VITEEE 2023 Question Paper PDF (Available) : Download Solution PDF with Answer Key

VITEEE 2023 Question Paper With Solution

PHYSICS

Question 1:

Light of wavelength \( \lambda_A \) and \( \lambda_B \) falls on two identical metal plates A and B respectively. The maximum kinetic energy of photoelectrons is \( K_A \) and \( K_B \) respectively. Given that \( \lambda_A = 2\lambda_B \), which one of the following relations is true?

• (A) \( K_A < \frac{K_B}{2} \)
• (B) \( 2K_A = K_B \)
• (C) \( K_A = 2K_B \)
• (D) \( K_A > 2K_B \)

Which one of the following curves represents the variation of impedance \( (Z) \) with frequency \( f \) in a series LCR circuit?

• (A)
• (B) 
• (C) 
• (D) 

A Carnot engine takes \( 3 \times 10^6 \) cal of heat from a reservoir at \( 627^\circ C \), and gives it to a sink at \( 27^\circ C \). The work done by the engine is:

• (A) \( 4.2 \times 10^6 \) J
• (B) \( 8.4 \times 10^6 \) J
• (C) \( 16.8 \times 10^6 \) J
• (D) \( 0 \)

An element of \( 0.05 \) m is placed at the origin, carrying a large current of \( 10 A \). The magnetic field at a perpendicular distance of \( 1 \) m is:

• (A) \( 4.5 \times 10^{-8} \) T
• (B) \( 5.5 \times 10^{-8} \) T
• (C) \( 5.0 \times 10^{-8} \) T
• (D) \( 7.5 \times 10^{-8} \) T

A sinusoidal voltage of amplitude \( 25 \) V and frequency \( 50 \) Hz is applied to a half-wave rectifier using a P-N junction diode. No filter is used, and the load resistor is \( 1000\Omega \). The forward resistance \( R_f \) of the ideal diode is \( 10\Omega \). The percentage rectifier efficiency is:

• (A) \( 40% \)
• (B) \( 20% \)
• (C) \( 30% \)
• (D) \( 15% \)

A flask contains a monoatomic and a diatomic gas in the ratio of \( 4:1 \) by mass at a temperature of \( 300K \). The ratio of average kinetic energy per molecule of the two gases is:

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

The potential energy of a particle \( U(x) \) executing simple harmonic motion is given by:

• (A) \( U(x) = \frac{k}{2} (x - a)^2 \)
• (B) \( U(x) = k_1 x + k_2 x^2 + k_3 x^3 \)
• (C) \( U(x) = A e^{-bx} \)
• (D) \( U(x) = \text{a constant} \)

Consider an electric field \( \mathbf{E} = E_0 \hat{x} \), where \( E_0 \) is a constant. The flux through the shaded area (as shown in the figure) due to this field is:

• (A) \( 2E_0 a^2 \)
• (B) \( \sqrt{2} E_0 a^2 \)
• (C) \( E_0 a^2 \)
• (D) \( \frac{E_0 a^2}{\sqrt{2}} \)

The equation of a wave on a string of linear mass density \( 0.04 \) kg/m is given by:
\[ y = 0.02 \sin 2\pi \left( \frac{t}{0.04} - \frac{x}{0.50} \right) \] The tension in the string is:

• (A) \( 4.0 N \)
• (B) \( 12.5 N \)
• (C) \( 0.5 N \)
• (D) \( 6.25 N \)

Equipotential surfaces are shown in the figure. The electric field strength will be:

• (A) \( 100 \) V/m along X-axis
• (B) \( 100 \) V/m along Y-axis
• (C) \( 200 \) V/m at an angle \( 120^\circ \) with X-axis
• (D) \( 50 \) V/m at an angle \( 120^\circ \) with X-axis

Water falls from a \( 40 \) m high dam at the rate of \( 9 \times 10^4 \) kg per hour. Fifty percent of gravitational potential energy can be converted into electrical energy. The number of \( 100W \) lamps that can be lit is:

• (A) \( 25 \)
• (B) \( 50 \)
• (C) \( 100 \)
• (D) \( 18 \)

An electron (mass = \( 9 \times 10^{-31} \) kg, charge = \( 1.6 \times 10^{-19} \) C) moving with a velocity of \( 10^6 \) m/s enters a magnetic field. If it describes a circle of radius \( 0.1 \) m, then the strength of the magnetic field must be:

• (A) \( 4.5 \times 10^{-5} \) T
• (B) \( 1.4 \times 10^{-5} \) T
• (C) \( 5.5 \times 10^{-5} \) T
• (D) \( 2.6 \times 10^{-5} \) T

If \( V_1 \) is the velocity of a body projected from point A and \( V_2 \) is the velocity of a body projected from point B, which is vertically below the highest point C, and if both the bodies collide, then:

• (A) \( V_1 = \frac{1}{2} V_2 \)
• (B) \( V_2 = \frac{1}{2} V_1 \)
• (C) \( V_1 = V_2 \)
• (D) \( V_1 = 3V_2 \)

A square frame of side \( 10 \) cm and a long straight wire carrying current \( 1A \) are in the plane of the paper. Starting from close to the wire, the frame moves towards the right with a constant speed of \( 10 \) m/s (see figure). The induced EMF at the time the left arm of the frame is at \( x = 10 \) cm from the wire is:

• (A) \( 2 \mu V \)
• (B) \( 1 \mu V \)
• (C) \( 0.75 \mu V \)
• (D) \( 0.5 \mu V \)

For the circuit shown in the figure, the current through the inductor is \( 0.9A \) while the current through the condenser is \( 0.4A \). Then:

• (A) Current drawn from source \( I = 1.13A \)
• (B) \( \omega = \frac{1}{1.5LC} \)
• (C) \( I = 0.5A \)
• (D) \( I = 0.6A \)

The ozone layer in the atmosphere absorbs:

• (A) Only the radio waves
• (B) Only the visible light
• (C) Only the \( \gamma \)-rays
• (D) X-rays and ultraviolet rays

The P-V diagram of a diatomic ideal gas system undergoing a cyclic process is shown in the figure. The work done during the adiabatic process \( CD \) is (Use \( \gamma = 1.4 \)):

• (A) \( -500J \)
• (B) \( 200J \)
• (C) \( -400J \)
• (D) \( 400J \)

In YDSE, how many maximas can be obtained on a screen, including central maxima, on both sides of the central fringe if \( \lambda = 3000\) Å, \( d = 5000\) Å?

• (A) 2
• (B) 5
• (C) 3
• (D) 1

A and B are two metals with threshold frequencies \( 1.8 \times 10^{14} \) Hz and \( 2.2 \times 10^{14} \) Hz. Two identical photons of energy \( 0.825 \) eV each are incident on them. Then photoelectrons are emitted in (Take \( h = 6.6 \times 10^{-34} \) Js):

• (A) B alone
• (B) A alone
• (C) Neither A nor B
• (D) Both A and B

A sinusoidal voltage of amplitude 25 V and frequency 50 Hz is applied to a half-wave rectifier using a P-N junction diode. No filter is used, and the load resistor is \( 1000\Omega \). The forward resistance \( R_f \) of the ideal diode is \( 10\Omega \). The percentage rectifier efficiency is:

• (A) \( 40% \)
• (B) \( 20% \)
• (C) \( 30% \)
• (D) \( 15% \)

The force between two short bar magnets with magnetic moments \( M_1 \) and \( M_2 \) whose centers are \( r \) meters apart is 8 N when their axes are in the same line. If the separation is increased to \( 2r \), the force between them is reduced to:

• (A) \( 4N \)
• (B) \( 2N \)
• (C) \( 1N \)
• (D) \( 0.5N \)

In a Rutherford scattering experiment, when a projectile of charge \( Z_1 \) and mass \( M_1 \) approaches a target nucleus of charge \( Z_2 \) and mass \( M_2 \), the distance of closest approach is \( r_0 \). The energy of the projectile is:

• (A) Directly proportional to \( Z_1Z_2 \)
• (B) Inversely proportional to \( Z_1 \)
• (C) Directly proportional to mass \( M_1 \)
• (D) Directly proportional to \( M_1 \times M_2 \)

What will be the maximum speed of a car on a road turn of radius 30m if the coefficient of friction between the tyres and the road is 0.4? (Take \( g = 9.8 \text{ m/s}^2 \))

• (A) \( 10.84 \text{ m/s} \)
• (B) \( 9.84 \text{ m/s} \)
• (C) \( 8.84 \text{ m/s} \)
• (D) \( 6.84 \text{ m/s} \)

A person aiming to reach the exactly opposite point on the bank of a stream is swimming with speed of \( 0.5 \) m/s at an angle of \( 120^\circ \) with the direction of flow of water. The speed of water in the stream is:

• (A) \( 1 \) m/s
• (B) \( 0.5 \) m/s
• (C) \( 0.25 \) m/s
• (D) \( 0.433 \) m/s

A car moves at a speed of \( 20 \text{ m/s} \) on a banked track and describes an arc of a circle of radius \( 40\sqrt{3} \) m. The angle of banking is: (Take \( g = 10 \text{ m/s}^2 \))

• (A) \( 25^\circ \)
• (B) \( 60^\circ \)
• (C) \( 45^\circ \)
• (D) \( 30^\circ \)

A force \( \mathbf{F} = \alpha \hat{i} + 3 \hat{j} + 6 \hat{k} \) is acting at a point \( \mathbf{r} = 2 \hat{i} - 6 \hat{j} - 12 \hat{k} \). The value of \( \alpha \) for which angular momentum about the origin is conserved is:

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

A convex lens has power \( P \). It is cut into two halves along its principal axis. Further, one piece (out of the two halves) is cut into two halves perpendicular to the principal axis (as shown in figure). Choose the incorrect option for the reported pieces.

• (A) Power of \( L_1 = \frac{P}{2} \)
• (B) Power of \( L_2 = \frac{P}{2} \)
• (C) Power of \( L_3 = \frac{P}{2} \)
• (D) Power of \( L_1 = P \)

A ball of radius \( r \) and density \( \rho \) falls freely under gravity through a distance \( h \) before entering water. The velocity of the ball does not change even on entering water. If the viscosity of water is \( \eta \), the value of \( h \) is given by:

• (A) \( \frac{2}{9} \frac{r^2 (1 - \rho)}{\eta g} \)
• (B) \( \frac{2}{81} \frac{r^2 (\rho - 1)}{\eta g} \)
• (C) \( \frac{2}{81} \frac{r^4 (\rho - 1)}{\eta^2 g} \)
• (D) \( \frac{2}{9} \frac{r^4 (\rho - 1)}{\eta^2 g} \)

The pressure inside a tyre is 4 times that of the atmosphere. If the tyre bursts suddenly at temperature \( 300K \), what will be the new temperature?

• (A) \( 300(4)^{7/2} \)
• (B) \( 300(4)^{2/7} \)
• (C) \( 300(2)^{7/2} \)
• (D) \( 300(4)^{-27} \)

A parallel plate air capacitor of capacitance \( C \) is connected to a cell of emf \( V \) and then disconnected from it. A dielectric slab of dielectric constant \( K \), which can just fill the air gap of the capacitor, is now inserted in it. Which of the following is incorrect?

• (A) The energy stored in the capacitor decreases \( K \) times.
• (B) The change in energy stored is \( \frac{1}{2} C V^2 (1 - \frac{1}{K}) \).
• (C) The charge on the capacitor is not conserved.
• (D) The potential difference between the plates decreases \( K \) times.

A given ray of light suffers minimum deviation in an equilateral prism \( P \). Additional prisms \( Q \) and \( R \) of identical shape and of the same material as \( P \) are now added as shown in the figure. The ray will now suffer:

• (A) Greater deviation
• (B) No deviation
• (C) Same deviation as before
• (D) Total internal reflection

If \( m \) is magnetic moment and \( B \) is the magnetic field, then the torque is given by:

• (A) \(\vec{m} \vec B \)
• (B) \( \frac{\vec m}{\vec B} \)
• (C) \(\vec m \times \vec B \)
• (D) \( |\vec m||\vec B| \)

An \( \alpha \)-particle of 10 MeV collides head-on with a copper nucleus (\( Z = 29 \)) and is deflected back. The minimum distance of approach between the centers of the two is:

• (A) \( 8.4 \times 10^{-15} \) cm
• (B) \( 8.4 \times 10^{-15} \) m
• (C) \( 4.2 \times 10^{-15} \) m
• (D) \( 4.2 \times 10^{-15} \) cm

A planet in a distant solar system is 10 times more massive than Earth and its radius is 10 times smaller. Given that the escape velocity from Earth's surface is 11 km/s, the escape velocity from the planet’s surface would be:

• (A) \( 1.1 \) km/s
• (B) \( 11 \) km/s
• (C) \( 110 \) km/s
• (D) \( 0.11 \) km/s

In the given figure, two equal positive point charges \( q_1 = q_2 = 2.0 \mu C \) interact with a third point charge \( Q = 4.0 \mu C \). The magnitude and direction of the net force on \( Q \) is:

• (A) \( 0.23 N \) in the \( +x \)-direction
• (B) \( 0.46 N \) in the \( +x \)-direction
• (C) \( 0.23 N \) in the \( -x \)-direction
• (D) \( 0.46 N \) in the \( -x \)-direction

Which of the following sets of quantum numbers is correct for an electron in a 4f orbital?

• (A) \( n = 4, \, l = 3, \, m = +1, \, s = +\frac{1}{2} \)
• (B) \( n = 4, \, l = 4, \, m = -4, \, s = -\frac{1}{2} \)
• (C) \( n = 4, \, l = 3, \, m = +4, \, s = +\frac{1}{2} \)
• (D) \( n = 3, \, l = 2, \, m = -2, \, s = +\frac{1}{2} \)

Arrange the following in increasing order of ionic radii: \( C^{4-}, N^{3-}, F^{-}, O^{2-} \).

• (A) \( C^{4-} < N^{3-} < O^{2-} < F^{-} \)
• (B) \( N^{3-} < C^{4-} < O^{2-} < F^{-} \)
• (C) \( F^{-} < O^{2-} < N^{3-} < C^{4-} \)
• (D) \( O^{2-} < F^{-} < N^{3-} < C^{4-} \)

The bond dissociation energies of \( X_2, Y_2, \) and \( XY \) are in the ratio of 1:0.5:1. If \( \Delta H \) for the formation of \( XY \) is -200 kJ mol\(^{-1}\), what is the bond dissociation energy of \( X_2 \)?

• (A) \( 200 \) kJ mol\(^{-1} \)
• (B) \( 100 \) kJ mol\(^{-1} \)
• (C) \( 400 \) kJ mol\(^{-1} \)
• (D) \( 800 \) kJ mol\(^{-1} \)

Values of dissociation constant \( K_a \) are given as follows:

Correct order of increasing base strength of the conjugate bases \( \text{CN}^-, \text{F}^- \) and \( \text{NO}_2^- \) is:

• (A) \( \text{F}^- < \text{CN}^- < \text{NO}_2^- \)
• (B) \( \text{NO}_2^- < \text{CN}^- < \text{F}^- \)
• (C) \( \text{F}^- < \text{NO}_2^- < \text{CN}^- \)
• (D) \( \text{NO}_2^- < \text{F}^- < \text{CN}^- \)

The product(s) formed when diborane (\( B_2H_6 \)) is hydrolyzed is/are:

• (A) \( B_2O_3 \) and \( H_3BO_3 \)
• (B) \( B_2O_3 \) only
• (C) \( H_3BO_3 \) and \( H_2 \)
• (D) \( H_3BO_3 \) only

The compounds \( CH_3CH=CHCH_3 \) and \( CH_3CH_2CH=CH_2 \):

• (A) are tautomers
• (B) are position isomers
• (C) contain the same number of sp\(^3\)-sp\(^3\), sp\(^3\)-sp\(^2\), and sp\(^2\)-sp\(^2\) carbon-carbon bonds
• (D) are chain isomers

Choose the correct option for the following reactions.

• (A) \( A { and } B \text{ are both Markovnikov addition products.} \)
• (B) \( A { is Markovnikov product and } B { is anti-Markovnikov product.} \)
• (C) \( A { and } B { are both anti-Markovnikov products.} \)
• (D) \( B { is Markovnikov and } A { is anti-Markovnikov product.} \)

Which of the following exhibits Frenkel defects?

• (A) Sodium chloride
• (B) Silver bromide
• (C) Graphite
• (D) Diamond

An element X has a body-centred cubic (bcc) structure with a cell edge of 200 pm. The density of the element is 5 g cm\(^{-3}\). The number of atoms present in 300g of the element X is:
Given: Avogadro Constant, \( N_A = 6.0 \times 10^{23} \) mol\(^{-1} \).

• (A) \( 5N_A \)
• (B) \( 6N_A \)
• (C) \( 15N_A \)
• (D) \( 25N_A \)

On passing current through two cells, connected in series, containing solutions of \( \text{AgNO}_3 \) and \( \text{CuSO}_4 \), 0.18 g of Ag is deposited. The amount of Cu deposited is:

• (A) \( 0.529 \) g
• (B) \( 10.623 \) g
• (C) \( 0.0529 \) g
• (D) \( 1.2708 \) g

The limiting molar conductivities of \( HCl \), \( CH_3COONa \), and \( NaCl \) are respectively 425, 90, and 125 mho cm\(^2\) mol\(^{-1}\) at 25°C. The molar conductivity of 0.1M \( CH_3COOH \) solution is 7.8 mho cm\(^2\) mol\(^{-1}\) at the same temperature. The degree of dissociation of 0.1M acetic acid solution at the same temperature is:

• (A) \( 0.10 \)
• (B) \( 0.02 \)
• (C) \( 0.15 \)
• (D) \( 0.03 \)

The rate law for a reaction between the substances A and B is given by: {Rate} = k[A]^m[B]^n On doubling the concentration of A and halving the concentration of B, the ratio of the new rate to the earlier rate of the reaction will be:

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

In a reaction, the threshold energy is equal to:

• (A) Activation energy + normal energy of reactants
• (B) Activation energy - normal energy of reactants
• (C) Normal energy of reactants - activation energy
• (D) Average kinetic energy of molecules of reactants

Which property of white phosphorus is common to red phosphorus?

• (A) It burns when heated in air.
• (B) It reacts with hot caustic soda solution to give phosphine.
• (C) It shows chemiluminescence.
• (D) It is soluble in carbon disulphide.

\( XeO_4 \) molecule is tetrahedral having:

• (A) Two \( p\pi - d\pi \) bonds
• (B) One \( p\pi - d\pi \) bond
• (C) Four \( p\pi - d\pi \) bonds
• (D) Three \( p\pi - d\pi \) bonds

Cuprous ion is colourless while cupric ion is coloured because:

• (A) Both have half-filled p- and d-orbitals.
• (B) Cuprous ion has an incomplete d-orbital and cupric ion has a complete d-orbital.
• (C) Both have unpaired electrons in the d-orbitals.
• (D) Cuprous ion has a complete d-orbital and cupric ion has an incomplete d-orbital.

The reason for the greater range of oxidation states in actinoids is attributed to:

• (A) Actinoid contraction
• (B) \( 5f, 6d \) and \( 7s \) levels having comparable energies
• (C) \( 4f \) and \( 5d \) levels being close in energies
• (D) The radioactive nature of actinoids

The geometry and magnetic behaviour of the complex \([Ni(CO)_4]\) are:

• (A) Square planar geometry and diamagnetic
• (B) Tetrahedral geometry and diamagnetic
• (C) Tetrahedral geometry and paramagnetic
• (D) Square planar geometry and paramagnetic

Indicate the complex ion which shows geometrical isomerism.

• (A) \([Cr(H_2O)_4Cl_2]^+\)
• (B) \([Pt(NH_3)_3Cl]_2^-\)
• (C) \([Co(NH_3)_6]^{3+}\)
• (D) \([Co(CN)(NC)]^{3-}\)

Reaction of \( C_6H_5CH_2Br \) with aqueous sodium hydroxide follows:

• (A) SN1 mechanism
• (B) SN2 mechanism
• (C) Any of the above two depending upon the temperature of reaction
• (D) Saytzeff rule

What is the correct order of reactivity of alcohols in the following reaction? \[ R - OH + HCl \rightarrow R - Cl + H_2O \]

• (A) \( 1^\circ > 2^\circ > 3^\circ \)
• (B) \( 3^\circ > 2^\circ > 1^\circ \)
• (C) \( 1^\circ < 2^\circ < 3^\circ \)
• (D) \( 3^\circ > 1^\circ > 2^\circ \)

Which of the following cannot be made by using Williamson's synthesis?

• (A) Methoxybenzene
• (B) Benzyl p-nitrophenyl ether
• (C) Methyl tertiary butyl ether
• (D) Di-tert-butyl ether

Which of the following reactions will yield benzaldehyde as a product?

 Choose the correct answer from the following options:

In Clemmensen reduction, carbonyl compounds are treated with:

• (A) Zinc amalgam + HCl
• (B) Sodium amalgam + HCl
• (C) Zinc amalgam + Nitric acid
• (D) Sodium amalgam + HNO\(_3\)

The correct increasing order of basic strength for the following compounds is: 

• (A) \( II < III < I \)
• (B) \( III < I < II \)
• (C) \( III < II < I \)
• (D) \( II < I < III \)

The major product of the following reaction is: 

• (A) 
• (B) 
• (C) 
• (D) 

Blister copper is:

• (A) Impure Cu
• (B) Cu alloy
• (C) Pure Cu
• (D) Cu having 1% impurity

\( P_A \) and \( P_B \) are the vapor pressures of pure liquid components A and B, respectively, in an ideal binary solution. If \( X_A \) represents the mole fraction of component A, the total pressure of the solution will be:

• (A) \( P_A + X_A (P_B - P_A) \)
• (B) \( P_B + X_A (P_B - P_A) \)
• (C) \( P_A + X_A (P_A - P_B) \)
• (D) \( P_B + X_A (P_A - P_B) \)

Which of the following complexes shows \( sp^3d^2 \) hybridization?

• (A) \([Cr(NO_2)_6]^{3-}\)
• (B) \([Fe(CN)_6]^{4-}\)
• (C) \([CoF_6]^{3-}\)
• (D) \([Ni(CO)_4]\)

2-Pentene contains:

• (A) 15 \( \sigma \)- and one \( \pi \)-bond
• (B) 14 \( \sigma \)- and one \( \pi \)-bond
• (C) 15 \( \sigma \)- and two \( \pi \)-bonds
• (D) 14 \( \sigma \)- and two \( \pi \)-bonds

For the below-given cyclic hemiacetal (X), the correct pyranose structure is: 

• (A)
• (B) 
• (C) 
• (D) 

Sucrose, which is dextrorotatory in nature, after hydrolysis gives glucose and fructose, among which: (i) Glucose is laevorotatory and fructose is dextrorotatory.
(ii) Glucose is dextrorotatory and fructose is laevorotatory.
(iii) The mixture is laevorotatory.
(iv) Both are dextrorotatory.

• (A) (i) and (iii)
• (B) (iii) and (iv)
• (C) (ii) and (iii)
• (D) (iii) only

The Allyl cyanide molecule contains:

• (A) 9 sigma bonds, 4 pi bonds, and no lone pair
• (B) 9 sigma bonds, 3 pi bonds, and one lone pair
• (C) 8 sigma bonds, 5 pi bonds, and one lone pair
• (D) 8 sigma bonds, 3 pi bonds, and two lone pairs

Which of the following pairs of compounds is isoelectronic and isostructural?

• (A) \( TeI_2, XeF_2 \)
• (B) \( IBr_2^-, XeF_2 \)
• (C) \( IF_5, XeF_5 \)
• (D) \( BeCl_2, XeF_2 \)

In which case does the change in entropy (\( \Delta S \)) become negative?

• (A) Evaporation of water
• (B) Expansion of a gas at constant temperature
• (C) Sublimation of solid to gas
• (D) \( 2H(g) \rightarrow H_2(g) \)

The argument of the complex number \[ \left( \frac{i}{2} - \frac{2}{i} \right) \] is equal to:

• (A) \( \frac{\pi}{4} \)
• (B) \( \frac{3\pi}{4} \)
• (C) \( \frac{\pi}{12} \)
• (D) \( \frac{\pi}{2} \)

The lines \[ p(p^2 + 1)x - y + q = 0 \quad \text{and} \quad (p^2 + 1)^2 x + (p^2 + 1)y + 2q = 0 \] are perpendicular to a common line for:

• (A) Exactly one value of \( p \)
• (B) Exactly two values of \( p \)
• (C) More than two values of \( p \)
• (D) No value of \( p \)

The probability that a card drawn from a pack of 52 cards will be a diamond or a king is:

• (A) \( \frac{1}{52} \)
• (B) \( \frac{2}{13} \)
• (C) \( \frac{4}{13} \)
• (D) \( \frac{1}{13} \)

If \( n(A) = 4 \) and \( n(B) = 7 \), then the difference between the maximum and minimum value of \( n(A \cup B) \) is:

• (A) 1
• (B) 2
• (C) 3
• (D) 4

The domain of the function \[ f(x) = \frac{1}{\sqrt{9 - x^2}} \] is:

• (A) \( -3 \leq x \leq 3 \)
• (B) \( -3 < x < 3 \)
• (C) \( -9 \leq x \leq 9 \)
• (D) \( -9 < x < 9 \)

If \[ \sin x + \cos x = \frac{1}{5} \] then \( \tan 2x \) is: \flushleft

• (A) \( \frac{25}{17} \)
• (B) \( \frac{7}{25} \)
• (C) \( \sqrt{\frac{25}{7}} \)
• (D) \( \frac{24}{7} \)

For the binary operation defined on \( \mathbb{R} - \{1\} \) such that: \[ a b = \frac{a}{b + 1} \] which of the following is true?

• (A) Not associative
• (B) Commutative
• (C) Not commutative
• (D) Both (A) and (B)

Evaluate: \[ \cos^{-1} \frac{1}{2} + \sin^{-1} (1) + \tan^{-1} \frac{1}{\sqrt{3}} \]

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

If \[ A = \begin{bmatrix} 1 & -1
2 & -1 \end{bmatrix}, \quad B = \begin{bmatrix} x & 1
y & -1 \end{bmatrix} \] and \[ (A + B)^2 = A^2 + B^2 \] then \( x + y \) is:

• (A) 2
• (B) 3
• (C) 4
• (D) 5

The determinant of the matrix: \[ \begin{bmatrix} -a^2 & ab & ac
ab & -b^2 & bc
ac & bc & -c^2 \end{bmatrix} \] is:

• (A) 0
• (B) \( abc \)
• (C) \( 4a^2b^2c^2 \)
• (D) None of these

If \[ A = \begin{bmatrix} \alpha & \beta
\gamma & \alpha \end{bmatrix} \] then \( \text{Adj} (A) \) is equal to:

• (A) \( \begin{bmatrix} \delta & -\gamma
  -\beta & \alpha \end{bmatrix} \)
• (B) \( \begin{bmatrix} \delta & -\beta
  -\gamma & \alpha \end{bmatrix} \)
• (C) \( \begin{bmatrix} -\delta & \beta
  \gamma & -\alpha \end{bmatrix} \)
• (D) \( \begin{bmatrix} -\delta & -\beta
  \gamma & \alpha \end{bmatrix} \)

If \[ \left| \frac{\sec(x - y)}{\sec(x + y)} \right| = a \] then \( \frac{dy}{dx} \) is:

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

The number of nonzero terms in the expansion of \[ (1 + 3\sqrt{2}x)^9 + (1 - 3\sqrt{2}x)^9 \] is:

• (A) 2
• (B) 3
• (C) 4
• (D) 5

If \[ \frac{a^n + b^n}{a^{n-1} + b^{n-1}} \] is the arithmetic mean (A.M.) between \( a \) and \( b \), then the value of \( n \) is:

• (A) 1
• (B) 2
• (C) 3
• (D) 4

The sum of the series \[ \frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \dots \] up to 15 terms is:

• (A) 1
• (B) 2
• (C) 3
• (D) 4

The equation of the circle with centre (0,2) and radius 2 is \[ x^2 + y^2 - my = 0. \] The value of \( m \) is:

• (A) 1
• (B) 2
• (C) 4
• (D) 3

The integral \[ \int x^n (1 + \log x) \, dx \] is equal to: \flushleft

• (A) \( x^n + C \)
• (B) \( x^{2x} + C \)
• (C) \( x^n \log x + C \)
• (D) \( \frac{1}{2}(1 + \log x)^2 + C \)

Evaluate the definite integral: \[ I = \int_{0}^{\frac{\pi}{2}} (\sqrt{\tan x} + \sqrt{\cot x})dx \]

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

The area of the region bounded by the ellipse \[ \frac{x^2}{16} + \frac{y^2}{9} = 1 \] is:

• (A) \( 12\pi \)
• (B) \( 3\pi \)
• (C) \( 24\pi \)
• (D) \( \pi \)

If the vertex of a parabola is \( (2, -1) \) and the equation of its directrix is \[ 4x - 3y = 21, \] then the length of its latus rectum is:

• (A) 2
• (B) 8
• (C) 12
• (D) 16

Eccentricity of ellipse \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] if it passes through point (9, 5) and (12, 4) is: \flushleft

• (A) \( \sqrt{\frac{3}{4}} \)
• (B) \( \sqrt{\frac{4}{5}} \)
• (C) \( \sqrt{\frac{5}{6}} \)
• (D) \( \sqrt{\frac{6}{7}} \)

In \triangle ABC \text{ the mid-point of the sides AB, BC and CA are respectively (1, 0, 0), (0, m, 0) \text{ and (0, 0, n). \text{ Then, \[ \frac{AB^2 + BC^2 + CA^2}{1^2 + m^2 + n^2} \text{ is equal to:} \] \flushleft

• (A) 8
• (B) 16
• (C) 9
• (D) 25

If \[ f(x) = \frac{x + |x|}{x} \] then the value of \[ \lim_{x \to 0} f(x) \] is:

• (A) 0
• (B) 2
• (C) Does not exist
• (D) None of these

Negation of the Boolean expression \[ p \Leftrightarrow (q \Rightarrow p) \] is:

• (A) \( \sim p \wedge q \)
• (B) \( p \wedge \sim q \)
• (C) \( \sim p \vee \sim q \)
• (D) \( \sim p \wedge \sim q \)

If \[ R = \{ (x, y) : x \text{ is exactly 7cm taller than } y \} \] then \( R \) is:

• (A) Not symmetric
• (B) Reflexive
• (C) Symmetric but not transitive
• (D) An equivalence relation

The particular solution of \[ \log \frac{dy}{dx} = 3x + 4y, \quad y(0) = 0 \] is:

• (A) \( e^{3x} + 3e^{-4y} = 4 \)
• (B) \( 4e^{3x} - 3e^{-4y} = 3 \)
• (C) \( 3e^{3x} + 4e^{4y} = 7 \)
• (D) \( 4e^{3x} + 3e^{-4y} = 7 \)

The general solution of the differential equation given by: \[ \tan^{-1} x + \tan^{-1} y = c \]

• (A) \( \frac{dy}{dx} = \frac{1 + y^2}{1 + x^2} \)
• (B) \( \frac{dy}{dx} = \frac{1 + x^2}{1 + y^2} \)
• (C) \( (1 + x^2) dy + (1 + y^2) dx = 0 \)
• (D) \( (1 + x^2) dx + (1 + y^2) dy = 0 \)

If \( |\mathbf{a}| = 3 \), \( |\mathbf{b}| = 4 \), then the value of \( \lambda \) for which \( \mathbf{a} + \lambda \mathbf{b} \) is perpendicular to \( \mathbf{a} - \lambda \mathbf{b} \) is:

• (A) \( \frac{9}{16} \)
• (B) \( \frac{3}{4} \)
• (C) \( \frac{3}{2} \)
• (D) \( \frac{4}{3} \)

The area of the parallelogram whose diagonals are \[ \mathbf{d_1} = \frac{3}{2} \hat{i} + \frac{1}{2} \hat{j} - \hat{k}, \quad \mathbf{d_2} = 2 \hat{i} - 6 \hat{j} + 8 \hat{k} \] is:

• (A) \( 5\sqrt{3} \)
• (B) \( 5\sqrt{2} \)
• (C) \( 25\sqrt{3} \)
• (D) \( 25\sqrt{2} \)

Bag P contains 6 red and 4 blue balls, and Bag Q contains 5 red and 6 blue balls. A ball is transferred from Bag P to Bag Q, and then a ball is drawn from Bag Q. What is the probability that the ball drawn is blue?

• (A) \( \frac{7}{15} \)
• (B) \( \frac{8}{15} \)
• (C) \( \frac{4}{19} \)
• (D) \( \frac{8}{19} \)

The mean and variance of a random variable \( X \) having binomial distribution are 4 and 2, respectively. Find \( P(X = 1) \).

• (A) \( \frac{1}{4} \)
• (B) \( \frac{1}{32} \)
• (C) \( \frac{1}{16} \)
• (D) \( \frac{1}{8} \)

Evaluate: \[ \tan (\cos^{-1} \frac{4}{5}) + \tan^{-1} \frac{2}{3} \]

• (A) \( \frac{6}{17} \)
• (B) \( \frac{7}{16} \)
• (C) \( \frac{16}{7} \)
• (D) None of these

If the function \[ f(x) = \begin{cases} 1, & x \leq 2
ax + b, & 2 < x < 4
7, & x \geq 4 \end{cases} \] is continuous at \( x = 2 \) and \( x = 4 \), then the values of \( a \) and \( b \) are:

• (A) \( a = 3, b = -5 \)
• (B) \( a = -5, b = 3 \)
• (C) \( a = -3, b = 5 \)
• (D) \( a = 5, b = -3 \)

The derivative of \[ \sin^{-1} \left(\frac{2x}{1 + x^2} \right) \] with respect to \[ \cos^{-1} \left(\frac{1 - x^2}{1 + x^2} \right) \] is equal to:

• (A) \( 1 \)
• (B) \( -1 \)
• (C) \( 2 \)
• (D) None of these

The number of distinct real roots of the equation: \[ x^7 - 7x - 2 = 0 \] is:

• (A) 5
• (B) 7
• (C) 1
• (D) 3

The minimum value of the function \[ y = x^4 - 2x^2 + 1 \] in the interval \( \left[\frac{1}{2}, 2 \right] \) is:

• (A) \( 0 \)
• (B) \( 2 \)
• (C) \( 8 \)
• (D) \( 9 \)

Evaluate the integral: \[ I = \int \frac{\sin^2 x - \cos^2 x}{\sin^2 x \cos^2 x} dx \]

• (A) \( \tan x + \cot x + C \)
• (B) \( \csc x + \sec x + C \)
• (C) \( \tan x + \sec x + C \)
• (D) \( \tan x + \csc x + C \)

Consider a curve \( y = y(x) \) in the first quadrant as shown in the figure. Let the area \( A_1 \) be twice the area \( A_2 \). The normal to the curve perpendicular to the line \[ 2x - 12y = 15 \] does NOT pass through which point?

• (A) \( (6, 21) \)
• (B) \( (8, 9) \)
• (C) \( (10, -4) \)
• (D) \( (12, -15) \)

The shortest distance between the lines \( x = y + 2 = 6z - 6 \) and \( x + 1 = 2y = -12z \) is:

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

The angle between the two lines: \[ \frac{x + 1}{2} = \frac{y + 3}{2} = \frac{z - 4}{-1} \] \[ \frac{x - 4}{1} = \frac{y + 4}{2} = \frac{z + 1}{2} \] is:

• (A) \( \cos^{-1} \frac{1}{9} \)
• (B) \( \cos^{-1} \frac{4}{9} \)
• (C) \( \cos^{-1} \frac{2}{9} \)
• (D) \( \cos^{-1} \frac{3}{9} \)

What is the approximate percentage increase in the production of Monopoly from 1993 to 1995?

• (A) \( 10 \)
• (B) \( 20 \)
• (C) \( 30 \)
• (D) \( 25 \)

For which toy category has there been a continuous increase in production over the years?

• (A) Ludo
• (B) Chess
• (C) Monopoly
• (D) Carrom

What is the percentage drop in the production of Ludo from 1992 to 1994?

• (A) \( 30 \)
• (B) \( 50 \)
• (C) \( 20 \)
• (D) \( 10 \)

Find the missing number in the sequence: \[ 285, 253, 221, 189, ? \]

• (A) \( 150 \)
• (B) \( 182 \)
• (C) \( 157 \)
• (D) \( 156 \)

In a certain code language, PRESENTATION is written as ENESTAITPRON. How would INTELLIGENCE be written in that code?

• (A) \( \text{TETGLLTNENCE} \)
• (B) \( \text{LUENLINTETG} \)
• (C) \( \text{LLKKTGTEEBTB} \)
• (D) \( \text{LLTEIGENINCE} \)

Ram moves from a point \( X \) to 20 metres towards North. Then he moves 40 metres towards West. Then he moves 20 metres North. Then he moves 40 metres towards East and then 10 metres towards right and he reaches a point \( Y \). Find the distance and direction of \( Y \) from \( X \)?

• (A) \( 30 \) metres, North
• (B) \( 40 \) metres, North
• (C) \( 30 \) metres, South
• (D) \( 40 \) metres, South

If the 5th date of a month is Tuesday, what date will be 3 days after the 3rd Friday in the month?

• (A) \( 17 \)
• (B) \( 22 \)
• (C) \( 19 \)
• (D) \( 18 \)

Statements:
I. Some cats are dogs.
II. No dog is a toy.
Conclusions:
I. Some dogs are cats.
II. Some toys are cats.
III. Some cats are not toys.
IV. All toys are cats.

• (A) Only Conclusions I and either II or III.
• (B) Only Conclusions II and III follow.
• (C) Only Conclusions I and II follow.
• (D) Only Conclusion I follows.

How is \( H \) related to \( B \)?
Statements:
I. \( H \) is married to \( P \). \( P \) is the mother of \( T \). \( T \) is married to \( D \). \( D \) is the father of \( B \).
II. \( B \) is the daughter of \( T \). \( T \) is the sister of \( N \). \( H \) is the father of \( N \).

• (A) Statement I alone is sufficient.
• (B) Statement II alone is sufficient.
• (C) Either statement I or II is sufficient.
• (D) Both statements together are necessary.

Among five persons \( D, E, F, G, H \), each having different heights, who is the second tallest? Statements:
I. \( D \) is taller than only \( G \) and \( E \). \( F \) is not the tallest.
II. \( H \) is taller than \( F \). \( G \) is taller than \( E \) but shorter than \( D \).

• (A) If the data in Statement I alone are sufficient to answer the question, while the data in Statement II alone are not sufficient.
• (B) If the data in Statement II alone are sufficient to answer the question, while the data in Statement I alone are not sufficient.
• (C) If the data in Statement I alone or in Statement II alone are sufficient to answer the question.
• (D) If the data in both the Statements I and II together are not sufficient.
• (E) If the data in both the Statements I and II together are necessary to answer the question.

If someone else's opinion makes us angry, it means that

• (A) we are subconsciously aware of having no good reason for becoming angry
• (B) there may be good reasons for his opinion but we are not consciously aware of them
• (C) our own opinion is not based on good reason and we know this subconsciously
• (D) we are not consciously aware of any reason for our own opinion

"Your own contrary conviction" refers to

• (A) the fact that you feel pity rather than anger
• (B) the opinion that two and two are four and that Iceland is a long way from the Equator
• (C) the opinion that two and two are five and that Iceland is on the Equator
• (D) the fact that you know so little about arithmetic or geography

Conviction means

• (A) persuasion
• (B) disbelief
• (C) strong belief
• (D) ignorance

The writer says if someone maintains that two and two are five, you feel pity because you

• (A) have sympathy
• (B) don't agree with him
• (C) want to help the person
• (D) feel sorry for his ignorance

The second sentence in the passage

• (A) builds up the argument of the first sentence by restating it from the opposite point of view
• (B) makes the main point which has only been introduced by the first sentence
• (C) simply adds a further point to the argument already stated in the first sentence
• (D) illustrates the point made in the first sentence