JEE Main 2015 B.E./B.Tech Question Paper with Answer Key PDF- Online (April 11, 2015)

Let \(A=\{x_1, x_2, \ldots, x_7\}\) and \(B=\{y_1, y_2, y_3\}\) be two sets containing seven and three distinct elements respectively. Then the total number of functions \(f: A \to B\) that are onto, if there exist exactly three elements \(x\) in \(A\) such that \(f(x)=y_2\), is equal to :

• (1) \(14\cdot {}^7C_2\)
• (2) \(16\cdot {}^7C_3\)
• (3) \(12\cdot {}^7C_2\)
• (4) \(14\cdot {}^7C_3\)

If \(z\) is a non-real complex number, then the minimum value of \(\dfrac{\operatorname{Im} z^5}{(\operatorname{Im} z)^5}\) is :

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

If the two roots of the equation, \((a-1)(x^4+x^2+1)+(a+1)(x^2+x+1)^2=0\) are real and distinct, then the set of all values of 'a' is :

• (1) \(\left(-\frac{1}{2}, 0\right)\)
• (2) \((-\infty, -2)\cup(2, \infty)\)
• (3) \(\left(-\frac{1}{2}, 0\right)\cup\left(0, \frac{1}{2}\right)\)
• (4) \(\left(0, \frac{1}{2}\right)\)

If A is a \(3\times3\) matrix such that \(|5\cdot \operatorname{adj}A|=5\), then \(|A|\) is equal to :

• (1) \(\pm\dfrac{1}{5}\)
• (2) \(\pm5\)
• (3) \(\pm1\)
• (4) \(\pm\dfrac{1}{25}\)

If \(\begin{vmatrix}x^{2}+x&x+1&x-2\\ 2x^{2}+3x-1&3x&3x-3\\ x^{2}+2x+3&2x-1&2x-1\end{vmatrix}=ax-12,\) then 'a' is equal to :

• (1) 12
• (2) 24
• (3) -12
• (4) -24

If in a regular polygon the number of diagonals is \(54\), then the number of sides of this polygon is :

• (1) 10
• (2) 12
• (3) 9
• (4) 6

The term independent of \(x\) in the binomial expansion of \(\left(1-\frac{1}{x}+3x^{5}\right)\left(2x^{2}-\frac{1}{x}\right)^{8}\) is :

• (1) 400
• (2) 496
• (3) -400
• (4) -496

The sum of the \(3^{rd}\) and the \(4^{th}\) terms of a G.P. is 60 and the product of its first three terms is 1000. If the first term of this G.P. is positive, then its \(7^{th}\) term is :

• (1) \(7290\)
• (2) \(320\)
• (3) \(640\)
• (4) \(2430\)

If \(\displaystyle\sum_{n=1}^{5}\dfrac{1}{n(n+1)(n+2)(n+3)}=\dfrac{k}{3}\), then k is equal to :

• (1) \(\dfrac{55}{336}\)
• (2) \(\dfrac{17}{105}\)
• (3) \(\dfrac{1}{6}\)
• (4) \(\dfrac{19}{112}\)

Let k be a non-zero real number. If

\( f(x) = \begin{cases} \dfrac{(e^x - 1)^2}{\sin\left(\frac{x}{k}\right)\log\left(1+\frac{x}{4}\right)}, & x \ne 0 \\[2ex] 12, & x = 0 \end{cases} \)

is a continuous function, then the value of \( k \) is:

• (1) 1
• (2) 2
• (3) 3
• (4) 4

The equation of a normal to the curve, \(\sin y=x\sin\left(\dfrac{\pi}{3}+y\right)\) at \(x=0\), is :

• (1) \(2x+\sqrt3\,y=0\)
• (2) \(2y-\sqrt3\,x=0\)
• (3) \(2y+\sqrt3\,x=0\)
• (4) \(2x-\sqrt3\,y=0\)

Let k and K be the minimum and the maximum values of the function \(f(x)=\dfrac{(1+x)^{0.6}}{1+x^{0.6}}\) in \([0,1]\) respectively, then the ordered pair \((k, K)\) is equal to :

• (1) \((1, 2^{0.6})\)
• (2) \((2^{-0.4}, 2^{0.6})\)
• (3) \((2^{-0.6}, 1)\)
• (4) \((2^{-0.4}, 1)\)

From the top of a 64 metres high tower, a stone is thrown upwards vertically with the velocity of 48 m/s. The greatest height (in metres) attained by the stone, assuming the value of the gravitational acceleration \(g=32\) m/s\(^2\), is :

• (1) \(100\)
• (2) \(88\)
• (3) \(128\)
• (4) \(112\)

If \(\displaystyle\int\dfrac{\log\left(t+\sqrt{1+t^2}\right)}{\sqrt{1+t^2}}\,dt=\dfrac12\left(g(t)\right)^2+C\), where C is a constant, then \(g(2)\) is equal to :

• (1) \(2\log\left(2+\sqrt5\right)\)
• (2) \(\log\left(2+\sqrt5\right)\)
• (3) \(\dfrac{1}{\sqrt5}\log\left(2+\sqrt5\right)\)
• (4) \(\dfrac12\log\left(2+\sqrt5\right)\)

Let \(f:\mathbf{R}\to\mathbf{R}\) be a function such that \(f(2-x)=f(2+x)\) and \(f(4-x)=f(4+x)\), for all \(x\in\mathbf{R}\) and \(\int_0^2 f(x)\,dx=5\). Then the value of \(\int_{10}^{50} f(x)\,dx\) is :

• (1) 80
• (2) 100
• (3) 125
• (4) 200

Let \(f:(-1,1)\to\mathbf{R}\) be a continuous function. If \(\int_0^{\sin x} f(t)\,dt=\frac{\sqrt{3}}{2}x\), then \(f\left(\frac{\sqrt{3}}{2}\right)\) is equal to :

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

The solution of the differential equation \(ydx - (x + 2y^2)dy = 0\) is \(x = f(y)\). If \(f(- 1) = 1\), then \(f(1)\) is equal to :

• (1) 4
• (2) 3
• (3) 2
• (4) 1

A straight line L through the point \((3,-2)\) is inclined at an angle of \(60^\circ\) to the line \(\sqrt{3}\,x+y=1\). If L also intersects the \(x\)-axis, then the equation of L is :

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

If the incentre of an equilateral triangle is \((1,1)\) and the equation of its one side is \(3x+4y+3=0\), then the equation of the circumcircle of this triangle is :

• (1) \(x^2+y^2-2x-2y-2=0\)
• (2) \(x^2+y^2-2x-2y-14=0\)
• (3) \(x^2+y^2-2x-2y+2=0\)
• (4) \(x^2+y^2-2x-2y-7=0\)

If a circle passing through the point \((-1,0)\) touches \(y\)-axis at \((0,2)\), then the length of the chord of the circle along the \(x\)-axis is :

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

If the distance between the foci of an ellipse is half the length of its latus rectum, then the eccentricity of the ellipse is :

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

Let \(PQ\) be a double ordinate of the parabola, \(y^2= - 4x\), where P lies in the second quadrant. If R divides \(PQ\) in the ratio \(2 : 1\), then the locus of R is :

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

The shortest distance between the \(z\)-axis and the line \(x+y+2z-3=0=2x+3y+4z-4\), is :

• (1) 1
• (2) 2
• (3) 3
• (4) 4

A plane containing the point \((3,2,0)\) and the line \(\frac{x-1}{1}=\frac{y-2}{5}=\frac{z-3}{4}\) also contains the point :

• (1) \((0,-3,1)\)
• (2) \((0,7,10)\)
• (3) \((0,7,-10)\)
• (4) \((0,3,1)\)

In a parallelogram ABCD, \(\left|\overrightarrow{AB}\right|=a\), \(\left|\overrightarrow{AD}\right|=b\) and \(\left|\overrightarrow{AC}\right|=c\), then \(\overrightarrow{DB}\cdot\overrightarrow{AB}\) has the value :

• (1) \(\frac{1}{2}\left(a^2-b^2+c^2\right)\)
• (2) \(\frac{1}{4}\left(a^2+b^2-c^2\right)\)
• (3) \(\frac{1}{3}\left(b^2+c^2-a^2\right)\)
• (4) \(\frac{1}{2}\left(a^2+b^2+c^2\right)\)

If the lengths of the sides of a triangle are decided by the three throws of a single fair die, then the probability that the triangle is of maximum area given that it is an isosceles triangle, is :

• (1) \(\frac{1}{26}\)
• (2) \(\frac{1}{27}\)
• (3) \(\frac{1}{21}\)
• (4) \(\frac{1}{15}\)

If the mean and the variance of a binomial variate \(X\) are \(2\) and \(1\) respectively, then the probability that \(X\) takes a value greater than or equal to one is :

• (1) \(\frac{1}{16}\)
• (2) \(\frac{9}{16}\)
• (3) \(\frac{3}{4}\)
• (4) \(\frac{15}{16}\)

If \(\cos\alpha+\cos\beta=\frac{3}{2}\) and \(\sin\alpha+\sin\beta=\frac{1}{2}\) and \(\theta\) is the arithmetic mean of \(\alpha\) and \(\beta\), then \(\sin2\theta+\cos2\theta\) is equal to :

• (1) \(\frac{3}{5}\)
• (2) \(\frac{4}{5}\)
• (3) \(\frac{7}{5}\)
• (4) \(\frac{8}{5}\)

Let 10 vertical poles standing at equal distances on a straight line, subtend the same angle of elevation \(\alpha\) at a point O on this line and all the poles are on the same side of O. If the height of the longest pole is 'h' and the distance of the foot of the smallest pole from O is 'a'; then the distance between two consecutive poles, is :

• (1) \(\frac{h\sin\alpha+a\cos\alpha}{9\sin\alpha}\)
• (2) \(\frac{h\cos\alpha-a\sin\alpha}{9\cos\alpha}\)
• (3) \(\frac{h\cos\alpha-a\sin\alpha}{9\sin\alpha}\)
• (4) \(\frac{h\sin\alpha+a\cos\alpha}{9\cos\alpha}\)

Consider the following statements :
P : Suman is brilliant.
Q : Suman is rich.
R : Suman is honest.
The negation of the statement, "Suman is brilliant and dishonest if and only if Suman is rich" can be equivalently expressed as :

• (1) \(\sim Q\leftrightarrow\ \sim P\wedge R\)
• (2) \(\sim Q\leftrightarrow\ \sim P\vee R\)
• (3) \(\sim Q\leftrightarrow P\vee\ \sim R\)
• (4) \(\sim Q\leftrightarrow P\wedge\ \sim R\)

If electronic charge e, electron mass m, speed of light in vacuum c and Planck's constant h are taken as fundamental quantities, the permeability of vacuum \(\mu_0\) can be expressed in units of :

• (1) \(\left(\frac{hc}{me^2}\right)\)
• (2) \(\left(\frac{h}{me^2}\right)\)
• (3) \(\left(\frac{h}{ce^2}\right)\)
• (4) \(\left(\frac{mc^2}{he^2}\right)\)

A vector \(\vec{A}\) is rotated by a small angle \(\Delta\theta\) radians \(\left(\Delta\theta<<1\right)\) to get a new vector \(\vec{B}.\) In that case \(\left|\vec{B}-\vec{A}\right|\) is :

• (1) \(0\)
• (2) \(\left|\vec{A}\right|\left(1-\frac{\Delta\theta^{2}}{2}\right)\)
• (3) \(\left|\vec{A}\right|\Delta\theta\)
• (4) \(\left|\vec{B}\right|\Delta\theta-\left|\vec{A}\right|\)

A large number (n) of identical beads, each of mass m and radius r are strung on a thin smooth rigid horizontal rod of length L (\(L \gg r\)) and are at rest at random positions. The rod is mounted between two rigid supports (see figure). If one of the beads is now given a speed \(v\), the average force experienced by each support after a long time is (assume all collisions are elastic) :

• (1) \(\frac{mv^2}{L-nr}\)
• (2) \(\frac{mv^2}{L-2nr}\)
• (3) \(\frac{mv^2}{2(L-nr)}\)
• (4) zero

A particle is moving in a circle of radius r under the action of a force \(F=\alpha r^2\) which is directed towards centre of the circle. Total mechanical energy (kinetic energy + potential energy) of the particle is (take potential energy = 0 for r = 0) :

• (1) \(\alpha r^3\)
• (2) \(\frac{1}{2}\,\alpha r^3\)
• (3) \(\frac{4}{3}\,\alpha r^3\)
• (4) \(\frac{5}{6}\,\alpha r^3\)

A uniform thin rod AB of length L has linear mass density \(\mu(x)=a+\frac{bx}{L}\), where x is measured from A. If the CM of the rod lies at a distance of \(\left(\frac{7}{12}L\right)\) from A, then a and b are related as :

• (1) \(a=b\)
• (2) \(a=2b\)
• (3) \(2a=b\)
• (4) \(3a=2b\)

A particle of mass 2 kg is on a smooth horizontal table and moves in a circular path of radius 0.6 m. The height of the table from the ground is 0.8 m. If the angular speed of the particle is 12 rad s^-1, the magnitude of its angular momentum about a point on the ground right under the centre of the circle is :

• (1) \(8.64\ \mathrm{kg\ m^2 s^{-1}}\)
• (2) \(11.52\ \mathrm{kg\ m^2 s^{-1}}\)
• (3) \(14.4\ \mathrm{kg\ m^2 s^{-1}}\)
• (4) \(20.16\ \mathrm{kg\ m^2 s^{-1}}\)

Which of the following most closely depicts the correct variation of the gravitation potential V(r) due to a large planet of radius R and uniform mass density \(\rho\) ? (figures are not drawn to scale)

• (1)
• (2)
• (3)
• (4)

A cylindrical block of wood (density = 650 kg m^-3), of base area 30 cm² and height 54 cm, floats in a liquid of density 900 kg m^-3. The block is depressed slightly and then released. The time period of the resulting oscillations of the block would be equal to that of a simple pendulum of length (nearly) :

• (1) 65 cm
• (2) 52 cm
• (3) 39 cm
• (4) 26 cm

A beaker contains a fluid of density \(\rho\ \mathrm{kg/m^3}\), specific heat \(S\ \mathrm{J/kg^\circ C}\) and viscosity \(\eta\). The beaker is filled up to height h. To estimate the rate of heat transfer per unit area \((\dot Q/A)\) by convection when beaker is put on a hot plate, a student proposes that it should depend on \(\eta\), \(\left(\frac{S\Delta\theta}{h}\right)\) and \(\left(\frac{1}{\rho g}\right)\) when \(\Delta\theta\) (in \(^\circ\)C) is the difference in the temperature between the bottom and top of the fluid. In that situation the correct option for \((\dot Q/A)\) is :

• (1) \(\eta\,\frac{S\Delta\theta}{h}\)
• (2) \(\eta\left(\frac{S\Delta\theta}{h}\right)\left(\frac{1}{\rho g}\right)\)
• (3) \(\frac{S\Delta\theta}{\eta h}\)
• (4) \(\left(\frac{S\Delta\theta}{\eta h}\right)\left(\frac{1}{\rho g}\right)\)

An experiment takes \(10\, min\) to raise the temperature of water in a container from \(0^{\circ} C\) to \(100^{\circ} C\) and another \(55\, min\) to convert it totally into steam by a heater supplying heat at a uniform rate. Neglecting the specific heat of the container and taking specific heat of water to be \(1\, cal / g { }^{\circ} C\), the heat of vaporisation according to this experiment will come out to be

• (1) 530 cal/ g
• (2) 540 cal/g
• (3) 550 cal/g
• (4) 560 cal/g

Using equipartition of energy, the specific heat (in J kg^-1 K^-1) of aluminium at room temperature can be estimated to be (atomic weight of aluminium = 27)

• (1) 25
• (2) 410
• (3) 925
• (4) 1850

A pendulum with time period of 1 s is losing energy due to damping. At certain time its energy is 45 J. If after completing 15 oscillations, its energy has become 15 J, its damping constant (in s^-1) is :

• (1) \(\frac{1}{30}\,\ln 3\)
• (2) \(\frac{1}{15}\,\ln 3\)
• (3) 2
• (4) \(\frac12\)

A source of sound emits sound waves at frequency \(f_0\). It is moving towards an observer with fixed speed \(v_s\) (\(v_s \lt v\), where v is the speed of sound in air). If the observer were to move towards the source with speed \(v_0\), one of the following two graphs (A and B) will give the correct variation of the frequency f heard by the observer as \(v_0\) is changed.

The variation of f with \(v_0\) is given correctly by :

• (1) graph A with slope \(=\frac{f_0}{(v-v_s)}\)
• (2) graph A with slope \(=\frac{f_0}{(v+v_s)}\)
• (3) graph B with slope \(=\frac{f_0}{(v-v_s)}\)
• (4) graph B with slope \(=\frac{f_0}{(v+v_s)}\)

A wire, of length L (= 20 cm), is bent into a semi-circular arc. If the two equal halves, of the arc, were each to be uniformly charged with charges \(\pm Q\), [\(|Q|=10^3\,\varepsilon_0\) Coulomb where \(\varepsilon_0\) is the permittivity (in SI units) of free space] the net electric field at the centre O of the semi-circular arc would be :

• (1) \((50\times10^3\ \mathrm{N/C})\ \hat{j}\)
• (2) \((25\times10^3\ \mathrm{N/C})\ \hat{i}\)
• (3) \((25\times10^3\ \mathrm{N/C})\ \hat{j}\)
• (4) \((50\times10^3\ \mathrm{N/C})\ \hat{i}\)

An electric field \(\vec E=(25\,\hat i+30\,\hat j)\ \mathrm{NC^{-1}}\) exists in a region of space. If the potential at the origin is taken to be zero then the potential at x = 2 m, y = 2 m is :

• (1) \(-130\ \mathrm{J}\)
• (2) \(-120\ \mathrm{J}\)
• (3) \(-140\ \mathrm{J}\)
• (4) \(-110\ \mathrm{J}\)

In the following figure is shown a system of four capacitors connected across a \(10\, V\) battery. Charge that will flow from switch \(S\) when it is closed is

• (1) 5 \(\mu\,C\) from \(b\) to \(a\)
• (2) 20 \(\mu\,C\) from \(a\) to \(b\)
• (3) 5 \(\mu\,C\) from \(a\) to \(b\)
• (4) \(zero\)

In the electric network shown, when no current flows through the \(4\, \Omega\) resistor in the arm \(EB\), the potential difference between the points \(A\) and \(D\) will be :

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

The value of the resistor, \(R_S\), needed in the dc voltage regulator circuit shown here, equals :

• (1) \((V_i-V_L)/n\,I_L\)
• (2) \((V_i+V_L)/n\,I_L\)
• (3) \((V_i-V_L)/(n+1)\,I_L\)
• (4) \((V_i+V_L)/(n+1)\,I_L\)

Two long straight parallel wires, carrying (adjustable) currents \(I_1\) and \(I_2\), are kept at a distance \(d\) apart. If the force 'F' between the two wires is taken as 'positive' when the wires repel each other and 'negative' when the wires attract each other, the graph showing the dependence of 'F', on the product \(I_1 I_2\), would be :

• (1)
• (2)
• (3)
• (4)

A wire carrying current \(I\) is tied between points P and Q and is in the shape of a circular arch of radius \(R\) due to a uniform magnetic field \(B\) (perpendicular to the plane of the paper, shown by xxx) in the vicinity of the wire. If the wire subtends an angle \(2\theta_0\) at the centre of the circle (of which it forms an arch) then the tension in the wire is :

• (1) \(IBR\)
• (2) \(\frac{IBR}{\sin\theta_0}\)
• (3) \(\frac{IBR}{2\sin\theta_0}\)
• (4) \(\frac{IBR\theta_0}{\sin\theta_0}\)

A short bar magnet is placed in the magnetic meridian of the earth with north pole pointing north. Neutral points are found at a distance of \(30 \,cm\) from the magnet on the East - West line, drawn through the middle point of the magnet. The magnetic moment of the magnet in \(Am^2\) is close to : (Given \(\frac{\mu_{0}}{4\pi}=10^{-7}\) in SI units and \(B_H\) = Horizontal component of earth's magnetic field = \(3.6 \times 10^{-5}\) Tesla.)

• (1) 9.7
• (2) 4.9
• (3) 19.4
• (4) 14.6

For the LCR circuit, shown here, the current is observed to lead the applied voltage. An additional capacitor \(C'\), when joined with the capacitor \(C\) present in the circuit, makes the power factor of the circuit unity. The capacitor \(C'\), must have been connected in :

• (1) series with \(C\) and has a magnitude \(\frac{1-\omega^2 LC}{\omega^2 L}\).
• (2) series with \(C\) and has a magnitude \(\frac{C}{(\omega^2 LC-1)}\).
• (3) parallel with \(C\) and has a magnitude \(\frac{C}{(\omega^2 LC-1)}\).
• (4) parallel with \(C\) and has a magnitude \(\frac{1-\omega^2 LC}{\omega^2 L}\).

For plane electromagnetic waves propagating in the \(z\) direction, which one of the following combination gives the correct possible direction for \(\vec{E}\) and \(\vec{B}\) field respectively ?

• (1) \(\left(\hat{i}+2\hat{j}\right)\) and \(\left(2\hat{i}-\hat{j}\right)\)
• (2) \(\left(\hat{-2i}-3\hat{j}\right)\) and \(\left(3\hat{i}-\hat{2j}\right)\)
• (3) \(\left(\hat{2i}+3\hat{j}\right)\) and \(\left(\hat{i}-\hat{2j}\right)\)
• (4) \(\left(\hat{3i}+4\hat{j}\right)\) and \(\left(4\hat{i}-\hat{3j}\right)\)

A thin convex lens of focal length ' \(f'\) is put on a plane mirror as shown in the figure. When an object is kept at a distance ' \(a\) ' from the lens-mirror combination, its image is formed at a distance \(\frac{a}{3}\) in front of the combination. The value of ' \(a\) ' is

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

In a Young's double slit experiment with light of wavelength \(\lambda\) the separation of slits is \(d\) and distance of screen is \(D\) such that \(D \gt\gt d \gt\gt \lambda\). If the Fringe width is \(\beta\), the distance from point of maximum intensity to the point where intensity falls to half of maximum intensity on either side is :

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

Unpolarized light of intensity \(I_0\) is incident on surface of a block of glass at Brewster's angle. In that case, which one of the following statements is true ?

• (1) transmitted light is partially polarized with intensity \(I_0/2\).
• (2) transmitted light is completely polarized with intensity less than \(I_0/2\).
• (3) reflected light is completely polarized with intensity less than \(I_0/2\).
• (4) reflected light is partially polarized with intensity \(I_0/2\).

The de - Broglie wavelength associated with the electron in the \(n = 4\) level is :

• (1) two times the de-Broglie wavelength of the electron in the ground state
• (2) four times the de-Broglie wavelength of the electron in the ground state
• (3) half of the de-Broglie wavelength of the electron in the ground state
• (4) 1/4th of the de-Broglie wavelength of the electron in the ground state

Let \(N_\beta\) be the number of \(\beta\) particles emitted by 1 gram of \(^{24}\mathrm{Na}\) radioactive nuclei (half life = 15 hrs) in 7.5 hours, \(N_\beta\) is close to (Avogadro number \(= 6.023\times10^{23}\)/g. mole) :

• (1) \(6.2\times10^{21}\)
• (2) \(7.5\times10^{21}\)
• (3) \(1.25\times10^{22}\)
• (4) \(1.75\times10^{22}\)

A 2V battery is connected across AB as shown in the figure. The value of the current supplied by the battery when in one case battery's positive terminal is connected to A and in other case when positive terminal of battery is connected to B will respectively be :

• (1) 0.2 A and 0.1 A
• (2) 0.4 A and 0.2 A
• (3) 0.1 A and 0.2 A
• (4) 0.2 A and 0.4 A

The AC voltage across a resistance can be measured using a :

• (1) potentiometer
• (2) moving coil galvanometer
• (3) moving magnet galvanometer
• (4) hot wire voltmeter

\(A +2 B +3 C \rightleftharpoons AB _{2} C _{3}\) Reaction of \(6.0\, g\) of \(A , 6.0 \times 10^{23}\) atoms of \(B\), and \(0.036\, mol\) of \(C\) yields \(4.8\, g\) of compound \(AB _{2} C _{3}\). If the atomic mass of \(A\) and \(C\) are 60 and 80 amu respectively, the atomic mass of \(B\) is (Avogadro no. \(=6 \times 10^{23}\) ):

• (1) 70 amu
• (2) 60 amu
• (3) 50 amu
• (4) 40 amu

When does a gas deviate the most from its ideal behaviour ?

• (1) At low pressure and low temperature
• (2) At low pressure and high temperature
• (3) At high pressure and low temperature
• (4) At high pressure and high temperature

At temperature T, the average kinetic energy of any particle is \(\frac{3}{2}\) kT. The de Broglie wavelength follows the order :

• (1) Thermal proton > Visible photon > Thermal electron
• (2) Thermal proton > Thermal electron > Visible photon
• (3) Visible photon > Thermal electron > Thermal neutron
• (4) Visible photon > Thermal neutron > Thermal electron

Molecule AB has a bond length of 1.617 \(\mathrm{\AA}\) and a dipole moment of 0.38 D. The fractional charge on each atom (absolute magnitude) is : (\(e_0 = 4.802\times10^{-10}\) esu)

• (1) 0
• (2) 0.05
• (3) 0.5
• (4) 1.0

For the equilibrium, \(\mathrm{A(g) \rightleftharpoons B(g)}\), \(\Delta H\) is \(-40\) kJ/mol. If the ratio of the activation energies of the forward (\(E_f\)) and reverse (\(E_b\)) reactions is \(\frac{2}{3}\) then :

• (1) \(E_f = 60\) kJ/mol ; \(E_b = 100\) kJ/mol
• (2) \(E_f = 30\) kJ/mol ; \(E_b = 70\) kJ/mol
• (3) \(E_f = 80\) kJ/mol ; \(E_b = 120\) kJ/mol
• (4) \(E_f = 70\) kJ/mol ; \(E_b = 30\) kJ/mol

Determination of the molar mass of acetic acid in benzene using freezing point depression is affected by :

• (1) dissociation
• (2) association
• (3) partial ionization
• (4) complex formation

The increase of pressure on ice \(\rightleftharpoons\) water system at constant temperature will lead to :

• (1) no effect on the equilibrium
• (2) a decrease in the entropy of the system
• (3) a shift of the equilibrium in the forward direction
• (4) an increase in the Gibbs energy of the system

At \(298\, K\), the standard reduction potentials are \(1.51\, V\) for \(MnO _{4}^{-} \mid Mn ^{2+} 1.36\, V\) for \(Cl _{2} \mid Cl ^{-} 1.07\, V\) for \(Br _{2} \mid Br ^{-}\), and \(0.54\, V\) for \(\left. H _{2}\right|^{-}\). At \(pH =3\), permanganate is expected to oxidize: \(\left(\frac{R T}{F}=0.059\, V \right)\)

• (1) \(Cl ^{-}, Br ^{-}\) and \(I ^{-}\)
• (2) \(Cl ^{-}\) and \(Br ^{-}\)
• (3) \(Br ^{-}\) and \(I ^{-}\)
• (4) \(I^-\) only

\(\mathrm{A + 2B \rightarrow C}\), the rate equation for this reaction is given as
Rate = k[A][B].
If the concentration of A is kept the same but that of B is doubled what will happen to the rate itself ?

• (1) halved
• (2) the same
• (3) doubled
• (4) quadrupled

Under ambient conditions, which among the following surfactants will form micelles in aqueous solution at lowest molar concentration ?

• (1) \(\mathrm{CH_3(CH_2)_{15}\overset{\oplus}{N}(CH_3)_3\overset{\ominus}{Br}}\)
• (2) \(\mathrm{CH_3-(CH_2)_{13}-OSO_3^-\ Na^+}\)
• (3) \(\mathrm{CH_3-(CH_2)_8-COO^{\ominus}\ Na^+}\)
• (4) \(\mathrm{CH_3(CH_2)_{11}\overset{\oplus}{N}(CH_3)_3Br^-}\)

Choose the incorrect formula out of the four compounds for an element X below :

• (1) \(X_2Cl_3\)
• (2) \(X_2O_3\)
• (3) \(X_2(SO_4)_3\)
• (4) \(XPO_4\)

Calamine is an ore of :

• (1) Aluminium
• (2) Copper
• (3) Iron
• (4) Zinc

Which physical property of dihydrogen is wrong ?

• (1) Colourless gas
• (2) Odourless gas
• (3) Tasteless gas
• (4) Non-inflammable gas

Which of the alkaline earth metal halides given below is essentially covalent in nature ?

• (1) \(\mathrm{MgCl_2}\)
• (2) \(\mathrm{BeCl_2}\)
• (3) \(\mathrm{SrCl_2}\)
• (4) \(\mathrm{CaCl_2}\)

Which of the following compounds has a \(\mathrm{P-P}\) bond ?

• (1) \(\mathrm{H_4P_2O_5}\)
• (2) \(\mathrm{H_4P_2O_6}\)
• (3) \(\mathrm{H_4P_2O_7}\)
• (4) \(\mathrm{(HPO_3)_3}\)

Chlorine water on standing loses its colour and forms :

• (1) \(HCl\) only
• (2) \(HOCl\) and \(HOCl_2\)
• (3) \(HCl\) and \(HOCl\)
• (4) \(HCl\) and \(HClO_2\)

Which of the following statements is false ?

• (1) \(\mathrm{CrO_4^{2-}}\) is tetrahedral in shape
• (2) \(\mathrm{Cr_2O_7^{2-}}\) has a Cr-O-Cr bond
• (3) \(\mathrm{Na_2Cr_2O_7}\) is a primary standard in volumetry
• (4) \(\mathrm{Na_2Cr_2O_7}\) is less soluble than \(\mathrm{K_2Cr_2O_7}\)

When concentrated HCl is added to an aqueous solution of \(\mathrm{CoCl_2}\), its colour changes from reddish pink to deep blue. Which complex ion gives blue colour in this reaction ?

• (1) \(\mathrm{[CoCl_6]^{4-}}\)
• (2) \(\mathrm{[CoCl_6]^{3-}}\)
• (3) \(\mathrm{[CoCl_4]^{2-}}\)
• (4) \(\mathrm{[Co(H_2O)_6]^{2+}}\)

Which of the following complex ions has electrons that are symmetrically filled in both \(t_{2g}\) and \(e_g\) orbitals ?

• (1) \(\mathrm{[CoF_6]^{3-}}\)
• (2) \(\mathrm{[Co(NH_3)_6]^{2+}}\)
• (3) \(\mathrm{[Mn(CN)_6]^{4-}}\)
• (4) \(\mathrm{[FeF_6]^{3-}}\)

Addition of phosphate fertilisers to water bodies causes :

• (1) enhanced growth of algae
• (2) increase in amount of dissolved oxygen in water
• (3) deposition of calcium phosphate
• (4) increase in fish population

• (1) (A) - (ii) ; (B) - (i) ; (C) - (iii)
• (2) (A) - (iii) ; (B) - (ii) ; (C) - (i)
• (3) (A) - (ii) ; (B) - (iii) ; (C) - (i)
• (4) (A) - (iii) ; (B) - (i) ; (C) - (ii)

Which of the following pairs of compounds are positional isomers?

• (1)
• (2)
• (3)
• (4)

The number of structural isomers for \(\mathrm{C_6H_{14}}\) is :

• (1) 3
• (2) 4
• (3) 5
• (4) 6

What is the major product expected from the following reaction ?

Where D is an isotope of Hydrogen.

• (1)
• (2)
• (3)
• (4)

In the reaction sequence
\(2\,\mathrm{CH_3CHO}\xrightarrow{\mathrm{OH^-}}\mathrm{A}\xrightarrow{\Delta}\mathrm{B}\) ; the product B is :

• (1) \(\mathrm{CH_3-CH_2-CH_2-CH_2-OH}\)
• (2) \(\mathrm{CH_3-CH=CH-CHO}\)
• (3) \(\mathrm{CH_3-CO-CH_3}\)
• (4) \(\mathrm{CH_3-CH_2-CH_2-CH_3}\)

Which compound exhibits maximum dipole moment among the following ?

• (1) Nitrobenzene (benzene ring bearing only one \(\mathrm{NO_2}\) group)
• (2) para-Nitroaniline (\(\mathrm{NO_2}\) and \(\mathrm{NH_2}\) at positions 1 and 4 of the benzene ring)
• (3) meta-Nitroaniline (\(\mathrm{NO_2}\) and \(\mathrm{NH_2}\) at positions 1 and 3 of the benzene ring)
• (4) ortho-Nitroaniline (\(\mathrm{NO_2}\) and \(\mathrm{NH_2}\) at positions 1 and 2 of the benzene ring)

Which one of the following structures represents the neoprene polymer ?

• (1) \(\left[\mathrm{CH_2-C(Cl)=CH-CH_2}\right]_n\)
• (2) \(\left[\mathrm{CH_2-CH(CN)}\right]_n\)
• (3) \(\left[\mathrm{CH_2-CH(Cl)}\right]_n\)
• (4) \(\left[\mathrm{CH(C_6H_5)-CH_2}\right]_n\)

Accumulation of which of the following molecules in the muscles occurs as a result of vigorous exercise ?

• (1) Glucose
• (2) Glycogen
• (3) L-lactic acid
• (4) Pyruvic acid

Which artificial sweetener contains chlorine ?

• (1) Aspartame
• (2) Saccharin
• (3) Sucralose
• (4) Alitame

A pink coloured salt turns blue on heating. The presence of which cation is most likely ?

• (1) \(\mathrm{Cu^{2+}}\)
• (2) \(\mathrm{Fe^{2+}}\)
• (3) \(\mathrm{Zn^{2+}}\)
• (4) \(\mathrm{Co^{2+}}\)