JEE Main 2019 B.E./B.Tech Question Paper with Answer Keys (April 10, 2019- Forenoon Session)

If a directrix of a hyperbola centred at the origin and passing through the point \((4, -2\sqrt{3})\) is \(5x = 4\sqrt{5}\) and its eccentricity is \(e\), then :

• (1) \(4e^4 - 24e^2 + 27 = 0\)
• (2) \(4e^4 - 24e^2 + 35 = 0\)
• (3) \(4e^4 - 12e^2 - 27 = 0\)
• (4) \(4e^4 + 8e^2 - 35 = 0\)

If \(\displaystyle\lim_{x \to 1} \frac{x^4 - 1}{x - 1} = \lim_{x \to k} \frac{x^3 - k^3}{x^2 - k^2}\), then \(k\) is :

• (1) \(\frac{3}{2}\)
• (2) \(\frac{8}{3}\)
• (3) \(\frac{4}{3}\)
• (4) \(\frac{3}{8}\)

If \(\alpha\) and \(\beta\) are the roots of the quadratic equation, \(x^2 + x\sin\theta - 2\sin\theta = 0\), \(\theta \in \left(0, \frac{\pi}{2}\right)\), then \(\dfrac{\alpha^{12} + \beta^{12}}{\left(\alpha^{-12} + \beta^{-12}\right)\cdot(\alpha - \beta)^{24}}\) is equal to :

• (1) \(\frac{2^{12}}{(\sin\theta - 8)^6}\)
• (2) \(\frac{2^6}{(\sin\theta + 4)^{12}}\)
• (3) \(\frac{2^{12}}{(\sin\theta + 8)^{12}}\)
• (4) \(\frac{2^{12}}{(\sin\theta - 4)^{12}}\)

If the length of the perpendicular from the point \((\beta, 0, \beta)\) \((\beta \ne 0)\) to the line, \(\frac{x}{1} = \frac{y-1}{0} = \frac{z+1}{-1}\) is \(\sqrt{\frac{3}{2}}\), then \(\beta\) is equal to :

• (1) \(2\)
• (2) \(1\)
• (3) \(-2\)
• (4) \(-1\)

If \(a_1, a_2, a_3, \ldots, a_n\) are in A.P. and \(a_1 + a_4 + a_7 + \ldots + a_{16} = 114\), then \(a_1 + a_6 + a_{11} + a_{16}\) is equal to :

• (1) \(38\)
• (2) \(98\)
• (3) \(76\)
• (4) \(64\)

• (1) \(3.0\)
• (2) \(2.8\)
• (3) \(2.5\)
• (4) \(3.2\)

If \(a > 0\) and \(z = \dfrac{(1+i)^2}{a - i}\), has magnitude \(\sqrt{\frac{2}{5}}\), then \(\bar{z}\) is equal to :

• (1) \(-\frac{1}{5} + \frac{3}{5}i\)
• (2) \(-\frac{1}{5} - \frac{3}{5}i\)
• (3) \(\frac{1}{5} - \frac{3}{5}i\)
• (4) \(-\frac{3}{5} - \frac{1}{5}i\)

If \(f(x) = \begin{cases} \dfrac{\sin(p+1)x + \sin x}{x}, & x < 0 \\ q, & x = 0 \\ \dfrac{\sqrt{x + x^2} - \sqrt{x}}{x^{3/2}}, & x > 0 \end{cases}\) is continuous at \(x = 0\), then the ordered pair \((p, q)\) is equal to

• (1) \(\left(-\frac{3}{2}, -\frac{1}{2}\right)\)
• (2) \(\left(-\frac{1}{2}, \frac{3}{2}\right)\)
• (3) \(\left(-\frac{3}{2}, \frac{1}{2}\right)\)
• (4) \(\left(\frac{5}{2}, \frac{1}{2}\right)\)

The value of \(\displaystyle\int_0^{2\pi} \left[\sin 2x\,(1 + \cos 3x)\right] dx\), where \([t]\) denotes the greatest integer function is :

• (1) \(2\pi\)
• (2) \(\pi\)
• (3) \(-2\pi\)
• (4) \(-\pi\)

If \(\Delta_1 = \begin{vmatrix} x & \sin\theta & \cos\theta \\ -\sin\theta & -x & 1 \\ \cos\theta & 1 & x \end{vmatrix}\) and \(\Delta_2 = \begin{vmatrix} x & \sin 2\theta & \cos 2\theta \\ -\sin 2\theta & -x & 1 \\ \cos 2\theta & 1 & x \end{vmatrix}\), \(x \ne 0\); then for all \(\theta \in \left(0, \frac{\pi}{2}\right)\) :

• (1) \(\Delta_1 - \Delta_2 = x(\cos 2\theta - \cos 4\theta)\)
• (2) \(\Delta_1 + \Delta_2 = -2x^3\)
• (3) \(\Delta_1 + \Delta_2 = -2(x^3 + x - 1)\)
• (4) \(\Delta_1 - \Delta_2 = -2x^3\)

Let \(f : R \to R\) be differentiable at \(c \in R\) and \(f(c) = 0\). If \(g(x) = |f(x)|\), then at \(x = c\), \(g\) is :

• (1) differentiable if \(f'(c) = 0\)
• (2) differentiable if \(f'(c) \ne 0\)
• (3) not differentiable
• (4) not differentiable if \(f'(c) = 0\)

The region represented by \(|x - y| \le 2\) and \(|x + y| \le 2\) is bounded by a :

• (1) rhombus of area \(8\sqrt{2}\) sq. units
• (2) square of side length \(2\sqrt{2}\) units
• (3) square of area 16 sq. units
• (4) rhombus of side length 2 units

Let \(f(x) = e^x - x\) and \(g(x) = x^2 - x\), \(\forall\, x \in R\). Then the set of all \(x \in R\), where the function \(h(x) = (f \circ g)(x)\) is increasing, is :

• (1) \([0, \infty)\)
• (2) \(\left[-1, -\frac{1}{2}\right] \cup \left[\frac{1}{2}, \infty\right)\)
• (3) \(\left[-\frac{1}{2}, 0\right] \cup [1, \infty)\)
• (4) \(\left[0, \frac{1}{2}\right] \cup [1, \infty)\)

The number of 6 digit numbers that can be formed using the digits 0, 1, 2, 5, 7 and 9 which are divisible by 11 and no digit is repeated is :

• (1) \(36\)
• (2) \(60\)
• (3) \(72\)
• (4) \(48\)

Let \(f(x) = x^2\), \(x \in R\). For any \(A \subseteq R\), define \(g(A) = \{x \in R : f(x) \in A\}\). If \(S = [0, 4]\), then which one of the following statements is not true ?

• (1) \(g(f(S)) \ne S\)
• (2) \(f(g(S)) = S\)
• (3) \(f(g(S)) \ne f(S)\)
• (4) \(g(f(S)) = g(S)\)

If the system of linear equations
\(x + y + z = 5\)
\(x + 2y + 2z = 6\)
\(x + 3y + \lambda z = \mu\), \((\lambda, \mu \in \mathbb{R})\), has infinitely many solutions, then the value of \(\lambda + \mu\) is :

• (1) 10
• (2) 9
• (3) 7
• (4) 12

Let \(A(3, 0, -1)\), \(B(2, 10, 6)\) and \(C(1, 2, 1)\) be the vertices of a triangle and \(M\) be the midpoint of \(AC\). If \(G\) divides \(BM\) in the ratio \(2 : 1\), then \(\cos(\angle GOA)\) (\(O\) being the origin) is equal to :

• (1) \(\frac{1}{\sqrt{15}}\)
• (2) \(\frac{1}{6\sqrt{10}}\)
• (3) \(\frac{1}{\sqrt{30}}\)
• (4) \(\frac{1}{2\sqrt{15}}\)

If \(\int \frac{dx}{(x^2 - 2x + 10)^2} = A\left(\tan^{-1}\left(\frac{x-1}{3}\right) + \frac{f(x)}{x^2 - 2x + 10}\right) + C\)
where \(C\) is a constant of integration then :

• (1) \(A = \frac{1}{54}\) and \(f(x) = 9(x-1)^2\)
• (2) \(A = \frac{1}{54}\) and \(f(x) = 3(x-1)\)
• (3) \(A = \frac{1}{81}\) and \(f(x) = 3(x-1)\)
• (4) \(A = \frac{1}{27}\) and \(f(x) = 9(x-1)^2\)

Assume that each born child is equally likely to be a boy or a girl. If two families have two children each, then the conditional probability that all children are girls given that at least two are girls is :

• (1) \(\frac{1}{11}\)
• (2) \(\frac{1}{17}\)
• (3) \(\frac{1}{10}\)
• (4) \(\frac{1}{12}\)

If \(y = y(x)\) is the solution of the differential equation
\(\frac{dy}{dx} = (\tan x - y)\sec^2 x\), \(x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\),
such that \(y(0) = 0\), then \(y\left(-\frac{\pi}{4}\right)\) is equal to :

• (1) \(\frac{1}{2} - e\)
• (2) \(e - 2\)
• (3) \(2 + \frac{1}{e}\)
• (4) \(\frac{1}{e} - 2\)

All the pairs \((x, y)\) that satisfy the inequality
\(2^{\sqrt{\sin^2 x - 2\sin x + 5}} \cdot \frac{1}{4^{\sin^2 y}} \le 1\)
also satisfy the equation

• (1) \(\sin x = |\sin y|\)
• (2) \(\sin x = 2\sin y\)
• (3) \(2\sin x = \sin y\)
• (4) \(2|\sin x| = 3\sin y\)

If the coefficients of \(x^2\) and \(x^3\) are both zero, in the expansion of the expression \((1 + ax + bx^2)(1 - 3x)^{15}\) in powers of \(x\), then the ordered pair \((a, b)\) is equal to :

• (1) \((28, 861)\)
• (2) \((28, 315)\)
• (3) \((-21, 714)\)
• (4) \((-54, 315)\)

Let \( P \) be the image of the point \( Q(7, -2, 5) \) in the line \( L: \frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z}{4} \), and let \( R(5, p, q) \) be a point on \( L \). Then the square of the area of \( \triangle PQR \) is:

The sum
\(\frac{3\times 1^3}{1^3} + \frac{5\times(1^3 + 2^3)}{1^2 + 2^2} + \frac{7\times(1^3 + 2^3 + 3^3)}{1^2 + 2^2 + 3^2} + \dots\)
upto 10 terms is:

• (1) 600
• (2) 660
• (3) 680
• (4) 620

Which one of the following Boolean expressions is a tautology?

• (1) \((p \vee q) \wedge (\sim p \vee \sim q)\)
• (2) \((p \vee q) \vee (p \vee \sim q)\)
• (3) \((p \wedge q) \vee (p \wedge \sim q)\)
• (4) \((p \vee q) \wedge (p \vee \sim q)\)

\(PQR\) 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) 100
• (2) 50
• (3) \(100 \sqrt{3}\)
• (4) \(50 \sqrt{2}\)

If the line \(x - 2y = 12\) is tangent to the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) at the point \(\left(3, -\frac{9}{2}\right)\), then the length of the latus rectum of the ellipse is :

• (1) 5
• (2) 9
• (3) \(8\sqrt{3}\)
• (4) \(12\sqrt{2}\)

\(\displaystyle\lim_{n\to\infty} \left(\frac{\left(n+1\right)^{\frac{1}{3}} }{n^{\frac{4}{3}}} + \frac{\left(n+2\right)^{\frac{1}{3}}}{n^{\frac{4}{3}}} + ..... + \frac{\left(2n\right)^{\frac{1}{3}}}{n^{\frac{4}{3}}}\right) \) equal to :

• (1) \(\frac{4}{3} \left(2\right)^{\frac{4}{3}} \)
• (2) \(\frac{3}{4} \left(2\right)^{\frac{4}{3}} - \frac{4}{3} \)
• (3) \(\frac{3}{4} \left(2\right)^{\frac{4}{3}} - \frac{3}{4} \)
• (4) \(\frac{4}{3} \left(2\right)^{\frac{3}{4}} \)

If the circles \(x^2 + y^2 + 5Kx + 2y + K = 0\) and \(2(x^2 + y^2) + 2Kx + 3y - 1 = 0\), \((K \in \mathbb{R})\), intersect at the points \(P\) and \(Q\), then the line \(4x + 5y - K = 0\) passes through \(P\) and \(Q\), for :

• (1) exactly two values of \(K\)
• (2) no value of \(K\)
• (3) exactly one value of \(K\)
• (4) infinitely many values of \(K\)

The line \(x = y\) touches a circle at the point \((1, 1)\). If the circle also passes through the point \((1, -3)\), then its radius is :

• (1) 3
• (2) 2
• (3) \(2\sqrt{2}\)
• (4) \(3\sqrt{2}\)

The ratio of surface tensions of mercury and water is given to be 7.5 while the ratio of their densities is 13.6. Their contact angles, with glass, are close to \(135^\circ\) and \(0^\circ\), respectively. It is observed that mercury gets depressed by an amount \(h\) in a capillary tube of radius \(r_1\), while water rises by the same amount \(h\) in a capillary tube of radius \(r_2\). The ratio, \((r_1/r_2)\), is then close to :

• (1) 2/5
• (2) 2/3
• (3) 3/5
• (4) 4/5

Two wires \(A\) & \(B\) are carrying currents \(I_1\) & \(I_2\) as shown in the figure. The separation between them is \(d\). A third wire \(C\) carrying a current \(I\) is to be kept parallel to them at a distance \(x\) from A such that the net force acting on it is zero. The possible values of \(x\) are :

• (1) \(x=\left(\frac{I_{1}}{I_{1} -I_{2}}\right)d\) and \( x = \frac{I_{2}}{\left(I_{1} + I_{2}\right)}d \)
• (2) \( x = \pm \frac{I_{1} d}{\left(I_{1} + I_{2}\right)} \)
• (3) \(x=\left(\frac{I_{1}}{I_{1} + I_{2}}\right)d\) and \( x = \frac{I_{2}}{\left(I_{1} - I_{2}\right)}d \)
• (4) \(x=\left(\frac{I_{1}}{I_{1} + I_{2}}\right)d\) and \( x = ( \frac{I_{2}}{I_{1} + I_{2}})d \)

A thin disc of mass \(M\) and radius \(R\) has mass per unit area \(\sigma (r) = kr^2\) where \(r\) is the distance from its centre. Its moment of inertia about an axis going through its centre of mass and perpendicular to its plane is :

• (1) \(\frac{MR^2}{6}\)
• (2) \(\frac{MR^2}{3}\)
• (3) \(\frac{2MR^2}{3}\)
• (4) \(\frac{MR^2}{2}\)

Figure shows charge (q) versus voltage (V) graph for series and parallel combination of two given capacitors. The capacitances are :

• (1) \(40\,\mu\mathrm{F}\) and \(10\,\mu\mathrm{F}\)
• (2) \(60\,\mu\mathrm{F}\) and \(40\,\mu\mathrm{F}\)
• (3) \(20\,\mu\mathrm{F}\) and \(30\,\mu\mathrm{F}\)
• (4) \(50\,\mu\mathrm{F}\) and \(30\,\mu\mathrm{F}\)

A \(25\times 10^{-3}\,\mathrm{m^3}\) volume cylinder is filled with 1 mol of \(\mathrm{O_2}\) gas at room temperature (300 K). The molecular diameter of \(\mathrm{O_2}\), and its root mean square speed, are found to be 0.3 nm, and 200 m/s, respectively. What is the average collision rate (per second) for an \(\mathrm{O_2}\) molecule ?

• (1) \(\sim 10^{13}\)
• (2) \(\sim 10^{12}\)
• (3) \(\sim 10^{11}\)
• (4) \(\sim 10^{10}\)

The electric field of a plane electromagnetic wave is given by
\(\vec{E} = E_0\hat{i}\cos(kz)\cos(\omega t)\)
The corresponding magnetic field \(\vec{B}\) is then given by

• (1) \(\vec{B} = \frac{E_0}{C}\hat{j}\sin(kz)\sin(\omega t)\)
• (2) \(\vec{B} = \frac{E_0}{C}\hat{j}\sin(kz)\cos(\omega t)\)
• (3) \(\vec{B} = \frac{E_0}{C}\hat{j}\cos(kz)\sin(\omega t)\)
• (4) \(\vec{B} = \frac{E_0}{C}\hat{k}\sin(kz)\cos(\omega t)\)

• (1) 4
• (2) 3
• (3) 2
• (4) 1

Two coaxial discs, having moments of inertia \(I_1\) and \(I_1/2\), are rotating with respective angular velocities \(\omega_1\) and \(\omega_1/2\), about their common axis. They are brought in contact with each other and thereafter they rotate with a common angular velocity. If \(E_f\) and \(E_i\) are the final and initial total energies, then \((E_f - E_i)\) is:

• (1) \(\dfrac{I_1\omega_1^2}{24}\)
• (2) \(\dfrac{I_1\omega_1^2}{12}\)
• (3) \(\dfrac{3}{8}I_1\omega_1^2\)
• (4) \(\dfrac{I_1\omega_1^2}{6}\)

In an experiment, the resistance of a material is plotted as a function of temperature (in some range). As shown in the figure, it is a straight line. One may conclude that :

• (1) \(R(T)=R_0e^{-T^2/T_0^2}\)
• (2) \(R(T)=\dfrac{R_0}{T^2}\)
• (3) \(R(T)=R_0e^{T^2/T_0^2}\)
• (4) \(R(T)=R_0e^{-T_0^2/T^2}\)

A proton, an electron, and a Helium nucleus, have the same energy. They are in circular orbits in a plane due to magnetic field perpendicualr to the plane. Let \(r_p\), \(r_e\) and \(r_{He}\) be their respective radii, then

• (1) \(r_e \lt r_p \lt r_{He}\)
• (2) \(r_e \lt r_p = r_{He}\)
• (3) \(r_e \gt r_p \gt r_{He}\)
• (4) \(r_e \gt r_p = r_{He}\)

Which of the following expressions give the value of acceleration due to gravity (g') at the altitude h above the surface of Earth.
(R = radius of Earth, g = acceleration due to gravity at surface of Earth)

• (1) \( g' = g \left( 1 - \frac{h^2}{2R^2} \right) \)
  
• (2) \( g' = g \left( 1 - \frac{2h}{R} \right) \)
• (3) \( g' = g \left( 1 - \frac{h}{2R} \right) \)
  
• (4) \( g' = g \left( 1 - \frac{2h^2}{R^2} \right) \)
  

A particle of mass m is moving along a trajectory given by
\(x = x_0 + a\cos\omega_1 t\)
\(y = y_0 + b\sin\omega_2 t\)
The torque, acting on the particle about the origin, at \(t = 0\) is :

• (1) Zero
• (2) \(+my_0a\,\omega_1^2\,\hat{k}\)
• (3) \(-m\left(x_0b\omega_2^2-y_0a\omega_1^2\right)\hat{k}\)
• (4) \(m\left(-x_0b+y_0a\right)\omega_1^2\,\hat{k}\)

Two particles, of masses \(M\) and \(2M\), moving, as shown, with speeds of \(10\, m/s\) and \(5\, m/s\), collide elastically at the origin. After the collision, they move along the indicated directions with speeds \(\upsilon_1\) and \(\upsilon_2\), respectively. The values of \(\upsilon_1\) and \(\upsilon_2\) are nearly :

• (1) 3.2 m/s and 6.3 m/s
• (2) 3.2 m/s and 12.6 m/s
• (3) 6.5 m/s and 6.3 m/s
• (4) 6.5 m/s and 3.2 m/s

One plano-convex and one plano-concave lens of same radius of curvature \('R'\) but of different materials are joined side by side as shown in the figure. If the refractive index of the material of \(1\) is \(\mu_1\) and that of \(2\) is \(\mu_2\), then the focal length of the combination is :

• (1) \(\frac{R}{2-\left(\mu_{1} -\mu_{2}\right)} \)
• (2) \(\frac{ 2 R}{\mu_{1} -\mu_{2} } \)
• (3) \(\frac{R}{2\left(\mu_{1} -\mu_{2}\right)} \)
• (4) \(\frac{ R}{\mu_{1} -\mu_{2} } \)

A moving coil galvanometer allows a full scale current of \(10^{-4}\) A. A series resistance of 2 M\(\Omega\) is required to convert the above galvanometer into a voltmeter of range 0-5 V. Therefore the value of shunt resistance required to convert the above galvanometer into an ammeter of range 0.10 mA is :

• (1) \(200\ \Omega\)
• (2) \(500\ \Omega\)
• (3) \(100\ \Omega\)
• (4) None of the options are correct

In a photoelectric effect experiment the threshold wavelength of the light is 380 nm. If the wavelentgh of incident light is 260 nm, the maximum kinetic energy of emitted electrons will be:
Given E (in eV) = 1237/\(\lambda\) (in nm)

• (1) 4.5 eV
• (2) 15.1 eV
• (3) 3.0 eV
• (4) 1.5 eV

A ball is thrown upward with an initial velocity \(V_0\) from the surface of the earth. The motion of the ball is affected by a drag force equal to \(m\gamma u^2\) (where m is mass of the ball, u is its instantaneous velocity and \(\gamma\) is a constant). Time taken by the ball to rise to its zenith is :

• (1) \(\dfrac{1}{\sqrt{\gamma g}}\tan^{-1}\left(\sqrt{\dfrac{\gamma}{g}}\,V_0\right)\)
• (2) \(\dfrac{1}{\sqrt{\gamma g}}\ln\left(1+\sqrt{\dfrac{\gamma}{g}}\,V_0\right)\)
• (3) \(\dfrac{1}{\sqrt{\gamma g}}\sin^{-1}\left(\sqrt{\dfrac{\gamma}{g}}\,V_0\right)\)
• (4) \(\dfrac{1}{\sqrt{2\gamma g}}\tan^{-1}\left(\sqrt{\dfrac{2\gamma}{g}}\,V_0\right)\)

A ray of light AO in vacuum is incident on a glass slab at angle \(60^\circ\) and refracted at angle \(30^\circ\) along OB as shown in the figure. The optical path length of light ray from A to B is:

• (1) \(2a+\dfrac{2b}{\sqrt3}\)
• (2) \(2a+\dfrac{2b}{3}\)
• (3) \(2a+2b\)
• (4) \(2\sqrt3\,a+2b\)

A transformer consisting of 300 turns in the primary and 150 turns in the secondary gives output power of 2.2 kW. If the current in the secondary coil is 10A, then the input voltage and current in the primary coil are :

• (1) 220 V and 20 A
• (2) 220 V and 10A
• (3) 440 V and 5A
• (4) 440 V and 20 A

A uniformly charged disc of radius R having surface charge density \(\sigma\) is placed in the xy plane with its center at the origin. Find the electric field intensity along the z-axis at a distance Z from origin :

• (1) \(E = \frac{\sigma}{2\epsilon_0} \left( 1 + \frac{Z}{(Z^2 + R^2)^{1/2}} \right)\)
  
• (2) \(E = \frac{\sigma}{2\epsilon_0} \left( 1 - \frac{Z}{(Z^2 + R^2)^{1/2}} \right)\)
  
• (3) \(E = \frac{2\epsilon_0}{\sigma} \left( \frac{1}{(Z^2 + R^2)^{1/2}} + Z \right)\)
  
• (4) \(E = \frac{\sigma}{2\epsilon_0} \left( \frac{1}{(Z^2 + R^2)} + \frac{1}{Z^2} \right)\)
  

In the given circuit, an ideal voltmeter connected across the \(10\, \Omega\) resistance reads \(2V\). The internal resistance \(r\), of each cell is :

• (1) \(1 \Omega\)
• (2) \(1.5 \Omega\)
• (3) \(0\Omega\)
• (4) \(0.5\Omega\)

n moles of an ideal gas with constant volume heat capcity \(C_V\) undergo an isobaric expansion by certain volume. The ratio of the work done in the process, to the heat supplied is :

• (1) \(\dfrac{nR}{C_V-nR}\)
• (2) \(\dfrac{4nR}{C_V-nR}\)
• (3) \(\dfrac{4nR}{C_V+nR}\)
• (4) \(\dfrac{nR}{C_V+nR}\)

A cylinder of fixed capacity of 44.8 litres contains helium gas at standard temperature and pressure. The amount of heat needed to raise the temperature of gas in the cylinder by 20.0°C will be (Given gas constant R = 8.3 JK^–1-mol^–1)

• (1) 249 J
• (2) 415 J
• (3) 498 J
• (4) 830 J

A current of 5 A passes through a copper conductor (resistivity = \(1.7\times10^{-8}\ \Omega\,\mathrm{m}\)) of radius of cross-section 5 mm. Find the mobility of the charges if their drift velocity is \(1.1\times10^{-3}\) m/s.

• (1) \(1.3\ \mathrm{m^2/Vs}\)
• (2) \(1.0\ \mathrm{m^2/Vs}\)
• (3) \(1.8\ \mathrm{m^2/Vs}\)
• (4) \(1.5\ \mathrm{m^2/Vs}\)

An npn transistor operates as a common emitter amplifier with a power gain of 10⁶. The input circuit resistance is 100 Ω and the output load resistance is 10 kΩ. The common emitter current gain 'β' will be __________. (Round off to the Nearest Integer)

The displacement of a damped harmonic oscillator is given by
\(x(t)=e^{-0.1t}\cos(10\pi t+\phi)\).
Here t is in seconds. The time taken for its amplitude of vibration to drop to half of its initial value is close to :

• (1) 27 s
• (2) 13 s
• (3) 7 s
• (4) 4 s

Two radioactive materials \(A\) and \(B\) have decay constants \(10 \lambda\) and \(\lambda\), respectively. It initially they have the same number of nuclei, then the ratio of the number of nuclei of \(A\) to that of \(B\) will be \(1/e\) after a time :

• (1) \(\frac{11}{10 \lambda}\)
• (2) \(\frac{1}{9 \lambda}\)
• (3) \(\frac{1}{10 \lambda}\)
• (4) \(\frac{1}{11 \lambda}\)

• (1) A-Q, B-S, C-R, D-P
• (2) A-Q, B-S, C-P, D-R
• (3) A-S, B-Q, C-R, D-P
• (4) A-R, B-P, C-S, D-Q

A stationary source emits sound waves of frequency 500 Hz. Two observers moving along a line passing through the source detect sound to be of frequencies 480 Hz and 530 Hz. Their respective speeds are, in ms^-1, (Given speed of sound = 300 m/s)

• (1) 12, 18
• (2) 16, 14
• (3) 12, 16
• (4) 8, 18

A message signal of frequency \(100\, MHz\) and peak voltage \(100\, V\) is used to execute amplitude modulation on a carrier wave of frequency \(300\, GHz\) and peak voltage \(400\, V\). The modulation index and difference between the two side band frequencies are :

• (1) \( 4 ; 1 \times 10^8 \; Hz \)
• (2) \( 0.25 ; 1 \times 10^8 \; Hz \)
• (3) \( 4 ; 2 \times 10^8 \; Hz \)
• (4) \( 0.25 ; 2 \times 10^8 \; Hz \)

At room temperature, a dilute solution of urea is prepared by dissolving 0.60 of urea in 360 g of water. If the vapour pressure of pure water at this temperature is 35 mm Hg, lowering of vapour pressure will be. (molar mass of urea = 60 g mol\(^{-1}\))

• (1) 0.031 mmHg
• (2) 0.017 mmHg
• (3) 0.028 mmHg
• (4) 0.027 mmHg

The isoelectronic set of ions is :

• (1) \(\ce{N^{3-}, Li^{+}, Mg^{2+}}\) and \(\ce{O^{2-}}\)
• (2) \(\ce{Li^{+}, Na^{+}, O^{2-} }\) and \(\ce{F^{-}}\)
• (3) \(\ce{F^{-}, Li^{+}, Na^{+} }\) and \(\ce{Mg^{2+}}\)
• (4) \(\ce{N^{3-}, O^{2-} , F^{-}}\) and \(\ce{Na^{+}}\)

The principle of column chromatography is :

• (1) Capillary action.
• (2) Gravitational force.
• (3) Differential adsorption of the substances on the solid phase.
• (4) Differential absorption of the substances on the solid phase.

The graph between \(|\psi|^2\) and r (radial distance) is shown below. This represents :

• (1) 3s orbital
• (2) 2s orbital
• (3) 2p orbital
• (4) 1s orbital

The major product of the following reaction is :

• (1)
• (2)
• (3)
• (4) \(\mathrm{CH_3CH=CH{-}CH_2NH_2}\)

The increasing order of the reactivity of the following compounds towards electrophilic aromatic substitution reactions is :

• (1) \(\mathrm{III} \lt \mathrm{II} \lt \mathrm{I}\)
• (2) \(\mathrm{III} \lt \mathrm{I} \lt \mathrm{II}\)
• (3) \(\mathrm{II} \lt \mathrm{I} \lt \mathrm{III}\)
• (4) \(\mathrm{I} \lt \mathrm{III} \lt \mathrm{II}\)

The species that can have a trans-isomer is :
(en = ehane-1, 2-diamine, ox = oxalate)

• (1) \([\mathrm{Cr(en)_2(ox)}]^{+}\)
• (2) \([\mathrm{Pt(en)Cl_2}]\)
• (3) \([\mathrm{Pt(en)_2Cl_2}]^{2+}\)
• (4) \([\mathrm{Zn(en)Cl_2}]\)

The major product of the following reaction is :

• (1)
• (2)
• (3)
• (4)

A process will be spontaneous at all temperatures if :

• (1) \(\Delta H \lt 0\) and \(\Delta S \gt 0\)
• (2) \(\Delta H \lt 0\) and \(\Delta S \lt 0\)
• (3) \(\Delta H \gt 0\) and \(\Delta S \lt 0\)
• (4) \(\Delta H \gt 0\) and \(\Delta S \gt 0\)

Amylopectin is composed of :

• (1) \(\alpha\)-D-glucose, \(C_1-C_4\) and \(C_1-C_6\) linkages
• (2) \(\alpha\)-D-glucose,\(C_1-C_4\) and \(C_2-C_6\) linkages
• (3) \(\beta\)-D-glucose, \(C_1-C_4\) and \(C_2-C_6\) linkages
• (4) \(\beta\)-D-Glucose, \(C_1-C_4\) and \(C_1-C_6\) linkages

Consider the following statements
(a) The pH of a mixture containing 400 mL of 0.1 M \(\mathrm{H_2SO_4}\) and 400 mL of 0.1 M NaOH will be approximately 1.3
(b) Ionic product of water is temperature dependent.
(c) A monobasic acid with Ka = \(10^{-5}\) has pH = 5. The degree of dissociation of this acid is 50 %.
(d) The Le Chatelier's principle is not applicable to common-ion effect.
The correct statements are :

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

The oxoacid of sulphur that does not contain bond between sulphur atoms is :

• (1) \(H_2 S_4 O_6\)
• (2) \(H_2S_2O_7\)
• (3) \(H_2S_2O_3\)
• (4) \(H_2S_2O_4\)

The major product of the following reaction is :

• (1)
• (2)
• (3)
• (4)

Three complexes,
\([\mathrm{CoCl(NH_3)_5}]^{2+}\) (I),
\([\mathrm{Co(NH_3)_5H_2O}]^{3+}\) (II) and
\([\mathrm{Co(NH_3)_6}]^{3+}\) (III)
absorb light in the visible region. The correct order of the wavelength of light absorbed by them is :

• (1) (III) \(\gt\) (I) \(\gt\) (II)
• (2) (III) \(\gt\) (II) \(\gt\) (I)
• (3) (I) \(\gt\) (II) \(\gt\) (III)
• (4) (II) \(\gt\) (I) \(\gt\) (III)

At 300 K and 1 atmospheric pressure, 10 mL of a hydrocarbon required 55 mL of \(\mathrm{O_2}\) for complete combustion, and 40 mL of \(\mathrm{CO_2}\) is formed. The formula of the hydrocarbon is :

• (1) \(\mathrm{C_4H_7Cl}\)
• (2) \(\mathrm{C_4H_6}\)
• (3) \(\mathrm{C_4H_8}\)
• (4) \(\mathrm{C_4H_{10}}\)

Increasing rate of \(\mathrm{S_N1}\) reaction in the following compounds is :

• (1) (B) \(\lt\) (A) \(\lt\) (D) \(\lt\) (C)
• (2) (A) \(\lt\) (B) \(\lt\) (C) \(\lt\) (D)
• (3) (A) \(\lt\) (B) \(\lt\) (D) \(\lt\) (C)
• (4) (B) \(\lt\) (A) \(\lt\) (C) \(\lt\) (D)

Major products of the following reaction are :

• (1)
• (2)
• (3)
• (4) \(\mathrm{CH_3OH}\) and \(\mathrm{HCO_2H}\)

Consider the hydrated ions of \(\mathrm{Ti^{2+}}\), \(\mathrm{V^{2+}}\), \(\mathrm{Ti^{3+}}\), and \(\mathrm{Sc^{3+}}\). The correct order of their spin-only magnetic moments is :

• (1) \(\mathrm{Sc^{3+}} \lt \mathrm{Ti^{3+}} \lt \mathrm{V^{2+}} \lt \mathrm{Ti^{2+}}\)
• (2) \(\mathrm{Sc^{3+}} \lt \mathrm{Ti^{3+}} \lt \mathrm{Ti^{2+}} \lt \mathrm{V^{2+}}\)
• (3) \(\mathrm{Ti^{3+}} \lt \mathrm{Ti^{2+}} \lt \mathrm{Sc^{3+}} \lt \mathrm{V^{2+}}\)
• (4) \(\mathrm{V^{2+}} \lt \mathrm{Ti^{2+}} \lt \mathrm{Ti^{3+}} \lt \mathrm{Sc^{3+}}\)

During the change of \(\mathrm{O_2}\) to \(\mathrm{O_2^-}\), the incoming electron goes to the orbital :

• (1) \(\pi 2p_y\)
• (2) \(\sigma^{*}2p_z\)
• (3) \(\pi^{*}2p_x\)
• (4) \(\pi 2p_x\)

A bacterial infection in an internal wound grows as \(N'(t) = N_0\exp(t)\), where the time \(t\) is in hours. A dose of antibiotic, taken orally, needs 1 hour to reach the wound. Once it reaches there, the bacterial population goes down as \(\frac{dN}{dt} = -5N^2\). What will be the plot of \(\frac{N_0}{N}\) vs. \(t\) after 1 hour?

• (1)
• (2)
• (3)
• (4)

Ethylamine \({(C2H5NH2)}\) can be obtained from N-ethylphthalimide on treatment with :

• (1) \({NaBH_4}\)
• (2) \({CaH_2}\)
• (3) \({H_2O}\)
• (4) \({NH_2NH_2}\)

Consider the statements S1 and S2
S1 : Conductivity always increases with decrease in the concentration of electrolyte.
S2 : Molar conductivity always increases with decrease in the concentration of electrolyte.
The correct option among the following is :

• (1) Both S1 and S2 are wrong
• (2) S1 is correct and S2 is wrong
• (3) Both S1 and S2 are correct
• (4) S1 is wrong and S2 is correct

A gas undergoes physical adsorption on a surface and follows the given Freundlich adsorption isotherm equation
\(\frac{x}{m} = kp^{0.5}\)
Adsorption of the gas increase with :

• (1) Decrease in p and increase in T
• (2) Increase in p and decrease in T
• (3) Decrease in p and decrease in T
• (4) Increase in p and increase in T

The regions of the atmosphere, where clouds form and where we live, respectively, are :

• (1) Stratosphere and Stratosphere
• (2) Stratosphere and Troposphere
• (3) Troposphere and Stratosphere
• (4) Troposphere and Troposphere

The synonym for water gas when used in the production of methanol is :

• (1) fuel gas
• (2) laughing gas
• (3) syn gas
• (4) natural gas

The alloy used in the construction of aircrafts is :

• (1) \(\mathrm{Mg-Zn}\)
• (2) \(\mathrm{Mg-Al}\)
• (3) \(\mathrm{Mg-Mn}\)
• (4) \(\mathrm{Mg-Sn}\)

• (1) (I)-(c) ; (II)-(a) ; (III)-(b) ; (IV)-(d)
• (2) (I)-(c) ; (II)-(d) ; (III)-(b) ; (IV)-(a)
• (3) (I)-(b) ; (II)-(d) ; (III)-(a) ; (IV)-(c)
• (4) (I)-(b) ; (II)-(c) ; (III)-(d) ; (IV)-(a)

Which of the following is a condensation polymer?

• (1) Buna - S
• (2) Nylon 6, 6
• (3) Teflon
• (4) Neoprene

• (1) Gas C will occupy more volume than gas A; gas B will be more compressible than gas D
• (2) Gas C will occupy lesser volume than gas A; gas B will be more compressible than gas D
• (3) Gas C will occupy lesser volume than gas A; gas B will be lesser compressible than gas D
• (4) Gas C will occupy more volume than gas A; gas B will be lesser compressible than gas D

The correct order of catenation is :

• (1) \(\mathrm{C > Sn > Si \approx Ge}\)
• (2) \(\mathrm{Si > Sn > C > Ge}\)
• (3) \(\mathrm{C > Si > Ge \approx Sn}\)
• (4) \(\mathrm{Ge > Sn > Si > C}\)