JEE Main 2019 B.E./B.Tech Question Paper with Answer Keys (April 10, 2019- Afternoon Session)

The number of real roots of the equation x|x-2| + 3|x-3| + 1 = 0 is:

• (1) 4
• (2) 2
• (3) 1
• (4) 3

The sum of the real roots of the equation
\(\begin{vmatrix} x & -6 & -1 \\ 2 & -3x & x-3 \\ -3 & 2x & x+2 \end{vmatrix}=0\)
is equal to :

• (1) \(-4\)
• (2) \(0\)
• (3) \(1\)
• (4) \(6\)

In an A.P., the sixth term a_6​=2. If the product a₁​a₄​a₅ is the greatest, then the common difference of the A.P. is equal to:

• (1) \(\frac{3}{2}\)
• (2) \(\frac{8}{5}\)
• (3) \(\frac{2}{3}\)
• (4) \(\frac{5}{8}\)

Minimum number of times a fair coin must be tossed so that the probability of getting at least one head is more than 99% is :

• (1) \(6\)
• (2) \(5\)
• (3) \(8\)
• (4) \(7\)

Suppose that \(20\) pillars of the same height have been erected along the boundary of a circular stadium. If the top of each pillar has been connected by beams with the top of all its non-adjacent pillars, then the total number beams is :

• (1) 210
• (2) 190
• (3) 170
• (4) 180

Let \(\lambda\) be a real number for which the system of linear equations \(x+y+z=6,\; 4x+\lambda y-\lambda z=\lambda-2,\; 3x+2y-4z=-5\) has infinitely many solutions. Then \(\lambda\) is a root of the quadratic equation :

• (1) \(\lambda^2+\lambda-6=0\)
• (2) \(\lambda^2-\lambda-6=0\)
• (3) \(\lambda^2-3\lambda-4=0\)
• (4) \(\lambda^2+3\lambda-4=0\)

If \(\displaystyle\lim_{x\to 1}\frac{x^2-ax+b}{x-1}=5\), then \(a+b\) is equal to :

• (1) \(1\)
• (2) \(-4\)
• (3) \(-7\)
• (4) \(5\)

If \(\displaystyle\int x^5e^{-x^2}\,dx=g(x)e^{-x^2}+c\), where \(c\) is a constant of integration, then \(g(-1)\) is equal to :

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

The distance of the point having position vector \(-\hat{i}+2\hat{j}+6\hat{k}\) from the straight line passing through the point \((2,3,-4)\) and parallel to the vector \(6\hat{i}+3\hat{j}-4\hat{k}\) is :

• (1) \(6\)
• (2) \(7\)
• (3) \(2\sqrt{13}\)
• (4) \(4\sqrt{3}\)

A spherical iron ball of radius 10 cm is coated with a layer of ice of uniform thickness that melts at a rate of 50 cm\(^3\)/min. When the thickness of the ice is 5 cm, then the rate at which the thickness (in cm/min) of the ice decreases, is :

• (1) \(\frac{5}{6\pi}\)
• (2) \(\frac{1}{9\pi}\)
• (3) \(\frac{1}{36\pi}\)
• (4) \(\frac{1}{18\pi}\)

• (1) \(arg \,z_2=(\frac{\pi}{4})\)
• (2) \(arg \,z_2=-(\frac{3\pi}{4})\)
• (3) \(arg \,z_1=(\frac{\pi}{4})\)
• (4) \(arg \,z_1=-(\frac{3\pi}{4})\)

Let \(f(x)=\log_e(\sin x)\), \((0\lt x\lt\pi)\) and \(g(x)=\sin^{-1}(e^{-x})\), \((x\ge 0)\). If \(\alpha\) is a positive real number such that \(a=(f\circ g)^{\prime}(\alpha)\) and \(b=(f\circ g)(\alpha)\), then :

• (1) \(a\alpha^2+b\alpha-a=-2\alpha^2\)
• (2) \(a\alpha^2+b\alpha+a=0\)
• (3) \(a\alpha^2-b\alpha-a=0\)
• (4) \(a\alpha^2-b\alpha-a=1\)

If both the mean and the standard deviation of 50 observations \(x_1,x_2,\ldots,x_{50}\) are equal to 16, then the mean of \((x_1-4)^2,(x_2-4)^2,\ldots,(x_{50}-4)^2\) is :

• (1) \(400\)
• (2) \(480\)
• (3) \(380\)
• (4) \(525\)

The \(4^{\text {th }}\) term of \(G P\) is \(500\) and its common ratio is \(\frac{1}{m}, m \in N\) Let \(S_n\) denote the sum of the first \(n\) terms of this GP If \(S_6>S_5+1\) and \(S_7< S_6+\frac{1}{2}\), then the number of possible values of \(m\) is

The integral \(\displaystyle\int_{\pi/6}^{\pi/3}\sec^{2/3}x\;\mathrm{cosec}^{4/3}x\,dx\) is equal to :

• (1) \(3^{5/3}-3^{1/3}\)
• (2) \(3^{5/6}-3^{2/3}\)
• (3) \(3^{4/3}-3^{1/3}\)
• (4) \(3^{7/6}-3^{5/6}\)

The area (in sq. units) of the region bounded by the curves \(y=2^x\) and \(y=|x+1|\), in the first quadrant is :

• (1) \(\frac12\)
• (2) \(\frac32\)
• (3) \(\frac32-\frac{1}{\log_e 2}\)
• (4) \(\log_e 2+\frac32\)

Let \(y=y(x)\) be the solution of the differential equation \(\dfrac{dy}{dx}+y\tan x=2x+x^2\tan x\), \(x\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\), such that \(y(0)=1\). Then :

• (1) \(y\left(\frac{\pi}{4}\right)-y\left(-\frac{\pi}{4}\right)=\sqrt2\)
• (2) \(y^{\prime}\left(\frac{\pi}{4}\right)-y^{\prime}\left(-\frac{\pi}{4}\right)=\pi-\sqrt2\)
• (3) \(y\left(\frac{\pi}{4}\right)+y\left(-\frac{\pi}{4}\right)=\frac{\pi^2}{2}+2\)
• (4) \(y^{\prime}\left(\frac{\pi}{4}\right)+y^{\prime}\left(-\frac{\pi}{4}\right)=-\sqrt2\)

If \(\cos^{-1}x-\cos^{-1}\dfrac{y}{2}=\alpha\), where \(-1\le x\le1\), \(-2\le y\le2\), \(x\le\dfrac{y}{2}\), then for all \(x,y\), \(4x^2-4xy\cos\alpha+y^2\) is equal to :

• (1) \(4\sin^2\alpha\)
• (2) \(2\sin^2\alpha\)
• (3) \(4\sin^2\alpha-2x^2y^2\)
• (4) \(4\cos^2\alpha+2x^2y^2\)

Lines are drawn parallel to the line \(4x-3y+2=0\), at a distance \(\dfrac35\) from the origin. Then which one of the following points lies on any of these lines ?

• (1) \(\left(\frac14,-\frac13\right)\)
• (2) \(\left(-\frac14,\frac23\right)\)
• (3) \(\left(-\frac14,-\frac23\right)\)
• (4) \(\left(\frac14,\frac13\right)\)

If \(5x+9=0\) is the directrix of the hyperbola \(16x^2-9y^2=144\), then its corresponding focus is :

• (1) \(\left(\frac53,0\right)\)
• (2) \((5,0)\)
• (3) \((-5,0)\)
• (4) \(\left(-\frac53,0\right)\)

The smallest natural number n, such that the coefficient of x in the expansion of \(\left(x^2+\frac{1}{x^3}\right)^n\) is \(^{n}\mathrm{C}_{23}\), is :

• (1) 23
• (2) 58
• (3) 38
• (4) 35

Let the tangent to the parabola S : \(y^2 = 2x\) at the point P(2, 2) meet the x-axis at Q and normal at it meet the parabola S at the point R. Then the area (in sq. units) of the triangle PQR is equal to :

• (1) \(\frac{25}{2}\)
• (2) \(\frac{35}{2}\)
• (3) \(\frac{15}{2}\)
• (4) 25

The negation of the Boolean expression \(\sim s \vee (\sim r \wedge s)\) is equivalent to :

• (1) \(\sim s \wedge \sim r\)
• (2) \(r\)
• (3) \(s \vee r\)
• (4) \(s \wedge r\)

The sum \(1+ \frac{1^{3} +2^{3}}{1+2} + \frac{1^{3}+2^{3}+3^{3}}{1+2+3} +.... + \frac{1^{3} +2^{3}+3^{3} +....+15^{3}}{1+2+3+...+15} - \frac{1}{2} \left(1+2+3+...+15\right)\)

• (1) 1240
• (2) 1860
• (3) 660
• (4) 620

The locus of the centres of the circles, which touch the circle, \(x^2+y^2=1\) externally, also touch the y-axis and lie in the first quadrant, is :

• (1) \(x=\sqrt{1+2y}\ ,\ y\ge 0\)
• (2) \(y=\sqrt{1+2x}\ ,\ x\ge 0\)
• (3) \(y=\sqrt{1+4x}\ ,\ x\ge 0\)
• (4) \(x=\sqrt{1+4y}\ ,\ y\ge 0\)

The angles A, B and C of a triangle ABC are in A.P. and \(a:b=1:\sqrt{3}\). If \(c=4\) cm, then the area (in sq. cm) of this triangle is :

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

A perpendicular is drawn from a point on the line \(\frac{x-1}{2}=\frac{y+1}{-1}=\frac{z}{1}\) to the plane \(x+y+z=3\) such that the foot of the perpendicular Q also lies on the plane \(x-y+z=3\). Then the co-ordinates of Q are :

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

If the tangent to the curve \(y=\frac{x}{x^2-3}\), \(x\in\mathbb{R},\ (x\ne\pm\sqrt3)\), at a point \((\alpha,\beta)\ne(0,0)\) on it is parallel to the line \(2x+6y-11=0\), then :

• (1) \(|6\alpha+2\beta|=9\)
• (2) \(|2\alpha+6\beta|=11\)
• (3) \(|2\alpha+6\beta|=19\)
• (4) \(|6\alpha+2\beta|=19\)

If the plane \(2x-y+2z+3=0\) has the distances \(\frac13\) and \(\frac23\) units from the planes \(4x-2y+4z+\lambda=0\) and \(2x-y+2z+\mu=0\), respectively, then the maximum value of \(\lambda+\mu\) is equal to :

• (1) 13
• (2) 9
• (3) 5
• (4) 15

If the line \(ax+y=c\), touches both the curves \(x^2+y^2=1\) and \(y^2=4\sqrt{2}\,x\), then \(|c|\) is equal to :

• (1) 2
• (2) \(\sqrt{2}\)
• (3) \(\frac{1}{\sqrt{2}}\)
• (4) \(\frac12\)

A spaceship orbits around a planet at a height of 20 km from its surface. Assuming that only gravitational field of the planet acts on the spaceship, what will be the number of complete revolutions made by the spaceship in 24 hours around the planet?
[Given ; Mass of planet = \(8\times10^{22}\) kg, Radius of planet = \(2\times10^{6}\) m, Gravitational constant \(G=6.67\times10^{-11}\ \mathrm{Nm^2/kg^2}\)]

• (1) 13
• (2) 9
• (3) 17
• (4) 11

One mole of an ideal gas passes through a process where pressure and volume obey the relation \(P =P_{o} \left[1- \frac{1}{2} \left(\frac{V_{0}}{V}\right)^{2}\right] \) .Here \(P_o\) and \(V_o\) are constants. Calculate the change in the temperature of the gas if its volume changes from \(V_o\) to \(2 V_o\) .

• (1) \(\frac{1}{2} \frac{P_{o}V_{o}}{R} \)
• (2) \(\frac{3}{4} \frac{P_{o}V_{o}}{R} \)
• (3) \(\frac{5}{4} \frac{P_{o}V_{o}}{R} \)
• (4) \(\frac{1}{4} \frac{P_{o}V_{o}}{R} \)

A ball projected from ground at an angle of \(45^\circ\) just clears a wall in front. If point of projection is 4 m from the foot of wall and ball strikes the ground at a distance of 6 m on the other side of the wall, the height of the wall is :

• (1) 4.4 m
• (2) 2.4 m
• (3) 3.6 m
• (4) 1.6 m

A particle of mass 1 mg and charge q is lying at the mid-point of two stationary particles kept at a distance '2 m' when each is carrying same charge 'q'. If the free charged particle is displaced from its equilibrium position through distance 'x' (\(x << 1\) m). The particle executes SHM. Its angular frequency of oscillation will be ________ \(\times 10^5\) rad/s if \(q^2 = 10 \text{ C}^2\).

Number of photons of equal energy emitted per second by a 6 mW laser source
operating at wavelength 663 nm is _______.

(Given: h = 6.63 × 10^-34 J·s and c = 3 × 10^8 m/s)

• (1) \(10\times10^{15}\)
  
• (2) \(5\times10^{16}\)
  
• (3) \(5\times10^{15}\)
  
• (4) \(2\times10^{16}\)

The time dependence of the position of a particle of mass \(m=2\) is given by \(\vec{r}(t)=2t\hat{i}-3t^2\hat{j}\). Its angular momentum, with respect to the origin, at time \(t=2\) is

• (1) \(36\hat{k}\)
• (2) \(-48\hat{k}\)
• (3) \(-34(\hat{k}-\hat{i})\)
• (4) \(48(\hat{i}+\hat{j})\)

A simple pendulum of mass 'm', length 'l' and charge '+q' suspended in the electric field produced by two conducting parallel plates as shown. The value of deflection of pendulum in equilibrium position will be: 

 

• (1) \(\tan^{-1}\left[ \frac{q}{mg} \times \frac{C_1(V_1 + V_2)}{(C_1 + C_2)(d-t)} \right]\)
  
• (2) \(\tan^{-1}\left[ \frac{q}{mg} \times \frac{C_2(V_1 + V_2)}{(C_1 + C_2)(d-t)} \right]\)
  
• (3) \(\tan^{-1}\left[ \frac{q}{mg} \times \frac{C_1(V_2 - V_1)}{(C_1 + C_2)(d-t)} \right]\)
  
• (4) \(\tan^{-1}\left[ \frac{q}{mg} \times \frac{C_2(V_2 - V_1)}{(C_1 + C_2)(d-t)} \right]\)
  

A square loop is carrying a steady current I and the magnitude of its magnetic dipole moment is m. if this square loop is changed to a circular loop and it carries the same current, the magnitude of the magnetic dipole moment of circular loop will be:

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

Space between two concentric conducting spheres of radii a and b (b > a) is filled with a medium of resistivity \(\rho\). The resistance between the two spheres will be :

• (1) \(\frac{\rho}{2\pi}\left(\frac1a+\frac1b\right)\)
• (2) \(\frac{\rho}{4\pi}\left(\frac1a+\frac1b\right)\)
• (3) \(\frac{\rho}{2\pi}\left(\frac1a-\frac1b\right)\)
• (4) \(\frac{\rho}{4\pi}\left(\frac1a-\frac1b\right)\)

A coil of self inductance 10 mH and resistance \(0.1\ \Omega\) is connected through a switch to a battery of internal resistance \(0.9\ \Omega\). After the switch is closed, the time taken for the current to attain 80% of the saturation value is: [take \(\ln5=1.6\)]

• (1) 0.324 s
• (2) 0.002 s
• (3) 0.103 s
• (4) 0.016 s

Water from a tap emerges vertically downwards with an initial speed of \(1.0\, ms^{-1}\). The crosssectional area of the tap is \(10^{-4} \; m^2\). Assume that the pressure is constant throughout the stream of water and that the flow is streamlined. The cross-sectional area of the stream, \(0.15\, m\) below the tap would be : (Take \(g \,= \,10\, ms^{-2}\))

• (1) \( 1 \times 10^{-5} \; m^2\)
• (2) \( 5 \times 10^{-5} \; m^2\)
• (3) \( 2 \times 10^{-5} \; m^2\)
• (4) \(5 \times 10^{-4} \; m^2\)

A metal coin of mass \(5\,g\) and radius \(1\, cm\) is fixed to a thin stick \(AB\) of negligible mass as shown in the figure. The system is initially at rest. The constant torque, that will make the system rotate about \(AB\) at \(25\) rotations per second in \(5\, s\), is close to :

• (1) \(4.0 \times 10^{-6} \) Nm
• (2) \(2.0 \times 10^{-5} \) Nm
• (3) \(1.6 \times 10^{-5} \) Nm
• (4) \(7.9 \times 10^{-6} \) Nm

The figure represents a voltage regulator circuit using a Zener diode. The breakdown voltage of the Zener diode is 6 V and the load resistance is, \(R_L=4\,\mathrm{k}\Omega\). The series resistance of the circuit is \(R_i=1\,\mathrm{k}\Omega\). If the battery voltage \(V_B\) varies from 8 V to 16 V, what are the minimum and maximum values of the current through Zener diode?

• (1) 0.5 mA; 8.5 mA
• (2) 1.5 mA; 8.5 mA
• (3) 1 mA; 8.5 mA
• (4) 0.5 mA; 6 mA

In a Young's double slit experiment, the ratio of the slit's width is \(4:1\). The ratio of the intensity of maxima to minima, close to the central fringe on the screen, will be :

• (1) \(25:9\)
• (2) \(4:1\)
• (3) \((\sqrt{3}+1)^4:16\)
• (4) \(9:1\)

A plane is inclined at an angle \(\alpha = 30^{\circ}\) with a respect to the horizontal. A particle is projected with a speed \(u = 2 ms^{-1}\) from the base of the plane, making an angle \(q = 15^{\circ}\) with respect to the plane as shown in the figure. The distance from the base, at which the particle hits the plane is close to : (Take \(g = 10 \; ms^{-2}\))

• (1) 14 cm
• (2) 20 cm
• (3) 18 cm
• (4) 26 cm

A submarine experiences a pressure of \(5.05\times10^{6}\ \mathrm{Pa}\) at a depth of \(d_1\) in a sea. When it goes further to a depth of \(d_2\), it experiences a pressure of \(8.08\times10^{6}\ \mathrm{Pa}\). Then \(d_2-d_1\) is approximately (density of water \(=10^{3}\ \mathrm{kg/m^3}\) and acceleration due to gravity \(=10\ \mathrm{ms^{-2}}\)) :

• (1) \(600\ \mathrm{m}\)
• (2) \(400\ \mathrm{m}\)
• (3) \(300\ \mathrm{m}\)
• (4) \(500\ \mathrm{m}\)

When heat \(Q\) is supplied to a diatomic gas of rigid molecules, at constant volume its temperature increases by \(\Delta T\). The heat required to produce the same change in temperature, at a constant pressure is :

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

In the formula, $X=3Y{Z}^2,X$ and $Z$ have dimensions of capacitance and magnetic field respectively. What are the dimensions of $\mathrm{Y}$ in the MKSQ system?

• (1) (A) $\left[{\mathrm{M}}^{-3}{\mathrm{L}}^{-1}{\mathrm{T}}^3{\mathrm{Q}}^4\right]$
  
• (2) (B) $[{\mathrm{M}}^{-3}{\mathrm{L}}^{-2}{\mathrm{T}}^4{\mathrm{Q}}^4$
  
• (3) (C) $\left[{\mathrm{M}}^{-2}{L}^{-2}{\mathrm{T}}^4{\mathrm{Q}}^4\right]$
  
• (4) (D) $\left[{\mathrm{M}}^{-3}{\mathrm{L}}^{-2}{\mathrm{T}}^4\mathrm{Q}\right]$
  

The graph shows how the magnification \(m\) produced by a thin lens varies with image distance \(v\). What is the focal length of the lens used ?

• (1) \(\frac{b^2 c}{a}\)
• (2) \(\frac{b^2}{ac}\)
• (3) \(\frac{a}{c}\)
• (4) \(\frac{b}{c}\)

In an experiment, brass and steel wires of length \(1\ \mathrm{m}\) each with areas of cross section \(1\ \mathrm{mm^2}\) are used. The wires are connected in series and one end of the combined wire is connected to a rigid support and other end is subjected to elongation. The stress required to produce a net elongation of \(0.2\ \mathrm{mm}\) is, [Given, the Young's Modulus for steel and brass are, respectively, \(120\times10^{9}\ \mathrm{N/m^2}\) and \(60\times10^{9}\ \mathrm{N/m^2}\)]

• (1) \(8.0\times10^{6}\ \mathrm{N/m^2}\)
• (2) \(1.2\times10^{6}\ \mathrm{N/m^2}\)
• (3) \(0.2\times10^{6}\ \mathrm{N/m^2}\)
• (4) \(1.8\times10^{6}\ \mathrm{N/m^2}\)

In \(Li^{++}\), electron in first Bohr orbit is excited to a level by a radiation of wavelength \(\lambda\). when the ion gets deexcited to the ground state in all possible ways (including intermediate emissions), a total of six spectral lines are observed. What is the value of \(\lambda\) ? (Given : \(h = 6.63 \times 10^{-34} \; Js ; c = 3 \times 10^8 \; ms^{-1})\) ;

• (1) 9.4 nm
• (2) 12.3 nm
• (3) 10.8 nm
• (4) 11.4 nm

A solid sphere of mass \(M\) and radius \(R\) is divided into two unequal parts. The first part has a mass of \(\frac{7M}{8}\) and is converted into a uniform disc of radius \(2R\). The second part is converted into a uniform solid sphere. Let \(I_1\) be the moment of inertia of the disc about its axis and \(I_2\) be the moment of inertia of the new sphere about its axis. The ratio \(I_1/I_2\) is given by :

• (1) 185
• (2) 65
• (3) 285
• (4) 140

Light is incident normally on a completely absorbing surface with an energy flux of \(25\ \mathrm{W\,cm^{-2}}\). If the surface has an area of \(25\ \mathrm{cm^2}\), the momentum transferred to the surface in \(40\) min time duration will be :

• (1) \(6.3\times10^{-4}\ \mathrm{Ns}\)
• (2) \(5.0\times10^{-3}\ \mathrm{Ns}\)
• (3) \(1.4\times10^{-6}\ \mathrm{Ns}\)
• (4) \(3.5\times10^{-6}\ \mathrm{Ns}\)

A cubical block of side \(0.5\ \mathrm{m}\) floats on water with \(30\%\) of its volume under water. What is the maximum weight that can be put on the block without fully submerging it under water? [Take, density of water \(=10^{3}\ \mathrm{kg/m^3}\)]

• (1) \(30.1\ \mathrm{kg}\)
• (2) \(87.5\ \mathrm{kg}\)
• (3) \(65.4\ \mathrm{kg}\)
• (4) \(46.3\ \mathrm{kg}\)

The magnitude of the magnetic field at the center of an equilateral triangular loop of side \(1\,m\) which is carrying a current of \(10\, A\) is : [Take \(\mu_0 = 4\pi \times 10^{-7} \; NA^{-2}]\)

• (1) \(18\, \mu T\)
• (2) \(3\, \mu T\)
• (3) \(1 \, \mu T\)
• (4) \(9 \, \mu T\)

The elastic limit of brass is \(379\, MPa\). What should be the minimum diameter of a brass rod if it is to support a \(400 \,N\) load without exceeding its elastic limit ?

• (1) \(1.16\, mm\)
• (2) \(0.90\, mm\)
• (3) \(1.36\, mm\)
• (4) \(1.00\, mm\)

Two blocks \(A\) and \(B\) of masses \(m_A = 1 \; kg\) and \(m_B = 3 \; kg\) are kept on the table as shown in figure. The coefficient of friction between \(A\) and \(B\) is \(0.2\) and between \(B\) and the surface of the table is also \(0.2.\) The maximum force \(F\) that can be applied on \(B\) horizontally, so that the block \(A\) does not slide over the block \(B\) is : (Take \(g = 10 \; m/s^2\))

• (1) 16 N
• (2) 40 N
• (3) 12 /n
• (4) 8 N

The correct figure that shows, schematically, the wave pattern produced by superposition of two waves of frequencies \(9 \,Hz\) and \(11\, Hz\) is :

• (1)
• (2)
• (3)
• (4)

Two radioactive materials A and B have decay constants 25λ and 16λ respectively. If initially they have the same number of nuclei, then the ratio of the number of nuclei of B to that of A will be ‘e’ after a time \(\frac{1}{(aλ)}\). The value of a is _____.

A source of sound S is moving with a velocity of \(50\ \mathrm{m/s}\) towards a stationary observer. The observer measures the frequency of the source as \(1000\ \mathrm{Hz}\). What will be the apparent frequency of the source when it is moving away from the observer after crossing him? (Take velocity of sound in air is \(350\ \mathrm{m/s}\))

• (1) \(750\ \mathrm{Hz}\)
• (2) \(857\ \mathrm{Hz}\)
• (3) \(807\ \mathrm{Hz}\)
• (4) \(1143\ \mathrm{Hz}\)

The major product 'Y' in the following reaction is :

• (1)
• (2)
• (3)
• (4)

Which one of the following graphs between molar conductivity (\(\Lambda_m\)) versus \(\sqrt{C}\) is correct ?

• (1)
• (2)
• (3)
• (4)

\(2\ g\) of a non-volatile non-electrolyte solute is dissolved in \(200\ g\) of two different solvents A and B whose ebullioscopic constants are in the ratio of \(1:8\). The elevation in boiling points of A and B are in the ratio \(\frac xy\ (x : y)\). The value of \(y\) is ______ . (Nearest Integer)

For the reaction \(\text{A} \to \text{B}\), the rate constant \(k\) (in \(\text{s}^{-1}\)) is given by \(\log_{10} k = 20.35 - \frac{2.47 \times 10^3}{\text{T}}\). The energy of activation in \(\text{kJ mol}^{-1}\) is ________. (Nearest integer) [Given : R = 8.314 J K\(^{-1}\) mol\(^{-1}\)]

The Crystal Field Stabilization Energy (CFSE) of \([\mathrm{CoF_3(H_2O)_3}]\) (\(\Delta_0 \lt P\)) is :

• (1) \(-0.8\,\Delta_0\)
• (2) \(-0.4\,\Delta_0\)
• (3) \(-0.8\,\Delta_0 + 2P\)
• (4) \(-0.4\,\Delta_0 + P\)

For the reaction,
\(\mathrm{2SO_2(g)+O_2(g)\rightleftharpoons 2SO_3(g)}\), \(\Delta H=-57.2\ \mathrm{kJ\,mol^{-1}}\) and \(K_C=1.7\times10^{16}\)
Which of the following statement is incorrect ?

• (1) The equilibrium will shift in forward direction as the pressure increase.
• (2) The addition of inert gas at constant volume will be not affect the equilibrium constant.
• (3) The equilibrium constant is large suggestive of reaction going to completion and so no catalyst is required.
• (4) The equilibrium constant decreases as the temperature increase.

The minimum amount of \(\mathrm{O_2(g)}\) consumed per gram of reactant is for the reaction :
(Given atomic mass : Fe = 56, O = 16, Mg = 24, P = 31, C = 12, H = 1)

• (1) \(\mathrm{4Fe(s)+3O_2(g)\rightarrow 2Fe_2O_3(s)}\)
• (2) \(\mathrm{P_4(s)+5O_2(g)\rightarrow P_4O_{10}(s)}\)
• (3) \(\mathrm{C_3H_8(g)+5O_2(g)\rightarrow 3CO_2(g)+4H_2O(l)}\)
• (4) \(\mathrm{2Mg(s)+O_2(g)\rightarrow 2MgO(s)}\)

The increasing order of nucleophilicity of the following nucleophiles is : (a) \({CH_3CO^{\ominus}_2}\) (b) \({H_2O} \) (c) \({CH_3SO^{\ominus}_3}\) (d) \({O^{\ominus}H}\)

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

The difference between \(\Delta H\) and \(\Delta U\) (\(\Delta H-\Delta U\)), when the combustion of one mole of heptane(l) is carried out at a temperature \(T\), is equal to :

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

The major product 'Y' in the following reactions is :

• (1)
• (2)
• (3)
• (4)

Which of these factors does not govern the stability of a conformation in acyclic compounds?

• (1) Angle strain
• (2) Torsional strain
• (3) Electrostatic forces of interaction
• (4) Steric interactions

The ratio of the shortest wavelength of two spectral series of hydrogen spectrum is found to be about \(9\). The spectral series are :

• (1) Paschen and Pfund
• (2) Balmer and Brackett
• (3) Lyman and Paschen
• (4) Brackett and Pfund

The highest possible oxidation states of uranium and plutonium, respectively are :

• (1) 4 and 6
• (2) 6 and 4
• (3) 7 and 6
• (4) 6 and 7

In chromatography, which of the following statements is incorrect for R_f ?

• (1) R_f value depends on the type of chromatography.
• (2) Higher R_f value means higher adsorption.
• (3) R_f value is dependent on the mobile phase.
• (4) The value of R_f can not be more than one.

Compound A(C₉H₁₀O) shows positive iodoform test. Oxidation of A with KMnO₄/KOH given acid B(C₈H₆O₄). Anhydride of B is used for the preparation of phenolphthalein. Compound A is:

• (1)
• (2)
• (3)
• (4)

The major product obtained in the given reaction is :

• (1)
• (2)
• (3)
• (4)

The pH of a 0.02M \(NH_4Cl\) solution will be [given \(K_b(NH_4OH) = 10^{-5}\) and log2=0.301]

• (1) 4.65
• (2) 5.35
• (3) 4.35
• (4) 2.65

The correct order of the first ionization enthalpies is:

• (1) Mn < Ti < Zn < Ni
• (2) Ti < Mn < Ni < Zn
• (3) Zn < Ni < Mn < Ti
• (4) Ti < Mn < Zn < Ni

The INCORRECT statement is :

• (1) the color of \(\mathrm{[CoCl(NH_3)_5]^{2+}}\) is violet as it absorbs the yellow light.
• (2) the gemstone, ruby, has \(\mathrm{Cr^{3+}}\) ions occupying the octahedral sites of beryl.
• (3) the spin-only magnetic moment of \(\mathrm{[Ni(NH_3)_4(H_2O)_2]^{2+}}\) is 2.83 BM.
• (4) the spin-only magnetic moments of \(\mathrm{[Fe(H_2O)_6]^{2+}}\) and \(\mathrm{[Cr(H_2O)_6]^{2+}}\) are nearly similar.

The noble gas that does NOT occur in the atmosphere is:

• (1) He
• (2) Ra
• (3) Ne
• (4) Kr

Which of the following is not a correct method of the preparation of benzylamine from cyanobenzene ?

• (1) (i) \(\mathrm{SnCl_2 + HCl(gas)}\)    (ii) \(\mathrm{NaBH_4}\)
• (2) \(\mathrm{H_2/Ni}\)
• (3) (i) \(\mathrm{LiAlH_4}\)    (ii) \(\mathrm{H_3O^+}\)
• (4) (i) \(\mathrm{HCl/H_2O}\)    (ii) \(\mathrm{NaBH_4}\)

Number of stereo centers present in linear and cyclic structures of glucose are respectively :

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

The correct option among the following is :

• (1) Addition of alum to water makes it unfit for drinking.
• (2) Colloidal medicines are more effective because they have small surface area.
• (3) Colloidal paraticles in lyophobic sols can be precipitated by electrophoresis.
• (4) Brownian motion in colloidal solution is faster if the viscosity of the solution is very high.

The correct statements among (a) to (b) are: (a) saline hydrides produce \(H_2\) gas when reacted with \({H2O}\). (b) reaction of \(LiAH_4\) with \(BF_3\) leads to \(B_2H_6\). (c) \(PH_3\) and \(CH_4\) are electron - rich and electronprecise hydrides, respectively. (d) HF and \(CH_4\) are called as molecular hydrides.

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

The type of pollution that gets increased during the day time and in the presence of O\(_3\) is :

• (1) Oxidising smog
• (2) Global warming
• (3) Acid rain
• (4) Reducing smog

A hydrated solid X on heating initially gives a monohydrated compound Y. Y upon heating above 373 K leads to an anhydrous white powder Z. X and Z, respectively, are :

• (1) Baking soda and dead burnt plaster.
• (2) Washing soda and soda ash
• (3) Washing soda and dead burnt plaster.
• (4) Baking soda and soda as.

The number of pentagons in \(\mathrm{C_{60}}\) and trigons (triangles) in white phosphorus, respectively, are :

• (1) 12 and 3
• (2) 20 and 3
• (3) 20 and 4
• (4) 12 and 4

Points I, II and III in the following plot respectively correspond to (\(V_{mp} :\) most probable velocity)

• (1) \({V_{mp} \; of \; N_2 (300K); V_{mp} \; of \; H_2(300K); V_{mp} \; of \; O_2(400K)}\)
• (2) \({V_{mp} \; of \; H_2 (300K); V_{mp} \; of \; N_2(300K); V_{mp} \; of \; O_2(400K)}\)
• (3) \({V_{mp} \; of \; O_2 (300K); V_{mp} \; of \; N_2(300K); V_{mp} \; of \; H_2(400K)}\)
• (4) \({V_{mp} \; of \; N_2 (300K); V_{mp} \; of \; O_2(300K); V_{mp} \; of \; H_2(400K)}\)

• (1) (a) \(\rightarrow\) (IV), (b) \(\rightarrow\) (II), (c) \(\rightarrow\) (I), (d) \(\rightarrow\) (III)
• (2) (a) \(\rightarrow\) (III), (b) \(\rightarrow\) (I), (c) \(\rightarrow\) (IV), (d) \(\rightarrow\) (II)
• (3) (a) \(\rightarrow\) (II), (b) \(\rightarrow\) (IV), (c) \(\rightarrow\) (I), (d) \(\rightarrow\) (III)
• (4) (a) \(\rightarrow\) (III), (b) \(\rightarrow\) (I), (c) \(\rightarrow\) (II), (d) \(\rightarrow\) (IV)

The correct statement is :

• (1) zincite is a carbonate ore.
• (2) Sodium cyanide cannot be used in the metallurgy of silver.
• (3) aniline is a froth stabilizer.
• (4) zone refining process is used for the refining of titanium.