JEE Main 2017 — 2 Apr Offline Set D (Previous Year Paper) with Answer Key

Question 1

Mathematics · 4 marks · −1

If S is the set of distinct values of 'b' for which the following system of linear equations $x + y + z = 1$ , $x + ay + z = 1$ , $ax + by + z = 0$ has no solution, then S is:

A.an empty set
B.an infinite set
C.a finite set containing two or more elements
D.a singleton

Correct answer: D

Explanation: $D=0$ forces $a=1$ ; then the equations are $x+y+z=1$ (twice) and $x+by+z=0$ , inconsistent only when $b=1$ . So S $=\{1\}$ .

Question 2

Mathematics · 4 marks · −1

The following statement $(p \to q) \to [(\sim p \to q) \to q]$ is:

A.a tautology
B.equivalent to $\sim p \to q$
C.equivalent to $p \to \sim q$
D.a fallacy

Correct answer: A

Explanation: $(\sim p\to q)\to q \equiv (p\vee q)\to q\equiv(\sim p\wedge\sim q)\vee q\equiv\sim p\vee q\equiv p\to q$ , so the statement is $(p\to q)\to(p\to q)$ , a tautology.

Question 3

Mathematics · 4 marks · −1

If $5(\tan^2 x - \cos^2 x) = 2\cos 2x + 9$ , then the value of $\cos 4x$ is:

A. $-\frac{3}{5}$
B. $\frac{1}{3}$
C. $\frac{2}{9}$
D. $-\frac{7}{9}$

Correct answer: D

Explanation: With $t=\cos^2x$ : $5(\frac1t-1-t)=2(2t-1)+9 \Rightarrow 9t^2+12t-5=0$ , $t=\frac13$ . $\cos2x=-\frac13$ , $\cos4x=2\cdot\frac19-1=-\frac79$ .

Question 4

Mathematics · 4 marks · −1

For three events A, B and C, P(Exactly one of A or B occurs) = P(Exactly one of B or C occurs) = P(Exactly one of C or A occurs) = $\frac{1}{4}$ and P(All the three events occur simultaneously) = $\frac{1}{16}$ . Then the probability that at least one of the events occurs, is:

A. $\frac{7}{32}$
B. $\frac{7}{16}$
C. $\frac{7}{64}$
D. $\frac{3}{16}$

Correct answer: B

Explanation: Adding $P(A)+P(B)-2P(AB)=\frac14$ etc.: $\sum P(A)-\sum P(AB)=\frac38$ . $P(A\cup B\cup C)=\frac38+\frac1{16}=\frac7{16}$ .

Question 5

Mathematics · 4 marks · −1

Let $\omega$ be a complex number such that $2\omega + 1 = z$ where $z = \sqrt{-3}$ . If $\begin{vmatrix} 1 & 1 & 1 \\ 1 & -\omega^2-1 & \omega^2 \\ 1 & \omega^2 & \omega^7 \end{vmatrix} = 3k$ , then k is equal to:

A. $-z$
B. $z$
C. $-1$
D.1

Correct answer: A

Explanation: $\omega$ is a cube root of unity, $-\omega^2-1=\omega$ , $\omega^7=\omega$ . The determinant is $3(\omega^2-\omega)=-3z$ , so $k=-z$ .

Question 6

Mathematics · 4 marks · −1

Let k be an integer such that the triangle with vertices $(k, -3k)$ , $(5, k)$ and $(-k, 2)$ has area 28 sq. units. Then the orthocentre of this triangle is at the point:

A. $\left(2, -\frac{1}{2}\right)$
B. $\left(1, \frac{3}{4}\right)$
C. $\left(1, -\frac{3}{4}\right)$
D. $\left(2, \frac{1}{2}\right)$

Correct answer: D

Explanation: Area condition gives $5k^2+13k-46=0$ , integer root $k=2$ . Vertices $(2,-6),(5,2),(-2,2)$ : altitude $x=2$ and altitude through $(5,2)$ with slope $\frac12$ meet at $(2,\frac12)$ .

Question 7

Mathematics · 4 marks · −1

Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq. m) of the flower-bed, is:

A.12.5
B.10
C.25
D.30

Correct answer: C

Explanation: $2r+r\theta=20$ , area $=\frac12r^2\theta=10r-r^2$ , maximum 25 at $r=5$ .

Question 8

Mathematics · 4 marks · −1

The area (in sq. units) of the region $\{(x, y) : x \ge 0,\ x + y \le 3,\ x^2 \le 4y \text{ and } y \le 1 + \sqrt{x}\}$ is:

A. $\frac{59}{12}$
B. $\frac{3}{2}$
C. $\frac{7}{3}$
D. $\frac{5}{2}$

Correct answer: D

Explanation: Area $=\int_0^1(1+\sqrt x)dx+\frac12\cdot3\cdot1$ (trapezium under $x+y=3$ from 1 to 2) $-\int_0^2\frac{x^2}{4}dx=\frac53+\frac32-\frac23=\frac52$ .

Question 9

Mathematics · 4 marks · −1

If the image of the point $P(1, -2, 3)$ in the plane, $2x + 3y - 4z + 22 = 0$ measured parallel to the line, $\frac{x}{1} = \frac{y}{4} = \frac{z}{5}$ is Q, then PQ is equal to:

A. $3\sqrt{5}$
B. $2\sqrt{42}$
C. $\sqrt{42}$
D. $6\sqrt{5}$

Correct answer: B

Explanation: Line through P: $(1+\lambda,-2+4\lambda,3+5\lambda)$ meets the plane at $\lambda=1$ , M $=(2,2,8)$ . $PM=\sqrt{42}$ , $PQ=2\sqrt{42}$ .

Question 10

Mathematics · 4 marks · −1

If for $x \in \left(0, \frac{1}{4}\right)$ , the derivative of $\tan^{-1}\left(\frac{6x\sqrt{x}}{1-9x^3}\right)$ is $\sqrt{x}\cdot g(x)$ , then $g(x)$ equals:

A. $\frac{9}{1+9x^3}$
B. $\frac{3x\sqrt{x}}{1-9x^3}$
C. $\frac{3x}{1-9x^3}$
D. $\frac{3}{1+9x^3}$

Correct answer: A

Explanation: The function is $2\tan^{-1}(3x^{3/2})$ ; derivative $\frac{9\sqrt x}{1+9x^3}$ .

Question 11

Mathematics · 4 marks · −1

If $(2 + \sin x)\frac{dy}{dx} + (y + 1)\cos x = 0$ and $y(0) = 1$ , then $y\left(\frac{\pi}{2}\right)$ is equal to:

A. $\frac{1}{3}$
B. $-\frac{2}{3}$
C. $-\frac{1}{3}$
D. $\frac{4}{3}$

Correct answer: A

Explanation: $\frac{d}{dx}[(y+1)(2+\sin x)]=0$ , so $(y+1)(2+\sin x)=4$ ; at $\pi/2$ : $y=\frac13$ .

Question 12

Mathematics · 4 marks · −1

Let a vertical tower AB have its end A on the level ground. Let C be the mid-point of AB and P be a point on the ground such that $AP = 2AB$ . If $\angle BPC = \beta$ , then $\tan\beta$ is equal to:

A. $\frac{6}{7}$
B. $\frac{1}{4}$
C. $\frac{2}{9}$
D. $\frac{4}{9}$

Correct answer: C

Explanation: With AB $=2x$ , AP $=4x$ : $\tan\angle APC=\frac14$ , $\tan\angle APB=\frac12$ . $\tan\beta=\frac{1/2-1/4}{1+1/8}=\frac29$ .

Question 13

Mathematics · 4 marks · −1

If $A = \begin{bmatrix} 2 & -3 \\ -4 & 1 \end{bmatrix}$ , then $\text{adj}(3A^2 + 12A)$ is equal to:

A. $\begin{bmatrix} 72 & -84 \\ -63 & 51 \end{bmatrix}$
B. $\begin{bmatrix} 51 & 63 \\ 84 & 72 \end{bmatrix}$
C. $\begin{bmatrix} 51 & 84 \\ 63 & 72 \end{bmatrix}$
D. $\begin{bmatrix} 72 & -63 \\ -84 & 51 \end{bmatrix}$

Correct answer: B

Explanation: $A^2=\begin{bmatrix}16&-9\\-12&13\end{bmatrix}$ , $3A^2+12A=\begin{bmatrix}72&-63\\-84&51\end{bmatrix}$ , adjoint $\begin{bmatrix}51&63\\84&72\end{bmatrix}$ .

Question 14

Mathematics · 4 marks · −1

For any three positive real numbers a, b and c, $9(25a^2 + b^2) + 25(c^2 - 3ac) = 15b(3a + c)$ . Then:

A.b, c and a are in G.P.
B.b, c and a are in A.P.
C.a, b and c are in A.P.
D.a, b and c are in G.P.

Correct answer: B

Explanation: The relation is $(15a-3b)^2+(15a-5c)^2+(3b-5c)^2=0$ , so $15a=3b=5c$ : $a=3\lambda,b=15\lambda,c=9\lambda$ , giving $2c=a+b$ .

Question 15

Mathematics · 4 marks · −1

The distance of the point $(1, 3, -7)$ from the plane passing through the point $(1, -1, -1)$ , having normal perpendicular to both the lines $\frac{x-1}{1} = \frac{y+2}{-2} = \frac{z-4}{3}$ and $\frac{x-2}{2} = \frac{y+1}{-1} = \frac{z+7}{-1}$ , is:

A. $\frac{20}{\sqrt{74}}$
B. $\frac{10}{\sqrt{83}}$
C. $\frac{5}{\sqrt{83}}$
D. $\frac{10}{\sqrt{74}}$

Correct answer: B

Explanation: Normal $=(1,-2,3)\times(2,-1,-1)=(5,7,3)$ ; plane $5x+7y+3z+5=0$ ; distance $=\frac{|5+21-21+5|}{\sqrt{83}}=\frac{10}{\sqrt{83}}$ .

Question 16

Mathematics · 4 marks · −1

Let $I_n = \int \tan^n x\,dx$ , $(n > 1)$ . If $I_4 + I_6 = a\tan^5 x + bx^5 + C$ , where C is a constant of integration, then the ordered pair $(a, b)$ is equal to:

A. $\left(-\frac{1}{5}, 1\right)$
B. $\left(\frac{1}{5}, 0\right)$
C. $\left(\frac{1}{5}, -1\right)$
D. $\left(-\frac{1}{5}, 0\right)$

Correct answer: B

Explanation: $I_n+I_{n-2}=\frac{\tan^{n-1}x}{n-1}$ , so $I_6+I_4=\frac{\tan^5x}{5}+C$ : $a=\frac15$ , $b=0$ .

Question 17

Mathematics · 4 marks · −1

The eccentricity of an ellipse whose centre is at the origin is $\frac{1}{2}$ . If one of its directrices is $x = -4$ , then the equation of the normal to it at $\left(1, \frac{3}{2}\right)$ is:

A. $2y - x = 2$
B. $4x - 2y = 1$
C. $4x + 2y = 7$
D. $x + 2y = 4$

Correct answer: B

Explanation: $a/e=4$ gives $a=2$ , $b^2=3$ : $\frac{x^2}{4}+\frac{y^2}{3}=1$ . Normal at $(1,\frac32)$ : $\frac{4x}{1}-\frac{3y}{3/2}=4-3$ , i.e. $4x-2y=1$ .

Question 18

Mathematics · 4 marks · −1

A hyperbola passes through the point $P(\sqrt{2}, \sqrt{3})$ and has foci at $(\pm2, 0)$ . Then the tangent to this hyperbola at P also passes through the point:

A. $(3\sqrt{2}, 2\sqrt{3})$
B. $(2\sqrt{2}, 3\sqrt{3})$
C. $(\sqrt{3}, \sqrt{2})$
D. $(-\sqrt{2}, -\sqrt{3})$

Correct answer: B

Explanation: $\frac{2}{a^2}-\frac{3}{4-a^2}=1$ gives $a^2=1$ (as $a^2<4$ ): $x^2-\frac{y^2}{3}=1$ . Tangent at P: $\sqrt2x-\frac{y}{\sqrt3}=1$ , passing through $(2\sqrt2,3\sqrt3)$ .

Question 19

Mathematics · 4 marks · −1

The function $f : \mathbb{R} \to \left[-\frac{1}{2}, \frac{1}{2}\right]$ defined as $f(x) = \frac{x}{1+x^2}$ , is:

A.invertible.
B.injective but not surjective.
C.surjective but not injective.
D.neither injective nor surjective.

Correct answer: C

Explanation: Range of $f$ is exactly $[-\frac12,\frac12]$ (extremes at $x=\pm1$ ), so onto; $f(x)=f(1/x)$ , so not one-one.

Question 20

Mathematics · 4 marks · −1

$\lim_{x\to\pi/2} \frac{\cot x - \cos x}{(\pi - 2x)^3}$ equals:

A. $\frac{1}{24}$
B. $\frac{1}{16}$
C. $\frac{1}{8}$
D. $\frac{1}{4}$

Correct answer: B

Explanation: Put $x=\frac\pi2+t$ : expression $=\frac{\tan t-\sin t}{8t^3}\to\frac18\cdot\frac12=\frac1{16}$ .

Question 21

Mathematics · 4 marks · −1

Let $\vec{a} = 2\hat{i} + \hat{j} - 2\hat{k}$ and $\vec{b} = \hat{i} + \hat{j}$ . Let $\vec{c}$ be a vector such that $|\vec{c} - \vec{a}| = 3$ , $|(\vec{a}\times\vec{b})\times\vec{c}| = 3$ and the angle between $\vec{c}$ and $\vec{a}\times\vec{b}$ be $30^\circ$ . Then $\vec{a}\cdot\vec{c}$ is equal to:

A. $\frac{25}{8}$
B.2
C.5
D. $\frac{1}{8}$

Correct answer: B

Question 22

Mathematics · 4 marks · −1

The normal to the curve $y(x - 2)(x - 3) = x + 6$ at the point where the curve intersects the y-axis passes through the point:

A. $\left(-\frac{1}{2}, -\frac{1}{2}\right)$
B. $\left(\frac{1}{2}, \frac{1}{2}\right)$
C. $\left(\frac{1}{2}, -\frac{1}{3}\right)$
D. $\left(\frac{1}{2}, \frac{1}{3}\right)$

Correct answer: B

Explanation: At $x=0$ , $y=1$ ; differentiating, $6y'-5=1$ so $y'=1$ . Normal: $x+y=1$ , which contains $(\frac12,\frac12)$ .

Question 23

Mathematics · 4 marks · −1

If two different numbers are taken from the set $\{0, 1, 2, 3, \dots, 10\}$ ; then the probability that their sum as well as absolute difference are both multiple of 4, is:

A. $\frac{6}{55}$
B. $\frac{12}{55}$
C. $\frac{14}{45}$
D. $\frac{7}{55}$

Correct answer: A

Explanation: Both numbers must be $\equiv0$ or both $\equiv2\pmod4$ : pairs from $\{0,4,8\}$ or $\{2,6,10\}$ : $\frac{3+3}{\binom{11}{2}}=\frac6{55}$ .

Question 24

Mathematics · 4 marks · −1

A man X has 7 friends, 4 of them are ladies and 3 are men. His wife Y also has 7 friends, 3 of them are ladies and 4 are men. Assume X and Y have no common friends. Then the total number of ways in which X and Y together can throw a party inviting 3 ladies and 3 men, so that 3 friends of each of X and Y are in this party, is:

A.485
B.468
C.469
D.484

Correct answer: A

Explanation: Cases (X's ladies, X's men): (3,0),(2,1),(1,2),(0,3): $\binom43\binom43+\binom42\binom31\binom31\binom42+\binom41\binom32\binom32\binom41+\binom33\binom33=16+324+144+1=485$ .

Question 25

Mathematics · 4 marks · −1

The value of $\left({}^{21}C_1 - {}^{10}C_1\right) + \left({}^{21}C_2 - {}^{10}C_2\right) + \left({}^{21}C_3 - {}^{10}C_3\right) + \dots + \left({}^{21}C_{10} - {}^{10}C_{10}\right)$ is:

A. $2^{21} - 2^{11}$
B. $2^{21} - 2^{10}$
C. $2^{20} - 2^9$
D. $2^{20} - 2^{10}$

Correct answer: D

Explanation: $\sum_{r=0}^{10}{}^{21}C_r=2^{20}$ and $\sum_{r=0}^{10}{}^{10}C_r=2^{10}$ ; the $r=0$ terms cancel, giving $2^{20}-2^{10}$ .

Question 26

Mathematics · 4 marks · −1

A box contains 15 green and 10 yellow balls. If 10 balls are randomly drawn, one-by-one, with replacement, then the variance of the number of green balls drawn is:

A. $\frac{12}{5}$
B.6
C.4
D. $\frac{6}{25}$

Correct answer: A

Explanation: Binomial with $n=10$ , $p=\frac35$ : variance $npq=\frac{12}{5}$ .

Question 27

Mathematics · 4 marks · −1

Let $a, b, c \in \mathbb{R}$ . If $f(x) = ax^2 + bx + c$ is such that $a + b + c = 3$ and $f(x + y) = f(x) + f(y) + xy$ , $\forall x, y \in \mathbb{R}$ , then $\sum_{n=1}^{10} f(n)$ is equal to:

A.330
B.165
C.190
D.255

Correct answer: A

Explanation: Comparing gives $2a=1$ , $c=0$ , so $f(x)=\frac{x^2}{2}+\frac52x$ . $\sum_{1}^{10}f(n)=\frac{385}{2}+\frac{5\cdot55}{2}=330$ .

Question 28

Mathematics · 4 marks · −1

If, for a positive integer n, the quadratic equation, $x(x + 1) + (x + 1)(x + 2) + \dots + (x + \overline{n-1})(x + n) = 10n$ has two consecutive integral solutions, then n is equal to:

A.12
B.9
C.10
D.11

Correct answer: D

Explanation: The equation is $x^2+nx+\frac{n^2-31}{3}=0$ . Roots $I, I+1$ : discriminant $=1$ , i.e. $n^2-\frac{4(n^2-31)}{3}=1$ , $n^2=121$ , $n=11$ .

Question 29

Mathematics · 4 marks · −1

The integral $\int_{\pi/4}^{3\pi/4} \frac{dx}{1 + \cos x}$ is equal to:

A. $-2$
B.2
C.4
D. $-1$

Correct answer: B

Explanation: $\frac{1}{1+\cos x}=\csc^2x-\csc x\cot x$ ; antiderivative $-\cot x+\csc x$ , giving $(1+\sqrt2)-(\sqrt2-1)=2$ .

Question 30

Physics · 4 marks · −1

A radioactive nucleus A with a half life T, decays into a nucleus B. At $t = 0$ , there is no nucleus B. At sometime t, the ratio of the number of B to that of A is 0.3. Then, t is given by :

A. $t = \frac{T}{\log(1.3)}$
B. $t = \frac{T}{2}\frac{\log 2}{\log 1.3}$
C. $t = T\frac{\log 1.3}{\log 2}$
D. $t = T\log(1.3)$

Correct answer: C

Explanation: $\frac{N_0-N}{N}=0.3\Rightarrow e^{\lambda t}=1.3$ , $t=\frac{\ln1.3}{\ln2}T$ .

Question 31

Physics · 4 marks · −1

The following observations were taken for determining surface tension T of water by capillary method: diameter of capillary, $D = 1.25\times10^{-2}$ m; rise of water, $h = 1.45\times10^{-2}$ m. Using $g = 9.80$ m/s $^2$ and the simplified relation $T = \frac{rhg}{2}\times10^3$ N/m, the possible error in surface tension is closest to :

A.10 %
B.0.15 %
C.1.5 %
D.2.4 %

Correct answer: C

Explanation: $\frac{\Delta T}{T}=\frac{\Delta D}{D}+\frac{\Delta h}{h}=\frac{0.01}{1.25}+\frac{0.01}{1.45}\approx1.5\%$ (least counts 0.01 in the last digit).

Question 32

Physics · 4 marks · −1

An electron beam is accelerated by a potential difference V to hit a metallic target to produce X-rays. It produces continuous as well as characteristic X-rays. If $\lambda_{min}$ is the smallest possible wavelength of X-ray in the spectrum, the variation of $\log\lambda_{min}$ with $\log V$ is correctly represented in:

A.Graph (1): horizontal line
B.Graph (2): straight line with negative slope
C.Graph (3): straight line through origin with positive slope
D.Graph (4): straight line with positive slope and positive intercept

Correct answer: B

Explanation: $\lambda_{min}=\frac{hc}{eV}$ , so $\log\lambda_{min}=\text{const}-\log V$ : a straight line of slope $-1$ .

Question 33

Physics · 4 marks · −1

The moment of inertia of a uniform cylinder of length $\ell$ and radius R about its perpendicular bisector is I. What is the ratio $\ell/R$ such that the moment of inertia is minimum ?

A. $\frac{3}{\sqrt{2}}$
B. $\sqrt{\frac{3}{2}}$
C. $\frac{\sqrt{3}}{2}$
D.1

Correct answer: B

Explanation: $I=\frac m{12}(3R^2+\ell^2)$ with $R^2=\frac{m}{\pi\rho\ell}$ (fixed mass). $\frac{dI}{d\ell}=0\Rightarrow2\ell=\frac{3R^2}{\ell}$ , $\frac\ell R=\sqrt{\frac32}$ .

Question 34

Physics · 4 marks · −1

A slender uniform rod of mass M and length $\ell$ is pivoted at one end so that it can rotate in a vertical plane (see figure). There is negligible friction at the pivot. The free end is held vertically above the pivot and then released. The angular acceleration of the rod when it makes an angle $\theta$ with the vertical is:

A. $\frac{2g}{3\ell}\cos\theta$
B. $\frac{3g}{2\ell}\sin\theta$
C. $\frac{2g}{3\ell}\sin\theta$
D. $\frac{3g}{2\ell}\cos\theta$

Correct answer: B

Explanation: $Mg\frac\ell2\sin\theta=\frac{M\ell^2}{3}\alpha\Rightarrow\alpha=\frac{3g}{2\ell}\sin\theta$ .

Question 35

Physics · 4 marks · −1

$C_p$ and $C_v$ are specific heats at constant pressure and constant volume respectively. It is observed that $C_p - C_v = a$ for hydrogen gas and $C_p - C_v = b$ for nitrogen gas. The correct relation between a and b is

A. $a = 28b$
B. $a = \frac{1}{14}b$
C. $a = b$
D. $a = 14b$

Correct answer: D

Explanation: For specific heats per unit mass, $c_p-c_v=R/M$ : $a=R/2$ , $b=R/28$ , so $a=14b$ .

Question 36

Physics · 4 marks · −1

A copper ball of mass 100 gm is at a temperature T. It is dropped in a copper calorimeter of mass 100 gm, filled with 170 gm of water at room temperature. Subsequently, the temperature of the system is found to be $75^\circ$ C. T is given by: (Given : room temperature = $30^\circ$ C, specific heat of copper = 0.1 cal/gm $^\circ$ C)

A. $825^\circ$ C
B. $800^\circ$ C
C. $885^\circ$ C
D. $1250^\circ$ C

Correct answer: C

Explanation: $100(0.1)(T-75)=100(0.1)(45)+170(1)(45)\Rightarrow T-75=810$ , $T=885^\circ$ C.

Question 37

Physics · 4 marks · −1

In amplitude modulation, sinusoidal carrier frequency used is denoted by $\omega_c$ and the signal frequency is denoted by $\omega_m$ . The bandwidth $(\Delta\omega_m)$ of the signal is such that $\Delta\omega_m \ll \omega_c$ . Which of the following frequencies is not contained in the modulated wave?

A. $\omega_c - \omega_m$
B. $\omega_m$
C. $\omega_c$
D. $\omega_m + \omega_c$

Correct answer: B

Explanation: The AM wave contains $\omega_c$ and the side bands $\omega_c\pm\omega_m$ , not $\omega_m$ itself.

Question 38

Physics · 4 marks · −1

The temperature of an open room of volume 30 m $^3$ increases from $17^\circ$ C to $27^\circ$ C due to the sunshine. The atmospheric pressure in the room remains $1\times10^5$ Pa. If $n_i$ and $n_f$ are the number of molecules in the room before and after heating, then $n_f - n_i$ will be :

A. $-2.5\times10^{25}$
B. $-1.61\times10^{23}$
C. $1.38\times10^{23}$
D. $2.5\times10^{25}$

Correct answer: A

Explanation: $n=\frac{PV}{kT}$ : $n_f-n_i=\frac{10^5\cdot30}{1.38\times10^{-23}}\left(\frac1{300}-\frac1{290}\right)\approx-2.5\times10^{25}$ .

Question 39

Physics · 4 marks · −1

In a Young's double slit experiment, slits are separated by 0.5 mm, and the screen is placed 150 cm away. A beam of light consisting of two wavelengths, 650 nm and 520 nm, is used to obtain interference fringes on the screen. The least distance from the common central maximum to the point where the bright fringes due to both the wavelengths coincide is

A.15.6 mm
B.1.56 mm
C.7.8 mm
D.9.75 mm

Correct answer: C

Explanation: $m\cdot650=n\cdot520\Rightarrow m=4,n=5$ . $y=\frac{4\cdot650\times10^{-9}\cdot1.5}{5\times10^{-4}}=7.8$ mm.

Question 40

Physics · 4 marks · −1

A particle A of mass m and initial velocity v collides with a particle B of mass $\frac{m}{2}$ which is at rest. The collision is head on, and elastic. The ratio of the de-Broglie wavelengths $\lambda_A$ to $\lambda_B$ after the collision is:

A. $\frac{\lambda_A}{\lambda_B} = \frac{1}{2}$
B. $\frac{\lambda_A}{\lambda_B} = \frac{1}{3}$
C. $\frac{\lambda_A}{\lambda_B} = 2$
D. $\frac{\lambda_A}{\lambda_B} = \frac{2}{3}$

Correct answer: C

Explanation: Elastic: $v_A=\frac v3$ , $v_B=\frac{4v}{3}$ . Momenta $\frac{mv}{3}$ and $\frac{2mv}{3}$ ; $\frac{\lambda_A}{\lambda_B}=\frac{p_B}{p_A}=2$ .

Question 41

Physics · 4 marks · −1

A magnetic needle of magnetic moment $6.7\times10^{-2}$ Am $^2$ and moment of inertia $7.5\times10^{-6}$ kg m $^2$ is performing simple harmonic oscillations in a magnetic field of 0.01 T. Time taken for 10 complete oscillations is:

A.8.76 s
B.6.65 s
C.8.89 s
D.6.98 s

Correct answer: B

Explanation: $T=2\pi\sqrt{\frac{I}{MB}}=2\pi\sqrt{\frac{7.5\times10^{-6}}{6.7\times10^{-4}}}\approx0.665$ s; 10 oscillations 6.65 s.

Question 42

Physics · 4 marks · −1

An electric dipole has a fixed dipole moment $\vec{p}$ , which makes angle $\theta$ with respect to x-axis. When subjected to an electric field $\vec{E}_1 = E\hat{i}$ , it experiences a torque $\vec{T}_1 = \tau\hat{k}$ . When subjected to another electric field $\vec{E}_2 = \sqrt{3}E_1\hat{j}$ it experiences a torque $\vec{T}_2 = -\vec{T}_1$ . The angle $\theta$ is:

A. $90^\circ$
B. $30^\circ$
C. $45^\circ$
D. $60^\circ$

Correct answer: D

Explanation: $|\tau_1|=pE\sin\theta$ , $|\tau_2|=\sqrt3pE\cos\theta$ ; equal magnitudes give $\tan\theta=\sqrt3$ , $\theta=60^\circ$ .

Question 43

Physics · 4 marks · −1

In a coil of resistance $100\,\Omega$ , a current is induced by changing the magnetic flux through it as shown in the figure. The magnitude of change in flux through the coil is:

A.275 Wb
B.200 Wb
C.225 Wb
D.250 Wb

Correct answer: D

Explanation: $\Delta\phi=R\int i\,dt=100\cdot\frac12\cdot10\cdot0.5=250$ Wb.

Question 44

Physics · 4 marks · −1

A time dependent force $F = 6t$ acts on a particle of mass 1 kg. If the particle starts from rest, the work done by the force during the first 1 sec. will be:

A.18 J
B.4.5 J
C.22 J
D.9 J

Correct answer: B

Explanation: Impulse $\int_0^16t\,dt=3=mv$ , so $v=3$ m/s; work $=\frac12(1)(9)=4.5$ J.

Question 45

Physics · 4 marks · −1

Some energy levels of a molecule are shown in the figure. The ratio of the wavelengths $r = \lambda_1/\lambda_2$ , is given by:

A. $r = \frac{1}{3}$
B. $r = \frac{4}{3}$
C. $r = \frac{2}{3}$
D. $r = \frac{3}{4}$

Correct answer: A

Explanation: $\lambda_1$ : $-E\to-2E$ ( $\Delta E=E$ ); $\lambda_2$ : $-E\to-\frac43E$ ( $\Delta E=\frac E3$ ). $\frac{\lambda_1}{\lambda_2}=\frac13$ .

Question 46

Physics · 4 marks · −1

In the given circuit, the current in each resistance is:

A.0 A
B.1 A
C.0.25 A
D.0.5 A

Correct answer: A

Explanation: Each $1\,\Omega$ resistor connects points whose potential difference is zero (the 2 V cells in the upper and lower rails cancel), so no current flows.

Question 47

Physics · 4 marks · −1

A body is thrown vertically upwards. Which one of the following graphs correctly represent the velocity vs time?

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

Correct answer: D

Explanation: $v=u-gt$ : a single straight line with negative slope crossing the time axis.

Question 48

Physics · 4 marks · −1

A capacitance of $2\,\mu$ F is required in an electrical circuit across a potential difference of 1.0 kV. A large number of $1\,\mu$ F capacitors are available which can withstand a potential difference of not more than 300 V. The minimum number of capacitors required to achieve this is:

A.32
B.2
C.16
D.24

Correct answer: A

Explanation: Need at least 4 in series (4 x 300 V > 1 kV); each series row gives $\frac14\,\mu$ F, so 8 parallel rows: 32 capacitors.

Question 49

Physics · 4 marks · −1

In the given circuit diagram when the current reaches steady state in the circuit, the charge on the capacitor of capacitance C will be:

A. $CE\frac{r_1}{(r_1+r)}$
B. $CE$
C. $CE\frac{r_1}{(r_2+r)}$
D. $CE\frac{r_2}{(r+r_2)}$

Correct answer: D

Explanation: In steady state no current flows through the $C$ - $r_1$ branch; current $\frac{E}{r+r_2}$ flows through $r_2$ , and the capacitor voltage equals the voltage across $r_2$ .

Question 50

Physics · 4 marks · −1

In a common emitter amplifier circuit using an n-p-n transistor, the phase difference between the input and the output voltages will be:

A. $180^\circ$
B. $45^\circ$
C. $90^\circ$
D. $135^\circ$

Correct answer: A

Explanation: CE amplifier output is inverted relative to input: $180^\circ$ .

Question 51

Physics · 4 marks · −1

Which of the following statements is false?

A.Kirchhoff's second law represents energy conservation.
B.Wheatstone bridge is the most sensitive when all the four resistances are of the same order of magnitude.
C.In a balanced wheatstone bridge if the cell and the galvanometer are exchanged, the null point is disturbed.
D.A rheostat can be used as a potential divider.

Correct answer: C

Explanation: Exchanging cell and galvanometer changes the balance condition to $\frac PR=\frac QS$ , equivalent to $\frac PQ=\frac RS$ ; balance is not disturbed.

Question 52

Physics · 4 marks · −1

A particle is executing simple harmonic motion with a time period T. At time $t = 0$ , it is at its position of equilibrium. The kinetic energy - time graph of the particle will look like:

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

Correct answer: A

Explanation: $KE\propto\cos^2\omega t$ : maximum at $t=0$ , zero at $T/4$ , maximum at $T/2$ , period $T/2$ .

Question 53

Physics · 4 marks · −1

An observer is moving with half the speed of light towards a stationary microwave source emitting waves at frequency 10 GHz. What is the frequency of the microwave measured by the observer? (speed of light = $3\times10^8$ ms $^{-1}$ )

A.15.3 GHz
B.10.1 GHz
C.12.1 GHz
D.17.3 GHz

Correct answer: D

Explanation: Relativistic Doppler: $f'=f\sqrt{\frac{1+\beta}{1-\beta}}=10\sqrt3\approx17.3$ GHz.

Question 54

Physics · 4 marks · −1

A man grows into a giant such that his linear dimensions increase by a factor of 9. Assuming that his density remains same, the stress in the leg will change by a factor of:

A. $\frac{1}{81}$
B.9
C. $\frac{1}{9}$
D.81

Correct answer: B

Explanation: Stress $=\frac{mg}{A}\propto\frac{L^3}{L^2}=L$ : factor 9.

Question 55

Physics · 4 marks · −1

When a current of 5 mA is passed through a galvanometer having a coil of resistance $15\,\Omega$ , it shows full scale deflection. The value of the resistance to be put in series with the galvanometer to convert it into a voltmeter of range 0 - 10V is:

A. $4.005\times10^3\,\Omega$
B. $1.985\times10^3\,\Omega$
C. $2.045\times10^3\,\Omega$
D. $2.535\times10^3\,\Omega$

Correct answer: B

Explanation: $R=\frac{V}{i_g}-G=\frac{10}{5\times10^{-3}}-15=1985\,\Omega$ .

Question 56

Physics · 4 marks · −1

The variation of acceleration due to gravity g with distance d from centre of the earth is best represented by (R = Earth's radius):

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

Correct answer: A

Explanation: $g\propto d$ inside ( $d<R$ ) and $g\propto\frac1{d^2}$ outside: linear rise to a peak at R, then inverse-square fall.

Question 57

Physics · 4 marks · −1

An external pressure P is applied on a cube at $0^\circ$ C so that it is equally compressed from all sides. K is the bulk modulus of the material of the cube and $\alpha$ is its coefficient of linear expansion. Suppose we want to bring the cube to its original size by heating. The temperature should be raised by:

A. $3PK\alpha$
B. $\frac{P}{3\alpha K}$
C. $\frac{P}{\alpha K}$
D. $\frac{3\alpha}{PK}$

Correct answer: B

Explanation: Compression $\frac{\Delta V}{V}=\frac PK$ must equal thermal expansion $3\alpha\Delta T$ : $\Delta T=\frac{P}{3\alpha K}$ .

Question 58

Physics · 4 marks · −1

A diverging lens with magnitude of focal length 25 cm is placed at a distance of 15 cm from a converging lens of magnitude of focal length 20 cm. A beam of parallel light falls on the diverging lens. The final image formed is:

A.real and at a distance of 6 cm from the convergent lens.
B.real and at a distance of 40 cm from convergent lens.
C.virtual and at a distance of 40 cm from convergent lens.
D.real and at a distance of 40 cm from the divergent lens.

Correct answer: B

Explanation: Diverging lens images at 25 cm before it, i.e. 40 cm before the converging lens. $\frac1v+\frac1{40}=\frac1{20}\Rightarrow v=40$ cm, real.

Question 59

Physics · 4 marks · −1

A body of mass $m = 10^{-2}$ kg is moving in a medium and experiences a frictional force $F = -kv^2$ . Its initial speed is $v_0 = 10$ ms $^{-1}$ . If, after 10 s, its energy is $\frac{1}{8}mv_0^2$ , the value of k will be:

A. $10^{-1}$ kg m $^{-1}$ s $^{-1}$
B. $10^{-3}$ kg m $^{-1}$
C. $10^{-3}$ kg s $^{-1}$
D. $10^{-4}$ kg m $^{-1}$

Correct answer: D

Explanation: $v_f=\frac{v_0}{2}$ . $m\left(\frac1{v_f}-\frac1{v_0}\right)=kt\Rightarrow\frac{m}{v_0}=10k$ , $k=10^{-4}$ kg/m.

Question 60

Chemistry · 4 marks · −1

1 gram of a carbonate $(M_2CO_3)$ on treatment with excess HCl produces 0.01186 mole of $CO_2$ . The molar mass of $M_2CO_3$ in g mol $^{-1}$ is:

A.84.3
B.118.6
C.11.86
D.1186

Correct answer: A

Explanation: Moles of carbonate = moles of $CO_2$ : $M=\frac{1}{0.01186}=84.3$ g/mol.

Question 61

Chemistry · 4 marks · −1

Given $C_{(graphite)} + O_2(g) \rightarrow CO_2(g)$ ; $\Delta_rH^\circ = -393.5$ kJ mol $^{-1}$ ; $H_2(g) + \frac{1}{2}O_2(g) \rightarrow H_2O(l)$ ; $\Delta_rH^\circ = -285.8$ kJ mol $^{-1}$ ; $CO_2(g) + 2H_2O(l) \rightarrow CH_4(g) + 2O_2(g)$ ; $\Delta_rH^\circ = +890.3$ kJ mol $^{-1}$ . Based on the above thermochemical equations, the value of $\Delta_rH^\circ$ at 298 K for the reaction $C_{(graphite)} + 2H_2(g) \rightarrow CH_4(g)$ will be:

A. $+144.0$ kJ mol $^{-1}$
B. $-74.8$ kJ mol $^{-1}$
C. $-144.0$ kJ mol $^{-1}$
D. $+74.8$ kJ mol $^{-1}$

Correct answer: B

Explanation: $\Delta H=-393.5+890.3+2(-285.8)=-74.8$ kJ/mol.

Question 62

Chemistry · 4 marks · −1

The freezing point of benzene decreases by $0.45^\circ$ C when 0.2 g of acetic acid is added to 20 g of benzene. If acetic acid associates to form a dimer in benzene, percentage association of acetic acid in benzene will be: ( $K_f$ for benzene = 5.12 K kg mol $^{-1}$ )

A.80.4 %
B.74.6 %
C.94.6 %
D.64.6 %

Correct answer: C

Explanation: $0.45=i\cdot5.12\cdot\frac{0.2/60}{0.02}\Rightarrow i=0.527$ ; for dimerisation $i=1-\frac\alpha2$ , $\alpha=0.946$ .

Question 63

Chemistry · 4 marks · −1

The most abundant elements by mass in the body of a healthy human adult are: Oxygen (61.4%); Carbon (22.9%), Hydrogen (10.0%); and Nitrogen (2.6%). The weight which a 75 kg person would gain if all $^1$ H atoms are replaced by $^2$ H atoms is:

A.37.5 kg
B.7.5 kg
C.10 kg
D.15 kg

Correct answer: B

Explanation: Hydrogen mass $=7.5$ kg; replacing by deuterium doubles it, a gain of 7.5 kg.

Question 64

Chemistry · 4 marks · −1

$\Delta U$ is equal to

A.Isobaric work
B.Adiabatic work
C.Isothermal work
D.Isochoric work

Correct answer: B

Explanation: For an adiabatic process $q=0$ , so $\Delta U=w$ .

Question 65

Chemistry · 4 marks · −1

The formation of which of the following polymers involves hydrolysis reaction?

A.Bakelite
B.Nylon 6, 6
C.Terylene
D.Nylon 6

Correct answer: D

Explanation: Nylon 6 is made by heating caprolactam with water: hydrolytic ring opening followed by polymerisation.

Question 66

Chemistry · 4 marks · −1

Given $E^\circ_{Cl_2/Cl^-} = 1.36$ V, $E^\circ_{Cr^{3+}/Cr} = -0.74$ V, $E^\circ_{Cr_2O_7^{2-}/Cr^{3+}} = 1.33$ V, $E^\circ_{MnO_4^-/Mn^{2+}} = 1.51$ V. Among the following, the strongest reducing agent is

A. $Mn^{2+}$
B. $Cr^{3+}$
C. $Cl^-$
D.Cr

Correct answer: D

Explanation: The species with the lowest reduction potential couple (Cr in $Cr^{3+}/Cr$ , $-0.74$ V) is the strongest reducing agent.

Question 67

Chemistry · 4 marks · −1

The Tyndall effect is observed only when following conditions are satisfied: (a) The diameter of the dispersed particles is much smaller than the wavelength of the light used. (b) The diameter of the dispersed particle is not much smaller than the wavelength of the light used. (c) The refractive indices of the dispersed phase and dispersion medium are almost similar in magnitude. (d) The refractive indices of the dispersed phase and dispersion medium differ greatly in magnitude.

A.(b) and (d)
B.(a) and (c)
C.(b) and (c)
D.(a) and (d)

Correct answer: A

Explanation: Scattering needs particles not much smaller than $\lambda$ and a large refractive-index difference.

Question 68

Chemistry · 4 marks · −1

In the following reactions, ZnO is respectively acting as a/an: (a) $ZnO + Na_2O \rightarrow Na_2ZnO_2$ (b) $ZnO + CO_2 \rightarrow ZnCO_3$

A.base and base
B.acid and acid
C.acid and base
D.base and acid

Correct answer: C

Explanation: With basic $Na_2O$ it acts as an acid; with acidic $CO_2$ it acts as a base (amphoteric).

Question 69

Chemistry · 4 marks · −1

Which of the following compounds will behave as a reducing sugar in an aqueous KOH solution?

A.Furanose whose anomeric carbon bears $CH_2OH$ and H (no anomeric OH)
B.Furanose with anomeric $-OCH_3$ (glycoside)
C.Pyranose with anomeric $-CH_2OCH_3$ group
D.Furanose with anomeric $-OCOCH_3$ (ester)

Correct answer: D

Explanation: In aqueous KOH the anomeric acetate ester is hydrolysed, liberating a free hemiacetal which reduces (glycosides and C-glycosides are stable to base).

Question 70

Chemistry · 4 marks · −1

The major product obtained in the following reaction is: (+)- $C_6H_5CH(Br)CH_2C_6H_5$ (optically active 1-bromo-1,2-diphenylethane) $\xrightarrow{{}^tBuOK,\ \Delta}$ ?

A. $C_6H_5CH=CHC_6H_5$
B. $(+)C_6H_5CH(O^tBu)CH_2C_6H_5$
C. $(-)C_6H_5CH(O^tBu)CH_2C_6H_5$
D. $(\pm)C_6H_5CH(O^tBu)CH_2C_6H_5$

Correct answer: A

Explanation: Bulky strong base with heat favours E2, giving stilbene.

Question 71

Chemistry · 4 marks · −1

Which of the following species is not paramagnetic?

A.CO
B. $O_2$
C. $B_2$
D.NO

Correct answer: A

Explanation: CO (14 electrons) has all electrons paired; $O_2$ and $B_2$ have two unpaired electrons, NO one.

Question 72

Chemistry · 4 marks · −1

On treatment of 100 mL of 0.1 M solution of $CoCl_3\cdot6H_2O$ with excess $AgNO_3$ ; $1.2\times10^{22}$ ions are precipitated. The complex is:

A. $[Co(H_2O)_3Cl_3]\cdot3H_2O$
B. $[Co(H_2O)_6]Cl_3$
C. $[Co(H_2O)_5Cl]Cl_2\cdot H_2O$
D. $[Co(H_2O)_4Cl_2]Cl\cdot2H_2O$

Correct answer: D

Explanation: 0.01 mol complex. $1.2\times10^{22}$ precipitated ions $=0.02$ mol of ions; the precipitate AgCl consists of $Ag^+$ and $Cl^-$ ions in equal number, i.e. 0.01 mol AgCl, so one ionisable chloride per formula: $[Co(H_2O)_4Cl_2]Cl\cdot2H_2O$ .

Question 73

Chemistry · 4 marks · −1

$pK_a$ of a weak acid (HA) and $pK_b$ of a weak base (BOH) are 3.2 and 3.4, respectively. The pH of their salt (AB) solution is:

A.6.9
B.7.0
C.1.0
D.7.2

Correct answer: A

Explanation: $pH=7+\frac{pK_a-pK_b}{2}=7-0.1=6.9$ .

Question 74

Chemistry · 4 marks · −1

The increasing order of the reactivity of the following halides for the $S_N1$ reaction is: (I) $CH_3CH(Cl)CH_2CH_3$ (II) $CH_3CH_2CH_2Cl$ (III) $p-H_3CO-C_6H_4-CH_2Cl$

A.(II) < (I) < (III)
B.(I) < (III) < (II)
C.(II) < (III) < (I)
D.(III) < (II) < (I)

Correct answer: A

Explanation: $S_N1$ rate follows carbocation stability: primary < secondary < p-methoxybenzyl.

Question 75

Chemistry · 4 marks · −1

Both lithium and magnesium display several similar properties due to the diagonal relationship; however, the one which is incorrect, is:

A.both form soluble bicarbonates
B.both form nitrides
C.nitrates of both Li and Mg yield $NO_2$ and $O_2$ on heating
D.both form basic carbonates

Correct answer: D

Explanation: Mg forms a basic carbonate ( $MgCO_3\cdot Mg(OH)_2$ ) but Li forms only the normal carbonate $Li_2CO_3$ ; the other three statements hold (bicarbonates exist in solution).

Question 76

Chemistry · 4 marks · −1

The correct sequence of reagents for the following conversion will be: 4-oxocyclohexane-1-carbaldehyde $\rightarrow$ 4-(2-hydroxypropan-2-yl)-1-methylcyclohexan-1-ol

A. $CH_3MgBr$ , $H^+/CH_3OH$ , $[Ag(NH_3)_2]^+OH^-$
B. $CH_3MgBr$ , $[Ag(NH_3)_2]^+OH^-$ , $H^+/CH_3OH$
C. $[Ag(NH_3)_2]^+OH^-$ , $CH_3MgBr$ , $H^+/CH_3OH$
D. $[Ag(NH_3)_2]^+OH^-$ , $H^+/CH_3OH$ , $CH_3MgBr$

Correct answer: D

Explanation: Tollens' oxidises CHO to COOH; esterify ( $H^+/CH_3OH$ ); excess $CH_3MgBr$ converts the ketone to a tertiary alcohol and the ester to $-C(CH_3)_2OH$ .

Question 77

Chemistry · 4 marks · −1

The products obtained when chlorine gas reacts with cold and dilute aqueous NaOH are:

A. $ClO_2^-$ and $ClO_3^-$
B. $Cl^-$ and $ClO^-$
C. $Cl^-$ and $ClO_2^-$
D. $ClO^-$ and $ClO_3^-$

Correct answer: B

Explanation: $Cl_2+2NaOH(\text{cold, dil})\rightarrow NaCl+NaOCl+H_2O$ .

Question 78

Chemistry · 4 marks · −1

Which of the following compounds will form significant amount of meta product during mono-nitration reaction?

A.Phenyl acetate ( $C_6H_5OCOCH_3$ )
B.Aniline ( $C_6H_5NH_2$ )
C.Acetanilide ( $C_6H_5NHCOCH_3$ )
D.Phenol ( $C_6H_5OH$ )

Correct answer: B

Explanation: In nitrating mixture aniline is protonated to anilinium ( $-NH_3^+$ , meta-directing), giving about 47% m-nitroaniline.

Question 79

Chemistry · 4 marks · −1

3-Methyl-pent-2-ene on reaction with HBr in presence of peroxide forms an addition product. The number of possible stereoisomers for the product is :

A.Zero
B.Two
C.Four
D.Six

Correct answer: C

Explanation: Anti-Markovnikov addition gives $CH_3CH(Br)CH(CH_3)C_2H_5$ with two stereocentres: $2^2=4$ stereoisomers.

Question 80

Chemistry · 4 marks · −1

Two reactions $R_1$ and $R_2$ have identical pre-exponential factors. Activation energy of $R_1$ exceeds that of $R_2$ by 10 kJ mol $^{-1}$ . If $k_1$ and $k_2$ are rate constants for reactions $R_1$ and $R_2$ respectively at 300 K, then $\ln(k_2/k_1)$ is equal to : ( $R = 8.314$ J mol $^{-1}$ K $^{-1}$ )

A.12
B.6
C.4
D.8

Correct answer: C

Explanation: $\ln\frac{k_2}{k_1}=\frac{E_1-E_2}{RT}=\frac{10000}{8.314\cdot300}\approx4$ .

Question 81

Chemistry · 4 marks · −1

Which of the following molecules is least resonance stabilized?

A.Furan
B.Pyridine
C.4,4-Dimethylcyclohexa-2,5-dien-1-one
D.Benzene

Correct answer: C

Explanation: Furan, pyridine and benzene are aromatic; the cross-conjugated dienone is not.

Question 82

Chemistry · 4 marks · −1

The group having isoelectronic species is:

A. $O^-, F^-, Na, Mg^+$
B. $O^{2-}, F^-, Na, Mg^{2+}$
C. $O^-, F^-, Na^+, Mg^{2+}$
D. $O^{2-}, F^-, Na^+, Mg^{2+}$

Correct answer: D

Explanation: $O^{2-},F^-,Na^+,Mg^{2+}$ all have 10 electrons.

Question 83

Chemistry · 4 marks · −1

The radius of the second Bohr orbit for hydrogen atom is: (Planck's Const. $h = 6.6262\times10^{-34}$ Js; mass of electron $= 9.1091\times10^{-31}$ kg; charge of electron $e = 1.60210\times10^{-19}$ C; permittivity of vacuum $\epsilon_0 = 8.854185\times10^{-12}$ kg $^{-1}$ m $^{-3}$ A $^2$ )

A.4.76 Å
B.0.529 Å
C.2.12 Å
D.1.65 Å

Correct answer: C

Explanation: $r_n=0.529n^2$ Å; $n=2$ : 2.12 Å.

Question 84

Chemistry · 4 marks · −1

The major product obtained in the following reaction is: a bicyclic γ-lactone fused to a cyclopentene ring that also carries a $-COOH$ group $\xrightarrow{DIBAL-H}$ ?

A.Cyclopentadiene ring bearing OH, $-CH_2CHO$ and $-CHO$
B.Cyclopentadiene ring bearing $-CH_2CHO$ and $-COOH$ (no OH)
C.Cyclopentadiene ring bearing $-CH_2CHO$ and $-CHO$ (no OH)
D.Cyclopentadiene ring bearing OH, $-CH_2CHO$ and $-COOH$

Correct answer: D

Explanation: DIBAL-H reduces the lactone to the lactol, which opens to a hydroxy-aldehyde ( $-OH$ and $-CH_2CHO$ ). Under controlled conditions the carboxylic acid is only deprotonated, not reduced, giving option 4. If DIBAL-H is also assumed to reduce $-COOH$ to $-CHO$ , option 1 results.

Question 85

Chemistry · 4 marks · −1

Which of the following reactions is an example of a redox reaction?

A. $XeF_2 + PF_5 \rightarrow [XeF]^+PF_6^-$
B. $XeF_6 + H_2O \rightarrow XeOF_4 + 2HF$
C. $XeF_6 + 2H_2O \rightarrow XeO_2F_2 + 4HF$
D. $XeF_4 + O_2F_2 \rightarrow XeF_6 + O_2$

Correct answer: D

Explanation: Xe goes from +4 to +6 and O from +1 to 0; the others involve no change of oxidation state.

Question 86

Chemistry · 4 marks · −1

A metal crystallises in a face centred cubic structure. If the edge length of its unit cell is 'a', the closest approach between two atoms in metallic crystal will be:

A. $2\sqrt{2}a$
B. $\sqrt{2}a$
C. $\frac{a}{\sqrt{2}}$
D. $2a$

Correct answer: C

Explanation: In fcc, atoms touch along the face diagonal: $4r=\sqrt2a$ , closest approach $2r=\frac{a}{\sqrt2}$ .

Question 87

Chemistry · 4 marks · −1

Sodium salt of an organic acid 'X' produces effervescence with conc. $H_2SO_4$ . 'X' reacts with the acidified aqueous $CaCl_2$ solution to give a white precipitate which decolourises acidic solution of $KMnO_4$ . 'X' is:

A. $HCOONa$
B. $CH_3COONa$
C. $Na_2C_2O_4$
D. $C_6H_5COONa$

Correct answer: C

Explanation: Oxalate gives $CO+CO_2$ with conc. $H_2SO_4$ and white $CaC_2O_4$ , which is oxidised by acidified $KMnO_4$ .

Question 88

Chemistry · 4 marks · −1

A water sample has ppm level concentration of following anions: $F^- = 10$ ; $SO_4^{2-} = 100$ ; $NO_3^- = 50$ . The anion/anions that make/makes the water sample unsuitable for drinking is/are:

A.both $SO_4^{2-}$ and $NO_3^-$
B.only $F^-$
C.only $SO_4^{2-}$
D.only $NO_3^-$

Correct answer: B

Explanation: Limits: $F^-$ about 1 ppm, $SO_4^{2-}$ 500 ppm, $NO_3^-$ 50 ppm; only fluoride exceeds.

Question 89

Chemistry · 4 marks · −1

Which of the following, upon treatment with tert-BuONa followed by addition of bromine water, fails to decolourize the colour of bromine?

A.1-bromo-1,2-diphenylethane type: $C_6H_5CH_2CH(Br)C_6H_5$
B.1-bromoethyl phenyl ether: $C_6H_5OCH(Br)CH_3$
C.2-bromo-1-cyclohexylpropan-1-one: $C_6H_{11}COCH(Br)CH_3$
D.Bromomethyl cyclohexyl ether: $C_6H_{11}OCH_2Br$

Correct answer: D

Explanation: The first three have a $\beta$ -hydrogen and give alkenes by elimination (which decolourise bromine water); $C_6H_{11}OCH_2Br$ has no $\beta$ -H on carbon, so no alkene forms.