MHT CET 2026 April 16 Shift 2 Question Paper- Download PDF with Solutions

The error in the measurement of length and mass of a cube is 3% and 4% respectively. The error in the measurement of its density will be

• (A) \(13 \%\)
• (B) \(6 \%\)
• (C) \(9 \%\)
• (D) \(15 \%\)

Vectors \(a\hat{i}+b\hat{j}+\hat{k}\) and \(2\hat{i}-3\hat{j}+4\hat{k}\) are perpendicular to each other when \(3a+2b = 7\), the ratio of a to b is \(x/2\). The value of x is

• (A) \(4\)
• (B) \(1\)
• (C) \(8\)
• (D) \(3\)

Two objects are projected with same velocity 'u' at different angles \(α\) and \(β\) with the horizontal. If \(α+β = 90^{\circ}\), the ratio of horizontal range of the first object to the second object will be \([sin(π-θ) = sinθ]\)

• (A) \(4:1\)
• (B) \(2:1\)
• (C) \(1:1\)
• (D) \(1:2\)

A body of mass 1kg begins to move under the action of a time dependent force \(\overset{⃗}{F} = (t\hat{i}+3t^2\hat{j}) \text{N}\) where \(\hat{i}\) and \(\hat{j}\) are the unit vectors along x and y axis. The power developed by the above force at time \(t = 2 \text{s}\) will be (in watt)

• (A) \(100\)
• (B) \(50\)
• (C) \(25\)
• (D) \(5\)

A solid sphere of mass M and a disc of mass \(\frac{M}{2}\) have the same radius. The ratio of moment of inertia of the disc about a tangent in its plane to the moment of inertia of the sphere about its tangent will be

• (A) \(15:8\)
• (B) \(25:56\)
• (C) \(12:7\)
• (D) \(16:9\)

A uniform solid cylinder with radius R and length L has moment of inertia \(I_1\) about the axis of the cylinder. A concentric solid cylinder of radius \(R/2\) and length \(L/2\) is carved out of the original cylinder. If \(I_2\) is the moment of inertia of the carved out portion of the cylinder then \(I_1/I_2\) is (Both \(I_1\) and \(I_2\) are about the axis of the cylinder)

• (A) \(4:1\)
• (B) \(8:1\)
• (C) \(16:1\)
• (D) \(32:1\)

Solid sphere is rotating about an axis as shown in figure. If the radius of the sphere is 5 cm, then radius of gyration about that axis will be \(\sqrt{x}\) cm. The value of x is

• (A) \(100\)
• (B) \(110\)
• (C) \(120\)
• (D) \(140\)

A body of mass m is taken from earth surface to a height h equal to twice the radius of earth, the increase in potential energy will be (\(g\) = acceleration due to gravity on earth's surface) (R - radius of earth)

• (A) \(\frac{2}{3}\text{mgR}\)
• (B) \(\frac{1}{3}\text{mgR}\)
• (C) \(\frac{1}{2}\text{mgR}\)
• (D) \(3\text{mgR}\)

One thousand small water drops of equal radii combine to form a big drop. The ratio of final surface energy to the total initial surface energy is

• (A) \(1:1\)
• (B) \(1:10\)
• (C) \(1:100\)
• (D) \(1:1000\)

Under isothermal condition, two soap bubbles of radii \(r_1\) and \(r_2\) combine to form a single soap bubble of radius R. The surface tension of the soap solution is (P = outside pressure)

• (A) \(\frac{P(R^3-r_1^3-r_2^3)}{4(r_1^2+r_2^2-R^2)}\)
• (B) \(\frac{P(R^3+r_1^3+r_2^3)}{2(r_1^2-r_2^2+R^2)}\)
• (C) \(\frac{P(r_1^3-r_2^3-R^3)}{(R^2+r^2+r^2)}\)
• (D) \(\frac{P(R^3-r_1^3+r_2^3)}{2(r_1^2+r_2^2-R^2)}\)

A horizontal pipe carries water in a streamline flow. At point along the pipe, where the cross-sectional area is \(A_1\), the velocity of water is \(V_1\) and the pressure is \(P_1\). What is the pressure of water at another point where the cross-sectional area is \(A_2\)?

• (A) \(P_1-\frac{ρV_2^2}{2A_1^2}(A_2^2-A_1^2)\)
• (B) \(P_1+\frac{ρV_1^2}{2A_2^2}(A_2^2-A_1^2)\)
• (C) \(P_1ρV_1^2(A_1^2-A_2^2)\)
• (D) \(P/ρV_1^2A_2^2\)

A bowl filled with very hot water cools from \(98^{\circ}\text{C}\) to \(86^{\circ}\text{C}\) in 2 minutes when the room temperature is \(22^{\circ}\text{C}\). How long it will take to cool from \(75^{\circ}\text{C}\) to \(69^{\circ}\text{C}\) ?

• (A) \(0.5\) minute
• (B) \(1.4\) minute
• (C) \(1\) minute
• (D) \(2\) minute

A Carnot engine with efficiency 50% takes heat from a source at 600 K. To increase the efficiency by 20%, keeping temperature of the sink same, the new temperature of the source will be

• (A) \(300 \text{K}\)
• (B) \(900 \text{K}\)
• (C) \(1000 \text{K}\)
• (D) \(360 \text{K}\)

One gram of a liquid is converted to vapour at \(3\times 10^5\) Pa pressure. If 10% of the heat supplied is used for increasing the volume by \(1600 \text{cm}^3\) during this phase change, then the increase in internal energy in the process will be

• (A) \(4.32\times 10^8 \text{J}\)
• (B) \(4800 \text{J}\)
• (C) \(4320 \text{J}\)
• (D) \(4.32\times 10^5 \text{J}\)

The average transnational kinetic energy of a molecule in a gas is \(E_1\). The kinetic energy of the electron (e) accelerated from rest through potential difference 'V' volt is \(E_2\). The temperature at which \(E_1 = E_2\) possible is (mass of molecule and electron is same) (N = number of molecules, V = Velocity, R = gas constant)

• (A) \(\frac{2VNe}{3R}\)
• (B) \(\frac{VNe}{2R}\)
• (C) \(\frac{3NeV}{2R}\)
• (D) \(\frac{5NeV}{3R}\)

A rigid diatomic gas having molar mass m is contained in an insulated container. The container is moving with velocity V. If it is stopped suddenly, the change in temperature is (R - gas constant)

• (A) \(\frac{mV^2}{R}\)
• (B) \(\frac{mV^2}{3R}\)
• (C) \(\frac{mV^2}{5R}\)
• (D) \(\frac{mV^2}{7R}\)

If a gas is compressed isothermally then the r.m.s. velocity of the molecules

• (A) increases
• (B) remains the same
• (C) decreases
• (D) first increases and then decreases

A mass \(m_1\) performs S.H.M. with amplitude A which is connected to horizontal spring. While mass \(m_1\) passing through mean position, another mass \(m_2\) (\(m_2 < m_1\)) is placed on it so that both masses move together with amplitude \(A_1\). The ratio of \(A/A_1\) is

• (A) \([\frac{m_1+m_2}{m_1}]^{\frac{1}{2}}\)
• (B) \([\frac{m_1}{m_1+m_2}]^{\frac{1}{2}}\)
• (C) \(\frac{m_1}{m_1+m_2}\)
• (D) \(\frac{m_2}{m_1+m_2}\)

The maximum potential energy of a block executing S.H.M. is E and amplitude of oscillation is 'A'. The kinetic energy of the block at amplitude A/2 is

• (A) \(\frac{3\text{E}}{4}\)
• (B) \(\text{E}\)
• (C) \(\frac{2\text{E}}{4}\)
• (D) \(\frac{5\text{E}}{4}\)

The time taken by simple pendulum for one oscillation is T on earth's surface. Its time period becomes xT when taken to a height R (equal to earth's radius) above the earth's surface. The value of x is

• (A) \(\frac{1}{4}\)
• (B) \(\frac{1}{2}\)
• (C) \(2\)
• (D) \(4\)

An air column in a pipe which is closed at one end will be in resonance with a vibrating tuning fork of frequency 415 Hz for various vibrating air columns. Which one of the following lengths is not in resonance ? (velocity of sound in air = 332 m/s) (Neglect end correction)

• (A) \(20 \text{cm}\)
• (B) \(40 \text{cm}\)
• (C) \(60 \text{cm}\)
• (D) \(100 \text{cm}\)

Two waves \(Y_1 = 0.25sin316t\) and \(Y_2 = 0.25sin310t\) are propagating along the same direction. The number of beats produced per second are

• (A) \(\frac{3}{π}\)
• (B) \(\frac{π}{3}\)
• (C) \(\frac{2}{π}\)
• (D) \(\frac{π}{2}\)

When open pipe is closed from one end then third overtone of closed pipe is higher in frequency by 150Hz than second overtone of open pipe. The fundamental frequency of open end pipe will be (Neglect end correction)

• (A) \(75 \text{Hz}\)
• (B) \(150 \text{Hz}\)
• (C) \(225 \text{Hz}\)
• (D) \(300 \text{Hz}\)

A person observes two moving trains. 'A' reaching the station and 'B' leaving the station with equal speed of 30m/s. If both trains emit sounds with frequency 300 Hz, (speed of sound = 330 m/s) difference of frequencies heard by the person will be

• (A) \(10 \text{Hz}\)
• (B) \(33 \text{Hz}\)
• (C) \(55 \text{Hz}\)
• (D) \(80 \text{Hz}\)

Two spherical hollow spheres of radii '\(R_1\)' and '\(R_2\)' are charged with same charge. Let '\(σ_1\)' and '\(σ_2\)' are their respective charge densities, then the ratio \(σ_1\) to \(σ_2\) is

• (A) \(R_1:R_2\)
• (B) \(R_2:R_1\)
• (C) \(R_2^2:R_1^2\)
• (D) \(R_1^2:R_2^2\)

An electric dipole of length \(0.5 μ\text{m}\) is placed with its axis making an angle of \(30^{\circ}\) with uniform electric field \(10^4 \text{V/m}\). If it experiences a torque of \(5\times 10^{-9} \text{Nm}\), the magnitude of the charge on the dipole is (\(sin30^{\circ} = 0.5\))

• (A) \(1 μ\text{C}\)
• (B) \(1.5 μ\text{C}\)
• (C) \(2 μ\text{C}\)
• (D) \(2.5 μ\text{C}\)

Two charges \(q_1\) and \(q_2\) are separated by a distance of 30cm. A third charge \(q_3\) initially at point C is shown in figure, is moved along the circular path of radius 40 cm from C to D. If the difference in potential energy due to movement of \(q_3\) from C to D is \(q_3k/4πε_0\), the value of K is (\(ε_0\) = permittivity of free space)

• (A) \(8q_2\)
• (B) \(8q_1\)
• (C) \(6q_2\)
• (D) \(6q_1\)

A capacitor has capacity C when it's parallel plates are separated by air medium of thickness 'd'. A slab of material of dielectric constant K having area equal to that of plates but thickness \(\frac{d}{2}\) is inserted between the plates. Capacitance of the capacitor in the presence of slab will be

• (A) \(\text{KC}\)
• (B) \(2\text{KC}\)
• (C) \(\frac{\text{KC}}{\text{K}+1}\)
• (D) \(\frac{2\text{KC}}{\text{K}+1}\)

A galvanometer has resistance G and range Vg. How much resistance is required to read voltage up to V volt?

• (A) \(GV_g\)
• (B) \(G(\frac{V-V_g}{V})\)
• (C) \(G(\frac{V}{V_g}-1)\)
• (D) \(G(\frac{V+V_g}{V})\)

If the potential difference between points B and D is zero in the given circuit, the value of x is \(\frac{1}{n} \Omega\). The value n is

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

The magnetic field intensity (H) at the centre of a long solenoid having 'n' turns per unit length and carrying a current I, when no material is kept in it is (\(μ_0\) = permeability of free space)

• (A) \(n/I\)
• (B) \(nI\)
• (C) \(μ_0I\)
• (D) \(μ_0/nI\)

The magnitude of magnetic induction at mid point 'O' due to current arrangement as shown in figure will be (\(μ_0\) = permeability of free space)

• (A) \(\frac{μ_0I}{2πa}\)
• (B) zero
• (C) \(\frac{μ_0I}{4πa}\)
• (D) \(\frac{μ_0I}{πa}\)

A circular loop of radius R is carrying current I. The ratio of magnetic field at the center of circular loop and at a distance R from the center of the loop on its axis is

• (A) \(1:\sqrt{2}\)
• (B) \(1:3\sqrt{2}\)
• (C) \(\sqrt{8}:1\)
• (D) \(3\sqrt{2}:1\)

A circular coil of radius 'r' is placed on another circular coil whose radius is 'R' and the current flowing through it is changing and their centers coincide. (\(R≫r\)). If both the coils are coplanar, then the mutual inductance between them is proportional to

• (A) \(\frac{r}{R}\)
• (B) \(\frac{R}{r}\)
• (C) \(\frac{R^2}{r}\)
• (D) \(\frac{r^2}{R}\)

A graph of magnetic flux (\(\Phi\)) versus current (I) is shown for four inductors A, B, C, D. The smallest value of self inductance is for inductor.

• (A) A
• (B) B
• (C) C
• (D) D

In the given circuit, if \(dI/dt = -1 \text{A/s}\) then the value of \(V_{AB}\) at this instant will be

• (A) \(30\text{V}\)
• (B) \(20\text{V}\)
• (C) \(10\text{V}\)
• (D) \(15\text{V}\)

Alternating current of peak value \((\frac{2}{π})\) A flows through the primary coil of a transformer. The coefficient of mutual inductance between primary and secondary coils is 1H. The peak value of induced e.m.f. in the secondary coil is
(Frequency of a.c. = 50 Hz)

• (A) \(100\text{V}\)
• (B) \(300\text{V}\)
• (C) \(200\text{V}\)
• (D) \(400\text{V}\)

Series LCR circuit containing an a.c. source of 100V has a inductor and a capacitor of reactance \(24 \Omega\) and \(16 \Omega\) respectively. If a resistance of \(6 \Omega\) is connected in series, then the potential difference across the series combination of inductor and capacitor only is

• (A) \(80\text{V}\)
• (B) \(8\text{V}\)
• (C) \(40\text{V}\)
• (D) \(20\text{V}\)

An inductor of inductance \(2 μ\text{H}\) is connected in series with a resistance, a variable capacitor and an a.c. source of 10 kHz. The value of capacitance for which maximum current is drawn in to the circuit is \(\frac{1}{x} \text{F}\), where the value of x is (Take \(π^2 = 10\))

• (A) \(8000\)
• (B) \(600\)
• (C) \(400\)
• (D) \(1600\)

A ray of light passes from the vacuum into a medium of refractive index 'n'. If the angle of incidence is twice the angle of refraction, then the angle of incidence in terms of refractive index is

• (A) \(sin^{-1}(\frac{n}{2})\)
• (B) \(2cos^{-1}(\frac{n}{2})\)
• (C) \(2sin^{-1}(\frac{n}{2})\)
• (D) \(cos^{-1}(\frac{n}{2})\)

A convex lens having refractive index 1.6 has focal length 12 cm, when it is in air. The focal length of that lens when placed in water is (refractive index of water = 1.28)

• (A) \(655 \text{mm}\)
• (B) \(288 \text{mm}\)
• (C) \(555 \text{mm}\)
• (D) \(355 \text{mm}\)

Three identical polaroids \(P_1\), \(P_2\) and \(P_3\) are placed one after another. The pass axis of \(P_2\) and \(P_3\) are inclined at angle of \(60^{\circ}\) and \(90^{\circ}\) with respect to axis of \(P_1\). The source has an intensity \(I_0\). The intensity of light finally coming out is

• (A) \(\frac{I_0}{2}cos^230^{\circ}cos^290^{\circ}\)
• (B) \(\frac{I_0}{2}cos^260^{\circ}cos^230^{\circ}\)
• (C) \(I_0cos^260^{\circ}cos^230^{\circ}\)
• (D) zero

In Young's double slit experiment, two slits are illuminated with a light of wavelength \(λ\). The line joining \(A_1P\) is perpendicular to \(A_1A_2\) as shown in figure. If the first minimum is detected at P, the value of slits separation 'a' will be
(D = distance between source and screen)

• (A) \(λ\text{D}\)
• (B) \(\sqrt{λ\text{D}}\)
• (C) \(\sqrt{\frac{λ}{\text{D}}}\)
• (D) \(\sqrt{\frac{\text{D}}{λ}}\)

When a light of wavelength '\(λ\)' falls on the emitter of a photosensitive surface, maximum speed of emitted photoelectrons is 'V'. If the incident wavelength is changed to '\(2λ/3\)', maximum speed of emitted photoelectrons will be

• (A) less than \(V(1.5)^{\frac{1}{2}}\)
• (B) greater than \(V(1.5)^{\frac{1}{2}}\)
• (C) less than \(V\)
• (D) less than \(\frac{V}{2}\)

A photoemissive substance is illuminated with a radiation of wavelength \(λ_i\) so that it releases electrons with de-Broglie wavelength \(λ_e\). The longest wavelength of radiation that can emit photoelectron is \(λ_0\). Expression for de-Broglie wavelength is (m = mass of electron, h = Planck's constant, C = Speed of light)

• (A) \((\text{h}λ_i/2\text{mc})^{\frac{1}{2}}\)
• (B) \((\text{h}λ_0/2\text{mc})^{\frac{1}{2}}\)
• (C) \([\text{h}/2\text{mc}(\frac{1}{λ_i}-\frac{1}{λ_0})]^{\frac{1}{2}}\)
• (D) \([\text{h}/[2\text{mc}(\frac{1}{λ_i}-\frac{1}{λ_0})]^{\frac{1}{2}}\)

An electron makes a transition from an excited state to the ground state of a hydrogen like atom. Out of the following statements which one is correct ?
(A) K.E, P. E. and T.E. decreases
(B) K.E. increases but P. E. and T.E. decreases
(C) K.E. and T.E. decreases but P.E. increases
(D) K.E. decreases, P.E. increases but T.E. remains the same

• (A) A
• (B) B
• (C) C
• (D) D

The ratio of the density of oxygen nucleus \((_8^{16}\text{O})\) and helium nucleus \((_2^4\text{He})\) is

• (A) \(1:1\)
• (B) \(2:1\)
• (C) \(4:1\)
• (D) \(8:1\)

Electrical conductivity of insulators is

• (A) exactly zero
• (B) extremely large
• (C) extremely small
• (D) sometimes small and sometimes large

Which of the following statement is NOT correct in the case of LED?
(A) It is a heavily doped p-n junction diode.
(B) It emits light only when it is forward biased.
(C) It emits light only when it is reverse biased.
(D) The energy of the light emitted is equal to or slightly less than the energy gap of the semiconductor used.

• (A) A
• (B) B
• (C) C
• (D) D

In which of the logic gate the following statement is true? "The output is high when either of the inputs is high but not if both inputs are high"

• (A) NAND
• (B) NOR
• (C) X-OR
• (D) OR

The ratio of mass of 'x' atoms of an element to mass of carbon atoms is 9:1. Find the mass of 1 mol of x if molar mass of carbon is 12 g/mol.

• (A) \(54 \text{g}\)
• (B) \(108 \text{g}\)
• (C) \(180 \text{g}\)
• (D) \(216 \text{g}\)

What is the number of unpaired electrons present in ground state of Silicon and Chromium, respectively ?

• (A) 2 and 4
• (B) 2 and 6
• (C) 0 and 6
• (D) 4 and 5

Identify the most electronegative element by the following.

• (A) Oxygen
• (B) Nitrogen
• (C) Fluorine
• (D) Chlorine

What is the total number of electrons present in bonding and antibonding orbitals respectively in \(\text{F}_2\) molecule according to MO theory ?

• (A) Bonding - 8, Antibonding - 10
• (B) Bonding - 6, Antibonding - 12
• (C) Bonding - 10, Antibonding - 8
• (D) Bonding - 12, Antibonding - 6

At constant temperature, a gas occupies a volume of 200 mL at a pressure of 500 mm Hg. What will be its volume at 800 mm Hg pressure?

• (A) \(280 \text{mL}\)
• (B) \(125 \text{mL}\)
• (C) \(100 \text{mL}\)
• (D) \(350 \text{mL}\)

Which of the following statements is not correct ?

• (A) \(\Delta G^{\circ}\) is an extensive property.
• (B) \(E^{\circ}\) cell is an intensive property.
• (C) For a chemical reaction to be spontaneous \(\Delta G\) must be \(-\text{ve}\).
• (D) Electrical work is equal to \(\Delta G\)

The heat of combustion of carbon to \(\text{CO}_2\) is \(-393.5 \text{kJ/mol}\). The heat released upon formation of 35.2 g of \(\text{CO}_2\) from carbon and oxygen gas is

• (A) \(+315 \text{kJ}\)
• (B) \(-315 \text{kJ}\)
• (C) \(-3.15 \text{kJ}\)
• (D) \(-630 \text{kJ}\)

Calculate the enthalpy change when 6 g. \(\text{CO(g)}\) reacts with sufficient \(\text{NO}_2\text{(g)}\) according to the following reaction
\(4\,\text{CO(g)}+2\,\text{NO}_2\text{(g)}\rightarrow 4\,\text{CO}_2\text{(g)}+2\text{N}_2\text{(g)}\,; \Delta _rH^0 = -1200 \text{kJ}\)

• (A) \(-16.08 \text{kJ}\).
• (B) \(-32.15 \text{kJ}\).
• (C) \(-64.29 \text{kJ}\)
• (D) \(-128.58 \text{kJ}\).

Identify the process that proceeds with no heat exchange between system and surrounding.

• (A) Isothermal
• (B) Isobaric
• (C) Isochoric
• (D) Adiabatic

The solubility product of sparingly soluble salt BA is \(4\times 10^{-13}\). Calculate the \([\text{A}^-]\) if \([\text{B}^+]\) is \(1\times 10^{-6} \text{M}\).

• (A) \(2\times 10^{-7} \text{M}\)
• (B) \(3\times 10^{-7} \text{M}\)
• (C) \(4\times 10^{-7} \text{M}\)
• (D) \(5\times 10^{-7} \text{M}\)

Which of the following salt undergoes anionic hydrolysis ?

• (A) \(\text{CuSO}_4\)
• (B) \(\text{Na}_2\text{CO}_3\)
• (C) \(\text{NH}_4\text{Cl}\)
• (D) \(\text{FeCl}_3\)

Identify from the following salts an example of a salt of strong acid and weak base.

• (A) Ammonium acetate.
• (B) Sodium acetate.
• (C) Copper (II) chloride.
• (D) Sodium carbonate.

What is oxidation state of chlorine in chlorous acid?

• (A) \(+2\)
• (B) \(+4\)
• (C) \(+3\)
• (D) \(+7\)

Which of the following phenomena represents oxidation?

• (A) Removal of oxygen
• (B) Addition of electronegative element
• (C) Addition of electropositive element
• (D) Gain of electron

What is the role of glycerol in the decomposition of hydrogen peroxide?

• (A) It increases the reaction rate.
• (B) It reduces the reaction rate.
• (C) It increases the enthalpy of the reaction.
• (D) It increases the rate of collisions between reactant molecules.

Which of the following statements is NOT correct regarding dihydrogen?

• (A) It is colorless and tasteless gas.
• (B) It burns with a pale blue flame.
• (C) It is a polar water-soluble gas.
• (D) It is lighter than air.

Which of following has the highest nucleophilicity ?

• (A) \(\text{F}^-\)
• (B) \(\text{OH}^-\)
• (C) \(\text{CH}_3^-\)
• (D) \(\text{NH}_2^-\)

Identify the catalyst used in the Friedel crafts acylation reaction.

• (A) Anhydrous \(\text{AlCl}_3\)
• (B) Anhydrous \(\text{FeCl}_3\)
• (C) \(\text{NaNO}_2\)
• (D) Alcoholic KOH

Identify the use of Baeyer's reagent by the following.

• (A) Detecting the presence of carbon carbon double bond.
• (B) Polymerisation
• (C) Oxidation
• (D) Reduction

What is the total contribution of all corner particles in bcc unit cell?

• (A) 1 particle
• (B) 2 particles
• (C) 3 particles
• (D) 4 particles

When silver crystallizes, it forms face centered cubic cells, if the volume of unit cells \(6.84\times 10^{-23} \text{cm}^3\). Calculate the density of silver. (Molar mass of silver is 108 g/mol, \(N_A = 6.022\times 10^{23}\))

• (A) \(12.49 \text{gm cm}^{-3}\)
• (B) \(10.49 \text{gm cm}^{-3}\)
• (C) \(16.89 \text{gm cm}^{-3}\)
• (D) \(20.49 \text{gm cm}^{-3}\)

What is the coordination number of Ag in a crystal lattice?

• (A) \(4\)
• (B) \(6\)
• (C) \(12\)
• (D) \(8\)

The normal boiling point of Ethyl acetate is \(77^{\circ}\text{C}\). A solution of non-volatile non-electrolyte solute in ethyl acetate boils at \(78^{\circ}\text{C}\). \(K_b\) for ethyl acetate is \(2.77 ^{\circ}\text{C kg mol}^{-1}\). What is the molality of the solution ?

• (A) \(0.361 \text{m}\)
• (B) \(0.052 \text{m}\)
• (C) \(0.075 \text{m}\)
• (D) \(0.25 \text{m}\)

18 g of glucose (molar mass = 180 g/mol) is dissolved in water to prepare 500 ml solution at \(15^{\circ}\text{C}\). Calculate the osmotic pressure of the solution. [\(R = 0.0821 \text{L atm K}^{-1} \text{mol}^{-1}\)]

• (A) \(1.65 \text{atm}\)
• (B) \(4.73 \text{atm}\)
• (C) \(5.57 \text{atm}\)
• (D) \(2.34 \text{atm}\)

What type of solution is a brass ?

• (A) solid in solid
• (B) liquid in solid
• (C) gas in solid
• (D) liquid in gas

What mass of silver (Atomic mass = 108 g/mol) deposited by a quantity of electricity which displaces 5600 mL of \(\text{O}_2\) at STP ?

• (A) \(5.4 \text{g}\)
• (B) \(10.8 \text{g}\)
• (C) \(54.0 \text{g}\)
• (D) \(108.0 \text{g}\)

Which of the following reactions takes place at anode during the electrolysis of molten NaCl?

• (A) \(\text{Na}_{(l)}^++\text{e}^-\rightarrow \text{Na}_{(l)}\)
• (B) \(\text{Na}_{(s)}\rightarrow \text{Na}_{(l)}^++\text{e}^-\)
• (C) \(\text{Cl}_{2(g)}+2\text{e}^-\rightarrow 2\text{Cl}_{(l)}^-\)
• (D) \(2\text{Cl}_{(l)}^-\rightarrow \text{Cl}_{2(g)}+2\text{e}^-\)

The rate law for the reaction \(A+B\rightarrow P\) is found to be rate \(= k[A]^2[B]\).
The rate constant of the reaction at 300 K is \(6.0 \text{M}^{-2}\text{s}^{-1}\). Calculate the rate of the reaction when \([A] = 1 \text{M}\) and \([B] = 0.2 \text{M}\)

• (A) \(0.6 \text{M s}^{-1}\).
• (B) \(1.2 \text{M s}^{-1}\).
• (C) \(1.8 \text{M s}^{-1}\).
• (D) \(2.4 \text{M s}^{-1}\).

If for a first-order reaction, \([A]_0 = 1.0 \text{M}\) and \([A]_t = 0.25 \text{M}\) after 276 s, find the value of the rate constant (k).

• (A) \(0.0021 \text{s}^{-1}\)
• (B) \(0.0050 \text{s}^{-1}\)
• (C) \(0.003 \text{s}^{-1}\)
• (D) \(0.006 \text{s}^{-1}\)

The equation for the rate constant is \(k = Ae^{-E_a/RT}\).
A chemical reaction will proceed more rapidly if there is a decrease in

• (A) \(k\)
• (B) \(A\)
• (C) \(E_a\)
• (D) \(T\)

What is the oxidation state of chromium in the final product when KI reacts with acidified potassium dichromate solution?

• (A) \(+2\)
• (B) \(+3\)
• (C) \(+6\)
• (D) \(+4\)

Which of the following series of transition elements has general electronic configuration, \([\text{Kr}]\,4d^{1-10}\,5s^{0-2}\)?

• (A) 3d series elements
• (B) 4d series elements
• (C) 5d series elements
• (D) 6d series elements

Identify the ionization isomer of \([\text{Cr}(\text{H}_2\text{O})_4\,\text{Cl}(\text{NO}_2)]\,\text{Cl}\) from the following.

• (A) \([\text{Cr}(\text{H}_2\text{O})_4\,(\text{O}_2\text{N})]\,\text{Cl}_2\)
• (B) \([\text{Cr}(\text{H}_2\text{O})_4\,\text{Cl}_2]\,(\text{NO}_2)\)
• (C) \([\text{Cr}(\text{H}_2\text{O})_4\,\text{Cl}(\text{ONO})]\,\text{Cl}\)
• (D) \([\text{Cr}(\text{H}_2\text{O})_4\,\text{Cl}_2(\text{NO}_2)]\,\text{H}_2\text{O}\)

The number of ligands which are directly bonded to the metal is known as

• (A) co-ordination number
• (B) oxidation number
• (C) co-ordination sphere
• (D) valency

Identify the reagent used for replacement of \(-\text{N}_2^+\text{Cl}^-\) from benzene diazonium chloride by iodine.

• (A) HI
• (B) KOI
• (C) KI
• (D) \(\text{PI}_3\)

What is the product P obtained in following reaction?
\(\text{CH}_3\text{CH}_2\text{Br}+\text{AgCN}_{(alc)}\rightarrow \text{P}\)

• (A) \(\text{CH}_3\text{CH}_2\text{NC}\)
• (B) \(\text{CH}_3\text{CH}_2\text{CN}\)
• (C) \(\text{CH}_3\text{CH}_2\text{CONH}_2\)
• (D) \(\text{CH}_3\text{CH}_2\text{NH}_2\)

What is action of \(\text{CH}_3\text{ONa}\) on \(\text{CH}_3\text{CH}_2\text{Br}\)
\(\text{CH}_3\text{ONa}+\text{CH}_3\text{CH}_2\text{Br}\rightarrow\) ?

• (A) Formation of \(\text{CH}_3\text{OCH}_2\text{CH}_3+\text{NaBr}\)
• (B) Formation of \(\text{CH}_3\text{OH}+\text{NaBr}\)
• (C) Formation of \(\text{CH}_3\text{CH}_2\text{OH}\)
• (D) Formation of \((\text{CH}_3)_2\text{O}\)

Which of the following reagents limits the oxidation of \(\text{R-CH}_2\text{OH}\) up to R-CHO only?

• (A) PCC
• (B) \(\text{CrO}_3\)
• (C) \(\text{K}_2\text{Cr}_2\text{O}_7\) / dil. \(\text{H}_2\text{SO}_4\)
• (D) \(\text{KMnO}_4\) (acidic)

Which of the following ether, on hydrolysis, gives two different products that are successive members of the homologous series?

• (A) Methoxymethane
• (B) Methoxypropane
• (C) Ethoxypropane
• (D) 1-Methoxypropane

Identify the product X in the following reaction.

• (A) 2,3-dimethyl-2-butene
• (B) 3,2-diethyl butene
• (C) 3,3-dimethyl-1-butene
• (D) 2,2-diethyl butene

Acetone on heating with \(\text{CrO}_3\) mainly forms,

• (A) Acetic acid
• (B) Ethyl alcohol
• (C) Toluene
• (D) Dimethyl ether

Identify the name of the above reaction.

• (A) Etard reaction
• (B) Gatterman-Koch formylation reaction
• (C) Stephen reaction
• (D) Rosenmund reaction

When sodium bicarbonate is added to an organic compound, brisk effervescences is evolved. This indicate the presence of which functional group.

• (A) \(-\text{CHO}\)
• (B) \(-\text{C} = \text{O}\)
• (C) \(-\text{COCl}\)
• (D) \(-\text{COOH}\)

Tertiary amines have the lowest boiling points because ,

• (A) They possess polar N-C bonds
• (B) They possess inter-molecular dipole-dipole attraction forces.
• (C) They possess inter-molecular H-bonding
• (D) They do not possess inter-molecular H- bonding

Which of the following molecules is capable of forming Zwitter ion?

• (A) \(\text{NH}_2\text{CH}_2\text{COOH}\)
• (B) \(\text{CH}_3\text{COOH}\)
• (C) \(\text{CH}_3\text{CH}_2\text{NH}_2\)
• (D) \(\text{CCl}_3\text{NO}_2\)

The letter 'D' in carbohydrates signifies

• (A) dextrorotatory
• (B) configuration
• (C) diamagnetic nature
• (D) mode of synthesis

Which of the following is found to raise the HDL level in the blood?

• (A) Omega-6 fatty acids
• (B) Omega-3 fatty acids
• (C) Trans fats
• (D) Cis fats

Identify the polymer used to prepare plastic bottles needed to store cooking oil.

• (A) PVC
• (B) PS
• (C) LDPE
• (D) PET

Which polymer is commonly used in nonstick cookwares ?

• (A) PVC
• (B) Teflon
• (C) SBR
• (D) Buna -S

Buna-S is prepared from

• (A) 1, 3- butadiene and vinyl cyanide
• (B) 1, 3- butadiene and styrene
• (C) 1,2 -butadiene and vinyl cyanide
• (D) 1, 2-butadiene and styrene

If \(n\) is a positive integer, then \((1+i\sqrt{3})^{2n}+(1-i\sqrt{3})^{2n}\) is equal to .........

• (A) \(2^{2n+1}cos(\frac{2nπ}{3})\)
• (B) \(2^{2n}cos(\frac{nπ}{3})\)
• (C) \(2^{2n+1}cos(\frac{nπ}{3})\)
• (D) \(2^ncos(\frac{2nπ}{3})\)

The number of 4-letter words formed from the English alphabet such that there are exactly 2 vowels and 2 consonants and no vowel is repeated, but consonants may be repeated is:

• (A) \(^4C_2\times ^5C_2\times (21)^2\)
• (B) \(^4C_2\times ^5C_2\times ^{21}C_2\)
• (C) \(^4C_2\times ^5P_2\times ^{21}P_2\)
• (D) \(^4C_2\times ^5P_2\times (21)^2\)

The value of \(cos(60^{\circ}-A)\cdot cosA\cdot cos(60^{\circ}+A)\) is

• (A) \(\frac{1}{4}cos3A\)
• (B) \(sin3A\)
• (C) \(cos3A\)
• (D) \(\frac{1}{4}sin3A\)

The general solution of the equation \(cotθ\cdot cot2θ = 1\) is...

• (A) \(θ = nπ\pm \frac{π}{6},n\in Z\)
• (B) \(θ = nπ\pm \frac{π}{3},n\in Z\)
• (C) \(θ = nπ\pm \frac{π}{4},n\in Z\)
• (D) \(θ = nπ\pm \frac{π}{8},n\in Z\)

The line \((2+k)x+(1+k)y = 5+7k\) passes through the fixed point for different values of k. If 'd' is the distance of a fixed point from the origin, then \(d^2 = \ldots\)

• (A) \(29\)
• (B) \(37\)
• (C) \(65\)
• (D) \(85\)

If \(θ\) is the acute angle between the lines \(2x^2+7xy+3y^2 = 0\), then the value of \(\frac{2cosθ-3sinθ}{4sinθ+5cosθ} = \ldots\)

• (A) \(1\)
• (B) \(\frac{5}{9}\)
• (C) \(-\frac{1}{9}\)
• (D) \(\frac{1}{9}\)

The combined equation of the pair of lines passing through the origin and making an angle \(\frac{π}{4}\) with the line \(3x+y-6 = 0\) is...

• (A) \(x^2+3xy-2y^2 = 0\)
• (B) \(2x^2-3xy+y^2 = 0\)
• (C) \(2x^2-xy+3y^2 = 0\)
• (D) \(2x^2+3xy-2y^2 = 0\)

Line \(l:x+y = 4\) intersects the circle \(x^2+y^2-2x-2y = 2\) at points \(A\) and \(B\). If C is the center of the circle, then the area of \(△ABC\) is...

• (A) \(\sqrt{2}\)
• (B) \(2\)
• (C) \(2\sqrt{2}\)
• (D) \(4\)

If \(\underset{x\rightarrow 0}{lim}\frac{45^x-9^x-5^x+1}{(k^x-1)(3^x-1)} = 2\), then the value of \(k\) is ...

• (A) \(45\)
• (B) \(9\)
• (C) \(5\)
• (D) \(3\)

If the truth value of the compound statement \([(p\leftrightarrow q)∧(q\rightarrow r)∧\sim r]\rightarrow (p∧\sim q)\) is false, then the truth values of the statement patterns \((p\rightarrow q)\leftrightarrow (q\rightarrow r)\) and \(\sim (p∨r)\rightarrow (q∧p)\) are, respectively ...

• (A) \((T,T)\)
• (B) \((T,F)\)
• (C) \((F,T)\)
• (D) \((F,F)\)

The negation of the contrapositive of the statement \((p∨\sim q)\rightarrow (p∧\sim q)\) is

• (A) \((p∧\sim q)∨(\sim p∧\sim q)\)
• (B) \((\sim p∧q)∨(p∧\sim q)\)
• (C) \((\sim p∨\sim q)∧(p∨q)\)
• (D) \((\sim p∨q)∧(p∨\sim q)\)

In triangle ABC, with usual notations, if \(a = 4,b = 5\) and \(c = 6\), then angle C is equal to...

• (A) \(A\)
• (B) \(2A\)
• (C) \(3A\)
• (D) \(\frac{A}{2}\)

If \(A = [\begin{array}{cc}2 & 3 \\ 5 & -2\end{array}]\), \(B^{-1} = \left[ \begin{array}{cc}\frac{1}{5} & \frac{2}{5} \\ \frac{2}{5} & -\frac{1}{5}\end{array} \right]\), then \((AB)^{-1} =\)

• (A) \(\frac{1}{95}[\begin{array}{cc}8 & -1 \\ -1 & 12\end{array}]\)
• (B) \(\frac{1}{95}[\begin{array}{cc}12 & -1 \\ -1 & 8\end{array}]\)
• (C) \(\frac{1}{95}[\begin{array}{cc}-12 & 1 \\ 1 & -8\end{array}]\)
• (D) \(\frac{1}{95}[\begin{array}{cc}-8 & 1 \\ 1 & -12\end{array}]\)

If \(A = [a_{ij}]_{3\times 3}\) is a matrix such that \(a_{ij} = |2i-5j|\), where \(|.|\) denotes the modulus function, then the element in the \(2^{\text{nd}}\) row and \(3^{\text{rd}}\) column of \(A^{-1}\) is ...

• (A) \(3\)
• (B) \(1\)
• (C) \(0\)
• (D) \(-1\)

If \(A = [\begin{array}{cc}2 & -1 \\ 0 & 2\end{array}]\) and \(A^2+xA+yI_2 = O_2\), where \(I_2\) and \(O_2\) are the identity matrix and null matrix of order 2 respectively, then:

• (A) \(x = 4, y = 4\)
• (B) \(x = -4, y = 4\)
• (C) \(x = -4, y = -2\)
• (D) \(x = 4, y = -4\)

The value of \(cot^{-1}[\frac{\sqrt{1-sinx}+\sqrt{1+sinx}}{\sqrt{1-sinx}-\sqrt{1+sinx}}]\), where \(x\in (0,\frac{π}{2})\) is...

• (A) \(π-x\)
• (B) \(2π-x\)
• (C) \(\frac{π}{2}-\frac{x}{2}\)
• (D) \(π-\frac{x}{2}\)

If \((tan^{-1}x)^2+(cot^{-1}x)^2 = \frac{5π^2}{8}\), then the value of \(x\) is equal to...

• (A) \(-1\)
• (B) \(-2\)
• (C) \(1\)
• (D) \(2\)

If \(f(\frac{x-1}{x+1}) = x+1\), then \(\int f(x)\,dx = \cdots\)

• (A) \(2log|1-x|+c\)
• (B) \(-2log|1-x|+c\)
• (C) \(2log|1+x|+c\)
• (D) \(-2log|1+x|+c\)

If \(e^x+e^{f(x)} = e\), then the domain of \(f(x)\) is

• (A) \((1,\infty )\)
• (B) \((-\infty ,1)\)
• (C) \((-\infty ,\infty )\)
• (D) \((-\infty ,0)\)

If the function f is continuous at \(x = π\), where \(f(x) = \frac{1-cos[7(x-π)]}{5(x-π)^2}\), for \(x\neq π\), then \(f(π) =\)

• (A) \(\frac{49}{4}\)
• (B) \(\frac{4}{49}\)
• (C) \(\frac{49}{10}\)
• (D) \(\frac{10}{49}\)

Let f(x) be a twice differentiable function such that \(f^{''}(x) = -f(x)\), \(f^'(x) = g(x)\) and \(h(x) = \{f(x)\}^2+\{g(x)\}^2\). If \(h(5) = 11\), then \(h(10)\) is equal to ...

• (A) \(11\)
• (B) \(22\)
• (C) \(0\)
• (D) Not defined

If \(cot[f(x)] = \frac{3x-x^3}{1-3x^2}\) and \(sin[g(x)] = \frac{1-x^2}{1+x^2}\), then \(\underset{x\rightarrow t}{lim}\frac{f(x)-f(t)}{g(x)-g(t)} = \ldots\)

• (A) \(\frac{3}{2(1+t^2)}\)
• (B) \(\frac{3}{2}\)
• (C) \(\frac{5}{2}\)
• (D) \(-\frac{5}{2(1+t^2)}\)

If \(y = cos^{-1}(\frac{1-4^x}{1+4^x})\), then \(\frac{dy}{dx}\) at \(x = 1\) is

• (A) \(\frac{4log2}{5}\)
• (B) \(\frac{log2}{5}\)
• (C) \(\frac{2log8}{5}\)
• (D) \(\frac{5log8}{2}\)

A cylindrical tank without a top lid is being manufactured to hold a fixed volume of \(125π\) cubic cm. The minimum surface area required to construct this tank is ............square cm

• (A) \(75π\)
• (B) \(50π\)
• (C) \(25π\)
• (D) \(125π\)

A spherical snow ball is melting so that its volume is decreasing at the rate of 8 c.c./sec then the rate of change of radius when the radius is 2 cm, is :

• (A) The radius is increasing at the rate of \(\frac{1}{2π} \text{cm/s}\)
• (B) The radius is decreasing at the rate of \(\frac{1}{2π} \text{cm/s}\)
• (C) The radius is increasing at the rate of \(\frac{1}{π} \text{cm/s}\)
• (D) The radius is decreasing at the rate of \(\frac{1}{π} \text{cm/s}\)

The equation of tangent to the curves \(x = 1-3t^2\) and \(y = t-3t^3\) at the point \((-2,2)\) is...

• (A) \(4x+3y+2 = 0\)
• (B) \(4x-3y+2 = 0\)
• (C) \(3x+4y+2 = 0\)
• (D) \(3x-4y+2 = 0\)

If \(f(x) = cosx\), then the value of \(\int \frac{e^{f(x)}(xsin^3x+f(x))}{1-(f(x))^2}dx =\)

• (A) \(e^{f(x)}(-\text{cosec} x-x)+c\)
• (B) \(e^{f(x)}(\text{cosec}\,x+x)+c\)
• (C) \(e^{f(x)}(cosx-x)+c\)
• (D) \(e^{f(x)}(cot^2x+x)+c\)

\(\int \frac{x^4+1}{x^6+1}dx =\)

• (A) \(tan^{-1}x-\frac{1}{3}tan^{-1}(x^3)+c\)
• (B) \(tan^{-1}x+\frac{1}{3}tan^{-1}(x^3)+c\)
• (C) \(tan^{-1}x-tan^{-1}(x^3)+c\)
• (D) \(tan^{-1}x+tan^{-1}(x^3)+c\)

If \(a > 0,b > 0\) and \(\int \frac{1}{ax^2+b}dx = \frac{1}{\sqrt{6}}tan^{-1}(\frac{\sqrt{2}x}{\sqrt{3}})+c\), then \(\int \frac{1}{bx^2+a}dx = \ldots\)

• (A) \(-\frac{1}{\sqrt{6}}tan^{-1}(\frac{\sqrt{2}x}{\sqrt{3}})+c\)
• (B) \(\frac{1}{\sqrt{6}}tan^{-1}(\frac{\sqrt{3}x}{\sqrt{2}})+c\)
• (C) \(-\sqrt{6}\,tan^{-1}(\frac{\sqrt{2}x}{\sqrt{3}})+c\)
• (D) \(\sqrt{6}\,tan^{-1}(\frac{\sqrt{3}x}{\sqrt{2}})+c\)

The value of integral \(\int _0^1cot^{-1}(1+x^2-x)\,dx\) is...

• (A) \(\frac{π}{2}-log2\)
• (B) \(π-log2\)
• (C) \(\frac{π}{4}-log2\)
• (D) \(2π-log2\)

The value of \(\int _{-π}^π\frac{2x(1+sinx)}{1+cos^2x}dx\) is...

• (A) \(-\sqrt{2}π^2\)
• (B) \(π^2\)
• (C) \(\frac{π^2}{\sqrt{2}}\)
• (D) \(-\frac{π^2}{\sqrt{2}}\)

The area of the region enclosed by the lines \(y = 2x,2y = x\) and \(x = 2\) (in sq.units) is...

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

The area of the region common to the parabolas \(4y^2 = 9x\) and \(3x^2 = 16y\) is...

• (A) 2 sq. units
• (B) 4 sq. units
• (C) 8 sq. units
• (D) 16 sq. units

For the differential equation \(\frac{d^3y}{dx^3}+cos(\frac{d^2y}{dx^2}) = 0\), which of the following is true ?

• (A) Order = 3, degree = 1
• (B) Order = 3, degree = 2
• (C) Order = 3, degree is not defined
• (D) Order = 2, degree is not defined

If water at \(100^{\circ}\text{C}\) cools in 10 minutes to \(80^{\circ}\text{C}\) and to \(65^{\circ}\text{C}\) in the next 10 minutes, then the room temperature will be ...

• (A) \(30^{\circ}\text{C}\)
• (B) \(15^{\circ}\text{C}\)
• (C) \(25^{\circ}\text{C}\)
• (D) \(20^{\circ}\text{C}\)

The equation of the curve passing through the point (0, 1) and having a slope of the tangent at point \(P(x,y)\) is equal to \(\frac{y}{x+y}\), is

• (A) \(x = e^{-\frac{y}{x}}\)
• (B) \(y = e^{-\frac{y}{x}}+1\)
• (C) \(x = e^{\frac{x}{y}}-1\)
• (D) \(y = e^{\frac{x}{y}}\)

If \(f(x)\) is a polynomial such that \(f(x) = [f^'(x)]^2\) and \(f(2) = 0\), then \(f(-2) = \ldots\)

• (A) \(1\)
• (B) \(-1\)
• (C) \(4\)
• (D) \(-4\)

The equation of a line in cartesian form passing through (0, 0, 0) and (4, 3, c) and parallel to \(\overset{⃗}{a}\times \overset{⃗}{b}\) where \(\overset{⃗}{a} = 2\hat{i}+\hat{j}+2\hat{k}\), \(\overset{⃗}{b} = 3\hat{i}-4\hat{j}\) is

• (A) \(\frac{x-4}{8} = \frac{y-3}{6} = \frac{z+\frac{11}{2}}{11}\)
• (B) \(\frac{x-4}{8} = \frac{y+3}{6} = \frac{z+\frac{11}{2}}{-11}\)
• (C) \(\frac{x-4}{8} = \frac{y-3}{6} = \frac{z+\frac{11}{2}}{-11}\)
• (D) \(\frac{x+4}{8} = \frac{y-3}{6} = \frac{z+\frac{11}{2}}{-11}\)

If \(\overset{̄}{a} = 4\hat{i}+\hat{j}+\hat{k}\), \(\overset{̄}{b} = 2\hat{i}+\hat{j}+2\hat{k}\) and \(\overset{̄}{c} = 3\hat{i}+4\hat{j}+5\hat{k}\), then \((\overset{̄}{a}+\overset{̄}{b})\cdot (\overset{̄}{b}+\overset{̄}{c}) =\)

• (A) \(30\)
• (B) \(21\)
• (C) \(61\)
• (D) \(10\)

If \(\overset{̄}{a} = 2\hat{i}+\hat{j}-\hat{k}\), \(\overset{̄}{b} = \hat{i}+3\hat{k}\) and \(\overset{̄}{c}\) is a unit vector, then the maximum value of the scalar triple product \([\overset{̄}{a} \overset{̄}{b} \overset{̄}{c}]\) is

• (A) \(\sqrt{10}+\sqrt{6}\)
• (B) \(\sqrt{10}\)
• (C) \(\sqrt{6}\)
• (D) \(\sqrt{59}\)

Let \(\overset{̄}{a},\overset{̄}{b}\) and \(\overset{̄}{c}\) be three coplanar unit vectors. A unit vector \(\overset{̄}{d}\) is perpendicular to them. If \((\overset{̄}{a}\times \overset{̄}{b})\times (\overset{̄}{c}\times \overset{̄}{d}) = \frac{3}{26}\hat{i}-\frac{2}{13}\hat{j}+\frac{6}{13}\hat{k}\) and the angle between \(\overset{̄}{a}\) and \(\overset{̄}{b}\) is \(30^{\circ}\), then \(\overset{̄}{c}\) is equal to...

• (A) \(\frac{3}{13}\hat{i}-\frac{4}{13}\hat{j}+\frac{12}{13}\hat{k}\)
• (B) \(\frac{3}{13}\hat{i}-\frac{2}{13}\hat{j}+\frac{6}{13}\hat{k}\)
• (C) \(\frac{3}{26}\hat{i}-\frac{4}{13}\hat{j}+\frac{12}{13}\hat{k}\)
• (D) \(\frac{3}{26}\hat{i}-\frac{3}{26}\hat{j}+\frac{5}{26}\hat{k}\)

The volume of a parallelopiped with coterminous edges \(\overset{̄}{a},\overset{̄}{b},\overset{̄}{c}\) is 3 cubic units. The volume (in cubic units) of a tetrahedron with coterminous edges \((\overset{̄}{a}\times \overset{̄}{b}),(\overset{̄}{a}\times 2\overset{̄}{c}),(\overset{̄}{b}\times 2\overset{̄}{c})\) is...

• (A) \(6\)
• (B) \(12\)
• (C) \(24\)
• (D) \(36\)

If the lines \(2x = ky = -z\) and \(6x = -y = -4z\) are perpendicular to each other then the value of \(k\) is ...

• (A) \(16\)
• (B) \(5\)
• (C) \(10\)
• (D) \(3\)

The vector equation of plane in parametric form, passing through the points (-1, 2, 0), (2, 2, -1) and parallel to the line \(\frac{x-1}{1} = \frac{2y+1}{2} = \frac{z+1}{-1}\) is

• (A) \(\overset{̄}{r} = (-\hat{i}+2\hat{j})+λ(3\hat{i}-\hat{k})+μ(\hat{i}+\hat{j}-\hat{k})\)
• (B) \(\overset{̄}{r} = (-\hat{i}+2\hat{j})+λ(3\hat{i}-\hat{k})+μ(\hat{i}+2\hat{j}-\hat{k})\)
• (C) \(\overset{̄}{r} = (\hat{i}-2\hat{j})+λ(3\hat{i}+\hat{k})+μ(\hat{i}+\hat{j}-\hat{k})\)
• (D) \(\overset{̄}{r} = (-\hat{i}+2\hat{j})+λ(3\hat{i}-\hat{k})+μ(\hat{i}-\hat{j}+\hat{k})\)

If \(d\) is the distance of point (2, 5, 10) from the plane containing the lines \(\overset{̄}{r} = (4\hat{j}-\hat{k})+λ(\hat{i}+2\hat{j}-2\hat{k})\) and \(\overset{̄}{r} = (2\hat{i}+\hat{j})+μ(\hat{i}+2\hat{j}-2\hat{k})\), then \(d^2 =\)

• (A) \(145\)
• (B) \(13\)
• (C) \(90\)
• (D) \(79\)

The values of p and q so that the line joining the points (7, p, 2) and (q, -2, 5) may be parallel to the line joining the points (2, -3, 5) and (-6, -15, 11) are

• (A) \(p = 4, q = -3\)
• (B) \(p = 4, q = 3\)
• (C) \(p = -4, q = 3\)
• (D) \(p = -4, q = -3\)

The minimum value of \(z = 3x+5y\), subject to constraints \(x\leq 80\), \(y\geq 60\), \(x+y\leq 200\) & \(x,y\geq 0\) occurs at the point...

• (A) \((0,200)\)
• (B) \((60,0)\)
• (C) \((0,60)\)
• (D) \((80,60)\)

If a fair coin is tossed 8 times, then the probability of getting at most 2 heads is...

• (A) \(\frac{37}{64}\)
• (B) \(\frac{37}{256}\)
• (C) \(\frac{37}{128}\)
• (D) \(\frac{37}{512}\)

If a random variable \(X\) has a probability mass function \(P(x) = \{\begin{array}{cc}kx^2, & \text{for }x = 1,2,3,4 \\ 0, & \text{otherwise}\end{array}\), then the mean of \(X\) is ...

• (A) \(3.232\)
• (B) \(2.221\)
• (C) \(3.333\)
• (D) \(2.222\)

If \(E_1\) and \(E_2\) are equally likely, mutually exclusive and exhaustive events and \(P(A|E_1) = 0.2\), \(P(A|E_2) = 0.3\), then \(P(E_1|A)\) equal to...

• (A) \(\frac{1}{5}\)
• (B) \(\frac{4}{5}\)
• (C) \(\frac{2}{3}\)
• (D) \(\frac{2}{5}\)