KEAM Question Paper 2024 June 8 Available- Download Solution PDF with Answer Key

In the travelling plane wave equation given by \( y = A \sin \omega \left( \frac{x}{v} - t \right) \), where \( \omega \) is the angular velocity and \( v \) is the linear velocity. The dimension of \( \omega t \) is:

• (A) \( LM^\circ{T}^{-1} \)
• (B) \( \text{L}^\circ\text{M}^\circ\text{T}^\circ \)
• (C) \( \text{L}^\circ\text{M}^\circ\text{T} \)
• (D) \( LMT \)
• (E) \( LMT^{-2} \)

Add \( 2.7 \times 10^{-5} \) to \( 4.5 \times 10^{-4} \) with due regard to significant figures

• (A) \( 4.8 \times 10^{-4} \)
• (B) \( 4.7 \times 10^{-5} \)
• (C) \( 4.8 \times 10^{-5} \)
• (D) \( 4.7 \times 10^{-4} \)
• (E) \( 5.0 \times 10^{-4} \)

The length of the second's hand in a watch is 1 cm. The magnitude of the change in the velocity of its tip in 30 seconds (in cm/s) is:

• (A) \( \frac{\pi}{30} \)
• (B) \( \frac{\sqrt{2\pi}}{30} \)
• (C) \( \frac{\sqrt{2\pi}}{15} \)
• (D) \( \frac{\pi}{15} \)
• (E) \( \frac{\pi}{30\sqrt{2}} \)

If the slope of the velocity-time graph of a moving particle is zero, then its acceleration is:

• (A) constant but not zero
• (B) zero
• (C) constant and in the direction of velocity
• (D) not a constant
• (E) constant and opposite to the direction of velocity

A projectile is projected with a velocity of \( 20 \, ms^{-1} \) at an angle of 45° to the horizontal. After some time its velocity vector makes an angle of 30° to the horizontal. Its speed at this instant (in \( ms^{-1} \)) is:

• (A) \( 10\sqrt{\frac{2}{3}} \)
• (B) \( \frac{20}{\sqrt{3}} \)
• (C) \( 20\sqrt{\frac{2}{3}} \)
• (D) \( 10\sqrt{2} \)
• (E) \( 10\sqrt{3} \)

A boy sitting in a bus moving at a constant velocity throws a ball vertically up in the air. The ball will fall:

• (A) in the bus in front of the boy
• (B) in the bus on the side of the boy
• (C) outside the bus
• (D) in the hands of the boy
• (E) in the bus behind the boy

A machine gun fires a bullet of mass 25 g with a velocity of 1000 ms\(^{-1}\). If the man holding the gun can exert a maximum force of 100 N on the gun, the maximum number of bullets that he can fire per second is:

• (A) 4
• (B) 12
• (C) 8
• (D) 6
• (E) 3

When a vehicle moving with kinetic energy \( K \) is stopped in a distance \( d \) by applying a stopping force \( F \), the relation between \( F \) and \( K \) is given by:

• (A) \( F = \frac{K}{d} \)
• (B) \( F = Kd \)
• (C) \( F = \frac{1}{Kd} \)
• (D) \( F = \frac{d}{K} \)
• (E) \( F = \frac{d}{K^2} \)

In moving a body of mass \( m \) down a smooth incline of inclination \( \theta \) with velocity \( v \), the power required is (g = acceleration due to gravity):

• (A) \( mgv \)
• (B) \( (mg \cos \theta) v \)
• (C) \( (mg \sin \theta) v \)
• (D) \( \frac{mg \sin \theta}{v} \)
• (E) \( \frac{mg \cos \theta}{v} \)

The torque required to increase the angular speed of a uniform solid disc of mass 10 kg and diameter 0.5 m from zero to 120 rotations per minute in 5 sec is:

• (A) \( \frac{\pi}{4} \, Nm \)
• (B) \( \pi \, Nm \)
• (C) \( \frac{\pi}{2} \, Nm \)
• (D) \( \frac{\pi}{3} \, Nm \)
• (E) \( \frac{3\pi}{4} \, Nm \)

Radius of gyration \( K \) of a hollow cylinder of mass \( M \) and radius \( R \) about its long axis of symmetry is:

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

The value of escape velocity \( v_e \) for a planet depends on:

• (A) the mass of the body thrown from the planet
• (B) the direction of projection of the body
• (C) the angle of projection
• (D) only on the mass of the planet
• (E) its mass \( M \), density \( \rho \), and radius of the planet

The slope of the graph plotted between the square of the time period of a planet \( T^2 \) and the cube of its mean distance from the sun \( r^3 \) is:

• (A) \( \frac{4\pi^2}{GM} \)
• (B) \( 4\pi GM \)
• (C) \( \frac{4\pi G}{M} \)
• (D) \( \frac{4\pi^2 M}{G} \)
• (E) Zero

If \( n \) small identical liquid drops, each having terminal velocity \( v \), merge together, then the terminal velocity of the bigger drop is:

• (A) \( n^2 v \)
• (B) \( n^{1/3} v \)
• (C) \( \frac{v}{n} \)
• (D) \( nv \)
• (E) \( n^{2/3} v \)

A fluid has stream line flow through a horizontal pipe of variable cross-sectional area. Then:

• (A) its velocity is minimum at the narrowest part of the tube and the pressure is minimum at the widest point
• (B) its velocity and pressure both are maximum at the widest point
• (C) its velocity and pressure both are minimum at the narrowest point
• (D) its velocity is maximum at the narrowest point and the pressure is maximum at the widest part
• (E) its velocity is maximum and pressure is minimum at the narrowest point

A metal rod of length 1 m at 20°C is made up of a material of coefficient of linear expansion \( 2 \times 10^{-5} \, ^\circ C^{-1} \). The temperature at which its length is increased by 1 mm is:

• (A) 45°C
• (B) 70°C
• (C) 65°C
• (D) 60°C
• (E) 50°C

The ends of a metallic rod are at temperatures \( T_1 \) and \( T_2 \), and the rate of flow of heat through it is \( Q \, J s^{-1} \). If all the dimensions of the rod are halved, keeping the end temperatures constant, the new rate of flow of heat will be:

• (A) \( 2Q \)
• (B) \( \frac{Q}{8} \)
• (C) \( \frac{Q}{4} \)
• (D) \( \frac{Q}{2} \)
• (E) \( Q \)

The rate of emission of a perfectly black body at temperature \( 27^\circ C \) is \( E_1 \). If the temperature of the body is raised to \( 627^\circ C \), its rate of emission becomes \( E_2 \). The ratio of \( \frac{E_1}{E_2} \) is:

• (A) \( \frac{1}{81} \)
• (B) \( \frac{1}{16} \)
• (C) \( \frac{1}{25} \)
• (D) \( \frac{1}{36} \)
• (E) \( \frac{1}{49} \)

A monoatomic ideal gas of \( n \) moles heated from temperature \( T_1 \) to \( T_2 \) under two different conditions (i) at constant pressure, (ii) at constant volume). The change in internal energy of the gas is:

• (A) more in process (ii)
• (B) more in process (i)
• (C) same in both the processes
• (D) zero
• (E) proportional to \( \frac{T_1 + T_2}{2} \)

The ratio between the root mean square velocities of \( O_2 \) and \( O_3 \) molecules at the same temperature is:

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

A particle is executing linear simple harmonic oscillation with an amplitude of \( A \). If the total energy of oscillation is \( E \), then its kinetic energy at a distance of \( 0.707A \) from the mean position is:

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

The equation of a stationary wave is given by \[ y = 5 \sin \frac{\pi}{2} \cos 10\pi t \, cm \]
The distance between two consecutive nodes (in cm) is:

• (A) 5
• (B) 2
• (C) 8
• (D) 1
• (E) 6

A thin spherical shell of radius 12 cm is charged such that the potential on its surface is 60 V. Then the potential at the centre of the sphere is:

• (A) 5 V
• (B) Zero
• (C) 30 V
• (D) 120 V
• (E) 60 V

A stationary body of mass 5 g carries a charge of 5 \(\mu\)C. The potential difference with which it should be accelerated to acquire a speed of 10 m/s is:

• (A) 4 kV
• (B) 25 kV
• (C) 50 kV
• (D) 40 kV
• (E) 2 kV

An electric dipole of dipole moment \( p \) is kept in a uniform electric field \( E \) such that it is aligned parallel to the field. The energy required to rotate it by 45° is:

• (A) \( pE \)
• (B) \( pE \left(\frac{\sqrt{2} + 1}{\sqrt{2}}\right) \)
• (C) \( pE \left(\frac{\sqrt{2} - 1}{\sqrt{2}}\right) \)
• (D) \( \frac{pE}{\sqrt{2}} \)
• (E) \( \sqrt{2} pE \)

A steady current of 2 A is flowing through a conducting wire. The number of electrons flowing per second in it is:

• (A) \( 1.25 \times 10^7 \)
• (B) \( 1.25 \times 10^{19} \)
• (C) \( 2.50 \times 10^{10} \)
• (D) \( 0.125 \times 10^{25} \)
• (E) \( 2.5 \times 10^{17} \)

If the voltage across a bulb rated 220V – 60W drops by 1.5% of its rated value, the percentage drop in the rated value of the power is:

• (A) 0.75%
• (B) 1.5%
• (C) 4.5%
• (D) 3%
• (E) 2.5%

The terminal potential difference of a cell in the open circuit is 2 V. When the cell is connected to a 10\(\omega\) resistor, the terminal potential difference falls to 1.5 V. The internal resistance of the cell is:

• (A) \( \frac{10}{3} \, \Omega \)
• (B) \( \frac{10}{9} \, \Omega \)
• (C) \( \frac{20}{7} \, \Omega \)
• (D) \( \frac{15}{6} \, \Omega \)
• (E) \( \frac{13}{2} \, \Omega \)

For a linear material, the relation between the relative magnetic permeability \( \mu_r \) and magnetic susceptibility \( \chi \) is:

• (A) \( \chi = \mu_r + 1 \)
• (B) \( \chi = \mu_r - 1 \)
• (C) \( \chi = \mu_r \mu \)
• (D) \( \mu - 1 \)
• (E) \( \mu = \mu_r + 1 \)

The magnetic field at the centre of a circular coil having a single turn of the wire carrying current \( I \) is \( B \). The magnetic field at the centre of the same coil with 4 turns carrying the same current is:

• (A) \( 16B \)
• (B) \( 8B \)
• (C) \( 4B \)
• (D) \( \frac{B}{2} \)
• (E) \( \frac{B}{4} \)

A current carrying square loop is suspended in a uniform magnetic field acting in the plane of the loop. If \( \vec{F} \) is the force acting on one arm of the loop, then the net force acting on the remaining three arms of the loop is:

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

If the magnetic field energy stored in an inductor changes from maximum to minimum value in 5 ms, when connected to an a.c. source, the frequency of the a.c. source is:

• (A) 200 Hz
• (B) 500 Hz
• (C) 50 Hz
• (D) 20 Hz
• (E) 100 Hz

In an LCR circuit, at resonance, the value of the power factor is:

• (A) 1
• (B) 0
• (C) 0.5
• (D) 0.75
• (E) infinity

An electromagnetic wave is propagating in a medium with velocity \( \vec{v} = v \hat{i} \). The instantaneous oscillating magnetic field of this electromagnetic wave is along positive \( z \)-direction. Then the direction of the oscillating electric field is in the:

• (A) positive \( x \)-direction
• (B) negative \( x \)-direction
• (C) positive \( y \)-direction
• (D) negative \( y \)-direction
• (E) negative \( z \)-direction

When light is reflected from an optically rarer medium:

• (A) its phase remains unchanged but its frequency increases
• (B) both its phase and frequency remain unchanged
• (C) its phase changes by \( \pi \) but the frequency remains unchanged
• (D) its phase remains the same but the frequency decreases
• (E) its phase changes by \( \frac{\pi}{2} \) but the frequency remains unchanged

Focal length of a convex lens of refractive index 1.5 is 3 cm. When the lens is immersed in water of refractive index \( \frac{4}{3} \), its focal length will be:

• (A) 3 cm
• (B) 10 cm
• (C) 12 cm
• (D) 1.5 cm
• (E) 6 cm

A narrow single slit of width \( d \) is illuminated by white light. If the first minimum for violet light (\( \lambda = 4500 \, Å \)) falls at \( \theta = 30^\circ \), the width of the slit \( d \) in microns is (1 micron = \( 10^{-6} \) m):

• (A) 0.4
• (B) 0.5
• (C) 0.3
• (D) 0.7
• (E) 0.9

Threshold frequency for photoelectric effect from a metallic surface corresponds to a wavelength of 6000 \(Å\). The photoelectric work function for the metal is \( h = 6.6 \times 10^{-34} \, Js \):

• (A) \( 1.5 \times 10^{-19} \, J \)
• (B) \( 2.7 \times 10^{-18} \, J \)
• (C) \( 5.4 \times 10^{-18} \, J \)
• (D) \( 4.5 \times 10^{-19} \, J \)
• (E) \( 3.3 \times 10^{-19} \, J \)

A proton and a photon have the same energy. Then the de-Broglie wavelength of proton \( \lambda_p \) and wavelength of photon \( \lambda_0 \) are related by:

• (A) \( \lambda_0 \propto \frac{1}{\sqrt{\lambda_p}} \)
• (B) \( \lambda_0 \propto \sqrt{\lambda_p} \)
• (C) \( \lambda_0 \propto \lambda_p \)
• (D) \( \lambda_0 \propto \lambda_p^2 \)
• (E) \( \lambda_0 \propto \frac{1}{\lambda_p} \)

Bohr atom model is invalid for:

• (A) Hydrogen atom
• (B) doubly ionized helium atom
• (C) deuteron atom
• (D) singly ionized helium atom
• (E) doubly ionized lithium atom

The energy equivalent of 1 g of a substance in joules is:

• (A) \( 9 \times 10^{13} \)
• (B) \( 4.5 \times 10^{13} \)
• (C) \( 1 \times 10^{13} \)
• (D) \( 0.5 \times 10^{13} \)
• (E) \( 2.25 \times 10^{13} \)

Mass numbers of two nuclei are in the ratio 2:3. The ratio of the nuclear densities would be:

• (A) \( 2:3^{1/3} \)
• (B) \( 3^{1/3}:2 \)
• (C) \( 2:3 \)
• (D) \( 3:2 \)
• (E) \( 1:1 \)

Four hydrogen atoms combine to form an \( ^4_2He \) atom with a release of 26.7 MeV of energy. This is:

• (A) fission reaction
• (B) \( \beta^+ \) emission
• (C) \( \beta^- \) emission
• (D) \( \gamma \) emission
• (E) fusion reaction

In the circuit given below, the current is:

• (A) 0.10 A
• (B) \( 10^{-3} \) A
• (C) 0.5 A
• (D) 1 A
• (E) 0 A

Electric conduction in a semiconductor is due to:

• (A) holes only
• (B) electrons only
• (C) neither holes nor electrons
• (D) both electrons and holes
• (E) recombination of electrons and holes

260 g of an aqueous solution contains 60 g of urea (Molar mass = 60 g mol\(^{-1}\)). The molality of the solution is:

• (A) 2m
• (B) 3m
• (C) 4m
• (D) 5m
• (E) 6m

Which of the following pair exhibits diagonal relationship?

• (A) Li and Mg
• (B) Li and Na
• (C) Mg and Al
• (D) B and P
• (E) C and Cl

The molecule which has a see-saw structure is:

• (A) NH\(_3\)
• (B) SF\(_4\)
• (C) CCl\(_4\)
• (D) SiCl\(_4\)
• (E) BrF\(_5\)

The quantum number which determines the shape of the subshell is:

• (A) Principal quantum number
• (B) Magnetic quantum number
• (C) Azimuthal quantum number
• (D) Spin quantum number
• (E) Principal and magnetic quantum number

The total enthalpy change when 1 mol of water at 100°C and 1 bar pressure is converted to ice at 0°C is:

• (A) -7.56 kJ mol\(^{-1}\)
• (B) -6.00 kJ mol\(^{-1}\)
• (C) -13.56 kJ mol\(^{-1}\)
• (D) -756 kJ mol\(^{-1}\)
• (E) -1.356 kJ mol\(^{-1}\)

The balanced ionic equation for the reaction of K\(_2\)Cr\(_2\)O\(_7\) with Na\(_2\)SO\(_3\) in an acid solution is:

• (A) Cr\(_2\)O\(_7^{2-}\)(aq) + SO\(_3^{2-}\)(aq) + 8H\(^+\)(aq) \( \to \) 2Cr\(^3+\)(aq) + SO\(_4^{2-}\)(aq) + 4H\(_2\)O(l)
• (B) Cr\(_2\)O\(_7^{2-}\)(aq) + 3SO\(_3^{2-}\)(aq) + 2H\(^+\)(aq) \( \to \) 2Cr\(^3+\)(aq) + 3SO\(_4^{2-}\)(aq) + H\(_2\)O(l)
• (C) 3Cr\(_2\)O\(_7^{2-}\)(aq) + 3SO\(_3^{2-}\)(aq) + 8H\(^+\)(aq) \( \to \) 6Cr\(^3+\)(aq) + 3SO\(_4^{2-}\)(aq) + H\(_2\)O(l)
• (D) 3Cr\(_2\)O\(_7^{2-}\)(aq) + 3SO\(_3^{2-}\)(aq) + 2H\(^+\)(aq) \( \to \) 3Cr\(^3+\)(aq) + 3SO\(_4^{2-}\)(aq) + H\(_2\)O(l)
• (E) Cr\(_2\)O\(_7^{2-}\)(aq) + 3SO\(_3^{2-}\)(aq) + 8H\(^+\)(aq) \( \to \) 2Cr\(^3+\)(aq) + 3SO\(_4^{2-}\)(aq) + 4H\(_2\)O(l)

The limiting molar conductances of NaCl, HCl and CH\(_3\)COONa at 300 K are 126.4, 425.9 and 91.0 S cm\(^2\) mol\(^-1\) respectively. The limiting molar conductance of acetic acid at 300 K is:

• (A) \(266 \, S cm^2 \, mol^{-1}\)
• (B) \(390.5 \, S cm^2 \, mol^{-1}\)
• (C) \(461.3 \, S cm^2 \, mol^{-1}\)
• (D) \(208 \, S cm^2 \, mol^{-1}\)
• (E) \(108 \, S cm^2 \, mol^{-1}\)

Which of the following liquid pairs shows negative deviation from Raoult’s law?

• (A) Phenol - Aniline
• (B) Acetone - Carbon disulphide
• (C) Benzene - Toluene
• (D) n-hexane — n-heptane
• (E) Bromoethane — Chloroethane

The half-life period of a first order reaction is 1000 seconds. Its rate constant is:

• (A) \(0.693 \, sec^{-1}\)
• (B) \(6.93 \times 10^{-2} \, sec^{-1}\)
• (C) \(6.93 \times 10^{-3} \, sec^{-1}\)
• (D) \(6.93 \times 10^{-4} \, sec^{-1}\)
• (E) \(6.93 \times 10^{-1} \, sec^{-1}\)

Which of the following material acts as a semiconductor at 298 K?

• (A) Iron
• (B) Copper oxide
• (C) Sodium
• (D) Graphite
• (E) Glass

The resistance of a conductivity cell filled with 0.02 M KCl solution is 520 ohm at 298 K. The conductivity of the solution at 298 K is (Cell constant = 130 cm\(^{-1}\)):

• (A) \(0.50 \, S cm^{-1}\)
• (B) \(1.25 \, S cm^{-1}\)
• (C) \(0.025 \, S cm^{-1}\)
• (D) \(0.25 \, S cm^{-1}\)
• (E) \(0.75 \, S cm^{-1}\)

For the equilibrium at 500 K, \( N_2 (g) + 3H_2 (g) \rightleftharpoons 2NH_3 (g) \), the equilibrium concentrations of \(N_2 (g)\), \(H_2 (g)\) and \(NH_3 (g)\) are respectively 4.0 M, 2.0 M and 2.0 M. The \(K_c\) for the formation of \(NH_3\) at 500 K is:

• (A) \( \frac{1}{16} \, mol^{-2} \, dm^6 \)
• (B) \( \frac{1}{32} \, mol^{-2} \, dm^6 \)
• (C) \( \frac{1}{8} \, mol^{-2} \, dm^6 \)
• (D) \( \frac{1}{4} \, mol^{-2} \, dm^6 \)
• (E) \( \frac{1}{2} \, mol^{-2} \, dm^6 \)

The molarity of a solution containing 8 g of NaOH (Molar mass = 40 g mol\(^{-1}\)) in 250 mL solution is:

• (A) \(0.8 \, M\)
• (B) \(0.4 \, M\)
• (C) \(0.2 \, M\)
• (D) \(0.5 \, M\)
• (E) \(0.6 \, M\)

Which of the following are the conditions for a reaction spontaneous at all temperatures?

• (A) \( \Delta_r H > 0 ; \Delta_r S > 0 \)
• (B) \( \Delta_r H < 0 ; \Delta_r S > 0 \)
• (C) \( \Delta_r H < 0 ; \Delta_r S < 0 \)
• (D) \( \Delta_r H = 0 ; \Delta_r S < 0 \)
• (E) \( \Delta_r H = 0 ; \Delta_r S = 0 \)

Transition elements act as catalyst because

• (A) their melting points are high
• (B) their ionization potential values are high
• (C) they have high density
• (D) they show variable oxidation state
• (E) they have high electronegativity

Lanthanides (Ln) burn in O\(_2\) to give

• (A) LnO
• (B) Ln(OH)\(_3\)
• (C) Ln\(_2\)O\(_3\)
• (D) LnO\(_2\)
• (E) LnO\(_3\)

The IUPAC name of the coordination compound Hg[Co(SCN)\(_4\)] is

• (A) Mercury (I) tetrathiocyanato-S-cobaltate (III)
• (B) Mercury (II) tetrathiocyanato-S-cobaltate (II)
• (C) Mercury (I) tetrathiocyanato-S-cobaltate (IV)
• (D) Mercury (II) tetraisocyanato-S-cobaltate (III)
• (E) Mercury (I) tetraisocyanato-N-cobaltate (III)

In a combustion reaction, heat change during the formation of 40 g of carbon dioxide from carbon and dioxygen gas is (Enthalpy of combustion of carbon = -396 kJ mol\(^1\))

• (A) 320 kJ
• (B) -320 kJ
• (C) -360 kJ
• (D) 360 kJ
• (E) 240 kJ

Which of the following statement is incorrect?

• (A) Hyperconjugation is a permanent effect.
• (B) Tertiary carbocation is relatively more stable than a secondary carbocation.
• (C) F has stronger -I effect than Cl.
• (D) Inductive effect decreases with increasing distance.
• (E) When inductive and electromeric effects operate in opposite directions, the inductive effect predominates.

Which of the following statement is incorrect with regard to ozonolysis?

• (A) It involves addition of ozone on alkene.
• (B) An unsymmetrical alkene gives two different carbonyl compounds.
• (C) It is used to identify the number of double bonds in the starting material.
• (D) It cannot be used to detect the position of the double bonds.
• (E) Ozonide will undergo cleavage by Zn-H\(_2\)O.

Which of the following statement is true?

• (A) Dehydration of alcohol takes place in presence of HCl/ZnCl\(_2\).
• (B) Formation of ethene from ethyl iodide occurs on heating with aqueous KOH.
• (C) Hydrogenation of an unsymmetrical alkyne in presence of Pd/C gives cis-alkene.
• (D) Hydrogenation of an unsymmetrical alkyne in presence of Na/liqu. NH\(_3\) gives cis-alkene.
• (E) The order of reactivity of hydrogen halides towards alkenes is HI < HBr < HCl.

An organic compound X (C\(_6\)H\(_5\)O) on reaction with zinc dust gives Y. The product Y reacts with CH\(_3\)COCl in presence of anhydrous AlCl\(_3\) to give Z (C\(_6\)H\(_5\)O). The compounds X, Y, and Z are respectively

• (A) benzaldehyde, benzene, methyl phenyl ketone
• (B) phenol, benzene, acetophenone
• (C) phenol, naphthalene, acetophenone
• (D) benzene, phenol, diphenyl ketone
• (E) cyclohexanol, cyclohexane, benzophenone

The percentage amylose in starch is about

• (A) 40-50 %
• (B) 80-85 %
• (C) 60-80 %
• (D) 50-60 %
• (E) 15-20 %

Which of the following statement is correct?

• (A) Bromination of phenol in CS\(_2\), at low temperature gives 2,4,6-tribromophenol.
• (B) Oxidation of phenol with chromic acid gives benzene.
• (C) Conversion of phenol into tribromophenol by bromine water is a nucleophilic substitution reaction.
• (D) p-Nitrophenol is steam volatile due to intermolecular hydrogen bonding.
• (E) The intermediate in Reimer-Tiemann reaction is substituted benzal chloride.

On heating an aldehyde with Fehling’s reagent, a reddish-brown precipitate is obtained due to the formation of

• (A) cupric oxide
• (B) cuprous oxide
• (C) carboxylic acid
• (D) silver
• (E) copper acetate

The decreasing order of basic strength of amines in aqueous medium is:

• (A) \( CH_3NH_2 > (CH_3)_2NH > (CH_3)_3N > NH_3 \)
• (B) \( (CH_3)_2NH > CH_3NH_2 > (CH_3)_3N > NH_3 \)
• (C) \( CH_3NH_2 > (CH_3)_3N > (CH_3)_2NH > NH_3 \)
• (D) \( (CH_3)_2NH > NH_3 > (CH_3)_3N > CH_3NH_2 \)
• (E) \( NH_3 > CH_3NH_2 > (CH_3)_3N > (CH_3)_2NH \)

Which of the following statement is correct?

• (A) Sucrose is laevorotatory.
• (B) Fructose is a disaccharide.
• (C) Sucrose on hydrolysis gives D(+) glucose only.
• (D) Sucrose is made up of a glycosidic linkage between C1 of \(\alpha\)-D-glucose and C2 of \(\beta\)-D-Fructose.
• (E) Sucrose is a reducing sugar.

The structure of MnO\(_4^-\) ion is:

• (A) square planar
• (B) octahedral
• (C) trigonal pyramid
• (D) pyramid
• (E) tetrahedral

When benzene diazonium fluoroborate is heated with aqueous sodium nitrite solution in the presence of copper, the product formed is:

• (A) fluorobenzene
• (B) benzene
• (C) aniline
• (D) nitrobenzene
• (E) phenol

A fibrous protein present in muscles is:

• (A) keratin
• (B) albumin
• (C) riboflavin
• (D) insulin
• (E) myosin

Let \( P \) and \( Q \) be two finite sets having 3 elements each. The total number of mappings from \( P \) to \( Q \) is

• (A) 32
• (B) 516
• (C) 6
• (D) 9
• (E) 27

If \( f(x) = \lfloor x \rfloor \), where \( \lfloor x \rfloor \) denotes the greatest integer function, and if the domain of \( f \) is \( \{-3.01, 2.99\} \), then the range of \( f \) is

• (A) \( \{-3, 3\} \)
• (B) \( \{-4, 3\} \)
• (C) \( \{-3, 2\} \)
• (D) \( \{-4, 2\} \)
• (E) \( \{-2, 3\} \)

The domain of the function \( f(x) = \sqrt{7 - 8x + x^2} \) is

• (A) \( (-\infty, 1) \cup (7, \infty) \)
• (B) \( (-\infty, 1] \cup [7, \infty) \)
• (C) \( (-\infty, 1) \cup [7, \infty) \)
• (D) \( (-\infty, -1) \cup (7, \infty) \)
• (E) \( (-\infty, -7] \cup [1, \infty) \)

The period of the function \( \sin\left( \frac{\pi x}{4} \right) \) is

• (A) 4
• (B) \( 4\pi \)
• (C) \( 8\pi \)
• (D) 8
• (E) \( 2\pi \)

If \( f(x) = x + 8 \), and \( g(x) = 2x^2 \), then \( (g \circ f)(x) \) is equal to

• (A) \( (2x + 8)^2 \)
• (B) \( 2(x + 8)^2 \)
• (C) \( 2x^2 + 8 \)
• (D) \( 2x^2 + 64 \)
• (E) \( 2x^3 + 8x \)

If \( f(x) = \frac{x}{1 - x} \), \( x \neq 1 \), then the inverse of \( f \) is

• (A) \( \frac{1 - x}{1 + x}, \, x \neq -1 \)
• (B) \( \frac{1}{1 + x}, \, x \neq -1 \)
• (C) \( \frac{1 - x}{x}, \, x \neq 0 \)
• (D) \( \frac{x}{1 + x}, \, x \neq -1 \)
• (E) \( \frac{1 + x}{1 - x}, \, x \neq 1 \)

If the complex numbers \( (2 + i)x + (1 - i)y + 2i - 3 \) and \( x + (-1 + 2i)y + 1 + i \) are equal, then \( (x, y) \) is

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

If \( x + iy = \frac{3 + 4i}{5 - 12i} \), then \( x + y \) is equal to

• (A) \( \frac{23}{169} \)
• (B) \( \frac{56}{169} \)
• (C) \( \frac{15}{169} \)
• (D) \( \frac{15}{169} \)
• (E) \( \frac{71}{169} \)

If \( z = 1 + i \), then the maximum value of \( |z + 12 + 9i| \) is

• (A) 225
• (B) 265
• (C) 269
• (D) 200
• (E) \( \sqrt{265} \)

If \( \frac{|z - 5i|}{|z - 5i|} = 1 \), then

• (A) \( Re(z) = 0 \)
• (B) \( |z| = 10 \)
• (C) \( |z| = 25 \)
• (D) \( |z| = 5 \)
• (E) \( Im(z) = 0 \)

The coefficient of \( x^7 \) in the expansion of \( \left( \frac{1}{x + x^2} \right)^8 \) is

• (A) 70
• (B) 28
• (C) 42
• (D) 56
• (E) 8

If \( a_1 = 3 \) and \( a_n = n \cdot a_{n-1} \), for \( n \geq 2 \), then \( a_6 \) is equal to

• (A) 72
• (B) 144
• (C) 720
• (D) 2160
• (E) 4320

If \( \frac{1}{\log_2 x} + \frac{1}{\log_3 x} + \frac{1}{\log_4 x} + \frac{1}{\log_5 x} + \frac{1}{\log_6 x} = 1 \), then the value of \( x \) is

• (A) 18
• (B) 36
• (C) 120
• (D) 360
• (E) 720

The common ratio of a G.P. is 10. Then the ratio between its 11^th term and its 6^th term is:

• (A) \( 10^6 : 1 \)
• (B) \( 10^5 : 1 \)
• (C) \( 10^4 : 1 \)
• (D) \( 10^{11} : 1 \)
• (E) \( 10^3 : 1 \)

Let \( a, b, c \) be positive numbers. If \( a + b + c \geq K \left[ (a + b)(b + c)(c + a) \right]^{1/3} \), then the maximum value of \( K \) is:

• (A) \( \frac{3}{2} \)
• (B) \( \frac{1}{2} \)
• (C) \( \frac{1}{4} \)
• (D) \( \frac{1}{8} \)
• (E) 1

If \( A = \begin{bmatrix} 4 & -1
12 & x \end{bmatrix} \) and \( A^2 = A \), then the value of \( x \) is:

• (A) -8
• (B) -3
• (C) 0
• (D) 3
• (E) 8

If \( A = \begin{bmatrix} 3 & 7
2 & 5 \end{bmatrix} \), then \( A^2 (adj A) \) is:

• (A) \( I \)
• (B) \( 4I \)
• (C) \( 2A \)
• (D) \( 3A \)
• (E) \( A \)

If \( |x - 2| \leq 4 \), then \( x \) lies in the interval:

• (A) \( (-\infty, -2) \)
• (B) \( (-\infty, 0) \)
• (C) \( [-2, 6] \)
• (D) \( (-2, \infty) \)
• (E) \( (-2, 4) \)

If \( \tan \left( \frac{\pi}{12} + 2x \right) = \cot 3x \), where \( 0 < x < \frac{\pi}{2} \), then the value of \( x \) is:

• (A) \( \frac{\pi}{12} \)
• (B) \( 3 \)
• (C) \( \frac{\pi}{4} \)
• (D) \( \frac{\pi}{6} \)
• (E) \( \frac{\pi}{24} \)

If \( \cos \theta + \sin \theta = \sqrt{2} \), then \( \cos \theta - \sin \theta \) is equal to:

• (A) 0
• (B) -\frac{1}{2}
• (C) \frac{1}{2}
• (D) \frac{1}{4}
• (E) 1

The value of \( \cos 26^\circ + \cos 54^\circ + \cos 126^\circ + \cos 206^\circ + \cos 240^\circ \) is:

• (A) 0
• (B) 1
• (C) -\frac{1}{2}
• (D) \frac{1}{2}
• (E) -1

If \( \cos x - \sin x = 0 \), \( 0 \leq x \leq \pi \), then the value(s) of \( x \) is/are:

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

If \( 2 \sin \left( \frac{\pi}{3} - 2x \right) - 1 = 0 \), \( 0 < x < \frac{\pi}{2} \), then the value of \( x \) is:

• (A) \( \frac{\pi}{4} \)
• (B) \( \frac{\pi}{3} \)
• (C) \( \frac{5\pi}{12} \)
• (D) \( \frac{\pi}{12} \)
• (E) \( \frac{\pi}{6} \)

Domain of the function \( \sin^{-1}(2x - 1) \) is:

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

If \( 3 \tan^{-1}(x) + \cot^{-1}(x) = \pi \), then \( \sin^{-1}(x) \) is:

• (A) \( \frac{\pi}{12} \)
• (B) \( \frac{\pi}{3} \)
• (C) \( \frac{\pi}{4} \)
• (D) \( \frac{\pi}{6} \)
• (E) \( \frac{\pi}{2} \)

tan\(^{-1} 2 - \) tan\(^{-1} \frac{1}{3}\) is equal to:

• (A) \( \frac{\pi}{2} \)
• (B) \( \frac{\pi}{3} \)
• (C) \( \frac{\pi}{4} \)
• (D) \( \frac{\pi}{6} \)
• (E) 0

sin\(^{-1} \left( \sin \left( \frac{5\pi}{6} \right) \right) \) is equal to:

• (A) \( \frac{5\pi}{6} \)
• (B) \( \frac{\pi}{6} \)
• (C) \( \frac{\pi}{3} \)
• (D) \( \frac{2\pi}{3} \)
• (E) \( \frac{\pi}{2} \)

If \( \sin x = \frac{3}{5} \), then the value of \( \sec x + \tan x \) is equal to:

• (A) -2
• (B) 3
• (C) 0
• (D) 2
• (E) -3

If \( P(-3, 4) \) and \( Q(3, 1) \) are points on a straight line, then the slope of the straight line perpendicular to PQ is:

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

The length of perpendicular from the origin to the line \( \frac{x}{5} - \frac{y}{12} = 1 \) is:

• (A) \( \frac{60}{13} \)
• (B) \( \frac{5}{12} \)
• (C) \( \frac{12}{5} \)
• (D) \( \frac{13}{12} \)
• (E) \( \frac{13}{60} \)

The equation of the straight line passing through the point \( (1, 1) \) and perpendicular to the line \( x + y = 5 \) is:

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

The area of the triangle formed by the coordinate axes and a line whose perpendicular from the origin makes an angle of 45° with the x-axis is 50 square units. Then the equation of the line is:

• (A) \( x + y = 10 \)
• (B) \( x + 2y = 10 \)
• (C) \( 2x + y = 5 \)
• (D) \( x + y = 25 \)
• (E) \( x + y = 5 \)

The equation of the straight line, intersecting the coordinate axes \( x \) and \( y \) are respectively 1 and 2, is:

• (A) \( x + y = 3 \)
• (B) \( x - 2y = -3 \)
• (C) \( 2x - y = 0 \)
• (D) \( 2x + y = 2 \)
• (E) \( x - y = -1 \)

If the sum of distances of a point from the origin and the line \( x = 3 \) is 8, then its locus is:

• (A) \( y^2 - 10x + 25 = 0 \)
• (B) \( y^2 + 10x + 25 = 0 \)
• (C) \( y^2 - 10x - 25 = 0 \)
• (D) \( y^2 - 25x + 10 = 0 \)
• (E) \( y^2 + 25x - 10 = 0 \)

If the point \( (2, k) \) lies on the circle \( (x - 2)^2 + (y + 1)^2 = 4 \), then the value of \( k \) is:

• (A) \( 1, 3 \)
• (B) \( 1, 2 \)
• (C) \( -1, 3 \)
• (D) \( 2, 3 \)
• (E) \( 1, -3 \)

The radius of the circle \( x^2 + y^2 - 2x - 4y - 4 = 0 \) is:

• (A) \( 2 \)
• (B) \( 3 \)
• (C) \( 4 \)
• (D) \( 5 \)
• (E) \( 6 \)

The eccentricity of an ellipse is \( \frac{1}{3} \) and its center is at the origin. If one of the directrices is \( x = 9 \), then the equation of the ellipse is:

• (A) \( 8x^2 + 9y^2 = 32 \)
• (B) \( 8x^2 + 9y^2 = 36 \)
• (C) \( 9x^2 + 8y^2 = 36 \)
• (D) \( 9x^2 + 8y^2 = 32 \)
• (E) \( 8x^2 + 9y^2 = 72 \)

If the parametric form of the circle is \( x = 3\cos\theta + 3 \) and \( y = 3\sin\theta \), then the Cartesian form of the equation of the circle is:

• (A) \( x^2 + y^2 - 6x = 0 \)
• (B) \( x^2 + y^2 - 6x = 9 \)
• (C) \( x^2 + y^2 + 6x = 9 \)
• (D) \( x^2 + y^2 - 6x = 0 \)
• (E) \( x^2 + y^2 - 2x - 2y = 9 \)

A line makes angles \( \alpha, \beta, \gamma \) with \( x, y \), and \( z \)-axes respectively. Then the value of \( \sin^2\alpha + \sin^2\beta - \cos^2\gamma \) is:

• (A) \( 3 \)
• (B) \( 2 \)
• (C) \( 1 \)
• (D) \( \frac{3}{2} \)
• (E) \( 0 \)

The direction ratios of the line joining the points \( (2, 3, 4) \) and \( (-1, 4, -3) \) is:

• (A) \( \pm (3, -1, 7) \)
• (B) \( \pm (-3, -1, 7) \)
• (C) \( \pm (3, 1, 7) \)
• (D) \( \pm (3, -1, -7) \)
• (E) \( \pm (-3, 1, 7) \)

Equation of the line parallel to the line \( \frac{x-2}{2} = \frac{y-2}{3} = \frac{z-1}{-2} \) and passing through the point \( (3, 2, -1) \) is:

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

If the lines \( \frac{x-1}{2} = \frac{y-2}{\alpha} = \frac{z-3}{2} \) and \( \frac{x-1}{2} = \frac{y-2}{1} = \frac{z-3}{-2} \) are perpendicular, then the value of \( \alpha \) is:

• (A) 6
• (B) 4
• (C) 3
• (D) -3
• (E) -2

If \( \vec{a} = 2\vec i + 4\vec j + 7\vec k \) and \( \vec {b} = 4\vec i + 7\vec j + 2\vec k \), then the angle between \( \vec{a} + \vec{b} \) and \( \vec{a} - \vec{b} \) is equal to:

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

A vector of magnitude 6 and perpendicular to \( \vec{a} = 2i + 2j + k \) and \( \vec{b} = i - 2j + 2k \), is:

• (A) \( \pm (2\vec i - \vec j - 2\vec k) \)
• (B) \( \pm 2(2\vec i - \vec j + 2\vec k) \)
• (C) \( \pm (2\vec i - \vec j + 2\vec k) \)
• (D) \( \pm 2(2\vec i + \vec j - 2\vec k) \)
• (E) \( \pm 2(2\vec i - \vec j - 2\vec k) \)

If \( \vec{a} \) and \( \vec{b} \) are non-collinear unit vectors and \( |\vec{a} + \vec{b}|^2 = 3 \), then \( (3\vec{a} + 2\vec{b}) \cdot (3\vec{a} - \vec{b}) \) is equal to:

• (A) \( \frac{32}{3} \)
• (B) \( \frac{17}{2} \)
• (C) 15
• (D) 7
• (E) \( \frac{17}{4} \)

If \( x_1, i=2, 3, \ldots, n \) are \( n \) observations such that \( \sum_{i=1}^{n} x_i^2 = 550 \), mean \( \bar{x} = 5 \) and variance is zero, then the number of observations is equal to:

• (A) 30
• (B) 25
• (C) 22
• (D) 16
• (E) 4

If the mean of five observations \( x, 2x+5, 13, 2x-7, \) and 9 is 22, then the value of \( x \) is:

• (A) 20
• (B) 15
• (C) 10
• (D) 12
• (E) 18

If \( A \) and \( B \) are two independent events such that \( P(A) = 0.4 \) and \( P(A \cup B) = 0.7 \), then \( P(B) \) is equal to:

• (A) 0.3
• (B) 0.4
• (C) 0.5
• (D) 0.6
• (E) 0.7

The probability that at least one of \( A \) or \( B \) occurs is 0.6. If \( A \) and \( B \) occur simultaneously with probability 0.2, then \( P(A') + P(B') \) is:

• (A) 0.7
• (B) 1.5
• (C) 1.1
• (D) 1.2
• (E) 0.3

The value of \( \lim_{x \to 0} \frac{\sin(5x)}{\sin(3x)} \) is:

• (A) \( \frac{3}{5} \)
• (B) \( \frac{5}{3} \)
• (C) 1
• (D) 0
• (E) 5

The value of \( \lim_{x \to 1} \frac{x^2 + 2x - 3}{x - 1} \) is:

• (A) 2
• (B) 4
• (C) 3
• (D) 1
• (E) 0

If \( f(x) = \frac{1}{2 - x} \) and \( g(x) = \frac{1}{1 - x} \), then the point(s) of discontinuity of the function \( g(f(x)) \) is (are):

• (A) \( x = 2 \)
• (B) \( x = 3 \)
• (C) \( x = 2, x = 3 \)
• (D) \( x = 2, x = 1 \)
• (E) \( x = 1, x = -2 \)

Let \( f(x) = \cos^{-1} \left( \frac{1 - \tan^2 x}{1 + \tan^2 x} \right) \). Then \( f\left( \frac{\pi}{2} \right) \) is equal to:

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

If \( x = r \cos \theta, y = r \sin \theta \), then \( \frac{dy}{dx} \) at \( \theta = \frac{\pi}{4} \), where \( r \) is a constant and \( \theta \) is a parameter, is equal to:

• (A) 0
• (B) 1
• (C) -1
• (D) \( \frac{\sqrt{2}}{2} \)
• (E) \( \frac{1}{\sqrt{2}} \)

If \( f(x) = \int_0^{x^3} (t + 4)^2 dt \), then \( f'(2) \) is equal to:

• (A) 288
• (B) 432
• (C) 144
• (D) 216
• (E) 24

The limit \( \lim_{x \to 0} \frac{3 \sin^2 2x}{x^2} \) is equal to:

• (A) 3
• (B) 2
• (C) 6
• (D) \( \frac{3}{2} \)
• (E) 12

The function \( f(x) = (x - 4)^2 (1 + x)^3 \) attains a local extremum at the point:

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

The derivative of \( t^2 + t \) with respect to \( t-1 \) at \( t = -2 \), is equal to:

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

If a continuous function \( f \) is defined as \[ f(x) = \left\{ \begin{array}{ll} ax + 1, & x < 2
x^2 + 7, & x \geq 2 \end{array} \right. \]
then the value of \( a \) is:

• (A) 7
• (B) 6
• (C) 5
• (D) 3
• (E) 2

If \( f(x) = x|x| \), then \( f'(-1) + f'(1) \) is equal to:

• (A) 2
• (B) -2
• (C) 0
• (D) -4
• (E) 4

The integral \( \int \frac{1 + x^2 + x^4}{(1 - x^3)(1 + x^3)} \, dx \) is equal to:

• (A) \( \tan^{-1}(x) + C \)
• (B) \( \tan^{-1}(1 + x^2) + C \)
• (C) \( \frac{1}{2} \log(1+x) - \log(1-x) + C \)
• (D) \( \log(1 + x^3) + C \)
• (E) \( \log(1 + x^2) + C \)

A train starts from X towards Y at 3 pm (time \( t = 0 \)) with velocity \( v(t) = 10t + 25 \) km per hour, where \( t \) is measured in hours. Then the distance covered by the train at 5 pm (in km) is:

• (A) 70
• (B) 140
• (C) 35
• (D) 60
• (E) 55

The integral \( \int \sqrt{1 + \sin 2x} \, dx \) is equal to:

• (A) \( \sin x - \cos x + C \)
• (B) \( \sin x - \csc x + C \)
• (C) \( \tan x - \cot x + C \)
• (D) \( \cos x - \sec x + C \)
• (E) \( \tan x - \sec x + C \)

The integral \( \int xe^x \, dx \) is equal to:

• (A) \( xe^x + e^x + C \)
• (B) \( e^x - xe^x + C \)
• (C) \( x + e^x + C \)
• (D) \( xe^x - e^x + C \)
• (E) \( xe^x - x^2e^x + C \)

The integral \( \int e^x \sec x (1 + \tan x) \, dx \) is equal to:

• (A) \( e^x \sec x + C \)
• (B) \( e^x \tan x + C \)
• (C) \( e^x (\sec x + \tan x) + C \)
• (D) \( e^x \sec x \tan x + C \)
• (E) \( e^x \sec x + \tan x + C \)

The value of \( \int_0^1 x(1 - x)^{10} \, dx \) is equal to:

• (A) \( \frac{1}{110} \)
• (B) \( \frac{1}{132} \)
• (C) \( \frac{1}{156} \)
• (D) \( \frac{1}{90} \)
• (E) \( \frac{5}{156} \)

The value of \( \int_{\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{\tan x + \sin x}{1 + \cos^2 x} \, dx \) is equal to:

• (A) 0
• (B) 2
• (C) \( \sqrt{2} \)
• (D) \( 2\sqrt{2} \)
• (E) \( -2\sqrt{2} \)

The integral \( \int_5^{10} \left\lfloor x \right\rfloor dx \) is equal to (where \( \left\lfloor x \right\rfloor \) denotes the greatest integer function):

• (A) 55
• (B) 45
• (C) 35
• (D) 26
• (E) 5

The integral \( \int_{-2}^4 x^2 |x| \, dx \) is equal to:

• (A) 72
• (B) 68
• (C) 64
• (D) 48
• (E) 37

The value of \( \int_{-1}^{1} x^2 \sin x \, dx \) is equal to:

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

The area of the region bounded by the curve \( y = 3x^2 \) and the x-axis, between \( x = -1 \) and \( x = 1 \), is:

• (A) 2 sq. units
• (B) 4 sq. units
• (C) \( \frac{55}{27} \) sq. units
• (D) \( \frac{55}{23} \) sq. units
• (E) \( \frac{1}{2} \) sq. units

The order and degree of the following differential equation: \( \frac{d^2 y}{dx^2} - 2x = \sqrt{y} + \frac{dy}{dx} \), respectively, are:

• (A) 2, 2
• (B) 2, 1
• (C) 1, 2
• (D) 4, 2
• (E) 1, 1

The solution of the differential equation \( x + y \frac{dy}{dx} = 0 \), given that at \( x = 0 \), \( y = 5 \), is:

• (A) \( x^2 + y^2 = 5y \)
• (B) \( x^2 + 5y^2 = 125 \)
• (C) \( x^2 + y = 5 \)
• (D) \( x^2 + y^2 = 25 \)
• (E) \( 2x^2 + y^2 = 25 \)

The general solution of the differential equation \( (x + y)^2 \frac{dy}{dx} = 1 \) is:

• (A) \( y = \frac{1}{2} \tan^{-1}(x + y) + c \)
• (B) \( y = -(x + y)^{-1} + c \)
• (C) \( y = \frac{1}{3}(x + y)^3 + c \)
• (D) \( y = \sin^{-1}(x + y) + c \)
• (E) \( y = \tan^{-1}(x + y) + c \)

The equation of the curve passing through \( (1, 0) \) and which has slope \( \left( 1 + \frac{y}{x} \right) \) at \( (x, y) \), is:

• (A) \( y = x e^x \)
• (B) \( y = x + \log x \)
• (C) \( y = x - \log x \)
• (D) \( y = x + 2 \log x \)
• (E) \( y = x \log x \)