MHT CET 2026 April 18 Shift 1 Question Paper- Download PDF with Solutions

The angle made by vector \(\overset{⃗}{A} = 2\hat{i}+3\hat{j}\) with x-axis and that with y-axis are respectively.

• (A) \(tan^{-1}2\sqrt{13}\) , \(tan^{-1}3\sqrt{13}\)
• (B) \(cos^{-1}\frac{2}{\sqrt{13}}\) , \(cos^{-1}\frac{3}{\sqrt{13}}\)
• (C) \(cos^{-1}\frac{1}{\sqrt{2}}\) , \(cos^{-1}\frac{1}{\sqrt{3}}\)
• (D) \(sin^{-1}\frac{1}{\sqrt{6}}\) , \(sin^{-1}\frac{1}{2\sqrt{3}}\)

If two resistors of resistances have values \(R_1 = (350\pm 3) \Omega\) and \(R_2 = (140\pm 4) \Omega\). The percentage error for the sum and difference of \(R_1\) and \(R_2\) are respectively

• (A) \(1.43\%\) , \(3.33\%\)
• (B) \(7\%\) , \(1\%\)
• (C) \(1\%\) , \(7\%\)
• (D) \(3.33\%\) , \(1.43\%\)

A body moves along a circular path of diameter \(30 \text{cm}\). It starts from one end of diameter, moves along the circular path and reaches the other end of diameter in 3 second. The angular speed of the body in radian per second is

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

A block of mass M is lying on a horizontal frictionless surface. One end of the uniform rope of half the mass of the block is fixed to the block which is pulled in the horizontal direction by applying a force F at the other end. The tension in the middle of the rope will be

• (A) \(\frac{F}{3}\)
• (B) \(\frac{3F}{4}\)
• (C) \(\frac{4F}{5}\)
• (D) \(\frac{5F}{6}\)

The angular momentum of a rotating body is 'L'. When the frequency of rotating body is tripled and its kinetic energy is made one-third, the new angular momentum becomes

• (A) \(9L\)
• (B) \(6L\)
• (C) \(\frac{L}{3}\)
• (D) \(\frac{L}{9}\)

A solid cylinder and a solid sphere having the same mass and radius roll down on the same smooth inclined plane. The ratio of the acceleration of the cylinder \((a_c)\) to that of the sphere \((a_s)\) is

• (A) \(\frac{14}{15}\)
• (B) \(\frac{15}{14}\)
• (C) \(\frac{13}{14}\)
• (D) \(\frac{11}{15}\)

The radius of gyration K of a hollow sphere of mass M and radius R about an axis XY is equal to R as shown in figure. The distance of that axis from the center of the sphere is h. The value of h is

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

The lengths of seconds pendulums on the surface of the earth and at an altitude '\(h\)' from the surface of the earth are \(l_s\) and \(l_h\) respectively. The radius of the earth is

• (A) \(\frac{h\sqrt{l_h}}{\sqrt{l_s}-\sqrt{l_h}}\)
• (B) \(\frac{h\sqrt{l_h}}{\sqrt{l_h}-\sqrt{l_s}}\)
• (C) \(\frac{\sqrt{l_h}}{h(\sqrt{l_s}-\sqrt{l_h})}\)
• (D) \(\frac{\sqrt{l_s}}{h(\sqrt{l_h}-\sqrt{l_s})}\)

Time period of a simple pendulum is \(T_1\) when on the earth's surface and \(T_2\) when taken to a height '2R' above the earth's surface, where 'R' is the radius of the earth. The ratio \(T_1:T_2\) is

• (A) \(1:2\)
• (B) \(1:3\)
• (C) \(1:4\)
• (D) \(1:5\)

The pressure at half the depth of a lake is equal to two-third pressure at the bottom of the lake. So the depth 'h' of the lake is (\(ρ\) = density of water in the lake, \(g\) = acceleration due to gravity, \(P_0\) = atmospheric pressure).

• (A) \(\frac{2P_0}{ρg}\)
• (B) \(\frac{P_0}{ρg}\)
• (C) \(\frac{2P_0}{3ρg}\)
• (D) \(\frac{P_0}{2ρg}\)

A spherical object is falling under gravity through a viscous fluid. The sphere attains the terminal velocity when

• (A) viscous force is zero.
• (B) buoyant force is equal to force due to gravity
• (C) viscous force plus force of gravity becomes equal to buoyant force.
• (D) buoyant force plus viscous force becomes equal to force due to gravity.

A piston of cross-sectional area \(2.5\times 10^{-2} \text{m}^2\) is used in a hydraulic lift to exert a force of \(250 \text{N}\) on water. The cross-sectional area of the other piston which supports a car of mass \(3000 \text{kg}\) is (\(g = 9.8 \text{m/s}^2\))

• (A) \(1.96 \text{m}^2\)
• (B) \(2.94 \text{m}^2\)
• (C) \(3.92 \text{m}^2\)
• (D) \(5.88 \text{m}^2\)

Two spherical black bodies of radii \(R_1\) and \(R_2\) having surface temperature \(T_1\) and \(T_2\) respectively, radiate the same power. The ratio of \(R_1\) to \(R_2\) will be

• (A) \((\frac{T_1}{T_2})^2\)
• (B) \((\frac{T_1}{T_2})^4\)
• (C) \((\frac{T_2}{T_1})^2\)
• (D) \((\frac{T_2}{T_1})^4\)

The ratio of thermal conductivity of two rods of different materials is \(4:3\). The two rods have same area of cross-section and same thermal resistance. The lengths of rods are in the ratio

• (A) \(5:4\)
• (B) \(4:3\)
• (C) \(1:5\)
• (D) \(7:1\)

In thermodynamic process for free expansion, select the 'WRONG' statement out of the following options.

• (A) Free expansions are adiabatic expansions and there is no exchange of heat between system and environment.
• (B) A free expansion can be plotted on a p-V diagram.
• (C) There is no work done on the system or by the system.
• (D) It is an instantaneous change and the system is not in thermodynamic equilibrium.

Which of the following graphs between pressure and volume of a gas correctly shows isochoric process in thermodynamics?

• (A) (D)
• (B) (A)
• (C) (C)
• (D) (B)

The r.m.s. speed of hydrogen at S.T.P. is '\(u\)' \(\text{m/s}\). If the gas is heated at constant pressure till its volume becomes three times, the final temperature of the gas and the r.m.s. speed are respectively

• (A) \(1092 \text{K}\) , \(3u \text{m/s}\)
• (B) \(1092 \text{K}\) , \(\frac{u}{3} \text{m/s}\)
• (C) \(819 \text{K}\) , \(\sqrt{3}\,u \text{m/s}\)
• (D) \(819 \text{K}\) , \(\frac{u}{\sqrt{3}} \text{m/s}\)

An ideal gas is heated from \(27^{\circ}\text{C}\) to \(627^{\circ}\text{C}\) at constant pressure. If initial volume of gas is \(4 \text{m}^3\), then the new volume of the gas will be

• (A) \(12 \text{m}^3\)
• (B) \(6 \text{m}^3\)
• (C) \(3 \text{m}^3\)
• (D) \(2 \text{m}^3\)

• (A) \(T_a = \sqrt{2}\,T_b\)
• (B) \(T_b = 2T_a\)
• (C) \(T_a = \frac{T_c}{\sqrt{2}}\)
• (D) \(T_b = 2T_c\)

A particle is performing simple harmonic motion about \(x = 0\) with an amplitude '\(a\)' and periodic time T. The speed of the particle at \(x = \frac{a}{3}\) will be

• (A) \(\frac{2πa}{T}\)
• (B) \(\frac{4πa}{3T}\)
• (C) \(\frac{4\sqrt{2}\,πa}{3T}\)
• (D) \(\frac{\sqrt{3}\,π^2a}{2T}\)

Two sources of sound are emitting progressive waves \(y_1 = 4sin708\,πt\) and \(y_2 = 3sin700\,πt\). The sources are placed close to each other. The number of beats heard per second and intensity ratio between waxing and wanning are respectively

• (A) \(4\) , \(16:9\)
• (B) \(8\) , \(16:9\)
• (C) \(8\) , \(49:1\)
• (D) \(4\) , \(49:1\)

Fifth overtone of an open pipe of length \(L_0\) is in unison with the fifth overtone of the pipe closed at one end of length \(L_c\).
The ratio \(L_0\) to \(L_c\) is

• (A) \(11:6\)
• (B) \(11:12\)
• (C) \(6:11\)
• (D) \(12:11\)

The lengths of the two pipes open at both ends are L and \((L+L_1)\). If they are sounded together, the beat frequency will be
(\(v\) = velocity of sound in air)

• (A) \(\frac{vL_1}{L(L+L_1)}\)
• (B) \(\frac{2vL_1}{L(L+L_1)}\)
• (C) \(\frac{2L(L+L_1)}{vL_1}\)
• (D) \(\frac{vL_1}{2L(L+L_1)}\)

The number of waves counted by an observer on the sea-coast in one minute is 54. If the wavelength of the waves is \(5 \text{m}\) then the velocity of the waves in \(\text{m/s}\) will be

• (A) \(4.5\)
• (B) \(5.4\)
• (C) \(6.3\)
• (D) \(7.2\)

When a dipole placed parallel to electric field is rotating through \(π^c\), the work done is W. The work done in rotating the dipole through \((\frac{π}{3})^c\) is
(\(cos0^{\circ} = 1\), \(cos60^{\circ} = \frac{1}{2}\), \(cos180^{\circ} = -1\))

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

The point charges \(+q\), \(-q\), \(-q\), \(+q\), \(+Q\) and \(-q\) are placed at the vertices of a regular hexagon ABCDEF as shown in the figure.

• (A) \(\frac{q}{3}\)
• (B) \(\frac{q}{4}\)
• (C) \(3q\)
• (D) \(4q\)

In an oscillating LC circuit, the maximum charge on the capacitor is Q. When the energy is stored equally between the electric and magnetic fields, the charge on the capacitor becomes

• (A) \(\frac{Q}{\sqrt{2}}\)
• (B) \(Q\sqrt{3}\)
• (C) \(\frac{Q}{2}\)
• (D) \(\frac{Q}{4}\)

Two circular plates each of radius '\(r\)' are kept parallel to each other distance '\(d\)' apart. The capacitance of the capacitor formed is '\(C_1\)'. If the radius of each of the plates is increased to \(\sqrt{3}\) times the earlier radius and their distance of separation decreased to half the initial value, the capacitance now becomes '\(C_2\)'. The ratio \(C_1:C_2\) is

• (A) \(1:2\)
• (B) \(1:4\)
• (C) \(1:6\)
• (D) \(6:1\)

A capacitor of capacitance C has charge Q and energy stored in it is W. If the charge is increased to 3Q, the energy stored in the capacitor W' will be

• (A) \(9W\)
• (B) \(3W\)
• (C) \(\frac{W}{3}\)
• (D) \(\frac{W}{6}\)

A potentiometer wire of length \(4 \text{m}\) and resistance \(5 \Omega\) is connected in series with a resistance of \(992 \Omega\) and a cell of e.m.f. \(4 \text{V}\) with internal resistance \(3 \Omega\). The length of \(0.75 \text{m}\) on potentiometer wire balances the e.m.f. of

• (A) \(2.50 \text{mV}\)
• (B) \(3 \text{mV}\)
• (C) \(3.75 \text{mV}\)
• (D) \(4 \text{mV}\)

By connecting a resistance of \(980 \Omega\) in series with a galvanometer, it is converted into a voltmeter of a certain range. When the resistance of \(470 \Omega\) is connected in series, the range is halved. The resistance of the galvanometer is

• (A) \(30 \Omega\)
• (B) \(40 \Omega\)
• (C) \(50 \Omega\)
• (D) \(60 \Omega\)

The region inside a current carrying toroid is filled with a material (Niobium) having susceptibility \(χ = 2.6\times 10^{-5}\). The percentage increase in the magnetic field in the presence of Niobium over that without it, is

• (A) \(2.6\times 10^{-5}\)
• (B) \(2.6\times 10^{-4}\)
• (C) \(2.6\times 10^{-3}\)
• (D) \(2.6\times 10^2\)

In the following figure, the magnitude of the magnetic field at point 'O' will be

• (A) \(\frac{μ_0}{4π}\frac{I}{r}(\frac{2}{π}+2)\)
• (B) \(\frac{μ_0}{4π}\frac{I}{r}(\frac{2}{π}-2)\)
• (C) \(\frac{μ_0}{4π}\frac{I}{r}(2+\frac{π}{2})\)
• (D) \(\frac{μ_0}{4π}\frac{I}{r}(2-\frac{π}{2})\)

The ratio of magnetic field at the centre of the current carrying circular loop and its magnetic moment is '\(x\)'. When both the current and radius are tripled, then the ratio will be

• (A) \(\frac{x}{27}\)
• (B) \(\frac{x}{9}\)
• (C) \(\frac{x}{3}\)
• (D) \(3x\)

In the circuit shown, when the current '\(i\)' is \(3\text{A}\) and increasing at the rate of \(1\text{A/s}\) the measurement of the potential difference between A and B is \(12 \text{V}\). But when the same current \(3\text{A}\) is decreasing at the rate of \(1\text{A/s}\), the measured potential difference \(V_{AB}\) between A and B is \(6\text{V}\). The value of 'R' in the circuit is

• (A) \(3 \Omega\)
• (B) \(4 \Omega\)
• (C) \(6 \Omega\)
• (D) \(8 \Omega\)

Two coils have a mutual inductance of \(0.005 \text{H}\). The current changes in the first coil according to equation \(I = I_0sinωt\), where \(I_0 = 10 \text{A}\) and \(ω = 60π \text{rad s}^{-1}\). The maximum values of e.m.f. in the second coil in volt will be

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

The magnetic potential energy stored in a certain inductor is \(49 \text{mJ}\) when the current in the inductor is \(70 \text{mA}\). The inductance of the inductor is

• (A) \(0.20 \text{H}\)
• (B) \(2.0 \text{H}\)
• (C) \(20 \text{H}\)
• (D) \(200 \text{H}\)

An a.c. source is applied to a series LR circuit with \(X_L = 3R\) and power factor is \(X_1\). Now a capacitor with \(X_c = R\) is added in series to LR circuit and power factor is \(X_2\). The ratio \(X_1\) to \(X_2\) is

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

In an a.c. circuit, a resistance R is connected in series with an inductance 'L'. If phase angle between voltage and current is \(45^{\circ}\), the value of inductive reactance will be
(\(sin45^{\circ} = \frac{1}{\sqrt{2}} = cos45^{\circ}\))

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

A simple microscope is a combination of two lenses in contact. Their powers are \(+20\text{D}\) and \(-4\text{D}\). The distance of distinct vision is \(25 \text{cm}\). When seen through the microscope, the size of the image of an object \(3 \text{mm}\) high is

• (A) \(1.8 \text{cm}\)
• (B) \(1.5 \text{cm}\)
• (C) \(1.2 \text{cm}\)
• (D) \(0.6 \text{cm}\)

When a glass plate of refractive index \(1.44\) is introduced in the path of one of the interfering beams, the fringes are displaced by a distance '\(y\)'. If this plate is replaced by another plate of same thickness but of refractive index \(1.66\), the fringes will be displaced by a distance

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

In a certain region of the screen, number of fringes formed is observed to be 8 in Young's double slit experiment when light of wavelength \(6600 \text{Å}\) is used. If the light of wavelength \(4400 \text{Å}\) is used, the number of fringes observed in the same region of the screen will be

• (A) \(10\)
• (B) \(12\)
• (C) \(16\)
• (D) \(18\)

In a Fraunhofer diffraction of a single slit, when a slit is illuminated by a light of wavelength \(6480 \text{Å}\), angular width of central maximum is measured. When the slit is illuminated by light of another wavelength '\(λ\)' the angular width decreases by \(25\%\). The value of \(λ\) in \(\text{Å}\) units is

• (A) \(4220\)
• (B) \(4530\)
• (C) \(4860\)
• (D) \(5230\)

If the kinetic energy of a particle is increased to 16 times, the percentage change in the de Broglie wavelength of a particle is

• (A) \(40 \%\)
• (B) \(60 \%\)
• (C) \(75 \%\)
• (D) \(80 \%\)

In case of photoelectric emission from certain metal, the cutoff frequency is \(ν\). If the radiation of frequency \(3ν\) is incident on the metal plate, the maximum possible velocity of the emitted electrons will be (\(m\) = mass of electron, \(h\) = Planck's constant)

• (A) \(\sqrt{\frac{hν}{2m}}\)
• (B) \(\sqrt{\frac{hν}{m}}\)
• (C) \(2(\sqrt{\frac{hν}{m}})\)
• (D) \(\sqrt{\frac{2hν}{m}}\)

The triply ionized beryllium (\(\text{Be}^{+++}\)) has the same electron orbital radius as that of the ground state of hydrogen. Hence, the energy state of triply ionized beryllium is (Given \(Z = 4\) for beryllium)

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

The angular momentum of the electron in the third Bohr orbit of a hydrogen atom is '\(l\)' so its angular momentum in the fourth Bohr orbit is

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

Which one of the following statements is 'WRONG' regarding LED?

• (A) LED's have long lifetime if properly manufactured.
• (B) Colours produced by LED's do not fade out.
• (C) LED's are energy efficient.
• (D) Brightness of light emitted by LED cannot be controlled.

Which of the following logic gates will have an output of '1' (one) for the given inputs?

• (A) D
• (B) B
• (C) A
• (D) C

A diode and resistance are connected as shown in figures.

• (A) D
• (B) B
• (C) A
• (D) C

How many moles of potassium chlorate are heated to produce \(22.4 \text{lit}\) of oxygen at STP ?

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

Which of the following colours of visible light has lowest energy?

• (A) Violet
• (B) Blue
• (C) Yellow
• (D) Red

What is formal charge on sulphur in sulphuric acid molecule?

• (A) \(-2\)
• (B) \(+2\)
• (C) \(-1\)
• (D) \(0\)

Which of the following compound has square pyramidal structure?

• (A) \(\text{XeF}_4\)
• (B) \(\text{XeF}_6\)
• (C) \(\text{XeO}_3\)
• (D) \(\text{XeOF}_4\)

Which of the following metal halides have more covalent character?

• (A) \(\text{SnCl}_2\)
• (B) \(\text{PbCl}_4\)
• (C) \(\text{SbCl}_3\)
• (D) \(\text{PbCl}_2\)

What ratio by mass of 'Ne' and \(\text{CH}_4\) should be mixed so that partial pressure exerted by each gas is same?

• (A) \(1:1\)
• (B) \(2:1\)
• (C) \(2:3\)
• (D) \(5:4\)

Calculate \(\Delta \text{H}\) for the following reaction at \(300 \text{K}\)
\(2\text{C}_{(s)}+3\text{H}_{2(g)}⟶\text{C}_2\text{H}_{6(g)}\) if \(\Delta \text{U}\) for the reaction is \(-80 \text{kJ}\) (\(\text{R} = 8.314 \text{JK}^{-1}\text{mol}^{-1}\))

• (A) \(-85.00 \text{kJ}\)
• (B) \(-43.00 \text{kJ}\)
• (C) \(-128.00 \text{kJ}\)
• (D) \(-170.00 \text{kJ}\)

For a certain reaction, \(\Delta \text{H}^{\circ} = 40 \text{kJ}\) and \(\Delta \text{S}^{\circ} = 80 \text{JK}^{-1}\). Find the temperature so that \(\Delta \text{G}^{\circ} = 0\)

• (A) \(500 \text{K}\)
• (B) \(400 \text{K}\)
• (C) \(300 \text{K}\)
• (D) \(600 \text{K}\)

Which from following is correct expression of first law of thermodynamics for isothermal process?

• (A) \(\text{W} = -\text{Q}\)
• (B) \(-\Delta \text{U} = -\text{W}\)
• (C) \(\Delta \text{U} = \text{Q}_{\text{v}}\)
• (D) \(\text{Q}_{\text{p}} = \Delta \text{U}+\text{P}_{\text{ext}}\Delta \text{V}\)

Which of the following is Lewis acid?

• (A) \(\text{BaCl}_2\)
• (B) \(\text{KCl}\)
• (C) \(\text{BeCl}_2\)
• (D) \(\text{LiCl}\)

The solubility of a sparingly soluble a salt \(\text{AB}_2\) is \(1\times 10^{-6} \text{mol/dm}^3\). Calculate its solubility product?

• (A) \(1\times 10^{-12}\)
• (B) \(2\times 10^{-12}\)
• (C) \(3\times 10^{-12}\)
• (D) \(4\times 10^{-18}\)

What is the molarity of \(\text{H}_2\text{SO}_4\) solution having \(\text{pH} = 4\) ?

• (A) \(4\times 10^{-1} \text{M}\)
• (B) \(4\times 10^{-2} \text{M}\)
• (C) \(5\times 10^{-5} \text{M}\)
• (D) \(1\times 10^{-4} \text{M}\)

Identify the conjugate bases of \(\text{H}_3\text{PO}_3\) and \(\text{H}_2\text{SO}_4\) respectively from following.

• (A) \(\text{HPO}_3^{2-}\) and \(\text{SO}_4^{2-}\)
• (B) \(\text{HPO}_3^{2-}\) and \(\text{HSO}_4^-\)
• (C) \(\text{H}_2\text{PO}_3^-\) and \(\text{HSO}_4^-\)
• (D) \(\text{PO}_3^{3-}\) and \(\text{SO}_4^{2-}\)

What happens in the oxidation number of Mn during working of dry cell.

• (A) Increases by 1
• (B) decreases by 1
• (C) Increases by 2
• (D) decreases by 2

What is the formula of Tin (IV) oxide?

• (A) \(\text{Sn}_2\text{O}_3\)
• (B) \(\text{SnO}_2\)
• (C) \(\text{SnO}_4\)
• (D) \(\text{SnO}\)

Identify the name of second higher homologue of formic acid?

• (A) Methanoic acid
• (B) Ethanoic acid
• (C) Propanoic acid
• (D) Butanoic acidaa

What is the of number of chiral carbon atoms present in 2-Bromo-3,4,5-trimethylhexane ?

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

Which of the following alkyl chlorides on Wurtz reaction gives 2, 2, 5, 5- tetramethylhexane ?

• (A) \((\text{CH}_3)_2\text{CH}-\text{CH}_2-\text{Cl}\)
• (B) \((\text{CH}_3)_3\text{C}-\text{CH}_2-\text{Cl}\)
• (C) \((\text{CH}_3)_2\text{CH}-\underset{\underset{\text{CH}_3}{|}}{\text{CH}}-\text{Cl}\)
• (D) \(\text{CH}_3-\text{CH}_2-\underset{\underset{\text{Cl}}{|}}{\text{CH}}-\text{CH}_3\)

Calculate the number of atoms present in \(1 \text{g}\) metal that forms simple unit cell structure if product of density and volume of unit cell is \(66.5\times 10^{-24} \text{g}\)

• (A) \(1.504\times 10^{22}\)
• (B) \(3.419\times 10^{22}\)
• (C) \(2.516\times 10^{22}\)
• (D) \(2.018\times 10^{22}\)

Calculate the void volume in fcc unit cell if total volume of a unit cell is \(6.4\times 10^{-23} \text{cm}^3\) ?

• (A) \(3.172\times 10^{-23} \text{cm}^3\)
• (B) \(3.911\times 10^{-23} \text{cm}^3\)
• (C) \(1.664\times 10^{-23} \text{cm}^3\)
• (D) \(2.051\times 10^{-23} \text{cm}^3\)

What is the total number of Bravais lattices in orthorhombic crystal system?

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

\(40\) gram nonelectrolyte solute having molar mass \(180 \text{g mol}^{-1}\) dissolved in water has osmotic pressure \(2 \text{atm}\) at \(300 \text{K}\). Calculate the volume of solution. (\(\text{R} = 0.0821 \text{atm mol}^{-1}\text{K}^{-1}\))

• (A) \(2.10 \text{dm}^3\)
• (B) \(2.34 \text{dm}^3\)
• (C) \(2.74 \text{dm}^3\)
• (D) \(3.40 \text{dm}^3\)

Calculate the molality of solution of nonvolatile solute if boiling point of solution, molal elevation constant for solvent are \(319.8 \text{K}\) and \(2.5 \text{K kg mol}^{-1}\) respectively [Boiling point of solvent = \(319.5 \text{K}\)]

• (A) \(0.12 \text{mol kg}^{-1}\)
• (B) \(0.14 \text{mol kg}^{-1}\)
• (C) \(0.17 \text{mol kg}^{-1}\)
• (D) \(0.21 \text{mol kg}^{-1}\)

Identify correct statements from following regarding \(0.2 \text{M}\) urea and \(0.2 \text{M}\) sucrose solutions.

• (A) The urea solution is hypertonic to sucrose solution.
• (B) The osmotic pressure of urea solution is higher than that of sucrose solution.
• (C) These solutions exhibit same osmotic pressure.
• (D) The sucrose solution is hypotonic to urea solution.

For a certain redox reaction in galvanic cell \(\text{X}_{(s)}+\text{Y}_{(aq)}^{2+}⟶\text{X}_{(aq)}^{2+}+\text{Y}_{(s)}\), \(\text{E}^{\circ}\) cell is \(0.0296 \text{V}\) at \(298 \text{K}\). What is equilibrium constant of reaction?

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

Which of the following net cell reactions occurs in a galvanic cell containing cadmium electrode and standard hydrogen electrode?
\(\text{E}_{(\text{Cd}_{(aq)}^{2+}|\text{Cd}_{(s)})}^{\circ} = -0.403 \text{V}\).

• (A) \(\text{H}_{2(g)}+\text{Cd}_{(aq)}^{2+}⟶2\text{H}_{(aq)}^++\text{Cd}_{(s)}\)
• (B) \(\text{Cd}_{(s)}+2\text{H}_{(aq)}^+⟶\text{Cd}_{(aq)}^{2+}+\text{H}_{2(g)}\)
• (C) \(2\text{H}_{2(g)}+\text{Cd}_{(aq)}^{2+}⟶4\text{H}_{(aq)}^++\text{Cd}_{(s)}\)
• (D) \(2\text{Cd}_{(s)}+2\text{H}_{(aq)}^+⟶2\text{Cd}_{(aq)}^{2+}+\text{H}_{2(g)}\)

In a first order reaction 20 millimole of reactant is lowered to 10 millimole in \(0.3010\) minute. Find rate constant of the reaction?

• (A) \(3.010 \text{minute}^{-1}\)
• (B) \(0.602 \text{minute}^{-1}\)
• (C) \(2.303 \text{minute}^{-1}\)
• (D) \(0.301 \text{minute}^{-1}\)

For the reaction \(2\text{NOBr}_{(g)}⟶2\text{NO}_{(g)}+\text{Br}_{2(g)}\), rate law is \(r = k[\text{NOBr}]^2\). Rate constant is \(1.62 \text{M s}^{-1}\) and concentration of NOBr is \(5\times 10^{-3} \text{M}\),
What is rate of reaction?

• (A) \(4.05\times 10^{-5} \text{M s}^{-1}\)
• (B) \(6.48\times 10^{-6} \text{M s}^{-1}\)
• (C) \(2.12\times 10^{-5} \text{M s}^{-1}\)
• (D) \(8.63\times 10^{-6} \text{M s}^{-1}\)

Which of the following reactions has of overall order \(1.5\) ?

• (A) \(2\text{H}_2\text{O}_{2(g)}⟶2\text{H}_2\text{O}_{(\ell )}+\text{O}_{2(g)}\)
• (B) \(\text{H}_{2(g)}+\text{I}_{2(g)}⟶2\text{HI}_{(g)}\)
• (C) \(\text{CH}_3\text{CHO}_{(g)}⟶\text{CH}_{4(g)}+\text{CO}_{(g)}\)
• (D) \(2\text{NO}_{(g)}+2\text{H}_{2(g)}⟶\text{N}_{2(g)}+2\text{H}_2\text{O}_{(\ell )}\)

Which from following anions has greater power for coagulation of positive sol?

• (A) \(\text{Cl}^-\)
• (B) \([\text{Fe(CN)}_6]^{4-}\)
• (C) \(\text{PO}_4^{2-}\)
• (D) \(\text{SO}_4^{2-}\)

Which actinoid from following has smallest ionic size in \(+3\) state?

• (A) Pa
• (B) Cm
• (C) Fm
• (D) Lr

What is the total number of transition series of d-block elements?

• (A) \(3\)
• (B) \(4\)
• (C) \(10\)
• (D) \(12\)

Identify a ligand with highest field strength among the following.

• (A) \(\text{H}_2\text{O}\)
• (B) \(\text{NH}_3\)
• (C) \(\text{S}^{2-}\)
• (D) \(\text{NCS}^-\)

Find EAN of Fe in \(\text{Fe(CO)}_5\)

• (A) \(35\)
• (B) \(36\)
• (C) \(26\)
• (D) \(30\)

The reaction of aryl halide with sodium metal in dry ether to give biphenyl is known as

• (A) Swartz reaction
• (B) Wurtz reaction
• (C) Fittig reaction
• (D) Wrutz-fittig reaction

Identify the product P obtained in following reaction

• (A)

• (B)

• (C)

• (D)

Which among the following has lowest boiling point?

• (A) n - Butyl alcohol
• (B) Isobutyl alcohol
• (C) sec-Butyl alcohol
• (D) tert-Butyl alcohol

Which from following is obtained by hydroboration - oxidation of propene?

• (A) Propanal
• (B) Propanone
• (C) Propan-1-ol
• (D) Propan-2-ol

Identify the reagent 'R' used in the following reaction
\(\text{C}_6\text{H}_5\text{COCl}\overset{\text{R}\,}{\rightarrow }\text{C}_6\text{H}_5\text{CHO}+\text{HCl}\)

• (A) \(\text{CO},\text{HCl}\)
• (B) \(\text{H}_2,\text{Pd}-\text{BaSO}_4\)
• (C) \(\text{DIBAl}-\text{H}\)
• (D) \(\text{H}_2\text{O}\)

Identify the product 'B' in the following reaction
\(\text{Isopropyl cyanide}\overset{\text{SnCl}_2,\text{HCl}\,}{\rightarrow }\text{A}\overset{\text{H}_3\text{O}^+\,}{\rightarrow }\text{B}+\text{NH}_4\text{Cl}\)

• (A) Propanal
• (B) Propanone
• (C) 2-Methylpropanal
• (D) 2-Methylpropanoic acid

Benzonitrile on reaction with a reagent in dry ether followed by hydrolysis forms benzophenone along with ammonia and hydroxymagnesium bromide. Identify the reagent used in above transformation.

• (A) \(\text{C}_6\text{H}_5\text{MgCl}\)
• (B) \(\text{C}_2\text{H}_5\text{MgBr}\)
• (C) \(\text{C}_6\text{H}_5\text{MgI}\)
• (D) \(\text{C}_6\text{H}_5\text{MgBr}\)

Propyne on reaction with water in presence of \(40\%\) sulphuric acid and \(1\%\) mercuric sulphate forms

• (A) Propan-1-ol
• (B) Propan-2-ol
• (C) Propanone
• (D) Propanal

Identify final products in the following sequence of reactions
\(\text{RCOOH}+\text{R’OH}\overset{\text{H}^+\,}{\rightarrow }\text{intermediate}\overset{\text{Ni},\Delta \,}{\rightarrow }^{\text{H}_2}\text{Products}\)

• (A) \(\text{R’COOH}+\text{RCH}_2\text{OH}\)
• (B) \(\text{RCH}_2\text{OH}+\text{R’}-\text{OH}\)
• (C) \(\text{R’CH}_2\text{OH}+\text{R}-\text{OH}\)
• (D) \(\text{R}-\text{OH}+\text{R’}-\text{OH}\)

Which from following amines does NOT form alkyl isocyanide when heated with alc. KOH and chloroform

• (A) Methanamine
• (B) Propan-2-amine
• (C) N, N-Diethylethanamine
• (D) Ethanamine

Which from following statements is NOT correct regarding Hofmann's exhaustive alkylation?

• (A) In this quaternary ammmonium salts are obtained.
• (B) Two moles of alkylhalide when heated with one mole primary amine forms tetraalkyl ammonium halide.
• (C) Depending on the quantity of alkyl halides treated with primary amine, the mixture of secondary and tertiary amines obtained.
• (D) The reaction is carried out in presence of \(\text{NaHCO}_3\)

Which from following is NOT a globular protein?

• (A) Serum albumin
• (B) Insulin
• (C) Egg albumin
• (D) Keratin

Which of the following has largest size?

• (A) Antibody
• (B) Virus
• (C) Bacteria
• (D) Cancer cell

What is the total mass of products obtained when one gram mole of sucrose is hydrolysed?

• (A) \(180 \text{g}\)
• (B) \(342 \text{g}\)
• (C) \(170 \text{g}\)
• (D) \(360 \text{g}\)

Which from following polymers is NOT obtained by addition polymerisation method?

• (A) Teflon
• (B) Polyacrolonitrile
• (C) LDP
• (D) PHBV

Which medicinal property is exhibited by methylsalicylate?

• (A) Analgesic
• (B) Antioxident
• (C) Antiseptic
• (D) Antimicrobial

The value of \((\frac{-1+i\sqrt{3}}{2})^{18}+(\frac{-1-i\sqrt{3}}{2})^{18}\) is

• (A) \(2\)
• (B) \(35\)
• (C) \(\sqrt{3}\)
• (D) \(\frac{\sqrt{3}}{2}\)

11 players of the Indian cricket team are sitting at a circular table. The number of ways they can sit so that two players, the wicket keeper and the captain never sit together is ___

• (A) \(10!\)
• (B) \(2\times 9!\)
• (C) \(8\times 9!\)
• (D) \(9\times 9!\)

In \(△ABC\), if \(\angle C = \frac{π}{3}\), then the value of \(cos^2A+cos^2B+cosAcosB\) is...

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

The line \(L_1\) given by \(\frac{x}{p}+\frac{y}{2} = 1\) passes through the point \((5,0)\). The line \(L_2\) given by \(\frac{x}{10}+\frac{y}{q} = 1\) is parallel to \(L_1\). Then the distance between the lines \(L_1\) and \(L_2\) is...

• (A) \(\frac{10}{\sqrt{29}}\)
• (B) \(\frac{19}{2\sqrt{29}}\)
• (C) \(\frac{5}{\sqrt{41}}\)
• (D) \(\frac{10}{\sqrt{41}}\)

Two lines are given by \(x^2-4xy+4y^2+kx-2ky = 0\), then the value of k so that the distance between them is 3 is

• (A) \(\sqrt{5}\)
• (B) \(3\sqrt{3}\)
• (C) \(\sqrt{3}\)
• (D) \(3\sqrt{5}\)

The area of the region (in sq. unit) bounded by x-axis, the tangent and normal to the circle \(x^2+y^2 = 4\), drawn at a point \((1,\sqrt{3})\) is

• (A) \(5\)
• (B) \(\sqrt{3}\)
• (C) \(2\)
• (D) \(2\sqrt{3}\)

The tangent to the circle \(x^2+y^2 = 10\) at the point \((3,1)\) touches the circle \(x^2+y^2-2\sqrt{10}\,x-20y+k = 0\), then the value of \(k\) is...

• (A) \(-109\)
• (B) \(109\)
• (C) \(-101\)
• (D) \(101\)

If the eccentricity of the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1,(a > b)\) is \(\frac{2}{3}\) and its focal chord is \(3x+2y-6 = 0\), then the value of \(a^2+b^2\) is...

• (A) \(11\)
• (B) \(12\)
• (C) \(13\)
• (D) \(14\)

If \(\underset{x\rightarrow \infty }{lim}[\frac{x^2+x+1}{x+1}-ax-b] = 3\), then \(a-b =\)...

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

The correct logical equivalence from the following is /are ___
(I) \(p\rightarrow (q\rightarrow r)\equiv (p∧q)\rightarrow r\)
(II) \((p\rightarrow q)\rightarrow r\equiv p\rightarrow (q∨r)\)
(III) \((p\rightarrow q)\rightarrow r\equiv (p\rightarrow r)∧(\sim q\rightarrow r)\)
(IV) \(p\rightarrow (q\rightarrow r)\equiv q\rightarrow (p\rightarrow r)\)

• (A) only (I) and (II)
• (B) only (III) and (IV)
• (C) only (II) and (IV)
• (D) only (I) and (IV)

The contrapositive of the statement pattern \([p∨(p\rightarrow q)]\rightarrow (p∧\sim q)\) is

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

The negation of the converse of \(p∨q\) is

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

In triangle ABC, with usual notations, if \((a+b+c)(a+b-c) = ab\), then the measure of angle C is...

• (A) \(\frac{π}{2}\)
• (B) \(\frac{2π}{3}\)
• (C) \(\frac{5π}{6}\)
• (D) \(\frac{3π}{4}\)

In \(△ABC\), with usual notation, if \(a = 13,b = 14,c = 15\), then the sum of the values of \(sin(\frac{A}{2})\) and \(sinA\) is....

• (A) \(\frac{14}{5\sqrt{5}}\)
• (B) \(\frac{\sqrt{5}+4}{5}\)
• (C) \(\frac{5+\sqrt{5}}{5}\)
• (D) \(\frac{2}{\sqrt{5}}+\frac{1}{2}\)

Let \(A = \left[ \begin{array}{ccc}cosα & -sinα & 0 \\ sinα & cosα & 0 \\ 0 & 0 & 1\end{array} \right]\). If \(B = \text{adj}\,A\), then the matrix \(B^{-1}\) is equal to...

• (A) \(I\)
• (B) \(A^{-1}\)
• (C) \(-A\)
• (D) \(A\)

Let \(A = [\begin{array}{cc}3 & -4 \\ 1 & -1\end{array}]\) and \(B = [\begin{array}{cc}6 & -13 \\ 5 & -10\end{array}]\) be two matrices. If the variables \(x\) and \(y\) satisfy the matrix equation \(((A^{-1})^2+B)[\begin{array}{c}x \\ y\end{array}] = [\begin{array}{c}0 \\ 0\end{array}]\), then the ordered pair \((x,y) =\)

• (A) \((3,5)\)
• (B) \((10,7)\)
• (C) \((4,6)\)
• (D) \((5,3)\)

If \(0\leq x\leq 1\), \(I_1 = \int sin^{-1}\sqrt{1-x^2}\,dx\) and \(I_2 = \int sin^{-1}x\,dx\), then which of the following is true?

• (A) \(I_1 = I_2\)
• (B) \(I_1 = \frac{π}{2}I_2\)
• (C) \(I_1+I_2 = \frac{π}{2}x\)
• (D) \(I_1+I_2 = \frac{π}{2}\)

Let \(t\in (0,1)\) and \(α\in (0,\frac{π}{4})\). If \(x = \text{cosec}^{-1}(\frac{1+t^2}{2t})\), \(y = cot^{-1}(\frac{\sqrt{1-t^2}}{t})\) and \(\frac{dy}{dx} = f(t)\), then the value of \(f(tanα)\) is

• (A) \(\frac{secα}{\sqrt{cos2α}}\)
• (B) \(\frac{secα}{2\sqrt{cos2α}}\)
• (C) \(\frac{cosα}{2\sqrt{cos2α}}\)
• (D) \(\frac{cosα}{\sqrt{cos2α}}\)

If \(y = tan^{-1}⎛⎝\frac{log(\frac{e}{x^3})}{logex^3}⎞⎠+tan^{-1}⎛⎝\frac{log(e^4x^3)}{log(\frac{e}{x^{12}})}⎞⎠\), \(x\in (e^{-\frac{1}{3}},e^{\frac{1}{12}})\) then \(\frac{dy}{dx}\) is equal to...

• (A) \(1\)
• (B) \(0\)
• (C) \(-1\)
• (D) \(\frac{1}{e}\)

If \(0\leq x\leq 1\) and \((sin^{-1}x)^3+(cos^{-1}x)^3 = aπ^3\) then

• (A) \(a\geq \frac{1}{32}\)
• (B) \(a\geq \frac{1}{16}\)
• (C) \(a\leq \frac{1}{32}\)
• (D) \(a\leq \frac{1}{16}\)

If \(\sum _{n = 1}^{2026}tan^{-1}(\frac{1}{n^2+n+1}) = tan^{-1}(1-\frac{1}{x})\), where \(x\neq 0\), then \(x =\)

• (A) \(2028\)
• (B) \(2026\)
• (C) \(1014\)
• (D) \(1013\)

Let \(f(x) = ax+b\) and \(g(x) = cx+d\). The condition \(f(g(x)) = g(f(x))\) holds for all \(x\) if and only if ...

• (A) \(f(a) = f(c)\)
• (B) \(f(b) = g(b)\)
• (C) \(f(d) = g(b)\)
• (D) \(f(c) = g(a)\)

The number of point / points where the function \(f(x) = \frac{1}{x^2-5|x|+6}\) is discontinuous is......

• (A) \(0\)
• (B) \(1\)
• (C) \(2\)
• (D) \(4\)

If \(x\,e^{xy} = y+sin^2x\), then the value of \(\frac{dy}{dx}\) at \(x = 0\) is equal to

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

If \(g(x) = (x^2+2x+1)\cdot f(x)\) such that \(f(0) = 5\) and \(\underset{x\rightarrow 0}{lim}\frac{f(x)-5}{x} = 4\) then \(g^'(0) =\)

• (A) \(20\)
• (B) \(12\)
• (C) \(18\)
• (D) \(14\)

If the line \(x+By+C = 0\) is the normal to the curve given by \(x = asin^3t\), \(y = bcos^3t\), (where \(a,b\neq 0\)) at a point \(t = \frac{π}{2}\), then \(B-C =\)

• (A) \(a\)
• (B) \(2a\)
• (C) \(-a\)
• (D) \(0\)

If the tangent to the curve \(xy+ax+by = 0\) at \((1,1)\) makes an angle of \(tan^{-1}2\) with positive direction of the \(x\)-axis, then the value of \(\frac{ab}{a+b}\) is...

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

If the side of an equilateral triangle increases at the rate of \(\sqrt{3} \text{cm/sec}\), then the rate of change of increase of its area when the side is \(12 \text{cm}\) is ____

• (A) \(18 \text{cm}^2/\text{sec}\)
• (B) \(10 \text{cm}^2/\text{sec}\)
• (C) \(12 \text{cm}^2/\text{sec}\)
• (D) \(3\sqrt{3} \text{cm}^2/\text{sec}\)

\(\int sin(logx)\,dx =\)

• (A) \(\frac{x}{2}[sin(logx)-cos(logx)]+c\)
• (B) \(\frac{x}{2}[sin(logx)+cos(logx)]+c\)
• (C) \(\frac{x}{2}[cos(logx)-sin(logx)]+c\)
• (D) \(\frac{x}{4}[cos(logx)-sin(logx)]+c\)

The value of \(\int \frac{n\sqrt{\text{cosec}^2x^n-1}}{x^{(1-n)}}\,dx\) is ...

• (A) \(log(sinx)+c\)
• (B) \(log(sinx^n)+c\)
• (C) \(log(cotx^n)+c\)
• (D) \(log(cosx^n)+c\)

\(\int \frac{(x+1)\,dx}{x(1+xe^x)} = \ldots \ldots\)

• (A) \(log\frac{xe^x}{1+xe^x}+c\)
• (B) \(log\frac{1+xe^x}{xe^x}+c\)
• (C) \(log\frac{(x+1)e^x}{xe^x}+c\)
• (D) \(log\frac{(x+1)e^x}{1+xe^x}+c\)

The value of the definite integral \(\int _0^{log_e5}\frac{e^x\sqrt{e^x-1}}{e^x+3}\,dx\) is equal to...

• (A) \(5+π\)
• (B) \(5-π\)
• (C) \(4+π\)
• (D) \(4-π\)

If \([\cdot ]\) denotes the greatest integer function, then \(\int _0^{π/3}[tanx]\,dx =\)

• (A) \(\frac{π}{12}\)
• (B) \(\frac{π}{6}\)
• (C) \(\frac{π}{4}\)
• (D) \(\frac{π}{3}\)

The area (in square units) bounded by the line \(y = x\), the X-axis and the lines \(x = -2\) and \(x = 4\) is...

• (A) \(6\)
• (B) \(\frac{15}{2}\)
• (C) \(\frac{17}{2}\)
• (D) \(10\)

The equation of the curve whose slope is \(\frac{y-1}{x^2+x}\) and which passes through the point \((1,0)\) is

• (A) \(xy-x-y-1 = 0\)
• (B) \((y-1)(x+1) = 2x\)
• (C) \(xy+x+y-1 = 0\)
• (D) \(y(x+1)-x+1 = 0\)

A body cools from \(100^{\circ}\text{C}\) to \(60^{\circ}\text{C}\) in 20 minutes, the temperature of the surroundings being \(20^{\circ}\text{C}\). The total time taken (in minutes) for the body to cool down to \(40^{\circ}\text{C}\) is ...

• (A) \(60\)
• (B) \(40\)
• (C) \(50\)
• (D) \(30\)

The general solution of the differential equation \(xsinx\frac{dy}{dx}+(xcosx+sinx)y = sinx\) is

• (A) \(xsinx+ycosx = c\)
• (B) \(ysinx+xcosx = c\)
• (C) \(xysinx-cosx = c\)
• (D) \(xysinx+cosx = c\)

If \(\overset{̄}{a} = \hat{i}-\hat{k}\), \(\overset{̄}{b} = x\hat{i}+\hat{j}+(1-x)\hat{k}\) and \(\overset{̄}{c} = y\hat{i}+x\hat{j}+(1+x-y)\hat{k}\) then \([\overset{̄}{a} \overset{̄}{b} \overset{̄}{c}]\) depends on

• (A) only x
• (B) neither x nor y
• (C) either x or y
• (D) only y

Let \(\overset{̄}{u},\overset{̄}{v},\overset{̄}{w}\) be three vectors such that \(|\overset{̄}{u}| = 1,|\overset{̄}{v}| = 2,|\overset{̄}{w}| = 3\). If the projection of \(\overset{̄}{v}\) along \(\overset{̄}{u}\) is equal to the projection of \(\overset{̄}{w}\) along \(\overset{̄}{u}\) and \(\overset{̄}{v},\overset{̄}{w}\) are perpendicular to each other, then \(|\overset{̄}{u}-\overset{̄}{v}+\overset{̄}{w}| =\)...

• (A) \(4\)
• (B) \(\sqrt{7}\)
• (C) \(2\)
• (D) \(\sqrt{14}\)

Let \(\overset{̄}{a},\overset{̄}{b},\overset{̄}{c}\) be three vectors of equal magnitude such that the angle between \(\overset{̄}{a}\) and \(\overset{̄}{b}\) is \(α\), \(\overset{̄}{b}\) and \(\overset{̄}{c}\) is \(β\), \(\overset{̄}{c}\) and \(\overset{̄}{a}\) is \(γ\).
Then the minimum value of \(cosα+cosβ+cosγ\) is ...

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

A vector which is orthogonal to the vector \(\overset{̄}{a} = \hat{i}+2\hat{j}+3\hat{k}\) and coplanar with the vectors \(\overset{̄}{b} = 3\hat{i}+2\hat{j}\) and \(\overset{̄}{c} = 2\hat{i}+\hat{j}+3\hat{k}\) is

• (A) \(25\hat{i}+19\hat{j}-21\hat{k}\)
• (B) \(-25\hat{i}+19\hat{j}-21\hat{k}\)
• (C) \(-25\hat{i}+19\hat{j}+21\hat{k}\)
• (D) \(25\hat{i}+19\hat{j}+21\hat{k}\)

The line \(\ell\) passes through the point \((2,1,1)\) and is parallel to the plane \(x+y+2z = 18\). If line \(\ell\) intersects the line \(\frac{x+2}{3} = \frac{y+1}{-1} = \frac{z-2}{1}\), then equation of the line \(\ell\) is...

• (A) \(\frac{x-2}{3} = \frac{y-1}{1} = \frac{z-1}{-2}\)
• (B) \(\frac{x-3}{1} = \frac{y-2}{3} = \frac{z-2}{-2}\)
• (C) \(\frac{x-2}{3} = \frac{y-1}{-1} = \frac{z-1}{-1}\)
• (D) \(\frac{x-3}{1} = \frac{y-4}{3} = \frac{z+1}{-2}\)

The acute angle \(θ\) between the \(xy\)-plane and the plane passing through the point \((1,2,4)\) and parallel to the vectors with direction ratios \(3,2,-1\) and \(1,-2,-2\) is...

• (A) \(cos^{-1}(\frac{8}{5\sqrt{5}})\)
• (B) \(cos^{-1}(\frac{5}{8\sqrt{5}})\)
• (C) \(sin^{-1}(\frac{8}{5\sqrt{5}})\)
• (D) \(\frac{π}{4}\)

If the plane \(\overset{̄}{r} = (λ+μ)\hat{i}+(2+μ)\hat{j}+(3λ+2μ)\hat{k}\), where \(λ\) and \(μ\) are parameters, intersects coordinate axes at points \((a,0,0)\), \((0,b,0)\) and \((0,0,c)\) then \(a+b+c =\)...

• (A) \(\frac{7}{2}\)
• (B) \(5\)
• (C) \(\frac{8}{3}\)
• (D) \(\frac{10}{3}\)

The coordinates of the point of intersection of the lines \(\frac{x-3}{1} = \frac{y-5}{2} = \frac{z-1}{-1}\) and \(\frac{x-4}{2} = \frac{y-2}{-1} = \frac{z-4}{2}\) are...

• (A) \((2,3,2)\)
• (B) \((-2,3,-2)\)
• (C) \((-2,-3,2)\)
• (D) \((2,3,-2)\)

If the plane \(2x+3y+z = 6\) cuts coordinate axes at A, B and C, then the volume of tetrahedron OABC (where O is the origin) is ......cubic units.

• (A) \(30\)
• (B) \(6\)
• (C) \(36\)
• (D) \(180\)

The maximum value of \(Z = 4x+5y\), subject to the constraints \(3x+y\leq 15,3x+4y\leq 24,x\geq 0,y\geq 0\) is

• (A) \(31\)
• (B) \(30\)
• (C) \(42\)
• (D) \(47\)

A fair coin is tossed 9 times. On each toss, a man predicts that the outcome will be heads. The probability that the number of successful predictions is strictly greater than the number of unsuccessful predictions is...

• (A) \(\frac{3}{4}\)
• (B) \(\frac{1}{6}\)
• (C) \(\frac{1}{2}\)
• (D) \(-\frac{1}{2}\)

Let \(F(x)\) be the cumulative distribution function (c.d.f.) of a continuous random variable \(X\). If \(F(b) = 0.7\) and \(P(X > a) = 0.4\), then the value of \(P(a < X < b)\) is ...

• (A) \(0.1\)
• (B) \(0.2\)
• (C) \(0.3\)
• (D) \(0.5\)

If the probabilities of a student succeeding in the entrance tests for institutes A, B and C are \(0.6\), \(0.5\) and \(0.4\) respectively, while the probability of succeeding in both A and B is \(0.3\), in both B and C is \(0.2\), in both A and C is \(0.2\), and in all three is \(0.1\), then the probability that the student succeeds in exactly one of these tests is......

• (A) \(0.3\)
• (B) \(0.4\)
• (C) \(0.5\)
• (D) \(0.6\)