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

A circle touching the x-axis at \((3, 0)\) and making an intercept of length \(8\) on the y-axis passes through the point :

• (1) (3, 10)
• (2) (2,3)
• (3) (1,5)
• (4) (3,5)

A person throws two fair dice. He wins Rs. 15 for throwing a doublet (same numbers on the two dice), wins Rs. 12 when the throw results in the sum of 9, and loses Rs. 6 for any other outcome on the throw. Then the expected gain/loss (in Rs.) of the person is :

• (1) \(\frac{1}{4}\) loss
• (2) \(\frac{1}{2}\) gain
• (3) \(\frac{1}{2}\) loss
• (4) 2 gain

Let \(a \in \left(0, \frac{\pi}{2}\right)\) be fixed. If the integral \(\int \frac{\tan x + \tan \alpha}{\tan x - \tan \alpha}\,dx = A(x)\cos 2\alpha + B(x)\sin 2\alpha + C\), where C is a constant of integration, then the functions A(x) and B(x) are respectively :

• (1) \(x-\alpha\) and \(\log_e|\cos(x-\alpha)|\)
• (2) \(x+\alpha\) and \(\log_e|\sin(x-\alpha)|\)
• (3) \(x+\alpha\) and \(\log_e|\sin(x+\alpha)|\)
• (4) \(x-\alpha\) and \(\log_e|\sin(x-\alpha)|\)

The term independent of x in the expansion of \(\bigg(\frac{1}{60} - \frac{x^8}{81}\bigg). \bigg(2x^2 - \frac{3}{x^2}\bigg)^6\) is equal to:

• (1) 36
• (2) -108
• (3) -72
• (4) -36

Let \(z \in C\) with Im(z) = 10 and it satisfies \(\frac{2z-n}{2z+n} = 2i -1\) for some natural number \(n\).Then :

• (1) n = 20 and Re(z) = - 10
• (2) n = 20 and Re(z) = 10
• (3) n = 40 and Re(z) = - 10
• (4) n = 40 and Re(z) = 10

A group of students comprises of 5 boys and n girls. If the number of ways, in which a team of 3 students can randomly be selected from this group such that there is at least one boy and at least one girl in each team, is 1750, then n is equal to :

• (1) 24
• (2) 25
• (3) 27
• (4) 28

If \(\alpha\), \(\beta\) and \(\gamma\) are three consecutive terms of a non-constant G.P. such that the equations \(\alpha x^2 + 2\beta x + \gamma = 0\) and \(x^2 + x - 1 = 0\) have a common root, then \(\alpha(\beta+\gamma)\) is equal to :

• (1) \(\alpha\gamma\)
• (2) 0
• (3) \(\beta\gamma\)
• (4) \(\alpha\beta\)

The general solution of the differential equation \((y^2 - x^3)\,dx - xy\,dy = 0\) \((x \ne 0)\) is : (where c is a constant of integration)

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

Let the area of the triangle formed by a straight line \( L: x + by + c = 0 \) with co-ordinate axes be 48 square units. If the perpendicular drawn from the origin to the line \( L \) makes an angle of \( 45^\circ \) with the positive x-axis, then the value of \( b^2 + c^2 \) is:

• (1) 90
• (2) 93
• (3) 97
• (4) 83
  

Let A, B and C be sets such that \(\phi \ne A \cap B \subseteq C\). Then which of the following statements is not true ?

• (1) If \((A - B) \subseteq C\), then \(A \subseteq C\)
• (2) \(B \cap C \ne \phi\)
• (3) \((C \cup A) \cap (C \cup B) = C\)
• (4) If \((A - C) \subseteq B\), then \(A \subseteq B\)

Let \(f(x) = 5 - |x-2|\) and \(g(x) = |x+1|\), \(x \in R\). If \(f(x)\) attains maximum value at \(\alpha\) and \(g(x)\) attains minimum value at \(\beta\), then \(\lim_{x \to -\alpha\beta} \frac{(x-1)(x^2-5x+6)}{x^2-6x+8}\) is equal to :

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

A value of \(\alpha\) such that \(\int_{\alpha}^{\alpha+1} \frac{dx}{(x+\alpha)(x+\alpha+1)} = \log_e\left(\frac{9}{8}\right)\) is :

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

An ellipse, with foci at (0, 2) and (0, -2) and minor axis of length 4, passes through which of the following points?

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

A value of \(\theta \ \ \in \ (0, \pi /3)\) for which \(\begin {vmatrix} 1 + cos^2 \theta & sin^2\theta & 4 cos 6\theta\)
\(cos^2 \theta & 1+sin^2 \theta & 4 cos6 \theta\)
\(cos^2 \theta & sin^2 \theta & 1+4 cos6 \theta \end {vmatrix}\) =0 , is :

• (1) \(\frac{7\pi}{24}\)
• (2) \(\frac{\pi}{18}\)
• (3) \(\frac{\pi}{9}\)
• (4) \(\frac{7\pi}{36}\)

If the area (in sq. units) bounded by the parabola \(y^2 = 4\lambda x\) and the line \(y = \lambda x\), \(\lambda \gt 0\), is \(\frac{1}{9}\), then \(\lambda\) is equal to :

• (1) \(4\sqrt{3}\)
• (2) \(2\sqrt{6}\)
• (3) 48
• (4) 24

\(\lim_{x \to 0} \frac{x + 2\sin x}{\sqrt{x^2 + 2\sin x + 1} - \sqrt{\sin^2 x - x + 1}}\) is :

• (1) 6
• (2) 1
• (3) 3
• (4) 2

The derivative of \(\tan^{-1}\left(\frac{\sin x - \cos x}{\sin x + \cos x}\right)\), with respect to \(\frac{x}{2}\), where \(\left(x \in \left(0, \frac{\pi}{2}\right)\right)\) is :

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

If \(a_1, a_2, a_3, \ldots, a_n, \ldots\) are in A.P. such that \(a_4 - a_7 + a_{10} = m\), then the sum of first 13 terms of this A.P., is :

• (1) \(10\,m\)
• (2) \(12\,m\)
• (3) \(13\,m\)
• (4) \(15\,m\)

The equation of common tangent to the curves \(y^2 = 16x\) and \(xy = -4\), is :

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

If [\(x\)] denotes the greatest integer \(\le x\), then the system of linear equations [\(sin\,\theta\)] x + [\(-cos\,\theta\)]y=0 [\(cot\,\theta\)] \(x + y = 0\)

• (1) have infinitely many solutions if \(\theta \in (\frac{\pi}{2} , \frac{2\pi}{3}) \cup (\pi , \frac{7 \pi}{6})\)
• (2) have infinitely many solutions if \(\theta \in (\frac{\pi}{2} , \frac{2\pi}{3}\)and has a unique solution if \(\theta \in (\pi , \frac{7\pi}{6})\)
• (3) has a unique solution if \(\theta \in (\frac{\pi}{2} , \frac{2\pi}{3})\)and have infinitely many solutions if \(\theta \in (\pi, \frac{7 \pi}{6})\)
• (4) has a unique solution if \(\theta \in (\frac{\pi}{2} , \frac{2\pi}{3}) \cup (\pi, \frac{7\pi}{6})\)

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

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

The angle of elevation of the top of a vertical tower standing on a horizontal plane is observed to be \(45^\circ\) from a point A on the plane. Let B be the point 30 m vertically above the point A. If the angle of elevation of the top of the tower from B be \(30^\circ\), then the distance (in m) of the foot of the tower from the point A is :

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

Let \(\alpha \in R\) and the three vectors \(\vec{a} = \alpha\hat{i} + \hat{j} + 3\hat{k}\), \(\vec{b} = 2\hat{i} + \hat{j} - \alpha\hat{k}\) and \(\vec{c} = \alpha\hat{i} - 2\hat{j} + 3\hat{k}\). Then the set \(S = \{\alpha : \vec{a}, \vec{b}\) and \(\vec{c}\) are coplanar\(\}\) :

• (1) contains exactly two numbers only one of which is positive
• (2) is singleton
• (3) contains exactly two positive numbers
• (4) is empty

The tangents to the curve \(y = (x-2)^2 - 1\) at its points of intersection with the line \(x - y = 3\), intersect at the point :

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

A plane which bisects the angle between the two given planes \(2x - y + 2z - 4 = 0\) and \(x + 2y + 2z - 2 = 0\), passes through the point :

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

A triangle has a vertex at \((1, 2)\) and the mid points of the two sides through it are \((-1, 1)\) and \((2, 3)\). Then the centroid of this triangle is :

• (1) \(\left(\frac{1}{3}, 2\right)\)
• (2) \(\left(\frac{1}{3}, \frac{5}{3}\right)\)
• (3) \(\left(1, \frac{7}{3}\right)\)
• (4) \(\left(\frac{1}{3}, 1\right)\)

If \({}^{20}C_1 + (2^2)\,{}^{20}C_2 + (3^2)\,{}^{20}C_3 + \ldots + (20^2)\,{}^{20}C_{20} = A(2^{\beta})\), then the ordered pair \((A, \beta)\) is equal to :

• (1) \((420, 19)\)
• (2) \((420, 18)\)
• (3) \((380, 18)\)
• (4) \((380, 19)\)

For an initial screening of an admission test, a candidate is given fifty problems to solve. If the probability that the candidate solve any problem is \(\frac{4}{5}\), then the probability that he is unable to solve less than two problems is :

• (1) \(\frac{164}{25}\left(\frac{1}{5}\right)^{48}\)
• (2) \(\frac{316}{25}\left(\frac{4}{5}\right)^{48}\)
• (3) \(\frac{201}{5}\left(\frac{1}{5}\right)^{49}\)
• (4) \(\frac{54}{5}\left(\frac{4}{5}\right)^{49}\)

Let S be the set of all \(\alpha \in \mathbb{R}\) such that the equation, \(\cos 2x + \alpha \sin x = 2\alpha - 7\) has a solution. Then S is equal to :

• (1) \([2, 6]\)
• (2) \([3, 7]\)
• (3) \([1, 4]\)
• (4) \(\mathbb{R}\)

The length of the perpendicular drawn from the point \((2, 1, 4)\) to the plane containing the lines \(\vec{r} = (\hat{i}+\hat{j}) + \lambda(\hat{i}+2\hat{j}-\hat{k})\) and \(\vec{r} = (\hat{i}+\hat{j}) + \mu(-\hat{i}+\hat{j}-2\hat{k})\) is :

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

Three identical polaroids \(P_1\), \(P_2\), and \(P_3\) are placed one after another. The pass axis of \(P_2\) and \(P_3\) are inclined at angles of \(60^\circ\) and \(90^\circ\) with respect to the axis of \(P_1\). The source \(S\) has an intensity of \(\frac{256 \, \text{W}}{\text{m}^2}\). The intensity of light at point \(O\) is _____ \(\frac{\text{W}}{\text{m}^2}\).

One kg of water, at \(20^\circ\mathrm{C}\), heated in an electric kettle whose heating element has a mean (temperature averaged) resistance of \(20\ \Omega\). The rms voltage in the mains is \(200\ \mathrm{V}\). Ignoring heat loss from the kettle, time taken for water to evaporate fully, is close to : [Specific heat of water = \(4200\ \mathrm{J/(kg\,^\circ C)}\), Latent heat of water = \(2260\ \mathrm{kJ/kg}\)]

• (1) 10 minutes
• (2) 22 minutes
• (3) 3 minutes
• (4) 16 minutes

A spring whose unstretched length is \(l\) has a force constant \(k\). The spring is cut into two pieces of unstretched lengths \(l_1\) and \(l_2\) where, \(l_1\) = \(nl_2\) and \(n\) is an integer. The ratio \(k_1/k_2\) of the corresponding force constants, \(k_1\) and \(k_2\) will be :

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

A diatomic gas with rigid molecules does \(10\ \mathrm{J}\) of work when expanded at constant pressure. What would be the heat energy absorbed by the gas, in this process ?

• (1) \(35\ \mathrm{J}\)
• (2) \(30\ \mathrm{J}\)
• (3) \(25\ \mathrm{J}\)
• (4) \(40\ \mathrm{J}\)

A transparent cube of side, made of a material of refractive index \(\mu_2\), is immersed in a liquid of refractive index \(\mu_1\ (\mu_1 \lt \mu_2)\). A ray is incident on the face AB at an angle \(\theta\) (shown in the figure). Total internal reflection takes place at point E on the face BC.

Then \(\theta\) must satisfy :

• (1) \(\theta \gt \sin^{-1}\sqrt{\frac{\mu_2^2}{\mu_1^2} - 1}\)
• (2) \(\theta \lt \sin^{-1}\sqrt{\frac{\mu_2^2}{\mu_1^2} - 1}\)
• (3) \(\theta \lt \sin^{-1}\frac{\mu_1}{\mu_2}\)
• (4) \(\theta \gt \sin^{-1}\frac{\mu_1}{\mu_2}\)

Three particles of masses \(50\, g, 100\, g\) and \(150\, g\) are placed at the vertices of an equilateral triangle of side \(1\, m\) (as shown in the figure). The \((x, y)\) coordinates of the centre of mass will be :

• (1) \(\bigg(\frac{7}{12} m , \frac{\sqrt{3}}{8}\bigg)\)
• (2) \(\bigg(\frac{\sqrt{3}}{4} m , \frac{\sqrt{5}}{12}\bigg)\)
• (3) \(\bigg(\frac{7}{12} m , \frac{\sqrt{3}}{4}\bigg)\)
• (4) \(\bigg(\frac{\sqrt{3}}{8} m , \frac{\sqrt{7}}{12}\bigg)\)

A small speaker delivers \(2\ \mathrm{W}\) of audio output. At what distance from the speaker will one detect \(120\ \mathrm{dB}\) intensity sound ? [Given reference intensity of sound as \(10^{-12}\ \mathrm{W/m^2}\)]

• (1) \(20\ \mathrm{cm}\)
• (2) \(10\ \mathrm{cm}\)
• (3) \(40\ \mathrm{cm}\)
• (4) \(30\ \mathrm{cm}\)

A tuning fork of frequency \(480\, Hz\) is used in an experiment for measuring speed of sound (n) in air by resonance tube method. Resonance is observed to occur at two successive lengths of the air column, \(l_1\) = 30 cm and \(l_2\) = \(70 \,cm\). Then, \(v\) is equal to :

• (1) \(332\, ms^{-1}\)
• (2) \(379 \,ms^{-1}\)
• (3) \(384\, ms^{-1}\)
• (4) \(338\, ms^{-1}\)

The electron in a hydrogen atom first jumps from the third excited state to the second excited state and subsequently to the first excited state. The ratio of the respective wavelengths, \(\frac{\lambda_1}{\lambda_2}\), of the photons emitted in this process is :

• (1) \(\frac{22}{5}\)
• (2) \(\frac{7}{5}\)
• (3) \(\frac{9}{7}\)
• (4) \(\frac{20}{7}\)

Figure shown a \(DC\) voltage regulator circuit, with a Zener diode of breakdown voltage = \(6V\). If the unregulated input voltage varies between \(10\, V\) to \(16\, V\), then what is the maximum Zener current ?

• (1) 2.5 mA
• (2) 3.5 mA
• (3) 7.5 mA
• (4) 1.5 mA

The ratio of the weights of a body on the Earth's surface to that on the surface of a planets is \(9 : 4\). The mass of the planet is \(\frac{1}{9}\)th of that of the Earth. If 'R' is the radius of the Earth, what is the radius of the planet ? (Take the planets to have the same mass density)

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

An electron, moving along the x-axis with an initial energy of \(100\, eV\), enters a region of magnetic field \(\vec{B} \, = (1.5 \times 10^{-3} T) \hat{k}\) at \(S\) (See figure). The field extends between \(x = 0\) and \(x = 2\, cm\). The electron is detected at the point \(Q\) on a screen placed \(8\, cm\) away from the point \(S\). The distance \(d\) between \(P\) and \(Q\) (on the screen) is : (electron's charge = \(1.6 \times 10^{-19}C.\) mass of electron = \(9.1 \times 10^{-31}kg)\)

• (1) 12.87 cm
• (2) 1.22 cm
• (3) 11.65 cm
• (4) 2.25 cm

Two particles are projected from the same point with the same speed \(u\) such that they have the same range \(R\), but different maximum heights, \(h_1\) and \(h_2\). Which of the following is correct ?

• (1) \(R^2 \, = \, 2 h_1 h_2\)
• (2) \(R^2 \, = \, 16 h_1 h_2\)
• (3) \(R^2 \, = \, 4 h_1 h_2\)
• (4) \(R^2 \, = \, h_1 h_2\)

A smooth wire of length \(2\pi r\) is bent into a circle and kept in a vertical plane. A bead can slide smoothly on the wire. When the circle is rotating with angular speed \(\omega\) about the vertical diameter AB, as shown in figure, the bead is at rest with respect to the circular ring at position P as shown. Then the value of \(\omega^2\) is equal to -

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

The number density of molecules of a gas depends on their distance r from the origin as, \(n(r) = n_0e^{-\alpha r^4}\) Then the total number of molecules is proportional to :

• (1) \(n_0 \alpha^{1/4}\)
• (2) \(n_0 \alpha^{-3}\)
• (3) \(n_0 \alpha^{-3/4}\)
• (4) \(\sqrt{n_0} \alpha^{1/2}\)

A moving coil galvanometer, having a resistance G, produces full scale deflection when a current \(I_g\) flows through it. This galvanometer can be converted into (i) an ammeter of range 0 to \(I_0\ (I_0 \gt I_g)\) by connecting a shunt resistance \(R_A\) to it and (ii) into a voltmeter of range 0 to V \((V = GI_0)\) by connecting a series resistance \(R_V\) to it. Then,

• (1) \(R_AR_V = G^2\) and \(\frac{R_A}{R_V} = \frac{I_g}{(I_0 - I_g)}\)
• (2) \(R_AR_V = G^2\left(\frac{I_g}{I_0 - I_g}\right)\) and \(\frac{R_A}{R_V} = \left(\frac{I_0 - I_g}{I_g}\right)^2\)
• (3) \(R_AR_V = G^2\left(\frac{I_0 - I_g}{I_g}\right)\) and \(\frac{R_A}{R_V} = \left(\frac{I_g}{I_0 - I_g}\right)^2\)
• (4) \(R_AR_V = G^2\) and \(\frac{R_A}{R_V} = \left(\frac{I_g}{I_0 - I_g}\right)^2\)

A plane electromagnetic wave having a frequency \(\nu = 23.9\ \mathrm{GHz}\) propagates along the positive z-direction in free space. The peak value of the Electric Field is \(60\ \mathrm{V/m}\). Which among the following is the acceptable magnetic field component in the electromagnetic wave ?

• (1) \(\vec{B} = 2\times 10^{-7}\sin(1.5\times 10^{2}\,x + 0.5\times 10^{11}\,t)\,\hat{j}\)
• (2) \(\vec{B} = 60\sin(0.5\times 10^{3}\,x + 0.5\times 10^{11}\,t)\,\hat{k}\)
• (3) \(\vec{B} = 2\times 10^{-7}\sin(0.5\times 10^{3}\,z + 1.5\times 10^{11}\,t)\,\hat{i}\)
• (4) \(\vec{B} = 2\times 10^{-7}\sin(0.5\times 10^{3}\,z - 1.5\times 10^{11}\,t)\,\hat{i}\)

Consider an electron in a hydrogen atom, revolving in its second excited state (having radius \(4.65\,?\)). The de-Broglie wavelength of this electron is :

• (1) 12.9 ?
• (2) 3.5 ?
• (3) 9.7 ?
• (4) 6.6 ?

Find the magnetic field at point \(P\) due to a straight line segment \(AB\) of length \(6\, cm\) carrying a current of \(5\, A\). (See figure) (\(\mu_0 \, = \, 4\pi \times \, 10^{-7} \, N-A^{-2})\)

• (1) \(3.0 \times 10^{-5} T\)
• (2) \(2.5 \times 10^{-5} T\)
• (3) \(2.0 \times 10^{-5} T\)
• (4) \(1.5 \times 10^{-5} T\)

A solid sphere, of radius R acquires a terminal velocity \(v_1\) when falling (due to gravity) through a viscous fluid having a coefficient of viscosity \(\eta\). The sphere is broken into 27 identical solid spheres. If each of these spheres acquires a terminal velocity, \(v_2\), when falling through the same fluid, the ratio \((v_1/v_2)\) equals :

• (1) \(\frac{1}{9}\)
• (2) \(\frac{1}{27}\)
• (3) 27
• (4) 9

A uniform cylindrical rod of length \(L\) and radius \(r\), is made from a material whose Young's modulus of Elasticity equals \(Y\). When this rod is heated by temperature \(T\) and simultaneously subjected to a net longitudinal compressional force \(F\), its length remains unchanged. The coefficient of volume expansion, of the material of the rod, is (nearly) equals to :

• (1) \(F /\left(3 \pi r ^{2} YT \right)\)
• (2) \(3 F /\left(\pi r ^{2} YT \right)\)
• (3) \(6 F /\left(\pi r ^{2} YT \right)\)
• (4) \(9 F /\left(\pi r ^{2} YT \right)\)

A block of mass 5 kg is (i) pushed in case (A) and (ii) pulled in case (B), by a force \(F = 20\ \mathrm{N}\), making an angle of \(30^\circ\) with the horizontal, as shown in the figures. The coefficient of friction between the block and floor is \(\mu = 0.2\). The difference between the accelerations of the blocks, in case (B) and case (A) will be : (\(g = 10\ \mathrm{ms^{-2}}\))

• (1) \(3.2\ \mathrm{ms^{-2}}\)
• (2) \(0.8\ \mathrm{ms^{-2}}\)
• (3) \(0\ \mathrm{ms^{-2}}\)
• (4) \(0.4\ \mathrm{ms^{-2}}\)

Consider the \(LR\) circuit shown in the figure. If the switch \(S\) is closed at \(t = 0\) then the amount of charge that passes through the battery between \(t = 0\) and \(t = \frac{L}{R}\) is:

• (1) \(\frac{EL}{7.3 R^2}\)
• (2) \(\frac{EL}{2.7 R^2}\)
• (3) \(\frac{7.3 EL}{ R^2}\)
• (4) \(\frac{2.7 EL}{ R^2}\)

In the given circuit, the charge on \(4\ \mu\mathrm{F}\) capacitor will be :

• (1) \(5.4\ \mu\mathrm{C}\)
• (2) \(9.6\ \mu\mathrm{C}\)
• (3) \(13.4\ \mu\mathrm{C}\)
• (4) \(24\ \mu\mathrm{C}\)

A particle is moving with speed \(v = b\sqrt{x}\) along positive x-axis. Calculate the speed of the particle at time \(t = \tau\) (assume that the particle is at origin at \(t = 0\)).

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

Let a total charge \(2Q\) be distributed in a sphere of radius \(R\), with the charge density given by \(\rho (r) = kr\), where r is the distance from the centre. Two charges \(A\) and \(B\), of \(-Q\) each, are placed on diametrically opposite points, at equal distance, a, from the centre. If \(A\) and \(B\) do not experience any force, then :

• (1) \(a = \frac{3R}{2^{1/4}}\)
• (2) \(a = R / \sqrt{3}\)
• (3) \(a = 8^{-1/4 R}\)
• (4) \(a = 2^{-1/4} R\)

A Carnot engine has an efficiency of \(\frac{1}{6}\). When the temperature of the sink is reduced by \(62^\circ\mathrm{C}\), its efficiency is doubled. The temperatures of the source and the sink are, respectively

• (1) \(99^\circ\mathrm{C},\ 37^\circ\mathrm{C}\)
• (2) \(124^\circ\mathrm{C},\ 62^\circ\mathrm{C}\)
• (3) \(37^\circ\mathrm{C},\ 99^\circ\mathrm{C}\)
• (4) \(62^\circ\mathrm{C},\ 124^\circ\mathrm{C}\)

In an amplitude modulator circuit, the carrier wave is given by, \(C(t) = 4\sin(20000\pi t)\) while modulating signal is given by, \(m(t) = 2\sin(2000\pi t)\). The values of modulation index and lower side band frequency are :

• (1) 0.3 and 9 kHz
• (2) 0.5 and 10 kHz
• (3) 0.4 and 10 kHz
• (4) 0.5 and 9 kHz

Two sources of sound \(S_1\) and \(S_2\) produce sound waves of same frequency 660 Hz. A listener is moving from source \(S_1\) towards \(S_2\) with a constant speed \(u\) m/s and he hears 10 beats/s. The velocity of sound is 330 m/s. Then, \(u\) equals :

• (1) 10.0 m/s
• (2) 5.5 m/s
• (3) 15.0 m/s
• (4) 2.5 m/s

Half-lives of two radioactive elements \(A\) and \(B\) are \(20\, \) minutes and \(40\,\) minutes, respectively. Initially, the samples have equal number of nuclei. After \(80\,\) minutes, the ratio of decayed numbers of \(A\) and \(B\) nuclei will be :

• (1) \(1 : 16\)
• (2) \(4 : 1\)
• (3) \(1 : 4\)
• (4) \(5 : 4\)

Among the following, the energy of 2s orbital is lowest in :

• (1) Li
• (2) H
• (3) Na
• (4) K

In comparison to boron, berylium has :

• (1) lesser nuclear charge and greater first ionisation enthalpy
• (2) lesser nuclear charge and lesser first ionisation enthalpy
• (3) greater nuclear charge and greater first ionisation enthalpy
• (4) greater nuclear charge and lesser first ionisation enthalpy

A solution is prepared by dissolving \(0.6\, g\) of urea (molar mass = \(60\, g\, mol^{-1})\) and \(1.8\, g\) of glucose (molar mass = \(180\, g\) mol\({-1}\)) in \(100\, mL\) of water at \(27^{\circ}C\). The osmotic pressure of the solution is : \((R = 0.08206 \; L \; atm \; K^{-1} \; mol^{-1} ) \)

• (1) 4.92 atm
• (2) 1.64 atm
• (3) 2.46 atm
• (4) 8.2 atm

The decreasing order of electrical conductivity of the following aqueous solutions is : 0.1 M Formic acid (A), 0.1 M Acetic acid (B) 0.1 M Benzoic acid (C)

• (1) C > B > A
• (2) A > B > C
• (3) A > C > B
• (4) C > A > B

Which one of the following is likely to give a precipitate with \(\mathrm{AgNO_3}\) solution ?

• (1) \(\mathrm{CHCl_3}\)
• (2) \(\mathrm{(CH_3)_3CCl}\)
• (3) \(\mathrm{CCl_4}\)
• (4) \(\mathrm{CH_2{=}CH{-}Cl}\)

Benzene diazonium chloride on reaction with aniline in the presence of dilute hydrochloric acid gives :

• (1)
• (2)
• (3)
• (4)

The compound used in the treatment of lead poisoning is :

• (1) \(EDTA\)
• (2) Cis-platin
• (3) D-penicillamine
• (4) desferrioxime \(B\)

The IUPAC name for the following compound is :

• (1) 3, 5-dimethyl-4-propylhept-1-en-6-yne
• (2) 3-methyl-4-(3-methylprop-1-enyl)-1-heptyne
• (3) 3-methyl-4-(1-methylprop-2-enyl)-1-heptene
• (4) 3,5-dimethyl-4-propylhept-6-en-1-yne

The INCORRECT match in the following is :

• (1) \(\Delta G^\circ = 0,\ K = 1\)
• (2) \(\Delta G^\circ \lt 0,\ K \lt 1\)
• (3) \(\Delta G^\circ \gt 0,\ K \lt 1\)
• (4) \(\Delta G^\circ \lt 0,\ K \gt 1\)

10 mL of gaseous hydrocarbon on combustion gives 40 mL of CO\(_2\)(g) and 50 mL of water vapour. The total number of carbon and hydrogen atoms in the hydrocarbon is ______ .

In which one of the following equilibria, \(K_p \neq K_C\) ?

• (1) \(\mathrm{2NO(g) \rightleftharpoons N_2(g) + O_2(g)}\)
• (2) \(\mathrm{2C(s) + O_2(g) \rightleftharpoons 2CO(g)}\)
• (3) \(\mathrm{2HI(g) \rightleftharpoons H_2(g) + I_2(g)}\)
• (4) \(\mathrm{NO_2(g) + SO_2(g) \rightleftharpoons NO(g) + SO_3(g)}\)

Consider the following reactions

'A' is :

• (1)
• (2)
• (3)
• (4)

Heating of 2-chloro-1-phenylbutane with EtOK/EtOH gives X as the major product. Reaction of X with \(\mathrm{Hg(OAc)_2/H_2O}\) followed by \(\mathrm{NaBH_4}\) gives Y as the major product. Y is :

• (1)
• (2)
• (3)
• (4)

The pair that has similar atomic radii is :

• (1) Sc and Ni
• (2) Ti and HF
• (3) Mo and W
• (4) Mn and Re

An 'Assertion' and a 'Reason' are given below. Choose the correct answer from the following options :
Assertion (A) : Vinyl halides do not undergo nucleophilic substitution easily.
Reason (R) : Even though the intermediate carbocation is stabilized by loosely held p-electrons, the cleavage is difficult because of strong bonding.

• (1) Both (A) and (R) are correct statements but (R) is not the correct explanation of (A)
• (2) Both (A) and (R) are correct statements and (R) is the correct explanation of (A)
• (3) (A) is correct statement but (R) is a wrong statement
• (4) Both (A) and (R) are wrong statements.

What will be the major product when m-cresol is reacted with propargyl bromide (HC\(\equiv\)C–CH₂Br) in presence of K₂CO₃ in acetone?

• (1)
• (2)
• (3)
• (4)

The coordination numbers of Co and Al in \({ [Co(Cl)(en)2]Cl}\) and \({K3[Al(C2O4)3]}\), respectively, are : (en=ethane-1,2-diamine)

• (1) 3 and 3
• (2) 6 and 6
• (3) 5 and 6
• (4) 5 and 3

The molar solubility of Cd(OH)₂ is \(1.84 \times 10^{-5}\) M in water. The expected solubility of Cd(OH)₂ in a buffer solution of pH = 12 is :

• (1) \(2.49 \times 10^{-10}\) M
• (2) \(1.84 \times 10^{-9}\) M
• (3) \(6.23 \times 10^{-11}\) M
• (4) \(\frac{2.49}{1.84} \times 10^{-9}\) M

\( {NO_2}\) required for a reaction is produced by the decomposition of \( {N2O5}\) in \( {CCl4}\) as per the equation \( { 2N2O5(g) -> 4NO2(g) + O2(g).}\) The initial concentration of \( {N2O5}\) is \(3.00\, mol\) \( {L^{-1}}\) and it is \(2.75\, mol\) \( {L^{-1}}\) after \(30\) minutes. The rate of formation of \( {NO2}\) is :

• (1) \(\ce{2.083 \times 10^{-3} \; mol \; L^{-1} \; min^{-1}}\)
• (2) \(\ce{4.167 \times 10^{-3} \; mol \; L^{-1} \; min^{-1}}\)
• (3) \(\ce{8.333 \times 10^{-3} \; mol \; L^{-1} \; min^{-1}}\)
• (4) \(\ce{1.667 \times 10^{-2} \; mol \; L^{-1} \; min^{-1}}\)

Thermal decomposition of a Mn compound (X) at 513 K results in compound Y, MnO₂ and gaseous product. MnO₂ reacts with NaCl and concentrated H₂O₄ to give a pungent gas Z. X, Y and Z, respectively, are :

• (1) KMnO₄, K₂MnO₄ and Cl₂
• (2) K₂MnO₄, KMnO₄ and SO₂
• (3) K₃MnO₄, K₂MnO₄ and Cl₂
• (4) K₂MnO₄, KMnO₄ and Cl₂

In the following skew conformation of ethane, \(H'-C-C-H''\) dihedral angle is :

• (1) \(120^{\circ}\)
• (2) \(58^{\circ}\)
• (3) \(149^{\circ}\)
• (4) \(151^{\circ}\)

Which of the given statements is INCORRECT about glycogen?

• (1) It is a straight chain polymer similar to amylose
• (2) It is present in animal cells
• (3) It is present in some yeast and fungi
• (4) Only \(\alpha\)-linkages are present in the molecule

The temporary hardness of a water sample is due to compound X. Boiling this sample converts X to compound Y. X and Y, respectively, are :

• (1) Mg(HCO₃)₂ and MgCO₃
• (2) Ca(HO₃)₂ and CaO
• (3) Ca(HCO₃)₂ and Ca(OH)₂
• (4) Mg(HCO₃)₂ and Mg(OH)₂

The correct statement is :

• (1) leaching of bauxite using concentrated NaOH solution gives sodium aluminate and sodium silicate
• (2) the Hall-Heroult process is used for the production of aluminium and iron
• (3) pig iron is obtained from cast iron
• (4) the blistered appearance of copper during the metallurgical process is due to the evolution of CO₂

The correct name of the following polymer is :

• (1) Polytert-butylene
• (2) Polyisobutane
• (3) Polyisobutylene
• (4) Polyisoprene

The INCORRECT statement is :

• (1) Lithium is least reactive with water among the alkali metals.
• (2) \(LiCl\) crystallises from aqueous solution as \(\ce{LiCl.2H2O.}\)
• (3) Lithium is the strongest reducing agent among the alkali metals.
• (4) \(\ce{LiNO3}\) decomposes on heating to give \(\ce{LiNO2}\) and \(\ce{O2}\).

The primary pollutant that leads to photochemical smog is :

• (1) sulphur dioxide
• (2) acrolein
• (3) ozone
• (4) nitrogen oxides

The ratio of number of atoms present in a simple cubic, body centered cubic and face centered cubic structure are, respectively :

• (1) 1:02:04
• (2) 8:01:06
• (3) 4:02:01
• (4) 4:02:03

The C–C bond length is maximum in :

• (1) C₆₀
• (2) diamond
• (3) graphite
• (4) C₇₀

Among the following, the INCORRECT statement about colloids is :

• (1) The range of diameters of colloidal particles is between 1 and to 1000 nm
• (2) they are larger than small molecules and have high molar mass
• (3) They can scatter light
• (4) The osmotic pressure of a colloidal solution is of higher order than the true solution at the same concentration.