KEAM 2024 Question Paper with Answer Key (June 9)

If the displacement of a body moving on a horizontal surface is 151.25 cm in a time interval of 2.25 s, then the velocity of the body in the correct number of significant figures in cm s\(^{-1}\) is:

• (A) 6722
• (B) 67.22
• (C) 67.222
• (D) 0.672
• (E) 67.2

The dimensions of the torque is:

• (A) \( [M L^3 T^{-2}] \)
• (B) \( [M L^3 T^{-1}] \)
• (C) \( [M^{-1} L^{-3} T^2] \)
• (D) \( [M L^2 T^{-2}] \)
• (E) \( [M^0 T^0] \)

A particle is projected at an angle \( \theta \) with the x axis in the xy plane with a velocity \( \mathbf{v} = 6\hat{i} - 4\hat{j} \). The velocity of the body on reaching the x axis again is:

• (A) \( 6\hat{i} - 4\hat{j} \)
• (B) \( 12\hat{i} - 8\hat{j} \)
• (C) \( 3\hat{i} - 2\hat{j} \)
• (D) \( 3\hat{i} + 2\hat{j} \)
• (E) \( 6\hat{i} + 4\hat{j} \)

The displacement (x) – time (t) graph for the motion of a body is a straight line making an angle 45\(^{\circ}\) with the time axis. Then the body is moving with:

• (A) uniform velocity
• (B) uniform acceleration
• (C) non-uniform acceleration
• (D) decreasing velocity
• (E) increasing velocity

A ball is thrown up vertically at a speed of 6.0 m/s. The maximum height reached by the ball (Take \( g = 10 m/s^2 \)) is:

• (A) 80 m
• (B) 100 m
• (C) 18 m
• (D) 1.8 m
• (E) 1 m

The INCORRECT statement is:

• (A) Forces in nature always occur between pair of bodies
• (B) Action and reaction forces are simultaneous forces
• (C) Coefficient of static friction is greater than the coefficient of kinetic friction
• (D) Force is always in the direction of motion
• (E) Centripetal force acts towards the centre of a circle

A bullet of 10 g, moving at 250 m/s, penetrates 5 cm into a tree limb before coming to rest. Assuming uniform force being exerted by the tree limb, the magnitude of the force is:

• (A) 12.5 N
• (B) 625 N
• (C) 62.5 N
• (D) 125 N
• (E) 6250 N

A block of mass \( M \) is kept on the floor of a lift at the centre. The acceleration with which the lift should descend so that the block exerts a force of \( \frac{Mg}{4} \) on the floor of the lift is:

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

A particle of mass 40 g executes simple harmonic motion of amplitude 2.0 cm. If the time period of oscillation is \( \frac{\pi}{20} \) s, then the total mechanical energy of the system is:

• (A) 128 J
• (B) 128 mJ
• (C) 12.8 mJ
• (D) 256 mJ
• (E) 2.56 mJ

The kinetic energy of a body is increased by 21%. The percentage increase in the magnitude of its linear momentum is:

• (A) 10%
• (B) 11%
• (C) 1%
• (D) 20%
• (E) 21%

A tennis ball of mass 50g thrown vertically up at a speed of 25 m s\(^{-1}\) reaches a maximum height of 25 m. The work done by the resistance forces on the ball is:

• (A) 12.5 J
• (B) 50 J
• (C) 62.5 J
• (D) 25 J
• (E) 31.25 J

The radius of gyration of a circular disc of radius \( R \), rotating about its diameter is:

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

For a smoothly running analog clock, the angular velocity of its second hand in rad s\(^{-1}\) is:

• (A) \(\frac{\pi}{1540}\)
• (B) \(\frac{\pi}{720}\)
• (C) \(\frac{\pi}{360}\)
• (D) \(\frac{\pi}{12}\)
• (E) \(\frac{\pi}{30}\)

If the acceleration due to gravity on the surface of a planet is 2.5 times that on Earth and radius, 10 times that of the Earth, then the ratio of the escape velocity on the surface of a planet to that on Earth is:

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

The time period of revolution of a planet around the sun in an elliptical orbit of semi-major axis \( a \) is \( T \). Then

• (A) \( T^2 \propto a^2 \)
• (B) \( T \propto a^3 \)
• (C) \( T^2 \propto a^3 \)
• (D) \( T \propto \frac{1}{a^3} \)
• (E) \( T^2 \propto \frac{1}{a^3} \)

In an incompressible liquid flow, mass conservation leads to:

• (A) Equation of continuity
• (B) Bernoulli's law
• (C) Stoke's law
• (D) Torricelli's law
• (E) Pascal's law

The maximum velocity of a fluid in a tube for which the flow remains streamlined is called its:

• (A) Terminal velocity
• (B) Critical velocity
• (C) Turbulent velocity
• (D) Streamlined velocity
• (E) Surface velocity

Coefficient of linear expansion of aluminum is \( 2.5 \times 10^{-5} \, K^{-1} \). Its coefficient of volume expansion in \( K^{-1} \) is:

• (A) \( 1.25 \times 10^{-5} \)
• (B) \( 5.0 \times 10^{-5} \)
• (C) \( 7.5 \times 10^{-5} \)
• (D) \( 1 \times 10^{-4} \)
• (E) \( 4.0 \times 10^{-5} \)

The efficiency of a Carnot engine operating between steam point and ice point is:

• (A) 100%
• (B) 50%
• (C) 77%
• (D) 27%
• (E) 11%

The type of processes represented by the curves X and Y are:

• (A) Isothermal and Isobaric
• (B) Isothermal and Adiabatic
• (C) Isobaric and Isochoric
• (D) Isochoric and Isobaric
• (E) Adiabatic and Isothermal

Two similar metallic rods of the same length \( l \) and area of cross section \( A \) are joined and maintained at temperatures \( T_1 \) and \( T_2 \) (\( T_1 > T_2 \)) at one of their ends as shown in the figure. If their thermal conductivities are \( K \) and \( \frac{K}{2} \) respectively. The temperature at the joining point in the steady state is:

• (A) \(\frac{T_1 + T_2}{2}\)
• (B) \(\frac{2(T_1 - T_2)}{3}\)
• (C) \(\frac{2T_1 + T_2}{3}\)
• (D) \(\frac{T_1 - T_2}{2}\)
• (E) \(\frac{3(T_1 - T_2)}{2}\)

According to the equipartition principle, the energy contributed by each translational degree of freedom and rotational degree of freedom at a temperature \( T \) are respectively (\( k_B = Boltzmann constant \)):

• (A) \(\frac{1}{2} k_B T, \frac{1}{2} k_B T\)
• (B) \(k_B T, \frac{1}{2} k_B T\)
• (C) \(k_B T, k_B T\)
• (D) \(\frac{1}{2} k_B T, k_B T\)
• (E) \(\frac{3}{2} k_B T, \frac{1}{2} k_B T\)

The kinetic energy of 3 moles of a diatomic gas molecules in a container at a temperature \( T \) is same as that of kinetic energy of \( n \) moles of monoatomic gas molecules in another container at the same temperature \( T \). The value of \( n \) is:

• (A) 3
• (B) 4
• (C) 2.5
• (D) 5
• (E) 3.5

A string of length \( L \) is fixed at both ends and vibrates in its fundamental mode. If the speed of waves on the string is \( v \), then the angular wave number of the standing wave is:

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

Ratio between the frequencies of the third harmonics in the closed organ pipe and open organ pipe of the same length is:

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

A tuning fork vibrating at 300 Hz, initially in air, is then placed in a trough of water. The ratio of the wavelength of the sound waves produced in air to that in water is (Given that the velocity of sound in water and in air at that place are 1500 m/s and 350 m/s respectively):

• (A) 1:1
• (B) 37:23
• (C) 30:7
• (D) 7:30
• (E) 23:37

The ratio of the magnitudes of electrostatic force between an electron and a proton separated by a distance \( r \) to that between a proton and an alpha particle separated by the same distance \( r \) is:

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

The electric field due to an infinitely long thin wire with linear charge density \( \lambda \) at a radial distance \( r \) is proportional to:

• (A) \(\frac{\lambda^2}{r}\)
• (B) \(\frac{\lambda}{r}\)
• (C) \(\frac{\lambda}{r^2}\)
• (D) \(\frac{\sqrt{\lambda}}{\sqrt{r}}\)
• (E) \(\frac{\lambda}{\sqrt{r}}\)

A spherical metal shell \( A \) of radius \( R_A \) and a solid metal sphere \( B \) of radius \( R_B \) (\( R_B < R_A \)) are kept far apart and each is given charge \( +Q \). If they are connected by a thin metal wire and \( Q_A \) and \( Q_B \) are the charge on \( A \) and \( B \), respectively, then:

• (A) \( Q_A = Q_B = 0 \)
• (B) \( Q_A = Q_B = Q \)
• (C) \( Q_A < Q_B \)
• (D) \( Q_A = -Q_B \)
• (E) \( Q_A > Q_B \)

If the number of electron-hole pairs per cm\(^3\) of an intrinsic Si wafer at temperature 300 K is \(1.1 \times 10^{10}\) and the mobilities of electrons and holes at 300 K are 1500 and 500 cm\(^2\) per volt, second, respectively, then the conductivity of the Si wafer at this temperature (in \(\mu\)mho cm\(^{-1}\)) is nearly:

• (A) 352
• (B) 35.2
• (C) 3.52
• (D) 70.4
• (E) 17.6

Magnitude of drift velocity per unit electric field is known as:

• (A) Displacement current
• (B) Mobility
• (C) Electric resistance
• (D) Electrical conductivity
• (E) Relaxation time

The y-intercept of the graph between the terminal voltage \(V\) with load resistance \(R\) along \(y\) and \(x\) – axis, respectively, of a cell with internal resistance \(r\), as shown, is:

• (A) \(\varepsilon\)
• (B) \(-\varepsilon\)
• (C) \(\frac{\varepsilon}{R}\)
• (D) \(\varepsilon R\)
• (E) \(-\varepsilon R\)

A charged particle will continue to move in the same direction in a region, where \(E\) - Electric field, \(B\) - Magnetic field:

• (A) \(E = 0, B = 0\)
• (B) \(E \neq 0, B \neq 0\)
• (C) \(E = 0, B \neq 0\)
• (D) \(E \neq 0, B = 0\)
• (E) \(E = B \neq 0\)

When an \( \alpha \) particle and a proton are projected into a perpendicular uniform magnetic field, they describe circular paths of the same radius. The ratio of their respective velocities is:

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

An electric appliance draws 3A current from a 200 V, 50 Hz power supply. The amplitude of the supply voltage is nearly:

• (A) 140 V
• (B) 200 V
• (C) 283 V
• (D) 67 V
• (E) 600 V

The oscillating magnetic field in a plane electromagnetic wave is given by \(B_y = (8 \times 10^{-6}) \sin[2 \pi \times 10^{11} t + 200 \pi x] \) tesla. Then the wavelength of the electromagnetic wave (in cm) is:

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

A path length of 1m in air is equal to a path length of \(x\) m in a medium of refractive index 1.5. Then the value of \(x\) (in meters) is:

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

A parallel beam of light is incident from air at an angle \(\alpha\) on the side PQ of a right-angled triangular prism of refractive index \(\mu = \sqrt{2} \approx 1.414\). The beam of light undergoes total internal reflection in the prism at the face PR when \(\alpha\) has a minimum value of 45\(^{\circ}\). The angle \(\theta\) of the prism is:

• (A) 15\(^{\circ}\)
• (B) 30\(^{\circ}\)
• (C) 45\(^{\circ}\)
• (D) 60\(^{\circ}\)
• (E) 90\(^{\circ}\)

The wavelength of the de Broglie wave (in meter) associated with a particle of mass \(m\) moving with \(\frac{1}{10}\) of the velocity of light is (h = Planck's constant, c = velocity of light):

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

For a given radioactive material of mean life \(\tau\) and half-life \(t_{1/2}\), the relationship between \(t_{1/2}\) and \(\tau\) is:

• (A) \( t_{1/2} = \frac{\ln 2}{\tau} \)
• (B) \( t_{1/2} = \tau \ln 2 \)
• (C) \( t_{1/2} = \tau \)
• (D) \( t_{1/2} = 2\tau \)
• (E) \( t_{1/2} = \frac{\tau}{\ln 2} \)

The constancy of the binding energy per nucleon in medium-sized nuclei is due to:

• (A) Short-range nature of nuclear force
• (B) Attractive nature of nuclear force
• (C) Saturation nature of nuclear force
• (D) Charge independent nature of nuclear forces
• (E) Strongest nature of nuclear forces

In a radioactive decay, the fraction of the number of atoms left undecayed after time \( t \) is:

• (A) \( e^{-\lambda t+1} \)
• (B) \( e^{-\lambda t} \)
• (C) \( e^{\lambda t} \)
• (D) \( e^{\lambda t-1} \)
• (E) \( e^{t+1} \)

In the electron emission process, \( _{Z}^{A}X \rightarrow _{Z+1}^{A}Y + e^{-} + \bar{\nu} \), the particle \( q \) emitted along with the electron is:

• (A) Neutron
• (B) Neutrino
• (C) Antineutrino
• (D) Proton
• (E) Positron

The current flowing from p to n side in a pn junction diode irrespective of biasing is termed:

• (A) Drift current
• (B) Diffusion current
• (C) Net current
• (D) Displacement current
• (E) Biasing current

The energy required by the electron to cross the forbidden band for Germanium is:

• (A) 0.72 eV
• (B) 1.1 eV
• (C) 0.5 eV
• (D) 1.5 eV
• (E) 0.65 eV

The molarity of sodium hydroxide in the solution prepared by dissolving 6 g in 600 mL of water is (molar mass of NaOH = 40 g mol\(^{-1}\)):

• (A) 0.5 M
• (B) 0.4 M
• (C) 0.25 M
• (D) 0.1 M
• (E) 0.2 M

The volume of ethanol required to prepare 3 L of 0.25 M aqueous solution is (density of ethanol = 0.36 kg L\(^{-1}\), molar mass = 60 g mol\(^{-1}\)):

• (A) 125 mL
• (B) 25 mL
• (C) 75 mL
• (D) 50 mL
• (E) 12.5 mL

Which of the following statement is incorrect about Bohr's model of the atom?

• (A) It fails to account for the finer details of the hydrogen atom spectrum.
• (B) Unable to explain the splitting of spectral lines in the presence of magnetic field.
• (C) The angular momentum of the electron is quantised.
• (D) The ability of atoms to form molecules by chemical bonds.
• (E) Unable to explain the splitting of spectral lines in the presence of electric field.

The decreasing order of first ionisation enthalpy of the following elements is:

• (A) \(N > O > C > Be\)
• (B) \(O > N > C > Be\)
• (C) \(Be > C > O > N\)
• (D) \(N > O > Be > C\)

The hybridisation involved in the metal atom of \([CrF_6]^{3-}\) is:

• (A) \(d^2sp^3\)
• (B) \(dsp^2\)
• (C) \(sp^3\)
• (D) \(sp^2\)
• (E) \(sp^3d^2\)

The valence electron MO configuration of \( C_2 \) (atomic number of C = 6) molecule is:

• (A) \( (\sigma 2s)^3 (\sigma^* 2s)^3 (\pi 2p)^2 \)
• (B) \( (\sigma 2s)^2 (\sigma^* 2s)^2 (\pi 2p)^4 \)
• (C) \( (\sigma 2s)^2 (\sigma^* 2s)^3 (\pi 2p)^3 \)
• (D) \( (\sigma 2s)^2 (\sigma^* 2s)^4 (\pi 2p)^2 \)
• (E) \( (\sigma 2s)^2 (\sigma^* 2s)^2 (\pi 2p)^5 \)

Which of the following is used as anode in mercury cell?

• (A) Paste of \( NH_4Cl \) and \( ZnCl_2 \)
• (B) Manganese dioxide and carbon
• (C) Paste of \( HgO \) and carbon
• (D) Paste of KOH and ZnO
• (E) Zinc-Mercury amalgam

Which of the following is true for a reaction that is spontaneous only at high temperature?

• (A) \( \Delta_r H^\circ < 0, \Delta_r S^\circ > 0, \Delta_r G^\circ < 0 \)
• (B) \( \Delta_r H^\circ > 0, \Delta_r S^\circ > 0, \Delta_r G^\circ > 0 \)
• (C) \( \Delta_r H^\circ > 0, \Delta_r S^\circ > 0, \Delta_r G^\circ < 0 \)
• (D) \( \Delta_r H^\circ > 0, \Delta_r S^\circ < 0, \Delta_r G^\circ < 0 \)
• (E) \( \Delta_r H^\circ < 0, \Delta_r S^\circ < 0, \Delta_r G^\circ < 0 \)

In a process, 600 J of heat is absorbed by a system and 375 J of work is done by the system. The change in internal energy of the process is:

• (A) 975 J
• (B) -225 J
• (C) -975 J
• (D) 985 J
• (E) 225 J

The value of \( K_c \) for the equilibrium reaction \[ 2 NO_2(g) \rightleftharpoons N_2O_4(g) \]
is \(2 \times 10^{-40} \, mol^{-1} \, dm^3\) at 298 K. If the equilibrium concentration of \(NO_2\) is \(2 \times 10^{-2}\) M, the concentration of \(N_2O_4\) is:

• (A) \(6 \times 10^{-42} \, M\)
• (B) \(12 \times 10^{-44} \, M\)
• (C) \(8 \times 10^{-44} \, M\)
• (D) \(2 \times 10^{-44} \, M\)
• (E) \(4 \times 10^{-44} \, M\)

The quantity of electricity required to produce 18 g of Al from molten Al\(_2\)O\(_3\) is (Atomic mass of Al = 27):

• (A) 2F
• (B) 4F
• (C) 5F
• (D) 6F
• (E) 1.5F

The average oxidation state of sulphur in the tetrathionate ion is:

• (A) +3
• (B) +2.5
• (C) +5
• (D) +3.5
• (E) +1.5

The mass percentage of glucose in acetonitrile when 6 g of glucose is dissolved in 294 g of acetonitrile is:

• (A) 6%
• (B) 10%
• (C) 8%
• (D) 4%
• (E) 2%

The rate constant of a first order reaction is \(4.606 \times 10^{-3} \, s^{-1}\). The time taken to reduce 20 g of reactant into 2 g is:

• (A) 300 s
• (B) 500 s
• (C) 150 s
• (D) 400 s
• (E) 250 s

The rate law for the reaction, A + B → Product, is:
\[ rate = [A][B]^{3/2} \]

The total order of the reaction is:

• (A) 3
• (B) 2.5
• (C) 3.5
• (D) 1.5
• (E) 2

Which of the following mixture forms azeotrope?

• (A) Phenol-aniline
• (B) Nitric acid-water
• (C) Ethanol-acetone
• (D) Chloroform-acetone
• (E) Cs\(_2\)-acetone

A coordination compound of cobalt acts as an antipericious anemia factor is:

• (A) Cyanocobalamine
• (B) Carboxypeptidase
• (C) \([Co(NH_3)_6]^{3+}\)
• (D) Haemoglobin
• (E) Myoglobin

The type of d-d transition of the electron occurs in \([Ti(H_2O)_6]^{3+}\) is:

• (A) \( t_{2g}^2 e_g^1 \rightarrow t_{2g}^3 e_g^0 \)
• (B) \( t_{2g}^1 e_g^2 \rightarrow t_{2g}^2 e_g^1 \)
• (C) \( t_{2g}^1 e_g^3 \rightarrow t_{2g}^2 e_g^2 \)
• (D) \( t_{2g}^1 e_g^2 \rightarrow t_{2g}^2 e_g^0 \)
• (E) \( t_{2g}^2 e_g^1 \rightarrow t_{2g}^3 e_g^1 \)

The increasing order of field strength of ligands in the spectrochemical series is:

• (A) \( CO < H_2O < Cl^- < I^- \)
• (B) \( Cl^- < H_2O < CO < I^- \)
• (C) \( H_2O < CO < F^- < Cl^- \)
• (D) \( H_2O < F^- < CO \)
• (E) \( I^- < Cl^- < H_2O < CO \)

The reaction, \( 2I^- + S_2O_8^{2-} \rightarrow I_2 + 2SO_4^{2-} \), is catalysed by:

• (A) Iron(II)
• (B) Manganese(VI)
• (C) Iron(III)
• (D) Vanadium(V)
• (E) Cobalt(III)

Which of the following is used in the treatment of lead poisoning?

• (A) EDTA
• (B) DMG
• (C) Cupron
• (D) \( \alpha \)-nitroso-\( \beta \)-naphthol
• (E) Myoglobin

The increasing order of acid strength of the following carboxylic acids is:

• (i) (CH_3)_3C-COOH \quad (ii) (CH_3)_2CH-COOH \quad (iii) CH_3CH_2COOH \]
• (A) \( (i) < (ii) < (iii) \)
• (B) \( (i) < (iii) < (ii) \)
• (C) \( (ii) < (i) < (iii) \)
• (D) \( (ii) < (iii) < (i) \)
• (E) \( (iii) < (ii) < (i) \)

The decreasing order of stability of the following carbocations is:

• (i) (CH_3)_3C^+ \quad (ii) (CH_3)_2C-CH_2^+ \quad (iii) CH_3CH_2-CH_2^+ \]
• (A) \( (i) > (ii) > (iii) \)
• (B) \( (ii) > (i) > (iii) \)
• (C) \( (iii) > (ii) > (i) \)
• (D) \( (i) > (iii) > (ii) \)
• (E) \( (i) > (ii) > (iii) \)

The number of unpaired electrons in \([CoF_6]^{3-}\) is:

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

One mole of an alkene on ozonolysis gives a mixture of one mole pentan-3-one and one mole methanal. The alkene is:

• (A) 3-ethylbut-1-ene
• (B) 2-methylpent-1-ene
• (C) 2-ethylbut-1-ene
• (D) 4-methylpent-1-ene
• (E) 4-methylpent-2-ene

A tertiary alkyl halide (X), C\(_4\)H\(_9\)Br, reacted with alc.KOH to give compound (Y). Compound (Y) reacted with HBr in presence of peroxide to give compound (Z). The compounds (Y) and (Z) are respectively:

• (A) Propene and tert-butylbromide
• (B) 2-methyl-1-propene and 1-bromo-2-methylpropane
• (C) but-1-ene and 2-bromopropane
• (D) but-2-ene and 2-methylpropane
• (E) but-2-ene and 3-methylpropane

The major products formed when one mole of \( CH_3CH_2CH(CH_3)CH_2OCH_2CH_3 \) is treated with one mole of HI are:

• (A) 2-methylbutan-1-ol and iodoethane
• (B) ethanol and 2-methylidobutane
• (C) 2-methylbutan-2-ol and iodomethane
• (D) 2-methylbutan-1-ol and iodomethane
• (E) 2-methylbutan-1-ol and ethene

The reagent used for the conversion of but-2-ene to ethanol is:

• (A) anhydrous CrO\(_3\)
• (B) DIBAL-H
• (C) PCC
• (D) O\(_3\)/H\(_2\)O-Zn dust
• (E) anhydrous AlCl\(_3\)

Which of the following is used as insect attractant?

• (A) Propan-1-amine
• (B) N,N-Dimethylmethanamine
• (C) Propan-2-amine
• (D) N,N-dimethylbutan-1-amine
• (E) Ethanamine

Lactose is composed of:

• (A) \( \alpha \)-D-glucose and \( \beta \)-D-galactose
• (B) two units of \( \alpha \)-D-glucose
• (C) \( \beta \)-D-galactose and \( \alpha \)-D-glucose
• (D) \( \alpha \)-D-glucose and \( \beta \)-D-fructose
• (E) two units of \( \beta \)-D-galactose

If A and B are two sets, such that A has 20 elements, \( A \cup B \) has 32 elements, and \( A \cap B \) has 10 elements, the number of elements in the set B is:

• (A) 22
• (B) 12
• (C) 32
• (D) 42
• (E) 52

Let a relation \( R \) on the set of natural numbers be defined by \( (x, y) \in R \) if and only if \( x^2 - 4xy + 3y^2 = 0 \) for all \( x, y \in \mathbb{N} \). Then the relation is:

• (A) reflexive
• (B) symmetric
• (C) transitive
• (D) reflexive and symmetric but not transitive
• (E) an equivalence relation

If \( f(x) = \begin{cases} x^2 & for x < 0
5x - 3 & for 0 \leq x \leq 2
x^2 + 1 & for x > 2 \end{cases} \), then the positive value of \( x \) for which \( f(x) = 2 \) is:

• (A) \( \frac{3}{5} \)
• (B) \( \frac{1}{2} \)
• (C) \( \frac{3}{4} \)
• (D) 1
• (E) 0

Let \( X \) and \( Y \) be subsets of \( \mathbb{R} \). If \( f : X \rightarrow Y \) given by \( f(x) = -8(x + 5)^2 \) is one-to-one, then the codomain \( Y \) is:

• (A) \( (-\infty, 0] \)
• (B) \( (-\infty, -5] \)
• (C) \( (-\infty, -5) \)
• (D) \( [0, -\infty) \)
• (E) \( (-\infty, \infty) \)

Let \( z \) be a complex number satisfying \( |z + 16| = 4|z + 1| \). Then:

• (A) \( |z| = 2 \)
• (B) \( |z| = 4 \)
• (C) \( |z| = 8 \)
• (D) \( |z| = 10 \)
• (E) \( |z| = 16 \)

If \( 2z = 7 + i\sqrt{3} \), then the value of \( z^2 - 7z + 4 \) is:

• (A) \( \frac{39}{4} \)
• (B) \( \frac{39}{4} \)
• (C) -9
• (D) 17
• (E) 9

If \( \left( \frac{1 - i}{1 + i} \right)^{10} = a + ib \), then the values of \( a \) and \( b \) are, respectively:

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

If \( z_1 \) and \( z_2 \) are two complex numbers with \( |z_1| = 1 \), then \( \left| \frac{z_1 - z_2}{1 - z_1 \overline{z_2}} \right| \) is equal to:

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

The second term of a G.P. is 4, then the product of the first three terms is:

• (A) 16
• (B) 32
• (C) 48
• (D) 64
• (E) 128

The common ratio of a G.P. is \( \frac{1}{2} \). If the product of the first three terms is 64, then the sum of the first 10 terms is:

• (A) \( \frac{1023}{128} \)
• (B) \( \frac{1023}{256} \)
• (C) \( \frac{511}{128} \)
• (D) \( \frac{511}{256} \)
• (E) \( \frac{511}{512} \)

The numbers \( a, b, c, d \) are in G.P. with common ratio \( r \). If \( \frac{1}{a^3 + b^3} + \frac{1}{b^3 + c^3} + \frac{1}{c^3 + d^3} \) are also in G.P., then the common ratio is:

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

The minimum value of \( f(x) = 7x^4 + 28x^3 + 31 \) is:

• (A) 12
• (B) 10
• (C) 38
• (D) 76
• (E) 56

Evaluate \( \binom{10}{1} + \binom{10}{2} + \dots + \binom{10}{10} \):

• (A) 1023
• (B) 1024
• (C) 511
• (D) 2047
• (E) 612

The coefficient of \( x^3 \) in the binomial expansion of \( \left( \frac{1}{\sqrt{x}} - x \right)^6 \) is:

• (A) 12
• (B) 15
• (C) 10
• (D) 30
• (E) 20

If \( _nP_r = 480 \) and \( _nC_r = 20 \), then the value of \( r \) is equal to:

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

The constant term in the expansion of \( \left( x^3 + \frac{1}{x^2} \right)^{10} \) is:

• (A) 210
• (B) 240
• (C) 140
• (D) 120
• (E) 320

If \[ \begin{bmatrix} 3 & 4
5 & x \end{bmatrix} + \begin{bmatrix} 1 & y
0 & 1 \end{bmatrix} = \begin{bmatrix} 7 & 0
10 & 5 \end{bmatrix}, \]
then the value of \( x - y \) is:

• (A) 1
• (B) 3
• (C) 5
• (D) 10
• (E) 20

If \( B = \begin{bmatrix} 1 & \alpha & 3
1 & 3 & 3
2 & 4 & 4 \end{bmatrix} \) is the adjoint of a \( 3 \times 3 \) matrix \( A \) and \( |A| = 4 \), then the value of \( \alpha \) is:

• (A) 4
• (B) 7
• (C) 9
• (D) 11
• (E) 13

If the points \( (2, -3), (\lambda, -1) \) and \( (0, 4) \) are collinear, then the value of \( \lambda \) is equal to:

• (A) 0
• (B) \( \frac{1}{7} \)
• (C) \( \frac{3}{10} \)
• (D) \( \frac{7}{10} \)
• (E) \( \frac{10}{7} \)

The solution set for the inequalities \( -5 \leq \frac{2 - 3x}{4} \leq 9 \) is:

• (A) \( \left( \frac{-34}{2}, \frac{-22}{3} \right) \)
• (B) \( \left( \frac{22}{34}, \frac{2}{3} \right) \)
• (C) \( \left( \frac{-34}{22}, \frac{3}{3} \right) \)
• (D) \( (-34, -22) \)
• (E) \( \left( \frac{11}{22}, \frac{3}{3} \right) \)

If the determinant of the matrix \( \begin{bmatrix} |x| & 1 & 2
4 & 1 & x
1 & -1 & 3 \end{bmatrix} \) equals -10, then the values of \( x \) are:

• (A) -2 and -6
• (B) 2 and 6
• (C) 1 and 4
• (D) -1 and -4
• (E) 2 and -6

Let \( A = (a_{ij}) \) be a square matrix of order 3 and let \( M_{ij} \) be the minors of \( a_{ij} \). If \( M_{11} = -40, M_{12} = -10, M_{13} = 35 \), and \( a_{11} = 1, a_{12} = 3, a_{13} = -2 \), then the value of \( |A| \) is equal to:

• (A) -100
• (B) -80
• (C) 0
• (D) 60
• (E) 80

If \( \frac{\sec^2 15^\circ - 1}{\sec^2 15^\circ} \) equals:

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

The value of \( \sin^2 \left( \frac{3\pi}{8} \right) + \sin^2 \left( \frac{7\pi}{8} \right) \) is:

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

If \( \sin \theta = \frac{b}{a} \), then \( \frac{\sqrt{a+b}} {\sqrt{a-b}} + \frac{\sqrt{a-b}} {\sqrt{a+b}} \) is equal to:

• (A) \( \frac{2}{\cos \theta} \)
• (B) \( \frac{1}{\cos \theta} \)
• (C) \( \frac{2}{\sqrt{\cos \theta}} \)
• (D) \( \frac{1}{\cos \theta} \)
• (E) 1

The period of \( 2 \sin 4x \cos 4x \) is:

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

The domain of the function \( f(x) = \frac{\sin^{-1} \left( x-3 \right)}{\sqrt{9 - x^2}} \) is:

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

If \( \alpha = \tan^2 x + \cot^2 x \), where \( x \in \left( 0, \frac{\pi}{2} \right) \), then \( \alpha \) lies in the interval:

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

The value of \( \tan \left[ \tan^{-1} \left( \frac{3}{4} \right) + \tan^{-1} \left( \frac{2}{3} \right) \right] \) is:

• (A) \( \frac{17}{6} \)
• (B) \( \frac{6}{17} \)
• (C) \( -\frac{17}{6} \)
• (D) \( -\frac{6}{11} \)
• (E) 1

If \( 3 \sin \theta + 5 \cos \theta = 5 \), then the value of \( 5 \sin \theta - 3 \cos \theta \) is:

• (A) 0
• (B) 1
• (C) 3
• (D) 5
• (E) \( \sqrt{10} \)

Evaluate \( \cos \left( \cot^{-1} \left( \frac{7}{24} \right) \right) \):

• (A) \( \frac{24}{25} \)
• (B) \( \frac{7}{24} \)
• (C) \( \frac{7}{27} \)
• (D) \( \frac{7}{25} \)
• (E) \( \frac{24}{27} \)

If \( \cos \theta = \frac{2 \cos \alpha + 1}{2 + \cos \alpha} \), then \( \tan^2 \left( \frac{\theta}{2} \right) \) is equal to:

• (A) \( \frac{1}{3} \tan^2 \left( \frac{\alpha}{2} \right) \)
• (B) \( \frac{1}{2} \tan^2 \left( \frac{\alpha}{2} \right) \)
• (C) \( \frac{1}{3} \cos^2 \left( \frac{\alpha}{2} \right) \)
• (D) \( \frac{1}{3} \cot^2 \left( \frac{\alpha}{2} \right) \)
• (E) \( 3 \cot^2 \left( \frac{\alpha}{2} \right) \)

If a vector makes angles \( \frac{\pi}{3}, \frac{\pi}{4} \) and \( \gamma \) with \( \hat{i}, \hat{j} \), and \( \hat{k} \), respectively, where \( \gamma \in \left( \frac{\pi}{2}, \pi \right) \), then the angle \( \gamma \) is:

• (A) \( \frac{3\pi}{4} \)
• (B) \( \frac{7\pi}{12} \)
• (C) \( \frac{11\pi}{12} \)
• (D) \( \frac{5\pi}{6} \)
• (E) \( \frac{2\pi}{3} \)

Let \( \mathbf{u}, \mathbf{v}, \mathbf{w} \) be vectors such that \( \mathbf{u} + \mathbf{v} + \mathbf{w} = \mathbf{0} \). If \( |\mathbf{u}| = 3 \), \( |\mathbf{v}| = 4 \), and \( |\mathbf{w}| = 5 \), then \( \mathbf{u} \cdot \mathbf{v} + \mathbf{w} \cdot \mathbf{u} \) is:

• (A) 47
• (B) -25
• (C) 26
• (D) -47
• (E) 0

Let \( \vec{a} = \hat{i} - \hat{j} \), \( \vec{b} = \hat{j} - \hat{k} \), and \( \vec{c} = \hat{k} - \hat{i} \), then the value of \( \vec{b} \cdot (\vec{a} + \vec{c}) \) is:

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

Let \( \vec{a}, \vec{b}, \vec{c} \) be three vectors with magnitudes 4, 4, and 2, respectively. If \( \vec{a} \) is perpendicular to \( (\vec{b} + \vec{c}) \), \( \vec{b} \) is perpendicular to \( (\vec{c} + \vec{a}) \), and \( \vec{c} \) is perpendicular to \( (\vec{a} + \vec{b}) \), then the value of \( |\vec{a} + \vec{b} + \vec{c}| \) is:

• (A) 3
• (B) 6
• (C) \( \sqrt{6} \)
• (D) \( \sqrt{6} \)
• (E) -6

If two vectors \( \vec{a} = \cos \alpha \hat{i} + \sin \alpha \hat{j} + \sin \frac{\alpha}{2} \hat{k} \) and \( \vec{b} = \sin \alpha \hat{i} - \cos \alpha \hat{j} + \cos \frac{\alpha}{2} \hat{k} \) are perpendicular, then the values of \( \alpha \) are:

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

If one end of a diameter of the circle \( x^2 + y^2 - 4x - 6y + 11 = 0 \) is \( (3, 4) \), then the coordinate of the other end of the diameter is:

• (A) \( (1, 1) \)
• (B) \( \left( \frac{1}{2}, \frac{1}{2} \right) \)
• (C) \( (1, 2) \)
• (D) \( (2, 1) \)
• (E) \( (2, 2) \)

If the focus of a parabola is \( (0, -3) \) and its directrix is \( y = 3 \), then its equation is:

• (A) \( x^2 = 12y \)
• (B) \( x^2 = -12y \)
• (C) \( y^2 = 12x \)
• (D) \( y^2 = -12x \)
• (E) \( y^2 = x \)

The length of the minor axis of the ellipse with foci \( (\pm 2, 0) \) and eccentricity \( \frac{1}{3} \) is:

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

The equation of the line passing through the point \( (1, 2) \) and perpendicular to the line \( x + y + 1 = 0 \) is:

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

The line \( \frac{x}{5} + \frac{y}{b} = 1 \) passes through the point \( (13, 32) \) and is parallel to the line \( \frac{x}{c} + \frac{y}{3} = 1 \). Then the values of \( b \) and \( c \) are, respectively:

• (A) \( -20, \frac{-3}{4} \)
• (B) \( 20, \frac{3}{4} \)
• (C) \( \frac{3}{4}, 20 \)
• (D) \( \frac{-3}{4}, 20 \)
• (E) \( -20, \frac{3}{4} \)

A ray of light passing through the point \( (1, 2) \) is reflected on the \( x \)-axis at a point \( P \) and passes through the point \( (5, 6) \). Then the abscissa of the point \( P \) is:

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

If the straight line \( \frac{x - a}{1} = \frac{y - b}{2} = \frac{z - 3}{-1} \) passes through \( (-1, 3, 2) \), then the values of \( a \) and \( b \) are, respectively:

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

The lines \( \frac{x + 3}{-2} = \frac{y}{1} = \frac{z - 4}{3} \) and \( \frac{x - 1}{\mu} = \frac{y - 1}{\mu + 1} = \frac{z}{\mu + 2} \) are perpendicular to each other. Then the value of \( \mu \) is:

• (A) \( \frac{-5}{3} \)
• (B) 3
• (C) 4
• (D) \( \frac{-1}{4} \)
• (E) \( \frac{-7}{2} \)

If the straight lines \( \frac{x - 3}{2} = \frac{y - 4}{3} = \frac{z - 6}{-1} \) and \( \frac{x - 2}{a} = \frac{y + 3}{b} = \frac{z + 4}{-1} \) are parallel, then \( a^2 + b^2 \) is:

• (A) 1
• (B) 13
• (C) 24
• (D) 17
• (E) 3

The angle between the lines \( \frac{x}{1} = \frac{y}{1} = \frac{z}{1} \) and \( \frac{x}{0} = \frac{y}{1} = \frac{z}{-1} \) is:

• (A) \( \frac{\pi}{2} \)
• (B) 0
• (C) \( \frac{\pi}{4} \)
• (D) \( \frac{\pi}{3} \)
• (E) \( \sin^{-1}(\sqrt{2}) \)

If three distinct numbers are chosen randomly from the first 50 natural numbers, then the probability that all of them are divisible by 2 and 3 is:

• (A) \( \frac{3}{350} \)
• (B) 3
• (C) \( \frac{2}{175} \)
• (D) \( \frac{1}{175} \)
• (E) \( \frac{1}{350} \)

If \( \frac{1 + 3p}{4}, \frac{1 - p}{3}, \frac{1 - 3p}{2} \) are the probabilities of three mutually exclusive and exhaustive events, then the value of \( p \) is:

• (A) \( \frac{1}{3} \)
• (B) \( \frac{12}{13} \)
• (C) \( \frac{2}{3} \)
• (D) \( \frac{1}{13} \)
• (E) \( \frac{2}{13} \)

The mean deviation of the numbers 3, 10, 10, 4, 7, 10 and 5 from the mean is:

• (A) 2
• (B) 2.5
• (C) 2.57
• (D) 3
• (E) 3.75

If \( g(x) = -\sqrt{25 - x^2} \), then \( g'(1) \) is:

• (A) \( -\sqrt{24} \)
• (B) \( \sqrt{24} \)
• (C) \( \frac{1}{24} \)
• (D) \( \frac{1}{\sqrt{24}} \)
• (E) \( \frac{-1}{\sqrt{24}} \)

Evaluate \( \lim_{x \to 0} \frac{\sin 2x + \sin 5x}{\sin 4x + \sin 6x} \):

• (A) \( \frac{2}{5} \)
• (B) \( \frac{7}{5} \)
• (C) \( \frac{3}{7} \)
• (D) \( \frac{7}{10} \)
• (E) \( \frac{5}{7} \)

If \( f(x) = \left\{ \begin{array}{ll} mx + 1, & when x \leq \frac{\pi}{2}
\sin x + n, & when x > \frac{\pi}{2} \end{array} \right. \) is continuous at \( x = \frac{\pi}{2} \), then the values of \( m \) and \( n \) are:

• (A) \( m = 1, n = 0 \)
• (B) \( m = 0, n = 1 \)
• (C) \( n = \frac{m\pi}{2} \)
• (D) \( m = \frac{n\pi}{2} \)
• (E) \( m = n = \frac{\pi}{2} \)

Let \( f(x) = x - \lfloor x \rfloor \), where \( \lfloor \cdot \rfloor \) denotes the greatest integer function and \( x \in (-1, 2) \). The number of points at which the function is not continuous is:

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

If \( f(x) = \cos x - \sin x \), and \( x \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right) \), then \( f' \left( \frac{\pi}{3} \right) \) is equal to:

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

If \( f(x) = \sin^{-1}(\cos x) \), then \( \frac{d^2 y}{dx^2} \) at \( x = \frac{\pi}{4} \) is:

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

If \( y = \tan^{-1} \left( \frac{\cos x - \sin x}{\cos x + \sin x} \right) \), \( \frac{-\pi}{2} < x < \frac{\pi}{2} \), then \( \frac{dy}{dx} \) is:

• (A) \( \tan x \)
• (B) \( \cos x \)
• (C) \( \sin x \)
• (D) \( -1 \)
• (E) 0

If \( y = \frac{x^2}{x - 1} \), then \( \frac{dy}{dx} \) at \( x = -1 \) is:

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

The function \( f(x) = 2x^3 + 9x^2 + 12x - 1 \) is decreasing in the interval:

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

The maximum value of \( y = 12 - |x - 12| \) in the range \( -11 \leq x \leq 11 \) is:

• (A) 12
• (B) 11
• (C) 10
• (D) 9
• (E) 35

The limit \( \lim_{x \to 10} \frac{x - 10}{\sqrt{x + 6} - 4} \) is equal to:

• (A) 4
• (B) 8
• (C) 10
• (D) 16
• (E) 12

The integral \( \int \frac{dx}{1 + e^x} \) is:

• (A) \( e^x + C \)
• (B) \( \log|1 + e^x| + C \)
• (C) \( \log|1 + e^{-x}| + C \)
• (D) \( \log |1 - e^{-x}| + C \)
• (E) \( \log |1 - e^x| + C \)

Evaluate \( \int x \cos x \, dx \):

• (A) \( \sin x - x \cos x + C \)
• (B) \( x \sin x - \cos x + C \)
• (C) \( \sin x + x \cos x + C \)
• (D) \( x \sin x + \cos x + C \)
• (E) \( \sin x + \cos x + C \)

Evaluate \( \int x e^{x^2} \, dx \):

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

If \[ \int \frac{dx}{\sqrt{16 - 9x^2}} = A \sin^{-1}(Bx) + C, where C is an arbitrary constant, then A + B = \]

Evaluate \( \int \frac{dx}{x^2 (x^4 + 1)^{3/4}} \):

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

Evaluate \( \int \frac{e^{6 \log x} - e^{5 \log x}}{e^{4 \log x} - e^{3 \log x}} \, dx \):

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

Evaluate \( \int_0^1 \log \left( \frac{1}{x - 1} \right) \, dx \):

• (A) \( \frac{1}{4} \)
• (B) \( \frac{1}{2} \)
• (C) \( -1 \)
• (D) \( 3 \)
• (E) \( 0 \)

Evaluate \( \int_{-\pi/2}^{\pi/2} \sin^9 x \cos^2 x \, dx \):

• (A) \( \frac{2}{3} \)
• (B) \( 1 \)
• (C) \( \frac{1}{11} \)
• (D) \( \frac{7\pi}{6} \)
• (E) \( 0 \)

Find the area bounded by the curves \( y = 2x \) and \( y = x^2 \):

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

Find the area of the smaller segment cut-off from the circle \( x^2 + y^2 = 25 \) by \( x = 3 \):

• (A) \( 75 \cos^{-1}\left( \frac{3}{5} \right) - 12 \)
• (B) \( 25 \cos^{-1}\left( \frac{3}{5} \right) - 24 \)
• (C) \( 25 \cos^{-1}\left( \frac{3}{5} \right) - 12 \)
• (D) \( 25 \cos^{-1}\left( \frac{3}{5} \right) - 6 \)
• (E) \( 50 \cos^{-1}\left( \frac{3}{5} \right) - 12 \)

The differential equation \( \frac{dy}{dx} + x = A \) (where A is constant) represents:

• (A) A family of circles having centre on the x-axis
• (B) A family of circles having centre on the y-axis
• (C) A family of all circles having centre at the origin
• (D) A family of ellipses
• (E) A family of hyperbolas

The general solution of \( \frac{dy}{dx} + y = 5 \) is:

• (A) \( -\log|5 - y| = x + C \)
• (B) \( -\log|5 - y| = e^x + C \)
• (C) \( (5 - y)^2 = 2x + C \)
• (D) \( y = \log|x| + C \)
• (E) \( \log|x| + C \)

The degree of the differential equation \( (y^m)^2 + (\sin y')^4 + y = 0 \) is:

• (A) 1
• (B) 2
• (C) 3
• (D) 4
• (E) Not defined

Given the Linear Programming Problem:

Maximize \( z = 11x + 7y \)

subject to the constraints: \( x \leq 3 \), \( y \leq 2 \), \( x, y \geq 0 \).

Then the optimal solution of the problem is:

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