JEE Main 2019 B.E./B.Tech Question Paper with Answer Keys (April 9, 2019- Forenoon Session)

Let \(f(x)=15-|x-10|\); \(x\in R\). Then the set of all values of \(x\), at which the function, \(g(x)=f(f(x))\) is not differentiable, is :

• (1) \(\{10,15\}\)
• (2) \(\{5,10,15,20\}\)
• (3) \(\{10\}\)
• (4) \(\{5,10,15\}\)

Let \(\vec{\alpha}=3\hat{i}+\hat{j}\) and \(\vec{\beta}=2\hat{i}-\hat{j}+3\hat{k}\). If \(\vec{\beta}=\vec{\beta}_1-\vec{\beta}_2\), where \(\vec{\beta}_1\) is parallel to \(\vec{\alpha}\) and \(\vec{\beta}_2\) is perpendicular to \(\vec{\alpha}\), then \(\vec{\beta}_1\times\vec{\beta}_2\) is equal to

• (1) \(3\hat{i}-9\hat{j}-5\hat{k}\)
• (2) \(\frac{1}{2}\left(-3\hat{i}+9\hat{j}+5\hat{k}\right)\)
• (3) \(-3\hat{i}+9\hat{j}+5\hat{k}\)
• (4) \(\frac{1}{2}\left(3\hat{i}-9\hat{j}+5\hat{k}\right)\)

If one end of a focal chord of the parabola, \(y^2 = 16x\) is at \((1, 4),\) then the length of this focal chord is

• (1) 25
• (2) 24
• (3) 20
• (4) 22

The value of cos \(^2 10^{\circ} - cos10^{\circ} cos50^{\circ} + cos^2 50^{\circ}\)

• (1) \(\frac{3}{2}(1+cos 20^{\circ})\)
• (2) \(\frac{3}{4}\)
• (3) \(\frac{3}{4} + cos 20^{\circ}\)
• (4) \(\frac{3}{2}\)

If the fourth term in the binomial expansion of \(\left(\frac{2}{x}+x^{\log_8 x}\right)^6\) (\(x\gt0\)) is \(20\times8^7\), then a value of \(x\) is :

• (1) \(8^{-2}\)
• (2) \(8^{2}\)
• (3) \(8^{3}\)
• (4) \(8\)

All the points in the set \(S=\left\{\frac{\alpha+i}{\alpha-i}:\alpha\in R\right\}\) (\(i=\sqrt{-1}\)) lie on a :

• (1) straight line whose slope is \(-1\)
• (2) straight line whose slope is 1.
• (3) circle whose radius is 1.
• (4) circle whose radius is \(\sqrt2\).

The integral \(\int sec^{2/3} x cosec^{4/3}x\, dx\) is equal to (Hence \(C\) is a constant of integration)

• (1) \(3 tan^{-1/3}x + C\)
• (2) \(-\frac{3}{4} tan^{-4/3} x +C\)
• (3) \(-3 cot^{-1/3}x +c\)
• (4) \(-3 tan^{-1/3}x +C\)

If \(\begin {bmatrix} 1 & 1\)
\(0 & 1 \end {bmatrix}\) . \(\begin {bmatrix} 1 & 2\)
\(0 & 1 \end {bmatrix}\) . \(\begin {bmatrix} 1 & 3\)
\(0 & 1 \end {bmatrix}\) ..... \(\begin {bmatrix} 1 & n-1\)
\(1 & 1 \end {bmatrix}\) = \(\begin {bmatrix} 1 & 78\)
\(0 & 1 \end {bmatrix}\), then the inverse of \(\begin {bmatrix} 1 & n\)
\(0 & 1 \end {bmatrix}\) is

• (1) \(\begin {bmatrix} 1 & -13\)
  \(0 & 1 \end {bmatrix}\)
• (2) \(\begin {bmatrix} 1 & 0\)
  \(12 & 1 \end {bmatrix}\)
• (3) \(\begin {bmatrix} 1 & -12\)
  \(0 & 1 \end {bmatrix}\)
• (4) \(\begin {bmatrix} 1 & 0\)
  \(13 & 1 \end {bmatrix}\)

Slope of a line passing through P(2, 3) and intersecting the line, x + y = 7 at a distance of 4 units from P, is

• (1) \(\frac{\sqrt{5}-1}{\sqrt{5}+1}\)
• (2) \(\frac{1- \sqrt{5}}{1+ \sqrt{5}}\)
• (3) \(\frac{1- \sqrt{7}}{1+ \sqrt{7}}\)
• (4) \(\frac{\sqrt{7}-1}{\sqrt{7}+1}\)

The solution of the differential equation \(x\frac{dy}{dx}+2y=x^2\) (\(x\ne0\)) with \(y(1)=1\), is :

• (1) \(y=\frac45x^3+\frac{1}{5x^2}\)
• (2) \(y=\frac34x^2+\frac{1}{4x^2}\)
• (3) \(y=\frac{x^2}{4}+\frac{3}{4x^2}\)
• (4) \(y=\frac{x^3}{5}+\frac{1}{5x^2}\)

Let \(\sum_{k=1}^{10}f(a+k)=16\left(2^{10}-1\right)\) where the function \(f\) satisfies \(f(x+y)=f(x)f(y)\) for all natural numbers \(x,y\) and \(f(1)=2\). then the natural number '\(a\)' is

• (1) 2
• (2) 16
• (3) 4
• (4) 3

Let \(\alpha\) and \(\beta\) be the roots of the equation \(x^2+x+1=0\). Then for \(y\ne0\) in \(R\), \(\begin{vmatrix}y+1&\alpha&\beta\\ \alpha&y+\beta&1\\ \beta&1&y+\alpha\end{vmatrix}\) is equal to

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

If the standard deviation of the numbers \(-1,0,1,k\) is \(\sqrt5\) where \(k\gt0\), then \(k\) is equal to

• (1) \(2\sqrt6\)
• (2) \(\sqrt6\)
• (3) \(2\sqrt{\frac56}\)
• (4) \(2\sqrt{\frac{10}{3}}\)

If \(f(x)\) is a non-zero polynomial of degree four, having local extreme points at \(x=-1,0,1\); then the set \(S=\{x\in R:f(x)=f(0)\}\) Contains exactly :

• (1) four rational numbers.
• (2) four irrational numbers.
• (3) two irrational and one rational number.
• (4) two irrational and two rational numbers.

The area (in sq. units) of the region \(A=\{(x,y):x^2\le y\le x+2\}\) is

• (1) \(\frac{31}{6}\)
• (2) \(\frac{10}{3}\)
• (3) \(\frac{13}{6}\)
• (4) \(\frac92\)

Let \(p,q\in R\). If \(2-\sqrt3\) is a root of the quadratic equation, \(x^2+px+q=0\), then :

• (1) \(p^2-4q-12=0\)
• (2) \(q^2-4p-16=0\)
• (3) \(q^2+4p+14=0\)
• (4) \(p^2-4q+12=0\)

If the function \(f:R-\{1,-1\}\to A\) defined by \(f(x)=\frac{x^2}{1-x^2}\), is surjective, then \(A\) is equal to

• (1) \(R-(-1,0)\)
• (2) \(R-\{-1\}\)
• (3) \(R-[-1,0)\)
• (4) \([0,\infty)\)

If the function \(f\) defined on \(\left(\frac{\pi}{6},\frac{\pi}{3}\right)\) by \(f(x)=\begin{cases}\dfrac{\sqrt2\cos x-1}{\cot x-1},&x\ne\dfrac{\pi}{4}\\ k,&x=\dfrac{\pi}{4}\end{cases}\) is continuous, then \(k\) is equal to

• (1) 1
• (2) \(1/\sqrt2\)
• (3) \(\frac12\)
• (4) 2

Let the sum of the first \(n\) terms of a non-constant A.P., \(a_1,a_2,a_3,\dots\) be \(50n+\frac{n(n-7)}{2}A\), where \(A\) is a constant. If \(d\) is the common difference of this A.P., then the ordered pair \((d,a_{50})\) is equal to

• (1) \((A,\,50+45A)\)
• (2) \((50,\,50+45A)\)
• (3) \((A,\,50+46A)\)
• (4) \((50,\,50+46A)\)

A committee of 4 persons is to be formed from 2 ladies, 2 old men and 4 young men such that it includes at least 1 lady, at least 1 old man and at most 2 young men. Then the total number of ways in which this committee can be formed is :

• (1) 40
• (2) 41
• (3) 16
• (4) 32

The value of \(\int_0^{\pi/2}\frac{\sin^3x}{\sin x+\cos x}\,dx\) is

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

A, B, C try to hit a target simultaneously but independently. Their respective probabilities of hitting the targets are \(\frac{3}{4},\frac{1}{2},\frac{5}{8}\). The probability that the target is hit by A or B but not by C is :

• (1) 21/64
• (2) 7/8
• (3) 7/32
• (4) 9/64

If the line \(y = mx + 7\sqrt{3}\) is normal to the hyperbola \(\frac{x^2}{24} - \frac{y^2}{18} = 1\), then a value of \(m\) is :

• (1) \(\frac{3}{\sqrt{5}}\)
• (2) \(\frac{\sqrt{15}}{2}\)
• (3) \(\frac{\sqrt{5}}{2}\)
• (4) \(\frac{2}{\sqrt{5}}\)

If the line, \(\frac{x-1}{2} = \frac{y+1}{3} = \frac{z-2}{4}\) meets the plane, \(x + 2y + 3z = 15\) at a point P, then the distance of P from the origin is :

• (1) \(\frac{\sqrt{5}}{2}\)
• (2) \(2\sqrt{5}\)
• (3) \(\frac{9}{2}\)
• (4) \(\frac{7}{2}\)

If the tangent to the curve, \(y = x^3 + ax - b\) at the point \((1, -5)\) is perpendicular to the line, \(-x + y + 4 = 0\), then which one of the following points lies on the curve ?

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

Let S be the set of all values of x for which the tangent to the curve \(y = f(x) = x^3 - x^2 - 2x\) at \((x, y)\) is parallel to the line segment joining the points \((1, f(1))\) and \((-1, f(-1))\), then S is equal to :

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

If the tangents drawn to the hyperbola \(4y^2 = x^2 + 1\) intersect the co-ordinate axes at the distinct points \(A\) and \(B\) , then the locus of the mid point of \(AB\) is :

• (1) \(x^2 - 4y^2 + 16x^2y^2 = 0\)
• (2) \(x^2 - 4y^2 - 16x^2y^2 = 0\)
• (3) \(4x^2 - y^2 + 16x^2y^2 = 0 \)
• (4) \(4x^2 - y^2 - 16x^2y^2 = 0\)

Let \(S = \{\theta \in [-2\pi, 2\pi] : 2\cos^2\theta + 3\sin\theta = 0\}\). Then the sum of the elements of S is

• (1) \(\pi\)
• (2) \(2\pi\)
• (3) \(\frac{13\pi}{6}\)
• (4) \(\frac{5\pi}{3}\)

For any two statements $p$ and $q$, the negation of the expression $p\vee (\sim p\wedge q)$ is:

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

A plane passing through the points \((0, -1, 0)\) and \((0, 0, 1)\) and making an angle \(\frac{\pi}{4}\) with the plane \(y - z + 5 = 0\), also passes through the point

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

For a given gas at 1 atm pressure, rms speed of the molecule is 200 m/s at \(127^\circ\mathrm{C}\). At 2 atm pressure and at \(227^\circ\mathrm{C}\), the rms speed of the molecules will be :

• (1) \(100\) m/s
• (2) \(100\sqrt{5}\) m/s
• (3) \(80\sqrt{5}\) m/s
• (4) \(80\) m/s

Taking the wavelength of first Balmer line in hydrogen spectrum (n = 3 to n = 2) as 660 nm, the wavelength of the 2nd Balmer line (n = 4 to n = 2) will be :

• (1) \(642.7\) nm
• (2) \(488.9\) nm
• (3) \(889.2\) nm
• (4) \(388.9\) nm

A uniform cable of mass \('M'\) and length \('L'\) is placed on a horizontal surface such that its \(\bigg(\frac{1}{n}\bigg)^{th}\) part is hanging below the edge of the surface. To lift the hanging part of the cable upto the surface, the work done should be :

• (1) \(\frac{MgL}{n^2}\)
• (2) \(\frac{MgL}{2n^2}\)
• (3) \(\frac{2MgL}{n^2}\)
• (4) \(nMgL\)

A string is clamped at both the ends and it is vibrating in its 4th harmonic. The equation of the stationary wave is \(Y = 0.3\sin(0.157x)\cos(200\pi t)\). The length of the string is : (All quantities are in SI units.)

• (1) \(60\) m
• (2) \(20\) m
• (3) \(80\) m
• (4) \(40\) m

Following figure shows two processes A and B for a gas. If \(\Delta Q_A\) and \(\Delta Q_B\) are the amount of heat absorbed by the system in two cases, and \(\Delta U_A\) and \(\Delta U_B\) are changes in internal energies, respectively, then :

• (1) \(\Delta Q_A > \Delta Q_B\) ; \(\Delta U_A > \Delta U_B\)
• (2) \(\Delta Q_A \lt \Delta Q_B\) ; \(\Delta U_A \lt \Delta U_B\)
• (3) \(\Delta Q_A > \Delta Q_B\) ; \(\Delta U_A = \Delta U_B\)
• (4) \(\Delta Q_A = \Delta Q_B\) ; \(\Delta U_A = \Delta U_B\)

The pressure wave, \(P = 0.01\sin[1000t - 3x]\ \mathrm{Nm^{-2}}\), corresponds to the sound produced by a vibrating blade on a day when atmospheric temperature is \(0^\circ\mathrm{C}\). On some other day, when temperature is T, the speed of sound produced by the same blade and at the same frequency is found to be \(336\ \mathrm{ms^{-1}}\). Approximate value of T is

• (1) \(12^\circ\mathrm{C}\)
• (2) \(15^\circ\mathrm{C}\)
• (3) \(4^\circ\mathrm{C}\)
• (4) \(11^\circ\mathrm{C}\)

A rigid square loop of side 'a' and carrying current I\(_2\) is lying on a horizontal surface near a long current I\(_1\) carrying wire in the same plane as shown in figure. The net force on the loop due to wire will be :

• (1) Attractive and equal to \(\frac{\mu_0 I_1 I_2}{3 \pi}\)
• (2) Repulsive and equal to \(\frac{\mu_0 I_1 I_2}{4 \pi}\)
• (3) Repulsive and equal to \(\frac{\mu_0 I_1 I_2}{2 \pi}\)
• (4) Zero

A concave mirror for face viewing has focal length of 0.4 m. The distance at which you hold the mirror from your face in order to see your image upright with a magnification of 5 is :

• (1) \(0.24\) m
• (2) \(0.32\) m
• (3) \(1.60\) m
• (4) \(0.16\) m

A rectangular coil (Dimension \(5\ \mathrm{cm} \times 2.5\ \mathrm{cm}\)) with 100 turns, carrying a current of 3 A in the clock-wise direction is kept centered at the origin and in the X-Z plane. A magnetic field of 1 T is applied along X-axis. If the coil is tilted through \(45^\circ\) about Z-axis, then the torque on the coil is :

• (1) \(0.42\) Nm
• (2) \(0.55\) Nm
• (3) \(0.38\) Nm
• (4) \(0.27\) Nm

A solid sphere of mass \('M'\) and radius \('a'\) is surrounded by a uniform concentric spherical shell of thickness \(2a\) and mass \(2M\). The gravitational field at distance \('3a'\) from the centre will be :

• (1) \(\frac{2GM}{9a^2}\)
• (2) \(\frac{GM}{3a^2}\)
• (3) \(\frac{GM}{9a^2}\)
• (4) \(\frac{2GM}{3a^2}\)

A system of three charges are placed as shown in the figure :

If \(D \gg d\), the potential energy of the system is best given by :

• (1) \(\frac{1}{4\pi\varepsilon_0}\left[\frac{q^2}{d} + \frac{qQd}{D^2}\right]\)
• (2) \(\frac{1}{4\pi\varepsilon_0}\left[-\frac{q^2}{d} - \frac{qQd}{2D^2}\right]\)
• (3) \(\frac{1}{4\pi\varepsilon_0}\left[-\frac{q^2}{d} - \frac{qQd}{D^2}\right]\)
• (4) \(\frac{1}{4\pi\varepsilon_0}\left[-\frac{q^2}{d} + \frac{2qQd}{D^2}\right]\)

If \('M'\) is the mass of water that rises in a capillary tube of radius \('r'\), then mass of water which will rise in a capillary tube of radius \('2r'\) is :

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

A moving coil galvanometer has resistance \(50\ \Omega\) and it indicates full deflection at 4 mA current. A voltmeter is made using this galvanometer and a \(5\ \mathrm{k\Omega}\) resistance. The maximum voltage, that can be measured using this voltmeter, will be close to :

• (1) \(15\) V
• (2) \(10\) V
• (3) \(40\) V
• (4) \(20\) V

A wire of resistance \(R\) is bent to form a square ABCD as shown in the figure. The effective resistance between E and C is : (E is mid-point of arm CD)

• (1) \(\frac{7}{64}R\)
• (2) \(\frac{1}{16}R\)
• (3) \(R\)
• (4) \(\frac{3}{4}R\)

If the rms velocity of hydrogen gas molecules is \(V_o\), find the rms velocity of oxygen molecules at same temperature :

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

The following bodies are made to roll up (without slipping) the same inclined plane from a horizontal plane. : (i) a ring of radius \(R\), (ii) a solid cylinder of radius \(R/2\) and (iii) a solid sphere of radius \(R/4\). If in each case, the speed of the centre of mass at the bottom of the incline is same, the ratio of the maximum heights they climb is :

• (1) 20 : 15 : 14
• (2) 4 : 3 : 2
• (3) 2 : 3 : 4
• (4) 10 : 15 : 7

In the density measurement of a cube, the mass and edge length are measured as \((10.00\pm0.10)\) kg and \((0.10\pm0.01)\) m, respectively. The error in the measurement of density is :

• (1) 0.01 kg/m³
• (2) 0.10 kg/m³
• (3) 0.31 kg/m³
• (4) 0.07 kg/m³

The figure shows a Young's double slit experimental setup. It is observed that when a thin transparent sheet of thickness \(t\) and refractive index \(\mu\) is put in front of one of the slits, the central maximum gest shifted by a distance equal to \(n\) fringe widths. If the wavelength of light used is \(\lambda\), \(t\) will be :

• (1) \(\frac{D\lambda}{a(\mu-1)}\)
• (2) \(\frac{2nD\lambda}{a(\mu-1)}\)
• (3) \(\frac{2D\lambda}{a(\mu-1)}\)
• (4) \(\frac{n\lambda}{(\mu-1)}\)

The electric field of light wave is given as
\(\vec{E}=10^{-3}\cos\left(\frac{2\pi x}{5\times10^{-7}}-2\pi\times6\times10^{14}t\right)\hat{x}\ \frac{\mathrm{N}}{C}\)
This light falls on a metal plate of work function 2 eV. The stopping potential of the photoelectrons is :
Given, \(E\) (in eV) \(=12375/\lambda\) (in \(\mathrm{\AA}\))

• (1) 2.48 V
• (2) 0.48 V
• (3) 0.72 V
• (4) 2.0 V

A stationary horizontal disc is free to rotate about its axis. When a torque is applied on it, its kinetic energy as a function of \(\theta\), where \(\theta\) is the angle by which it has rotated, is given as \(k\theta^2\). If its moment of inertia is \(I\) then the angular acceleration of the disc is :

• (1) \(\frac{k}{4I}\theta\)
• (2) \(\frac{k}{I}\theta\)
• (3) \(\frac{k}{2I}\theta\)
• (4) \(\frac{2k}{I}\theta\)

A ball is thrown vertically upwards with a velocity of \(19.6 ms^{–1}\) from the top of a tower. The ball strikes the ground after \(6 s\). The height from the ground up to which the ball can rise will be \((k/5) m\). The value of k is ______ (use \(g = 9.8 m/s^2\))

A simple pendulum oscillating in air has period \(T\). The bob of the pendulum is completely immersed in a non-viscous liquid. The density of the liquid is 1/16 th of the material of the bob. If the bob is inside liquid all the time, its period of oscillation in this liquid is :

• (1) \(2T\sqrt{\frac{1}{10}}\)
• (2) \(4T\sqrt{\frac{1}{14}}\)
• (3) \(4T\sqrt{\frac{1}{15}}\)
• (4) \(2T\sqrt{\frac{1}{14}}\)

The magnetic field of a plane electromagnetic wave is given by \(\vec{B} = B_0 \hat{i} [cos (kz - \omega t)] + B_1 \hat{j} cps (kz + \omega t).\) where B\(_0 = 3 � 10^{-5} T\) and B\(_1 = 2 � 10^{-6}\) T. The rms value of the force experienced by a stationary charge Q = 10 C at z = 0 is closest to :

• (1) \(0.9 \,N\)
• (2) \(0.1 \,N\)
• (3) \(3 \times 10^-2\, N\)
• (4) \(0.6\, N\)

A capacitor with capacitance \(5\mu\mathrm{F}\) is charged to \(5\mu\mathrm{C}\). If the plates are pulled apart to reduce the capacitance to \(2\mu\mathrm{F}\), how much work is done ?

• (1) \(2.16\times10^{-6}\) J
• (2) \(2.55\times10^{-6}\) J
• (3) \(3.75\times10^{-6}\) J
• (4) \(6.25\times10^{-6}\) J

A body of mass 2 kg moving with a speed of 4 m/s makes an elastic collision with another body at rest and continues to move in the original direction but with one fourth of its initial speed. The speed of the two body centre of mass is \(\frac{x}{10}\) m/s. Then the value of \(x\) is ________.

Determine the charge on the capacitor in the following circuit :

• (1) 2?C
• (2) 60?C
• (3) 200?C
• (4) 10?C

The stream of a river is flowing with a speed of 2 km/h. A swimmer can swim at a speed of 4 km/h. What should be the direction of the swimmer with respect to the flow of the river to cross the river straight ?

• (1) \(150^\circ\)
• (2) \(120^\circ\)
• (3) \(60^\circ\)
• (4) \(90^\circ\)

The total number of turns and cross-section area in a solenoid is fixed. However, its length \(L\) is varied by adjusting the separation between windings. The inductance of solenoid will be proportional to :

• (1) \(1/L^2\)
• (2) \(1/L\)
• (3) \(L\)
• (4) \(L^2\)

A signal \(A\cos\omega t\) is transmitted using \(v_0\sin\omega_0 t\) as carrier wave. The correct amplitude modulated (AM) signal is

• (1) \(v_0\sin\omega_0t+A\cos\omega t\)
• (2) \((v_0+A)\cos\omega t\,\sin\omega_0t\)
• (3) \(v_0\sin[\omega_0(1+0.01A\sin\omega t)t]\)
• (4) \(v_0\sin\omega_0t+\frac{A}{2}\sin(\omega_0-\omega)t+\frac{A}{2}\sin(\omega_0+\omega)t\)

An NPN transistor is used in common emitter configuration as an amplifier with \(1\ \mathrm{k}\Omega\) load resistance. Signal voltage of 10 mV is applied across the base-emitter. This produces a 3 mA change in the collector current and \(15\mu\mathrm{A}\) change in the base current of the amplifier. The input resistance and voltage gain are :

• (1) 0.67 k\(\Omega\), 200
• (2) 0.33 k\(\Omega\), 1.5
• (3) 0.67 k\(\Omega\), 300
• (4) 0.33 k\(\Omega\), 300

The major product of the following reaction is :

• (1)
• (2)
• (3)
• (4)

The element having the highest first ionization enthalpy is

• (1) Si
• (2) Al
• (3) N
• (4) C

Liquid 'M' and liquid 'N' form an ideal solution. The vapour pressures of pure liquids 'M' and 'N' are 450 and 700 mmHg, respectively, at the same temperature. Then correct statement is:
(\(x_M\) = Mole fraction of 'M' in solution ;
\(x_N\) = Mole fraction of 'N' in solution ;
\(y_M\) = Mole fraction of 'M' in vapour phase ;
\(y_N\) = Mole fraction of 'N' in vapour phase)

• (1) \(\frac{x_M}{x_N}\lt\frac{y_N}{y_N}\)
• (2) \((x_M-y_M)\lt(x_N-y_N)\)
• (3) \(\frac{x_M}{x_N}=\frac{y_N}{y_N}\)
• (4) \(\frac{x_M}{x_N}\gt\frac{y_M}{y_N}\)

The degenerate orbitals of \([\mathrm{Cr(H_2O)_6}]^{3+}\) are :

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

The major product of the following reaction is :
\(\mathrm{CH_3CH=CHCO_2CH_3}\ \xrightarrow{\ \mathrm{LiAlH_4}\ }\ ?\)

• (1) \(\mathrm{CH_3CH_2CH_2CHO}\)
• (2) \(\mathrm{CH_3CH_2CH_2CO_2CH_3}\)
• (3) \(\mathrm{CH_3CH=CHCH_2OH}\)
• (4) \(\mathrm{CH_3CH_2CH_2CH_2OH}\)

The given plots represent the variation of the concentration of a reactant R with time for two different reactions (i) and (ii). The respective orders of the reactions are :

• (1) 0, 1
• (2) 1, 0
• (3) 0, 2
• (4) 1, 1

• (1)
• (2)
• (3)
• (4)

Among the following, the molecule expected to be stabilized by anion formation is :
\(\mathrm{C_2,\ O_2,\ NO,\ F_2}\)

• (1) \(\mathrm{C_2}\)
• (2) \(\mathrm{NO}\)
• (3) \(\mathrm{O_2}\)
• (4) \(\mathrm{F_2}\)

Which of the following statements is not true about sucrose?

• (1) The glycosidic linkage is present between \(\mathrm{C_1}\) of \(\alpha\)-glucose and \(\mathrm{C_1}\) of \(\beta\)-fructose
• (2) It is also named as invert sugar
• (3) It is a non reducing sugar
• (4) On hydrolysis, it produces glucose and fructose

The correct IUPAC name of the following compound is:

• (1) 4-methyl-2-nitro-5-oxohept-3-enal
• (2) 4-methyl-5-oxo-2-nitrohept-3-enal
• (3) 4-methyl-6-nitro-3-oxohept-4-enal
• (4) 6-formyl-4-methyl-2-nitrohex-3-enal

The correct order of the oxidation states of nitrogen in \(NO, N_2O, NO_2\) and \(N_2 O_3\) is :

• (1) \(NO_2 < N_2O_3 < NO < N_2O\)
• (2) \(NO_2 < NO < N_2O_3 < N_2O\)
• (3) \(N_2O < N_2 O_3 < NO < NO_2\)
• (4) \(N_2O < NO < N_2 O_3 < NO_2\)

The major product of the following reaction is :

• (1)
• (2)
• (3)
• (4)

The major product of the following reaction is :

• (1)
• (2)
• (3)
• (4)

The one that will show optical activity is :
(en = ethane-1,2-diamine)

• (1)
• (2)
• (3)
• (4)

For a reaction, \(N_2(g) + 3H_2(g) \rightarrow 2NH_3(g)\); identify dihydrogen \((H_2)\) as a limiting reagent in the following reaction mixtures.

• (1) \(14g\) of \(N_2\) + \(4g\) of \(H_2\)
• (2) \(28g\) of \(N_2\) + \(6g\) of \(H_2\)
• (3) \(56g\) of \(N_2\) + \(10g\) of \(H_2\)
• (4) \(35g\) of \(N_2\) + \(8g\) of \(H_2\)

The standard Gibbs energy for the given cell reaction in kJ mol^-1 at 298 K is :
\(\mathrm{Zn(s) + Cu^{2+}(aq) \rightarrow Zn^{2+}(aq) + Cu(s)}\),
\(E^\circ = 2\ \mathrm{V}\) at 298 K
(Faraday's constant, \(F = 96000\ \mathrm{C\ mol^{-1}}\))

• (1) 384
• (2) \(-192\)
• (3) \(-384\)
• (4) 192

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

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

Which among the following represents reagent ‘A’?

• (1)
• (2)
• (3)
• (4)

For any given series of spectral lines of atomic hydrogen, let \(\Delta\bar{\nu} = \Delta\bar{\nu}_{max} - \Delta\bar{\nu}_{min}\) be the difference in maximum and minimum frequencies in \(\mathrm{cm^{-1}}\). The ratio \(\dfrac{\Delta\bar{\nu}_{Lyman}}{\Delta\bar{\nu}_{Balmer}}\) is :

• (1) 9 : 4
• (2) 4 : 1
• (3) 27 : 5
• (4) 5 : 4

Aniline dissolved in dilute HCl is reacted with sodium nitrite at \(0^\circ\mathrm{C}\). This solution was added dropwise to a solution containing equimolar mixture of aniline and phenol in dil. HCl. The structure of the major product is

• (1)
• (2)
• (3)
• (4)

The major product of the following reaction is :
\(\mathrm{CH_3-C\equiv CH}\ \xrightarrow[\mathrm{(ii)\ DI}]{\mathrm{(i)\ DCl\ (1\ equiv.)}}\)

• (1) \(\mathrm{CH_3CD_2CH(Cl)(I)}\)
• (2) \(\mathrm{CH_3CD(Cl)CHD(I)}\)
• (3) \(\mathrm{CH_3C(I)(Cl)CHD_2}\)
• (4) \(\mathrm{CH_3CD(I)CHD(Cl)}\)

The osmotic pressure of a dilute solution of an ionic compound XY in water is four times that of a solution of 0.01 M \(\mathrm{BaCl_2}\) in water. Assuming complete dissociation of the given ionic compounds in water, the concentration of XY (in mol \(\mathrm{L^{-1}}\)) in solution is :

• (1) \(4\times10^{-4}\)
• (2) \(6\times10^{-2}\)
• (3) \(4\times10^{-2}\)
• (4) \(16\times10^{-4}\)

Excessive release of \(\mathrm{CO_2}\) into the atmosphere results in :

• (1) depletion of ozone
• (2) polar vortex
• (3) global warming
• (4) formation of smog

Magnesium powder burns in air to give :

• (1) \(\mathrm{MgO}\) and \(\mathrm{Mg_3N_2}\)
• (2) \(\mathrm{MgO}\) only
• (3) \(\mathrm{Mg(NO_3)_2}\) and \(\mathrm{Mg_3N_2}\)
• (4) \(\mathrm{MgO}\) and \(\mathrm{Mg(NO_3)_2}\)

The number of water molecule(s) not coordinated to copper ion directly in \(CuSO_4.5H_2O\), is :

• (1) 4
• (2) 3
• (3) 1
• (4) 2

The aerosol is a kind of colloid in which :

• (1) gas is dispersed in solid
• (2) solid is dispersed in gas
• (3) liquid is dispersed in water
• (4) gas is dispersed in liquid

The ore that contains the metal in the form of fluoride is :

• (1) sphalerite
• (2) magnetite
• (3) malachite
• (4) cryolite

\(C_{60}\), an allotrope of carbon contains :

• (1) 20 hexagons and 12 pentagons
• (2) 12 hexagons and 20 pentagons
• (3) 18 hexagons and 14 pentagons
• (4) 16 hexagons and 16 pentagons

• (1) (A)-(iii); (B)-(i); (C)-(ii); (D)-(iv)
• (2) (A)-(ii); (B)-(iii); (C)-(i); (D)-(iv)
• (3) (A)-(iii); (B)-(iv); (C)-(i); (D)-(ii)
• (4) (A)-(iv); (B)-(iii); (C)-(ii); (D)-(i)

• (1) Ne
• (2) Kr
• (3) Xe
• (4) Ar