JEE Main 2014 B.E./B.Tech Question Paper with Answer Key PDF- Code E (April 6, 2014)

If \(X = \{4^n- 3 n - 1: n \in N \}\) and \(Y = \{9 (n - 1): n \in N \}\) ,where \(N\) is the set of natural numbers, then \( X \cup Y\) is equal to

• (1) N
• (2) Y-X
• (3) X
• (4) Y

If \(z\) is a complex number such that \(|z|\geq 2\), then the minimum value of \(\left|z+\frac{1}{2}\right|\) :

• (1) is strictly greater than \(\frac{5}{2}\)
• (2) is strictly greater than \(\frac{3}{2}\) but less than \(\frac{5}{2}\)
• (3) is equal to \(\frac{5}{2}\)
• (4) lies in the interval \((1, 2)\)

If \(a\in\mathbf{R}\) and the equation \(-3(x-[x])^2+2(x-[x])+a^2=0\) (where \([x]\) denotes the greatest integer \(\leq x\)) has no integral solution, then all possible values of \(a\) lie in the interval :

• (1) \((-2, -1)\)
• (2) \((-\infty, -2)\cup(2, \infty)\)
• (3) \((-1, 0)\cup(0, 1)\)
• (4) \((1, 2)\)

Let \(\alpha\) and \(\beta\) be the roots of the equation \(px^2+qx+r=0\), \(p\neq0\). If \(p, q, r\) are in A.P. and \(\frac{1}{\alpha}+\frac{1}{\beta}=4\), then the value of \(|\alpha-\beta|\) is :

• (1) \(\frac{\sqrt{34}}{9}\)
• (2) \(\frac{2\sqrt{13}}{9}\)
• (3) \(\frac{\sqrt{61}}{9}\)
• (4) \(\frac{2\sqrt{17}}{9}\)

If \(\alpha,\beta\neq0\), \(f(n)=\alpha^n+\beta^n\) and \(\begin{vmatrix}3&1+f(1)&1+f(2)\\1+f(1)&1+f(2)&1+f(3)\\1+f(2)&1+f(3)&1+f(4)\end{vmatrix}=K(1-\alpha)^2(1-\beta)^2(\alpha-\beta)^2\), then \(K\) is equal to :

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

If \(A\) is a \(3\times3\) non-singular matrix such that \(AA'=A'A\) and \(B=A^{-1}A'\), then \(BB'\) equals :

• (1) \(B^{-1}\)
• (2) \((B^{-1})'\)
• (3) \(I+B\)
• (4) \(I\)

If the coefficients of \(x^3\) and \(x^4\) in the expansion of \((1+ax+bx^2)(1-2x)^{18}\) in powers of \(x\) are both zero, then \((a,b)\) is equal to :

• (1) \(\left(14,\frac{272}{3}\right)\)
• (2) \(\left(16,\frac{272}{3}\right)\)
• (3) \(\left(16,\frac{251}{3}\right)\)
• (4) \(\left(14,\frac{251}{3}\right)\)

If \((10)^9 +2(11)^1 (10)^8 + 3(11)^2 (10)^7 +............+10(11)^9 = k (10)^9,\) then k is equal to

• (1) \(\frac{121}{10}\)
• (2) \(\frac{441}{100}\)
• (3) \(100\)
• (4) \(110\)

Three positive numbers form an increasing \(G.P.\) If the middle term in this \(G.P.\) is doubled, then new numbers are in \(A.P.\) Then, the common ratio of the \(G.P.\) is

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

\(\displaystyle\lim_{x\to0}\frac{\sin(\pi\cos^2x)}{x^2}\) is equal to :

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

If \(g\) is the inverse of a function \(f\) and \(f'(x)=\frac{1}{1+x^5}\), then \(g'(x)\) is equal to :

• (1) \(\frac{1}{1+\{g(x)\}^5}\)
• (2) \(1+\{g(x)\}^5\)
• (3) \(1+x^5\)
• (4) \(5x^4\)

If \(f\) and \(g\) are differentiable functions in \((0, 1)\) satisfying \(f (0) = 2 = g (1), g (0) = 0\) and \(f (1) = 6\), then for some \(c \in ] 0, 1 [\)

• (1) \(2f ' (c) = g ' (c)\)
• (2) \(2f ' (c) = 3 g ' (c)\)
• (3) \(f ' (c) = g ' (c)\)
• (4) \(f ' (c) = 2 g ' (c)\)

If \(x=-1\) and \(x=2\) are extreme points of \(f(x)=\alpha\log|x|+\beta x^2+x\), then :

• (1) \(\alpha=2,\ \beta=-\frac12\)
• (2) \(\alpha=2,\ \beta=\frac12\)
• (3) \(\alpha=-6,\ \beta=\frac12\)
• (4) \(\alpha=-6,\ \beta=-\frac12\)

The integral \(\displaystyle\int\left(1+x-\frac1x\right)e^{x+\frac1x}\,dx\) is equal to :

• (1) \((x+1)e^{x+\frac1x}+c\)
• (2) \(-x\,e^{x+\frac1x}+c\)
• (3) \((x-1)e^{x+\frac1x}+c\)
• (4) \(x\,e^{x+\frac1x}+c\)

The integral \(\displaystyle\int_0^{\pi}\sqrt{1+4\sin^2\frac x2-4\sin\frac x2}\,dx\) equals :

• (1) \(4\sqrt3-4\)
• (2) \(4\sqrt3-4-\frac{\pi}{3}\)
• (3) \(\pi-4\)
• (4) \(\frac{2\pi}{3}-4-4\sqrt3\)

The area (in sq units) of the region described by \(A=\left\{(x, y): x^{2}+y^{2} \leq 1\right.\) and \(\left.y^{2} \leq 1-x\right\}\) is:

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

Let the population of rabbits surviving at a time \(t\) be governed by the differential equation \(\frac {dp(t)}{dt}=\frac {1}{2} p(t)-200.\) If \(p(0)=100,\) then \(p(t)\) is equal to

• (1) \(400-300\,e^{t/2}\)
• (2) \(300-200\,e^{-t/2}\)
• (3) \(600-500\,e^t/2\)
• (4) \(400-300\,e^t/2\)

Let \(PS\) be the median of the triangle with vertices \(P(2,2)\), \(Q(6,-1)\) and \(R(7,3)\). The equation of the line passing through \((1,-1)\) and parallel to \(PS\) is :

• (1) \(4x+7y+3=0\)
• (2) \(2x-9y-11=0\)
• (3) \(4x-7y-11=0\)
• (4) \(2x+9y+7=0\)

Let \(a ,b,c\) and \(d\) be non-zero numbers. If the point of intersection of the lines \(4ax+2ay+c = 0\) and \(5bx + 2by+ d = 0\) lies in the fourth quadrant and is equidistant from the two axes, then

• (1) \(2bc - 3ad = 0\)
• (2) \(2bc + 3ad = 0\)
• (3) \( 3bc - 2ad= 0\)
• (4) \(3bc + 2ad = 0\)

The locus of the foot of perpendicular drawn from the centre of the ellipse \(x^2+3y^2=6\) on any tangent to it is :

• (1) \((x^2+y^2)^2=6x^2+2y^2\)
• (2) \((x^2+y^2)^2=6x^2-2y^2\)
• (3) \((x^2-y^2)^2=6x^2+2y^2\)
• (4) \((x^2-y^2)^2=6x^2-2y^2\)

Let C be the circle with centre at (1, 1) and radius = 1. If T is the circle centred at (0, y) passing through origin and touching the circle \(C\) externally, then the radius of\( T \)is equal to

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

The slope of the line touching both the parabolas \(y^2 = 4x\) and \(x^2 = - 32y\) is :

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

The image of the line \(\frac{x-1}{3}=\frac{y-3}{1}=\frac{z-4}{-5}\) in the plane \(2 x - y + z+ 3 = 0\) is the line.

• (1) \(\frac{x+3}{3}=\frac{y-5}{1}=\frac{z-2}{-5}\)
• (2) \(\frac{x+3}{-3}=\frac{y-5}{-1}=\frac{z+2}{5}\)
• (3) \(\frac{x-3}{3}=\frac{y+5}{1}=\frac{z-2}{-5}\)
• (4) \(\frac{x-3}{-3}=\frac{y+5}{-1}=\frac{z-2}{5}\)

The angle between the lines whose direction cosines satisfy the equations \(l + m + n = 0 \) and \( l^2= m^2 + n^2 \)is

• (1) \(\pi/6\)
• (2) \(\pi/2\)
• (3) \(\pi/3\)
• (4) \(\pi/4\)

If \(\left[\vec{a}\times\vec{b}\ \ \vec{b}\times\vec{c}\ \ \vec{c}\times\vec{a}\right]=\lambda\left[\vec{a}\ \ \vec{b}\ \ \vec{c}\right]^2\) then \(\lambda\) is equal to :

• (1) 0
• (2) 1
• (3) 2
• (4) 3

Let \(A\) and \(B\) be two events such that \(P\left(\overline{A\cup B}\right)=\frac{1}{6}, P\left(\overline{A \cap B}\right)=\frac{1}{4}\) and \(P\left(\bar{A}\right)=\frac{1}{4}\) where \(\overline{A}\) stands for the complement of the event \(A\). Then , the events \(A\) and \(B\) are

• (1) independent but not equally likely
• (2) independent and equally likely
• (3) mutually exclusive and independent
• (4) equally likely but not independent

The variance of first \(50\) even natural numbers is

• (1) \( \frac{ 833}{ 4}\)
• (2) \(833\)
• (3) \(437\)
• (4) \( \frac{437}{4}\)

Let \(f_k(x)=\frac{1}{k}\left(\sin^k x+\cos^k x\right)\) where \(x\in\mathbf{R}\) and \(k\ge1\). Then \(f_4(x)-f_6(x)\) equals :

• (1) \(\frac14\)
• (2) \(\frac{1}{12}\)
• (3) \(\frac16\)
• (4) \(\frac13\)

A bird is sitting on the top of a vertical pole 20 m high and its elevation from a point O on the ground is \(45^\circ\). It flies off horizontally straight away from the point O. After one second, the elevation of the bird from O is reduced to \(30^\circ\). Then the speed (in m/s) of the bird is :

• (1) \(20\sqrt2\)
• (2) \(20(\sqrt3-1)\)
• (3) \(40(\sqrt2-1)\)
• (4) \(40(\sqrt3-\sqrt2)\)

The statement \(\sim(p\leftrightarrow\sim q)\) is :

• (1) a tautology
• (2) a fallacy
• (3) equivalent to \(p\leftrightarrow q\)
• (4) equivalent to \(\sim p\leftrightarrow q\)

The current voltage relation of diode is given by \(I=\left(e^{1000V/T}-1\right)\) mA, where the applied voltage V is in volts and the temperature T is in degree Kelvin. If a student makes an error measuring \(\pm0.01\) V while measuring the current of 5 mA at 300 K, what will be the error in the value of current in mA ?

• (1) 0.2 mA
• (2) 0.02 mA
• (3) 0.5 mA
• (4) 0.05 mA

From a tower of height \(H\), a particle is thrown vertically upwards with a speed \(u\). The time taken by the particle to hit the ground, is \(n\) times that taken by it to reach the highest point of its path. The relation between \(H, u\) and \(n\) is

• (1) \(2gH = n^2u^2\)
• (2) \( gH = (n-2)^2 u^2\)
• (3) \(2gH = nu^2 (n-2)\)
• (4) \( gH = (n-2)u^{2}\)

A mass 'm' is supported by a massless string wound around a uniform hollow cylinder of mass m and radius R. If the string does not slip on the cylinder, with what acceleration will the mass fall on release ?

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

A block of mass m is placed on a surface with a vertical cross-section given by \(y = \frac{x^3}{6}\) If the coefficient of friction is \(0.5\), the maximum height above the ground at which the block can be placed without slipping is

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

When a rubber -band is stretched by a distance x, it exerts a restoring force of magnitude \( F = ax + bx^2\), where a and b are constants. The work done in stretching the unstretched rubber-band by \(L\) is :

• (1) \(aL^2 + b L^3\)
• (2) \( \frac{1}{2}(aL^2 + b L^3)\)
• (3) \(\frac{a L^2}{2} + \frac{ bL^3}{ 3}\)
• (4) \(\frac{1}{2}\left(\frac{a L^{2}}{2}+\frac{b L^{3}}{3}\right)\)

A bob of mass m attached to an inextensible string of length l is suspended from a vertical support. The bob rotates in a horizontal circle with an angular speed \(\omega \,rad/s\) about the vertical support. About the point of suspension

• (1) Angular momentum is conserved
• (2) Angular momentum changes in magnitude but not in direction
• (3) Angular momentum changes in direction but not in magnitude
• (4) Angular momentum changes both in direction and magnitude

Four particles, each of mass M and equidistant from each other, move along a circle of radius R under the action of their mutual gravitational attraction, the speed of each particle is

• (1) \(\sqrt{\frac{GM}{R}}\)
• (2) \(\sqrt{ 2 \sqrt 2 \frac{GM}{R}}\)
• (3) \(\sqrt{\frac{GM}{R} (1+2 \sqrt2)}\)
• (4) \( \frac{1}{2} \sqrt{\frac{GM}{R} (1+2 \sqrt2)}\)

The pressure that has to be applied to the ends of a steel wire of length 10 cm to keep its length constant when its temperature is raised by \(100^\circ\mathrm{C}\) is :
(For steel Young's modulus is \(2\times10^{11}\ \mathrm{N\,m^{-2}}\) and coefficient of thermal expansion is \(1.1\times10^{-5}\ \mathrm{K^{-1}}\))

• (1) \(2.2\times10^{8}\) Pa
• (2) \(2.2\times10^{9}\) Pa
• (3) \(2.2\times10^{7}\) Pa
• (4) \(2.2\times10^{6}\) Pa

There is a circular tube in a vertical plane. Two liquids which do not mix and of densities \(d_1\) and \(d_2\) are filled in the tube. Each liquid subtends \(90^\circ\) angle at centre. Radius joining their interface makes an angle a with vertical. Ratio \(\frac{d_{1}}{d_{2}}\) is :

• (1) \(\frac{1+sin\alpha}{1-sin\alpha}\)
• (2) \(\frac{1+cos\alpha}{1-cos\alpha}\)
• (3) \(\frac{1+tan\alpha}{1-tan\alpha}\)
• (4) \(\frac{1+sin\alpha}{1-cos\alpha}\)

On heating water, bubbles being formed at the bottom of the vessel detatch and rise. Take the bubbles to be spheres of radius \(R\) and making a circular contact of radius r with the bottom of the vessel. If \(r << R\), and the surface tension of water is T, value of r just before bubbles detatch is (density of water is \(\rho_{w}\) )

• (1) \(R^{2}\sqrt{\frac{2 \rho_{w}\,g}{3T}}\)
• (2) \(R^{2}\sqrt{\frac{\rho_{w}\,g}{6T}}\)
• (3) \(R^{2}\sqrt{\frac{\rho_{w}\,g}{T}}\)
• (4) \(R^{2}\sqrt{\frac{3\rho_{w}\,g}{T}}\)

Three rods of copper , brass and steel are welded together to form a Y-shaped structure .Area of cross -section of each rod is \(4cm^2\).End of copper rod is maintained at \(100^{\circ}C\) whereas ends of brass and steel are kept at \(0^{\circ C}\) Lengths of the copper, brass and steel rods are 46,13 and 12 cm respectively The rods are thermally insulated from surroundings except at ends.Thermal conductivities of copper , brass and steel are 092 , 0.26 and 0.12 in CGS units , respectively .Rate of heat flow throught cooper rod is

• (1) 1.2 cal /s
• (2) 2.4 cal/s
• (3) 4.8 cal/s
• (4) 6.0 cal/s

One mole of diatomic ideal gas undergoes a cyclic process ABC as shown in figure. The process BC is adiabatic. The temperatures at A, B and C are 400 K, 800 K and 600 K respectively. Choose the correct statement :

• (1) The change in internal energy in whole cyclic process is \(250\ R\).
• (2) The change in internal energy in the process CA is \(700\ R\).
• (3) The change in internal energy in the process AB is \(-350\ R\).
• (4) The change in internal energy in the process BC is \(-500\ R\).

An open glass tube is immersed in mercury in such a way that a length of 8 cm extends above the mercury level. The open end of the tube is then closed and sealed and the tube is raised vertically up by additional 46 cm What will be length of the air column above mercury in the tube now ? (Atmospheric pressure = \(76\, cm\) of Hg)

• (1) 16 cm
• (2) 22 cm
• (3) 38 cm
• (4) 6 cm

A particle moves with simple harmonic motion in a straight line. In first \(\tau\) sec, after starting from rest it travels a distance a and in next \(\tau\) sec, it travels \(2a\), in same direction, then

• (1) amplitude of motion is 3a
• (2) time period of oscillations is \(8\,\tau\)
• (3) amplitude of motion is \(4a\)
• (4) time period of oscillations is \(6\tau\)

A pipe of length \(85\ \mathrm{cm}\) is closed from one end. Find the number of possible natural oscillations of air column in the pipe whose frequencies lie below \(1250\ \mathrm{Hz}\). The velocity of sound in air is \(340\ \mathrm{m/s}\).

• (1) 12
• (2) 8
• (3) 6
• (4) 4

Assume that an electric field \(\vec{E}=30 x^{2} \hat{i}\) exists in space. Then the potential difference \(V_{A}-V_{O'}\) where \(V_{O}\) is the potential at the origin and \(V_{A}\) the potential at \(x=2 \,m\) is

• (1) 120 J
• (2) -120 J
• (3) -80 J
• (4) 80 J

A parallel plate capacitor is made of two circular plates separated by a distance of 5 mm and with a dielectric of dielectric constant 2.2 between them. When the electric field in the dielectric is \(3 \times 10^4 \,V/m\), the charge density of the positive plate will be close to

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

In a large building, there are 15 bulbs of 40\, W, 5 bulbs of 100 W, 5 fans of \(80\, W\) and 1 heater of \(1\, kW\). The voltage of the electric mains is \(220 \,V\). The minimum capacity of the main fuse of the building will be :

• (1) 8 A
• (2) 10 A
• (3) 12 A
• (4) 14 A

A conductor lies along the \(z\) -axis at \(-1.5 \leq z<1.5 \,m\) and carries a fixed current of \(10.0 A\) in \(-\hat{a}_{z}\) direction (see figure). For a field \(\vec{B}=3.0 \times 10^{-4} e^{-0.2 x} \hat{a}_{y} T\), find the power required to move the conductor at constant speed to \(x=2.0 m , y=0 \,m\) in \(5 \times 10^{-3} \,s\). Assume parallel motion along the \(x\) -axis

• (1) 1.57 W
• (2) 2.97 W
• (3) 14.85 W
• (4) 29.7 W

The coercivity of a small magnet where the ferromagnet gets demagnetized is \(3\times10^{3}\ \mathrm{A\,m^{-1}}\). The current required to be passed in a solenoid of length \(10\ \mathrm{cm}\) and number of turns \(100\), so that the magnet gets demagnetized when inside the solenoid, is :

• (1) 30 mA
• (2) 60 mA
• (3) 3 A
• (4) 6 A

In the circuit shown here, the point 'C' is kept connected to point 'A' till the current flowing through the circuit becomes constant. Afterward, suddenly, point 'C' is disconnected from point 'A' and connected to point 'B' at time \(t=0\). Ratio of the voltage across resistance and the inductor at \(t=L/R\) will be equal to :

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

During the propagation of electromagnetic waves in a medium,

• (1) electric energy density is double of the magnetic energy density
• (2) electric energy density is half of the magnetic energy density
• (3) electric energy density is equal to the magnetic energy density
• (4) Both electric and magnetic energy densities are zero

A thin convex lens made from crown glass \(\left(\mu=\frac{3}{2}\right)\) has focal length \(f\). When it is measured in two different liquids having refractive indices \(\frac{4}{3}\) and \(\frac{5}{3}\), it has the focal lengths \(f_{1}\) and \(f_{2}\) respectively. The correct relation between the focal lengths is

• (1) f_1 - f_2
• (2) \(f_2 >\, f\) and \(f_2\) becomes negative
• (3) \(f_2 >\, f\) and \(f_1\) becomes negative
• (4) \(f_1 \) and \(f_2 \) both become negative

A green light is incident from the water to the air - water interface at the critical angle(\(\theta\)). Select the correct statement.

• (1) The entire spectrum of visible light will come out of the water at an angle of \(90^\circ\) to the normal.
• (2) The spectrum of visible light whose frequency is less than that of green light will come out to the air medium.
• (3) The spectrum of visible light whose frequency is more than that of green light will come out to the air medium.
• (4) The entire spectrum of visible light will come out of the water at various angles to the normal.

Two beams, A and B, of plane polarised light with mutually perpendicular planes of polarisation are seen through a Polaroid. From the position when the beam A has maximum intensity (and beam B has zero intensity), a rotation of Polaroid through 30\(^{\circ}\) makes the two beams appear equally bright. If the initial intensities of the two beams are \(I_A \, and \, I_B\) respectively, then \(I_A / I_B\) equals

• (1) 3
• (2) 44622
• (3) 1
• (4) 44564

The radiation corresponding to \(3\to2\) transition of hydrogen atom falls on a metal surface to produce photoelectrons. These electrons are made to enter a magnetic field of \(3\times10^{-4}\ \mathrm{T}\). If the radius of the largest circular path followed by these electrons is \(10.0\ \mathrm{mm}\), the work function of the metal is close to :

• (1) 1.8 eV
• (2) 1.1 eV
• (3) 0.8 eV
• (4) 1.6 eV

Hydrogen (\({}_1\mathrm{H}^1\)), Deuterium (\({}_1\mathrm{H}^2\)), singly ionised Helium (\(({}_2\mathrm{He}^4)^+\)) and doubly ionised lithium (\(({}_3\mathrm{Li}^6)^{++}\)) all have one electron around the nucleus. Consider an electron transition from \(n=2\) to \(n=1\). If the wave lengths of emitted radiation are \(\lambda_1,\ \lambda_2,\ \lambda_3\) and \(\lambda_4\) respectively then approximately which one of the following is correct ?

• (1) \(4\lambda_1=2\lambda_2=2\lambda_3=\lambda_4\)
• (2) \(\lambda_1=2\lambda_2=2\lambda_3=\lambda_4\)
• (3) \(\lambda_1=\lambda_2=4\lambda_3=9\lambda_4\)
• (4) \(\lambda_1=2\lambda_2=3\lambda_3=4\lambda_4\)

The forward biased diode connection is :

• (1) Diode (anode on the left) in series with a resistor; left end at \(+2\ \mathrm{V}\), right end at \(-2\ \mathrm{V}\).
• (2) Diode (anode on the left) in series with a resistor; left end at \(-3\ \mathrm{V}\), right end at \(-3\ \mathrm{V}\).
• (3) Diode (anode on the left) in series with a resistor; left end at \(2\ \mathrm{V}\), right end at \(4\ \mathrm{V}\).
• (4) Diode (anode on the left) in series with a resistor; left end at \(-2\ \mathrm{V}\), right end at \(+2\ \mathrm{V}\).

Match List -1 (Electromagnetic wave type) with List - II (Its association/application) and select the correct option from the choices given below the lists :

• (1) (a)-(iv), (b)-(iii), (c)-(ii), (d)-(i)
• (2) (a)-(i), (b)-(ii), (c)-(iv), (d)-(iii)
• (3) (a)-(iii), (b)-(ii), (c)-(i), (d)-(iv)
• (4) (a)-(i), (b)-(ii), (c)-(iii), (d)-(iv)

A student measured the length of a rod and wrote it as \(3.50\ \mathrm{cm}\). Which instrument did he use to measure it ?

• (1) A meter scale.
• (2) A vernier calliper where the 10 divisions in vernier scale matches with 9 division in main scale and main scale has 10 divisions in 1 cm.
• (3) A screw gauge having 100 divisions in the circular scale and pitch as 1 mm.
• (4) A screw gauge having 50 divisions in the circular scale and pitch as 1 mm.

The correct set of four quantum numbers for the valence electrons of rubidium atom \((Z = 37)\) is

• (1) \(5,0,0,+\frac{1}{2}\)
• (2) \(5,1,0,+\frac{1}{2}\)
• (3) \(5,1,1,+\frac{1}{2}\)
• (4) \(5,0,1,+\frac{1}{2}\)

If Z is a compressibility factor, van der Waals equation at low pressure can be written as

• (1) \(Z=1+\frac{RT}{pb}\)
• (2) \(Z=1-\frac{a}{VRT}\)
• (3) \(Z=1-\frac{Pb}{RT}\)
• (4) \(Z=1+\frac{Pb}{RT}\)

\(CsCl\) crystallises in body centred cubic lattice. If 'a' is its edge length, then which of the following expressions is correct?

• (1) \( r_{Cs^+} + r_{Cl^-} = 3a \)
• (2) \( r_{Cs^+} + r_{Cl^-} = \frac{3a}{2} \)
• (3) \( r_{Cs^+} + r_{Cl^-} = \frac{\sqrt 3}{2} a \)
• (4) \( r_{Cs^+} + r_{Cl^-} = \sqrt 3 a \)

For the estimation of nitrogen, 1.4 g of an organic compound was digested by Kjeldahl method and the evolved ammonia was absorbed in 60 mL of \(\frac{M}{10}\) sulphuric acid. The unreacted acid required 20 mL of \(\frac{M}{10}\) sodium hydroxide for complete neutralization. The percentage of nitrogen in the compound is :

• (1) 6%
• (2) 10%
• (3) 3%
• (4) 5%

Resistance of \(0.2\, M\) solution of an electrolyte is \(50 \,\Omega\). The specific conductance of the solution of \(0.5 \,M\) solution of same electrolyte is \(1.4\, S \, m^{-1}\) and resistance of same solution of the same electrolyte is \(280 \,\Omega\) . The molar conductivity of \(0.5\, M\) solution of the electrolyte in \(S\,m^2 \,mol^{-1}\) is

• (1) \(5 \times10^{-4}\)
• (2) \(5 \times10^{-3}\)
• (3) \(5 \times10^{3}\)
• (4) \(5 \times10^{2}\)

For the complete combustion of ethanol, \(C_2H_5OH_{(l)} + 3O_{2(g)} \rightarrow 2CO_{2(g)} + 3H_2O_{(l)},\) the amount of heat produced as measured in bomb calorimeter, is \(1364.47 \,kJ \,moI^{-1}\) at \(25^{\circ}\)C. Assuming ideality the enthalpy of combustion, \(\Delta_ CH\), for the reaction will be \((R = 8.314\, J \,K^{-1}\) \(mol^{-1})\)

• (1) \(- 1366.95 \,kJ \,moI^{-1}\)
• (2) \(- 1361.95 \, kJ \, moI^{-1}\)
• (3) \(- 1460.50 \, kJ \,moI^{-1}\)
• (4) \(- 1350.50\, kJ \, moI^{-1}\)

The equivalent conductance of \(NaCl\) at concentration \(C\) and at infinite dilution are \(\lambda_C\) and \(\lambda_\infty\) respectively. The correct relationship between \(\lambda_C\) and \(\lambda_\infty\) is given as (where, the constant \(B\) is positive)

• (1) \(\lambda_C = \lambda_\infty+(B)C\)
• (2) \(\lambda_C = \lambda_\infty - (B)C\)
• (3) \(\lambda_C = \lambda_\infty - (B)\sqrt C\)
• (4) \(\lambda_C = \lambda_\infty +(B)\sqrt C\)

Consider separate solutions of 0.500 M \(\mathrm{C_2H_5OH(aq)}\), 0.100 M \(\mathrm{Mg_3(PO_4)_2(aq)}\), 0.250 M \(\mathrm{KBr(aq)}\) and 0.125 M \(\mathrm{Na_3PO_4(aq)}\) at 25\(^\circ\)C. Which statement is true about these solutions, assuming all salts to be strong electrolytes ?

• (1) They all have the same osmotic pressure.
• (2) 0.100 M \(\mathrm{Mg_3(PO_4)_2(aq)}\) has the highest osmotic pressure.
• (3) 0.125 M \(\mathrm{Na_3PO_4(aq)}\) has the highest osmotic pressure.
• (4) 0.500 M \(\mathrm{C_2H_5OH(aq)}\) has the highest osmotic pressure.

For the reaction, \( SO_2 ( g) + \frac{1}{2} O_2 \, ( g) \rightleftharpoons SO_3 \, ( g) \) if \( K_P = K_C \, ( RT)^x \) X where, the symbols have usual meaning, then the value of x is (assuming ideality)

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

• (1) \(\frac{dc}{dt} = k[A][B]\)
• (2) \(\frac{dc}{dt} = k[A]^2[B]\)
• (3) \(\frac{dc}{dt} = k[A][B]^2\)
• (4) \(\frac{dc}{dt} = k[A]\)

Among the following oxoacids, the correct decreasing order of acidic strength is

• (1) \(HOCl > HClO_2 > HClO_3 > HClO_4\)
• (2) \(HClO_4 > HOCl > HClO_2 > HClO_3\)
• (3) \(HClO_4 > HClO_3 > HClO_2 > HOCl\)
• (4) \(HClO_2 > HClO_4 > HClO_3 > HOCl\)

The metal that cannot be obtained by electrolysis of an aqueous solution of its salts is

• (1) \(Ag\)
• (2) \(Ca\)
• (3) \(Cu\)
• (4) \(Cr\)

The octahedral complex of a metal ion \(M^{3+} \) with four monodentate ligands \(L_1,L_2,L_3 \, and \, L_4 \) absorb wavelengths in the region of red, green, yellow and blue, respectively. The increasing order of ligand strength of the four ligands is

• (1) \(L_4 < L_3, L_2 < L_1 \)
• (2) \(L_1 < L_3 < L_2 < L_4 \)
• (3) \(L_3 < L_2 < L_4 < L_1 \)
• (4) \(L_1 < L_2 < L_4 < L_3 \)

Which one of the following properties is not shown by NO ?

• (1) It is diamagnetic in gaseous state
• (2) It is a neutral oxide
• (3) It combines with oxygen to form nitrogen dioxide
• (4) It's bond order is 2.5

In which of the following reactions \( H_2O_2 \) acts as a reducing agent? (a) \(H_2O_2 + 2H^+ + 2e^- \rightarrow 2H_2O \) (d) \( H_2O_2 - 2e^- \rightarrow O_2 + 2H^+ \) (c) \(H_2O_2 + 2e^- \rightarrow 2OH^- \) (d) \(H_2O_2 + 2OH^- - 2e^- \rightarrow O_2 + 2H_2O \)

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

The correct statement for the molecule, \(CsI_3\) is

• (1) it is a covalent molecule
• (2) it contains \({Cs}^+\) and \({I_3}^-\) ions
• (3) it contains \({Cs}^{3+}\) and \({I}^-\) ions
• (4) it contains \({Cs}^{+}\), \({I}^-\) and lattice \({I}_2\) molecule

The ratio mass of oxygen and nitrogen of a particular gaseous mixture is \(1 : 4\) The ratio of number of their molecule is

• (1) \(1 : 4\)
• (2) \(7 : 32\)
• (3) \(1 : 8\)
• (4) \(3 : 16\)

Given below are the half-cell reactions \(Mn ^{2+}+2 e ^{-} \rightarrow Mn ; E ^{\circ}=-1.18 \,V\) \(\left( Mn ^{3+}+ e ^{-} \rightarrow Mn ^{2+}\right) ; E ^{\circ}=+1.51 \,V\) The \( E ^{\circ} \) for \(3 \,Mn ^{2+} \rightarrow Mn +2 Mn ^{3+}\) will be

• (1) \(-2.69\, V\); the reaction will not occur
• (2) \(- 2.69\, V\); the reaction will occur
• (3) \(-2.69 \,V\); the reaction will occur
• (4) \(- 0.33\, V\); the reaction will occur

Which series of reactions correctly represents chemical relations related to iron and its compound?

• (1) \(Fe {->[dil. H_2SO_4]} FeSO_4 {->[H_2SO_4, O_2]} Fe_2(SO_4)_3 {->[\text{heat}]}Fe\)
• (2) \(Fe {->[O_2, \text{heat}]} FeO {->[dil. H_2SO_4]}Fe_2(SO_4) {->[\text{heat}]} Fe\)
• (3) \(Fe {->[Cl_2, \text{heat}]} FeCl_3 {->[\text{heat, air}]} FeCl_2 {->[Zn]}Fe\)
• (4) \(Fe {->[O_2, \text{heat}]} FeO_4 {->[CO, 600^{\circ}C]} FeO {->[CO, 700^{\circ}C]} Fe\)

The equation which is balanced and represents the correct product(s) is :

• (1) \(\mathrm{Li_2O + 2KCl \rightarrow 2LiCl + K_2O}\)
• (2) \(\mathrm{[CoCl(NH_3)_5]^+ + 5H^+ \rightarrow Co^{2+} + 5NH_4^+ + Cl^-}\)
• (3) \(\mathrm{[Mg(H_2O)_6]^{2+} + (EDTA)^{4-} \xrightarrow{excess\ NaOH} [Mg(EDTA)]^{2+} + 6H_2O}\)
• (4) \(\mathrm{CuSO_4 + 4KCN \rightarrow K_2[Cu(CN)_4] + K_2SO_4}\)

In S\(_N\)2 reactions, the correct order of reactivity for the following compounds \(CH_3Cl,CH_3CH_2Cl, (CH_3 )_2CHCl\) and (CH\(_3 )_3CCl\) is

• (1) \(CH_3Cl > (CH_3)_2CHCl > CH_3CH_2Cl > (CH_3)_3CCl\)
• (2) \(CH_3Cl > CH_3CH_2 Cl > (CH_3)_2 CHCl >(CH_3)_3CCl\)
• (3) \(CH_3CH_2Cl > CH_3Cl > (CH_3)_2CHCl > (CH_3)_3CCl\)
• (4) \((CH_3)_2CHCl > CH_3CH_2Cl > CH_3Cl > (CH_3)_3CCl\)

On heating an aliphatic primary amine with chloroform and ethanolic potassium hydroxide, the organic compound formed is :

• (1) an alkanol
• (2) an alkanediol
• (3) an alkyl cyanide
• (4) an alkyl isocyanide

The most suitable reagent for the conversion of \(RCH_2OH \to RCHO\) is

• (1) \(KMnO_4\)
• (2) \(K_2Cr_2O_7\)
• (3) \(CrO_3\)
• (4) \(PCC\) (pyridine chloro chromate)

The major organic compound formed by the reaction of 1, 1, 1 - trichloroethane with silver powder is :

• (1) Acetylene
• (2) Ethene
• (3) 2 - Butyne
• (4) 2 - Butene

Sodium phenoxide when heated with \(\mathrm{CO_2}\) under pressure at \(125^\circ\mathrm{C}\) yields a product which on acetylation produces C.

The major product C would be :

• (1)
• (2)
• (3)
• (4)

Considering the basic strength of amines in aqueous solution, which one has the smallest \(pK_b\) value?

• (1) \((CH_3)_2NH\)
• (2) \(CH_3NH_2\)
• (3) \((CH_3)_3 N\)
• (4) \(C_6H_5NH_2\)

For which of the following molecule significant \(\mu \neq 0\) ?

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

Which one is classified as a condensation polymer ?

• (1) Dacron
• (2) Neoprene
• (3) Teflon
• (4) Acrylonitrile

Which one of the following bases is not present in DNA ?

• (1) Quinoline
• (2) Adenine
• (3) Cytosine
• (4) Thymine

In the reaction,
\(\mathrm{CH_3COOH \xrightarrow{LiAlH_4} A \xrightarrow{PCl_5} B \xrightarrow{Alc.\ KOH} C}\),
the product C is :

• (1) Acetaldehyde
• (2) Acetylene
• (3) Ethylene
• (4) Acetyl chloride