JEE Main 2014 — 6 Apr Offline Set H (Previous Year Paper)

The pressure that has to be applied to the ends of a steel wire of length 10 cm to keep its length constant when its temperature is raised by $100^\circ$C is : (For steel Young's modulus is $2\times10^{11}$ N m$^{-2}$ and coefficient of thermal expansion is $1.1\times10^{-5}$ K$^{-1}$)

1. A.$2.2\times10^7$ Pa
2. B.$2.2\times10^6$ Pa
3. C.$2.2\times10^8$ Pa✓
4. D.$2.2\times10^9$ Pa

Correct answer: C

A conductor lies along the z-axis at $-1.5 \le z < 1.5$ m and carries a fixed current of 10.0 A in $-\hat{a}_z$ direction (see figure). For a field $\vec{B} = 3.0\times10^{-4}e^{-0.2x}\,\hat{a}_y$ T, find the power required to move the conductor at constant speed to $x = 2.0$ m, $y = 0$ m in $5\times10^{-3}$ s. Assume parallel motion along the x-axis

1. A.14.85 W
2. B.29.7 W
3. C.1.57 W
4. D.2.97 W✓

Correct answer: D

A bob of mass m attached to an inextensible string of length $\ell$ is suspended from a vertical support. The bob rotates in a horizontal circle with an angular speed $\omega$ rad/s about the vertical. About the point of suspension :

1. A.angular momentum changes in direction but not in magnitude.✓
2. B.angular momentum changes both in direction and magnitude.
3. C.angular momentum is conserved.
4. D.angular momentum changes in magnitude but not in direction.

Correct answer: A

The current voltage relation of diode is given by $I = (e^{1000V/T} - 1)$ mA, where the applied voltage V is in volts and the temperature T is in degree Kelvin. If a student makes an error measuring $\pm0.01$ V while measuring the current of 5 mA at 300 K, what will be the error in the value of current in mA ?

1. A.0.5 mA
2. B.0.05 mA
3. C.0.2 mA✓
4. D.0.02 mA

Correct answer: C

An open glass tube is immersed in mercury in such a way that a length of 8 cm extends above the mercury level. The open end of the tube is then closed and sealed and the tube is raised vertically up by additional 46 cm. What will be length of the air column above mercury in the tube now? (Atmospheric pressure = 76 cm of Hg)

1. A.38 cm
2. B.6 cm
3. C.16 cm✓
4. D.22 cm

Correct answer: C

Match List-I (Electromagnetic wave type) with List-II (Its association/application): (a) Infrared waves, (b) Radio waves, (c) X-rays, (d) Ultraviolet rays; (i) To treat muscular strain, (ii) For broadcasting, (iii) To detect fracture of bones, (iv) Absorbed by the ozone layer of the atmosphere

1. A.(a)-(iii), (b)-(ii), (c)-(i), (d)-(iv)
2. B.(a)-(i), (b)-(ii), (c)-(iii), (d)-(iv)✓
3. C.(a)-(iv), (b)-(iii), (c)-(ii), (d)-(i)
4. D.(a)-(i), (b)-(ii), (c)-(iv), (d)-(iii)

Correct answer: B

A parallel plate capacitor is made of two circular plates separated by a distance of 5 mm and with a dielectric of dielectric constant 2.2 between them. When the electric field in the dielectric is $3\times10^4$ V/m, the charge density of the positive plate will be close to :

1. A.$3\times10^4$ C/m$^2$
2. B.$6\times10^4$ C/m$^2$
3. C.$6\times10^{-7}$ C/m$^2$✓
4. D.$3\times10^{-7}$ C/m$^2$

Correct answer: C

A student measured the length of a rod and wrote it as 3.50 cm. Which instrument did he use to measure it?

1. A.A screw gauge having 100 divisions in the circular scale and pitch as 1 mm.
2. B.A screw gauge having 50 divisions in the circular scale and pitch as 1 mm.
3. C.A meter scale.
4. D.A vernier calliper where the 10 divisions in vernier scale matches with 9 division in main scale and main scale has 10 divisions in 1 cm.✓

Correct answer: D

Four particles, each of mass M and equidistant from each other, move along a circle of radius R under the action of their mutual gravitational attraction. The speed of each particle is :

1. A.$\sqrt{\frac{GM}{R}(1+2\sqrt{2})}$
2. B.$\frac{1}{2}\sqrt{\frac{GM}{R}(1+2\sqrt{2})}$✓
3. C.$\sqrt{\frac{GM}{R}}$
4. D.$\sqrt{2\sqrt{2}\frac{GM}{R}}$

Correct answer: B

In a large building, there are 15 bulbs of 40 W, 5 bulbs of 100 W, 5 fans of 80 W and 1 heater of 1kW. The voltage of the electric mains is 220 V. The minimum capacity of the main fuse of the building will be :

1. A.12 A✓
2. B.14 A
3. C.8 A
4. D.10 A

Correct answer: A

A particle moves with simple harmonic motion in a straight line. In first $\tau$ s, after starting from rest it travels a distance a, and in next $\tau$ s it travels 2a, in same direction, then :

1. A.amplitude of motion is 4a
2. B.time period of oscillations is $6\tau$✓
3. C.amplitude of motion is 3a
4. D.time period of oscillations is $8\tau$

Correct answer: B

The coercivity of a small magnet where the ferromagnet gets demagnetized is $3\times10^3$ Am$^{-1}$. The current required to be passed in a solenoid of length 10 cm and number of turns 100, so that the magnet gets demagnetized when inside the solenoid, is:

1. A.3A✓
2. B.6A
3. C.30 mA
4. D.60 mA

Correct answer: A

The forward biased diode connection is:

1. A.Anode at 2V, cathode side (through resistor) at 4V
2. B.Anode at $-2$V, cathode side at $+2$V
3. C.Anode at $+2$V, cathode side at $-2$V✓
4. D.Anode at $-3$V, cathode side at $-3$V

Correct answer: C

During the propagation of electromagnetic waves in a medium:

1. A.Electric energy density is equal to the magnetic energy density.✓
2. B.Both electric and magnetic energy densities are zero.
3. C.Electric energy density is double of the magnetic energy density.
4. D.Electric energy density is half of the magnetic energy density.

Correct answer: A

In the circuit shown here, the point 'C' is kept connected to point 'A' till the current flowing through the circuit becomes constant. Afterward, suddenly, point 'C' is disconnected from point 'A' and connected to point 'B' at time $t = 0$. Ratio of the voltage across resistance and the inductor at $t = L/R$ will be equal to :

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

Correct answer: A

A mass 'm' is supported by a massless string wound around a uniform hollow cylinder of mass m and radius R. If the string does not slip on the cylinder, with what acceleration will the mass fall on release?

1. A.$\frac{5g}{6}$
2. B.g
3. C.$\frac{2g}{3}$
4. D.$\frac{g}{2}$✓

Correct answer: D

One mole of diatomic ideal gas undergoes a cyclic process ABC as shown in figure. The process BC is adiabatic. The temperatures at A, B and C are 400 K, 800 K and 600 K respectively. Choose the correct statement:

1. A.The change in internal energy in the process AB is $-350$R.
2. B.The change in internal energy in the process BC is $-500$R.✓
3. C.The change in internal energy in whole cyclic process is 250R.
4. D.The change in internal energy in the process CA is 700R.

Correct answer: B

From a tower of height H, a particle is thrown vertically upwards with a speed u. The time taken by the particle, to hit the ground, is n times that taken by it to reach the highest point of its path. The relation between H, u and n is:

1. A.$2gH = nu^2(n-2)$✓
2. B.$gH = (n-2)u^2$
3. C.$2gH = n^2u^2$
4. D.$gH = (n-2)^2u^2$

Correct answer: A

A thin convex lens made from crown glass $\left(\mu = \frac{3}{2}\right)$ has focal length f. When it is measured in two different liquids having refractive indices $\frac{4}{3}$ and $\frac{5}{3}$, it has the focal lengths $f_1$ and $f_2$ respectively. The correct relation between the focal lengths is:

1. A.$f_2 > f$ and $f_1$ becomes negative
2. B.$f_1$ and $f_2$ both become negative
3. C.$f_1 = f_2 < f$
4. D.$f_1 > f$ and $f_2$ becomes negative✓

Correct answer: D

Three rods of Copper, Brass and Steel are welded together to form a Y-shaped structure. Area of cross-section of each rod = 4 cm$^2$. End of copper rod is maintained at $100^\circ$C where as ends of brass and steel are kept at $0^\circ$C. Lengths of the copper, brass and steel rods are 46, 13 and 12 cms respectively. The rods are thermally insulated from surroundings except at ends. Thermal conductivities of copper, brass and steel are 0.92, 0.26 and 0.12 CGS units respectively. Rate of heat flow through copper rod is :

1. A.4.8 cal/s✓
2. B.6.0 cal/s
3. C.1.2 cal/s
4. D.2.4 cal/s

Correct answer: A

A pipe of length 85 cm is closed from one end. Find the number of possible natural oscillations of air column in the pipe whose frequencies lie below 1250 Hz. The velocity of sound in air is 340 m/s.

1. A.6✓
2. B.4
3. C.12
4. D.8

Correct answer: A

There is a circular tube in a vertical plane. Two liquids which do not mix and of densities $d_1$ and $d_2$ are filled in the tube. Each liquid subtends $90^\circ$ angle at centre. Radius joining their interface makes an angle $\alpha$ with vertical. Ratio $d_1/d_2$ is

1. A.$\frac{1+\tan\alpha}{1-\tan\alpha}$✓
2. B.$\frac{1+\sin\alpha}{1-\cos\alpha}$
3. C.$\frac{1+\sin\alpha}{1-\sin\alpha}$
4. D.$\frac{1+\cos\alpha}{1-\cos\alpha}$

Correct answer: A

A green light is incident from the water to the air - water interface at the critical angle $(\theta)$. Select the correct statement

1. A.The spectrum of visible light whose frequency is more than that of green light will come out to the air medium.
2. B.The entire spectrum of visible light will come out of the water at various angles to the normal.
3. C.The entire spectrum of visible light will come out of the water at an angle of $90^\circ$ to the normal.
4. D.The spectrum of visible light whose frequency is less than that of green light will come out to the air medium.✓

Correct answer: D

Hydrogen $(_1H^1)$, Deuterium $(_1H^2)$, singly ionised Helium $(_2He^4)^+$ and doubly ionised lithium $(_3Li^6)^{++}$ all have one electron around the nucleus. Consider an electron transition from n = 2 to n = 1. If the wave lengths of emitted radiation are $\lambda_1$, $\lambda_2$, $\lambda_3$ and $\lambda_4$ respectively then approximately which one of the following is correct?

1. A.$\lambda_1 = \lambda_2 = 4\lambda_3 = 9\lambda_4$✓
2. B.$\lambda_1 = 2\lambda_2 = 3\lambda_3 = 4\lambda_4$
3. C.$4\lambda_1 = 2\lambda_2 = 2\lambda_3 = \lambda_4$
4. D.$\lambda_1 = 2\lambda_2 = 2\lambda_3 = \lambda_4$

Correct answer: A

The radiation corresponding to $3 \to 2$ transition of hydrogen atom falls on a metal surface to produce photoelectrons. These electrons are made to enter a magnetic field of $3\times10^{-4}$ T. If the radius of the largest circular path followed by these electrons is 10.0 mm, the work function of the metal is close to :

1. A.0.8 eV
2. B.1.6 eV
3. C.1.8 eV
4. D.1.1 eV✓

Correct answer: D

A block of mass m is placed on a surface with a vertical cross section given by $y = \frac{x^3}{6}$. If the coefficient of friction is 0.5, the maximum height above the ground at which the block can be placed without slipping is :

1. A.$\frac{1}{3}$ m
2. B.$\frac{1}{2}$ m
3. C.$\frac{1}{6}$ m✓
4. D.$\frac{2}{3}$ m

Correct answer: C

When a rubber-band is stretched by a distance x, it exerts a restoring force of magnitude $F = ax + bx^2$ where a and b are constants. The work done in stretching the unstretched rubber band by L is :

1. A.$\frac{aL^2}{2}+\frac{bL^3}{3}$✓
2. B.$\frac{1}{2}\left(\frac{aL^2}{2}+\frac{bL^3}{3}\right)$
3. C.$aL^2 + bL^3$
4. D.$\frac{1}{2}(aL^2 + bL^3)$

Correct answer: A

On heating water, bubbles being formed at the bottom of the vessel detach and rise. Take the bubbles to be spheres of radius R and making a circular contact of radius r with the bottom of the vessel. If $r \ll R$, and the surface tension of water is T, value of r just before bubbles detach is : (density of water is $\rho_w$)

1. A.$R^2\sqrt{\frac{\rho_wg}{T}}$
2. B.$R^2\sqrt{\frac{3\rho_wg}{T}}$
3. C.$R^2\sqrt{\frac{\rho_wg}{3T}}$✓
4. D.$R^2\sqrt{\frac{\rho_wg}{6T}}$

Correct answer: C

Two beams, A and B, of plane polarized light with mutually perpendicular planes of polarization are seen through a polaroid. From the position when the beam A has maximum intensity (and beam B has zero intensity), a rotation of Polaroid through $30^\circ$ makes the two beams appear equally bright. If the initial intensities of the two beams are $I_A$ and $I_B$ respectively, then $I_A/I_B$ equals :

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

Correct answer: B

Assume that an electric field $\vec{E} = 30x^2\hat{i}$ exists in space. Then the potential difference $V_A - V_O$, where $V_O$ is the potential at the origin and $V_A$ the potential at $x = 2$ m is :

1. A.$-80$ J✓
2. B.80 J
3. C.120 J
4. D.$-120$ J

Correct answer: A

The image of the line $\frac{x-1}{3} = \frac{y-3}{1} = \frac{z-4}{-5}$ in the plane $2x - y + z + 3 = 0$ is the line

1. A.$\frac{x+3}{3} = \frac{y-5}{1} = \frac{z-2}{-5}$✓
2. B.$\frac{x+3}{-3} = \frac{y-5}{-1} = \frac{z+2}{5}$
3. C.$\frac{x-3}{3} = \frac{y+5}{1} = \frac{z-2}{-5}$
4. D.$\frac{x-3}{-3} = \frac{y+5}{-1} = \frac{z-2}{5}$

Correct answer: A

If the coefficients of $x^3$ and $x^4$ in the expansion of $(1 + ax + bx^2)(1 - 2x)^{18}$ in powers of x are both zero, then $(a, b)$ is equal to

1. A.$\left(16, \frac{251}{3}\right)$
2. B.$\left(14, \frac{251}{3}\right)$
3. C.$\left(14, \frac{272}{3}\right)$
4. D.$\left(16, \frac{272}{3}\right)$✓

Correct answer: D

If $a \in \mathbb{R}$ and the equation $-3(x - [x])^2 + 2(x - [x]) + a^2 = 0$ (where [x] denotes the greatest integer $\le x$) has no integral solution, then all possible values of a lie in the interval

1. A.$(-1, 0) \cup (0, 1)$✓
2. B.$(1, 2)$
3. C.$(-2, -1)$
4. D.$(-\infty, -2) \cup (2, \infty)$

Correct answer: A

If $[\vec{a}\times\vec{b}\ \ \vec{b}\times\vec{c}\ \ \vec{c}\times\vec{a}] = \lambda[\vec{a}\ \vec{b}\ \vec{c}]^2$, then $\lambda$ is equal to

1. A.2
2. B.3
3. C.0
4. D.1✓

Correct answer: D

The variance of first 50 even natural numbers is

1. A.$\frac{833}{4}$
2. B.833✓
3. C.437
4. D.$\frac{437}{4}$

Correct answer: B

A bird is sitting on the top of a vertical pole 20 m high and its elevation from a point O on the ground is $45^\circ$. It flies off horizontally straight away from the point O. After one second, the elevation of the bird from O is reduced to $30^\circ$. Then the speed (in m/s) of the bird is

1. A.$40(\sqrt{2}-1)$
2. B.$40(\sqrt{3}-\sqrt{2})$
3. C.$20\sqrt{2}$
4. D.$20(\sqrt{3}-1)$✓

Correct answer: D

The integral $\int_0^\pi\sqrt{1 + 4\sin^2\frac{x}{2} - 4\sin\frac{x}{2}}\,dx$ equals

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

Correct answer: D

The statement $\sim(p \leftrightarrow \sim q)$ is

1. A.equivalent to $p \leftrightarrow q$✓
2. B.equivalent to $\sim p \leftrightarrow q$
3. C.a tautology
4. D.a fallacy

Correct answer: A

If A is an $3\times3$ non-singular matrix such that $AA' = A'A$ and $B = A^{-1}A'$, then $BB'$ equals

1. A.$I + B$
2. B.$I$✓
3. C.$B^{-1}$
4. D.$(B^{-1})'$

Correct answer: B

The integral $\int\left(1 + x - \frac{1}{x}\right)e^{x+\frac{1}{x}}\,dx$ is equal to

1. A.$(x-1)e^{x+\frac{1}{x}}+c$
2. B.$xe^{x+\frac{1}{x}}+c$✓
3. C.$(x+1)e^{x+\frac{1}{x}}+c$
4. D.$-xe^{x+\frac{1}{x}}+c$

Correct answer: B

If z is a complex number such that $|z| \ge 2$, then the minimum value of $\left|z + \frac{1}{2}\right|$

1. A.is equal to $\frac{5}{2}$
2. B.lies in the interval (1, 2)✓
3. C.is strictly greater than $\frac{5}{2}$
4. D.is strictly greater than $\frac{3}{2}$ but less than $\frac{5}{2}$

Correct answer: B

If g is the inverse of a function f and $f'(x) = \frac{1}{1+x^5}$, then $g'(x)$ is equal to

1. A.$1 + x^5$
2. B.$5x^4$
3. C.$\frac{1}{1+\{g(x)\}^5}$
4. D.$1 + \{g(x)\}^5$✓

Correct answer: D

If $\alpha, \beta \ne 0$, and $f(n) = \alpha^n + \beta^n$ and $\begin{vmatrix} 3 & 1+f(1) & 1+f(2) \\ 1+f(1) & 1+f(2) & 1+f(3) \\ 1+f(2) & 1+f(3) & 1+f(4) \end{vmatrix} = K(1-\alpha)^2(1-\beta)^2(\alpha-\beta)^2$, then K is equal to

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

Correct answer: C

Let $f_k(x) = \frac{1}{k}\left(\sin^k x + \cos^k x\right)$ where $x \in \mathbb{R}$ and $k \ge 1$. Then $f_4(x) - f_6(x)$ equals

1. A.$\frac{1}{6}$
2. B.$\frac{1}{3}$
3. C.$\frac{1}{4}$
4. D.$\frac{1}{12}$✓

Correct answer: D

Let $\alpha$ and $\beta$ be the roots of equation $px^2 + qx + r = 0$, $p \ne 0$. If p, q, r are in A.P. and $\frac{1}{\alpha} + \frac{1}{\beta} = 4$, then the value of $|\alpha - \beta|$ is

1. A.$\frac{\sqrt{61}}{9}$
2. B.$\frac{2\sqrt{17}}{9}$
3. C.$\frac{\sqrt{34}}{9}$
4. D.$\frac{2\sqrt{13}}{9}$✓

Correct answer: D

Let A and B be two events such that $P(\overline{A\cup B}) = \frac{1}{6}$, $P(A\cap B) = \frac{1}{4}$ and $P(\bar{A}) = \frac{1}{4}$, where $\bar{A}$ stands for the complement of the event A. Then the events A and B are

1. A.mutually exclusive and independent
2. B.equally likely but not independent
3. C.independent but not equally likely✓
4. D.independent and equally likely

Correct answer: C

If f and g are differentiable functions in [0, 1] satisfying $f(0) = 2 = g(1)$, $g(0) = 0$ and $f(1) = 6$, then for some $c \in ]0, 1[$

1. A.$2f'(c) = g'(c)$
2. B.$2f'(c) = 3g'(c)$
3. C.$f'(c) = g'(c)$
4. D.$f'(c) = 2g'(c)$✓

Correct answer: D

Let the population of rabbits surviving at a time t be governed by the differential equation $\frac{dp(t)}{dt} = \frac{1}{2}p(t) - 200$. If $p(0) = 100$, then $p(t)$ equals

1. A.$400 - 300e^{t/2}$✓
2. B.$300 - 200e^{-t/2}$
3. C.$600 - 500e^{t/2}$
4. D.$400 - 300e^{-t/2}$

Correct answer: A

Let C be the circle with centre at (1, 1) and radius = 1. If T is the circle centred at (0, y), passing through origin and touching the circle C externally, then the radius of T is equal to

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

Correct answer: D

The area of the region described by $A = \{(x, y) : x^2 + y^2 \le 1 \text{ and } y^2 \le 1 - x\}$ is

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

Correct answer: A

Let a, b, c and d be non-zero numbers. If the point of intersection of the lines $4ax + 2ay + c = 0$ and $5bx + 2by + d = 0$ lies in the fourth quadrant and is equidistant from the two axes then

1. A.$2bc - 3ad = 0$
2. B.$2bc + 3ad = 0$
3. C.$3bc - 2ad = 0$✓
4. D.$3bc + 2ad = 0$

Correct answer: C

Let PS be the median of the triangle with vertices P(2, 2), Q(6, $-1$) and R(7, 3). The equation of the line passing through $(1, -1)$ and parallel to PS is

1. A.$4x - 7y - 11 = 0$
2. B.$2x + 9y + 7 = 0$✓
3. C.$4x + 7y + 3 = 0$
4. D.$2x - 9y - 11 = 0$

Correct answer: B

$\lim_{x\to0}\frac{\sin(\pi\cos^2 x)}{x^2}$ is equal to

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

Correct answer: D

If $X = \{4^n - 3n - 1 : n \in \mathbb{N}\}$ and $Y = \{9(n - 1) : n \in \mathbb{N}\}$, where $\mathbb{N}$ is the set of natural numbers, then $X \cup Y$ is equal to

1. A.$\mathbb{N}$
2. B.$Y - X$
3. C.X
4. D.Y✓

Correct answer: D

The locus of the foot of perpendicular drawn from the centre of the ellipse $x^2 + 3y^2 = 6$ on any tangent to it is

1. A.$(x^2 - y^2)^2 = 6x^2 + 2y^2$
2. B.$(x^2 - y^2)^2 = 6x^2 - 2y^2$
3. C.$(x^2 + y^2)^2 = 6x^2 + 2y^2$✓
4. D.$(x^2 + y^2)^2 = 6x^2 - 2y^2$

Correct answer: C

Three positive numbers form an increasing G.P. If the middle term in this G.P. is doubled, the new numbers are in A.P. Then the common ratio of the G.P. is

1. A.$\sqrt{2}+\sqrt{3}$
2. B.$3+\sqrt{2}$
3. C.$2-\sqrt{3}$
4. D.$2+\sqrt{3}$✓

Correct answer: D