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

Let \(a_1, a_2 , ....... , a_{30}\) be an \(A. P\)., \(S =\displaystyle \sum^{30}_{i=1} a_i \) and \(T = \displaystyle\sum^{15}_{i=1} a_{(2i -1)}\). If \(a_5 = 27\) and \(S - 2T = 75, \) then \(a_{10}\) is equal to :

• (1) 57
• (2) 47
• (3) 42
• (4) 52

If \(a, b, c\) be three distinct real numbers in G.P. and \(a + b + c = xb\), then \(x\) cannot be

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

For any \(\theta \in \left(\frac{\pi}{4}, \frac{\pi}{2}\right)\), the expression
\(3(\cos\theta - \sin\theta)^4 + 6(\sin\theta + \cos\theta)^2 + 4\sin^6\theta\)
equals :

• (1) \(13 - 4\cos^2\theta + 6\sin^2\theta\cos^2\theta\)
• (2) \(13 - 4\cos^6\theta\)
• (3) \(13 - 4\cos^2\theta + 6\cos^2\theta\)
• (4) \(13 - 4\cos^4\theta + 2\sin^2\theta\cos^2\theta\)

Let \(\alpha\) and \(\beta\) be two roots of the equation \(x^2 + 2x + 2 = 0\), then \(\alpha^{15} + \beta^{15}\) is equal to :

• (1) \(-256\)
• (2) \(512\)
• (3) \(-512\)
• (4) \(256\)

For \(x \in R - \{0, 1\}\), Let \(f_1(x) = \frac{1}{x}\), \(f_2(x) = 1 - x\) and \(f_3(x) = \frac{1}{1 - x}\) be three given functions. If a function, \(J(x)\) satisfies \((f_2 \circ J \circ f_1)(x) = f_3(x)\) then \(J(x)\) is equal to :

• (1) \(f_1(x)\)
• (2) \(\frac{1}{x} f_3(x)\)
• (3) \(f_2(x)\)
• (4) \(f_3(x)\)

If the fractional part of the number \(\frac{2^{403}}{15}\) is \(\frac{k}{15}\), then \(k\) is equal to :

• (1) 14
• (2) 6
• (3) 4
• (4) 8

For \(x^2 \ne n\pi + 1,\ n \in N\) (the set of natural numbers), the integral
\(\displaystyle\int x\sqrt{\frac{2\sin(x^2 - 1) - \sin 2(x^2 - 1)}{2\sin(x^2 - 1) + \sin 2(x^2 - 1)}}\,dx\)
is equal to :
(where c is a constant of integration)

• (1) \(\log_e\left|\frac12\sec^2(x^2 - 1)\right| + c\)
• (2) \(\frac12\log_e\left|\sec(x^2 - 1)\right| + c\)
• (3) \(\frac12\log_e\left|\sec^2\left(\frac{x^2 - 1}{2}\right)\right| + c\)
• (4) \(\log_e\left|\sec\left(\frac{x^2 - 1}{2}\right)\right| + c\)

The value of \(\displaystyle\int_0^{\pi}|\cos x|^3\,dx\) is :

• (1) \(\frac{4}{3}\)
• (2) \(-\frac{4}{3}\)
• (3) \(0\)
• (4) \(\frac{2}{3}\)

Consider a class of \(5\) girls and \(7\) boys. The number of different teams consisting of \(2\) girls and \(3\) boys that can be formed from this class, if there are two specific boys \(A\) and \(B\), who refuse to be the members of the same team, is :

• (1) 200
• (2) 300
• (3) 500
• (4) 350

\(\displaystyle\lim_{y \to 0}\frac{\sqrt{1 + \sqrt{1 + y^4}} - \sqrt2}{y^4}\)

• (1) exists and equals \(\frac{1}{2\sqrt2}\)
• (2) exists and equals \(\frac{1}{4\sqrt2}\)
• (3) exists and equals \(\frac{1}{2\sqrt2(1 + \sqrt2)}\)
• (4) does not exists

5 students of a class have an average height 150 cm and variance 18 cm². A new student, whose height is 156 cm, joined them. The variance (in cm²) of the height of these six students is :

• (1) 16
• (2) 22
• (3) 20
• (4) 18

If \(\cos^{-1}\left(\frac{2}{3x}\right) + \cos^{-1}\left(\frac{3}{4x}\right) = \frac\pi2\) \(\left(x \gt \frac34\right)\), then \(x\) is equal to :

• (1) \(\frac{\sqrt{145}}{10}\)
• (2) \(\frac{\sqrt{145}}{11}\)
• (3) \(\frac{\sqrt{145}}{12}\)
• (4) \(\frac{\sqrt{146}}{12}\)

The area of the region bounded by the parabola \((y-2)^2 = (x-1)\), the tangent to it at the point whose ordinate is 3 and the x-axis is :

• (1) 6
  
• (2) 9
  
• (3) 10
  
• (4) 4
  

If \(A = \begin{bmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{bmatrix}\), then the matrix \(A^{-50}\) when \(\theta = \frac{\pi}{12}\), is equal to :

• (1) \(\begin{bmatrix}\frac{\sqrt3}{2} & -\frac12\\ \frac12 & \frac{\sqrt3}{2}\end{bmatrix}\)
• (2) \(\begin{bmatrix}\frac12 & -\frac{\sqrt3}{2}\\ \frac{\sqrt3}{2} & \frac{-1}{2}\end{bmatrix}\)
• (3) \(\begin{bmatrix}\frac{\sqrt3}{2} & \frac12\\ -\frac12 & \frac{\sqrt3}{2}\end{bmatrix}\)
• (4) \(\begin{bmatrix}\frac12 & \frac{\sqrt3}{2}\\ -\frac{\sqrt3}{2} & \frac12\end{bmatrix}\)

Let \(f : R \to R\) be a function defined as
\(f(x) = \begin{cases}5 & ; x \le 1\\ a + bx & ; 1 \lt x \lt 3\\ b + 5x & ; 3 \le x \lt 5\\ 30 & ; x \ge 5\end{cases}\)
Then, f is

• (1) continuous if \(a = 0\) and \(b = 5\)
• (2) continuous if \(a = -5\) and \(b = 10\)
• (3) continuous if \(a = 5\) and \(b = 5\)
• (4) not continuous for any values of \(a\) and \(b\)

The maximum volume (in \(cu. m\)) of the right circular cone having slant height \(3\,m\) is :

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

If \(y = y(x)\) is the solution of the differential equation,
\(x\dfrac{dy}{dx} + 2y = x^2\), satisfying \(y(1) = 1\), then \(y\left(\frac12\right)\) is equal to :

• (1) \(\frac{7}{64}\)
• (2) \(\frac{49}{16}\)
• (3) \(\frac14\)
• (4) \(\frac{13}{16}\)

Let \(A = \left\{\theta \in \left(-\frac\pi2, \pi\right) : \frac{3 + 2i\sin\theta}{1 - 2i\sin\theta}\ \text{is purely imaginary}\right\}\). Then the sum of the elements in A is :

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

Axis of a parabola lies along \(x\)-axis. If its vertex and focus are at distances \(2\) and \(4\) respectively from the origin, on the positive \(x\)-axis then which of the following points does not lie on it ?

• (1) \((4, -4)\)
• (2) \((5 , 2 \sqrt{6})\)
• (3) \((8, 6)\)
• (4) \( 6 , 4 \sqrt{2}\)

If \(\theta\) denotes the acute angle between the curves, \(y = 10 - x^2\) and \(y = 2 + x^2\) at a point of their intersection, the \(|\tan\theta|\) is equal to :

• (1) \(\frac{8}{15}\)
• (2) \(\frac49\)
• (3) \(\frac{7}{17}\)
• (4) \(\frac{8}{17}\)

The system of linear equations
\(x + y + z = 2\)
\(2x + 3y + 2z = 5\)
\(2x + 3y + (a^2 - 1)z = a + 1\)
then

• (1) has infinitely many solutions for \(a = 4\)
• (2) has a unique solution for \(|a| = \sqrt3\)
• (3) is inconsistent when \(|a| = \sqrt3\)
• (4) is inconsistent when \(a = 4\)

Let \(0 \lt \theta \lt \frac{\pi}{2}\). If the eccentricity of the hyperbola \(\frac{x^2}{\cos^2\theta}-\frac{y^2}{\sin^2\theta}=1\) is greater than 2, then the length of its latus rectum lies in the interval :

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

Equation of a common tangent to the circle, \(x^2+y^2-6x=0\) and the parabola, \(y^2=4x\) is :

• (1) \(2\sqrt{3}\,y=12x+1\)
• (2) \(\sqrt{3}\,y=x+3\)
• (3) \(2\sqrt{3}\,y=-x-12\)
• (4) \(\sqrt{3}\,y=3x+1\)

The plane through the intersection of the planes \(x + y + z = 1\) and \(2x + 3y - z + 4 = 0\) and parallel to y-axis also passes through the point :

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

Three circles of radii a, b, c \((a \lt b \lt c)\) touch each other externally. If they have x-axis as a common tangent, then :

• (1) \(a, b, c\) are in A.P.
• (2) \(\sqrt{a},\sqrt{b},\sqrt{c}\) are in A.P
• (3) \(\frac{1}{\sqrt{b}}+\frac{1}{\sqrt{c}}=\frac{1}{\sqrt{a}}\)
• (4) \(\frac{1}{\sqrt{b}}=\frac{1}{\sqrt{a}}+\frac{1}{\sqrt{c}}\)

The equation of the line passing through \((-4, 3, 1)\), parallel to the plane \(x+2y-z-5=0\) and intersecting the line \(\frac{x+1}{-3}=\frac{y-3}{2}=\frac{z-2}{-1}\) is :

• (1) \(\frac{x+4}{3}=\frac{y-3}{-1}=\frac{z-1}{1}\)
• (2) \(\frac{x+4}{1}=\frac{y-3}{1}=\frac{z-1}{3}\)
• (3) \(\frac{x+4}{-1}=\frac{y-3}{1}=\frac{z-1}{1}\)
• (4) \(\frac{x-4}{2}=\frac{y+3}{1}=\frac{z+1}{4}\)

Let \(\vec{a}=\hat{i}-\hat{j}\), \(\vec{b}=\hat{i}+\hat{j}+\hat{k}\) and \(\vec{c}\) be a vector such that \(\vec{a}\times\vec{c}+\vec{b}=\vec{0}\) and \(\vec{a}\cdot\vec{c}=4\), then \(|\vec{c}|^2\) is equal to :

• (1) 8
• (2) \(\frac{19}{2}\)
• (3) 9
• (4) \(\frac{17}{2}\)

If the Boolean expression \((p\oplus q)\wedge(\sim p\odot q)\) is equivalent to \(p\wedge q\), where \(\oplus,\odot\in\{\wedge,\vee\}\), then the ordered pair \((\oplus,\odot)\) is :

• (1) \((\vee,\wedge)\)
• (2) \((\vee,\vee)\)
• (3) \((\wedge,\vee)\)
• (4) \((\wedge,\wedge)\)

Consider the set of all lines \(px+qy+r=0\) such that \(3p+2q+4r=0\). Which one of the following statements is true?

• (1) The lines are not concurrent
• (2) The lines are concurrent at the point \(\left(\frac{3}{4},\frac{1}{2}\right)\)
• (3) The lines are all parallel
• (4) Each line passes through the origin

Two cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Let X denote the random variable of number of aces obtained in the two drawn cards. Then P(X = 1) + P(X = 2) equals :

• (1) 52/169
• (2) 25/169
• (3) 49/169
• (4) 24/169

Consider a tank made of glass(refractive index 1.5) with a thick bottom. It is filled with a liquid of refractive index \(\mu\). A student finds that, irrespective of what the incident angle i (see figure) is for a beam of light entering the liquid, the light reflected from the liquid glass interface is never completely polarized. For this to happen, the minimum value of \(\mu\) is :

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

A bar magnet is demagnetized by inserting it inside a solenoid of length 0.2 m, 100 turns, and carrying a current of 5.2 A. The corecivity of the bar magnet is :

• (1) 285 A/m
• (2) 2600 A/m
• (3) 520 A/m
• (4) 1200 A/m

A rod, of length \(L\) at room temperature and uniform area of cross section \(A\), is made of a metal having coefficient of linear expansion \(\alpha / {^{\circ}C}\). It is observed that an external compressive force \(F\), is applied on each of its ends, prevents any change in the length of the rod, when its temperature rises by \(\Delta T K\). Young's modulus, \(Y\), for this metal is :

• (1) \(\frac{F}{2A\alpha\Delta T} \)
• (2) \(\frac{F}{A\alpha \left(\Delta T - 273\right)} \)
• (3) \(\frac{F}{A\alpha \Delta T}\)
• (4) \(\frac{2F}{A\alpha \Delta T}\)

A copper wire is stretched to make it \(0.5 \%\) longer. The percentage change in its electrical resistance if its volume remains unchanged is:

• (1) 2.50%
• (2) 0.50%
• (3) 1.00%
• (4) 2.00%

A gas can be taken from \(A\) to \(B\) via two different processes \(ACB\) and \(ADB\). When path \(ACB\) is used \(60\, J\) of heat flows into the system and \(30\, J\) of work is done by the system. If path \(ADB\) is used work done by the system is \(10\, J\). The heat Flow into the system in path \(ADB\) is :

• (1) 80 J
• (2) 20 J
• (3) 100 J
• (4) 40 J

A plane electromagnetic wave of frequency \(50\, MHz\) travels in free space along the positive \(x\)-direction. At a particular point in space and time, \(\vec{E} = 6.3 \hat{j} V/m\) . The corresponding magnetic field \(\vec{B}\) , at that point will be:

• (1) \(18.9 \times 10^{-8} \hat{k} T \)
• (2) \(6.3 \times 10^{-8} \hat{k} T \)
• (3) \(2.1 \times 10^{-8} \hat{k} T \)
• (4) \(18.9 \times 10^{8} \hat{k} T \)

A block of mass \(m\), lying on a smooth horizontal surface, is attached to a spring (of negligible mass) of spring constant \(k\). The other end of the spring is fixed, as shown in the figure. The block is initally at rest in its equilibrium position. If now the block is pulled with a constant force \(F\), the maximum speed of the block is :

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

A particle is moving with a velocity \(\vec{v}=K(y\hat{i}+x\hat{j})\), where K is a constant. The general equation for its path is :

• (1) \(y=x^2+\text{constant}\)
• (2) \(y^2=x+\text{constant}\)
• (3) \(y^2=x^2+\text{constant}\)
• (4) \(xy=\text{constant}\)

Two masses \(m\) and \(\frac{m}{2}\) are connected at the two ends of a massless rigid rod of length \(l\). The rod is suspended by a thin wire of torsional constant \(k\) at the centre of mass of the rod-mass system(see figure). Because of torsional constant \(k\), the restoring torque is \(\tau =k\theta\) for angular displacement \(\theta\). If the rod is rota ted by \(\theta_0\) and released, the tension in it when it passes through its mean position will be:

• (1) \(\frac{3k\theta^{2}_{0}}{l}\)
• (2) \(\frac{k\theta^{{2}}_{0}}{2l}\)
• (3) \(\frac{2k\theta^{{2}}_{0}}{l}\)
• (4) \(\frac{k\theta^{{2}}_{0}}{l}\)

For a uniformly charged ring of radius \(R\), the electric field on its axis has the largest magnitude at a distance \(h\) from its centre. Then value of \(h\) is :

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

A parallel plate capacitor is made of two square plates of side 'a', separated by a distance d \((d \ll a)\). The lower triangular portion is filled with a dielectric of dielectric constant K, as shown in the figure. Capacitance of this capacitor is :

• (1) \(\frac{K\epsilon_0a^2}{2d(K+1)}\)
• (2) \(\frac{K\epsilon_0a^2}{d(K-1)}\ln K\)
• (3) \(\frac{K\epsilon_0a^2}{d}\ln K\)
• (4) \(\frac12\frac{K\epsilon_0a^2}{d}\)

Drift speed of electrons, when \(1.5\, A\) of current flows in a copper wire of cross section \(5\, mm^2\), is \(v\). If the electron density in copper is \(9 \times 10^{28} /m^3\) the value of \(v\) in mm/s is close to (Take charge of electron to be = \(1.6 \times 10^{-19}C\))

• (1) 0.2
• (2) 3
• (3) 2
• (4) 0.02

An infinitely long current carrying wire and a small current carrying loop are in the plane of the paper as shown. The radius of the loop is a and distance of its centre from the wire is d \((d \gg a)\). If the loop a applies a force F on the wire then :

• (1) \(F=0\)
• (2) \(F\propto\left(\frac{a}{d}\right)\)
• (3) \(F\propto\left(\frac{a^2}{d^3}\right)\)
• (4) \(F\propto\left(\frac{a}{d}\right)^2\)

A heavy ball of mass M is suspendeed from the ceiling of a car by a light string of mass m \((m \ll M)\). When the car is at rest, the speed of transverse waves in the string is 60 ms^-1. When the car has acceleration a, the wave-speed increases to 60.5 ms^-1. The value of a, in terms of gravitational acceleration g, is closest to :

• (1) \(\frac{g}{30}\)
• (2) \(\frac{g}{5}\)
• (3) \(\frac{g}{10}\)
• (4) \(\frac{g}{20}\)

A conducting circular loop made of a thin wire, has area \(3.5\times10^{-3}\ \mathrm{m^2}\) and resistance \(10\ \Omega\). It is placed perpendicular to a time dependent magnetic field \(B(t)=(0.4\,\mathrm{T})\sin(50\pi t)\). The field is uniform in space. Then the net charge flowing through the loop during \(t=0\ \mathrm{s}\) and \(t=10\ \mathrm{ms}\) is close to :

• (1) 0.14 mC
• (2) 0.7 mC
• (3) 0.21 mC
• (4) 0.6 mC

Three point charges q, –2q and 2q are placed on x-axis at a distance x = 0, x = \(\frac{3}{ 4}\) R and x = R respectively from origin as shown. If q = 2 × 10^-6 C and R = 2 cm, the magnitude of net force experienced by the charge –2q is ________ N.

A current loop, having two circular arcs joined by two radial lines is shown in the figure. It carries a current of \(10\, A\). The magnetic field at point \(O\) will be close to :

• (1) \(1.0 \times 10^{-5} T\)
• (2) \(1.5 \times 10^{-5} T\)
• (3) \(1.0 \times 10^{-7} T\)
• (4) \(1.5 \times 10^{-7} T\)

Temperature difference of \(120^{\circ}C\) is maintained between two ends of a uniform rod \(AB\) of length \(2L\). Another bent rod \(PQ\), of same cross section as \(AB\) and length \(\frac{3L}{2}\) , is connected across \(AB\) (See figure). In steady state, temperature difference between \(P\) and \(Q\) will be close to :

• (1) \(60^{\circ} C\)
• (2) \(75^{\circ} C\)
• (3) \(35^{\circ} C\)
• (4) \(45^{\circ} C\)

A block of mass \(10\, kg\) is kept on a rough inclined plane as shown in the figure. A force of \(3\, N\) is applied on the block. The coefficient of static friction between the plane and the block is \(0.6\). What should be the minimum value of force \(P\), such that the block doesnot move downward ? (take \(g = 10 \; ms^{-2}\))

• (1) 32 N
• (2) 25 N
• (3) 23 N
• (4) 18 N

Three blocks \(A, B\) and \(C\) are lying on a smooth horizontal surface, as shown in the figure. \(A\) and \(B\) have equal masses, \(m\) while \(C\) has mass \(M\). Block \(A\) is given an brutal speed \(v\) towards \(B\) due to which it collides with \(B\) perfectly inelastically. The combined mass collides with \(C\), also perfectly inelastically \(\frac{5}{6}\) th of the initial kinetic energy is lost in whole process. What is value of \(M/m\) ?

• (1) 4
• (2) 5
• (3) 3
• (4) 2

If the angular momentum of a planet of mass \(m\), moving around the Sun in a circular orbit is \(L\), about the center of the Sun, its areal velocity is :

• (1) \(\frac{4 L}{m}\)
• (2) \(\frac{L}{m}\)
• (3) \(\frac{ L}{ 2 m}\)
• (4) \(\frac{2L}{m}\)

Mobility of electrons in a semiconductor is defined as the ratio of their drift velocity to the applied electric field. If, for an n-type semiconductor, the density of electrons is \(10^{19}\ \mathrm{m^{-3}}\) and their mobility is \(1.6\ \mathrm{m^2/(V.s)}\) then the resistivity of the semiconductor (since it is an n-type semiconductor contribution of holes is ignored) is close to :

• (1) \(2\ \Omega\mathrm{m}\)
• (2) \(4\ \Omega\mathrm{m}\)
• (3) \(0.4\ \Omega\mathrm{m}\)
• (4) \(0.2\ \Omega\mathrm{m}\)

Surface of certain metal is first illuminated with light of wavelength \(\lambda_1 =350 \; nm\) and then, by light of wavelength \(\lambda_2=54\, \; nm\). It is found that the maximum speed of the photo electrons in the two cases differ by a factor of \(2\). The work function of the metal (in eV) is close to : (Energjr of photon = \(\frac{1240}{\lambda (in \; nm)} eV\))

• (1) 1.8
• (2) 1.4
• (3) 2.5
• (4) 5.6

A mixture of 2 moles of helium gas (atomic mass = 4 u), and 1 mole of argon gas (atomic mass = 40 u) is kept at 300 K in a container. The ratio of their rms speeds \(\left[\frac{V_{rms}(\text{helium})}{V_{rms}(\text{argon})}\right]\), is close to :

• (1) 3.16
• (2) 0.32
• (3) 0.45
• (4) 2.24

A convex lens is put \(10 \,cm\) from a light source and it makes a sharp image on a screen, kept \(10\, cm\) from the lens. Now a glass block (refractive index \(1.5\)) of \(1.5\, cm\) thickness is placed in contact with the light source. To get the sharp image again, the screen is shifted by a distance \(d\). Then \(d\) is :

• (1) \(0.55 \,cm\) away from the lens
• (2) \(1.1 \,cm\) away from the lens
• (3) \(0.55 \,cm\) towards the lens
• (4) 0

Two coherent sources produce waves of different intensities which interfere. After interference, the ratio of the maximum intensity to the minimum intensity is \(16\). The intensity of the waves are in the ratio:

• (1) 4:01
• (2) 25:09:00
• (3) 16:09
• (4) 5:03

When the switch \(S\), in the circuit shown, is closed, then the value of current \(i\) will be :

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

An L-shaped object, made of thin rods of uniform mass density, is suspended with a string as shown in figure. If \(AB = BC\), and the angle made by \(AB\) with downward vertical is \(\theta\), then :

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

A sample of radioactive material \(A\), that has an activity of \(10 \; mCi (1 \; Ci = 3.7 \times 10^{10}\) decays/s), has twice the number of nuclei as another sample of a different radioactive maternal \(B\) which has an activity of \(20\, mCi\). The correct choices for hall-lives of \(A\) and \(B\) would then be respectively :

• (1) 20 days and 5 days
• (2) 20 days and 10 days
• (3) 5 days and 10 days
• (4) 10 days and 40 days

A resistance is shown in the figure. Its value and tolerance are given respectively by :

• (1) \(27\, K \Omega , 20 \%\)
• (2) \(270\, K\Omega , 5 \%\)
• (3) \(270\, K\Omega , 10 \%\)
• (4) \(27\, K\Omega , 10 \%\)

In general, the properties that decrease and increase down a group in the periodic table respectively are:

• (1) electronegativity and electron gain enthalpy.
• (2) electronegativity and atomic radius.
• (3) atomic radius and electronegativity.
• (4) electron gain enthalpy and electronegativity.

Which one of the following statements regarding Henry's law not correct ?

• (1) The value of \(K_H\) increases with function of the nature of the gas
• (2) Higher the value of \(K_H\) at a given pressure, higher is the solubility of the gas in the liquids.
• (3) The partial of the gas in vapour phase is proportional to the mole fraction of the gas in the solution.
• (4) Different gases have different \(K_H\) (Henry's law constant) values at the same temperature.

The highest value of the calculated spin only magnetic moment (in BM) among all the transition metal complexes is :

• (1) 5.92
• (2) 6.93
• (3) 3.87
• (4) 4.90

For emission line of atomic hydrogen from \(n_i = 8\) to \(n_f \) = the plot of wave number \((\bar{v})\) against \(( \frac{1}{n^2})\) will be (The Ry dberg constant, \(R_H\) is in wave number unit).

• (1) Linear with slope - \(R_H\)
• (2) Linear with intercept - \(R_H\)
• (3) Non linear
• (4) Linear with slope \(R_H\)

• (1) A \(\rightarrow\) R; B \(\rightarrow\) P; C \(\rightarrow\) S; D \(\rightarrow\) Q
• (2) A \(\rightarrow\) Q; B \(\rightarrow\) S; C \(\rightarrow\) P; D \(\rightarrow\) R
• (3) A \(\rightarrow\) R; B \(\rightarrow\) S; C \(\rightarrow\) P; D \(\rightarrow\) Q
• (4) A \(\rightarrow\) Q; B \(\rightarrow\) P; C \(\rightarrow\) S; D \(\rightarrow\) R

The major product of following reaction is :

• (1) \(\mathrm{RCOOH}\)
• (2) \(\mathrm{RCONH_2}\)
• (3) \(\mathrm{RCHO}\)
• (4) \(\mathrm{RCH_2NH_2}\)

The increasing order of pka of the following amino acids in aqueous solution is :
Gly, Asp, Lys, Arg

• (1) Asp < Gly < Arg < Lys
• (2) Gly < Asp < Arg < Lys
• (3) Asp < Gly < Lys < Arg
• (4) Arg < Lys < Gly < Asp

Which amongst the following is the strongest acid ?

• (1) \(\({CHI_3}\)\)
• (2) \(\({CHCI_3}\)\)
• (3) \(\({CHBr_3}\)\)
• (4) \(\({CH(CN)_3}\)\)

The correct decreasing order for acid strength is :

• (1) \(NO_2CH_2COOH > NCCH_2COOH > FCH_2COOH > CICH_2COOH\)
• (2) \(FCH_2COOH > NCCH_2COOH > NO_2CHCOOH > CICH_2COOH\)
• (3) \(NO_2CH_2COOH > FCH_2COOH > CNCH_2COOH > CICH_2COOH\)
• (4) \(CNCH_2COOH > O_2NCH_2COOH > FCH_2COOH > CICH_2COOH\)

Arrange the following amines in the decreasing order of basicity:

• (1) I > II > III
• (2) III > II > I
• (3) I > III > II
• (4) III > I > II

Consider the reversible isothermal expansion of an ideal gas in a closed system at two different temperatures \(T_1\) and \(T_2 (T_1 < T_2)\). The correct graphical depiction of the dependence of work done \((w)\) on the final volume \((V)\) is:

• (1)
• (2)
• (3)
• (4)

Two complexes \(\mathrm{[Cr(H_2O)_6]Cl_3}\) (A) and \(\mathrm{[Cr(NH_3)_6]Cl_3}\) (B) are violet and yellow coloured, respectively. The incorrect statement regarding them is :

• (1) \(\Delta_0\) values of (A) and (B) are calculated from the energies of violet and yellow light, respectively
• (2) both are paramagnetic with three unpaired electrons
• (3) both absorb energies corresponding to their complementrary colors.
• (4) \(\Delta_0\) value for (A) is less than that of (B).

The major product of the following reaction is :

• (1)
• (2)
• (3)
• (4)

A solution of sodium sulfate contains \(92\, g\) of \({Na^{+}}\) ions per kilogram of water. The molality of \({Na^{+}}\) ions in that solution in \({mol \; kg^{-1}}\) is:

• (1) 16
• (2) 8
• (3) 4
• (4) 12

The anodic half-cell of lead-acid battery is recharged unsing electricity of \(0.05\) Faraday. The amount of \( {PbSO_4}\) electrolyzed in g during the process in : (Molar mass of \( {PbSO_4 = 303 \; g \; mol^{-1}}\))

• (1) 22.8
• (2) 15.2
• (3) 7.6
• (4) 11.4

20 mL of 0.1 M \(\mathrm{H_2SO_4}\) solution is added to 30 mL of of 0.2 M \(\mathrm{NH_4OH}\) solution. The pH of the resultant mixture is : [\(\mathrm{pk_b}\) of \(\mathrm{NH_4OH}\) = 4.7].

• (1) 5.2
• (2) 9.0
• (3) 5.0
• (4) 9.4

• (1) 5
• (2) 10
• (3) 1
• (4) 100

According to molecular orbital theory, which of the following is true with respect to \(\mathrm{Li_2^+}\) and \(\mathrm{Li_2^-}\) ?

• (1) \(\mathrm{Li_2^+}\) is unstable and \(\mathrm{Li_2^-}\) is stable
• (2) \(\mathrm{Li_2^+}\) is stable and \(\mathrm{Li_2^-}\) unstable
• (3) Both are stable
• (4) Both are unstable

Aluminium is usually found in \(+3\) oxidation stagte. In contarast, thallium exists in \(+1\) and \(+3\) oxidation states. This is due to :

• (1) lanthanoid contraction
• (2) lattice effect
• (3) diagonal relationship
• (4) inert pair effect

The compounds A and B in the following reaction are, respectively :

• (1) A = Benzyl alcohol, B = Benzyl cyanide
• (2) A = Benzyl chloride, B = Benzyl cyanide
• (3) A = Benzyl alcohol, B = Benzyl isocyanide
• (4) A = Benzyl chloride, B = Benzyl isocyanide

The major product of the following reaction is :

• (1)
• (2)
• (3)
• (4)

The one that is extensively used as a piezoelectric material is :

• (1) tridymite
• (2) amorphous silica
• (3) quartz
• (4) mica

Adsorption of a gas follows Freundlich adsorption isotherm. If x is the mass of the gas adsorbed on mass m of the adsorbent, the correct plot of \(\frac{x}{m}\) versus p is :

• (1)
• (2)
• (3)
• (4)

The ore that contains both iron and copper is:

• (1) malachite
• (2) dolomite
• (3) azurite
• (4) copper pyrites

A water sample has ppm level concentration of the following metals: \(Fe = 0.2; Mn = 5.0; Cu = 3.0; Zn = 5.0.\) The metal that makes the water sample unsuitable drinking is

• (1) Zn
• (2) Fe
• (3) Mn
• (4) Cu

Correct statements among \(a\) to \(d\) regarding silicones are : (a) They are polymers with hydrophobic character (b) They are biocompatible. (c) In general, they have high thermal stability and low dielectric strenth. (d) Usually, they are resistant to oxidation and used as greases.

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

Major product of the following reaction is -

• (1)
• (2)
• (3)
• (4)

The isotopes of hydrogen are :

• (1) Tritium and protium only
• (2) Protium and Deuterium only
• (3) Protium, deuterium and tritium
• (4) Deuterium and tritium only

0.5 moles of gas A and x moles of gas B exert a pressure of 200 Pa in a container of volume 10 m³ at 1000 K. Given R is the gas constant in \(\mathrm{JK^{-1}mol^{-1}}\), x is :

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

The alkaline earth metal nitrate that does not crystallise with water molecules, is :

• (1) \(\mathrm{Mg(NO_3)_2}\)
• (2) \(\mathrm{Sr(NO_3)_2}\)
• (3) \(\mathrm{Ca(NO_3)_2}\)
• (4) \(\mathrm{Ba(NO_3)_2}\)