 2 drops / second
3. 1 drop / second
4. 1 drop /7 seconds

Answer: (c)

Question 50

Physics · Gravitation · Single correct

The minimum and maximum distances of a planet revolving around the Sun are $x_1$ and $x_2$. If the minimum speed of the planet on its trajectory is $v_0$ then its maximum speed will be:

1. $\frac{v_0 x_1^2}{x_2^2}$
2. $\frac{v_0 x_2^2}{x_1^2}$
3. $\frac{v_0 x_1}{x_2}$
4. $\frac{v_0 x_2}{x_1}$

Answer: (d)

Question 51

Physics · System of Particles and Rotational Motion · Numerical

A body of mass 2 kg moving with a speed of 4 m/s. makes an elastic collision with another body at rest and continues to move in the original direction but with one fourth of its initial speed. The speed of the two body centre of mass is $\frac{x}{10} \, \mathrm{m/s}$. Then the value of x is

Answer: 25

Question 52

Physics · Experimental Physics · Fill in the blank

Student A and Student B used two screw gauges of equal pitch and 100 equal circular divisions to measure the radius of a given wire. The actual value of the radius of the wire is 0.322 cm. The absolute value of the difference between the final circular scale readings observed by the students A and B is Given pitch = 0.1 cm.

Answer: 13

Question 53

Physics · Alternating Current · Numerical

An inductor of 10 mH is connected to a 20 V battery through a resistor of 10 kΩ and a switch. After a long time, when maximum current is set up in the circuit, the current is switched off. The current in the circuit after 1 μs is $\frac{x}{100}$ mA. Then x is equal to (Take $e^{-1} = 0.37$)

Answer: 74

Question 54

Physics · Electromagnetic Induction · Numerical

A circular conducting coil of radius 1 m is being heated by the change of magnetic field $\vec{B}$ passing perpendicular to the plane in which the coil is laid. The resistance of the coil is $2 \, \mu \Omega$. The magnetic field is slowly switched off such that its magnitude changes in time as $B = \frac{4}{\pi} \times 10^{-3} \, \mathrm{T} \left( 1 - \frac{t}{100} \right)$ The energy dissipated by the coil before the magnetic field is switched off completely is $E = mJ$.

Answer: 80

Question 55

Physics · Oscillations · Numerical

In the reported figure, two bodies A and B of masses 200 g and 800 g are attached with the system of springs. Springs are kept in a stretched position with some extension when the system is released. The horizontal surface is assumed to be frictionless. The angular frequency will be rad/s when $k = 20 \, \mathrm{N/m}$.

Answer: 10

Question 56

Physics · Magnetism and Matter · Numerical

The value of aluminium susceptibility is $2.2 \times 10^{-5}$. The percentage increase in the magnetic field if space within a current carrying toroid is filled with aluminium is $\frac{x}{10^4}$. Then the value of $x$ is

Answer: 22

Question 57

Physics · Electric Charges and Fields · Numerical

A particle of mass 1 mg and charge $q$ is lying at the mid-point of two stationary particles kept at a distance '2 m' when each is carrying same charge '$q$'. If the free charged particle is displaced from its equilibrium position through distance '$x$' $(x \ll 1 \, \mathrm{m})$. The particle executes SHM. Its angular frequency of oscillation will be $\times 10^5 \, \mathrm{rad/s}$ if $q^2 = 10 \, \mathrm{C}^2$

Answer: 6

Question 58

Physics · Current Electricity · Numerical

An electric bulb rated as 200 $\mathrm{W}$ at 100 $\mathrm{V}$ is used in a circuit having 200 $\mathrm{V}$ supply. The resistance 'R' that must be put in series with the bulb so that the bulb delivers the same power is $\Omega$.

Answer: 50

Question 59

Physics · Oscillations · Numerical

A pendulum bob has a speed of $3 \, \mathrm{m/s}$ at its lowest position. The pendulum is $50 \, \mathrm{cm}$ long. The speed of bob, when the length makes an angle of $60^\circ$ to the vertical will be $(g = 10 \, \mathrm{m/s^2})$ m/s

Answer: 2

Question 60

Physics · System of Particles and Rotational Motion · Numerical

A particle of mass ' m ' is moving in time ' t ' on a trajectory given by $$\vec{r} = 10 \alpha t^2 \hat{i} + 5 \beta (t - 5) \hat{j}$$ Where $\alpha$ and $\beta$ are dimensional constants. The angular momentum of the particle becomes the same as it was for $t = 0$ at time $t =$ seconds.

Answer: 10

Question 61

Chemistry · Polymers · Single correct

is a repeating unit for:

1. Novolac
2. Buna-N
3. Acrilan
4. Neoprene

Answer: (a)

Question 62

Chemistry · Co-ordination Compounds · Single correct

Which one of the following species responds to a external magnetic field?

1. $[\mathrm{Fe(H_2O)_6}]^{3+}$
2. $[\mathrm{Ni(CN)_4}]^{2-}$
3. $[\mathrm{Co(CN)_6}]^{3-}$
4. $\mathrm{Ni(CO)_4}$

Answer: (a)

Question 63

Chemistry · Aldehydes, Ketones and Carboxylic Acids · Single correct

Consider the above reaction, the major product 'P' is:

Answer: (c)

Question 64

Chemistry · Chemistry in Everyday Life · Single correct

Sodium stearate $CH_3(CH_2)_{16}COO^-Na^+$ is an anionic surfactant which forms micelles in oil. Choose the correct statement for it from the following:

1. It forms spherical micelles with $\mathrm{CH_3(CH_2)_{16}}$ group pointing towards the centre of sphere.
2. It forms non-spherical micelles with $\mathrm{-COO^-}$ group pointing outwards on the surface.
3. It forms spherical micelles with $\mathrm{CH_3(CH_2)_{16}}$ group pointing outwards on the surface of sphere.
4. It forms non-spherical micelles with $\mathrm{CH_3(CH_2)_{16}}$ group pointing towards the centre.

Answer: (a)

Question 65

Chemistry · Biomolecules · Single correct

The water soluble protein is

1. Fibrin
2. Albumin
3. Myosin
4. Collagen

Answer: (b)

Question 66

Chemistry · Hydrogen · Single correct

At $298.2 \, \mathrm{K}$ the relationship between enthalpy of bond dissociation (in $\mathrm{kJ \, mol^{-1}}$) for hydrogen $(E_H)$ and its isotope, deuterium $(E_D)$, is best described by:

1. $E_H = \frac{1}{2} E_D$
2. $E_H = E_D$
3. $E_H = E_D - 7.5$
4. $E_H = 2E_D$

Answer: (c)

Question 67

Chemistry · Aldehydes, Ketones and Carboxylic Acids · Single correct

Consider the given reaction, the product 'X' is:

Answer: (d)

Question 68

Chemistry · Alcohols, Phenols and Ethers · Multiple correct

The given reaction can occur in the presence of: (a) Bromine water (b) Br$_2$ in CS$_2$, 273 K (c) Br$_2$/FeBr$_3$ (d) Br$_2$ in CHCl$_3$, 273 K

1. (b) and (d) only
2. (a) and (c) only
3. (b), (c) and (d) only
4. (a), (b) and (d) only

Answer: (c)

Question 69

Chemistry · Amines · Single correct

Given below are two statements, one is labelled as Assertion (A) and other is labelled as Reason (R). Assertion (R) : Gabriel phthal imide synthesis cannot be used to prepare aromatic primary amines. Reason : Aryl halides do not undergo nucleophilic substitution reaction. In the light of the above statements, choose the correct answer from the options given below:

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

Answer: (c)

Question 70

Chemistry · Chemical Kinetics and Nuclear Chemistry · Single correct

For the following graphs, Choose from the options given below, the correct one regarding order of reaction is:

1. (b )zero order ($c$) and (e) First order (b) Zero order
2. (a) and (e) First order
3. (b) and (d) Zero order (e) First order
4. (a) and (b) Zero order $(c)$ and (e) First order

Answer: (a)

Question 71

Chemistry · Amines · Single correct

Which one of the products of the following reactions does not react with Hinsberg reagent to form sulphonamide?

Answer: (b)

Question 72

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

The ionic radii of $\mathrm{K}^+$, $\mathrm{Na}^+$, $\mathrm{Al}^{3+}$ and $\mathrm{Mg}^{2+}$ are in the order:

1. $\mathrm{Na}^+ < \mathrm{K}^+ < \mathrm{Mg}^{2+} < \mathrm{Al}^{3+}$
2. $\mathrm{Al}^{3+} < \mathrm{Mg}^{2+} < \mathrm{K}^+ < \mathrm{Na}^+$
3. $\mathrm{Al}^{3+} < \mathrm{Mg}^{2+} < \mathrm{Na}^+ < \mathrm{K}^+$
4. $\mathrm{K}^+ < \mathrm{Al}^{3+} < \mathrm{Mg}^{2+} < \mathrm{Na}$

Answer: (c)

Question 73

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

Which one of the following compounds of Group- 14 elements is not known?

1. $[\mathrm{GeCl}_6]^{2-}$
2. $[\mathrm{Sn(OH)}_6]^{2-}$
3. $[\mathrm{SiCl}_6]^{2-}$
4. $[\mathrm{SiF}_6]^{2-}$

Answer: (c)

Question 74

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

Which one among the following resonating structures is not correct?

Answer: (a)

Question 75

Chemistry · The s-Block Elements · Single correct

Given below are two statements Statement I : None of the alkaline earth metal hydroxides dissolve in alkali. Statement II : Solubility of alkaline earth metal hydroxides in water increases down the group. In the light of the above statements, choose the most appropriate answer from the options given below

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

Answer: (b)

Question 76

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

The correct order of following 3 d metal oxides, according to their oxidation numbers is: (a) $\mathrm{CrO}_3$ (b) $\mathrm{Fe}_2\mathrm{O}_3$ (c ) $\mathrm{MnO}_2$ (d) $\mathrm{V}_2\mathrm{O}_5$ (e) $\mathrm{Cu}_2\mathrm{O}$

1. (d) > (a) > (b) > (c ) > (e)
2. (a) > (c ) > (d) > (b) > (e)
3. (a) > (d) > (c ) > (b) > (e)
4. (c ) > (a) > (d) > (e) > (b)

Answer: (c)

Question 77

Chemistry · Environmental Chemistry · Single correct

Which one of the following chemical agent is not being used for dry-cleaning of clothes?

1. H_2O_2
2. CCl_4
3. Liquid CO_2
4. Cl_2C = CCl_2

Answer: (b)

Question 78

Chemistry · Amines · Single correct

Which one of the following compounds will liberate $CO_2$, when treated with $NaHCO_3$?

1. $(CH_3)_3NHCl
2. $(CH_3)_4N^+OH^-
3. $CH_3NH_2

Answer: (a)

Question 79

Chemistry · General Principles and Processes of Isolation of Elements · Single correct

In the leaching of alumina from bauxite, the ore expected to leach out in the process by reacting with NaOH is :

1. $\mathrm{TiO}_{2}$
2. $\mathrm{Fe}_{2}\mathrm{O}_{3}$
3. $\mathrm{ZnO}$
4. $\mathrm{SiO}_{2}$

Answer: (d)

Question 80

Chemistry · Hydrocarbons · Single correct

An organic compound ' $A'C_4H_8$ on treatment with $KMnO_4/H^+$ yields compound ' $B'C_3H_6O$. Compound 'A' also yields compound 'B' an ozonolysis. Compound 'A' is :

1. 2-Methylpropene
2. 1-Methylcyclopropane
3. But-2-ene
4. Cyclobutane

Answer: (a)

Question 81

Chemistry · Hydrocarbons · Numerical

The number of sigma bonds in is

Answer: 10

Question 82

Chemistry · Co-ordination Compounds · Numerical

Three moles of $\mathrm{AgCl}$ are precipitated when one mole of an octahedral coordination compound with empirical formula $\mathrm{CrCl_3\cdot3NH_3\cdot3H_2O}$ reacts with excess $\mathrm{AgNO_3}$. The number of chloride ions satisfying the secondary valency of the metal ion is $\underline{\qquad}$.

Answer: 0

Question 83

Chemistry · Structure of Atom · Numerical

A source of monochromatic radiation of wavelength 400 $\mathrm{nm}$ provides 1000 $\mathrm{J}$ of energy in 10 seconds. When this radiation falls on the surface of sodium, x $\times 10^{20}$ electrons are ejected per second. Assume that wavelength 400 $\mathrm{nm}$ is sufficient for ejection of electron from the surface of sodium metal. The value of x is (Nearest integer) $( h = 6.626$ $\times 10^{-34}\mathrm{Js}$)"

Answer: 2

Question 84

Chemistry · Solutions · Numerical

$CO_2$ gas is bubbled through water during a soft drink manufacturing process at 298 $\mathrm{K}$. If $CO_2$ exerts a partial pressure of 0.835 $\mathrm{bar}$ then x mmol of $CO_2$ would dissolve in 0.9 $\mathrm{L}$ of water. The value of x is . (Nearest integer) (Henry's law constant for $CO_2$ at 298 $\mathrm{K}$ is 1.67 $\times 10^3$ $\mathrm{bar}$)

Answer: 25

Question 85

Chemistry · Equilibrium · Numerical

For the reaction $$\mathrm{A + B \rightleftharpoons 2C}$$ the value of equilibrium constant is 100 at 298 $\mathrm{K}$. If the initial concentration of all the three species is 1 $\mathrm{M}$ each, then the equilibrium concentration of $\mathrm{C}$ is $x \times 10^{-1}$ $\mathrm{M}$. The value of $x$ is (Nearest integer)

Answer: 25

Question 86

Chemistry · Electrochemistry · Numerical

Consider the cell at $25^\circ$C $\text{Zn}\left|\text{Zn}^{2+}\text{(aq), (1M)}\middle|\middle|\text{Fe}^{3+}\text{(aq), Fe}^{2+}\text{(aq)}\right|\text{Pt(s)}$ The fraction of total iron present as $\text{Fe}^{3+}$ ion at the cell potential of 1.500 V is $x \times 10^{-2}$. The value of $x$ (Nearest integer) $\left(\text{Given: } E^0_{\text{Fe}^{2+}/\text{Fe}^{2+}} = 0.77\,\text{V},\ E^0_{\text{Za}^2/\text{Zan}} = -0.76\,\text{V}\right)$

Answer: 24

Question 87

Chemistry · Thermodynamics · Numerical

At 298 K, the enthalpy of fusion of a solid (X) is $2.8 \, \mathrm{kJ \, mol^{-1}}$ and the enthalpy of vaporisation of the liquid (X) is $98.2 \, \mathrm{kJ \, mol^{-1}}$. The enthalpy of sublimation of the substance (X) in $\mathrm{kJ \, mol^{-1}}$ is (in nearest integer)

Answer: 101

Question 88

Chemistry · States of Matter · Numerical

A home owner uses $4.00 \times 10^3\,\mathrm{m}^3$ of methane ($\mathrm{CH_4}$) gas (assume $\mathrm{CH_4}$ is an ideal gas) in a year to heat his home. Under the pressure of $1.0\,\mathrm{atm}$ and $300\,\mathrm{K}$, mass of gas used is $x \times 10^5\,\mathrm{g}$. The value of $x$ is _____. (Nearest integer) (Given: $R = 0.083\,\mathrm{L\,atm\,K}^{-1}\,\mathrm{mol}^{-1}$)

Answer: 26

Question 89

Chemistry · Redox Reactions · Numerical

When $10\,\mathrm{mL}$ of an aqueous solution of $\mathrm{Fe^{2+}}$ ions was titrated in the presence of dil $\mathrm{H_2SO_4}$ using diphenylamine indicator, $15\,\mathrm{mL}$ of $0.02\,\mathrm{M}$ solution of $\mathrm{K_2Cr_2O_7}$ was required to get the end point. The molarity of the solution containing $\mathrm{Fe^{2+}}$ ions is $x \times 10^{-2}\,\mathrm{M}$. The value of $x$ is (Nearest integer)

Answer: 18

Question 90

Chemistry · Some Basic Concepts of Chemistry · Numerical

Consider the complete combustion of butane, the amount of butane utilized to produce 72.0 g of water is x $10^{-1}$ g. (in nearest integer)

Answer: 464