JEE Main 2019 B.E./B.Tech Question Paper with Answer Keys (April 12, 2019- Forenoon Session)

If \(\int_0^{\pi/2} \frac{\cot x}{\cot x + \operatorname{cosec} x}\,dx = m(\pi + n)\), then \(m \cdot n\) is equal to

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

Consider the differential equation, \(y^2\,dx + \left(x - \frac{1}{y}\right)dy = 0\), If value of y is 1 when x = 1, then the value of x for which y = 2, is :

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

If the angle of intersection at a point where the two circles with radii 5 cm and 12 cm intersect is \(90^{\circ}\), then the length (in cm) of their common chord is :

• (1) \(\frac{60}{13}\)
• (2) \(\frac{120}{13}\)
• (3) \(\frac{13}{2}\)
• (4) \(\frac{13}{5}\)

If three of the six vertices of a regular hexagon are chosen at random, then the probability that the triangle formed with these chosen vertices is equilateral is :

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

Let \(S_n\) denote the sum of the first \(n\) terms of an \(A.P\). If \(S_4 = 16\) and \(S_6 = -48\), then \(S_{10}\) is equal to :

• (1) -320
• (2) -260
• (3) -380
• (4) -410

The value of \(\sin^{-1}\left(\frac{12}{13}\right) - \sin^{-1}\left(\frac{3}{5}\right)\) is equal to :

• (1) \(\pi - \sin^{-1}\left(\frac{63}{65}\right)\)
• (2) \(\frac{\pi}{2} - \sin^{-1}\left(\frac{56}{65}\right)\)
• (3) \(\frac{\pi}{2} - \cos^{-1}\left(\frac{9}{65}\right)\)
• (4) \(\pi - \cos^{-1}\left(\frac{33}{65}\right)\)

If m is the minimum value of k for which the function \(f(x) = x\sqrt{kx - x^2}\) is increasing in the interval [0,3] and M is the maximum value of f in [0, 3] when k = m, then the ordered pair (m, M) is equal to :

• (1) \(\left(5, 3\sqrt{6}\right)\)
• (2) \(\left(4, 3\sqrt{3}\right)\)
• (3) \(\left(4, 3\sqrt{2}\right)\)
• (4) \(\left(3, 3\sqrt{3}\right)\)

The integral \( \int \frac{2x^{3} -1}{x^{4} +x} dx \) is equal to : (Here C is a constant of integration)

• (1) \(\log_{e} \left|\frac{x^{3} + 1}{x}\right|+C \)
• (2) \(\frac{1}{2} \log_{e} \frac{\left(x^{3} +1\right)^{2}}{\left|x^{3}\right|} +C \)
• (3) \(\frac{1}{2} \log_{e} \frac{\left|x^{3} +1\right|^{2}}{x^{3}} +C \)
• (4) \( \log_{e} \frac{\left|x^{3} +1\right|^{2}}{x^{3}} +C \)

If A is a symmetric matrix and B is a skew-symmetric matrix such that A + B = \(\begin{bmatrix} 2 & 3 \\ 5 & -1 \end{bmatrix}\), then AB is equal to :

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

Let \(\vec{p} = 2\hat{i} + 3\hat{j} + \hat{k}\) and \(\vec{q} = \hat{i} + 2\hat{j} + \hat{k}\) be two vectors. If a vector \(\vec{r} = \alpha\hat{i} + \beta\hat{j} + \gamma\hat{k}\) is perpendicular to each of the vectors \((\vec{p} + \vec{q})\) and \((\vec{p} - \vec{q})\), and \(|\vec{r}| = \sqrt{3}\), then \(|\alpha| + |\beta| + |\gamma|\) is equal to __________.

If the data \(x_1, x_2, \ldots\ldots, x_{10}\) is such that the mean of first four of these is 11, the mean of the remaining six is 16 and the sum of squares of all of these is 2,000 ; then the standard deviation of this data is :

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

If \(B = \begin{bmatrix} 5 & 2\alpha & 1 \\ 0 & 2 & 1 \\ \alpha & 3 & -1 \end{bmatrix}\) is the inverse of a \(3 \times 3\) matrix A, then the sum of all values of \(\alpha\) for which det(A) + 1 = 0, is :

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

The number of ways of choosing 10 objects out of 31 objects of which 10 are identical and the remaining 21 are distinct, is :

• (1) \(2^{20} - 1\)
• (2) \(2^{20}\)
• (3) \(2^{20} + 1\)
• (4) \(2^{21}\)

If \(\alpha\) and \(\beta\) are the roots of the equation \(375x^2 - 25x - 2 = 0\), then \(\lim_{n\to\infty}\sum_{r=1}^{n}\alpha^r + \lim_{n\to\infty}\sum_{r=1}^{n}\beta^r\) is equal to :

• (1) \(\frac{7}{116}\)
• (2) \(\frac{29}{348}\)
• (3) \(\frac{1}{12}\)
• (4) \(\frac{21}{346}\)

The equation \(|z - i| = |z - 1|\), \(i = \sqrt{-1}\), represents :

• (1) a circle of radius \(1\)
• (2) the line through the origin with slope \(-1\)
• (3) a circle of radius \(\frac{1}{2}\)
• (4) the line through the origin with slope \(1\)

The coefficient of \(x^{18}\) in the product \((1 + x)(1 - x)^{10}(1 + x + x^2)^9\) is :

• (1) \(126\)
• (2) \(-84\)
• (3) \(-126\)
• (4) \(84\)

If \(e^y + xy = e\), the ordered pair \(\left(\frac{dy}{dx}, \frac{d^2y}{dx^2}\right)\) at \(x = 0\) is equal to :

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

For \(x \in (0, 3/2)\), let \(f(x) = \sqrt{x}\), \(g(x) = \tan x\) and \(h(x) = \frac{1 - x^2}{1 + x^2}\). If \(\phi(x) = ((h\circ f)\circ g)(x)\), then \(\phi\left(\frac{\pi}{3}\right)\) is equal to :

• (1) \(\tan\frac{7\pi}{12}\)
• (2) \(\tan\frac{11\pi}{12}\)
• (3) \(\tan\frac{\pi}{12}\)
• (4) \(\tan\frac{5\pi}{12}\)

If the area (in sq. units) of the region \(\{(x, y) : y^2 \le 4x,\ x + y \le 1,\ x \ge 0,\ y \ge 0\}\) is \(a\sqrt{2} + b\), then \(a - b\) is equal to :

• (1) \(\frac{8}{3}\)
• (2) \(-\frac{2}{3}\)
• (3) \(6\)
• (4) \(\frac{10}{3}\)

Let \(f:\mathbb{R}\to\mathbb{R}\) be a continuously differentiable function such that \(f(2)=6\) and \(f'(2)=\frac{1}{48}\). If
\(\int_{6}^{f(x)} 4t^{3}\,dt=(x-2)\,g(x)\),
then \(\lim_{x\to 2} g(x)\) is equal to :

• (1) 18
• (2) 36
• (3) 12
• (4) 24

The equation \(y=\sin x\,\sin(x+2)-\sin^{2}(x+1)\) represents a straight line lying in :

• (1) first, second and fourth quadrants
• (2) first, third and fourth quadrants
• (3) second and third quadrants only
• (4) third and fourth quadrants only

A 2 m ladder leans against a vertical wall. If the top of the ladder begins to slide down the wall at the rate 25 cm/sec, then the rate (in cm/sec.) at which the bottom of the ladder slides away from the wall on the horizontal ground when the top of the ladder is 1 m above the ground is :

• (1) \(\frac{25}{3}\)
• (2) 25
• (3) \(25\sqrt{3}\)
• (4) \(\frac{25}{\sqrt{3}}\)

If the volume of parallelopiped formed by the vectors \(\hat{i}+\lambda\hat{j}+\hat{k}\), \(\hat{j}+\lambda\hat{k}\) and \(\lambda\hat{i}+\hat{k}\) is minimum, then \(\lambda\) is equal to :

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

If the normal to the ellipse \(3x^2+4y^2=12\) at a point P on it is parallel to the line, \(2x+y=4\) and the tangent to the ellipse at P passes through Q(4,4) then PQ is equal to :

• (1) \(\frac{\sqrt{61}}{2}\)
• (2) \(\frac{\sqrt{221}}{2}\)
• (3) \(\frac{\sqrt{157}}{2}\)
• (4) \(\frac{5\sqrt{5}}{2}\)

If the truth value of the statement \(P \to ( \sim p \vee r)\) is false(F), then the truth values of the statements \(p, q, r\) are respectively :

• (1) F, T, T
• (2) T, F, F
• (3) T, T, F
• (4) T, F, T

For \(x\in\mathbb{R}\), let [x] denote the greatest integer \(\le x\), then the sum of the series
\(\left[-\frac{1}{3}\right]+\left[-\frac{1}{3}-\frac{1}{100}\right]+\left[-\frac{1}{3}-\frac{2}{100}\right]+\ldots+\left[-\frac{1}{3}-\frac{99}{100}\right]\)
is :

• (1) - 153
• (2) - 135
• (3) - 133
• (4) - 131

The number of solutions of the equation
\(1+\sin^{4}x=\cos^{2}3x\),
\(x\in\left[-\frac{5\pi}{2},\frac{5\pi}{2}\right]\)
is :

• (1) 5
• (2) 3
• (3) 7
• (4) 4

Let P be the point of intersection of the common tangents to the parabola \(y^2=12x\) and the hyperbola \(8x^2-y^2=8\). If S and S' denote the foci of the hyperbola where S lies on the positive x-axis then P divides SS' in a ratio :

• (1) 14 : 13
• (2) 13 : 11
• (3) 5 : 4
• (4) 2 : 1

If the line
\(\frac{x-2}{3}=\frac{y+1}{2}=\frac{z-1}{-1}\)
intersects the plane \(2x+3y-z+13=0\) at a point P and the plane \(3x+y+4z=16\) at a point Q, then PQ is equal to :

• (1) \(2\sqrt{7}\)
• (2) 14
• (3) \(2\sqrt{14}\)
• (4) \(\sqrt{14}\)

Let a random variable X have a binomial distribution with mean 8 and variance 4. If
\(P(X\le2)=\frac{k}{2^{16}}\)
, then k is equal to :

• (1) 17
• (2) 1
• (3) 137
• (4) 121

An excited He\(^+\) ion emits two photons in succession, with wavelengths 108.5 nm and 30.4 nm, in making a transition to ground state. The quantum number n, corresponding to its initial excited state is (for photon of wavelength \(\lambda\), energy
\(E=\frac{1240\,\mathrm{eV}}{\lambda\,(\text{in nm})}\)
) :

• (1) n = 4
• (2) n = 7
• (3) n = 5
• (4) n = 6

Two identical parallel plate capacitors, of capacitance \(C\) each, have plates of area \(A\), separated by a distance \(d\). The space between the plates of the two capacitors, is filled with three dielectrics, of equal thickness and dielectric constants \(K_1, K_2\) and \(K_3\). The first capacitor is filled as shown in fig. \(I\), and the second one is filled as shown in fig. \(II\). If these two modified capacitors are charged by the same potential \(V\), the ratio of the energy stored in the two, would be (\(E_1\) refers to capacitor (I) and \(E_2\) to capacitor (II) :

• (1) \(\frac{E_{1}}{E_{2}} = \frac{9K_{1}K_{2}K_{3}}{\left(K_{1} +K_{2}+K_{3}\right)\left(K_{2}K_{3} + K_{3}K_{1} +K_{1}K_{2}\right)} \)
• (2) \(\frac{E_{1}}{E_{2}} = \frac{K_{1}K_{2}K_{3}}{\left(K_{1} +K_{2}+K_{3}\right)\left(K_{2}K_{3} + K_{3}K_{1} +K_{1}K_{2}\right)} \)
• (3) \(\frac{E_{1}}{E_{2}} = \frac{\left(K_{1} +K_{2}+K_{3}\right)\left(K_{2}K_{3} + K_{3}K_{1} +K_{1}K_{2}\right)}{K_{1}K_{2}K_{3}} \)
• (4) \(\frac{E_{1}}{E_{2}} = \frac{\left(K_{1} +K_{2}+K_{3}\right)\left(K_{2}K_{3} + K_{3}K_{1} +K_{1}K_{2}\right)}{9K_{1}K_{2}K_{3}} \)

An electromagnetic wave is represented by the electric field
\(\vec{E}=E_0\hat{n}\sin\left[\omega t+(6y-8z)\right]\)
. Taking unit vectors in x, y and z directions to be \(\hat{i},\hat{j},\hat{k}\), the direction of propagation \(\hat{s}\), is :

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

A sample of an ideal gas is taken through the cyclic process abca as shown in the figure. The change in the internal energy of the gas along the path ca is \(-180\,J\). The gas absorbs \(250\, J\) of heat along the path ab and \(60\, J\) along the path \(bc\). The work done by the gas along the path \(abc\) is :

• (1) 100 J
• (2) 120 J
• (3) 140 J
• (4) 130 J

A person of mass M is, sitting on a swing of length L and swinging with an angular amplitude \(\theta_0\). If the person stands up when the swing passes through its lowest point, the work done by him, assuming that his center of mass moves by a distance \(\ell\) (\(\ell\ \lt\lt\ L\)), is close to;

• (1) \(mg\ell\left(1+\theta_0^{2}\right)\)
• (2) \(mg\ell\)
• (3) \(mg\ell\left(1+\frac{\theta_0^{2}}{2}\right)\)
• (4) \(mg\ell\left(1-\theta_0^{2}\right)\)

A galvanometer of resistance 100 \(\Omega\) has 50 divisions on its scale and has sensitivity of 20 \(\mu\)A/division. It is to be converted to a voltmeter with three ranges of 0-2V, 0-10 V and 0-20 V. The appropriate circuit to do so is

• (1)
• (2)
• (3)
• (4)

A man (mass = \(50\, kg\)) and his son (mass = \(20\, kg\)) are standing on a frictionless surface facing each other. The man pushes his son so that he starts moving at a speed of \(0.70\, ms^{-1}\) with respect to the man. The speed of the man with respect to the surface is :

• (1) \(0.20\, ms ^{-1}\)
• (2) \(0.14\, ms ^{-1}\)
• (3) \(0.47\, ms ^{-1}\)
• (4) \(0.28\, ms ^{-1}\)

To verify Ohm's law, a student connects the voltmeter across the battery as, shown in the figure. The measured voltage is plotted as a function of the current, and the following graph is obtained: If \(V_0\) is almost zero, identify the correct statement:

• (1) The value of the resistance \(R\) is \(1.5 \,\Omega\)
• (2) The emf of the battery is \(1.5\, V\) and the value of \(R\) is \(1.5 \, \Omega\)
• (3) The emf of the battery is \(1.5\, V\) and its internal resistance is \(1.5\) \(\Omega\)
• (4) The potential difference across the battery is \(1.5\, V\) when it sends a current of \(1000\, mA\)

As shown in the figure, a point charge \(Q\) is placed at the centre of conducting spherical shell of inner radius \(a\) and outer radius \(b\). The electric field due to charge \(Q\) in three different regions I, II and III is given by: (I: r \(<\) a, II: \(a \)<\( r \)<\( b\), III: \(r\)>\(b)\)

• (1) \(E_I=0, E_{I I}=0, E_{I I I}=0\)
• (2) \(E_I \neq 0, E_{I I}=0, E_{I I}=0\)
• (3) \(E_I=0, E_{I I}=0, E_{I I I} \neq 0\)
• (4) \(E_I \neq 0, E_{I I}=0, E_{I I I} \neq 0\)

A concave mirror has radius of curvature of \(40\, cm\). It is at the bottom of a glass that has water filled up to \(5\, cm\) (see figure). If a small particle is floating on the surface of water, its image as seen, from directly above the glass, is at a distance d from the surface of water. The value of d is close to : (Refractive index of water \(= 1.33\))

• (1) \(8.8 \,cm\)
• (2) \(11.7 \,cm\)
• (3) \(6.7 \,cm\)
• (4) \(13.4 \,cm\)

The figure shows a square loop \(L\) of side \(5\, cm\) which is connected to a network of resistances. The whole setup is moving towards right with a constant speed of \(1\, cms^{-1}\). At some instant, a part of \(L\) is in a uniform magnetic field of \(1T\), perpendicular to the plane of the loop. If the resistance of \(L\) is \(1.7\, \Omega\) , the current in the loop at that instant will be close to :

• (1) \(115\, \mu A\)
• (2) \(170 \, \mu A\)
• (3) \(60\, \mu A\)
• (4) \(150\, \mu A\)

For the network shown below, the value of V_B– V_A is ________V.

A progressive wave travelling along the positive x-direction is represented by \(y(x,t)=A\sin(kx-\omega t+\phi)\). Its snapshot at t = 0 is given in the figure.

For this wave, the phase \(\phi\) is :

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

A thin ring of 10 cm radius carries a uniformly distributed charge. The ring rotates at a constant angular speed of \(40\pi\) rad s\(^{-1}\) about its axis, perpendicular to its plane. If the magnetic field at its centre is \(3.8 \times 10^{-9}\) T, then the charge carried by the ring is close to (\(\mu_0 = 4\pi \times 10^{-7}\) N/A\(^2\)).

• (1) \(7 \times 10^{-6}\) C
• (2) \(4 \times 10^{-5}\) C
• (3) \(2 \times 10^{-6}\) C
• (4) \(3 \times 10^{-5}\) C

When \(M_1\) gram of ice at \(-10^{\circ}C\) (specific heat = \(0.5 \; cal \; g^{-1 \circ } C^{-1}\)) is added to \(M_2\) gram of water at \(50^{\circ}C\), finally no ice is left and the water is at \(0^{\circ}C\). The value of latent heat of ice, in cal \(g^{-1}\) is:

• (1) \(\frac{5M_{1}}{M_{2}} - 50 \)
• (2) \(\frac{50M_{2}}{M_{1}} \)
• (3) \(\frac{50M_{2}}{M_{1}} - 5 \)
• (4) \(\frac{5M_{2}}{M_{1}} - 5 \)

The truth table for the circuit given in the fig. is:

• (1)

  A	B	Y
  0	0	1
  0	1	1
  1	0	1
  1	1	1

• (2)

  A	B	Y
  0	0	1
  0	1	0
  1	0	0
  1	1	0

• (3)

  A	B	Y
  0	0	1
  0	1	1
  1	0	0
  1	1	0

• (4)

  A	B	Y
  0	0	0
  0	1	0
  1	0	1
  1	1	1

Which of the following combinations has the dimension of electrical resistance (\(\in_0\) is the permittivity of vacuum and \(\mu_0\) is the permeability of vacuum)?

• (1) \(\sqrt{\dfrac{\in_0}{\mu_0}}\)
• (2) \(\dfrac{\in_0}{\mu_0}\)
• (3) \(\sqrt{\dfrac{\mu_0}{\in_0}}\)
• (4) \(\dfrac{\mu_0}{\in_0}\)

A point dipole \(\vec{p} = -p_0 \hat{x}\) is kept at the origin. The potential and electric field due to this dipole on the y-axis at a distance d are, respectively: (Take V = 0 at infinity)

• (1) \(\dfrac{|\vec{p}|}{4\pi \in_0 d^2},\ \dfrac{-\vec{p}}{4\pi \in_0 d^3}\)
• (2) \(0,\ \dfrac{\vec{p}}{4\pi \in_0 d^3}\)
• (3) \(\dfrac{|\vec{p}|}{4\pi \in_0 d^2},\ \dfrac{\vec{p}}{4\pi \in_0 d^3}\)
• (4) \(0,\ \dfrac{-\vec{p}}{4\pi \in_0 d^3}\)

Two moles a monoatomic gas is mixed with six moles of a diatomic gas. The molar specific heat of the mixture at constant volume is :

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

Mass per unit area of a circular disc of radius \( a \) depends on the distance r from its centre as \( \sigma \left( r \right) \) = A + Br . The moment of inertia of the disc about the axis, perpendicular to the plane and assing through its centre is:

• (1) \( 2\pi {a^4}\left( {{A \over 4} + {{aB} \over 5}} \right) \)
• (2) \( \pi {a^4}\left( {{A \over 4} + {{aB} \over 5}} \right) \)
• (3) \( 2\pi {a^4}\left( {{{aA} \over 4} + {B \over 5}} \right) \)
• (4) \( 2\pi {a^4}\left( {{A \over 4} + {B \over 5}} \right) \)

The trajectory of a projectile near the surface of the earth is given as \(y = 2x - 9x^2\). If it were launched at an angle \(\theta_0\) with speed \(v_0\) then (\(g = 10\) ms\(^{-2}\)) :

• (1) \(\theta_0 = \cos^{-1}\left(\dfrac{1}{\sqrt{5}}\right)\) and \(v_0 = \dfrac{5}{3}\) ms\(^{-1}\)
• (2) \(\theta_0 = \cos^{-1}\left(\dfrac{2}{\sqrt{5}}\right)\) and \(v_0 = \dfrac{3}{5}\) ms\(^{-1}\)
• (3) \(\theta_0 = \sin^{-1}\left(\dfrac{2}{\sqrt{5}}\right)\) and \(v_0 = \dfrac{3}{5}\) ms\(^{-1}\)
• (4) \(\theta_0 = \sin^{-1}\left(\dfrac{1}{\sqrt{5}}\right)\) and \(v_0 = \dfrac{5}{3}\) ms\(^{-1}\)

The stopping potential \(V_0\) (in volt) as a function of frequency (\(\upsilon\)) for a sodium emitter, is shown in the figure. The work function of sodium, from the data plotted in the figure, will be:
(Given: Planck's constant (h) = \(6.63 \times 10^{-34}\) Js, electron charges e = \(1.6 \times 10^{-19}\) C)

• (1) 1.95 eV
• (2) 2.12 eV
• (3) 1.82 eV
• (4) 1.66 eV

A metallic rod of length ' \(L\) ' is rotated with an angular speed of ' \(\omega\) ' normal to a uniform magnetic field ' \(B\) ' about an axis passing through one end of rod as shown in figure. The induced emf will be :

• (1) \(\frac{1}{2} B ^2 L ^2 \omega\)
• (2) \(\frac{1}{4} BL ^2 \omega\)
• (3) \(\frac{1}{2} BL ^2 \omega\)
• (4) \(\frac{1}{4} B ^2 L \omega\)

In a double slit experiment, when a thin film of thickness t having refractive index \(\mu\). is introduced in front of one of the slits, the maximum at the centre of the fringe pattern shifts by one fringe width. The value of t is (\(\lambda\) is the wavelength of the light used) :

• (1) \(\dfrac{\lambda}{2(\mu-1)}\)
• (2) \(\dfrac{\lambda}{(2\mu-1)}\)
• (3) \(\dfrac{2\lambda}{(\mu-1)}\)
• (4) \(\dfrac{\lambda}{(\mu-1)}\)

A shell is fired from a fixed artillery gun with an initial speed u such that it hits the target on the ground at a distance R from it. If t\(_1\) and t\(_2\) are the values of the time taken by it to hit the target in two possible ways, the product t\(_1\)t\(_2\) is -

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

Figure represents the extension (\(\Delta l\)) of a wire of length \(l\), suspended from the ceiling of the room at one end and with a load \(W\) connected to the other end. If the cross-sectional area of the wire is \(10^{-5}\,\text{m}^2\) then the Young’s modulus of the wire is _____ N/m\(^2\).

• (1) \(1.0\times10^{11}\)
  
• (2) \(2.0\times10^{10}\)
• (3) \(1.0\times10^{10}\)
• (4) \(2.0\times10^{11}\)

The value of numerical aperature of the objective lens of a microscope is 1.25. If light of wavelength 5000 \(\mathring{\mathrm{A}}\) is used, the minimum separation between two points, to be seen as distinct, will be :

• (1) 0.12 \(\mu\)m
• (2) 0.38 \(\mu\)m
• (3) 0.24 \(\mu\)m
• (4) 0.48 \(\mu\)m

A magnetic compass needle oscillates \(30\) times per minute at a place where the dip is \(45^{\circ}\), and \(40\) times per minute where the dip is \(30^{\circ}\). If \(B_1\) and \(B_2\) are respectively the total magnetic field due to the earth at the two places, then the ratio \(B_1/B_2\) is best given by :

• (1) 2.2
• (2) 1.8
• (3) 0.7
• (4) 3.6

The transfer characteristic curve of a transistor, having input and output resistance 100 \(\Omega\) and 100 k\(\Omega\) respectively is shown in the figure. The voltage and power gain, are respectively:

• (1) \(5 \times 10^{4},\ 5 \times 10^{5}\)
• (2) \(5 \times 10^{4},\ 5 \times 10^{6}\)
• (3) \(5 \times 10^{4},\ 2.5 \times 10^{6}\)
• (4) \(2.5 \times 10^{4},\ 2.5 \times 10^{6}\)

A submarine (A) travelling at 18 km/hr is being chased along the line of its velocity by another submarine (B) travelling at 27 km/hr. B sends a sonar signal of 500 Hz to detect A and receives a reflected sound of frequency \(\upsilon\). The value of \(\upsilon\) is close to: (Speed of sound in water = 1500 ms\(^{-1}\))

• (1) 507 Hz
• (2) 502 Hz
• (3) 499 Hz
• (4) 504 Hz

The mole fraction of a solvent in aqueous solution of a solute is 0.8. The molality (in mol \(kg^{-1}\)) of the aqueous solution is

• (1) \(13.88 \times 10^{-1}\)
• (2) \(13.88 \times 10^{-2}\)
• (3) \(13.88\)
• (4) \(13.88 \times 10^{-3}\)

The major product(s) obtained in the following reaction is/are :

• (1)
• (2)
• (3)
• (4)

The major product of the following reaction is :

• (1)
• (2)
• (3)
• (4)

The group number, number of valence electrons, and valency of an element with atomic number 15, respectively, are:

• (1) 16, 6 and 3
• (2) 15, 6 and 12
• (3) 16, 5 and 2
• (4) 15, 5 and 3

What is the molar solubility of \(Al(OH)_3\) in \(0.2\, M\, NaOH\) solution ? Given that, solubility product of \(Al(OH)_3 = 2.4 \times 10^{-24}\) :

• (1) \( 12 \times 10^{- 23}\)
• (2) \( 12 \times 10^{- 21 }\)
• (3) \( 3 \times 10^{- 19}\)
• (4) \( 3 \times 10^{- 22}\)

An ideal gas is allowed to expand form 1 L to 10 L against a constant external pressure of 1 bar. The work done in kJ is :

• (1) +10.0
• (2) \(-0.9\)
• (3) \(-2.0\)
• (4) \(-9.0\)

\(5\) moles of \(AB_2\) weigh \(125 \times 10^{-3}\; kg\) and \(10\) moles of \(A_2B_2\) weigh \(300 \times 10^{-3} \; kg\). The molar mass of \(A(M_A)\) and molar mass of \(B(M_B) \) in \(kg\, mol^{-1}\) are :

• (1) \(M_A = 50 \times 10^{-3} \) and \(M_B = 25 \times 10^{-3}\)
• (2) \(M_A = 25\times 10^{-3} \) and \(M_B = 50\times 10^{-3}\)
• (3) \(M_A = 5 \times 10^{-3} \) and \(M_B = 10 \times 10^{-3}\)
• (4) \(M_A = 10 \times 10^{-3} \) and \(M_B = 5 \times 10^{-3}\)

The major product of the following addition reaction is :
\(\mathrm{H_3C{-}CH{=}CH_2 \xrightarrow{Cl_2/H_2O}}\)

• (1)
• (2)
• (3)
• (4)

But-2-ene on reaction with alkaline \(\mathrm{KMnO_4}\) at elevated temperature followed by acidification will give :

• (1) one molecule of \(\mathrm{CH_3CHO}\) and one molecule of \(\mathrm{CH_3COOH}\)
• (2) 2 molecules of \(\mathrm{CH_3COOH}\)
• (3)
• (4) 2 molecules of \(\mathrm{CH_3CHO}\)

An organic compound \('A'\) is oxidized with \({Na2O2}\) followed by boiling with \({HNO3}\). The resultant solution is then treated with ammonium molybdate to yield a yellow precipitate. Based on above observation, the element present in the given compound is :

• (1) Sulphur
• (2) Nitrogen
• (3) Fluorine
• (4) Phosphorus

An example of a disproportionation reaction is :

• (1) \(\mathrm{2NaBr + Cl_2 \to 2NaCl + Br_2}\)
• (2) \(\mathrm{2KMnO_4 \to K_2MnO_4 + MnO_2 + O_2}\)
• (3) \(\mathrm{2CuBr \to CuBr_2 + Cu}\)
• (4) \(\mathrm{2MnO_4^- + 10I^- + 16H^+ \to 2Mn^{2+} + 5I_2 + 8H_2O}\)

Glucose and Galactose are having identical configuration in all the positions except position.

• (1) \(C - 3\)
• (2) \(C - 2\)
• (3) \(C - 4\)
• (4) \(C - 5\)

The complex ion that will lose its crystal field stabilization energy upon oxidation of its metal to +3 state is :
(Phen = and Ignore pairing energy)

• (1) \(\mathrm{[Fe(phen)_3]^{2+}}\)
• (2) \(\mathrm{[Zn(phen)_3]^{2+}}\)
• (3) \(\mathrm{[Co(phen)_3]^{2+}}\)
• (4) \(\mathrm{[Ni(phen)_3]^{2+}}\)

The electrons are more likely to be found :

• (1) only in the region a
• (2) in the region a and c
• (3) only in the region c
• (4) in the region a and b

Complete removal of both the axial ligands (along the z-axis) from an octahedral complex leads to which of the following splitting patterns? (relative orbital energies not on scale).

• (1)
• (2)
• (3)
• (4)

The increasing order of the \(\mathrm{pK_b}\) of the following compound is :

• (1) (C) < (A) < (D) < (B)
• (2) (B) < (D) < (C) < (A)
• (3) (B) < (D) < (A) < (C)
• (4) (A) < (C) < (D) < (B)

The major products of the following reaction are :

(1) \(\mathrm{CHCl_3}\)/aq. NaOH
(2) HCHO, NaOH (conc)
(3) \(\mathrm{H_3O^+}\)

• (1)
• (2)
• (3)
• (4)

Enthalpy of sublimation of iodine is \( {24 \; cal \; g^{-1}}\) at \(200^{\circ}C\). If specific heat of \(I_2(s)\) and \(I_2(vap)\) are \(0.055\) and \(0.031\, cal\, g^{-1}K^{-1}\) respectively, then enthalpy of sublimation of iodine at \(250^{\circ}C\) in \(cal\, g^{-1}\) is :

• (1) 2.85
• (2) 11.4
• (3) 5.7
• (4) 22.8

Given
\(\mathrm{Co^{3+} + e^- \to Co^{2+}}\) ; \(\mathrm{E^o}\) = + 1.81 V
\(\mathrm{Pb^{4+} + 2e^- \to Pb^{2+}}\) ; \(\mathrm{E^o}\) = + 1.67 V
\(\mathrm{Ce^{4+} + e^- \to Ce^{3+}}\) ; \(\mathrm{E^o}\) = + 1.61 V
\(\mathrm{Bi^{3+} + 3e^- \to Bi}\) ; \(\mathrm{E^o}\) = + 0.20 V
Oxidizing power of the species will increase in the order :

• (1) \(\mathrm{Co^{3+} \lt Ce^{4+} \lt Bi^{3+} \lt Pb^{4+}}\)
• (2) \(\mathrm{Co^{3+} \lt Pb^{4+} \lt Ce^{4+} \lt Bi^{3+}}\)
• (3) \(\mathrm{Ce^{4+} \lt Pb^{4+} \lt Bi^{3+} \lt Co^{3+}}\)
• (4) \(\mathrm{Bi^{3+} \lt Ce^{4+} \lt Pb^{4+} \lt Co^{3+}}\)

Which of the following statements is not true about RNA?

• (1) It has always double stranded a-helix structure.
• (2) It usually does not replicate.
• (3) It is present in the nucleus of the cell.
• (4) It controls the synthesis of protein.

The correct statement among the following is

• (1) \({(SiH3)3N}\) is pyramidal and more basic than \({(CH3)3N}\)
• (2) \({(SiH3)3N}\) is planar and more basic than \({(CH3)3N}\)
• (3) \({(SiH3)3N}\) is pyramidal and less basic than \({(CH3)3N}\)
• (4) \({(SiH3)3N}\) is planar and less basic than \({(CH3)3N}\)

In the following reaction; \(\mathrm{xA \to yB}\)
\(\log_{10}\left[-\frac{d[A]}{dt}\right] = \log_{10}\left[\frac{d[B]}{dt}\right] + 0.3010\)
'A' and 'B' respectively can be :

• (1) n-Butane and Iso-butane
• (2) \(\mathrm{C_2H_4}\) and \(\mathrm{C_4H_8}\)
• (3) \(\mathrm{C_2H_4}\) and \(\mathrm{C_6H_6}\)
• (4) \(\mathrm{N_2O_4}\) and \(\mathrm{NO_2}\)

An element has a face-centred cubic (fcc ) structure with a cell edge of a. The distance between the centres of two nearest tetrahedral voids in the lattice is :

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

The metal that gives hydrogen gas upon treatment with both acid as well as base is :

• (1) mercury
• (2) zinc
• (3) iron
• (4) magnesium

The idea of froth floatation method came from a person \(X\) and this method is related to the process \(Y\) of ores. \(X\) and \(Y\), respectively, are:

• (1) fisher woman and concentration
• (2) washer man and reduction
• (3) washer woman and concentration
• (4) fisher man and reduction

The correct sequence of thermal stability of the following carbonates is

• (1) \({BaCO3 < CaCO3 < SrCO3 < MgCO3}\)
• (2) \({MgCO3 < CaCO3 < SrCO3 < BaCO3}\)
• (3) \({BaCO3 < SrCO3 < CaCO3 < MgCO3}\)
• (4) \({MgCO3 < SrCO3 < CaCO3 < BaCO3}\)

The basic structural unit of feldspar, zeolites, mica, and asbestos is :

• (1)
• (2) \(\mathrm{(SiO_4)^{4-}}\)
• (3) \(\mathrm{(SiO_3)^{2-}}\)
• (4) \(\mathrm{SiO_2}\)

Correct statement about photo-chemical smog is:

• (1) It occurs in humid climate.
• (2) It is a mixture of smoke, fog and SO2.
• (3) It is reducing smog.
• (4) It results from reaction of unsaturated hydrocarbons

Peptization is a :

• (1) process of converting a colloidal solution into precipitate
• (2) process of converting precipitate into colloidal solution
• (3) process of converting soluble particles to form colloidal solution
• (4) process of bringing colloidal molecule into solution

Which of the following is a thermosetting polymer?

• (1) Buna-N
• (2) PVC
• (3) Bakelite
• (4) Nylon 6