JEE Main 2018 B.E./B.Tech Question Paper with Answer Key PDF- April 15 (Afternoon Session)

If the system of linear equations
\(x + ay + z = 3\)
\(x + 2y + 2z = 6\)
\(x + 5y + 3z = b\)
has no solution, then :

• (1) \(a = -1,\ b = 9\)
• (2) \(a = -1,\ b \neq 9\)
• (3) \(a \neq -1,\ b = 9\)
• (4) \(a = 1,\ b \neq 9\)

Let \(f(x) = \begin{cases} (x-1)^{\frac{1}{2-x}}, & x > 1,\ x \neq 2 \\ k, & x = 2 \end{cases}\)
The value of k for which f is continuous at \(x = 2\) is :

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

If \(|z - 3 + 2i| \leq 4\) then the difference between the greatest value and the least value of \(|z|\) is :

• (1) \(2\sqrt{13}\)
• (2) \(8\)
• (3) \(4+\sqrt{13}\)
• (4) \(\sqrt{13}\)

Suppose A is any \(3 \times 3\) non-singular matrix and \((A - 3I)(A - 5I) = O\) where \(I = I_3\) and \(O = O_3\). If \(\alpha A + \beta A^{-1} = 4I\), then \(\alpha + \beta\) is equal to :

• (1) \(8\)
• (2) \(7\)
• (3) \(13\)
• (4) \(12\)

Tangents drawn from the point \((-8, 0)\) to the parabola \(y^2 = 8x\) touch the parabola at \(P\) and \(Q\) . If \(F\) is the focus of the parabola, then the area of the triangle \(PFQ\) (in s units) is equal to :

• (1) 24
• (2) 32
• (3) 48
• (4) 64

The sides of a rhombus ABCD are parallel to the lines, \(x - y + 2 = 0\) and \(7x - y + 3 = 0\). If the diagonals of the rhombus intersect P(1, 2) and the vertex A (different from the origin) is on the y-axis, then the coordinate of A is :

• (1) \(\dfrac{5}{2}\)
• (2) \(\dfrac{7}{4}\)
• (3) \(2\)
• (4) \(\dfrac{7}{2}\)

If a, b, c are in A.P. and \(a^2, b^2, c^2\) are in G.P. such that \(a < b < c\) and \(a + b + c = \dfrac{3}{4}\), then the value of a is :

• (1) \(\dfrac14 - \dfrac{1}{4\sqrt2}\)
• (2) \(\dfrac14 - \dfrac{1}{3\sqrt2}\)
• (3) \(\dfrac14 - \dfrac{1}{2\sqrt2}\)
• (4) \(\dfrac14 - \dfrac{1}{\sqrt2}\)

The number of four letter words that can be formed using the letters of the word BARRACK is :

• (1) \(120\)
• (2) \(144\)
• (3) \(264\)
• (4) \(270\)

If \(f(x) = \sin^{-1}\left(\dfrac{2 \times 3^x}{1 + 9^x}\right)\), then \(f'\left(-\dfrac12\right)\) equals :

• (1) \(-\sqrt3 \log_e \sqrt3\)
• (2) \(\sqrt3 \log_e \sqrt3\)
• (3) \(-\sqrt3 \log_e 3\)
• (4) \(\sqrt3 \log_e 3\)

A player X has a biased coin whose probability of showing heads is p and a player Y has a fair coin. They start playing a game with their own coins and play alternately. The player who throws a head first is a winner. If X starts the game, and the probability of winning the game by both the players is equal, then the value of 'p' is :

• (1) \(\dfrac15\)
• (2) \(\dfrac13\)
• (3) \(\dfrac25\)
• (4) \(\dfrac14\)

Let \(f: A \to B\) be a function defined as \(f(x) = \dfrac{x-1}{x-2}\), where \(A = \mathbf{R} - \{2\}\) and \(B = \mathbf{R} - \{1\}\). Then f is :

• (1) invertible and \(f^{-1}(y) = \dfrac{3y-1}{y-1}\)
• (2) invertible and \(f^{-1}(y) = \dfrac{2y-1}{y-1}\)
• (3) invertible and \(f^{-1}(y) = \dfrac{2y+1}{y-1}\)
• (4) not invertible

Let α be the angle between the lines whose direction cosines satisfy the equations l + m - n = 0 and l² + m² - n² = 0. Then the value of sin⁴α + cos⁴α is :

• (1) 5/8
  
• (2) 3/4
  
• (3) 1/2
  
• (4) 3/4

The curve satisfying the differential equation, \((x^2 - y^2) dx + 2xydy = 0\) and passing through the point \((1, 1)\) is :

• (1) a circle of radius one.
• (2) a hyperbola.
• (3) an ellipse.
• (4) a circle of radius two.

If \(\int \frac{d\theta}{\cos^2\theta\,(\tan 2\theta + \sec 2\theta)} = \lambda \tan\theta + 2\log_e |f(\theta)| + C\) where C is a constant of integration, then the ordered pair \((\lambda, f(\theta))\) is equal to :

• (1) \((-1,\ 1 - \tan\theta)\)
• (2) \((1,\ 1 + \tan\theta)\)
• (3) \((-1,\ 1 + \tan\theta)\)
• (4) \((1,\ 1 - \tan\theta)\)

The foot of the perpendicular drawn from the origin, on the line, \(3x + y = \lambda\ (\lambda \neq 0)\) is P. If the line meets x-axis at A and y-axis at B, then the ratio BP : PA is :

• (1) \(1 : 3\)
• (2) \(3 : 1\)
• (3) \(1 : 9\)
• (4) \(9 : 1\)

If \(I_1 = \int_0^1 e^{-x}\cos^2 x\,dx;\ I_2 = \int_0^1 e^{-x^2}\cos^2 x\,dx\) and \(I_3 = \int_0^1 e^{-x^3}\,dx\); then

• (1) \(I_2 \gt I_3 \gt I_1\)
• (2) \(I_2 \gt I_1 \gt I_3\)
• (3) \(I_3 \gt I_2 \gt I_1\)
• (4) \(I_3 \gt I_1 \gt I_2\)

If f(x) is a quadratic expression such that \(f(1) + f(2) = 0\), and \(-1\) is a root of \(f(x) = 0\), then the other root of \(f(x) = 0\) is :

• (1) \(-\dfrac58\)
• (2) \(-\dfrac85\)
• (3) \(\dfrac58\)
• (4) \(\dfrac85\)

The value of integral \(\displaystyle\int_{\pi/4}^{3\pi/4} \dfrac{x}{1+\sin x}\,dx\) is :

• (1) \(\pi\sqrt2\)
• (2) \(\pi\left(\sqrt2 - 1\right)\)
• (3) \(\dfrac\pi2\left(\sqrt2 + 1\right)\)
• (4) \(2\pi\left(\sqrt2 - 1\right)\)

If the mean of the data : 7, 8, 9, 7, 8, 7, \(\lambda\), 8 is 8, then the variance of this data is :

• (1) \(\frac{7}{8}\)
• (2) 1
• (3) \(\frac{9}{8}\)
• (4) 2

If the position vectors of the vertices A, B and C of a \(\Delta\) ABC are respectively \(4\hat{i}+7\hat{j}+8\hat{k}\), \(2\hat{i}+3\hat{j}+4\hat{k}\), and \(2\hat{i}+5\hat{j}+7\hat{k}\), then the position vectors of the point, where the bisector of \(\angle A\) meets BC is :

• (1) \(\frac{1}{2}\left(4\hat{i}+8\hat{j}+11\hat{k}\right)\)
• (2) \(\frac{1}{3}\left(6\hat{i}+11\hat{j}+15\hat{k}\right)\)
• (3) \(\frac{1}{3}\left(6\hat{i}+13\hat{j}+18\hat{k}\right)\)
• (4) \(\frac{1}{4}\left(8\hat{i}+14\hat{j}+19\hat{k}\right)\)

\(\displaystyle\lim_{x \to0} \frac{x \tan2x - 2x \tan x}{\left(1-\cos2x\right)^{2}} \) equals :

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

Let \( f : \mathbb{R} \to \mathbb{R} \) be a polynomial function of degree four having extreme values at \( x = 4 \) and \( x = 5 \). If \lim_{x \to 0} \frac{f(x)}{x^2} = 5, then f(2) is equal to:

• (1) 12
• (2) 10
• (3) 8
• (4) 14

A normal to the hyperbola, \(4x^2-9y^2=36\) meets the co-ordinate axes \(x\) and \(y\) at A and B, respectively. If the parallelogram OABP (O being the origin) is formed, then the locus of P is :

• (1) \(4x^2+9y^2=121\)
• (2) \(9x^2+4y^2=169\)
• (3) \(4x^2-9y^2=121\)
• (4) \(9x^2-4y^2=169\)

If one of the diameters of the circle \(x^2 + y^2 - 2x - 6y + 6 = 0\) is a chord of another circle 'C', whose center is at (2, 1), then its radius is ________

The coefficient of \(x^{10}\) in the expansion of \((1+x)^2(1+x^2)^3(1+x^3)^4\) is equal to :

• (1) 52
• (2) 56
• (3) 50
• (4) 44

Let \(A_n=\left(\frac{3}{4}\right)-\left(\frac{3}{4}\right)^2+\left(\frac{3}{4}\right)^3-\ldots+(-1)^{n-1}\left(\frac{3}{4}\right)^n\) and \(B_n=1-A_n\).
Then, the least odd natural number p, so that \(B_n>A_n\), for all \(n\ge p\), is :

• (1) 9
• (2) 7
• (3) 11
• (4) 5

A plane bisects the line segment joining the points (1, 2, 3) and (-3, 4, 5) at right angles. Then this plane also passes through the point :

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

The number of solutions of \(\sin \, 3x = \cos \, 2x\), in the interval \(\left( \frac{\pi}{2} , \pi \right)\) is :

• (1) 1
• (2) 2
• (3) 3
• (4) 4

A tower \(T_1\) of height 60 m is located exactly opposite to a tower \(T_2\) of height 80 m on a straight road. From the top of \(T_1\), if the angle of depression of the foot of \(T_2\) is twice the angle of elevation of the top of \(T_2\), then the width (in m) of the road between the feet of the towers \(T_1\) and \(T_2\) is :

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

Consider the following two statements : The value of \(\sin \, 120^{\circ}\) can be derived by taking \(\theta =240^{\circ}\) in the equation \(2 \sin \, \frac{\theta}{2} = \sqrt{1 + \sin \, \theta} - \sqrt{1 - \sin \, \theta}\). The angles A, B, C and D of any quadrilateral ABCD satisfy the equation \(\cos \left( \frac{1}{2} (A + C) \right) + \cos \left( \frac{1}{2} (B + D) \right) = 0 \) Then the truth values of p and q are respectively :

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

Muon (\(\mu^-\)) is a negatively charged (\(|q|=|e|\)) particle with a mass \(m_\mu=200\,m_e\), where \(m_e\) is the mass of the electron and e is the electronic charge. If \(\mu^-\) is bond to a proton to form a hydrogen like atom, identify the correct statements.
(A) Radius of the muonic orbit is 200 times smaller than that of the electron.
(B) The speed of the \(\mu^-\) in the \(n^{th}\) orbit is \(\frac{1}{200}\) times that of the electron in the \(n^{th}\) orbit.
(C) The ionization energy of muonic atom is 200 times more than of an hydrogen atom.
(D) The momentum of the muon in the \(n^{th}\) orbit is 200 times more than that of the electron.

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

An unstable heavy nucleus at rest breaks into two nuclei which move away with velocities in the ratio of \(8 : 27\). The ratio of the radii of the nuclei (assumed to be spherical) is :

• (1) 8:27
• (2) 4:09
• (3) 3:02
• (4) 2:03

A convergent doublet of separated lenses, corrected for spherical aberration, has resultant focal length of \(10 \,cm\). The separation between the two lenses is \(2 \,cm\). The focal lengths of the component lenses are :

• (1) 10 cm, 12 cm
• (2) 12 cm, 14 cm
• (3) 16 cm, 18 cm
• (4) 18 cm, 20 cm

A constant voltages is applied between two ends of a metallic wire. If the length is halved and the radius of the wire is doubled, the rate of heat developed in the wire will be :

• (1) Doubled
• (2) Halved
• (3) Unchanged
• (4) Increased 8 times

A plane polarized monochromatic \(EM \) wave is traveling in vacuum along \(z\) direction such that at \(t = t_1\) it is found that the electric field is zero at a spatial point \(z_1\). The next zero that occurs in its neighbourhood is at \(z_2\). The frequency of the electromagnetic wave is :

• (1) \(\frac{3 \times 10^8}{|z_2 - z_1|}\)
• (2) \(\frac{1.5 \times 10^8}{|z_2 - z_1|}\)
• (3) \(\frac{6 \times 10^8}{|z_2 - z_1|}\)
• (4) \(\frac{1}{t_1 + \frac{|z_2 - z_1|}{3 \times 10^8}} \)

A thin rod \(MN\), free to rotate in the vertical plane about the fixed end \(N\), is held horizontal. When the end \(M\) is released the speed of this end, when the rod makes an angle \(\alpha\) with the horizontal, will be proportional to : (see figure)

• (1) \(\sqrt{\sin \, \alpha}\)
• (2) \({\sin \, \alpha}\)
• (3) \(\sqrt{\cos\, \alpha}\)
• (4) \({\cos\, \alpha}\)

A solid ball of radius R has a charge density \(\rho\) given by \(\rho=\rho_0\left(1-\frac{r}{R}\right)\) for \(0\le r\le R\). The electric field outside the ball is :

• (1) \(\frac{\rho_0R^3}{\varepsilon_0r^2}\)
• (2) \(\frac{\rho_0R^3}{12\,\varepsilon_0r^2}\)
• (3) \(\frac{4\rho_0R^3}{3\,\varepsilon_0r^2}\)
• (4) \(\frac{3\rho_0R^3}{4\,\varepsilon_0r^2}\)

A body of mass \(2\, kg\) slides down with an acceleration of \(3 \, m/s^2\) on a rough inclined plane having a slope of \(30^{\circ}\). The external force required to take the same body up the plane with the same acceleration will be : \((g = 10 \, m/s^2)\)

• (1) 14 N
• (2) 20 N
• (3) 6 N
• (4) 4 N

A plane polarized light is incident on a polariser with its pass axis making angle \(?\) with \(x\)-axis, as shown in the figure. At four different values of \(\theta, \theta = 8^{\circ}, 38^{\circ} , 188^{\circ} \) and \(218^{\circ}\), the observed intensities are same. What is the angle between the direction of polarization and x-axis ?

• (1) \(98^{\circ}\)
• (2) \(128^{\circ}\)
• (3) \(203^{\circ}\)
• (4) \(45^{\circ}\)

A capacitor \(C_1=1.0\ \mu\mathrm{F}\) is charged up to a voltage V = 60 V by connecting it to battery B through switch (1). Now \(C_1\) is disconnected from battery and connected to a circuit consisting of two uncharged capacitors \(C_2=3.0\ \mu\mathrm{F}\) and \(C_3=6.0\ \mu\mathrm{F}\) through switch (2), as shown in the figure. The sum of final charges on \(C_2\) and \(C_3\) is :

• (1) \(40\ \mu\mathrm{C}\)
• (2) \(36\ \mu\mathrm{C}\)
• (3) \(20\ \mu\mathrm{C}\)
• (4) \(54\ \mu\mathrm{C}\)

A copper rod of mass m slides under gravity on two smooth parallel rails, with separation \(l \) and set at an angle of \(\theta\) with the horizontal. At the bottom, rails are joined by a resistance \(R\). There is a uniform magnetic field \(B\) normal to the plane of the rails, as shown in the figure. The terminal speed of the copper rod is :

• (1) \(\frac{mg \, R \, \tan \, \theta}{B^2 l^2}\)
• (2) \(\frac{mg \, R \, \cot\, \theta}{B^2 l^2}\)
• (3) \(\frac{mg \, R \, \sin\, \theta}{B^2 l^2}\)
• (4) \(\frac{mg \, R \, \cos \, \theta}{B^2 l^2}\)

A man in a car at location \(Q\) on a straight highway is moving with speed \(v\). He decides to reach a point \(P\) in a field at a distance d from the highway (point \(M\)) as shown in the figure. Speed of the car in the field is half to that on the highway. What should be the distance \(RM\), so that the time taken to reach \(P\) is minimum ?

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

A copper rod of cross-sectional area A carries a uniform current I through it. At temperature T, if the volume charge density of the rod is \(\rho\), how long will the charges take to travel a distance d ?

• (1) \(\frac{2\rho\,d\,A}{\mathrm{I}}\)
• (2) \(\frac{2\rho\,d\,A}{\mathrm{I}\,T}\)
• (3) \(\frac{\rho\,d\,A}{\mathrm{I}}\)
• (4) \(\frac{\rho\,d\,A}{\mathrm{I}\,T}\)

As shown in the figure, forces of \(10^5\, N\) each are applied in opposite directions, on the upper and lower faces of a cube of side \(10\, cm\), shifting the upper face parallel to itself by \(0.5 \,cm\). If the side of another cube of the same material is \(20\, cm\), then under similar conditions as above, the displacement will be :

• (1) 0.25 cm
• (2) 0.37 cm
• (3) 0.75 cm
• (4) 1.00 cm

A disc rotates about its axis of symmetry in a horizontal plane at a steady rate of \(3.5\) revolutions per second. A coin placed at a distance of \(1.25\, cm\) from the axis of rotation remains at rest on the disc. The coefficient of friction between the coin and the disc is : \((g = 10 \, m/s^2)\)

• (1) 0.5
• (2) 0.3
• (3) 0.7
• (4) 0.6

At the center of a fixed large circular coil of radius R, a much smaller circular coil of radius r is placed. The two coils are concentric and are in the same plane. The larger coil carries a current I. The smaller coil is set to rotate with a constant angular velocity \(\omega\) about an axis along their common diameter. Calculate the emf induced in their smaller coil after a time t of its start of rotation.

• (1) \(\frac{\mu_0\mathrm{I}}{2R}\,\omega\,\pi r^2\sin\omega t\)
• (2) \(\frac{\mu_0\mathrm{I}}{4R}\,\omega\,\pi r^2\sin\omega t\)
• (3) \(\frac{\mu_0\mathrm{I}}{4R}\,\omega\,r^2\sin\omega t\)
• (4) \(\frac{\mu_0\mathrm{I}}{2R}\,\omega\,r^2\sin\omega t\)

When an air bubble of radius \(r\) rises from the bottom to the surface of a lake, its radius becomes \(\frac{5r}{4}\). Taking the atmospheric pressure to be equal to \(10\, m\) eight of water column, the depth of the lake would approximately be (ignore the surface tension and the effect of temperature) :

• (1) 11.2 m
• (2) 8.7 m
• (3) 9.5 m
• (4) 10.5 m

The characteristic distance at which quantum gravitational effects are significant, the Planck length, can be determined from a suitable combination of the fundamental physical constants \(G, h\) and \(c\). Which of the following correctly gives the Planck length ?

• (1) \(G \hbar^2 \, c^3 \)
• (2) \(G^2 \hbar \, c \)
• (3) \(G^{1/2} \hbar^2 \, c\)
• (4) \(\left( \frac{G \hbar}{c^3} \right)^{1/2}\)

Two simple harmonic motions, as shown below, are at right angles. They are combined to form Lissajous figures.
\(x(t)=A\sin(at+\delta)\)
\(y(t)=B\sin(bt)\)

Identify the correct match below.

• (1) Parameters \(A\ne B,\ a=b;\ \delta=0\);
  Curve Parabola
• (2) Parameters \(A=B,\ a=b;\ \delta=\frac{\pi}{2}\)
  Curve Line
• (3) Parameters \(A\ne B,\ a=b;\ \delta=\frac{\pi}{2}\)
  Curve Ellipse
• (4) Parameters \(A=B,\ a=2b;\ \delta=\frac{\pi}{2}\)
  Curve Circle

As shown in the figure, a current of \(2 A\) flowing in an equilateral triangle of side \(4 \sqrt{3} cm\) The magnetic field at the centroid \(O\) of the triangle is

 (Neglect the effect of earth's magnetic field)

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

A thin uniform bar of length \(L\) and mass \(8m\) lies on a smooth horizontal table. Two point masses \(m\) and \(2m\) are moving in the same horizontal plane from opposite sides of the bar with speeds \(2v\) and \(v\) respectively. The masses stick to the bar after collision at a distance \(\frac{L}{3}\) and \(\frac{L}{6}\) respectively from the centre of the bar. If the bar starts rotating about its center of mass as a result of collision, the angular speed of the bar will be :

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

A parallel plate capacitor with plates of area 1 m² each, are at a separation of 0.1 m. If the electric field between the plates is 100 N/C, the magnitude of charge on each plate is :
(Take \(\varepsilon_0=8.85\times10^{-12}\ \frac{\mathrm{C^2}}{\mathrm{N\,m^2}}\))

• (1) \(9.85\times10^{-10}\) C
• (2) \(8.85\times10^{-10}\) C
• (3) \(6.85\times10^{-10}\) C
• (4) \(7.85\times10^{-10}\) C

The value closest to the thermal velocity of a Helium atom at room temperature \((300\, K)\) in \(ms^{-1}\) is :\( [k_B = 1.4 \times 10^{-23} \, J/K; m_{He} = 7 \times 10^{-27} \, kg ]\)

• (1) \(1.3 \times 10^4\)
• (2) \(1.3 \times 10^3\)
• (3) \(1.3 \times 10^5\)
• (4) \(1.3 \times 10^2\)

A proton of mass m collides elastically with a particle of unknown mass at rest. After the collision, the proton and the unknown particle are seen moving at an angle of \(90^{\circ}\) with respect to each other. The mass of unknown particle is :

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

Truth table for the following digital circuit will be :

• (1)

  x	y	z
  0	0	0
  0	1	0
  1	0	0
  1	1	1

• (2)

  x	y	z
  0	0	1
  0	1	1
  1	0	1
  1	1	0

• (3)

  x	y	z
  0	0	1
  0	1	1
  1	0	1
  1	1	1

• (4)

  x	y	z
  0	0	0
  0	1	1
  1	0	1
  1	1	1

If the de Broglie wavelengths associated with a proton and an \(\alpha\)-particle are equal, then the ratio of velocities of the proton and the \(\alpha\)-particle will be :

• (1) 4 : 1
• (2) 2 : 1
• (3) 1 : 2
• (4) 1 : 4

A tuning fork resonates with a sonometer wire of length 1 m stretched with a tension of 6 N. When the tension in the wire is changed to 54 N, the same tuning fork produces 12 beats per second with it. The frequency of the tuning fork is _______ Hz.

Two Carnot engines A and B are operated in series. Engine A receives heat from a reservoir at 600 K and rejects heat to a reservoir at temperature T. Engine B receives heat rejected by engine A and in turn rejects it to a reservoir at 100 K. If the efficiencies of the two engines A and B are represented by \(\eta_A\) and \(\eta_B\), respectively, then what is the value of \(\dfrac{\eta_B}{\eta_A}\) ?

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

The carrier frequency of a transmitter is provided by a tank circuit of a coil of inductance \( 49\,\mu H\) and a capacitance of \(2.5 \,nF\). It is modulated by an audio signal of \(12 \,kHz\). The frequency range occupied by the side bands is:

• (1) 13482 kHz - 13494 kHz
• (2) 442 kHz - 466 kHz
• (3) 63 kHz - 75 kHz
• (4) 18 kHz - 30 kHz

A body takes \(10\) minutes to cool from \(60^{\circ}C\) to \(50^{\circ}C\). The temperature of surroundings is constant at \(25^{\circ}C\). Then, the temperature of the body after next \(10\) minutes will be approximately :

• (1) \(47^{\circ} C \)
• (2) \(41^{\circ} C \)
• (3) \(45^{\circ} C \)
• (4) \(43^{\circ} C \)

The increasing order of diazotisation of the following compounds is :

• (1) (a) < (b) < (c) < (d)
• (2) (a) < (d) < (b) < (c)
• (3) (a) < (d) < (c) < (b)
• (4) (d) < (c) < (b) < (a)

For a first order reaction, \(A \to P, t_{1/2}\) (half life) is 10 days. The time required for \(\frac{1^{th}}{4}\) conversion of A (in days) is : (ln 2=0.693, ln 3=1.1)

• (1) 5
• (2) 3.2
• (3) 4.1
• (4) 2.5

The correct order of electron affinity is :

• (1) F > Cl > O
• (2) F > O > Cl
• (3) Cl > F > O
• (4) O > F > Cl

Which of the following best describes the diagram below of a molecular orbital ?

• (1) A non-bonding orbital
• (2) An antibonding \(\sigma\) orbital
• (3) A bonding \(\pi\) orbital
• (4) An antibonding \(\pi\) orbital

The dipeptide, Gln-Gly, on treatment with CH₃COCl followed by aqueous work up gives :

• (1)
• (2)
• (3)
• (4)

The major product formed in the following reaction is :

• (1)
• (2)
• (3)
• (4)

Two 5 molal solutions are prepared by dissolving a non-electrolyte non-volatile solute separately in the solvents X and Y. The molecular weights of the solvents are \(M_x\) and \(M_y\), respectively where \(M_x = \frac{3}{4}M_y\). The relative lowering of vapor pressure of the solution in X is "m" times that of the solution in Y. Given that the number of moles of solute is very small in comparison to that of the solvent, the value of "m" is :

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

At a certain temperature in a 5 L vessel, 2 moles of carbon monoxide and 3 moles of chlorine were allowed to reach equilibrium according to the reaction,\(CO + Cl_{2} <=> COCl_2 \) At equilibrium, if one mole of CO is present then equilibrium constant \((K_c)\) for the reaction is :

• (1) 2
• (2) 2.5
• (3) 3
• (4) 4

Two compounds I and II are eluted by column chromatography (adsorption of I > II). Which one of following is a correct statement ?

• (1) I moves faster and has higher \(R_f\) value than \(II\)
• (2) II moves faster and has higher \(R_f\) value than \(I\)
• (3) I moves slower and has higher \(R_f\) value than \( II\)
• (4) II moves slower and has higher \(R_f\) value than \(I\)

For per gram of reactant, the maximum quantity of N₂ gas is produced in which of the following thermal decomposition reactions ?
(Given : Atomic wt. - Cr = 52 u, Ba = 137 u)

• (1) \((\mathrm{NH_4})_2\mathrm{Cr_2O_7}(s) \to \mathrm{N_2}(g) + 4\mathrm{H_2O}(g) + \mathrm{Cr_2O_3}(s)\)
• (2) \(2\mathrm{NH_4NO_3}(s) \to 2\mathrm{N_2}(g) + 4\mathrm{H_2O}(g) + \mathrm{O_2}(g)\)
• (3) \(\mathrm{Ba(N_3)_2}(s) \to \mathrm{Ba}(s) + 3\mathrm{N_2}(g)\)
• (4) \(2\mathrm{NH_3}(g) \to \mathrm{N_2}(g) + 3\mathrm{H_2}(g)\)

The total number of optically active compounds formed in the following reaction is :

• (1) Two
• (2) Four
• (3) Six
• (4) Zero

The major product formed in the following reaction is :

• (1)
• (2)
• (3)
• (4)

When 2-butyne is treated with H₂/Lindlar's catalyst, compound X is produced as the major product and when treated with Na/liq. NH₃ it produces Y as the major product. Which of the following statements is correct ?

• (1) X will have higher dipole moment and higher boiling point than Y.
• (2) Y will have higher dipole moment and higher boiling point than X.
• (3) X will have lower dipole moment and lower boiling point than Y.
• (4) Y will have higher dipole moment and lower boiling point than X.

The increasing order of the acidity of the following carboxylic acids is :

• (1) I < III < II < IV
• (2) IV < II < III < I
• (3) II < IV < III < I
• (4) III < II < IV < I

The correct order of spin-only magnetic moments among the following is : (Atomic number : Mn=25, Co=27, Ni=28, Zn=30)

• (1) \(\left[ ZnCl _{4}\right]^{2-}>\left[ NiCl _{4}\right]^{2-}>\left[ CoCl _{4}\right]^{2-}>\left[ MnCl _{4}\right]^{2}\)
• (2) \(\left[ CoCl _{4}\right]^{2^{2}}>\left[ MnCl _{4}\right]^{2^{-}}>\left[ NiCl _{4}\right]^{2^{2}}>\left[ ZnCl _{4}\right]^{2-}\)
• (3) \(\left[ NiCl _{4}\right]^{2}>\left[ CoCl _{4}\right]^{2-}>\left[ MnCl _{4}\right]^{2}>\left[ ZnCl _{4}\right]^{2-}\)
• (4) \(\left[ MnCl _{4}\right]^{2^{2}}>\left[ CoCl _{4}\right]^{2-}>\left[ NiCl _{4}\right]^{2}>\left[ ZnCl _{4}\right]^{2-}\)

Given
(i) 2Fe₂O₃(s) → 4Fe(s) + 3O₂(g); \(\Delta_r G^\circ = +1487.0\ \mathrm{kJ\ mol^{-1}}\)
(ii) 2CO(g) + O₂(g) → 2CO₂(g); \(\Delta_r G^\circ = -514.4\ \mathrm{kJ\ mol^{-1}}\)
Free energy change, \(\Delta_r G^\circ\) for the reaction
2Fe₂O₃(s) + 6CO(g) → 4Fe(s) + 6CO₂(g) will be :

• (1) \(-112.4\ \mathrm{kJ\ mol^{-1}}\)
• (2) \(-56.2\ \mathrm{kJ\ mol^{-1}}\)
• (3) \(-168.2\ \mathrm{kJ\ mol^{-1}}\)
• (4) \(-208.0\ \mathrm{kJ\ mol^{-1}}\)

In \(KO_2\), the nature of oxygen species and the oxidation state of oxygen atom are, respectively :

• (1) Oxide and -2
• (2) Superoxide and -1/2
• (3) Peroxide and -1/2
• (4) Superoxide and -1

Following four solutions are prepared by mixing different volumes of NaOH and HCl of different concentrations, pH of which one of them will be equal to 1 ?

• (1) 100 mL \(\frac{M}{10}\) HCl + 100 mL \(\frac{M}{10}\) NaOH
• (2) 75 mL \(\frac{M}{5}\) HCl + 25 mL \(\frac{M}{5}\) NaOH
• (3) 60 mL \(\frac{M}{10}\) HCl + 40 mL \(\frac{M}{10}\) NaOH
• (4) 55 mL \(\frac{M}{10}\) HCl + 45 mL \(\frac{M}{10}\) NaOH

On treatment of the following compound with a strong acid, the most susceptible site for bond cleavage is :

• (1) C1 - O2
• (2) O2 - C3
• (3) C4 - O5
• (4) O5 - C6

The total number of isomers for a square planar complex [M(F)(Cl)(SCN)(NO\(_2\))] is

• (1) 16
• (2) 8
• (3) 12
• (4) 4

\(\Delta_{r} G^{\circ}\) at \(500 \,K\) for substance 'S' in liquid state and gascous state are \(+100.7 \,kcal\, mol ^{-1} \) and \(+103 \,kcal \,mol ^{-1}\), respectively. Vapour pressure of liquid ' S ' at \(500\, K\) is approximately equal to \(\left(R=2 \,cal K ^{-1} \,mol ^{-1}\right)\)

• (1) 0.1 atm
• (2) 1 atm
• (3) 10 atm
• (4) 100 atm

The de-Broglie's wavelength of electron present in first Bohr orbit of 'H' atom is :

• (1) \(0.529\ \mathring{\mathrm{A}}\)
• (2) \(2\pi \times 0.529\ \mathring{\mathrm{A}}\)
• (3) \(\frac{0.529}{2\pi}\ \mathring{\mathrm{A}}\)
• (4) \(4 \times 0.529\ \mathring{\mathrm{A}}\)

In the leaching method, bauxite ore is digested with a concentrated solution of NaOH that produces ?X?. When \(CO_2\) gas is passed through the aqueous solution of ?X?, a hydrated compound ?Y? is precipitated. ?X? and ?Y? respectively are :

• (1) \(NaAlO_2 \) and \(Al_2(CO3)_3?x H_2O\)
• (2) \(Al(OH)_3 \) and \(Al2O_3?x H_2O\)
• (3) \(Na[Al(OH)_4] \) and \(Al_2O_3?x H_2O\)
• (4) \( Na[Al(OH)4]\) and \(Al_2(CO3)_3?x H_2O\)

The number of \(P - O\) bonds in \(P_4O_6\) is :

• (1) 6
• (2) 9
• (3) 12
• (4) 18

All of the following share the same crystal structure except :

• (1) LiCl
• (2) NaCl
• (3) RbCl
• (4) CsCl

If x gram of gas is adsorbed by m gram of adsorbent at pressure P, the plot of \(\log \frac{x}{m}\) versus \(\log P\) is linear. The slope of the plot is :
(n and k are constants and n > 1)

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

Which of the following statements is not true ?

• (1) Step growth polymerisation requires a bifunctional monomer.
• (2) Nylon 6 is an example of step-growth polymerisation.
• (3) Chain growth polymerisation includes both homopolymerisation and copylymerisation.
• (4) Chain growth polymerisation involves homopolymerisation only.

Biochemical Oxygen Demand (BOD) value can be a measure of water pollution caused by the organic matter. Which of the following statements is correct ?

• (1) Aerobic bacteria decrease the BOD value
• (2) Anaerobic bacteria increase the BOD value
• (3) Clean water has BOD value higher than 10 ppm
• (4) Polluted water has BOD value higher than 10 ppm

In \(XeO_3F_2\), the number of bond pair(s), \(\pi\)-bond(s) and lone pair(s) on \(Xe\) atom respectively are :

• (1) 5,3,0
• (2) 5,2,0
• (3) 4,2,2
• (4) 4,4,0

Lithium aluminium hydride reacts with silicon tetrachloride to form :

• (1) \(LiCl, AlH_3\) and \(SiH_4 \)
• (2) \(LiCl, AlCl_3\) and \(SiH_4\)
• (3) \( LiH, AlCl_3\) and \(SiCl_2 \)
• (4) \(LiH, AlH_3 \) and \(SiH_4 \)