VITEEE 2024 Question Paper Available - Download Solution PDF with Answer Key

Force between two point charges \(q_1\) and \(q_2\) placed in vacuum at \(r\) cm apart is \(F\). Force between them when placed in a medium having dielectric \(K = 5\) at \(r/5\) cm apart will be:

(a) \( \frac{F}{25} \)

(b) \( 5F \)

(c) \( \frac{F}{5} \)

(d) \( 25F \)

A parallel plate capacitor has 1 \(\mu\)F capacitance. One of its two plates is given \(+2 \mu C\) charge and the other plate, \(+4 \mu C\) charge. The potential difference developed across the capacitor is:

(a) 3 V

(b) 1 V

(c) 5 V

(d) 2 V

A body of mass \(M\) at rest explodes into three pieces, in the ratio of masses 1:1:2. Two smaller pieces fly off perpendicular to each other with velocities of 30 m/s and 40 m/s respectively. The velocity of the third piece will be:

(a) 15 m/s

(b) 25 m/s

(c) 35 m/s

(d) 50 m/s

A charge particle moving in magnetic field \( B \), has components of velocity along \( B \) as well as perpendicular to \( B \). The path of the charge particle will be:

(a) Helical path with the axis perpendicular to the direction of magnetic field \( B \)

(b) Straight along the direction of magnetic field \( B \)

(c) Helical path with the axis along magnetic field \( B \)

(d) Circular path

In meter bridge or Wheatstone bridge for measurement of resistance, the known and the unknown resistance are interchanged. The error so removed is:

(a) End error

(b) Index error

(c) Due to temperature effect

(d) Random error

Spherical insulating ball and a spherical metallic ball of same size and mass are dropped from the same height. Choose the correct statement out of the following (Assume negligible air friction):

(a) Time taken by them to reach the earth's surface will be independent of the properties of their materials

(b) Insulating ball will reach the earth's surface earlier than the metal ball

(c) Both will reach the earth's surface simultaneously

(d) Metal ball will reach the earth's surface earlier than the insulating ball

Two objects A and B are placed at 15 cm and 25 cm from the pole in front of a concave mirror having radius of curvature 40 cm. The distance between images formed by the mirror is:

(a) 60 cm

(b) 40 cm

(c) 160 cm

(d) 100 cm

A bob of mass \( m \) is suspended by a light string of length \( L \). It is imparted a minimum horizontal velocity at the lowest point A such that it just completes half a circle, reaching the topmost position B. The ratio of kinetic energies \(\left(\frac{K.E.}{K.E.}\right)_A \, to \, \left(\frac{K.E.}{K.E.}\right)_B\) is:

(a) 3:2

(b) 5:1

(c) 2:5

(d) 1:5

For a group of positive charges, which of the following statements is correct?

(a) Net potential of the system cannot be zero at a point, but net electric field can be zero at that point.

(b) Net potential of the system at a point can be zero, but net electric field can't be zero at that point.

(c) Both the net potential and the net electric field can be zero at a point.

(d) Both the net potential and the net electric field cannot be zero at a point.

The work function of a substance is 4.0 eV. The longest wavelength of light that can cause photoelectron emission from this substance is approximately:

(a) 310 nm

(b) 400 nm

(c) 540 nm

(d) 220 nm

Magnetic field at the centre of a circular coil of radius \( r \), through which a current \( I \) flows is:

(a) directly proportional to \( r \)

(b) inversely proportional to \( I \)

(c) directly proportional to \( I \)

(d) directly proportional to \( I^2 \)

The refractive index \( \mu \) of the material of a prism is given by:

Formula for Refractive Index \[ \mu = \frac{\sin \left( \frac{A + \delta_m}{2} \right)}{\sin \left( \frac{A}{2} \right)} \]

where \( A \) is the apex angle of the prism and \( \delta_m \) is the angle of minimum deviation.

(a) \( \delta_m = 180^{\circ} - 4A \)

(b) \( \delta_m = 180^{\circ} - 3A \)

(c) \( \delta_m = 180^{\circ} - 2A \)

(d) \( \delta_m = 180^{\circ} - A \)

If \( \mathbf{A} = 4\hat{i} + 3\hat{j} \) and \( \mathbf{B} = 3\hat{i} + 4\hat{j} \), then the cosine of the angle between \( \vec{A} \) and \( \vec{A} + \vec{B} \) is:

(a) \( \frac{9\sqrt{2}}{5} \)

(b) \( \frac{7}{5\sqrt{2}} \)

(c) \( \frac{5\sqrt{2}}{49} \)

(d) \( \frac{5\sqrt{2}}{28} \)

Two light beams of intensities in the ratio of 9:4 are allowed to interfere. The ratio of the intensity of maxima and minima will be:

(a) 2:3

(b) 16:81

(c) 25:169

(d) 25:1

A heavy box of mass 50 kg is moving on a horizontal surface. If the coefficient of kinetic friction between the box and the horizontal surface is 0.3, then the force of kinetic friction is:

(a) 14.7 N

(b) 147 N

(c) 1.47 N

(d) 1470 N

When unpolarized light is incident at an angle of 60° on a transparent medium from air, the reflected ray is completely polarized. The angle of refraction in the medium is:

(a) 30°

(b) 60°

(c) 90°

(d) 45°

The waves emitted when a metal target is bombarded with high energy electrons are:

(a) Microwaves

(b) X rays

(c) Infrared rays

(d) Radio waves

Two bodies of mass 4 g and 25 g are moving with equal kinetic energies. The ratio of magnitude of their linear momentum is:

(a) 3 : 5

(b) 5 : 4

(c) 2 : 5

(d) 4 : 5

Five charges \( +q, +5q, -2q, +3q \) and \( -4q \) are situated as shown in the figure. The electric flux due to this configuration through the surface S is:



(a) \( \frac{5q}{\epsilon_0} \)

(b) \( \frac{4q}{\epsilon_0} \)

(c) \( \frac{3q}{\epsilon_0} \)

(d) \( \frac{q}{\epsilon_0} \)

Mass numbers of two nuclei are in the ratio of 4:3. Their nuclear densities will be in the ratio of:

(a) \( 4:3 \)

(b) \( \left( \frac{3}{4} \right)^{1/3} \)

(c) \( 1:1 \)

(d) \( \left( \frac{4}{3} \right)^{1/3} \)

Under the same load, wire A having length 5.0 m and cross-section \( 2.5 \times 10^{-5} \, m^2 \) stretches uniformly by the same amount as another wire B of length 6.0 m and a cross-section \( 3.0 \times 10^{-5} \, m^2 \) stretches. The ratio of the Young's modulus of wire A to that of wire B will be:

(a) 1:4

(b) 1:1

(c) 1:10

(d) 1:2

A current of 10 A exists in a wire of cross
sectional area of \( 5 \times 10^{-6} \, m^2 \) with a drift velocity of \( 2 \times 10^{-3} \, m/s \). The number of free electrons in each cubic meter of the wire is:

(a) \( 2 \times 10^{25} \)

(b) \( 2 \times 10^{23} \)

(c) \( 625 \times 10^{25} \)

(d) \( 2 \times 10^{6} \)

A long straight wire of circular cross-section (radius \( a \)) is carrying a steady current \( I \). The current \( I \) is uniformly distributed across this cross-section. The magnetic field is:

(a) Zero in the region \( r < a \) and inversely proportional to \( r \) in the region \( r > a \)

(b) Inversely proportional to \( r \) in the region \( r < a \) and uniform throughout in the region \( r > a \)

(c) Directly proportional to \( r \) in the region \( r < a \) and inversely proportional to \( r \) in the region \( r > a \)

(d) Uniform in the region \( r < a \) and inversely proportional to distance \( r \) from the axis, in the region \( r > a \)

A small ball of mass \( M \) and density \( \rho \) is dropped in a viscous liquid of density \( \rho_0 \). After some time, the ball falls with a constant velocity. What is the viscous force on the ball?

(a) \( F = Mg \left( 1
\frac{\rho_0}{\rho} \right) \)

(b) \( F = Mg \left( 1 + \frac{\rho}{\rho_0} \right) \)

(c) \( F = Mg \left( 1 + \frac{\rho_0}{\rho} \right) \)

(d) \( F = Mg \left( 1 \pm \rho \rho_0 \right) \)

In the circuit shown, find \( C \) if the effective capacitance of the whole circuit is to be 0.5 \(\mu F\). All values in the circuit are in \(\mu F\).



(a) \( \frac{7}{11} \, \mu F \)

(b) \( \frac{6}{5} \, \mu F \)

(c) \( 4 \, \mu F \)

(d) \( \frac{7}{10} \, \mu F \)

Choose the correct option from the following options given below:

(a) In the ground state of Rutherford's model electrons are in stable equilibrium. While in Thomson's model electrons always experience a net
force.

(b) An atom has a nearly continuous mass distribution in Rutherford's model but has a highly non
uniform mass distribution in Thomson's model.

(c) A classical atom based on Rutherford's model is doomed to collapse.

(d) The positively charged part of the atom possesses most of the mass in Rutherford's model but not in Thomson's model.

A cylinder of height 20 m is completely filled with water. The velocity of efflux of water (in ms\(^{-1}\)) through a small hole on the side wall of the cylinder near its bottom is:

• (A) \( 10 \)
• (B) \( 20 \)
• (C) \( 25.5 \)
• (D) \( 5 \)

The resistivity (\( \rho \)) of a semiconductor varies with temperature. Which of the following curves represents the correct behavior?

(A) Curve (a)

(B) Curve (b)

(C) Curve (c)

(D) Curve (d)

Two parallel, long wires are kept 0.20 m apart in vacuum, each carrying current of \( x \) A in the same direction. If the force of attraction per meter of each wire is \( 2 \times 10^{-6} \, N \), then the value of \( x \) is approximately:

(A) \(1\)

(B) \(2.4\)

(C) \(1.4\)

(D) \(2\)

The output \( Y \) of the following logic circuit for given inputs is:

(A) \( A \cdot B (A + B) \)

• (B) \( A \cdot B \)
• (C) \( 0 \)
• (D) \( A \cdot B \)

If you are provided a set of resistances (\(2 \, \Omega\), \(4 \, \Omega\), \(6 \, \Omega\), and \(8 \, \Omega\)), connect these resistances so as to obtain an equivalent resistance of \(\frac{46}{3} \, \Omega\).

• (A) \(4 \, \Omega\) and \(6 \, \Omega\) are in parallel with \(2 \, \Omega\) and \(8 \, \Omega\) in series
• (B) \(6 \, \Omega\) and \(8 \, \Omega\) are in parallel with \(2 \, \Omega\) and \(4 \, \Omega\) in series
• (C) \(2 \, \Omega\) and \(6 \, \Omega\) are in parallel with \(4 \, \Omega\) and \(8 \, \Omega\) in series
• (D) \(2 \, \Omega\) and \(4 \, \Omega\) are in parallel with \(6 \, \Omega\) and \(8 \, \Omega\) in series

Two cells of the same emf but different internal resistances \( r_1 \) and \( r_2 \) are connected in series with a resistance \( R \). The value of resistance \( R \), for which the potential difference across the second cell is zero, is:

• (A) \( r_2 - r_1 \)
• (B) \( r_1 - r_2 \)
• (C) \( r_1 \)
• (D) \( r_2 \)

A cylinder of fixed capacity of 44.8 litres contains helium gas at standard temperature and pressure. The amount of heat needed to raise the temperature of gas in the cylinder by 20.0°C will be:

• (A) 249 J
• (B) 415 J
• (C) 498 J
• (D) 830 J

In the given circuit, the rms value of current (\( I_{rms} \)) through the resistor \( R \) is:

• (A) \(2A\)
• (B) \(\frac{1}{2}A\)
• (C) \(20A\)
• (D) \(2\sqrt{2}A\)

At what temperature should a gold ring of diameter 6.230 cm be heated so that it can be fitted on a wooden bangle of diameter 6.241 cm? Both the diameters have been measured at room temperature (27 °C).

Given: Coefficient of linear thermal expansion of gold \( \alpha = 1.4 \times 10^{-5} \, K^{-1} \).

(A) 125.7°C

(B) 91.7°C

(C) 425.7°C

(D) 152.7°C

The wavelength (in cm) of the second line in the Lyman series of the hydrogen atomic spectrum is (Rydberg constant \( R \, cm^{-1})\).

(A) \(\frac{8R}{9}\)

(B) \(\frac{9}{8R}\)

(C) \(\frac{4}{3R}\)

(D) \(\frac{3R}{4}\)

Polarizability of halide ions increases in the order:

(A) \( F^- < I^- < Br^- < Cl^- \)

(B) \( Cl^- < Br^- < I^- < F^- \)

(C) \( I^- < Br^- < Cl^- < F^- \)

(D) \( F^- < Cl^- < Br^- < I^- \)

The statement that is not correct for periodic classification of elements is:

• (A) The properties of elements are periodic function of their atomic numbers.
• (B) Non-metallic elements are less in number than metallic elements.
• (C) For transition elements, the \(3d\)-orbitals are filled with electrons after \(3p\)-orbitals and before \(4s\)-orbitals.
• (D) The first ionisation enthalpies of elements generally increase with increase in atomic number as we go along a period.

PCl\textsubscript{5} is dissociating 50% at 250°C at a total pressure of \( P \) atm. If the equilibrium constant is \( K_p \), then which of the following relation is numerically correct?

(A) \( K_p = 3P \)

(B) \( P = 3K_p \)

(C) \( P = \frac{2K_p}{3} \)

(D) \( K_p = \frac{2P}{3} \)

Which of the following statements is false?

(A) Cannizzaro reaction is given by aldehydes in presence of alkali.

(B) Aldol condensation is given by aldehydes in presence of alkali.

(C) Aldol condensation is given by aldehydes and ketones in presence of acids.

(D) None of the above.

How many electrons are involved in the following redox reaction?
\[ Cr_2O_7^{2-} + Fe^{2+} + C_2O_4^{2-} \rightarrow Cr^{3+} + Fe^{3+} + CO_2 \quad (Unbalanced) \]

(A) \( 3 \)

(B) \( 4 \)

(C) \( 6 \)

(D) \( 5 \)

The IUPAC name for

• (A) 1-chloro-2-nitro-4-methylbenzene
• (B) 1-chloro-4\-methyl-2-nitrobenzene
• (C) 2-chloro-1-nitro-5-methylbenzene
• (D) m-nitro-p-chlorotoluene

The decreasing order of reactivity towards dehydrohalogenation (\(E_1\)) reaction of the following compounds is:

(A) \( D > B > C > A \)

(B) \( B > D > A > C \)

(C) \( B > D > C > A \)

(D) \( B > A > D > C \)

Specific conductance of 0.1 M HNO\(_3\) is \(6.3 \times 10^{-2} \, ohm^{-1} \, cm^{-1}\). The molar conductance of the solution is:

• (A) \(100 \, ohm^{-1} \, cm^{2} \, mol^{-1}\)
• (B) \(515 \, ohm^{-1} \, cm^{2} \, mol^{-1}\)
• (C) \(630 \, ohm^{-1} \, cm^{2} \, mol^{-1}\)
• (D) \(6300 \, ohm^{-1} \, cm^{2} \, mol^{-1}\)

Which of the following compounds does not show Lassaigne’s test for nitrogen?

(A) Urea

(B) Hydrazine

(C) Phenylhydrazine

(D) Azobenzene

The strongest acid among the following is:

• (A) Salicylic acid
• (B) \( m \)-hydroxybenzoic acid
• (C) \( p \)-hydroxybenzoic acid
• (D) Benzoic acid

At 25°C and 1 atm pressure, the enthalpy of combustion of benzene (l) and acetylene (g) are \(3268 \, kJ mol^{-1}\) and \(1300 \, kJ mol^{-1}\), respectively. The change in enthalpy for the reaction \(3 \, C_2H_2(g) \rightarrow C_6H_6(l)\), is

(A) \( +324 \, kJ mol^{-1} \)

(B) \( +632 \, kJ mol^{-1} \)

(C) \( -632 \, kJ mol^{-1} \)

(D) \( -732 \, kJ mol^{-1} \)

How many conformations are possible for ethane?

• (A) 2
• (B) 3
• (C) Infinite
• (D) One

For which one of the following sets of four quantum numbers, an electron will have the highest energy?

 

Using VSEPR theory, predict the species which has square pyramidal shape.

• (A) SnCl2
• (B) CCl4
• (C) SO3
• (D) BrF5

Which of the following relation is not correct?

• (A) \( \Delta H = \Delta U - P \Delta V \)
• (B) \( \Delta U = q + W \)
• (C) \( \Delta S_{sys} + \Delta S_{surr} \geq 0 \)
• (D) \( \Delta G = \Delta H - T \Delta S \)

The acidic, basic and amphoteric oxides, respectively, are:

• (A) \( Na_2O, SO_3, Al_2O_3 \)
• (B) \( Cl_2O, CaO, P_4O_{10} \)
• (C) \( N_2O_3, Li_2O, Al_2O_3 \)
• (D) \( MgO, Cl_2O, Al_2O_3 \)

100 mL of 0.04 N HCl aqueous solution is mixed with 100 mL of 0.02 N NaOH solution. The pH of the resulting solution is:

• (A) 1.0
• (B) 1.7
• (C) 2.0
• (D) 2.3

Which of the following tests is positive for the given compound [A]?

(A) Br2 water test

(B) Tollen’s test

(C) Victor Meyer test

(D) Lucas test

The value of van't Hoff factors for KCl, NaCl and \( K_2SO_4 \) respectively are ......

• (A) 2, 2 and 2
• (B) 2, 2 and 3
• (C) 1, 1 and 2
• (D) 1, 1 and 1

A first order reaction is half completed in 45 minutes. How long does it need for 99.9% of the reaction to be completed?

• (A) 5 hours
• (B) 7.5 hours
• (C) 10 hours
• (D) 20 hours

The greater the valence of the flocculating ion added, the greater is its power to cause precipitation of a colloid. This rule is:

• (A) Hund's rule
• (B) Pauling rule
• (C) Henry's rule
• (D) Hardy—Schulze rule

The number of hydrogen bonds formed by a water molecule at normal conditions is:

(a) 1

(b) 2

(c) 3

(d) 4

Element not showing variable oxidation state is:

(a) Bromine

(b) Iodine

(c) Chlorine

(d) Fluorine

Which of the following arrangements does not represent the correct order of the property stated against it?

(a) \( V^{2+} < Cr^{2+} < Fe^{2+} < Mn^{2+} \): paramagnetic behaviour

(b) \( Ni^{2+} < Co^{2+} < Fe^{2+} < Mn^{2+} \): ionic size

(c) \( Co^{3+} < Fe^{3+} < Cr^{3+} < Sc^{3+} \): stability in aqueous solution

(d) \( Sc < Ti < Cr < Mn \): number of oxidation states

Lanthanoid which has the smallest size in \( +3 \) state is:

(a) Tb

(b) Er

(c) Ce

(d) Lu

Oxidation number of H in NaH, CaH$_2$, and LiH, respectively is:

(a) +1, +1, -1

(b) -1, +1, +1

(c) +1, +1, +1

(d) -1, -1, -1

Product of the following reaction is:

Reaction Diagram

Which of the following alkyl halides will undergo \( S_N1 \) reaction most readily?

(a) \( (CH_3)_3 C \, F \)

(b) \( (CH_3)_3 C \, Cl \)

(c) \( (CH_3)_3 C \, Br \)

(d) \( (CH_3)_3 C \, I \)

Phenol does not undergo nucleophilic substitution reaction easily due to:

(a) acidic nature of phenol

(b) partial double bond character of C
OH bond

(c) partial double bond character of C
C bond

(d) instability of phenoxide ion

The reaction





(a) Rosenmund's reaction

(b) Stephen's reaction

(c) Cannizzaro's reaction

(d) Gattermann-Koch reaction

Hoffmann bromamide degradation reaction is shown by which of the following?





(a) ArNH\(_2\)

(b) ArCONH\(_2\)

(c) ArNO\(_2\)

(d) ArCH\(_2\)NH\(_2\)

Which of the following acids is a vitamin?

(a) Aspartic acid

(b) Ascorbic acid

(c) Adipic acid

(d) Saccharic acid

Intermolecular forces in nylon
6, 6 are:

(a) Dipole
dipole interactions

(b) Hydrogen bonding

(c) van der Waals' forces

(d) Ionic bonds

Which of the following does not form a chelate?

(a) EDTA

(b) Oxalate

(c) Pyridine

(d) Ethylenediamine

Let \( n(A) = m \) and \( n(B) = n \), if the number of subsets of \( A \) is 56 more than that of subsets of \( B \), then \( m + n \) is equal to:

(a) 9

(b) 13

(c) 8

(d) 10

If \( f(x) = \cos^{
1} \left( \frac{\sqrt{2x^2 + 1}}{x^2 + 1} \right) \), then the range of \( f(x) \) is:

(a) \( [0, \pi] \)

(b) \( \left[ 0, \frac{\pi}{4} \right] \)

(c) \( \left[ 0, \frac{\pi}{3} \right] \)

(d) \( \left[ 0, \frac{\pi}{2} \right] \)

A, P, B are \( 3 \times 3 \) matrices. If \( |B| = 5 \), \( | BA^T | = 15 \), \( | P^T AP | = -27 \), then one of the values of \( | P | \) is:

(a) 3

(b) -5

(c) 9

(d) 6

If \( f(x) \) defined as given below, is continuous on \( R \), then the value of \( a + b \) is equal to:

Function Definition \[ f(x) = \begin{cases} \sin x, & x \leq 0
x^2 + a, & 0 < x < 1
bx + 3, & 1 \leq x \leq 3
-3, & x > 3 \end{cases} \]

(a) 0

(b) 2

(c) -2

(d) 3

Let \( f(x) \) be defined as:
\[ f(x) = \begin{cases} 3 - x, & x < -3
6, & -3 \leq x \leq 3
3 + x, & x > 3 \end{cases} \]

Let \( \alpha \) be the number of points of discontinuity of \( f(x) \) and \( \beta \) be the number of points where \( f(x) \) is not differentiable. Then, \( \alpha + \beta \) is:
(a) 6

(b) 3

(c) 2

(d) 0

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

The number of students who take both the subjects mathematics and chemistry is 30. This represents 10% of the enrolment in mathematics and 12% of the enrolment in chemistry. How many students take at least one of these two subjects?

(a) 520

(b) 490

(c) 560

(d) 480

If \( A \) and \( B \) are the two real values of \( k \) for which the system of equations \( x + 2y + z = 1 \), \( x + 3y + 4z = k \), \( x + 5y + 10z = k^2 \) is consistent, then \( A + B = \):

(a) 3

(b) 4

(c) 5

(d) 7

The radius of the base of a cone is increasing at the rate of 3 cm/minute and the altitude is decreasing at the rate of 4 cm/minute. The rate of change of lateral surface when the radius is 7 cm and altitude is 24 cm is:

(a) \( 54 \pi \, cm^2/min \)

(b) \( 7 \pi \, cm^2/min \)

(c) \( 27 \pi \, cm^2/min \)

(d) None of these

The maximum area of a right-angled triangle with hypotenuse \( h \) is:

(a) \( \frac{h^2}{2\sqrt{2}} \)

(b) \( \frac{h^2}{2} \)

(c) \( \frac{h^2}{\sqrt{2}} \)

(d) \( \frac{h^2}{4} \)

If \( f(x) \) is continuous and \( \int_0^9 f(x) \, dx = 4 \), then the value of the integral \( \int_0^3 x \cdot f(x^2) \, dx \) is:

(a) 2

(b) 18

(c) 16

(d) 4

The area bounded by \( y - 1 = |x| \) and \( y + 1 = |x| \) is:

(a) \( \frac{1}{2} \)

(b) 1

(c) 2

(d) 0

Let the vectors \( \overrightarrow{AB} = 2\hat{i} + 2\hat{j} + \hat{k} \) and \( \overrightarrow{AC} = 2\hat{i} + 4\hat{j} + 4\hat{k} \) be two sides of a triangle ABC. If \( G \) is the centroid of \( \triangle ABC \), then \( \frac{22}{7} |\overrightarrow{AG}|^2 + 5 = \):

(a) 25

(b) 38

(c) 47

(d) 52

If the two lines \( l_1: \frac{x - 2}{3} = \frac{y + 1}{-2} = \frac{z - 2}{0} \) and \( l_2: \frac{x - 1}{1} = \frac{y + 3}{\alpha} = \frac{z + 5}{2} \) are perpendicular, then the angle between the lines \( l_2 \) and \( l_3: \frac{x - 1}{-3} = \frac{y - 2}{-2} = \frac{z - 0}{4} \) is:

(a) \( \cos^{-1} \left( \frac{29}{4} \right) \)

(b) \( \sec^{-1} \left( \frac{29}{4} \right) \)

(c) \( \cos^{-1} \left( \frac{2}{29} \right) \)

(d) \( \cos^{-1} \left( \frac{2}{\sqrt{29}} \right) \)

If \( f(x) = \ln \left( \frac{x^2 + e}{x^2 + 1} \right) \), then the range of \( f(x) \) is:

(a) \( (0, 1) \)

(b) \( (0, 1] \)

(c) \( [0, 1] \)

(d) \( \{0, 1\} \)

If \( f(x) = \frac{\log(\pi + x)}{\log(e + x)} \), then the function is:

(a) Increasing in \( [0, \infty) \)

(b) Decreasing in \( [0, \infty) \)

(c) Decreasing in \( [0, \frac{\pi}{e}] \) and increasing in \( [\frac{\pi}{e}, \infty) \)

(d) Increasing in \( [0, \pi] \) and decreasing in \( [\pi, \infty) \)

Evaluate the integral: \[ \int_{
-\pi}^{\pi} x^2 \sin(x) \, dx \]

(a) \( \pi^2 \)

(b) \( \frac{\pi^2}{2} \)

(c) 0

(d) \( 2\pi^2 \)

The solution of the differential equation: \[ x^4 \frac{dy}{dx} + x^3 y + \csc(xy) = 0 \]
is equal to:

(a) \( x^{-2} + 2 \cos(xy) = c \)

(b) \( y^{-2} + 2 \cos(xy) = c \)

(c) \( x^{-2} + 2 \sin(xy) = c \)

(d) \( y^{-2} + 2 \sin(xy) = c \)

If the solution of

\[ \left( 1 + 2e^\frac{x}{y} \right) dx + 2e^\frac{x}{y} \left( 1 - \frac{x}{y} \right) dy = 0 \]

is

\[ x + \lambda y e^\frac{x}{y} = c \quad \text{(where \(c\) is an arbitrary constant)}, \quad \text{then} \quad \lambda \text{ is:} \]

The probability distribution of a random variable is given below:

The length of the perpendicular from the point \( (1, -2, 5) \) on the line passing through \( (1, 2, 4) \) and parallel to the line given by \( x + y - z = 0 \) and \( x - 2y + 3z - 5 = 0 \) is:

(a) \( \frac{\sqrt{21}}{2} \)

(b) \( \frac{\sqrt{9}}{2} \)

(c) \( \frac{\sqrt{73}}{2} \)

(d) 1

The range of \( 2 \left| \sin x + \cos x \right| - \sqrt{2} \) is:

(a) \( \left[ -\sqrt{2}, \sqrt{2} \right] \)

(b) \( \left[ -3\sqrt{2}, \sqrt{2} \right] \)

(c) \( \left( -3\sqrt{2}, \sqrt{2} \right) \)

(d) None of these

The integral \( I = \int_{\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{x + \frac{\pi}{4}}{2 - \cos 2x} \, dx \) is equal to:

(a) \( \frac{8\pi \sqrt{3}}{5} \)

(b) \( \frac{2\pi \sqrt{3}}{9} \)

(c) \( \frac{4\pi^2 \sqrt{3}}{9} \)

(d) \( \frac{\pi^2}{6\sqrt{3}} \)

A and B are independent events of a random experiment if and only if:

(a) \( P(A | B) \neq P(A \cap B) \)

(b) \( P(A | B) = P(B | A) \)

(c) \( P(A | B) \neq P(A | B^c) \)

(d) \( P(A | B) = P(A | B^c) \)

Correct Answer: (d) \( P(A | B) = P(A | B^c) \ )

The equation of a common tangent to the parabolas \( y = x^2 \) and \( y = -(x - 2)^2 \) is:

(a) \( y = 4(x - 2) \)

(b) \( y = 4(x - 1) \)

(c) \( y = 4(x + 1) \)

(d) \( y = 4(x + 2) \)

Negation of the statement \( (p \land r) \rightarrow (r \lor q) \) is:

(a) \( \sim (p \land r) \rightarrow \sim (r \lor q) \)

(b) \( (\sim p \lor \sim r) \lor (r \lor q) \)

(c) \( (p \land r) \land (r \land q) \)

(d) \( (p \land r) \land (\sim r \land \sim q) \)

The number of different permutations of all the letters of the word "PERMUTATION" such that any two consecutive letters in the arrangement are neither both vowels nor both identical is:

(a) \( 63 \times 6! \times 5! \)

(b) \( 57 \times 5! \times 5! \)

(c) \( 33 \times 6! \times 5! \)

(d) \( 7 \times 7! \times 5! \)

The coefficient of \( x^{50} \) in \( (1 + x)^{101} (1 - x + x^2)^{100} \) is:

(a) 1

(b) -1

(c) 0

(d) 2

If \( \frac{1}{q + r}, \frac{1}{r + p}, \frac{1}{p + q} \) are in A.P., then:

(a) \( p, q, r \) are in A.P.

(b) \( p^2, q^2, r^2 \) are in A.P.

(c) \( \frac{1}{p}, \frac{1}{q}, \frac{1}{r} \) are in A.P.

(d) \( p + q + r \) are in A.P.

The coordinates of the foot of perpendicular from the point \( (2, 3) \) on the line \( y = 3x + 4 \) is given by:

(a) \( \left( \frac{37}{10}, \frac{-1}{10} \right) \)

(b) \( \left( \frac{-1}{10}, \frac{37}{10} \right) \)

(c) \( \left( \frac{10}{37}, -10 \right) \)

(d) \( \left( \frac{2}{2}, \frac{-1}{3} \right) \)

The circle touching the y axis at a distance 4 units from the origin and cutting off an intercept 6 from the x axis is:

(A) \(x^2 + y^2 \pm 10x - 8y + 16 = 0\)

• (B) \(x^2 + y^2 \pm 5x - 8y + 16 = 0\)
• (C) \(x^2 + y^2 \pm 5x - 2y - 8 = 0\)
• (D) \(x^2 + y^2 \pm 2x - y - 12 = 0\)

The points A(4, -2, 1), B(7, -4, 7), C(2, -5, 10), and D(-1, -3, 4) are the vertices of a:

• (A) Tetrahedron
• (B) Parallelogram
• (C) Rhombus
• (D) Square

If A, B, C, D are the angles of a quadrilateral, then \[ \frac{\tan A + \tan B + \tan C + \tan D}{\cot A + \cot B + \cot C + \cot D} = \]

• (A) \(\cot A\) \(\cot B\) \(\cot C\) \(\cot D\)
• (B) \(\tan A\) \(\tan B\) \(\tan C\) \(\tan D\)
• (C) \(
  \tan A\) \(\tan B\) \(\tan C\) \(\tan D\)
• (D) \(
  \cot A\) \(\cot B\) \(\cot C\) \(\cot D\)

Let \( z \neq 1 \) be a complex number and let \( \omega = x + iy, y \neq 0 \). If \[ \frac{\omega
\overline{\omega}z}{1
z} \]
is purely real, then \( |z| \) is equal to

• (A) \( |\omega| \)
• (B) \( |\omega|^2 \)
• (C) \( \frac{1}{|\omega|^2} \)
• (D) 1

If the roots of the quadratic equation \[ (a^2 + b^2) \, x^2 - 2 \, (bc + ad) \, x + (c^2 + d^2) = 0 \]
are equal, then:

(a) \( \frac{a}{b} = \frac{c}{d} \)

(b) \( \frac{a}{c} + \frac{b}{d} = 0 \)

(c) \( \frac{a}{d} = \frac{b}{c} \)

(d) \( a + b = c + d \)

Evaluate the limit: \[ L = \lim_{x \to 0} \frac{35^x - 7^x - 5^x + 1}{(e^x - e^{-x}) \ln(1 - 3x)} \]

(a) \( \frac{\ln 35}{6} \)

(b) \( \frac{\ln 35}{6} \)

(c) \( \frac{\ln 2}{6} \)

(d) \( \frac{\ln (5) \cdot \ln 7}{-6} \)

If \( z_r = \cos \frac{r\alpha}{n^2} + i \sin \frac{r\alpha}{n^2} \), where \( r = 1, 2, 3, ..., n \), then the value of \( \lim_{n \to \infty} z_1 z_2 z_3 ... z_n \) is:

(a) 0

(b) \( e^{\frac{i \alpha}{2}} \)

(c) \( e^{\frac{i \alpha}{2}} \)

(d) \( e^{i \alpha} \)

Let \( f(x) \) be a polynomial function satisfying
\[ f(x) \cdot f\left(\frac{1}{x}\right) = f(x) + f\left(\frac{1}{x}\right). \]

If \( f(4) = 65 \) and \( I_1, I_2, I_3 \) are in GP, then \( f'(I_1), f'(I_2), f'(I_3) \) are in:

(a) AP

(b) GP

(c) Both

(d) None of these

In four schools \( B_1, B_2, B_3, B_4 \), the number of students is given as follows:
\[ B_1 = 12, \quad B_2 = 20, \quad B_3 = 13, \quad B_4 = 17 \]

A student is selected at random from any of the schools. The probability that the student is from school \( B_2 \) is:

(a) \( \frac{6}{31} \)

(b) \( \frac{10}{31} \)

(c) \( \frac{13}{62} \)

(d) \( \frac{17}{62} \)

For real numbers \(x\) and \(y\), we define \(x R y\) iff \(x - y + \sqrt{5}\) is an irrational number. Then, relation \(R\) is:

(a) Reflexive

(b) Symmetric

(c) Transitive

(d) None of these

For a person, whose monthly salary is Rs 6,000 p.m., how many items are there on which he has to spend more than Rs 1,000 p.m.?

• (A) 1
• (B) 2
• (C) 3
• (D) 4

The annual saving for such a person will be approximately:

• (A) 5,000
• (B) 10,000
• (C) 15,000
• (D) 16,560

The monthly salary for a person who follows the same expense pattern, but has a petrol expense of Rs 500 p.m., is:

• (A) 2,500
• (B) 3,000
• (C) 5,000
• (D) 6,500

If NATION is coded as 467234 and EARN is coded as 1654, then ATTENTION should be coded as:

• (A) 432769561
• (B) 956143654
• (C) 766412743
• (D) 677147234

In the following figure, how many educated people are employed?

(a) 18

(b) 20

(c) 15

(d) 9

A man is facing west. He runs 45° in the clockwise direction and then another 180° in the same direction and then 270° in the anticlockwise direction. Which direction is he facing now?

(a) South

(b) North-West

(c) West

(d) South-West

A is the brother of B. A is the brother of C. To determine the relation between B and C, what minimum information is necessary?

(a) Only (i)

(b) Only (ii)

(c) Either (i) or (ii)

(d) Both (i) and (ii)

In a class of 20 students, Alisha’s rank is 15th from the top. Manav is 4 ranks above Alisha. What is Manav’s rank from the bottom?

(a) 10th

(b) 11th

(c) 9th

(d) 12th

In this question, there are three statements followed by conclusions numbered I and II. You have to take the given statements to be true even if they seem to be at variance from commonly known facts and then decide which of the given conclusions logically follow from the three statements.

Statements:
All books are ledgers.
All pens are keys.
Some pens are books.


Conclusions:
I. Some ledgers are keys.
II. Some keys are books.

Options:
(a) if only conclusion I follows

(b) if only conclusion II follows

(c) if neither I nor II follows

(d) if both I and II follow

If it was a Friday on 1 January 2016, what was the day of the week on 31 December 2016?

(a) Saturday

(b) Friday

(c) Monday

(d) Sunday