JEE Main 2018 — 8 Apr Offline Set D (Previous Year Paper)

If the curves $y^2 = 6x$, $9x^2 + by^2 = 16$ intersect each other at right angles, then the value of $b$ is :

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

Correct answer: A

Let $\vec{u}$ be a vector coplanar with the vectors $\vec{a} = 2\hat{i} + 3\hat{j} - \hat{k}$ and $\vec{b} = \hat{j} + \hat{k}$. If $\vec{u}$ is perpendicular to $\vec{a}$ and $\vec{u}\cdot\vec{b} = 24$, then $|\vec{u}|^2$ is equal to :

1. A.84
2. B.336✓
3. C.315
4. D.256

Correct answer: B

For each $t \in \mathbb{R}$, let $[t]$ be the greatest integer less than or equal to $t$. Then $\lim_{x\to 0^+} x\left(\left[\frac{1}{x}\right] + \left[\frac{2}{x}\right] + \dots + \left[\frac{15}{x}\right]\right)$

1. A.does not exist (in $\mathbb{R}$)
2. B.is equal to 0
3. C.is equal to 15
4. D.is equal to 120✓

Correct answer: D

If $L_1$ is the line of intersection of the planes $2x - 2y + 3z - 2 = 0$, $x - y + z + 1 = 0$ and $L_2$ is the line of intersection of the planes $x + 2y - z - 3 = 0$, $3x - y + 2z - 1 = 0$, then the distance of the origin from the plane, containing the lines $L_1$ and $L_2$, is :

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

Correct answer: C

The value of $\int_{-\pi/2}^{\pi/2} \frac{\sin^2 x}{1+2^x}\,dx$ is :

1. A.$\frac{\pi}{4}$✓
2. B.$\frac{\pi}{8}$
3. C.$\frac{\pi}{2}$
4. D.$4\pi$

Correct answer: A

Let $g(x) = \cos x^2$, $f(x) = \sqrt{x}$, and $\alpha, \beta$ $(\alpha < \beta)$ be the roots of the quadratic equation $18x^2 - 9\pi x + \pi^2 = 0$. Then the area (in sq. units) bounded by the curve $y = (g\circ f)(x)$ and the lines $x = \alpha$, $x = \beta$ and $y = 0$, is :

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

Correct answer: B

If sum of all the solutions of the equation $8\cos x\left(\cos\left(\frac{\pi}{6}+x\right)\cos\left(\frac{\pi}{6}-x\right)-\frac{1}{2}\right)=1$ in $[0,\pi]$ is $k\pi$, then $k$ is equal to :

1. A.$\frac{20}{9}$
2. B.$\frac{2}{3}$
3. C.$\frac{13}{9}$✓
4. D.$\frac{8}{9}$

Correct answer: C

Let $f(x) = x^2 + \frac{1}{x^2}$ and $g(x) = x - \frac{1}{x}$, $x \in \mathbb{R} - \{-1, 0, 1\}$. If $h(x) = \frac{f(x)}{g(x)}$, then the local minimum value of $h(x)$ is :

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

Correct answer: A

The integral $\int \frac{\sin^2 x\cos^2 x}{\left(\sin^5 x+\cos^3 x\sin^2 x+\sin^3 x\cos^2 x+\cos^5 x\right)^2}\,dx$ is equal to : (where $C$ is a constant of integration)

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

Correct answer: C

A bag contains 4 red and 6 black balls. A ball is drawn at random from the bag, its colour is observed and this ball along with two additional balls of the same colour are returned to the bag. If now a ball is drawn at random from the bag, then the probability that this drawn ball is red, is :

1. A.$\frac{3}{4}$
2. B.$\frac{3}{10}$
3. C.$\frac{2}{5}$✓
4. D.$\frac{1}{5}$

Correct answer: C

Let the orthocentre and centroid of a triangle be $A(-3, 5)$ and $B(3, 3)$ respectively. If $C$ is the circumcentre of this triangle, then the radius of the circle having line segment $AC$ as diameter, is :

1. A.$\frac{3\sqrt{5}}{2}$
2. B.$\sqrt{10}$
3. C.$2\sqrt{10}$
4. D.$3\sqrt{\frac{5}{2}}$✓

Correct answer: D

If the tangent at $(1, 7)$ to the curve $x^2 = y - 6$ touches the circle $x^2 + y^2 + 16x + 12y + c = 0$ then the value of $c$ is :

1. A.95✓
2. B.195
3. C.185
4. D.85

Correct answer: A

If $\alpha, \beta \in \mathbb{C}$ are the distinct roots of the equation $x^2 - x + 1 = 0$, then $\alpha^{101} + \beta^{107}$ is equal to :

1. A.2
2. B.$-1$
3. C.0
4. D.1✓

Correct answer: D

PQR is a triangular park with $PQ = PR = 200$ m. A T.V. tower stands at the mid-point of QR. If the angles of elevation of the top of the tower at P, Q and R are respectively $45^\circ$, $30^\circ$ and $30^\circ$, then the height of the tower (in m) is :

1. A.$50\sqrt{2}$
2. B.100✓
3. C.50
4. D.$100\sqrt{3}$

Correct answer: B

If $\sum_{i=1}^{9}(x_i - 5) = 9$ and $\sum_{i=1}^{9}(x_i - 5)^2 = 45$, then the standard deviation of the 9 items $x_1, x_2, \dots, x_9$ is :

1. A.3
2. B.9
3. C.4
4. D.2✓

Correct answer: D

The sum of the co-efficients of all odd degree terms in the expansion of $\left(x+\sqrt{x^3-1}\right)^5+\left(x-\sqrt{x^3-1}\right)^5$, $(x > 1)$ is :

1. A.2✓
2. B.$-1$
3. C.0
4. D.1

Correct answer: A

Tangents are drawn to the hyperbola $4x^2 - y^2 = 36$ at the points P and Q. If these tangents intersect at the point $T(0, 3)$ then the area (in sq. units) of $\triangle PTQ$ is :

1. A.$36\sqrt{5}$
2. B.$45\sqrt{5}$✓
3. C.$54\sqrt{3}$
4. D.$60\sqrt{3}$

Correct answer: B

From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. The number of such arrangements is :

1. A.at least 750 but less than 1000
2. B.at least 1000✓
3. C.less than 500
4. D.at least 500 but less than 750

Correct answer: B

If the system of linear equations $x + ky + 3z = 0$, $3x + ky - 2z = 0$, $2x + 4y - 3z = 0$ has a non-zero solution $(x, y, z)$, then $\frac{xz}{y^2}$ is equal to :

1. A.30
2. B.$-10$
3. C.10✓
4. D.$-30$

Correct answer: C

If $\begin{vmatrix} x-4 & 2x & 2x \\ 2x & x-4 & 2x \\ 2x & 2x & x-4 \end{vmatrix} = (A + Bx)(x - A)^2$, then the ordered pair $(A, B)$ is equal to :

1. A.$(4, 5)$
2. B.$(-4, -5)$
3. C.$(-4, 3)$
4. D.$(-4, 5)$✓

Correct answer: D

Two sets A and B are as under : $A = \{(a, b) \in \mathbb{R}\times\mathbb{R} : |a - 5| < 1 \text{ and } |b - 5| < 1\}$; $B = \{(a, b) \in \mathbb{R}\times\mathbb{R} : 4(a - 6)^2 + 9(b - 5)^2 \le 36\}$. Then :

1. A.neither $A \subset B$ nor $B \subset A$
2. B.$B \subset A$
3. C.$A \subset B$✓
4. D.$A \cap B = \phi$ (an empty set)

Correct answer: C

Tangent and normal are drawn at $P(16, 16)$ on the parabola $y^2 = 16x$, which intersect the axis of the parabola at A and B, respectively. If C is the centre of the circle through the points P, A and B and $\angle CPB = \theta$, then a value of $\tan\theta$ is :

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

Correct answer: C

Let $S = \{t \in \mathbb{R} : f(x) = |x - \pi|\cdot(e^{|x|} - 1)\sin|x| \text{ is not differentiable at } t\}$. Then the set S is equal to :

1. A.$\{0, \pi\}$
2. B.$\phi$ (an empty set)✓
3. C.$\{0\}$
4. D.$\{\pi\}$

Correct answer: B

The Boolean expression $\sim(p \vee q) \vee (\sim p \wedge q)$ is equivalent to :

1. A.$\sim q$
2. B.$\sim p$✓
3. C.$p$
4. D.$q$

Correct answer: B

A straight line through a fixed point $(2, 3)$ intersects the coordinate axes at distinct points P and Q. If O is the origin and the rectangle OPRQ is completed, then the locus of R is :

1. A.$3x + 2y = 6xy$
2. B.$3x + 2y = 6$
3. C.$2x + 3y = xy$
4. D.$3x + 2y = xy$✓

Correct answer: D

Let A be the sum of the first 20 terms and B be the sum of the first 40 terms of the series $1^2 + 2\cdot2^2 + 3^2 + 2\cdot4^2 + 5^2 + 2\cdot6^2 + \dots$. If $B - 2A = 100\lambda$, then $\lambda$ is equal to :

1. A.496
2. B.232
3. C.248✓
4. D.464

Correct answer: C

Let $y = y(x)$ be the solution of the differential equation $\sin x\frac{dy}{dx} + y\cos x = 4x$, $x \in (0, \pi)$. If $y\left(\frac{\pi}{2}\right) = 0$, then $y\left(\frac{\pi}{6}\right)$ is equal to :

1. A.$-\frac{4}{9}\pi^2$
2. B.$\frac{4}{9\sqrt{3}}\pi^2$
3. C.$\frac{-8}{9\sqrt{3}}\pi^2$
4. D.$-\frac{8}{9}\pi^2$✓

Correct answer: D

The length of the projection of the line segment joining the points $(5, -1, 4)$ and $(4, -1, 3)$ on the plane, $x + y + z = 7$ is :

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

Correct answer: A

Let $S = \{x \in \mathbb{R} : x \ge 0 \text{ and } 2\left|\sqrt{x}-3\right| + \sqrt{x}\left(\sqrt{x}-6\right) + 6 = 0\}$. Then S :

1. A.contains exactly four elements
2. B.is an empty set
3. C.contains exactly one element
4. D.contains exactly two elements✓

Correct answer: D

Let $a_1, a_2, a_3, \dots, a_{49}$ be in A.P. such that $\sum_{k=0}^{12} a_{4k+1} = 416$ and $a_9 + a_{43} = 66$. If $a_1^2 + a_2^2 + \dots + a_{17}^2 = 140m$, then $m$ is equal to :

1. A.33
2. B.66
3. C.68
4. D.34✓

Correct answer: D

Three concentric metal shells A, B and C of respective radii a, b and c $(a < b < c)$ have surface charge densities $+\sigma$, $-\sigma$ and $+\sigma$ respectively. The potential of shell B is :

1. A.$\frac{\sigma}{\epsilon_0}\left[\frac{b^2-c^2}{c}+a\right]$
2. B.$\frac{\sigma}{\epsilon_0}\left[\frac{a^2-b^2}{a}+c\right]$
3. C.$\frac{\sigma}{\epsilon_0}\left[\frac{a^2-b^2}{b}+c\right]$✓
4. D.$\frac{\sigma}{\epsilon_0}\left[\frac{b^2-c^2}{b}+a\right]$

Correct answer: C

Seven identical circular planar disks, each of mass M and radius R are welded symmetrically as shown. The moment of inertia of the arrangement about the axis normal to the plane and passing through the point P is :

1. A.$\frac{181}{2}MR^2$✓
2. B.$\frac{19}{2}MR^2$
3. C.$\frac{55}{2}MR^2$
4. D.$\frac{73}{2}MR^2$

Correct answer: A

From a uniform circular disc of radius R and mass 9M, a small disc of radius $\frac{R}{3}$ is removed as shown in the figure. The moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through centre of disc is :

1. A.$\frac{37}{9}MR^2$
2. B.$4MR^2$✓
3. C.$\frac{40}{9}MR^2$
4. D.$10MR^2$

Correct answer: B

The reading of the ammeter for a silicon diode in the given circuit is :

1. A.13.5 mA
2. B.0
3. C.15 mA
4. D.11.5 mA✓

Correct answer: D

Unpolarized light of intensity I passes through an ideal polarizer A. Another identical polarizer B is placed behind A. The intensity of light beyond B is found to be $\frac{I}{2}$. Now another identical polarizer C is placed between A and B. The intensity beyond B is now found to be $\frac{I}{8}$. The angle between polarizer A and C is :

1. A.$60^\circ$
2. B.$0^\circ$
3. C.$30^\circ$
4. D.$45^\circ$✓

Correct answer: D

For an RLC circuit driven with voltage of amplitude $v_m$ and frequency $\omega_0 = \frac{1}{\sqrt{LC}}$ the current exhibits resonance. The quality factor, Q is given by :

1. A.$\frac{CR}{\omega_0}$
2. B.$\frac{\omega_0 L}{R}$✓
3. C.$\frac{\omega_0 R}{L}$
4. D.$\frac{R}{\omega_0 C}$

Correct answer: B

Two masses $m_1 = 5$ kg and $m_2 = 10$ kg, connected by an inextensible string over a frictionless pulley, are moving as shown in the figure. The coefficient of friction of horizontal surface is 0.15. The minimum weight m that should be put on top of $m_2$ to stop the motion is :

1. A.10.3 kg
2. B.18.3 kg
3. C.27.3 kg✓
4. D.43.3 kg

Correct answer: C

In a collinear collision, a particle with an initial speed $v_0$ strikes a stationary particle of the same mass. If the final total kinetic energy is 50% greater than the original kinetic energy, the magnitude of the relative velocity between the two particles, after collision, is :

1. A.$\frac{v_0}{\sqrt{2}}$
2. B.$\frac{v_0}{4}$
3. C.$\sqrt{2}\,v_0$✓
4. D.$\frac{v_0}{2}$

Correct answer: C

A particle is moving with a uniform speed in a circular orbit of radius R in a central force inversely proportional to the $n^{th}$ power of R. If the period of rotation of the particle is T, then :

1. A.$T \propto R^{n/2}$
2. B.$T \propto R^{3/2}$ for any n.
3. C.$T \propto R^{\frac{n}{2}+1}$
4. D.$T \propto R^{(n+1)/2}$✓

Correct answer: D

Two batteries with e.m.f 12 V and 13 V are connected in parallel across a load resistor of $10\,\Omega$. The internal resistances of the two batteries are $1\,\Omega$ and $2\,\Omega$ respectively. The voltage across the load lies between :

1. A.11.7 V and 11.8 V
2. B.11.6 V and 11.7 V
3. C.11.5 V and 11.6 V✓
4. D.11.4 V and 11.5 V

Correct answer: C

In an a.c. circuit, the instantaneous e.m.f. and current are given by $e = 100\sin 30t$, $i = 20\sin\left(30t - \frac{\pi}{4}\right)$. In one cycle of a.c., the average power consumed by the circuit and the wattless current are, respectively :

1. A.50, 0
2. B.50, 10
3. C.$\frac{1000}{\sqrt{2}}$, 10✓
4. D.$\frac{50}{\sqrt{2}}$, 0

Correct answer: C

An EM wave from air enters a medium. The electric fields are $\vec{E}_1 = E_{01}\hat{x}\cos\left[2\pi\nu\left(\frac{z}{c} - t\right)\right]$ in air and $\vec{E}_2 = E_{02}\hat{x}\cos[k(2z - ct)]$ in medium, where the wave number k and frequency $\nu$ refer to their values in air. The medium is non-magnetic. If $\epsilon_{r_1}$ and $\epsilon_{r_2}$ refer to relative permittivities of air and medium respectively, which of the following options is correct ?

1. A.$\frac{\epsilon_{r_1}}{\epsilon_{r_2}} = \frac{1}{2}$
2. B.$\frac{\epsilon_{r_1}}{\epsilon_{r_2}} = 4$
3. C.$\frac{\epsilon_{r_1}}{\epsilon_{r_2}} = 2$
4. D.$\frac{\epsilon_{r_1}}{\epsilon_{r_2}} = \frac{1}{4}$✓

Correct answer: D

A telephonic communication service is working at carrier frequency of 10 GHz. Only 10% of it is utilized for transmission. How many telephonic channels can be transmitted simultaneously if each channel requires a bandwidth of 5 kHz ?

1. A.$2\times10^6$
2. B.$2\times10^3$
3. C.$2\times10^4$
4. D.$2\times10^5$✓

Correct answer: D

A granite rod of 60 cm length is clamped at its middle point and is set into longitudinal vibrations. The density of granite is $2.7\times10^3$ kg/m$^3$ and its Young's modulus is $9.27\times10^{10}$ Pa. What will be the fundamental frequency of the longitudinal vibrations ?

1. A.7.5 kHz
2. B.5 kHz✓
3. C.2.5 kHz
4. D.10 kHz

Correct answer: B

It is found that if a neutron suffers an elastic collinear collision with deuterium at rest, fractional loss of its energy is $p_d$; while for its similar collision with carbon nucleus at rest, fractional loss of energy is $p_c$. The values of $p_d$ and $p_c$ are respectively :

1. A.(0, 1)
2. B.(.89, .28)✓
3. C.(.28, .89)
4. D.(0, 0)

Correct answer: B

The density of a material in the shape of a cube is determined by measuring three sides of the cube and its mass. If the relative errors in measuring the mass and length are respectively 1.5% and 1%, the maximum error in determining the density is:

1. A.6%
2. B.2.5%
3. C.3.5%
4. D.4.5%✓

Correct answer: D