JEE Main 2019 B.E B.Tech Question Paper with Answer Key January 12 Forenoon Session

The product of three consecutive terms of a G.P. is 512. If 4 is added to each of the first and the second of these terms, the three terms now form an A.P. Then the sum of the original three terms of the given G.P. is :

• (1) 36
• (2) 28
• (3) 32
• (4) 24

Let \(y = y(x)\) be the solution of the differential equation, \(x\frac{dy}{dx} + y = x\log_e x,\ (x > 1)\). If \(2y(2) = \log_e 4 - 1\), then \(y(e)\) is equal to :

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

Let \(S = \{1, 2, 3, \ldots, 100\}\). The number of non-empty subsets \(A\) of \(S\) such that the product of elements in \(A\) is even is :

• (1) \(2^{50} - 1\)
• (2) \(2^{50}(2^{50} - 1)\)
• (3) \(2^{100} - 1\)
• (4) \(2^{50} + 1\)

In a random experiment, a fair die is rolled until two fours are obtained in succession. The probability that the experiment will end in the fifth throw of the die is equal to :

• (1) \(\frac{150}{6^5}\)
• (2) \(\frac{175}{6^5}\)
• (3) \(\frac{200}{6^5}\)
• (4) \(\frac{225}{6^5}\)

A ratio of the 5th term from the beginning to the 5th term from the end in the binomial expansion of \(\left(2^{1/3} + \frac{1}{2(3)^{1/3}}\right)^{10}\) is :

• (1) \(1 : 2(6)^{1/3}\)
• (2) \(1 : 4(6)^{1/3}\)
• (3) \(2(36)^{1/3} : 1\)
• (4) \(4(36)^{1/3} : 1\)

Let \(f\) and \(g\) be continuous functions on \([0, a]\) such that \(f(x) = f(a - x)\) and \(g(x) + g(a - x) = 4\), then \(\int_0^a f(x)g(x)\,dx\) is equal to :

• (1) \(4\int_0^a f(x)\,dx\)
• (2) \(-3\int_0^a f(x)\,dx\)
• (3) \(\int_0^a f(x)\,dx\)
• (4) \(2\int_0^a f(x)\,dx\)

If \(\frac{z-\alpha}{z+\alpha} \left(\alpha\in R\right) \) is a purely imaginary number and \(|z| = 2\), then a value of \(\alpha\) is :

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

Let \(S\) be the set of all points in \((-\pi, \pi)\) at which the function, \(f(x) = \min\{\sin x, \cos x\}\) is not differentiable. Then \(S\) is a subset of which of the following ?

• (1) \(\left\{-\frac{\pi}{2}, -\frac{\pi}{4}, \frac{\pi}{4}, \frac{\pi}{2}\right\}\)
• (2) \(\left\{-\frac{3\pi}{4}, -\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{4}\right\}\)
• (3) \(\left\{-\frac{\pi}{4}, 0, \frac{\pi}{4}\right\}\)
• (4) \(\left\{-\frac{3\pi}{4}, -\frac{\pi}{4}, \frac{3\pi}{4}, \frac{\pi}{4}\right\}\)

Let \(P(4, -4)\) and \(Q(9, 6)\) be two points on the parabola, \(y^2 = 4x\) and let \(x\) be any point on the arc \(POQ\) of this parabola, where \(O\) is the vertex of this parabola, such that the area of \(\Delta PXQ\) is maximum. Then this maximum area (in sq. units) is :

• (1) \(\frac{625}{4}\)
• (2) \(\frac{125}{4}\)
• (3) \(\frac{75}{2}\)
• (4) \(\frac{125}{2}\)

Consider three boxes, each containing \(10\) balls labelled \(1,2,....,10\). Suppose one ball is randomly drawn from each of the boxes. Denote by \(n_i\), the label of the ball drawn from the \(i^{th}\) box, (\(i = 1, 2, 3\)). Then, the number of ways in which the balls can be chosen such that \(n_1 < n_2 < n_3\) is :

• (1) 82
• (2) 240
• (3) 164
• (4) 120

The integral \(\int \cos( \log \; x)dx\) is equal to : (where C is a constant of integration)

• (1) \(\frac{x}{2} \left[\sin\left(\log_{e} x\right)-\cos\left(\log_{e}x\right)+C\right]\)
• (2) \(\frac{x}{2}\left[\cos\left(\log_{e}x\right) + \sin\left(\log_{e} x\right)\right]+C\)
• (3) \(x [\cos (\log_e x) + \sin (\log_e x )] + C \)
• (4) \(x [\cos (\log_e x) - \sin (\log_e x )] + C \)

An ordered pair \((\alpha, \beta)\) for which the system of linear equations
\((1 + \alpha)x + \beta y + z = 2\)
\(\alpha x + (1 + \beta)y + z = 3\)
\(\alpha x + \beta y + 2z = 2\)
has a unique solution, is :

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

Let \(P = \begin{bmatrix} 1 & 0 & 0\\ 3 & 1 & 0\\ 9 & 3 & 1\end{bmatrix}\) and \(Q = [q_{ij}]\) be two \(3 \times 3\) matrices such that \(Q - P^5 = I_3\). Then \(\frac{q_{21} + q_{31}}{q_{32}}\) is equal to :

• (1) 15
• (2) 9
• (3) 135
• (4) 10

If \(\lambda\) be the ratio of the roots of the quadratic equation in \(x\), \(3m^2x^2 + m(m - 4)x + 2 = 0\), then the least value of \(m\) for which \(\lambda + \frac{1}{\lambda} = 1\), is

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

The maximum value of \(3\cos\theta + 5\sin\left(\theta - \frac{\pi}{6}\right)\) for any real value of \(\theta\) is :

• (1) \(\sqrt{34}\)
• (2) \(\sqrt{31}\)
• (3) \(\sqrt{19}\)
• (4) \(\frac{\sqrt{79}}{2}\)

Considering only the principal values of inverse functions, the set \(A = \{ x \ge 0 : \tan^{-1} (2x) + \tan^{-1} (3x) = \frac{\pi}{4} \}\)

• (1) is an empty set
• (2) Contains more than two elements
• (3) Contains two elements
• (4) is a singleton

\(\displaystyle\lim_{x \to \pi/4} \frac{\cot^3 x - \tan x}{\cos\left(x + \frac{\pi}{4}\right)}\) is :

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

For \(x > 1\), if \((2x)^{2y} = 4e^{2x - 2y}\), then \(\left( 1 +\log_{e} 2x\right)^{2} \frac{dy}{dx} \) is equal to :

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

If the vertices of a hyperbola be at \((-2, 0)\) and \((2, 0)\) and one of its foci be at \((-3, 0)\), then which one of the following points does not lie on this hyperbola?

• (1) \((6, 5\sqrt{2})\)
• (2) \((2\sqrt{6}, 5)\)
• (3) \((-6, 2\sqrt{10})\)
• (4) \((4, \sqrt{15})\)

If the sum of the deviations of \(50\) observations from \(30\) is \(50\), then the mean of these observation is :

• (1) 50
• (2) 51
• (3) 30
• (4) 31

If the foot of the perpendicular drawn from the point (1, 0, 3) on a line passing through (\( \alpha \), 7, 1) is \( \left( {{5 \over 3},{7 \over 3},{{17} \over 3}} \right) \), then \( \alpha \) is equal to______.

The area (in sq. units) of the region bounded by the parabola, \(y = x^2 + 2\) and the lines, \(y = x + 1\), \(x = 0\) and \(x = 3\), is

• (1) \(\frac{15}{4}\)
• (2) \(\frac{15}{2}\)
• (3) \(\frac{21}{2}\)
• (4) \(\frac{17}{4}\)

The maximum area (in sq. units) of a rectangle having its base on the x-axis and its other two vertices on the parabola, \(y = 12 - x^2\) such that the rectangle lies inside the parabola, is :

• (1) \(36\)
• (2) \(20\sqrt{2}\)
• (3) \(18\sqrt{3}\)
• (4) \(32\)

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

• (1) \(q \Rightarrow p\)
• (2) \(p \Rightarrow q\)
• (3) \(\sim q \Rightarrow p\)
• (4) \(p \Rightarrow \sim q\)

A tetrahedron has vertices \(P(1, 2, 1)\), \(Q(2, 1, 3)\), \(R(-1, 1, 2)\) and \(O(0, 0, 0)\). The angle between the faces OPQ and PQR is :

• (1) \(\cos^{-1}\left(\frac{17}{31}\right)\)
• (2) \(\cos^{-1}\left(\frac{9}{35}\right)\)
• (3) \(\cos^{-1}\left(\frac{19}{35}\right)\)
• (4) \(\cos^{-1}\left(\frac{7}{31}\right)\)

Let \(C_1\) and \(C_2\) be the centres of the circles \(x^2 + y^2 - 2x - 2y - 2 = 0\) and \(x^2 + y^2 - 6x - 6y + 14 = 0\) respectively. If P and Q are the points of intersection of these circles, then the area (in sq. units) of the quadrilateral \(PC_1QC_2\) is :

• (1) \(4\)
• (2) \(6\)
• (3) \(9\)
• (4) \(8\)

The sum of the distinct real values of \(\mu\), for which the vectors, \(\mu\hat{i} + \hat{j} + \hat{k},\ \hat{i} + \mu\hat{j} + \hat{k},\ \hat{i} + \hat{j} + \mu\hat{k}\) are co-planar, is :

• (1) \(2\)
• (2) \(-1\)
• (3) \(0\)
• (4) \(1\)

The perpendicular distance from the origin to the plane containing the two lines, \(\frac{x+2}{3} = \frac{y-2}{5} = \frac{z+5}{7}\) and \(\frac{x-1}{1} = \frac{y-4}{4} = \frac{z+4}{7}\), is :

• (1) \(6\sqrt{11}\)
• (2) \(\frac{11}{\sqrt{6}}\)
• (3) \(11\)
• (4) \(11\sqrt{6}\)

If a variable line, \(3x + 4y - \lambda = 0\) is such that the two circles \(x^2 + y^2 - 2x - 2y + 1 = 0\) and \(x^2 + y^2 - 18x - 2y + 78 = 0\) are on its opposite sides, then the set of all values of \(\lambda\) is the interval :

• (1) \((23, 31)\)
• (2) \((2, 17)\)
• (3) \([13, 23]\)
• (4) \([12, 21]\)

Let \(S_k = \frac{1 + 2 + 3 + \dots + k}{k}\). If \(S_1^2 + S_2^2 + \dots + S_{10}^2 = \frac{5}{12}A\), then A is equal to :

• (1) \(283\)
• (2) \(156\)
• (3) \(301\)
• (4) \(303\)

A particle of mass \(m\) moves in a circular orbit under the central potential field, \(U(r) = -\frac{C}{r}\), where \(C\) is a positive constant. The correct radius – velocity graph of the particle's motion is: 

• (1) A
• (2) B
• (3) C
• (4) D

There is a uniform spherically symmetric surface charge density at a distance \(R_0\) from the origin. The charge distribution is initially at rest and starts expanding because of mutual repulsion. The figure that represents best the speed \(V(R(t))\) of the distribution as a function of its instantaneous radius \(R(t)\) is :

• (1)
• (2)
• (3)
• (4)

Two light identical springs of spring constant k are attached horizontally at the two ends of a uniform horizontal rod AB of length \(\ell\) and mass m. The rod is pivoted at its centre 'O' and can rotate freely in horizontal plane. The other ends of the two springs are fixed to rigid supports as shown in figure. The rod is gently pushed through a small angle and released. The frequency of resulting oscillation is :

• (1) \(\frac{1}{2\pi}\sqrt{\frac{3k}{m}}\)
• (2) \(\frac{1}{2\pi}\sqrt{\frac{6k}{m}}\)
• (3) \(\frac{1}{2\pi}\sqrt{\frac{k}{m}}\)
• (4) \(\frac{1}{2\pi}\sqrt{\frac{2k}{m}}\)

A proton and an \(\alpha\)-particle (with their masses in the ratio of \(1 : 4\) and charges in the ratio of \(1 : 2\)) are accelerated from rest through a potential difference \(V\). If a uniform magnetic field (B) is set up perpendicular to their velocities, the ratio of the radii \(r_p : r_\alpha\) of the circular paths described by them will be :

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

A passenger train of length \(60\,m\) travels at a speed of \(80\, km/hr\). Another freight train of length \(120\, m\) travels at a speed of \(30\, km/hr\). The ratio of times taken by the passenger train to completely cross the freight train when : (i) they are moving in the same direction, and (ii) in the opposite directions is :

• (1) 44683
• (2) 25-Nov
• (3) 44622
• (4) 44870

The galvanometer deflection, when key \(K_1\) is closed but \(K_2\) is open, equals \(\theta_0\) (see figure). On closing \(K_2\) also and adjusting \(R_2\) to \(5\,\Omega\), the deflection in galvanometer becomes \(\frac{\theta_0}{5}\). The resistance of the galvanometer is, then, given by [Neglect the internal resistance of battery] :

• (1) \(5\,\Omega\)
• (2) \(25\,\Omega\)
• (3) \(12\,\Omega\)
• (4) \(22\,\Omega\)

What is the position and nature of image formed by lens combination shown in figure? \((f_1, f_2\) are focal lengths)

• (1) \(70 \,cm\) from point \(B\) at left; virtual
• (2) \(40\, cm\) from point \(B\) at right; real
• (3) \(20/3 \,cm\) from point \(B\) at right , real
• (4) \(70\,cm\) from point \(B\) at right, real

A straight rod of length \(L\) extends from \(x = a\) to \(x = L + a\). The gravitational force is exerts on a point mass \('m'\) at \(x = 0\), if the mass per unit length of the rod is \(A + Bx^2\), is given by:

• (1) \(Gm\left[A\left(\frac{1}{a + L} - \frac{1}{a}\right) - BL\right]\)
• (2) \(Gm\left[A\left(\frac{1}{a} - \frac{1}{a + L}\right) + BL\right]\)
• (3) \(Gm\left[A\left(\frac{1}{a + L} - \frac{1}{a}\right) + BL\right]\)
• (4) \(Gm\left[A\left(\frac{1}{a} - \frac{1}{a + L}\right) - BL\right]\)

The position vector of the centre of mass \(\vec{r}_{cm}\) of an asymmetric uniform bar of negligible area of crosssection as shown in figure is :

• (1) \(\vec{r}_{cm} = \frac{11}{8}L\,\hat{x} + \frac{8}{8}L\,\hat{y}\)
• (2) \(\vec{r}_{cm} = \frac{5}{8}L\,\hat{x} + \frac{13}{8}L\,\hat{y}\)
• (3) \(\vec{r}_{cm} = \frac{13}{8}L\,\hat{x} + \frac{5}{8}L\,\hat{y}\)
• (4) \(\vec{r}_{cm} = \frac{3}{8}L\,\hat{x} + \frac{11}{8}L\,\hat{y}\)

A travelling harmonic wave is represented by the equation \(y (x, t) = 10^{-3} sin (50 t + 2x)\), where \(x\) and \(y\) are in meter and \(t\) is in seconds. Which of the following is a correct statement about the wave? The wave is propagating along the

• (1) negative x-axis with speed \(25\,ms^{-1}\)
• (2) The wave is propagating along the positive x-axis with speed \(25\,ms^{-1}\)
• (3) The wave is propagating along the positive x-axis with speed \(100\, ms^{-1}\)
• (4) The wave is propagating along the negative x-axis with speed \(100\, ms^{-1}\)

For the given cyclic process \(CAB\) as shown for a gas, the work done is :

• (1) 1 J
• (2) 5 J
• (3) 10 J
• (4) 30 J

A point source of light, S is placed at distance L in front of the centre of plane mirror of width d which is hanging vertically on a wall. A man walks in front of the mirror along a line parallel to the mirror, at a distance 2L as shown below. The distance over which the man can see the image of the light source in the mirror is

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

An ideal gas occupies a volume of \(2\,m^3\) at a pressure of \(3 \times 10^6\, Pa\). The energy of the gas is:

• (1) \(3 \times 10^6 J\)
• (2) \( 10^8 J\)
• (3) \(6 \times 10^4 J\)
• (4) \(9 \times 10^{-6} J \)

The least count of the main scale of a screw gauge is 1 mm. The minimum number of divisions on its circular scale required to measure 5 \(\mu\)m diameter of a wire is :

• (1) \(500\)
• (2) \(100\)
• (3) \(200\)
• (4) \(50\)

A simple pendulum, made of a string of length \(l\) and a bob of mass m, is released from a small angle \(\theta_0\). It strikes a block of mass \(M\), kept on a horizontal surface at its lowest point of oscillations, elastically. It bounces back and goes up to an angle \(\theta_1\). Then M is given by :

• (1) \(\frac{m}{2}\left(\frac{\theta_{0} - \theta_{1}}{\theta_{0} + \theta_{1}}\right)\)
• (2) \(\frac{m}{2}\left(\frac{\theta_{0} + \theta_{1}}{\theta_{0} - \theta_{1}}\right)\)
• (3) \(m\left(\frac{\theta_{0} + \theta_{1}}{\theta_{0} - \theta_{1}}\right)\)
• (4) \(m\left(\frac{\theta_{0} - \theta_{1}}{\theta_{0} + \theta_{1}}\right)\)

In the figure shown, a circuit contains two identical resistors with resistance \(R = 5\,\Omega\) and an inductance with \(L = 2\,\text{mH}\). An ideal battery of 15 V is connected in the circuit. What will be the current through the battery long after the switch is closed ?

• (1) \(6\ \text{A}\)
• (2) \(7.5\ \text{A}\)
• (3) \(3\ \text{A}\)
• (4) \(5.5\ \text{A}\)

A light wave is incident normally on a glass slab of refractive index 1.5. If 4 % of light gets reflected and the amplitude of the electric field of the incident light is 30 V/m, then the amplitude of the electric field for the wave propagating in the glass medium will be :

• (1) \(6\ \text{V/m}\)
• (2) \(10\ \text{V/m}\)
• (3) \(30\ \text{V/m}\)
• (4) \(24\ \text{V/m}\)

A person standing on an open ground hears the sound of a jet aeroplane, coming from north at an angle \(60^\circ\) with ground level. But he finds the aeroplane right vertically above his position. If \(\upsilon\) is the speed of sound, speed of the plane is :

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

The output of the given logic circuit is :

• (1) \(\bar{A}B\)
• (2) \(A\bar{B}\)
• (3) \(AB + \bar{AB}\)
• (4) \(A\bar{B} + \bar{A}B\)

As shown in the figure, two infinitely long, identical wires are bent by \(90^\circ\) and placed in such a way that the segments LP and QM are along the x-axis, while segments PS and QN are parallel to the y-axis. If OP = OQ = 4 cm, and the magnitude of the magnetic field at O is \(10^{-4}\) T, and the two wires carry equal currents (see figure), the magnitude of the current in each wire and the direction of the magnetic field at O will be (\(\mu_0 = 4\pi \times 10^{-7}\ \mathrm{NA^{-2}}\)) :

• (1) 40 A, perpendicular into the page
• (2) 40 A, perpendicular out of the page
• (3) 20 A, perpendicular into the page
• (4) 40 A, perpendicular out of the page

In the figure shown, after the switch \('S'\) is turned from position \('A'\) to position \('B'\), the energy dissipated in the circuit in terms of capacitance \('C'\) and total charge \('Q'\) is:

• (1) \(\frac{3}{8}\frac{Q^2}{C}\)
• (2) \(\frac{4}{3}\frac{Q^2}{C}\)
• (3) \(\frac{1}{8}\frac{Q^2}{C}\)
• (4) \(\frac{5}{8}\frac{Q^2}{C}\)

Let the moment of inertia of a hollow cylinder of length \(30\, cm\) (inner radius \(10\, cm\) and outer radius \(20\, cm\)), about its axis be 1. The radius of a thin cylinder of the same mass such that its moment of inertia about its axis is also \(I\), is:

• (1) 12 cm
• (2) 18 cm
• (3) 16 cm
• (4) 14 cm

Determine the electric dipole moment of the system of three charges, placed on the vertices of an equilateral triangle, as shown in the figure:

• (1) \(\left(ql \right)\frac{\hat{i} + \hat{j}}{\sqrt{2}}\)
• (2) \(\sqrt{3}ql\frac{\hat{j} - \hat{I}}{\sqrt{2}}\)
• (3) \(-\sqrt{3}ql\hat{j}\)
• (4) \(2ql\hat{j}\)

A particle A of mass 'm' and charge 'q' is accelerated by a potential difference of 50 V. Another particle B of mass '4 m' and charge 'q' is accelerated by a potential difference of 2500 V. The ratio of de-Broglie wavelengths \(\dfrac{\lambda_A}{\lambda_B}\) is close to :

• (1) 4.47
• (2) 10.00
• (3) 14.14
• (4) 0.07

A satellite of mass \(M\) is in a circular orbit of radius \(R\) about the centre of the earth. A meteorite of the same mass, falling towards the earth, collides with the satellite completely inelastically. The speeds of the satellite and the meteorite are the same, just before the collision. The subsequent motion of the combined body will be :

• (1) in a circular orbit of a different radius
• (2) in the same circular orbit of radius R
• (3) in an elliptical orbit
• (4) such that it escapes to infinity

In a meter bridge, the wire of length \(1\, m\) has a non-uniform cross-section such that, the variation \(\frac{dR}{dl}\) of its resistance \(R\) with length \(l\) is \(\frac{dR}{dl} \propto \frac{1}{\sqrt{l}}\). Two equal resistances are connected as shown in the figure. The galvanometer has zero deflection when the jockey is at point \(P\). What is the length \(AP\) ?

• (1) 0.25 m
• (2) 0.3 m
• (3) 0.35 m
• (4) 0.2 m

Two electric bulbs, rated at (25 W, 220 V) and (100 W, 220 V), are connected in series across a 220 V voltage source. If the 25 W and 100 W bulbs draw powers \(P_1\) and \(P_2\) respectively, then :

• (1) \(P_1 = 4\ \mathrm{W},\ P_2 = 16\ \mathrm{W}\)
• (2) \(P_1 = 16\ \mathrm{W},\ P_2 = 4\ \mathrm{W}\)
• (3) \(P_1 = 9\ \mathrm{W},\ P_2 = 16\ \mathrm{W}\)
• (4) \(P_1 = 16\ \mathrm{W},\ P_2 = 9\ \mathrm{W}\)

A cylinder of radius R is surrounded by a cylindrical shell of inner radius R and outer radius 2R. The thermal conductivity of the material of the inner cylinder is \(K_1\) and the of the outer cylinder is \(K_2\). Assuming no loss of heat, the effective thermal conductivity of the system for heat flowing along the length of the cylinder is :

• (1) \(K_1 + K_2\)
• (2) \(\dfrac{K_1 + 3K_2}{4}\)
• (3) \(\dfrac{K_1 + K_2}{2}\)
• (4) \(\dfrac{2K_1 + 3K_2}{5}\)

A 100 V carrier wave is made to vary between 160 V and 40 V by a modulating signal. What is the modulation index ?

• (1) 0.5
• (2) 0.6
• (3) 0.4
• (4) 0.3

An ideal battery of \(4\, V\) and resistance \(R\) are connected in series in the primary circuit of a potentiometer of length \(1\, m\) and resistance \(5\Omega\). The value of \(R\), to give a potential difference of \(5 \,mV\) across \(10\, cm\) of potentiometer wire, is :

• (1) \(490\Omega\)
• (2) \(480\Omega\)
• (3) \(395\Omega\)
• (4) \(495\Omega\)

Two solids dissociate as follows \( {A(s) <=> B(g) + C(g) ; K_{p_1} = x \; atm^2}\) \( {D(s) <=> C(g) + E(g) ; K_{p_2} = y \; atm^2}\) The total pressure when both the solids dissociate simultaneously is :

• (1) \(x^2 + y^2 \; atm\)
• (2) \( 2 x^2 + y^2 \; atm\)
• (3) \(2 (\sqrt{x + y} ) atm\)
• (4) \(\sqrt{x + y} \,atm\)

In the following reaction, products A and B are \(-\)

• (1)
• (2)
• (3)
• (4)

Decomposition of X exhibits a rate constant of \(0.05\ \mu\mathrm{g/year}\). How many year are required for the decomposition of \(5\ \mu\mathrm{g}\) of X into \(2.5\ \mu\mathrm{g}\)?

• (1) 50
• (2) 20
• (3) 25
• (4) 40

Freezing point of a \(4\%\) aqueous solution of \(X\) is equal to freezing point of \(12\%\) aqueous solution of \(Y\). If molecular weight of \(X\) is \(A\), then molecular weight of \(Y\) is :

• (1) A
• (2) 3A
• (3) 4A
• (4) 2A

\( {Mn_2(CO)_{10}}\) is an organometallic compound due to the presence of :

• (1) Mn - Mn bond
• (2) Mn - C bond
• (3) Mn - O bond
• (4) C - O bond

The hardness of a water sample containing \(10^{-3}\) M \(\mathrm{MgSO_4}\) expressed as \(\mathrm{CaCO_3}\) equivalents (in ppm) is ______.
(molar mass of \(\mathrm{MgSO_4}\) is 120.37 g/mol)

The increasing order of reactivity of the following compounds towards reaction with alkyl halides directly is

(The figure shows four structures labelled (A) benzamide, (B) phthalimide, (C) 2-aminobenzonitrile, (D) aniline.)

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

• (1) \(\mathrm{CH_3CHO}\) and MeOH
• (2) HCHO and MeOH
• (3) \(\mathrm{CH_3CHO}\) and tBuOH
• (4) HCHO and tBuOH

Among the following compounds most basic amino acid is :

• (1) Lysine
• (2) Asparagine
• (3) Serine
• (4) Histidine

can not be prepared by \(-\)
(The structure shown is \(\mathrm{CH_3CH_2C(OH)(Ph)CH_3}\), i.e. 2-phenylbutan-2-ol.)

• (1) \(\mathrm{CH_3CH_2COCH_3 + PhMgX}\)
• (2) \(\mathrm{PhCOCH_3 + CH_3CH_2MgX}\)
• (3) \(\mathrm{PhCOCH_2CH_3 + CH_3MgX}\)
• (4) \(\mathrm{HCHO + PhCH(CH_3)CH_2MgX}\)

The pair of metal ions that can give a spin only magnetic moment of 3.9 BM for the complex \(\mathrm{[M(H_2O)_6]Cl_2}\), is \(-\)

• (1) \(\mathrm{V^{2+}}\) and \(\mathrm{Fe^{2+}}\)
• (2) \(\mathrm{V^{2+}}\) and \(\mathrm{Co^{2+}}\)
• (3) \(\mathrm{Co^{2+}}\) and \(\mathrm{Fe^{2+}}\)
• (4) \(\mathrm{Cr^{2+}}\) and \(\mathrm{Mn^{2+}}\)

The element with Z = 120 (not yet discovered) will be an/a \(-\)

• (1) Alkaline earth metal
• (2) Alkali metal
• (3) Transition metal
• (4) Inner transition metal

What is the work function of the metal if the light of wavelength \(4000\ \mathrm{\AA}\) generates photoelectrons of velocity \(6\times 10^{5}\ \mathrm{ms^{-1}}\) from it ?
(Mass of electron = \(9\times 10^{-31}\) kg;
Velocity of light = \(3\times 10^{8}\ \mathrm{ms^{-1}}\)
Plank's constant = \(6.626\times 10^{-34}\) Js;
Charge of electron = \(1.6\times 10^{-19}\ \mathrm{JeV^{-1}}\))

• (1) 4.0 eV
• (2) 0.9 eV
• (3) 2.1 eV
• (4) 3.1 eV

Among the following four aromatic compounds, which one will have the lowest melting point ?

• (1)
• (2)
• (3)
• (4)

\(50\, mL\) of \(0.5\, M\) oxalic acid is needed to neutralize \(25\, mL\) of sodium hydroxide solution. The amount of \(NaOH\) in \(50\, mL\) of the given sodium hydroxide solution is :

• (1) 40 g
• (2) 20 g
• (3) 80 g
• (4) 4 g

The major product of the following reaction is \(-\)

• (1)
• (2)
• (3)
• (4)

In reaction \(A + 2B {\rightleftharpoons} 2C + D\), initial concentration of \(B\) was \(1.5\) times of \([A]\), but at equilibrium the concentrations of \(A\) and \(B\) became equal. The equilibrium constant for the reaction is :

• (1) 8
• (2) 4
• (3) 12
• (4) 6

For diatomic ideal gas in a closed system, which of the following plots does not correctly describe the relation between various thermodynamic quantities ?

• (1)
• (2)
• (3)
• (4)

The correct order for acid strength of compounds :
\(\mathrm{CH\equiv CH}\), \(\mathrm{CH_3{-}C\equiv CH}\) and \(\mathrm{CH_2{=}CH_2}\) is as follows

• (1) \(\mathrm{CH\equiv CH \gt CH_2{=}CH_2 \gt CH_3{-}C\equiv CH}\)
• (2) \(\mathrm{CH_3{-}C\equiv CH \gt CH_2{=}CH_2 \gt HC\equiv CH}\)
• (3) \(\mathrm{HC\equiv CH \gt CH_3{-}C\equiv CH \gt CH_2{=}CH_2}\)
• (4) \(\mathrm{CH_3{-}CH\equiv CH \gt CH\equiv CH \gt CH_2{=}CH_2}\)

The major product of the following reaction is

• (1)
• (2)
• (3)
• (4)

The standard electrode potential \(E^\circ\) and its temperature coefficient \(\left(\frac{dE^\circ}{dT}\right)\) for a cell are 2 V and \(-5\times10^{-4}\ \mathrm{VK^{-1}}\) at 300 K respectively. The cell reaction is
\(\mathrm{Zn(s) + Cu^{2+}(aq) \rightarrow Zn^{2+}(aq) + Cu(s)}\)
The standard reaction enthalpy \((\Delta_rH^\circ)\) at 300 K in \(\mathrm{kJ\ mol^{-1}}\) is, [Use \(R = 8\ \mathrm{JK^{-1}mol^{-1}}\) and \(F = 96{,}000\ \mathrm{C\ mol^{-1}}\)]

• (1) \(-412.8\)
• (2) \(-384.0\)
• (3) \(192.0\)
• (4) \(206.4\)

The metal d-orbitals that are directly facing the ligands in \( {K_3[Co(CN)_6]}\) are :

• (1) \(d_{xz}, d_{yz}\) and \(d_{z^2}\)
• (2) \(d_{xy}, d_{xz}\) and \(d_{yz}\)
• (3) \(d_{xy}\) and \(d_{x^2-y^2}\)
• (4) \(d_{x^2-y^2}\) and \(d_{z^2}\)

The volume of gas A is twice than that of gas B. The compressibility factor of gas A is thrice than that of gas B at same temperature. The pressures of the gases for equal number of moles are -

• (1) \(2P_A = 3P_B\)
• (2) \(3P_A = 2P_B\)
• (3) \(P_A = 3P_B\)
• (4) \(P_A = 2P_B\)

• (1) He
• (2) CH₄
• (3) CO₂
• (4) NH₃

In the Hall-Heroult process, aluminium is formed at the cathode. The cathode is made out of :

• (1) Platinum
• (2) Carbon
• (3) Pure aluminium
• (4) Copper

Poly-\(\beta\)-hydroxybutyrate-co-\(\beta\) hydroxyvalerate (PHBV) is a copolymer of___.

• (1) 3-hydroxybutanoic acid and 4-hydroxypentanoic acid
• (2) 2-hydroxybutanoic acid and 3-hydroxypentanoic acid
• (3) 3-hydroxybutanoic acid and 2-hydroxypentanoic acid
• (4) 3-hydroxybutanoic acid and 3-hydroxypentanoic acid

Water samples with BOD values of 4 ppm and 18 ppm, respectively are :

• (1) Clean and Highly polluted
• (2) Highly polluted and Clean
• (3) Highly polluted and Highly polluted
• (4) Clean and Clean

A metal on combustion in excess air forms X. X upon hydrolysis with water yields \(\mathrm{H_2O_2}\) and \(\mathrm{O_2}\) along with another product. The metal is :

• (1) Rb
• (2) Mg
• (3) Na
• (4) Li

Iodine reacts with concentrated \({HNO_3}\) to yield \(Y\) along with other products. The oxidation state of iodine in \(Y\), is :

• (1) 5
• (2) 3
• (3) 1
• (4) 7

The molecule that has minimum /no role in the formation of photochemical smog, is :

• (1) \(\mathrm{CH_2{=}O}\)
• (2) \(\mathrm{O_3}\)
• (3) \(\mathrm{N_2}\)
• (4) \(\mathrm{NO}\)