KEAM 2024 Question Paper with Answer Key (June 6)

If the time period \( T \) of a satellite revolving close to the earth is given as \( T = 2\pi R^a g^b \), then the value of \( a \) and \( b \) are respectively (where \( R \) is the radius of the earth):

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

The angle between \( \vec{A} \times \vec{B} \) and \( \vec{B} \times \vec{A} \) is:

• (A) \( 90^\circ \)
• (B) \( 60^\circ \)
• (C) \( 180^\circ \)
• (D) \( 0^\circ \)
• (E) \( 270^\circ \)

If the initial speed of the car moving at constant acceleration is halved, then the stopping distance \( S \) becomes:

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

When a cricketer catches a ball in 30 s, the force required is 2.5 N. The force required to catch that ball in 50 s is:

• (A) 1.5 N
• (B) 1 N
• (C) 2.5 N
• (D) 3 N
• (E) 5 N

A ball is thrown vertically upwards with an initial speed of 20 ms\(^{-1}\). The velocity (in ms\(^{-1}\)) and acceleration (in ms\(^{-2}\)) at the highest point of its motion are respectively:

• (A) 20 and 9.8
• (B) 0 and 9.8
• (C) 0 and 0
• (D) 10 and 9.8
• (E) 0 and 4.9

Which one is an INCORRECT statement?

• (A) Forces always occur in pairs
• (B) Impulsive force is a force that acts for a shorter duration
• (C) Impulse is the change in momentum of the body
• (D) Momentum and change in momentum both have the same direction
• (E) Action and reaction forces act on different bodies

Impending motion is opposed by:

• (A) Static friction
• (B) Fluid friction
• (C) Sliding friction
• (D) Kinetic friction
• (E) Rolling friction

A block of 50 g mass is connected to a spring of spring constant 500 Nm\(^{-1}\). It is extended to the maximum and released. If the maximum speed of the block is 3 ms\(^{-1}\), then the length of extension is:

• (A) 4 cm
• (B) 1 cm
• (C) 2.5 cm
• (D) 3 cm
• (E) 5 cm

A particle is displaced from P \( (3i + 2j - k) \) to Q \( (2i + 2j + 2k) \) by a force \( \mathbf{F} = i + j + k \). The work done on the particle (in J) is:

• (A) 2
• (B) 1
• (C) 2.5
• (D) 3
• (E) 5

The motion of a cylinder on an inclined plane is:

• (A) Rotational but not translational
• (B) Translational but not rotational
• (C) Translational but not rolling
• (D) Rotational, translational and rolling motion
• (E) Rotational and rolling but not translational motion

A flywheel ensures a smooth ride on the vehicle because of its:

• (A) Larger speed
• (B) Zero moment of inertia
• (C) Large moment of inertia
• (D) Lesser mass with smaller radius
• (E) Small moment of inertia

The escape speed of the moon when compared with escape speed of the earth is approximately:

• (A) Twice smaller
• (B) Thrice smaller
• (C) 4 times smaller
• (D) 5 times smaller
• (E) 6 times smaller

The force of gravity is a:

• (A) Strong force
• (B) Noncentral force
• (C) Nonconservative force
• (D) Contact force
• (E) Conservative force

The terminal velocity of a small steel ball falling through a viscous medium is:

• (A) Directly proportional to the radius of the ball
• (B) Inversely proportional to the radius of the ball
• (C) Directly proportional to the square of the radius of the ball
• (D) Directly proportional to the square root of the radius of the ball
• (E) Inversely proportional to the square of the radius of the ball

The stress required to produce a fractional compression of 1.5% in a liquid having bulk modulus of \( 0.9 \times 10^9 \, Nm^{-2} \) is:

• (A) \( 2.48 \times 10^7 \, Nm^{-2} \)
• (B) \( 0.26 \times 10^7 \, Nm^{-2} \)
• (C) \( 3.72 \times 10^7 \, Nm^{-2} \)
• (D) \( 1.35 \times 10^7 \, Nm^{-2} \)
• (E) \( 4.56 \times 10^7 \, Nm^{-2} \)

When heat is supplied to the gas in an isochoric process, the supplied heat changes its:

• (A) Volume only
• (B) Internal energy and volume
• (C) Internal energy only
• (D) Internal energy and temperature
• (E) Temperature only

1 g of ice at 0°C is converted into water by supplying a heat of 418.72 J. The quantity of heat that is used to increase the temperature of water from 0°C is (Latent heat of fusion of ice = \( 3.35 \times 10^5 \, Jkg^{-1} \)):

• (A) 83.72 J
• (B) 33.52 J
• (C) 335.72 J
• (D) 837.24 J
• (E) 418.72 J

All real gases behave like an ideal gas at:

• (A) High pressure and low temperature
• (B) Low temperature and low pressure
• (C) High pressure and high temperature
• (D) At all temperatures and pressures
• (E) Low pressure and high temperature

0.5 mole of \( N_2 \) at 27°C is mixed with 0.5 mole of \( O_2 \) at 42°C. The temperature of the mixture is:

• (A) 42°C
• (B) 34.5°C
• (C) 32.5°C
• (D) 37.5°C
• (E) 27°C

A wave with a frequency of 600 Hz and wavelength of 0.5 m travels a distance of 200 m in a time of:

• (A) 1.67 s
• (B) 0.67 s
• (C) 1 s
• (D) 0.33 s
• (E) 1.33 s

If the fundamental frequency of the stretched string of length 1 m under a given tension is 3 Hz, then the fundamental frequency of the stretched string of length 0.75 m under the same tension is:

• (A) 1 Hz
• (B) 2 Hz
• (C) 6 Hz
• (D) 4 Hz
• (E) 5 Hz

The product of the total electric flux emanating from a closed surface enclosing a charge \( q \) in free space is (\( \epsilon_0 \) - electrical permittivity of free space):

• (A) 1
• (B) \( \frac{q}{\epsilon_0} \)
• (C) \( q \)
• (D) \( q\epsilon_0 \)
• (E) \( \epsilon_0 \)

Three capacitances 1 \(\mu\)F, 4 \(\mu\)F, and 5 \(\mu\)F are connected in parallel with a supply voltage. If the total charge flowing through the capacitors is 50 \(\mu\)C, then the supply voltage is:

• (A) 2 V
• (B) 10 V
• (C) 6 V
• (D) 3 V
• (E) 5 V

The resistance of a wire at 0 °C is 4 \(\Omega\). If the temperature coefficient of resistance of the material of the wire is \(5 \times 10^{-3} / ^\circ C\), then the resistance of a wire at 50 °C is:

• (A) 20 \(\Omega\)
• (B) 10 \(\Omega\)
• (C) 6 \(\Omega\)
• (D) 8 \(\Omega\)
• (E) 5 \(\Omega\)

n number of electrons flowing in a copper wire for 1 minute constitute a current of 0.5 A. Twice the number of electrons flowing through the same wire for 20 s will constitute a current of:

• (A) 0.25 A
• (B) 3 A
• (C) 1 A
• (D) 1.25 A
• (E) 2.25 A

If a cell of 12 V emf delivers 2 A current in a circuit having a resistance of 5.8 \(\Omega\), then the internal resistance of the cell is:

• (A) 1 \(\Omega\)
• (B) 0.2 \(\Omega\)
• (C) 0.3 \(\Omega\)
• (D) 0.6 \(\Omega\)
• (E) 0.8 \(\Omega\)

Torque on a coil carrying current \( I \) having \( N \) turns and area of cross section \( A \) when placed with its plane perpendicular to a magnetic field \( B \) is:

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

A long straight wire carrying a current 3 A produces a magnetic field \( B \) at a certain distance. The current that flows through the same wire will produce a magnetic field \( \frac{B}{3} \) at the same distance is:

• (A) 1.5 A
• (B) 1 A
• (C) 2.5 A
• (D) 3 A
• (E) 5 A

Which one of the following statement is INCORRECT?

• (A) Isolated magnetic poles do not exist
• (B) Magnetic field lines do not intersect
• (C) Moving charges do not produce magnetic field in the surrounding space
• (D) Magnetic field lines always form closed loops
• (E) Magnetic force on a negative charge is opposite to that on a positive charge

When a current passing through a coil changes at a rate of 30 A s\(^{-1}\), the emf induced in the coil is 12 V. If the current passing through this coil changes at a rate of 20 A s\(^{-1}\), the emf induced in this coil is:

• (A) 8 V
• (B) 10 V
• (C) 2.5 V
• (D) 3 V
• (E) 5 V

The reactance of an induction coil of 4 H for a dc current (in \( \Omega \)) is:

• (A) zero
• (B) \( 4\pi \)
• (C) \( 40\pi \)
• (D) \( 400\pi \)
• (E) infinity

If the total momentum delivered to a surface by an EM wave is \(3 \times 10^{-4}\) kg m/s, then the total energy transferred to this surface is:

• (A) \( 3 \times 10^4 \, J \)
• (B) \( 4.5 \times 10^4 \, J \)
• (C) \( 6 \times 10^4 \, J \)
• (D) \( 2 \times 10^4 \, J \)
• (E) \( 9 \times 10^4 \, J \)

The radiations used in LASIK eye surgery are:

• (A) IR radiations
• (B) micro waves
• (C) radio waves
• (D) gamma rays
• (E) UV radiations

When two coherent sources each of individual intensity \( I_0 \) interfere, the resultant intensity due to constructive and destructive interference are respectively

• (A) \( 4I_0 and 0 \)
• (B) \( I_0 and 2I_0 \)
• (C) \( 0 and 2I_0 \)
• (D) \( 2I_0 and I_0 \)
• (E) \( 2I_0 and 0 \)

If the power of a lens is +4 D, then the lens is a

• (A) convex lens of focal length 25 cm
• (B) concave lens of focal length 25 cm
• (C) concave lens of focal length 40 cm
• (D) convex lens of focal length 50 cm
• (E) concave lens of focal length 20 cm

In a single slit diffraction experiment, the width of the slit and the wavelength of the light are respectively 5 mm and 500 nm. If the focal length of the lens is 20 cm, then the size of the central bright fringe will be

• (A) \( 5 \times 10^{-5} \, m \)
• (B) \( 3 \times 10^{-5} \, m \)
• (C) \( 2.5 \times 10^{-5} \, m \)
• (D) \( 2 \times 10^{-5} \, m \)
• (E) \( 1 \times 10^{-5} \, m \)

A particle having mass 2000 times that of an electron travels with a velocity thrice that of the electron. The ratio of the de Broglie wavelength of the particle to that of the electron is

• (A) \( \frac{1}{3000} \)
• (B) \( \frac{1}{2000} \)
• (C) \( \frac{1}{6000} \)
• (D) \( \frac{1}{8000} \)
• (E) \( \frac{1}{1500} \)

The process by which the electrons can come out of the metal in a spark plug is:

• (A) field emission
• (B) ionic emission
• (C) secondary emission
• (D) thermionic emission
• (E) photoelectric emission

The energy required to excite the hydrogen atom from its first excited state to second excited state is:

• (A) 12.09 eV
• (B) 1.89 eV
• (C) 10.2 eV
• (D) 3.40 eV
• (E) 1.51 eV

If the maximum number of neighbours of a nucleon within the range of nuclear force is \( p \) and \( k \) is a constant, then the binding energy per nucleon is approximately:

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

In gamma emission, the nucleus emits

• (A) a photon
• (B) a neutron
• (C) a neutrino
• (D) an electron
• (E) a positron

If the initial decay rate of a radioactive sample is \( R_0 \), then the decay rate after a half-life time \( T_{1/2} \) is

• (A) \( 2R_0 \)
• (B) \( R_0 \)
• (C) \( \sqrt{R_0} \)
• (D) \( 3R_0 \)
• (E) \( \frac{R_0}{2} \)

An external voltage \( V \) is supplied to a semiconductor diode having built-in potential \( V_0 \). The effective barrier height under forward bias is

• (A) \( V_0 + V \)
• (B) \( \frac{V_0 + V}{2} \)
• (C) \( V_0 - V \)
• (D) \( \frac{V_0 - V}{2} \)
• (E) \( 2V_0 + V \)

If the conductivity of the material lies in the range \(10^2 - 10^8 \, \Omega^{-1}m^{-1}\), then it is a

• (A) insulator
• (B) semiconductor
• (C) superconductor
• (D) dielectric
• (E) metal

The thickness of the depletion layer on either side of the p-n junction is of the order of

• (A) \( \mu m \)
• (B) cm
• (C) mm
• (D) nm
• (E) m

The unit of an universal constant is cm\(^{-1}\). What is the constant?

• (A) Planck's constant
• (B) Boltzmann constant
• (C) Rydberg constant
• (D) Avogadro constant
• (E) Molar gas constant

Which of the following molecule has the most polar bond?

• (A) Cl\(_2\)
• (B) HCl
• (C) PCl\(_3\)
• (D) N\(_2\)
• (E) HF

ΔS would be negative for which of the following reactions?

• (I) \( CaCO_3(s) \rightarrow CaO(s) + CO_2(g) \)
• (A) I and III only
• (B) II and III only
• (C) I only
• (D) III only
• (E) I, II, and III

Equal volumes of pH 3, 4, and 5 are mixed in a container. The concentration of \( H^+ \) in the mixture is (Assume there is no change in the volume during mixing):

• (A) \( 1 \times 10^{-3} \, M \)
• (B) \( 3.7 \times 10^{-4} \, M \)
• (C) \( 1 \times 10^{-4} \, M \)
• (D) \( 3.7 \times 10^{-5} \, M \)
• (E) \( 3 \times 10^{-5} \, M \)

The reaction \( H_2O(g) + Cl_2O(g) \rightleftharpoons 2 HOCl(g) \) is allowed to attain equilibrium at 400K. At equilibrium, the partial pressure of \( H_2O(g) \) is 300 mm of Hg, and those of \( Cl_2O(g) \) and \( HOCl(g) \) are 20 mm and 60 mm respectively. The value of \( K_p \) for the reaction at 300K is:

• (A) 36
• (B) 6.0
• (C) 60
• (D) 3.6
• (E) 0.60

Strong intra-molecular hydrogen bond is present in:

• (A) water
• (B) hydrogen fluoride
• (C) o-cresol
• (D) o-nitrophenol
• (E) ammonia

Which of the following molecule has a Lewis structure that does not obey the octet rule?

• (A) HCN
• (B) CS\(_2\)
• (C) NO
• (D) CCl\(_4\)
• (E) PF\(_3\)

The rate and the rate constant of a reaction has the same units. The order of the reaction is

• (A) one
• (B) two
• (C) three
• (D) zero
• (E) half

For the reaction \( 2A + B \rightarrow 2C + D \), the following kinetic data were obtained for three different experiments performed at the same temperature.





The total order and order in [B] for the reaction are respectively

• (A) 2,1
• (B) 1,1
• (C) 1,2
• (D) 2,2
• (E) 2,0

The standard molar entropies of \( SO_2(g) \), \( SO_3(g) \), and \( O_2(g) \) are 250 J/K·mol, 257 J/K·mol, and 205 J/K·mol respectively. Calculate standard molar entropy change for the reaction \( 2SO_2(g) + O_2(g) \rightarrow 2SO_3(g) \).

• (A) -198 J/K·mol
• (B) -191 J/K·mol
• (C) 198 J/K·mol
• (D) 191 J/K·mol
• (E) -1219 J/K·mol

An aqueous solution contains 20g of a non-volatile strong electrolyte \( A_2B \) (Molar mass = 60 g mol\(^{-1}\)) in 1 kg of water. If the electrolyte is 100% dissociated at this concentration, what is the boiling point of the solution? (Kb of water is 0.52 K kg mol\(^{-1}\))

• (A) 372.482 K
• (B) 374.56 K
• (C) 373.52 K
• (D) 371.44 K
• (E) 374.02 K

An organic compound contains 37.5% C, 12.5% H and the rest oxygen. What is the empirical formula of the compound?

• (A) \( CH_4O \)
• (B) \( C_2H_3O \)
• (C) \( CH_3O_2 \)
• (D) \( C_2H_4O \)
• (E) \( CH_3O \)

How many grams of HCl will completely react with 17.4g of pure MnO\(_2\) (s) to liberate Cl\(_2\) (g)? (Atomic mass Mn = 55.0; H = 1; Cl = 35.5)

• (A) 14.6 g
• (B) 7.3 g
• (C) 21.9 g
• (D) 29.2 g
• (E) 34.8 g

What is the quantity of current required to liberate 16g of O\(_2\) (g) during electrolysis of water? (Given 1F = 96500C)

• (A) \(4.825 \times 10^4 \, C\)
• (B) \(9.65 \times 10^4 \, C\)
• (C) \(2.895 \times 10^5 \, C\)
• (D) \(4.825 \times 10^5 \, C\)
• (E) \(1.93 \times 10^5 \, C\)

Co-ordination compounds exhibit different types of isomerism. Some complexes are given in Column I and type of isomerism is given in Column II.

• (A) (a)-(ii), (b)-(iv), (c)-(i), (d)-(iii)
• (B) (a)-(iv), (b)-(i), (c)-(ii), (d)-(iii)
• (C) (a)-(iv), (b)-(iii), (c)-(ii), (d)-(ii)
• (D) (a)-(iv), (b)-(ii), (c)-(iii), (d)-(ii)
• (E) (a)-(iii), (b)-(ii), (c)-(iv), (d)-(i)

Which of the following amines will not undergo carblyamine reaction?

• (A) N-methylthanamine
• (B) Phenylmethanamine
• (C) Aniline
• (D) Ethanamine
• (E) Propan-2-amine

The 3d block metal having positive standard electrode potential (M\(^{2+}\)/M) is

• (A) Titanium
• (B) Vanadium
• (C) Iron
• (D) Copper
• (E) Chromium

Which of the following statement is incorrect with regard to interstitial compounds of transition elements?

• (A) They have high melting points.
• (B) They are very hard.
• (C) They have metallic conductivity.
• (D) They are chemically inert.
• (E) They are stoichiometric compounds.

The alloy containing about 95% lanthanoids, 5% iron and traces of S, C, Ca and Al which is used in producing Mg-based bullets is:

• (A) bell metal
• (B) monel metal
• (C) misch metal
• (D) bronze
• (E) german silver

The IUPAC name of the complex \([Cr(NH_3)_3(H_2O)_3]Cl_3\) is:

• (A) \(triaquatriamminechromium(III) chloride \)
• (B) \(triammnetriaquachromium(III) chloride \)
• (C) \(triaquatriamminechromium(II) chloride \)
• (D) \(triammnetriaquachromium(II) chloride \)
• (E) \(triaquatriamminechromium(III) trichloride \)

In the Carius method of estimation of halogen, 0.4g of an organic compound gave 0.188g of AgBr. What is the percentage of bromine in the organic compound? (The atomic mass of Ag = 108 g mol\(^{-1}\) \& Br = 80 g mol\(^{-1}\))

• (A) \(20%\)
• (B) \(10%\)
• (C) \(15%\)
• (D) \(25%\)
• (E) \(30%\)

Which one of the following compounds can exhibit both optical isomerism and geometrical isomerism?

• (A) \(2-chloropent-2-ene \)
• (B) \(5-chloropent-2-ene \)
• (C) \(4-chloropent-2-ene \)
• (D) \(3-chloropent-1-ene \)
• (E) \(3-chloropent-2-ene \)

Which one of the following nucleophiles is an ambident nucleophile?

• (A) \(CH_3O^-\)
• (B) \(HO^-\)
• (C) \(CH_3COO^-\)
• (D) \(H_2O\)
• (E) \(CN^-\)

Choose the achiral molecule in the following:

• (A) \(2-bromobutane \)
• (B) \(3-nitropentane \)
• (C) \(3-chlorobut-1-ene \)
• (D) \(1-bromoethanol \)
• (E) \(2-hydroxypropanoic acid \)

Phenol can be converted to salicylaldehyde by:

• (A) \(Kolbe reaction \)
• (B) \(Williamson reaction \)
• (C) \(Etard reaction \)
• (D) \(Reimer-Tiemann reaction \)
• (E) \(Stephen reaction \)

The order of decreasing acid strength of carboxylic acids is:

• (A) \(FCH_2COOH > ClCH_2COOH > NO_2CH_2COOH > CNCH_2COOH \)
• (B) \(CNCH_2COOH > FCH_2COOH > NO_2CH_2COOH > ClCH_2COOH \)
• (C) \(NO_2CH_2COOH > FCH_2COOH > ClCH_2COOH > CNCH_2COOH \)
• (D) \(FCH_2COOH > NO_2CH_2COOH > ClCH_2COOH > CNCH_2COOH \)
• (E) \(NO_2CH_2COOH > CNCH_2COOH > FCH_2COOH > ClCH_2COOH \)

Chlorophenylmethane is treated with ethanolic NaCN and the product obtained is reduced with H\(_2\) in the presence of finely divided nickel to give:

• (A) Phenylmethanamine
• (B) 1-phenylethanamine
• (C) 2-phenylethanamine
• (D) 1-methyl-2-phenylethanamine
• (E) phenylmethanamine

A reagent that can be used to reduce benzene diazonium chloride to benzene is:

• (A) ethanol
• (B) methanol
• (C) methanoic acid
• (D) acetone
• (E) phosphorous acid

Which one of the following is not an essential amino acid?

• (A) Lysine
• (B) Tyrosine
• (C) Threonine
• (D) Tryptophan
• (E) Methionine

14g of cyclopropane burnt completely in excess oxygen. The number of moles of water formed is:

• (A) 1.4 moles
• (B) 2.8 moles
• (C) 2.0 moles
• (D) 1.0 mole
• (E) 4 moles

Let \( f(x) = \log_e(x) \) and let \( g(x) = \frac{x - 2}{x^2 + 1} \). Then the domain of the composite function \( f \circ g \) is:

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

Let \( S \) denote the set of all subsets of integers containing more than two numbers. A relation \( R \) on \( S \) is defined by \[ R = \{ (A, B) : the sets A and B have at least two numbers in common \}. \]
Then the relation \( R \) is:

• (A) reflexive, symmetric and transitive
• (B) reflexive and symmetric but not transitive
• (C) not reflexive, not symmetric and not transitive
• (D) not reflexive but symmetric and transitive
• (E) reflexive but not symmetric and transitive

For two sets \( A \) and \( B \), we have \( n(A \cup B) = 50 \), \( n(A \cap B) = 12 \), and \( n(A - B) = 15 \). Then \( n(B - A) \) is equal to:

• (A) 27
• (B) 35
• (C) 38
• (D) 29
• (E) 23

The value of \[ \left(\frac{10i}{(2-i)(3-i)}\right)^{2024} \]
is equal to:

• (A) \(2^{2024}\)
• (B) \(2^{1012}\)
• (C) \(4^{2024}\)
• (D) \(\left(\frac{1}{2}\right)^{2024}\)
• (E) \(\left(\frac{1}{2}\right)^{1012}\)

The period of the function \( f(x) = \sin\left( \frac{3x}{2} \right) \) is equal to:

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

The value of \( \alpha \) for which the complex number \( \frac{2 - \alpha i}{\alpha - i} \) is purely imaginary, is:

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

The centre of a square is at the origin of the complex plane. If one of the vertices is at \( -3i \), then the area of the square is:

• (A) 9
• (B) 12
• (C) 18
• (D) 24
• (E) 27

The modulus of the complex number \[ \frac{(1 + i)^{10} (2 - i)^6}{(2i - 4)^4} \]
is equal to:

• (A) 8
• (B) 10
• (C) 16
• (D) 30
• (E) 32

If \( 0 \leq x \leq 5 \), then the greatest value of \( \alpha \) and the least value of \( \beta \) satisfying the inequalities \( \alpha \leq 3x + 5 \leq \beta \) are, respectively,

• (A) \(0,5\)
• (B) \(10,15\)
• (C) \(5,10\)
• (D) \(5,15\)
• (E) \(5,20\)

Let \( A = \begin{pmatrix} 3 & -2 & 1
-1 & 3 & -1 \end{pmatrix} \) and \( B = \begin{pmatrix} 1
\alpha
-1 \end{pmatrix} \). If \( AB = \begin{pmatrix} -2
6 \end{pmatrix} \), then the value of \( \alpha \) is equal to:

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

If \( 2 \) is a solution of the inequality \( \frac{x-a}{a-2x} < -3 \), then \( a \) must lie in the interval:

• (A) \( (4,5) \)
• (B) \( (2,5) \)
• (C) \( (4,10) \)
• (D) \( (2,10) \)
• (E) \( (0,10) \)

The coefficient of \( x^{14}y \) in the expansion of \( (x^2 + \sqrt{y})^9 \) is:

• (A) 84
• (B) 36
• (C) 63
• (D) 252
• (E) 128

The value of \( x \) that satisfies the equation

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

The sum of the series \( \frac{1}{2^{10}} + \frac{1}{2^{11}} + \cdots + \frac{1}{2^{19}} \) is equal to:

• (A) \( \frac{2^{10} - 1}{2^{21}} \)
• (B) \( \frac{2^9 - 1}{2^{20}} \)
• (C) \( \frac{2^{10} - 1}{2^{19}} \)
• (D) \( \frac{2^9 - 1}{2^{19}} \)
• (E) \( \frac{2^{10} - 1}{2^{20}} \)

Let \( A \) and \( B \) be two sets each containing more than one element. If \( n(A \times B) = 155 \), then \( n(A) \) is equal to:

• (A) 5
• (B) 3
• (C) 7
• (D) 15
• (E) 25

There are 3 different mathematics books and 4 different physics books in a shelf. Then the number of ways these books can be arranged so that the mathematics books are together is:

• (A) 144
• (B) 120
• (C) 520
• (D) 720
• (E) 620

11\( (10P_7) \) =

• (A) \( 11P_7 \)
• (B) \( 10P_8 \)
• (C) \( 11P_8 \)
• (D) \( 11P_9 \)
• (E) \( 10P_9 \)

The value of the sum \( 15 C_6 + 14 C_6 + 13 C_6 + 12 C_6 + 11 C_6 + 10 C_6 \) is equal to:

• (A) \( 15 C_7 - 10 C_6 \)
• (B) \( 15 C_7 - 10 C_7 \)
• (C) \( 16 C_7 - 10 C_7 \)
• (D) \( 16 C_7 - 10 C_6 \)
• (E) \( 16 C_7 - 11 C_6 \)

Let

and let \( B = \frac{1{|A|} A }\).
Then the value of \( |B| \) is equal to:

• (A) \( \frac{1}{9} \)
• (B) \( \frac{1}{11} \)
• (C) \( \frac{1}{81} \)
• (D) \( \frac{1}{121} \)
• (E) \( 1 \)

Let \( f(x) = 2 - 7 \sin{\left( \frac{2x}{7} \right)} \). Then the maximum value of \( f(x) \) is:

• (A) -5
• (B) 5
• (C) 4
• (D) 9
• (E) -9

The second term of a G.P. is \( \frac{1}{2} \). If the product of first five terms is 32, then the common ratio of the G.P. is:

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

The first term and the 6^th term of a G.P. are 2 and \( \frac{64}{243} \) respectively. Then the sum of first 10 terms of the G.P. is:

• (A) \( 6 - \frac{2^{11}}{3^9} \)
• (B) \( 1 - \frac{2^{11}}{3^9} \)
• (C) \( 6 - \frac{2^{10}}{3^9} \)
• (D) \( 1 - \frac{2^{10}}{3^9} \)
• (E) \( 6 - \frac{2^{11}}{3^{10}} \)

An assignment of probabilities for outcomes of the sample space \( S = \{1, 2, 3, 4, 5, 6\} \) is as follows:

If this assignment is valid, then the value of \( k \) is:

• (A) \( \frac{1}{34} \)
• (B) \( \frac{1}{35} \)
• (C) \( \frac{1}{38} \)
• (D) \( \frac{1}{37} \)
• (E) \( \frac{1}{36} \)

Three coins are tossed simultaneously. Then the probability that exactly two tails appear is:

• (A) \( \frac{1}{8} \)
• (B) \( \frac{1}{4} \)
• (C) \( \frac{3}{8} \)
• (D) \( \frac{1}{2} \)
• (E) \( \frac{5}{8} \)

A bag contains 10 green balls and 5 red balls. If two balls are selected randomly, then the probability that both are green balls, is:

• (A) \( \frac{9}{35} \)
• (B) \( \frac{2}{7} \)
• (C) \( \frac{3}{7} \)
• (D) \( \frac{5}{27} \)
• (E) \( \frac{2}{15} \)

Let \( A, B, C \) be three mutually and exhaustive events of an experiment. If \( 2P(A) = 3P(B) = 4P(C) \), then \( P(C) \) is equal to:

• (A) \( \frac{3}{13} \)
• (B) \( \frac{4}{13} \)
• (C) \( \frac{5}{13} \)
• (D) \( \frac{6}{13} \)
• (E) \( \frac{7}{13} \)

Two circles \(C_1\) and \(C_2\) have radii 18 and 12 units, respectively. If an arc of length \( \ell \) of \(C_1\) subtends an angle 80° at the centre, then the angle subtended by an arc of same length \( \ell \) of \(C_2\) at the centre is:

• (A) 90°
• (B) 100°
• (C) 110°
• (D) 120°
• (E) 135°

Given that: \[ \frac{1}{\tan A - \tan B} = \]

• (A) \( \frac{\sin A \sin B}{\cos(A - B)} \)
• (B) \( \frac{\sin A \sin B}{\sin(A - B)} \)
• (C) \( \frac{\cos A - \cos B}{\sin A - \sin B} \)
• (D) \( \cot A - \cot B \)
• (E) \( \frac{\cos A \cos B}{\sin(A - B)} \)

\[ \cos^{-1} \left( \cos \left( \frac{-7\pi}{9} \right) \right) = \]

• (A) \(\frac{-7\pi}{9}\)
• (B) \(\frac{7\pi}{9}\)
• (C) \(\frac{2\pi}{9}\)
• (D) \(\frac{-2\pi}{9}\)
• (E) \(\frac{-4\pi}{9}\)

The value of \[ \frac{\cos^{-1}(0) + \sin^{-1}\left( \frac{\sqrt{3}}{2} \right) + \cos^{-1}\left( \frac{1}{2} \right)}{\sin^{-1}(1) + \cos^{-1}\left( \frac{\sqrt{3}}{2} \right) + \sin^{-1}\left( \frac{1}{\sqrt{2}} \right)} \]
is equal to:

• (A) \( \frac{7}{11} \)
• (B) \( \frac{11}{12} \)
• (C) \( \frac{7}{10} \)
• (D) \( \frac{14}{11} \)
• (E) \( \frac{7}{5} \)

If \(\sec \theta + \tan \theta = 2 + \sqrt{3}\), then \(\sec \theta - \tan \theta\) is:

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

If \( a = \frac{1 + \tan \theta + \sec \theta}{2 \sec \theta} \) and \( b = \frac{\sin \theta}{1 - \sec \theta + \tan \theta} \), then \( \frac{a}{b} \) is equal to:

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

If \[ \frac{1}{1 - \tan x} = \frac{3 + \sqrt{3}}{2}, \quad 0 \leq x \leq \frac{\pi}{2}, \]
then the value of \( x \) is equal to:

• (A) \( \frac{\pi}{3} \)
• (B) \( \frac{\pi}{5} \)
• (C) \( \frac{\pi}{6} \)
• (D) \( \frac{\pi}{8} \)
• (E) \( \frac{\pi}{12} \)

If \( a = \tan^{-1}\left(\frac{4}{3}\right) \) and \( b = \tan^{-1}\left(\frac{1}{3}\right) \), where \( 0 < a, b < \frac{\pi}{2} \), then \( a - b \) is:

• (A) \(\tan^{-1}(3)\)
• (B) \(\tan^{-1}\left(\frac{3}{13}\right)\)
• (C) \(\tan^{-1}(5)\)
• (D) \(\tan^{-1}\left(\frac{9}{13}\right)\)
• (E) \(\tan^{-1}\left(\frac{5}{13}\right)\)

If \( 0 \leq \alpha \leq \frac{\pi}{2} \) and \(\sin \left(\alpha - \frac{\pi}{12}\right) = \frac{1}{2}\), then \(\alpha\) is equal to:

• (A) \(\frac{\pi}{6}\)
• (B) \(\frac{\pi}{4}\)
• (C) \(\frac{\pi}{3}\)
• (D) \(\frac{5\pi}{12}\)
• (E) \(\frac{7\pi}{12}\)

The equation of the line passing through the point \((-9,5)\) and parallel to the line \(5x - 13y = 19\) is:

• (A) \(5x - 13y + 110 = 0\)
• (B) \(5x - 13y + 100 = 0\)
• (C) \(5x - 13y + 65 = 0\)
• (D) \(5x - 13y - 110 = 0\)
• (E) \(5x - 13y - 100 = 0\)

The radius of the circle with centre at \((-4, 0)\) and passing through the point \((2, 8)\) is:

• (A) 6
• (B) 8
• (C) 10
• (D) 12
• (E) 14

The axis of a parabola is parallel to the y-axis and its vertex is at \((5, 0)\). If it passes through the point \((2, 3)\), then its equation is:

• (A) \(y^2 = 3(x - 5)\)
• (B) \(3y = (x - 5)^2\)
• (C) \(3y^2 = x - 5\)
• (D) \(y = 3(x - 5)^2\)
• (E) \(y = 9(x - 5)^2\)

The foci of the ellipse \(\frac{x^2}{49} + \frac{y^2}{24} = 1\) are:

• (A) \( (7,0) \) and \( (-7,0) \)
• (B) \( (6,0) \) and \( (-6,0) \)
• (C) \( (4,0) \) and \( (-4,0) \)
• (D) \( (5,0) \) and \( (-5,0) \)
• (E) \( (3,0) \) and \( (-3,0) \)

The line \(y = 5x + 7\) is perpendicular to the line joining the points \((2, 12)\) and \((12, k)\). Then the value of \(k\) is equal to:

• (A) 12
• (B) -12
• (C) 8
• (D) -8
• (E) 10

The centre of the hyperbola \(16x^2 - 4y^2 + 64x - 24y - 36 = 0\) is at the point:

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

The focus of the parabola \(y^2 + 4y - 8x + 20 = 0\) is at the point:

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

For a hyperbola, the vertices are at \( (6, 0) \) and \( (-6, 0) \). If the foci are at \( (2\sqrt{10}, 0) \) and \( -2\sqrt{10}, 0) \), then the equation of the hyperbola is:

• (A) \(\frac{x^2}{36} - \frac{y^2}{76} = 1\)
• (B) \(\frac{x^2}{76} - \frac{y^2}{36} = 1\)
• (C) \(\frac{x^2}{6} - \frac{y^2}{2} = 1\)
• (D) \(\frac{x^2}{4} - \frac{y^2}{36} = 1\)
• (E) \(\frac{x^2}{36} - \frac{y^2}{4} = 1\)

If a line makes angles \(\alpha\), \(\beta\), and \(\gamma\) with the positive directions of the x, y, and z-axis respectively, then \(\cos 2\alpha + \cos 2\beta + \cos 2\gamma\) equals:

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

Let \( \vec{a}, \vec{b}, \vec{c} \) be three vectors. The angle between \( \vec{a} \) and \( \vec{b} \) is \( 30^\circ \), the angle between \( \vec{a} \) and \( \vec{b} + \vec{c} \) is \( 45^\circ \). If \( |\vec{b}| = \sqrt{6} \) and \( |\vec{c}| = 2\sqrt{2} \), then \( |\vec{b} + \vec{c}| \) is:

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

The vectors \(\vec{a} = 4\mathbf{i} - 3\mathbf{j} - \mathbf{k}\) and \(\vec{b} = 3\mathbf{i} + 2\mathbf{j} + \lambda\mathbf{k}\) are perpendicular to each other. Then the value of \(\lambda\) is equal to:

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

The centre of a circle lies on the y-axis. If it passes through the points \( (-4, 3) \) and \( (3, -4) \), then its radius is:

• (A) \( 7\sqrt{2} \)
• (B) 4
• (C) \( 4\sqrt{2} \)
• (D) 5
• (E) \( 5\sqrt{2} \)

The point of intersection of the lines \(\frac{x-3}{2} = \frac{y-2}{2} = \frac{z-6}{1}\) and \(\frac{x-2}{3} = \frac{y-4}{2} = \frac{z-1}{3}\) is:

• (A) \( (3,4,3) \)
• (B) \( (7,6,6) \)
• (C) \( (4,3,3) \)
• (D) \( (10,11,10) \)
• (E) \( (11,10,10) \)

The angle between the lines \[ \frac{x-1}{6} = \frac{y-5}{8} = \frac{z-3}{10} \quad and \quad \frac{x+1}{2} = \frac{2y+3}{2} = \frac{z+3}{2} \]
is:

• (A) \( \cos^{-1} \left( \frac{\sqrt{2}}{6} \right) \)
• (B) \( \cos^{-1} \left( \frac{2\sqrt{2}}{3} \right) \)
• (C) \( \cos^{-1} \left( \frac{\sqrt{2}}{3} \right) \)
• (D) \( \cos^{-1} \left( \frac{1}{\sqrt{2}} \right) \)
• (E) \( \cos^{-1} \left( \frac{\sqrt{3}}{2} \right) \)

The angle between \(\vec{a}\) and \(\vec{b}\) is \(\frac{\pi}{3}\). If \(\|\vec{a}\| = 5\) and \(\|\vec{b}\| = 10\), then \(\|\vec{a} + \vec{b}\|\) is equal to:

• (A) \(7\sqrt{5}\)
• (B) \(5\sqrt{5}\)
• (C) 15
• (D) \(5\sqrt{3}\)
• (E) \(5\sqrt{7}\)

Let \(f(x) = a^{3x}\) and \(a^5 = 8\). Then the value of \(f(5)\) is equal to:

• (A) 64
• (B) 128
• (C) 256
• (D) 512
• (E) 1024

Let \( f(x) = \begin{cases} x^2 - \alpha, & if x < 1
\beta x - 3, & if x \geq 1 \end{cases} \). If \( f \) is continuous at \( x = 1 \), then the value of \( \alpha + \beta \) is:

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

The integral \(\int e^x \sqrt{e^x} \, dx\) equals:

• (A) \(\frac{3}{2} e^x \sqrt{e^x} + C\)
• (B) \(\frac{2}{3} e^x \sqrt{e^x} + C\)
• (C) \(\frac{5}{2} e^{2x} \sqrt{e^x} + C\)
• (D) \(\frac{2}{5} e^{2x} \sqrt{e^x} + C\)
• (E) \(\frac{2}{3} e^{2x/3} + C\)

The area bounded by the parabola \(y = x^2 + 2\) and the lines \(y = x\), \(x = 1\) and \(x = 2\) (in square units) is:

• (A) \(\frac{31}{6}\)
• (B) \(\frac{29}{6}\)
• (C) \(\frac{25}{6}\)
• (D) \(\frac{17}{6}\)
• (E) \(\frac{13}{6}\)

Let \( f(x) = x \sin(x^4) \). Then \( f'(x) \) at \( x = \sqrt[4]{\pi} \) is equal to:

• (A) \( 4\pi + 1 \)
• (B) \( 4\pi \)
• (C) \( -4\pi \)
• (D) \( 4\pi - 1 \)
• (E) \( 4\pi + 4 \)

For \(1 \leq x < \infty\), let \(f(x) = \sin^{-1}\left(\frac{1}{x}\right) + \cos^{-1}\left(\frac{1}{x}\right)\). Then \(f'(x) =\)

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

The value of the limit \(\lim_{t \to 0} \frac{(5-t)^2 - 25}{t}\) is equal to:

• (A) -10
• (B) -5
• (C) 10
• (D) 5
• (E) 0

A particle is moving along the curve \( y = 8x + \cos y \), where \( 0 \leq y \leq \pi \). If at a point the ordinate is changing 4 times as fast as the abscissa, then the coordinates of the point are:

• (A) \(\left(\frac{\pi}{16}, \frac{\pi}{2}\right)\)
• (B) \(\left(-\frac{1}{8}, 0\right)\)
• (C) \(\left(\frac{1}{8}, 0\right)\)
• (D) \(\left(-\frac{\pi}{2}, -\frac{\pi}{16}\right)\)
• (E) \(\left(\frac{\pi}{2}, \frac{9\pi}{16}\right)\)

The value of the limit \(\lim_{x \to 0} \frac{(2 + \cos 3x) \sin^2 x}{x \tan(2x)}\) is equal to:

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

Evaluate the integral: \[ \int_{\frac{\pi}{5}}^{\frac{3\pi}{10}} \frac{\sqrt{\tan x}}{1 + \sqrt{\tan x}} \, dx \]

• (A) \( \frac{\pi}{4} \)
• (B) \( \frac{\pi}{5} \)
• (C) \( \frac{\pi}{10} \)
• (D) \( \frac{\pi}{20} \)
• (E) \( \frac{\pi}{2} \)

Let \[ f(x) = \begin{cases} x\left( \frac{\pi}{2} + x \right), & if x \geq 0
x\left( \frac{\pi}{2} - x \right), & if x < 0 \end{cases} \]
Then \( f'(-4) \) is equal to:

• (A) \( \frac{\pi - 8}{2} \)
• (B) \( \frac{16 + \pi}{2} \)
• (C) \( \frac{8 + \pi}{2} \)
• (D) \( \frac{\pi - 16}{2} \)
• (E) \( \pi - 16 \)

Let \[ f(x) = \frac{|5 - x|(x + 5)}{\tan(x - 5)} \quad for \quad x \neq 5. \]
Then \[ \lim_{x \to 5} f(x) is equal to: \]

• (A) 10
• (B) -10
• (C) 5
• (D) -5
• (E) 0

The function \[ f(x) = x^{3/5}(5x - 12) \]
is increasing in the set:

• (A) \( \left( \frac{5}{12}, \infty \right) \)
• (B) \( (-\infty, 0) \cup (9, \infty) \)
• (C) \( (-\infty, 0) \cup \left( \frac{5}{12}, \infty \right) \)
• (D) \( \left( 0, \frac{9}{10} \right) \)
• (E) \( \left( \frac{9}{10}, \infty \right) \)

The value of \[ \lim_{x \to 1} \frac{\frac{1}{2x + 1} - \frac{1}{3}}{x - 1} \]
is equal to:

• (A) \( \frac{-2}{9} \)
• (B) \( \frac{2}{9} \)
• (C) \( \frac{-2}{3} \)
• (D) \( \frac{2}{3} \)
• (E) 0

The critical points of the function \( f(x) = (x-3)^3(x+2)^2 \) are:

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

The integrating factor of the differential equation \[ x \frac{dy}{dx} + 2y = x e^x \]
is:

• (A) \( \log_e x \)
• (B) \( \log_e 2x \)
• (C) \( x \)
• (D) \( x^2 \)
• (E) \( 2x \)

The minimum value of the function \( f(x) = x^4 - 4x - 5 \), where \( x \in \mathbb{R} \), is:

• (A) -7
• (B) 7
• (C) 8
• (D) -8
• (E) 0

\[ \int_0^{\frac{\pi}{4}} (\tan^3 x + \tan^5 x) \, dx \]

• (A) \(\frac{5}{12}\)
• (B) \(\frac{1}{3}\)
• (C) \(\frac{1}{4}\)
• (D) \(\frac{1}{6}\)
• (E) \(\frac{1}{12}\)

Let \( I = \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{\tan^2 x}{1+5^x} \, dx \). Then:

• (A) \( I = \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \tan^2 x \, dx \)
• (B) \( 2I = \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \tan^2 x \, dx \)
• (C) \( I = \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{1}{1+5^x} \, dx \)
• (D) \( 2I = \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} 5 \tan^2 x \, dx \)
• (E) \( 2I = \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{1}{1+5^x} \, dx \)

\[ \int \left( \frac{\log_e t}{1+t} + \frac{\log_e t}{t(1+t)} \right) dt \]

• (A) \(\frac{(\log_e t)^2}{2} + C\)
• (B) \(\frac{t^2 (\log_e t)^2}{2} + C\)
• (C) \(\frac{(1+\log_e t)^2}{2} + C\)
• (D) \(\frac{(\log_e t)^2}{2t^2} + C\)
• (E) \(\frac{(\log_e t)^2}{2} + \frac{1}{(1+t)^2} + C\)

Evaluate the integral: \[ \int \frac{x^2 - 1}{x^4 + 3x^2 + 1} \, dx \]

• (A) \( \frac{1}{\sqrt{3}} \tan^{-1} \left( \frac{x^2 + 1}{\sqrt{3}x} \right) + C \)
• (B) \( \tan^{-1} \left( x^2 - 1 \right) + C \)
• (C) \( \tan^{-1} \left( \frac{x - 1}{x} \right) + C \)
• (D) \( \frac{1}{\sqrt{5}} \tan^{-1} \left( \frac{x^2 + 1}{\sqrt{5}x} \right) + C \)
• (E) \( \tan^{-1} \left( \frac{x + 1}{x} \right) + C \)

Evaluate the integral:

• (A) \( \frac{1}{2} \sin \left( \sqrt{4x^2 + 7} \right) + C \)
• (B) \( \frac{7}{2} \sin \left( \sqrt{4x^2 + 7} \right) + C \)
• (C) \( \sin \left( \sqrt{4x^2 + 7} \right) + C \)
• (D) \( \frac{1}{4} \sin \left( \sqrt{4x^2 + 7} \right) + C \)
• (E) \( \frac{7}{4} \sin \left( \sqrt{4x^2 + 7} \right) + C \)

The general solution of the differential equation \( \frac{dy}{dx} = xy - 2x - 2y + 4 \) is:

• (A) \(\frac{1}{(y-2)^2} = \frac{(x-2)^2}{2} + C\)
• (B) \(\log_e|y-2| = \frac{(x-2)^2}{2} + C\)
• (C) \((y-2)^2 = \frac{(x-2)^2}{2} + C\)
• (D) \(\log_e|y-2| = C\)
• (E) \(\log_e|y-2| = (x-2)^2 + C\)

Let \( f(x) = \frac{x^2 + 40}{7x} \), \( x \neq 0 \), \( x \in [4,5] \). The value of \( c \) in \( [4,5] \) at which \( f'(c) = -\frac{1}{7} \) is equal to:

• (A) \( 3\sqrt{2} \)
• (B) \( 2\sqrt{5} \)
• (C) \( \frac{49}{\sqrt{3}} \)
• (D) \( \sqrt{21} \)
• (E) \( 2\sqrt{6} \)

If \( f'(x) = 4x\cos^2(x) \sin\left(\frac{x}{4}\right) \), then \( \lim_{x \to 0} \frac{f(\pi + x) - f(\pi)}{x} \) is equal to:

• (A) \( 4\pi \)
• (B) \( \sqrt{2}\pi \)
• (C) \( 2\pi \)
• (D) \( 2\sqrt{2}\pi \)
• (E) \( 0 \)