KEAM 2024 Question Paper with Answer Key (June 7)

Choose the INCORRECT dimensions:

• (A) Linear momentum: MLT-1
• (B) Angular momentum: ML2T-1
• (C) Speed of Light: M0L T-2
• (D) Kinetic energy: ML2T-2
• (E) Angular frequency: M0L0T-1

The length of the side of a cube is 1.1 * 10-2 m. Its volume in m3 up to correct significant figures is

• (A) 1.4 * 10-6
• (B) 1.33 * 10-6
• (C) 1.23 * 10-6
• (D) 1.42 * 10-6
• (E) 1.3 * 10-6

A person travels in a car from p to q with uniform speed u and returns to p with uniform speed v. The average speed for his round trip is

• (A) \(\dfrac{u+v}{2}\)
• (B) \(\dfrac{uv}{u+v}\)
• (C) \(\dfrac{\sqrt{uv}}{u+v}\)
• (D) \(\dfrac{2uv}{u+v}\)
• (E) \(\dfrac{uv}{u+v}\)

If \(\vec{a} = 0.4\hat{i} + 0.3\hat{j} + b\hat{k}\) is a unit vector, then the value of b is

• (A) \({\sqrt{3}}\)
• (B) \(\dfrac{2}{\sqrt{5}}\)
• (C) \(\dfrac{\sqrt{5}}{2}\)
• (D) \(\dfrac{1}{\sqrt{3}}\)
• (E) \(\dfrac{\sqrt{3}}{2}\)

The velocity (v)-time (t) graph for the motion of a body is a straight line making an angle 60° with the time axis. Then the body is moving with an acceleration (in m s\textsuperscript{-2}) of

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

A body of weight \( W \) is suspended from the ceiling of a room through a chain of weight \( w \). The ceiling pulls the chain by a force.

• (A) \( w \)
• (B) \( {Wg} \)
• (C) \( \dfrac{w + W}{2g} \)
• (D) \( \dfrac{w - W}{2} \)
• (E) \( w + W \)

The coefficient of friction between the road and the tyres of a cyclist is 0.1. The maximum speed with which he can take a circular turn of radius 2 m without skidding is (g = 10 m/s^{2})

• (A) \( \sqrt{2} \, ms^{-1} \)
• (B) \( \sqrt{3} \, ms^{-1} \)
• (C) \( \sqrt{5} \, ms^{-1} \)
• (D) \( 2 \, ms^{-1} \)
• (E) \( 3 \, ms^{-1} \)

A person standing in an elevator experiences weight loss when the elevator

• (A) moves down with uniform velocity
• (B) moves upward with constant acceleration
• (C) moves downward with constant acceleration
• (D) moves upward with uniform velocity
• (E) moves down with variable acceleration

The ratio of the maximum kinetic energy to the maximum potential energy of a bob of a simple pendulum executing small oscillations is

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

A constant force of 6 N acting on a stationary body displaces it by 3 m in 2 s. The average power delivered is

• (A) 18 W
• (B) 15 W
• (C) 12 W
• (D) 9 W
• (E) 6 W

A block of mass 3 kg executes simple harmonic motion under the restoring force of a spring. The amplitude and the time period of the motion are 0.1 m and 3.14 s respectively. The maximum force exerted by the spring on the block is

• (A) 1.2 N
• (B) 3 N
• (C) 12 N
• (D) 30 N
• (E) 90 N

The principle involved in the performance of a circus acrobat is the conservation of

• (A) translational energy
• (B) linear momentum
• (C) angular momentum
• (D) mass
• (E) rotational energy

For a smoothly running analog clock, the ratio of the angular velocity of the minute hand to the angular velocity of the hour hand is

• (A) 2
• (B) 12
• (C) 24
• (D) 60
• (E) 360

The height above the surface of the earth at which the acceleration due to gravity becomes half of that on the surface of the earth is (R is the radius of earth)

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

A particle of 100 g mass is projected vertically up with a kinetic energy of 20 J. The maximum height reached by the particle is (g = 10 m/s^{2}) (neglecting air resistance)

• (A) 5 m
• (B) 10 m
• (C) 15 m
• (D) 20 m
• (E) 25 m

A ball is projected in still air. With respect to the ball the streamlines appear as shown in the figure. If speed of air passing through the region 1 and 2 are \( v_1 \) and \( v_2 \), respectively and the respective pressures, \( P_1 \) and \( P_2 \), respectively, then

• (A) \( v_1 = v_2 ; P_1 = P_2 \)
• (B) \( v_1 > v_2 ; P_1 > P_2 \)
• (C) \( v_1 < v_2 ; P_1 < P_2 \)
• (D) \( v_1 > v_2 ; P_1 < P_2 \)
• (E) \( v_1 < v_2 ; P_1 > P_2 \)

If the radii of two soap bubbles are respectively 2 cm and 3 cm, then the ratio of the excess pressures inside the soap bubbles is

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

The elastic energy stored per unit volume in a stretched wire is \((Y = Young's modulus of the material of the wire; S = stress acting on the wire)\)

• (A) \( \dfrac{1}{2} \left( \dfrac{S}{Y} \right) \)
• (B) \( \dfrac{1}{2} \left( \dfrac{S}{Y^2} \right) \)
• (C) \( \dfrac{1}{2} \left( \dfrac{S^2}{Y} \right) \)
• (D) \( \dfrac{1}{2} \left( \dfrac{S^2}{Y^2} \right) \)
• (E) \( \dfrac{1}{2} (SY) \)

The zeroth law of thermodynamics leads to the concept of

• (A) Carnot engine
• (B) Work
• (C) Temperature
• (D) Heat
• (E) Internal energy

If \( m_a \) and \( m_i \) are the slopes of the adiabatic and isothermal curves for an ideal gas, then

• (A) \( m_a = \gamma m_i \)
• (B) \( m_i = \gamma m_a \)
• (C) \( m_a m_i = \gamma \)
• (D) \( m_a m_i = \gamma^2 \)
• (E) \( \sqrt{\frac{m_a}{m_i}} = \gamma \)

The work done by a gas on the system is zero in

• (A) adiabatic process
• (B) isothermal compression
• (C) isochores process
• (D) isobaric process
• (E) isothermal expansion

If \( c_p \), \( c_v \), and \( f \) are the specific heat capacity at constant pressure, specific heat capacity at constant volume, and number of degrees of freedom for a polyatomic gaseous system, then the ratio \( \frac{c_p}{c_v} \) is equal to

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

When the number of molecules per unit volume of an ideal gas is \( 0.8 \times 10^{24} \), the mean free path length for its molecules is \( 2.2 \times 10^{-5} \, m \). If the number of molecules per unit volume is \( 1.0 \times 10^{24} \), then the mean free path is

• (A) \( 17.6 \times 10^{-5} \, m \)
• (B) \( 1.76 \times 10^{-5} \, m \)
• (C) \( 3.52 \times 10^{-5} \, m \)
• (D) \( 35.2 \times 10^{-5} \, m \)
• (E) \( 8.8 \times 10^{-5} \, m \)

A particle executes a linear SHM with an amplitude \(a\) and angular velocity \(\omega\). The ratio between its acceleration amplitude and displacement amplitude is

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

Speed of a transverse wave on a stretched string under tension \(T\) and linear density \(\mu\) is

• (A) \( \sqrt{\frac{\mu}{T}} \)
• (B) \( \sqrt{\frac{T}{\mu}} \)
• (C) \( \sqrt{\mu T} \)
• (D) \( \mu T \)
• (E) \( \frac{\mu}{T} \)

The lowest frequency of the air column in an open pipe of length \( L \) is \( v \) (velocity of sound in air)

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

If \( E \) is the electric field intensity between the plates of a charged parallel plate capacitor, energy stored per unit volume in it is (permittivity of free space = \( \epsilon_0 \))

• (A) \( \epsilon_0 E^2 \)
• (B) \( \frac{1}{2} \epsilon_0 E^2 \)
• (C) \( \frac{1}{8} \epsilon_0 E^2 \)
• (D) \( \frac{1}{4} \epsilon_0 E^2 \)
• (E) \( \frac{1}{16} \epsilon_0 E^2 \)

Two like charges kept in air medium experience a force \( F \), when they are separated by a certain distance \( r \). When the same charges are kept in a dielectric medium at the same distance of the separation, the force between them is 0.5F. The dielectric constant of the medium is

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

The energy stored in the capacitor after closing the key K is

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

Masses of three copper wires are in the ratio 1:3:5 and their lengths are in the ratio 5:3:1. Then the ratio of their electric resistances is

• (A) \( 125 : 15 : 1 \)
• (B) \( 5 : 3 : 1 \)
• (C) \( 1 : 25 : 125 \)
• (D) \( 1 : 3 : 5 \)
• (E) \( 5 : 21 : 25 \)

Mobility \( \mu \) of an electron is related to average collision time \( \tau \) as
\textit{(e = electronic charge, m = mass of the electron)

• (A) \( \frac{1}{\tau} = m \mu \)
• (B) \( \mu =\frac{m \tau}{e} \)
• (C) \( \frac{1}{\mu} = \tau \)
• (D) \( \mu = \frac{e \tau}{m} \)
• (E) \( \mu \tau = e m \)

The electric power delivered by a transmission cable of resistance \( R_c \) at a voltage \( V \) is \( P \). The power dissipated is

• (A) \( \frac{PV}{R_c} \)
• (B) \( \frac{PR_c}{V} \)
• (C) \( PVR_c \)
• (D) \( \frac{P^2 R_c}{V^2} \)
• (E) \( \frac{P^2 R_c^2}{V} \)

The ratio of radii of the circular paths of a proton and a deuteron when projected perpendicular to the direction of a uniform magnetic field with the same speed is

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

An alternative form of Biot-Savart's law is

• (A) Gauss's law
• (B) Ohm's law
• (C) Coulomb's law
• (D) Ampere's circuital law
• (E) Joule's law

In an LCR series resonance circuit driven by the alternating voltage \( V = V_0 \sin \omega t \), inductance \( L = 1 \, \mu H \), capacitance \( C = 1 \, \mu F \) and resistance \( R = 1 \, k\Omega \). The resonant angular frequency (in rad/s) is:

• (A) \( 10^6 \)
• (B) \( 10^{-6} \)
• (C) \( 10^{12} \)
• (D) \( 10^{16} \)
• (E) \( 10^{10} \)

Electromagnetic waves of frequency \( 5 \times 10^{14} \, Hz \) lie in the

• (A) ultraviolet region
• (B) infrared region
• (C) visible region
• (D) radio region
• (E) microwave region

Whenever light travels from rarer medium into denser medium its

• (A) frequency increases
• (B) wavelength increases
• (C) frequency decreases
• (D) wavelength decreases
• (E) wavelength remains unchanged

Young’s double-slit experiment is carried out by using green, red and blue lights, one at a time. The fringe widths recorded are \( \beta_G \), \( \beta_R \) and \( \beta_B \) respectively. Then

• (A) \( \beta_G < \beta_R < \beta_B \)
• (B) \( \beta_B < \beta_R < \beta_G \)
• (C) \( \beta_G < \beta_B < \beta_R \)
• (D) \( \beta_B < \beta_G < \beta_R \)
• (E) \( \beta_G = \beta_R = \beta_B \)

The number of de Broglie waves associated with Bohr electron when it completes one revolution in its third orbit is

• (A) 1
• (B) 3
• (C) 5
• (D) 6
• (E) \( \infty \)

The particle which is expected to be emitted along with \( Y \) in the following nuclear reaction is \[ ^{198}_{80}X \rightarrow ^{197}_{79}Y + ? \]

• (A) \( \alpha \)-particle
• (B) \( \beta^+ \)-particle
• (C) \( \beta^- \)-particle
• (D) proton
• (E) neutron

In a nuclear fusion process, the masses of the fusing nuclei are \( M_A \) and \( M_B \). Then the mass of the product nucleus \( M_C \) is related to \( M_A \) and \( M_B \) as

• (A) \( M_C < M_A + M_B \)
• (B) \( M_C > M_A + M_B \)
• (C) \( M_C = |M_A - M_B| \)
• (D) \( M_C = M_A + M_B \)
• (E) \( M_C = \frac{M_A + M_B}{2} \)

The electron concentration \( n_e \) and hole concentration \( n_h \) in semiconductor are related to the number of intrinsic charge concentration \( n_i \) as

• (A) \( n_e n_h = n_i^2 \)
• (B) \( n_e + n_h = n_i^2 \)
• (C) \( n_e + n_h = 2n_i^2 \)
• (D) \( n_e n_h = 2n_i \)
• (E) \( n_e n_h^2 = n_i \)

The half-life period of a radioactive element is 2 days. If \( \frac{1}{32} \) part of the initial amount remains undecayed after a time \( t \), then the value of \( t \) in days is

• (A) 8
• (B) 10
• (C) 6
• (D) 12
• (E) 4

An intrinsic semiconductor at \( T = 0 \, K \) behaves like

• (A) insulator
• (B) n-type semiconductor
• (C) p-type semiconductor
• (D) conductor
• (E) superconductor

When a diode is reverse biased

• (A) applied voltage in the p-side is positive
• (B) the depletion layer width decreases
• (C) the applied voltage is in the opposite direction of barrier potential
• (D) minority carriers are not allowed to cross the barrier
• (E) the barrier height increases

10 g of alcohol is dissolved in 90 g of water. The percentage of alcohol in the solution is

• (A) 10\(% \)
• (B) 90\(% \)
• (C) 20\(% \)
• (D) 100\(% \)
• (E) 1\(% \)

Which of the following set of quantum numbers is possible?

• (A) \( n = 3, l = 2, m_l = -4, m_s = \frac{1}{2} \)
• (B) \( n = 2, l = 2, m_l = 0, m_s = \frac{1}{2} \)
• (C) \( n = 2, l = 2, m_l = -1, m_s = 1 \)
• (D) \( n = 3, l = 2, m_l = -2, m_s = \frac{1}{2} \)
• (E) \( n = 3, l = 3, m_l = -2, m_s = \frac{1}{2} \)

The electronic configuration of Pd (Z = 46) is

• (A) [Kr] \( 4d^8 5s^2 5p^0 \)
• (B) [Kr] \( 4d^9 5s^1 5p^0 \)
• (C) [Kr] \( 4d^{10} 5s^0 5p^0 \)
• (D) [Kr] \( 4d^5 5s^2 2p^3 \)
• (E) [Kr] \( 4d^6 5s^2 5p^2 \)

Which of the following has square planar structure?

• (A) NH\(_4^+\)
• (B) XeF\(_4\)
• (C) CCl\(_4\)
• (D) SiCl\(_4\)
• (E) CH\(_4\)

Which of the following molecule is paramagnetic?

• (A) \( O_2 \)
• (B) \( C_2 \)
• (C) \( N_2 \)
• (D) \( F_2 \)
• (E) \( H_2 \)

The vapour pressure of H\(_2\)O at 323K is 95 mm of Hg. 176g of sucrose (Molar mass = 342 gmol\(^{-1}\)) is added to 900g of H\(_2\)O at 323K. The vapour pressure of solution is about

• (A) 93.94 mm
• (B) 92.88 mm
• (C) 96.06 mm
• (D) 95.33 mm
• (E) 94.06 mm

Which of the following statement is incorrect?

• (A) The greater the disorder in an isolated system, the higher is the entropy.
• (B) The crystalline solid state of a substance is the state of lowest entropy.
• (C) Entropy is not the measure of average chaotic motion of particles in the system.
• (D) The gaseous state of a substance is state of highest entropy.
• (E) \( \Delta S \) is related to \( q \) and \( T \) for a reversible reaction as \( \Delta S = \frac{q_{rev}}{T} \)

PCl\(_5\)(g), PCl\(_3\)(g) and Cl\(_2\)(g) are at equilibrium at 500 K. The equilibrium concentrations of PCl\(_3\), Cl\(_2\) and PCl\(_5\) are respectively 4.0 M, 4.0 M and 2.0 M. Calculate \( K_c \) for the reaction: \[ PCl_5(g) \rightleftharpoons PCl_3(g) + Cl_2(g) \]

• (A) 2 mol dm\(^3\)
• (B) 4 mol dm\(^3\)
• (C) 6 mol dm\(^3\)
• (D) 8 mol dm\(^3\)
• (E) 10 mol dm\(^3\)

Which of the following statement is true with regard to Daniell cell?

• (A) Oxidation occurs at cathode
• (B) Reduction occurs at anode
• (C) \( E^\circ_{cell} \) is 1.1 V
• (D) Electrical energy produces chemical reaction
• (E) Electrolytes are aqueous solutions of CuSO\(_4\) and FeSO\(_4\).

The conductivity of 0.02 mol L\(^{-1}\) KCl solution is 0.248 S m\(^{-1}\). Its molar conductivity is

• (A) 20 S m\(^2\) mol\(^{-1}\)
• (B) \( 1.24 \times 10^{-3} \) S m\(^2\) mol\(^{-1}\)
• (C) \( 1.24 \times 10^{-4} \) S m\(^2\) mol\(^{-1}\)
• (D) \( 2.48 \times 10^{-2} \) S m\(^2\) mol\(^{-1}\)
• (E) \( 1.24 \times 10^{-2} \) S m\(^2\) mol\(^{-1}\)

Which of the following compound has the lowest boiling point?

• (A) Carbon disulfide
• (B) Water
• (C) Ethanol
• (D) Benzene
• (E) Chloroform

Radioactive decay follows

• (A) first order
• (B) second order
• (C) third order
• (D) zero order
• (E) Pseudo first order

In which of the following system, the number of moles of the substance present at equilibrium not be shifted by change in the volume of the system at constant temperature?

• (A) \( N_2(g) + 3H_2(g) \rightleftharpoons 2NH_3(g) \)
• (B) \( PCl_3(g) + Cl_2(g) \rightleftharpoons PCl_5(g) \)
• (C) \( CO(g) + 3H_2(g) \rightleftharpoons CH_4(g) \)
• (D) \( 2SO_2(g) + O_2(g) \rightleftharpoons 2SO_3(g) \)
• (E) \( NO_2(g) + SO_2(g) \rightleftharpoons SO_3(g) + NO(g) + H_2O(g) \)

Which of the following has the least atomic radius?

• (A) B
• (B) C
• (C) N
• (D) O
• (E) F

Which of the following tripositive ion has smallest size?

• (A) Ce\(^{3+}\)
• (B) Nd\(^{3+}\)
• (C) La\(^{3+}\)
• (D) Sm\(^{3+}\)
• (E) Gd\(^{3+}\)

Lanthanides (Ln) when heated with carbon at 2773K form product with general formula

• (A) \( LnC \)
• (B) \( Ln_2C_3 \)
• (C) \( LnC_3 \)
• (D) \( LnC_2 \)
• (E) \( Ln_3C_2 \)

Which of the following is an acidic oxide?

• (A) CrO\(_3\)
• (B) CrO
• (C) V\(_2\)O\(_4\)
• (D) V\(_2\)O\(_5\)
• (E) V\(_2\)O\(_3\)

The catalyst used in the Wacker process is

• (A) V\(_2\)O\(_5\)
• (B) PdCl\(_2\)
• (C) TiCl\(_4\) with Al(CH\(_3\))\(_3\)
• (D) Fe
• (E) Mo

The coordination number of Pt and Fe in the complexes [PtCl\(_6\)]\(^{2-}\) and [Fe(C\(_2\)O\(_4\))\(_3\)]\(^{3-}\) are respectively

• (A) 4 and 6
• (B) 6 and 6
• (C) 4 and 4
• (D) 6 and 8
• (E) 4 and 8

The IUPAC name of HOCH\(_2\)(CH\(_2\))\(_3\)CH\(_2\)COCH\(_3\) is

• (A) 2-oxo-heptan-7-ol
• (B) 7-hydroxyheptan-2-one
• (C) hydroxyheptan-6-one
• (D) 2-oxo-heptan-7-ol
• (E) hydroxy pentyl methyl ketone

Which of the following statement is incorrect with Kolbe’s electrolytic method?

• (A) It gives an alkane with even number of carbon atoms at the anode.
• (B) At anode decarboxylation and formation of methyl radical occurs.
• (C) Methane cannot be prepared by this method.
• (D) At anode acetate ion accepts electrons to give acetate free radical.
• (E) At cathode hydrogen gas is liberated.

Which of the following substitution reaction with methane requires HIO\(_3\) as an oxidising agent?

• (A) Chlorination
• (B) Bromination
• (C) Iodination
• (D) Fluorination
• (E) Friel-Crafts acylation

The reagents and conditions (X) required for the following conversion

• (A) \( X = H_2O, 623 K, 300 atm \& H^+ \)
• (B) \( X = KOH, 443 K, 100 atm \& H^+ \)
• (C) \( X = NaOH, 368 K, 300 atm \& H^+ \)
• (D) \( X = warm, H_2O \& H^+ \)
• (E) \( X = NaOH, 623 K, 300 atm \& H^+ \)

Which of the following statement is incorrect?

• (A) \( (-) \)-2-bromooctane reacts with NaOH gives (+)-octan-2-ol by \( S_N2 \) reaction.
• (B) 2-Bromobutane reacts with NaOH gives racemic mixture by \( S_N1 \) reaction.
• (C) \( \beta \)-elimination of 2-bromopentane gives pent-1-ene as major product.
• (D) The hybridization of the carbon in the intermediate formed in \( S_N1 \) reaction is sp\(^2\).
• (E) Primary alkyl halide undergoes \( S_N2 \) faster than secondary alkyl halide.

Compound ‘X’ (C\(_6\)H\(_6\)O) reacts with aqueous NaOH to give compound ‘Y’. ‘Y’ reacts with CO\(_2\) followed by acidification to give compound ‘Z’. The compounds X, Y and Z are respectively

• (A) benzene, phenol, salicylaldehyde
• (B) phenol, benzene, benzoquinone
• (C) phenol, sodium phenoxide, benzoquinone
• (D) benzaldehyde, sodium phenoxide, salicylic acid
• (E) phenol, sodium phenoxide, salicylic acid

The decreasing order of basic strength in aqueous solution of amines is

• (A) Dimethylamine \( > \) Methylamine \( > \) Trimethylamine \( > \) Ammonia
• (B) Methylamine \( > \) Dimethylamine \( > \) Trimethylamine \( > \) Ammonia
• (C) Trimethylamine \( > \) Dimethylamine \( > \) Methylamine \( > \) Ammonia
• (D) Ammonia \( > \) Trimethylamine \( > \) Dimethylamine \( > \) Methylamine
• (E) Ammonia \( > \) Dimethylamine \( > \) Trimethylamine \( > \) Methylamine

The melting point of \( \beta \)-form of crystalline glucose is

• (A) 473 K
• (B) 303 K
• (C) 423 K
• (D) 371 K
• (E) 503 K

Kjeldahl method can be used to estimate nitrogen in

• (A) azobenzene
• (B) aniline
• (C) o-nitrophenol
• (D) nitrobenzene
• (E) pyridine

Which of the following vitamin deficiency causes increased fragility of RBCs and muscular weakness?

• (A) Vitamin A
• (B) Vitamin B12
• (C) Riboflavin
• (D) Vitamin D
• (E) Vitamin E

Which of the following is the most reactive in aromatic electrophilic substitution reaction?

• (A) Benzene
• (B) Chlorobenzene
• (C) Phenol
• (D) Benzaldehyde
• (E) Nitrobenzene

Let A, B, C denote the set of students in a college who play football, basketball, and cricket respectively. If \( n(A) = 60 \), \( n(B) = 55 \), \( n(C) = 70 \), \( n(A \cup B \cup C) = 100 \) and \( n(A \cap B \cap C) = 20 \), then the number of students who play exactly two of these sports is

• (A) 40
• (B) 45
• (C) 60
• (D) 75
• (E) 85

Let \( f(x) = \sqrt{4 - x^2} \), \( g(x) = \sqrt{x^2 - 1} \). Then the domain of the function \( h(x) = f(x) + g(x) \) is equal to

• (A) \( (-\infty, -1] \cup [1, \infty) \)
• (B) \( (-\infty, -2] \cup [2, \infty) \)
• (C) \( [-2, -1] \cup [1, 2] \)
• (D) \( [-2, 1] \cup [1, 2] \)
• (E) \( [1, 2] \)

The range of the function \( f(x) = 8 + \sqrt{x - 5} \) is

• (A) \( (-\infty, 5] \)
• (B) \( [5, \infty) \)
• (C) \( (-\infty, 5] \cup [8, \infty) \)
• (D) \( [5, 8] \)
• (E) \( [8, \infty) \)

If \( x \) satisfies the inequality \( -3 < \frac{1}{2} + \frac{-3x}{2} \leq 6 \), then \( x \) lies in the interval

• (A)
• (B)
• (C) \( \left[ \frac{7}{3}, \frac{11}{3} \right] \)
• (D) \( \left[ \frac{-10}{3}, \frac{7}{3} \right] \)
• (E) \( \left[ \frac{7}{3}, \frac{10}{3} \right] \)

Let \( f(x) = 6x^2 + 9x + 10 \) and \( g(x) = x^2 - 9x - 9 \). Then the value of \( (f \circ g)(10) \) is

• (A) 10
• (B) 15
• (C) 25
• (D) 35
• (E) 45

If the complex number \( \frac{2 + i}{\lambda + i} \) lies on the line \( y = x \) of the first quadrant, then the value of \( \lambda \) is equal to

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

Let \( z = x + iy \), where \( y > 0 \). If \( z + \overline{z} = 6 \) and \( |z| + | \overline{z} | = 10 \), then \( z = \)

• (A) \( 3 + 2i \)
• (B) \( 3 + 5i \)
• (C) \( 3 + 3i \)
• (D) \( 3 + 4i \)
• (E) \( 3 + i\sqrt{5} \)

If the complex number \( 2 + i \) is rotated through an angle \( 90^\circ \) in the anti-clockwise direction about the origin in the complex plane, then the resulting complex number is

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

The number of positive integers that have at most seven digits and contain only the digits 0 and 9 is

• (A) 112
• (B) 127
• (C) 136
• (D) 142
• (E) 150

The sum of the first 20 terms of the G.P. \( \sqrt{3} + \frac{-1}{\sqrt{3}} + \frac{1}{3\sqrt{3}} + \cdots \) is equal to

• (A) \( \frac{\sqrt{3}(3^{20} - 1)}{4 \cdot 3^{19}} \)
• (B) \( \frac{\sqrt{3}(3^{20} - 1)}{2 \cdot 3^{19}} \)
• (C) \( \frac{\sqrt{3}(3^{20} - 1)}{3^{20}} \)
• (D) \( \frac{\sqrt{3}(3^{20} - 1)}{3^{20}} \)
• (E) \( \frac{\sqrt{3}(3^{20} - 1)}{2 \cdot 3^{20}} \)

Let \( A = \{ 1, 3, 5, 7, \dots, 21 \} \). The number of ways 4 numbers, containing always 11, can be selected from the set A is equal to

• (A) 120
• (B) 160
• (C) 240
• (D) 260
• (E) 320

The relation \( R \) in the set of integers \( \mathbb{Z} \) is given by \( R = \{(a, b) : b = 2a + 3\} \). Then the relation \( R \) is

• (A) reflexive, symmetric and transitive
• (B) neither reflexive nor symmetric nor transitive
• (C) not reflexive but symmetric and transitive
• (D) reflexive and symmetric but not transitive
• (E) reflexive but not symmetric and transitive

The value of the sum \[ \sum_{k=0}^{48} \frac{1}{(k + 1)(k + 2)} \]
is equal to

• (A) \( \frac{51}{50} \)
• (B) \( \frac{51}{49} \)
• (C) \( \frac{49}{50} \)
• (D) \( \frac{48}{49} \)
• (E) \( \frac{50}{49} \)

If the G.M. of the numbers 2 and \( \alpha \) is 16, then the A.M. of these two numbers is equal to

• (A) 10
• (B) 20
• (C) 45
• (D) 50
• (E) 65

Let \[ a_n = \frac{n(n - 5)}{n + 2}, \quad n = 1,2,3, \dots \]
If \( a_m = \frac{12}{5} \) for some \( m \), then the value of \( m \) is equal to

• (A) 6
• (B) 7
• (C) 8
• (D) 9
• (E) 10

In the binomial expansion of \[ \left( \sqrt{x} - \frac{3}{x^3} \right)^7 \]
the constant term is :

• (A) 21
• (B) -21
• (C) 14
• (D) -14
• (E) 7

\[ 23 \binom{50}{23} = \]

• (A) \quad 50 \( \binom{49}{27}\)
• (B) \quad 49 \( \binom{23}{23}\)
• (C) \quad 50 \( \binom{22}{22}\)
• (D) \quad 27 \( \binom{50}{23}\)
• (E) \quad 49 \( \binom{27}{27}\)

Let \[ p(x) = (1 + x + x^2 + \dots + x^{10}) (1 - x + x^2 - x^3 + \dots + x^{10}) \]
Then the sum of all coefficients of \( p(x) \) is equal to

• (A) 121
• (B) 66
• (C) 11
• (D) 10
• (E) 0

Let \[ A = \begin{bmatrix} a_1 & b_1 & c_1
a_2 & b_2 & c_2
a_3 & b_3 & c_3 \end{bmatrix} \quad and \quad B = \begin{bmatrix} a_1 & 2b_1 & 4c_1
2a_2 & 4b_2 & 8c_2
4a_3 & 8b_3 & 16c_3 \end{bmatrix} \]
If \( |B| = 16 \), then the value of \( |A| \) is equal to

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

If \( A \) is an invertible matrix and satisfies the equation
\[ 5A^2 - 4A - 7I = 0 \]

where \( I \) is the identity matrix and 0 is the zero matrix, then
\[ 7 A^{-1} = \]

• (A) \( 5A - 4I \)
• (B) \( 4A - 7I \)
• (C) \( 7A - 5I \)
• (D) \( 4A - 5I \)
• (E) \( 5A - 7I \)

Let \( A \) be a \( 3 \times 3 \) matrix with \( |A| = 7 \). If \( B = 3A \), then the value of
\[ \left| \frac{adj A}{B} \right| \]

is equal to

• (A) \( \frac{7}{3} \)
• (B) \( \frac{7}{9} \)
• (C) \( \frac{49}{9} \)
• (D) \( \frac{7}{27} \)
• (E) \( \frac{49}{27} \)

If \[ A = \begin{bmatrix} -7 & 3
3 & -1 \end{bmatrix} \]

then \( \det(A^5) \) is equal to

• (A) \( 81 \)
• (B) \( -81 \)
• (C) \( 243 \)
• (D) \( -243 \)
• (E) \( -32 \)

The means of two samples of size 30 and 40 are 35 and 42 respectively. Then the mean of the combined sample of size 70 is

• (A) \( 36 \)
• (B) \( 37 \)
• (C) \( 38 \)
• (D) \( 39 \)
• (E) \( 40 \)

The standard deviation of a data set \( x_1, x_2, \dots, x_6 \) (\( x_i > 0 \)) is 2. If
\[ \sum_{i=1}^{9} x_i^2 = 360, \]

then the mean of the data set is

• (A) \( 4 \)
• (B) \( 6 \)
• (C) \( 8 \)
• (D) \( 10 \)
• (E) \( 12 \)

If two dice are rolled simultaneously, then the probability that the difference of the numbers on the two dice equals to zero is

• (A) \( \frac{1}{12} \)
• (B) \( \frac{1}{9} \)
• (C) \( \frac{5}{36} \)
• (D) \( \frac{7}{36} \)
• (E) \( \frac{1}{6} \)

Let \( A \) and \( B \) be two events. If \( P(A) = 0.49 \), \( P(B) = 0.3 \) and \( P(A | B^c) = 0.4 \), then \( P(A | B) \) is equal to

• (A) \( 0.45 \)
• (B) \( 0.28 \)
• (C) \( 0.4 \)
• (D) \( 0.7 \)
• (E) \( 0.3 \)

Simplify: \( \tan x - \cot x + \csc x \sec x \)

• (A) \( 2 \tan x \)
• (B) \( 2 \csc x \sec x \)
• (C) \( 2 \tan x \sec x \)
• (D) \( 2 \cot x \)
• (E) \( 2 \cot x \csc x \)

The value of \( \tan \left( \cos^{-1} \left( \frac{-24}{25} \right) \right) \) is equal to

• (A) \( \frac{7}{24} \)
• (B) \( \frac{-7}{24} \)
• (C) \( \frac{-7}{25} \)
• (D) \( \frac{-24}{7} \)
• (E) \( \frac{24}{7} \)

If \( \sin t + \cos t = \sqrt{2} \), then \( \tan t + \cot t \) is equal to

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

The value of \( \csc x + \cot x \) is

• (A) \( \tan \left( \frac{x}{2} \right) \)
• (B) \( \sec \left( \frac{x}{2} \right) \)
• (C) \( \cot \left( \frac{x}{2} \right) \)
• (D) \( \cos \left( \frac{x}{2} \right) \)
• (E) \( \sin \left( \frac{x}{2} \right) \)

The value of \( \sin \left( 2 \cos^{-1} \left( \frac{5}{12} \right) + \sin^{-1} \left( \frac{5}{12} \right) \right) \) is equal to

• (A) \( \frac{5}{12} \)
• (B) \( \frac{12}{13} \)
• (C) \( \frac{5}{13} \)
• (D) \( \frac{10}{13} \)
• (E) \( \frac{5}{6} \)

The value of \[ \tan^{-1} \left( \frac{1}{3} \right) + \tan^{-1} \left( \frac{2}{3} \right) + \cot^{-1} \left( \frac{9}{7} \right) \]
is equal to

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

Let \[ \sum_{k=1}^{15} \sin (t_k) = 0 \quad and \quad \sum_{k=1}^{15} \sin (3t_k) = \frac{-24}{5}, \]
where \( t_1, t_2, t_3, \dots \) are real numbers. Then the value of the sum \[ \sum_{k=1}^{15} \sin^3 (t_k) \]
is equal to

• (A) \( \frac{4}{5} \)
• (B) \( \frac{6}{5} \)
• (C) \( \frac{3}{10} \)
• (D) \( \frac{24}{5} \)
• (E) \( \frac{96}{5} \)

If \[ 7 \cos^2 x + 3 \sin^2 x = 6, \]
then the value of \( \cos 2x \) is equal to

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

Evaluate \[ \frac{\csc^2(\theta) - 1}{\csc^2(\theta)} - \frac{\sec^2(\theta) - 1}{\sec^2(\theta)} \]

• (A) \( 2 \cos^2 \theta \)
• (B) \( 2 \cos \theta \)
• (C) \( 2 \sin^2 \theta \)
• (D) \( \cos 2\theta \)
• (E) \( 2 \sin \theta \)

Find the equation of the line perpendicular to the line \[ 7x - 5y = 11 \]
and passing through the point \( (7, -9) \).

• (A) \( 5x + 7y + 28 = 0 \)
• (B) \( 5x + 7y - 28 = 0 \)
• (C) \( 5x + 7y + 38 = 0 \)
• (D) \( 5x + 7y - 38 = 0 \)
• (E) \( 5x - 7y + 28 = 0 \)

Find the values of \( \alpha \) for which the circle \[ x^2 + y^2 + \alpha x - 8y + 56 = 0 \]
has radius 3.

• (A) \( 7, -7 \)
• (B) \( 9, -9 \)
• (C) \( 12, -12 \)
• (D) \( 18, -18 \)
• (E) \( 14, -14 \)

The coordinates of the vertex of the parabola \[ y = 2x^2 - 12x + 26 \]
are

• (A) \( (6,13) \)
• (B) \( (3,-8) \)
• (C) \( (3,8) \)
• (D) \( (6,-13) \)
• (E) \( (3,11) \)

Find the equation of the parabola with focus at \( (3,1) \) and vertex at \( (5,1) \).

• (A) \( (y - 1)^2 = -8(x - 5) \)
• (B) \( (y - 1)^2 = 8(x - 5) \)
• (C) \( (y - 1)^2 = 8(x - 3) \)
• (D) \( (y - 1)^2 = -8(x - 3) \)
• (E) \( (y - 1)^2 = -4(x - 5) \)

The eccentricity of the ellipse \[ p x^2 + 5y^2 = 80, \quad where p > 5, \]
is \( \frac{\sqrt{3}}{2} \). Then the value of \( p \) is

• (A) \( \frac{5}{8} \)
• (B) \( 16 \)
• (C) \( \frac{5}{4} \)
• (D) \( 20 \)
• (E) \( 25 \)

For an ellipse, the foci are \( F(3,0) \) and \( F'(-3,0) \). If the length of the minor axis is 8, then the length of the major axis is equal to

• (A) \( 16 \)
• (B) \( 15 \)
• (C) \( 14 \)
• (D) \( 12 \)
• (E) \( 10 \)

If \( (a,-6) \) lies on the perpendicular bisector of the line segment joining \( (-2,-1) \) and \( (4,-13) \), then the value of \( a \) is equal to

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

If \( (3,2) \) and \( (5,6) \) are end points of a diameter of a circle, then the equation of the circle is

• (A) \( x^2 + y^2 - 6x + 4y + 3 = 0 \)
• (B) \( x^2 + y^2 - 8x - 4y + 3 = 0 \)
• (C) \( x^2 + y^2 - 8x - 4y - 3 = 0 \)
• (D) \( x^2 + y^2 - 6x + 4y + 17 = 0 \)
• (E) \( x^2 + y^2 - 8x - 4y - 17 = 0 \)

Let \( \alpha, \beta, \gamma \) be the direction cosines of a vector \( \vec{a} = x \hat{i} + y \hat{j} + z \hat{k} \), where \( z < 0 \). If \( \alpha = \frac{-4}{\sqrt{105}} \) and \( \beta = \frac{\sqrt{5}}{\sqrt{21}} \), then \( \gamma \) is equal to

• (A) \( \frac{-8}{\sqrt{105}} \)
• (B) \( \frac{-\sqrt{8}}{\sqrt{105}} \)
• (C) \( \frac{-5}{\sqrt{105}} \)
• (D) \( \frac{-5}{\sqrt{21}} \)
• (E) \( \frac{-8}{\sqrt{21}} \)

Let \( A(0,3,-3) \), \( B(1,1,1) \) and \( C(2,0,3) \) be three points in space. Then the projection of \( \overrightarrow{AB} \) on \( \overrightarrow{AC} \) is equal to

• (A) \( \frac{26}{7} \)
• (B) \( \frac{32}{7} \)
• (C) \( \frac{34}{7} \)
• (D) \( \frac{24}{7} \)
• (E) \( \frac{20}{7} \)

If \( \vec{a} = 5\hat{i} - 7\hat{j} + 9\hat{k} \) and \( \vec{b} = -5\hat{i} + 7\hat{j} - 9\hat{k} \), then \( \vec{a} \cdot (\vec{a} \times \vec{b}) + (\vec{a} + \vec{b}) \cdot \hat{b} \) is equal to

• (A) \( 50 \)
• (B) \( -50 \)
• (C) \( 49 \)
• (D) \( -49 \)
• (E) \( 0 \)

The line joining the points \( (2,2,2) \) and \( (6,6,6) \) meets the line
\[ \frac{x - 1}{3} = \frac{y - 2}{2} = \frac{z - 5}{-1} \]

at the point

• (A) \( (1,1,1) \)
• (B) \( (2,2,2) \)
• (C) \( (3,3,3) \)
• (D) \( (4,4,4) \)
• (E) \( (6,6,6) \)

The angle between the vectors \( \vec{a} \) and \( \vec{b} \) is \( \frac{\pi}{3} \). If \( | \vec{a} \cdot \vec{b} |^2 = 15 \), then \( | \vec{a} \times \vec{b} |^2 \) is equal to

• (A) \( 5 \)
• (B) \( 15\sqrt{3} \)
• (C) \( \frac{15}{\sqrt{3}} \)
• (D) \( 5\sqrt{3} \)
• (E) \( 45 \)

The symmetric equation of the straight line passing through the points \( (-1, 4, 2) \) and \( (-3, 0, 5) \) is

• (A) \( \frac{x-1}{-2} = \frac{y+4}{-4} = \frac{z+2}{3} \)
• (B) \( \frac{x+1}{2} = \frac{y-4}{4} = \frac{z-2}{5} \)
• (C) \( \frac{x+1}{-2} = \frac{y-4}{-4} = \frac{z-2}{3} \)
• (D) \( \frac{x-3}{-2} = \frac{y+1}{-4} = \frac{z+5}{3} \)
• (E) \( \frac{x+1}{4} = \frac{y-4}{-4} = \frac{z-2}{3} \)

The angle between the lines
\[ \frac{x-1}{2} = \frac{2y+3}{4} = \frac{z+5}{-2} \quad and \quad \frac{x-3}{4} = \frac{y+1}{-4} = \frac{z+3}{-4} \]

is equal to

• (A) \( \cos^{-1} \left(\frac{1}{8} \right) \)
• (B) \( \cos^{-1} \left(\frac{1}{3} \right) \)
• (C) \( \cos^{-1} \left(\frac{1}{4} \right) \)
• (D) \( \cos^{-1} \left(\frac{1}{12} \right) \)
• (E) \( \cos^{-1} \left(\frac{1}{\sqrt{3}} \right) \)

If the function
\[ f(x) = \begin{cases} x^2, & for x < 4
5x - k, & for x \geq 4 \end{cases} \]

is continuous at \( x = 4 \), then the value of \( k \) is equal to

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

If
\[ f(x) = \sqrt[3]{x^2} + \sqrt{x}, \]

then the value of \( f'(64) \) is equal to

• (A) \( \frac{11}{48} \)
• (B) \( \frac{9}{48} \)
• (C) \( \frac{7}{48} \)
• (D) \( \frac{5}{48} \)
• (E) \( \frac{1}{16} \)

Ice is coated uniformly around a sphere of radius 15 cm. If ice is melting at the rate of \[ 80 cm^3/min \]
when the thickness is 5 cm, then the rate of change of thickness of ice is

• (A) \( \frac{1}{10\pi} \) cm/min
• (B) \( \frac{1}{50\pi} \) cm/min
• (C) \( \frac{1}{80\pi} \) cm/min
• (D) \( \frac{1}{40\pi} \) cm/min
• (E) \( \frac{1}{20\pi} \) cm/min

Evaluate the integral
\[ \int \frac{e^x}{2^x} dx. \]

• (A) \( \frac{e^x}{(\log_2 2) 2^x} + C \)
• (B) \( \frac{e^x}{2(2^x)} + C \)
• (C) \( \frac{2 \left(\frac{e}{2} \right)^{x-1}}{e} + C \)
• (D) \( \frac{e^x}{(1 - \log_2 2)2^x} + C \)
• (E) \( \frac{e^x}{2^x} + C \)

The area bounded by the parabola \[ y = x^2 + 4 \]
and the straight line passing through the points \[ (-1,2) \quad and \quad (1,6) \]
is (in square units)

• (A) \( \frac{20}{3} \)
• (B) \( \frac{4}{3} \)
• (C) \( \frac{8}{3} \)
• (D) \( \frac{16}{3} \)
• (E) \( \frac{14}{3} \)

Let \[ g(x) = 4x + 3 \quad and \quad f(g(x)) = x^2 + 9. \]
Then the value of \( f(7) \) is equal to

• (A) \( 7 \)
• (B) \( 9 \)
• (C) \( 10 \)
• (D) \( 12 \)
• (E) \( 14 \)

The range of the function \( f(x) = 7\cos(10x + 4\pi) \) is

• (A) \( [-1,1] \)
• (B) \( [-4\pi, 4\pi] \)
• (C) \( [-10,10] \)
• (D) \( [-7,7] \)
• (E) \( [-2\pi, 2\pi] \)

Let \( f(x) = \log_e \left( \frac{x^2 + 30}{11x} \right) \), for \( x \in [5,6] \). Then the point \( c \in (5,6) \) at which \( f'(c) = 0 \) is:

• (A) \( \sqrt{30} \)
• (B) \( 4\sqrt{2} \)
• (C) \( 2\sqrt{7} \)
• (D) \( \sqrt{35} \)
• (E) \( \sqrt{26} \)

Let \( f(x) = ax^3 + bx^2 + cx + d \). If \( f \) has a local maximum value 21 at \( x = -1 \) and a local minimum value 7 at \( x = 1 \), then \( f(0) \) is equal to:

• (A) \( 10 \)
• (B) \( 11 \)
• (C) \( 12 \)
• (D) \( 13 \)
• (E) \( 14 \)

The value of \( \int_{-2}^{2} x |x| \,dx \) is:

• (A) \( \frac{1}{8} \)
• (B) \( \frac{1}{4} \)
• (C) \( -\frac{1}{4} \)
• (D) \( -\frac{1}{8} \)
• (E) \( 0 \)

Evaluate the integral
\[ \int x^5 e^{x^3} \,dx. \]

• (A) \( \frac{e^{x^3}}{3} (x^3 - 1) + C \)
• (B) \( \frac{e^{x^3}}{5} (x^5 - 1) + C \)
• (C) \( \frac{e^{x^3}}{4} (x^4 - 1) + C \)
• (D) \( \frac{e^{x^3}}{3} (x^5 - 1) + C \)
• (E) \( \frac{x^3 e^{x^3}}{3} + C \)

Evaluate the limit:
\[ \lim\limits_{x \to 6} \frac{\sqrt{x^2 + 13} - 7}{x^2 - 36}. \]

• (A) \( \frac{1}{7} \)
• (B) \( \frac{1}{13} \)
• (C) \( \frac{13}{36} \)
• (D) \( \frac{1}{14} \)
• (E) \( \frac{1}{36} \)

If \[ x^4 + 2\sqrt{y} + 1 = 3, \]
then \( \frac{dy}{dx} \) at \( (1,0) \) is equal to

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

If
\[ \lim_{x \to 9} f(x) = 6 \quad and \quad \lim_{x \to 9} g(x) = 3, \]

then
\[ \lim_{x \to 9} \frac{f(x) - 2g(x)}{g(x)} \]

is equal to

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

For the curve
\[ y = \alpha x^2 + \cos y + \beta, \]

the value of \( \frac{dy}{dx} \) at \( (1,0) \) is 2. Then the value of \( \alpha \beta \) is equal to

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

Evaluate the limit:
\[ \lim_{x \to 4} \left( \frac{1}{x - 4} - \frac{5}{x^2 - 3x - 4} \right) \]

is equal to

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

If
\[ y = \log_e \left( \frac{1 + 2x^2}{1 - 3x^2} \right), \]

then \( \frac{dy}{dx} \) is:

• (A) \( \frac{10x}{1 - x^2 - 6x^4} \)
• (B) \( \frac{12x^3}{1 - x^2 - 6x^4} \)
• (C) \( \frac{10x}{1 - 6x^4} \)
• (D) \( \frac{-10x}{1 - x^2 - 6x^4} \)
• (E) \( \frac{-12x^3}{1 - x^2 - 6x^4} \)

Let \( \alpha \) and \( \beta \) be real numbers such that \( f(x) \) is defined as:
\[ f(x) = \begin{cases} 2x^2 + 4x + \alpha, & if x < 1
\beta x^2 + 5, & if x \geq 1 \end{cases} \]

and is differentiable at \( x = 1 \). Then \( \alpha + \beta \) is equal to:

• (A) \( 5 \)
• (B) \( 6 \)
• (C) \( 7 \)
• (D) \( 8 \)
• (E) \( 9 \)

If \( f(x) = x^2 + 2x f'(1) + f''(2) \) for all \( x \), then \( f(0) \) is equal to:

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

The function \( f(x) = 6x^4 - 3x^2 - 5 \) is increasing in the set:

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

The general solution of the differential equation
\[ 2y \tan x + \frac{dy}{dx} = 5 \sin x \]

is:

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

Evaluate the integral:
\[ \int \frac{\sin \theta \sin 2\theta}{1 - \cos 2\theta} d\theta. \]

• (A) \( 1 + \cos \theta + C \)
• (B) \( 1 + \sin \theta + C \)
• (C) \( \sin \theta + C \)
• (D) \( 1 + \cos 2\theta + C \)
• (E) \( 1 + \sin 2\theta + C \)

Evaluate the integral:
\[ \int \frac{6x^3 + 9x^2}{x^4 + 3x^3 - 9x^2} dx. \]

• (A) \( 3x \log |x^2 + 3x - 9| + C \)
• (B) \( 6x \log |x^2 + 3x - 9| + C \)
• (C) \( 6 \log |x^2 + 3x - 9| + C \)
• (D) \( x \log |x^2 + 3x - 9| + C \)
• (E) \( 3 \log |x^2 + 3x - 9| + C \)

Evaluate the integral:
\[ \int_{0}^{3} |x - 2| \,dx \]

is equal to

• (A) \( \frac{2}{3} \)
• (B) \( \frac{3}{2} \)
• (C) \( \frac{5}{2} \)
• (D) \( \frac{2}{5} \)
• (E) \( \frac{9}{2} \)

Find the integrating factor of the differential equation:
\[ (3 \sin x \cos x) \, dy = (1 + 3y \sin^2 x) \, dx, \quad where \quad 0 < x < \frac{\pi}{2} \]

is

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