Solved papers for CEE Kerala Engineering CEE Kerala Engineering Solved Paper-2011

done CEE Kerala Engineering Solved Paper-2011

• question_answer1) A radioactive sample at any instant has its disintegration rate 5000 disintegrations per minute. After 5 min, the rate becomes 1250 disintegration per minute. Then, its decay constant (per minute) is

A) $0.8\,{{\log }_{e}}2$

B) $0.4\,{{\log }_{e}}2$

C) $0.2\,{{\log }_{e}}2$

D) $0.1\,{{\log }_{e}}2$

E) $0.6\,{{\log }_{e}}2$

• question_answer2) The distance of closest approach of an $\alpha$-particle fired towards a nucleus with momentum p, is r. If the momentum of the a-particle is 2p, the corresponding distance of closest approach is

A) $\frac{r}{2}$

B) 2r

C) 4r

D) $\frac{r}{8}$

E) $\frac{r}{4}$

• question_answer3) If the binding energy per nucleon of deuteron is 1.115 MeV, its mass defect in atomic mass unit is

A) 0.0048

B) 0.0024

C) 0.0012

D) 0.0006

E) 2.230

• question_answer4) The circuit diagram shows a logic combination with the states of outputs X, Y and Z given for inputs P, Q, R and S all at state 1. When inputs P and R change to state 0 with inputs Q and S still at 1, the states of outputs X, Y and Z change to A) 1, 0, 0

B) 1, 1, 1

C) 0, 1, 0

D) 0, 0, 1

E) 0, 1, 1

• question_answer5) In a common emitter transistor amplifier, the output resistance is 500 k$\Omega$ and the current gain $\beta =49$. If the power gain of the amplifier is $5\times {{10}^{6}},$ the input resistance is

A) 325 $\Omega$

B) 165$\Omega$

C) 198 $\Omega$

D) 225$\Omega$

E) 240 $\Omega$

• question_answer6) In the circuit given the current through the zener diode is A) 10 mA

B) 6.67 mA

C) 5 mA

D) 3.33 m A

E) zero

• question_answer7) A transistor oscillator is

 (i) an amplifier with positive feedback (ii) an amplifier with reduced gain (iii) the one in which DC supply energy is converted into AC output energy. Then

A) All (i), (ii) and (iii) are correct

B) (i) and (ii) are correct

C) (i) and (iii) are correct

D) (ii) and (iii) are correct

E) (ii) is correct

• question_answer8) The distance of coverage of a transmitting antenna is 12.8 km. Then, the height of the antenna is (Given that radius of earth = 6400 km)

A) 6.4 m

B) 12.8 m

C) 3.2 m

D) 16 m

E) 25.6 m

• question_answer9) If ${{E}_{c}}=20\,\,\sin \,{{10}^{5}}\,\pi t$ and ${{E}_{m}}=10\sin 400\pi t$it are carrier and modulating signals, the modulation index is

A) 56%

B) 30%

C) 50%

D) 48%

E) 60%

• question_answer10) Which one of the following is incorrect statement in the transmission of electro- magnetic waves?

A) Ground wave propagation is for high frequency transmission.

B) Sky wave propagation is facilitated by ionospheric layers.

C) Space wave is of high frequency and is suitable for line of sight communication.

D) Space wave is used for satellite communication.

E) Very high frequency waves cannot be reflected by the ionospheric layers.

• question_answer11) 1000 kHz carrier wave is amplitude modulated by the signal frequency 200-4000 Hz. The channel width of this case is

A) 8 kHz

B) 4 kHz

C) 7.6 kHz

D) 3.8 kHz

E) 400 kHz

• question_answer12) The mass and volume of a body are found to be $5.\,00\,\pm \,0.05\,kg$ and $1.00\,\pm \,0.05\,{{m}^{3}}$ respectively. Then the maximum possible percentage error in its density is

A) 6%

B) 3%

C) 10%

D) 5%

E) 7%

• question_answer13) If F denotes force and t time, then in the equation $F=a{{t}^{-1}}+b{{t}^{2}}$, the dimensions of a and b respectively are

A) $[L{{T}^{-4}}]\,and\,[L{{T}^{-1}}]$

B) $[L{{T}^{-1}}]\,and\,[L{{T}^{-4}}]$

C) $[ML{{T}^{-4}}]\,and\,[ML{{T}^{-1}}]$

D) $[MLT]\,and\,[ML{{T}^{-4}}]$

E) $[ML{{T}^{-3}}]\,and\,[ML{{T}^{-2}}]$

• question_answer14) A car moves a distance of 200 m. It covers first half of the distance at speed $60\,\,km{{h}^{-1}}$ and the second half at speed v. If the average speed is $40\,\,km{{h}^{-1}},$ the value of v is

A) $30\,km{{h}^{-1}}$

B) $13\,km{{h}^{-1}}$

C) $60\,km{{h}^{-1}}$

D) $40\,km{{h}^{-1}}$

E) $20\,km{{h}^{-1}}$

• question_answer15) A bus begins to move with an acceleration of $1\,\,m{{s}^{-2}}$. A man who is 48 m behind the bus starts running at 10 ms to catch the bus. The man will be able to catch the bus after

A) 6s

B) 5 s

C) 3 s

D) 7 s

E) 8s

• question_answer16) A particle is moving with constant acceleration from A to Bin a straight line AB. If u and v are the velocities at A and B respectively then its velocity at the midpoint C will be

A) ${{\left( \frac{{{u}^{2}}+{{v}^{2}}}{2u} \right)}^{2}}$

B) $\frac{u+v}{2}$

C) $\frac{v-u}{2}$

D) $\sqrt{\frac{{{u}^{2}}+{{v}^{2}}}{2}}$

E) $\sqrt{\frac{{{v}^{2}}-{{u}^{2}}}{2}}$

• question_answer17) An aircraft is flying at a height of 3400 m above the ground. If the angle subtended at a ground observation point by the aircraft positions 10s apart is 30?, then the speed of the aircraft is

A) $19.63\,m{{s}^{-1}}$

B) $1963\,m{{s}^{-1}}$

C) $108\,m{{s}^{-1}}$

D) $196.3\,m{{s}^{-1}}$

E) $10.8\,m{{s}^{-1}}$

• question_answer18) Two projectiles A and B thrown with speeds in the ratio 1 : $\sqrt{2}$ acquired the same heights. If A is thrown at an angle of $45{}^\circ$ with the horizontal, the angle of projection of B will be

A) $0{}^\circ$

B) $60{}^\circ$

C) $30{}^\circ$

D) $45{}^\circ$

E) $15{}^\circ$

• question_answer19) A particle crossing the origin of co-ordinates at time t = 0, moves in the xy-plane with a constant acceleration a in the y-direction. If its equation of motion is $y=b{{x}^{2}}$ (b is a constant), its velocity component in the x-direction is

A) $\sqrt{\frac{2b}{a}}$

B) $\sqrt{\frac{A}{2b}}$

C) $\sqrt{\frac{a}{b}}$

D) $\sqrt{\frac{b}{a}}$

E) $\sqrt{ba}$

• question_answer20) A stationary tomb explodes into three pieces. One piece of 2 kg mass moves with a velocity of $8\,m{{s}^{-1}}$ at right angles to the other piece of mass 1 kg moving with a velocity of $12\,m{{s}^{-1}}$. If the mass of the third piece of 0.5 kg, then its velocity is

A) $10\,m{{s}^{-1}}$

B) $20\,m{{s}^{-1}}$

C) $30\,m{{s}^{-1}}$

D) $40\,m{{s}^{-1}}$

E) $50\,m{{s}^{-1}}$

• question_answer21) A block at rest slides down a smooth inclined plane which makes an angle 60? with the vertical and it reaches the ground in ${{t}_{1}}$ seconds. Another block is dropped vertically from the same point and reaches the ground in ${{t}_{2}}$ seconds. Then the ratio of ${{t}_{1}}:{{t}_{2}}$ is

A) 1 : 2

B) 2 : 1

C) 1 : 3

D) 1: $\sqrt{2}$

E) 3 : 1

• question_answer22) A bridge is in the form of a semi-circle of radius 40 m. The greatest speed with which a motor cycle can cross the bridge without leaving the ground at the highest point is $(g=10\,m{{s}^{-2}})$ (Frictional force is negligibly small)

A) $40\,m{{s}^{-1}}$

B) $20\,m{{s}^{-1}}$

C) $30\,m{{s}^{-1}}$

D) $15\,m{{s}^{-1}}$

E) $25\,m{{s}^{-1}}$

• question_answer23) A ball of mass m is dropped from a height h on a platform fixed at the top of a vertical spring, as shown in figure. The platform is depressed by a distance x. Then the spring constant is A) $\frac{mg}{(h+x)}$

B) $\frac{mg}{(h+2x)}$

C) $\frac{2mg(h+x)}{{{x}^{2}}}$

D) $\frac{mg}{(2h+x)}$

E) $\frac{2mg}{(h+x)}$

• question_answer24) A ball dropped from a height of 2 m rebounds to a height of 1.5 m after hitting the ground. Then the percentage of energy lost is

A) 25

B) 30

C) 50

D) 100

E) 200

• question_answer25) A particle of mass m is moving in a horizontal circle of radius r, under a centripetal force$F=\frac{k}{{{r}^{2}}},$ where k is a constant.

A) The potential energy of the particle is zero

B) The potential energy of the particle is $\frac{k}{r}$

C) The total energy of the particle is $-\frac{k}{2r}$

D) The kinetic energy of the particle is $-\frac{k}{r}$

E) The potential energy of the particle is $-\frac{k}{2r}$

• question_answer26) A ring starts to roll down the inclined plane of height h without slipping. The velocity with which it reaches the ground is

A) $\sqrt{\frac{10gh}{7}}$

B) $\sqrt{\frac{4gh}{7}}$

C) $\sqrt{\frac{4gh}{3}}$

D) $\sqrt{2gh}$

E) $\sqrt{gh}$

• question_answer27) The angular momentum of a particle describing uniform circular motion is L. If its kinetic energy is halved and angular velocity doubled, its new angular momentum is

A) 4L

B) $\frac{L}{4}$

C) $\frac{L}{2}$

D) 2L

E) $\frac{L}{8}$

• question_answer28) Two masses m1 = 1 kg and m2 = 2 kg are connected by a light inextensible string and suspended by means of a weightless pulley as shown in the figure. Assuming that both the masses start from rest, the distance travelled by the centre of mass in 2s is (Take$g=10\text{ }m{{s}^{-2}}$)

A) $\frac{20}{9}m$

B) $\frac{40}{9}m$

C) $\frac{2}{3}m$

D) $\frac{1}{3}m$

E) 4m

• question_answer29) The average depth of Indian ocean is about 3000 m. The fractional compression,$\frac{\Delta V}{V}$of water at the bottom of the ocean (given that the bulk modulus of the water$=2.2\times {{10}^{2}}N{{m}^{-2}}$ and$g=10m{{s}^{-2}}$) is

A) 0.82%

B) 0.91%

C) 1.36%

D) 1.24%

E) 1.52%

• question_answer30) A satellite is launched into a circular orbit of radius R around the earth. A second satellite is launched into an orbit of radius 4R. The ratio of their respective periods is

A) 4 : 1

B) 1 : 8

C) 8 : 1

D) 1 : 4

E) 1 : 2

• question_answer31) A body is projected with a velocity of$2\times 11.2$ $km{{s}^{-1}}$from the surface of earth. The velocity of the body when it escapes the gravitational pull of earth is

A) $\sqrt{3}\times 11.2km{{s}^{-1}}$

B) $11.2\,km{{s}^{-1}}$

C) $\sqrt{2}\times 11.2\,km{{s}^{-1}}$

D) $6.5\times 11.2\,km{{s}^{-1}}$

E) $2\times 11.2\,km{{s}^{-1}}$

• question_answer32) The terminal speed of a sphere of gold (density$=19.5\text{ }kg-{{m}^{-3}})$) is$0.2\,m{{s}^{-1}}$ in a viscous liquid (density$=1.5\,kg-{{m}^{-3}})$. Then the terminal speed of a sphere of silver (density$=10.5\,kg$$-{{m}^{-3}})$ of the same size in the same liquid is

A) $0.1\,m{{s}^{-1}}$

B) $1.133\,\,m{{s}^{-1}}$

C) $0.4\,m{{s}^{-1}}$

D) $0.2\,\,m{{s}^{-1}}$

E) $0.3\,\,m{{s}^{-1}}$

• question_answer33) A large open tank has two holes in its wall. One is a square hole of side a at a depth of$x$ from the top and the other is a circular hole of radius r at a depth$4x$ from the top. When the tank is completely filled with water, the quantities of water flowing out per second from both holes are the same. Then r is equal to

A) $2\pi a$

B) $a$

C) $\frac{a}{2\pi }$

D) $\frac{a}{\pi }$

E) $\frac{a}{\sqrt{2\pi }}$

• question_answer34) Ice pieces are floating in a beaker A containing water and also in a beaker B containing miscible liquid of specific gravity 1.2. When ice melts, the level of

A) water increases in A

B) water decreases in A

C) liquid in B decreases

D) liquid in B increases

E) water in A and liquid in B remains unaltered

• question_answer35) Identify the incorrect statement.

A) Youngs modulus and shear modulus are relevant only for solids.

B) Bulk modulus is relevant for solids, liquids and gases.

C) Alloys have larger values of Youngs modulus than metals.

D) Metals have larger values of Youngs modulus than elastomers.

E) Stress is not a vector quantity.

• question_answer36) A Cannot engine whose efficiency is 40%, receives heat at 500 K. If the efficiency is to be 50%, the source temperature for the same exhaust temperature is

A) 900 K

B) 600 K

C) 700 K

D) 800 K

E) 550 K

• question_answer37) The ratio of the molar heat capacities of a diatomic gas at constant pressure to that at constant volume is

A) $\frac{7}{2}$

B) $\frac{3}{2}$

C) $\frac{3}{5}$

D) $\frac{7}{5}$

E) $\frac{5}{2}$

• question_answer38) The thermodynamic process in which no work is done on or by the gas is

A) isothermal process

C) cyclic process

D) isobaric process

E) isochoric process

• question_answer39) A lead bullet strikes against a steel plate with a velocity$200\text{ }m{{s}^{-1}}$. If the impact is perfectly inelastic and the heat produced is equally shared between the bullet and the target, then the rise in temperature of the bullet is (Specific heat capacity of lead$=125\text{ }J\text{ }k{{g}^{-1}}{{K}^{-1}}$)

A) ${{80}^{o}}C$

B) ${{60}^{o}}C$

C) ${{160}^{o}}C$

D) ${{40}^{o}}C$

E) ${{120}^{o}}C$

• question_answer40) A body of mass 4.9 kg hangs from a spring and oscillates with a period 0.5 s. On the removal of the body, the spring is shortened by (Take $g=10\text{ m}{{s}^{-2}},{{n}^{2}}=10$)

A) 6.3 m

B) 0.63 m

C) 6.25 cm

D) 63 cm

E) 0.625 cm

• question_answer41) The amplitude of a damped oscillator becomes $\left( \frac{1}{3} \right)$rd in 2s. If its amplitude after 6 s is$\frac{1}{n}$times the original amplitude, the value of n is

A) ${{3}^{2}}$

B) $\sqrt{2}$

C) $\sqrt{3}$

D) ${{2}^{3}}$

E) ${{3}^{3}}$

• question_answer42) If two springs A and B with spring constants 2k and k, are stretched separately by same suspended weight, then the ratio between the work done in stretching A and B is

A) 1 : 2

B) 1 : 4

C) 1 : 3

D) 4 : 1

E) 2 : 1

• question_answer43) Tube A has both ends open while tube B has one end closed. Otherwise they are identical. Their fundamental frequencies are in the ratio

A) 4 : 1

B) 2 : 1

C) 1 : 4

D) 1 : 2

E) 2 : 3

• question_answer44) The speed of sound in a gas of density p at a pressure p is proportional to

A) ${{\left( \frac{p}{\rho } \right)}^{2}}$

B) ${{\left( \frac{p}{\rho } \right)}^{\frac{3}{2}}}$

C) $\sqrt{\frac{\rho }{p}}$

D) $\sqrt{\frac{p}{\rho }}$

E) ${{\left( \frac{\rho }{p} \right)}^{2}}$

• question_answer45) A tuning fork of frequency 330 Hz resonates with an air column of length 120 cm in a cylindrical tube, in the fundamental mode. When water is slowly poured in it, the minimum height of water required for observing resonance once again is (Velocity of sound$330\text{ }m{{s}^{-1}}$)

A) 75 cm

B) 60 cm

C) 50 cm

D) 30 cm

E) 45 cm

• question_answer46) Electric charge is uniformly distributed along a long straight wire of radius 1 mm. The charge per cm length of the wire is Q coulomb. Another cylindrical surface of radius 50 cm and length 1 m symmetrically encloses the wire. The total electric flux passing through the cylindrical surface is

A) $\frac{Q}{{{\varepsilon }_{0}}}$

B) $\frac{100Q}{{{\varepsilon }_{0}}}$

C) $\frac{10Q}{\pi {{\varepsilon }_{0}}}$

D) $\frac{100Q}{\pi {{\varepsilon }_{0}}}$

E) $\frac{Q}{100{{\varepsilon }_{0}}}$

• question_answer47) A charged particle q is shot towards another charged particle Q which is fixed, with a speed v. It approaches Q upto a closest distance r and then returns. If q is shot with speed 2v, the closest distance of approach would be

A) $\frac{r}{4}$

B) $\frac{r}{2}$

C) $2r$

D) $r$

E) $\frac{3}{2}r$

• question_answer48) A dipole of electric dipole moment p is placed in a uniform electric field of strength E. If $\theta$ is the angle between positive directions of p and E, then the potential energy of the electric dipole is largest when $\theta$ is

A) $\frac{\pi }{4}$

B) $\frac{\pi }{2}$

C) $\pi$

D) zero

E) $\frac{2}{3}\pi$

• question_answer49) Two conducting spheres of radii 3 cm and 1 cm are separated by a distance of 10 cm in free space. If the spheres are charged to same potential of 10 V each, the force of repulsion between them is

A) $\left( \frac{1}{3} \right)\times {{10}^{-9}}N$

B) $\left( \frac{2}{9} \right)\times {{10}^{-9}}N$

C) $\left( \frac{1}{9} \right)\times {{10}^{-9}}N$

D) $\left( \frac{4}{3} \right)\times {{10}^{-9}}N$

E) $\left( \frac{2}{3} \right)\times {{10}^{-9}}N$

• question_answer50) If${{q}_{1}}+{{q}_{2}}=q,$then the value of the ratio$\frac{{{q}_{1}}}{q},$for which the force between${{q}_{1}}$and${{q}_{2}}$is maximum is

A) 0.25

B) 0.75

C) 1

D) 0.5

E) 1.5

• question_answer51) The resistance of a 10 m long wire is$10\,\Omega$. Its length is increased by 25% by stretching the wire uniformly. Then the resistance of the wire will be

A) $12.5\,\,\Omega$

B) $14.5\,\,\Omega$

C) $15.6\,\,\Omega$

D) $16.6\,\,\Omega$

E) $18.6\,\,\Omega$

• question_answer52) If 2 A of current is passed through$CuS{{O}_{4}}$ solution for 32 s, then the number of copper ions deposited at the cathode will be

A) $4\times {{10}^{20}}$

B) $2\times {{10}^{20}}$

C) $4\times {{10}^{19}}$

D) $2\times {{10}^{19}}$

E) $1.6\times {{10}^{19}}$

• question_answer53) In a potentiometer experiment, when three cells A, B and C are connected in series the balancing length is found to be 740 cm. If A and B are connected in series balancing length is 440 cm and for B and C connected in series that is 540 cm. Then the emf of${{E}_{A}},{{E}_{B}}$and${{E}_{C}}$are respectively (in volts)

A) 1, 1.2 and 1.5

B) 1, 2 and 3

C) 1.5, 2 and 3

D) 1.5, 2.5 and 3.5

E) 1.2, 1.5 and 3.5

• question_answer54) Find the true statement.

A) Ohms law is applicable to all conductors of electricity.

B) In an electrolyte solution, the electric current is mainly due to the movement of electrons.

C) The resistance of an incandescent lamp is lesser when the lamp is switched on.

D) Specific resistance of a wire depends upon its dimension.

E) The resistance of carbon decreases with the increase of temperature.

• question_answer55) The tolerance level of a resistor with the colour code red, blue, orange, gold is

A) ? 5%

B) ? 10%

C) ? 20%

D) ? 40%

E) ? 30%

• question_answer56) An electron moving around the nucleus with an angular momentum I has a magnetic moment

A) $\frac{e}{m}l$

B) $\frac{e}{2m}l$

C) $\frac{2e}{m}l$

D) $\frac{e}{2\pi m}l$

E) $\frac{e}{4\pi m}l$

• question_answer57) The force between two parallel current carrying wires is independent of

A) their distance of separation

B) the length of the wires

C) the magnitude of currents

D) the radii of the wires

E) the medium in which they are placed

• question_answer58) A magnetic needle lying parallel to a magnetic field requires W units of work to turn it through${{60}^{o}}$. The torque required to keep the needle in this position will be

A) $2W$

B) $W$

C) $\frac{W}{\sqrt{2}}$

D) $\frac{W}{\sqrt{3}}$

E) $\sqrt{3}W$

• question_answer59) Two identical magnetic dipoles of magnetic moment $2\,A{{m}^{2}}$ are placed at a separation of 2 m with their axis perpendicular to each other in air. The resultant magnetic field at a midpoint between the dipoles is

A) $4\sqrt{5}\times {{10}^{-5}}T$

B) $2\sqrt{5}\times {{10}^{-5}}T$

C) $4\sqrt{5}\times {{10}^{-7}}T$

D) $2\sqrt{5}\times {{10}^{-7}}T$

E) $4\sqrt{2}\times {{10}^{-7}}T$

• question_answer60) A proton, a deuteron and an a-particle having the same kinetic energy are moving in circular trajectories in a constant magnetic field. If${{r}_{p}},{{r}_{d}}$and${{r}_{\alpha }}$denote respectively the radii of the trajectories of these particles, then

A) ${{r}_{\alpha }}={{r}_{d}}>{{r}_{p}}$

B) ${{r}_{\alpha }}={{r}_{d}}={{r}_{p}}$

C) ${{r}_{\alpha }}<{{r}_{d}}<{{r}_{p}}$

D) ${{r}_{\alpha }}={{r}_{p}}>{{r}_{d}}$

E) ${{r}_{\alpha }}>{{r}_{d}}>{{r}_{p}}$

• question_answer61) A metal conductor of length 1 m rotates vertically about one of its ends at angular velocity$5\text{ }rad{{s}^{-1}}$. If the horizontal component of earths magnetic field is$0.2\times {{10}^{-4}}T,$then the emf developed between the ends of the conductor is

A) $5\mu V$

B) $5mV$

C) $50\mu V$

D) $50mV$

E) $0.5\text{ }mV$

• question_answer62) If$E=100sin(100t)$volt and$I=100\sin \left( 100t+\frac{\pi }{3} \right)mA$are the instantaneous values of voltage and current, then the rms values of voltage and current are respectively

A) $70.7\,V,70.7\,mA$

B) $70.7\,V,70.7A$

C) $141.4\,V,141.4mA$

D) $141.4\,V,\,141.4\,A$

E) $100\,\,V,100\,mA$

• question_answer63) The core of a transformer is laminated to reduce

A) flux leakage

B) output power

C) hysteresis

D) copper loss

E) eddy current

• question_answer64) If${{E}_{0}}$is the peak emf,${{I}_{0}}$is the peak current and$\phi$is the phase difference between them, then the average power dissipation in the circuit is

A) $\frac{1}{2}{{E}_{0}}{{I}_{0}}$

B) $\frac{{{E}_{0}}{{I}_{0}}}{\sqrt{2}}$

C) $\frac{1}{2}{{E}_{0}}{{I}_{0}}\sin \phi$

D) $\frac{1}{2}{{E}_{0}}{{I}_{0}}\cos \phi$

E) $\frac{1}{2}{{E}_{0}}{{I}_{0}}\tan \phi$

• question_answer65) The electric field of an electromagnetic wave travelling through vacuum is given by the equation$E={{E}_{0}}\sin (kx-\omega t)$. The quantity that is independent of wavelength is

A) $\frac{k}{\omega }$

B) $k\omega$

C) $\omega$

D) $k$

E) ${{k}^{2}}\omega$

• question_answer66) The electric field of a plane electromagnetic wave varies with time of amplitude$2\text{ }V{{m}^{-1}}$propagating along z-axis. The average energy density of the magnetic field is (in$J{{m}^{-3}}$)

A) $13.29\times {{10}^{-12}}$

B) $8.86\times {{10}^{-12}}$

C) $17.72\times {{10}^{-12}}$

D) $4.43\times {{10}^{-12}}$

E) $2.22\times {{10}^{-12}}$

• question_answer67) In a Youngs double slit experiment, the intensity at a point where the path difference is $\frac{\lambda }{6}(\lambda =$wavelength of the light) is$I$. If${{I}_{0}}$denotes the maximum intensity, then$\frac{I}{{{I}_{0}}}$is equal to

A) $\frac{1}{2}$

B) $\frac{\sqrt{3}}{2}$

C) $\frac{1}{\sqrt{2}}$

D) $\frac{3}{4}$

E) $\frac{3}{4}$

• question_answer68) The focal length of the lens of refractive index ($\mu ~=1.5$)in air is 10 cm. If air is replaced by water of$\mu ~=\frac{4}{3},$its focal length is

A) 20 cm

B) 30 cm

C) 40 cm

D) 25 cm

E) 35 cm

• question_answer69) A beam of natural light falls on a system of 5 polaroids, which are arranged in succession such that the pass axis of each polaroid is turned through${{60}^{o}}$with respect to the preceding one. The fraction of the incident light intensity that passes through the system is

A) $\frac{1}{64}$

B) $\frac{1}{32}$

C) $\frac{1}{256}$

D) $\frac{1}{128}$

E) $\frac{1}{512}$

• question_answer70) A glass prism of refractive index 1.5 is immersed in water$\left( \mu =\frac{4}{3} \right)$. Refer figure. A light beam incident normally on the face AB is totally reflected to reach the face BC, if

A) $2/3<\sin \theta <8/9$

B) $\sin \theta \le 2/3$

C) $\cos \theta \ge 8/9$

D) $\sin \theta >8/9$

E) $\cos \theta \le 8/9$

• question_answer71) A narrow slit of width 2 mm is illuminated by monochromatic light of wavelength 500 nm. The distance between the first minima on either side on a screen at a distance of 1 m is

A) 5 mm

B) 0.5 mm

C) 1 mm

D) 10 mm

E) 2.5 mm

• question_answer72) If e/m of electron is$1.76\times {{10}^{11}}C\text{ }k{{g}^{-1}}$ and the stopping potential is 0.71 V, then the maximum velocity of the photo-electron is

A) $150\text{ }km\,\,{{s}^{-1}}$

B) $200\text{ }km\,\,{{s}^{-1}}$

C) $500\text{ }km\,\,{{s}^{-1}}$

D) $250\text{ }km\,\,{{s}^{-1}}$

E) $100\text{ }km\,\,{{s}^{-1}}$

• question_answer73) Phenol can be converted to o-hydroxybenzaldehyde by

A) Kolbes reaction

B) Reimer-Tiemann reaction

C) Wurtz reaction

D) Cannizaro reaction

E) Sandmeyers reaction

• question_answer74) n-butylamine (I), diethylamine (II) and N, N-dimethylethylamine(III) have the same molar mass. The increasing order of their boiling point is

A) $III<II<I$

B) $I<II<III$

C) $II<III<I$

D) $II<I<III$

E) $III<I<II$

• question_answer75) Choose the incorrect statement.

A) Primary amines show intermolecular hydrogen bonds

B) Teyt-butylamine is a primary amine

C) Tertiary amines do not show intermolecular hydrogen bonds

D) Isopropylamine is a secondary amine

E) Amines have lower boiling points as compared to those of alcohols of a comparable molecular mass

• question_answer76) The monomers used for their preparation of nylon 2-nylon 6 is/are

A) caprolactam

B) alanine and amino caproic acid

C) glycine and amino caproic acid

D) hexamethylenediamine and adipic acid

E) glycine and amino valeric acid

• question_answer77) Zeigler-Natta catalyst is used in the preparation of

A) low density polythene

B) high density polythene

C) Dacron

D) Teflon

E) PVC

• question_answer78) The cationic detergent that is used in hair conditioners is

A) sodium dodecylbenzene sulphonate

B) sodium lauryl sulphate

C) tetramethyl ammonium chloride

D) sodium stearyl sulphate

E) cetyltrimethyl ammonium bromide

• question_answer79) Salts of sorbic acid and propionic acid are used as

A) antioxidants

B) flavouring agents

C) food preservatives

D) nutritional supplements

E) detergents

• question_answer80) Arrange the following in the order of increasing mass (atomic mass$~O=16,Cu=63,$ $N=14$)

 I. One atom of oxygen II. One atom of nitrogen III. $1\times {{10}^{-10}}$mole of oxygen IV.$1\times {{10}^{-10}}$mole of copper

A) II < I < III < IV

B) I < II < III < IV

C) III < II < IV < I

D) IV < II < III < I

E) II < IV < I < III

• question_answer81) Which transition in the hydrogen atomic spectrum will have the same wavelength as the transition,$n=4$to$n=2$of$H{{e}^{+}}$spectrum?

A) $n=4$to$n=3$

B) $n=3$to$n=2$

C) $n=4$ to$n=2$

D) $n=3$to $n=1$

E) $~n=2$ to$n=1$

• question_answer82) Which of the following is not correct with respect to bond length of the species?

A) ${{C}_{2}}>C_{2}^{2-}$

B) $B_{2}^{+}>{{B}_{2}}$

C) $Li_{2}^{+}>L{{i}_{2}}$

D) $N_{2}^{+}>{{N}_{2}}$

E) ${{O}_{2}}>O_{2}^{-}$

• question_answer83) Intramolecular hydrogen bond is present in

A) water

B) o-nitrophenol

C) p-nitrophenol

D) methylamine

E) ethanol

• question_answer84) A mixture of ethane and ethene occupies 41 L at 1 atm and 500 K. The mixture reacts completely with$\frac{10}{3}$mole of${{O}_{2}}$to produce $C{{O}_{2}}$and ${{H}_{2}}O$. The mole fraction of ethane and ethene in the mixture are respectively (R = 0.082 L atm${{K}^{-1}}mo{{l}^{-1}}$)

A) 0.50, 0.50

B) 0.75, 0.25

C) 0.67, 0.33

D) 0.25, 0.75

E) 0.33, 0.67

• question_answer85) Substance which is weakly repelled by a magnetic field is

A) ${{O}_{2}}$

B) ${{H}_{2}}O$

C) $Cr{{O}_{2}}$

D) $F{{e}_{3}}{{O}_{4}}$

E) $ZnF{{e}_{2}}{{O}_{4}}$

• question_answer86) The correct decreasing order of first ionization enthalpies of five elements of the second period is

A) $Be>B>C>N>F$

B) $N>F>C>B>Be$

C) $F>N>C>Be>B$

D) $N>F>B>C>Be$

E) $F>C>N>B>Be$

• question_answer87) In the reaction ${{H}_{2}}{{O}_{2}}\xrightarrow{{}}S+2{{H}_{2}}O$

A) ${{H}_{2}}S$ is an acid and${{H}_{2}}{{O}_{2}}$is a base

B) ${{H}_{2}}S$is a base and${{H}_{2}}{{O}_{2}}$is an acid

C) ${{H}_{2}}S$is an oxidizing agent and${{H}_{2}}{{O}_{2}}$is a reducing agent

D) ${{H}_{2}}S$ is a reducing agent and${{H}_{2}}{{O}_{2}}$is an oxidizing agent

E) ${{H}_{2}}S$is hydrolyzed to S

• question_answer88) Be and$Al$exhibit diagonal relationship. Which of the following statements about them is/are not true? I. Both react with$HCl$to liberate${{H}_{2}}$ II. They are made passive by$HN{{O}_{3}}$ III. Their carbides give acetylene on treatment with water IV. Their oxides are amphoteric

A) III and IV

B) I and III

C) I only

D) II and III

E) III only

• question_answer89) Which one of the following on hydrolysis, gives the corresponding metallic hydroxide, ${{H}_{2}}{{O}_{2}}$and${{O}_{2}}$?

A) $L{{i}_{2}}O$

B) $N{{a}_{2}}{{O}_{2}}$

C) $Na{{O}_{2}}$

D) $N{{a}_{2}}O$

E) $BeO$

• question_answer90) The least stable hydride of 15th group elements is

A) $N{{H}_{3}}$

B) $P{{H}_{3}}$

C) $As{{H}_{3}}$

D) $Sb{{H}_{3}}$

E) $Bi{{H}_{3}}$

• question_answer91) Which one of the following oxides of nitrogen dimerises into colorless is solid/liquid on cooling?

A) ${{N}_{2}}O$

B) $NO$

C) ${{N}_{2}}{{O}_{3}}$

D) $N{{O}_{2}}$

E) ${{N}_{2}}{{O}_{5}}$

• question_answer92) The bonds present in the structure of dichromate ion are

A) four equivalent$Cr-O$bonds only

B) si equivalent$Cr-O$bonds and one $O-O$bond

C) six equivalent$Cr-O$bonds and one $Cr-Cr$bond

D) eight equivalent$Cr-O$bonds

E) six equivalent$Cr-O$bonds and one $Cr-O-Cr$bond

• question_answer93) Consider the following statements.

 I. $La{{(OH)}_{3}}$is the least basic among hydroxides of lanthanides. II. $Z{{r}^{4+}}$and$H{{f}^{4+}}$possess almost the same ionic radii. III. $C{{e}^{4+}}$can act as an oxidizing agent.
Which of the above is/are true?

A) (I) and (III)

B) (II) and (III)

C) (II) Only

D) (I) and (II)

E) (I)-only

• question_answer94) Molar heat capacity of aluminium is$25\text{ }J{{K}^{-1}}$ $mo{{l}^{-1}}$. The heat necessary to raise the temperature of 54 g of aluminium (atomic mass$27\text{ }g\text{ }mo{{l}^{-1}}.$ from$30{}^\circ C$to$50{}^\circ C$is

A) 1.5 kJ

B) 0.5 kJ

C) 1.0 kJ

D) 2.5 kJ

E) 2.0 kJ

• question_answer95) The solubility product$({{K}_{sp}})$of the following compounds are given at$25{}^\circ C$ Compounds ${{K}_{sp}}$ $AgCl$ $1.1\times {{10}^{-10}}$ $AgI$ $1.0\times {{10}^{-16}}$ $PbCr{{O}_{4}}$ $4.0\times {{10}^{-14}}$ $A{{g}_{2}}C{{O}_{3}}$ $8.0\times {{10}^{-12}}$ The most soluble and least soluble compounds are

A) $AgCl$and$PbCr{{O}_{4}}$

B) $AgI$ and$A{{g}_{2}}C{{O}_{3}}$

C) $AgCl$and$A{{g}_{2}}C{{O}_{3}}$

D) $A{{g}_{2}}C{{O}_{3}}$and $AgI$

E) $A{{g}_{2}}C{{O}_{3}}$and $PbCr{{O}_{4}}$

• question_answer96) A solution containing 1.8 g of a compound (empirical formula$C{{H}_{2}}O$) in 40 g of water is observed to freeze at -0.465?C. The molecular formula of the compound is (${{k}_{f}}$of water =$1.86\text{ }kg\,K\,mo{{l}^{-1}}$)

A) ${{C}_{2}}{{H}_{4}}{{O}_{2}}$

B) ${{C}_{3}}{{H}_{6}}{{O}_{3}}$

C) ${{C}_{4}}{{H}_{8}}{{O}_{4}}$

D) ${{C}_{5}}{{H}_{10}}{{O}_{5}}$

E) ${{C}_{6}}{{H}_{12}}{{O}_{6}}$

• question_answer97) In the disproportionation reaction $3HCl{{O}_{3}}\xrightarrow{{}}HCl{{O}_{4}}+C{{l}_{2}}+2{{O}_{2}}+{{H}_{2}}O,$the equivalent mass of the oxidizing agent is (molar mass of $HCl{{O}_{3}}=84.45$)

A) 16.89

B) 32.22

C) 84.45

D) 28.15

E) 29.7

• question_answer98) The rate of the reaction$A\xrightarrow{{}}$products, at the initial concentration of$3.24\times {{10}^{-2}}M$is nine times its rate at another initial concentration of$1.2\times {{10}^{-3}}M$. The order of the reaction is

A) $\frac{1}{2}$

B) $\frac{3}{4}$

C) $\frac{3}{2}$

D) $\frac{2}{3}$

E) $\frac{1}{3}$

• question_answer99) Associated colloid among the following is

A) enzymes

B) proteins

C) cellulose

D) starch

E) sodium stearate

• question_answer100) The correct statement with respect to the complexes$[Ni{{(CO)}_{4}}]$and${{[Ni{{(CN)}_{4}}]}^{2-}}$is

A) nickel is in the same oxidation state in both

B) both have tetrahedral geometry

C) both have square planar geometry

D) have square planar and tetrahedral geometry respectively

E) have tetrahedral and square planar geometry respectively

• question_answer101) Four moles of$PC{{l}_{5}}$are heated in a closed 4 $d{{m}^{3}}$container to reach equilibrium at 400 K. At equilibrium 50% of$PC{{l}_{5}}$ is dissociated. What is the value of${{K}_{c}}$for the dissociation of $PC{{l}_{5}}$into$PC{{l}_{3}}$and$C{{l}_{2}}$at$400K$?

A) 0.50

B) 1.00

C) 1.25

D) 0.05

E) 0.25

• question_answer102) At$25{}^\circ C,$at 5% aqueous solution of glucose (molecular weight$=180\,g\,mo{{l}^{-1}}$) is isotonic with a 2% aqueous solution containing and unknown solute. What is the molecular weight of the unknown solute?

A) 60

B) 80

C) 72

D) 63

E) 98

• question_answer103) A weak monobasic acid is 1% ionized in 0.1 M solution at$25{}^\circ C$. The percentage of ionisation in its 0.025 M solution is

A) 1

B) 2

C) 3

D) 4

E) 5

• question_answer104) Consider the following statements in respect of zero order reaction.

 I. The rate of the reaction is independent of reactant concentration. II. The rate of the reaction is independent of temperature. III. The rater constant of the reaction is independent of temperature. IV. The rate constant of the reaction is independent of reactant concentration.
Choose the correct statement/s

A) I only

B) I and II only

C) III and IV only

D) I and III only

E) I and IV only

• question_answer105) The complex ion which has the highest magnetic moment among the following is

A) ${{[Co{{F}_{6}}]}^{3-}}$

B) ${{[Co{{(N{{H}_{3}})}_{6}}]}^{3+}}$

C) ${{[Ni{{(N{{H}_{3}})}_{4}}]}^{2+}}$

D) ${{[Ni{{(CN)}_{4}}]}^{2-}}$

E) ${{[Fe{{(CN)}_{6}}]}^{4-}}$

• question_answer106) The standard redox potentials for the reactions$M{{n}^{2+}}+2{{e}^{-}}\xrightarrow{{}}Mn$and$M{{n}^{3+}}+{{e}^{-}}\xrightarrow{{}}M{{n}^{2+}}$are$-1.18\text{ }V$and$1.51\text{ }V$ respectively. What is the redox potential for the reaction$M{{n}^{3+}}+3{{e}^{-}}\to Mn$

A) 0.33V

B) 1.69V

C) $-\text{ }0.28\text{ }V$

D) $-\text{ }0.85\text{ }V$

E) 0.85 V

• question_answer107) The limiting molar conductivities of$HCl,$ $C{{H}_{3}}COONa$ and$NaCl$are respectively 425, 90 and$125\text{ }mho\text{ }c{{m}^{2}}\text{ }mo{{l}^{-1}}$at$25{}^\circ C$. The molar conductivity of$0.1M\,C{{H}_{3}}OOH$solution is 7.8 mho$c{{m}^{2}}mo{{l}^{-1}}$at the same temperature. The degree of dissociation of 0.1 M acetic acid solution at the same temperature is

A) 0.10

B) 0.02

C) 0.15

D) 0.03

E) 0.20

• question_answer108) When 0.01 mole of a cobalt complex is treated with excess silver nitrate solution, 4.305 g of silver chloride is precipitated. The formula of the complex is

A) $[Co{{(N{{H}_{3}})}_{3}}C{{l}_{3}}]$

B) $[Co{{(N{{H}_{3}})}_{5}}Cl]C{{l}_{2}}$

C) $[Co{{(N{{H}_{3}})}_{6}}]C{{l}_{3}}$

D) $[Co{{(N{{H}_{3}})}_{4}}C{{l}_{2}}]N{{O}_{3}}$

E) $[Co{{(N{{H}_{3}})}_{4}}C{{l}_{2}}]Cl$

• question_answer109) The IUPAC name of the compound $C{{H}_{3}}-CH(C{{H}_{3}})-CO-C{{H}_{3}},$is

A) 3-metyl 2-butanone

B) 2-methyl 3-butanone

C) isopropyl methyl ketone

D) methyl isopropyl ketone

E) 1, 1-dimethyl acetone

• question_answer110) Two organic compounds X and Y on analysis gave the same percentage composition namely;$C=(12/13)\times 100%$and$H=(1/13)\times 100%$. However, compound X decolourises bromine water while compound Y does not. The two compounds X and Y may be respectively

A) acetylene and ethylene

B) acetylene and benzene

C) ethylene and benzene

D) toluene and benzene

E) benzene and styrene

• question_answer111) The correct order of boiling points of 2, 2-dimethylpropane, 2-methylbutane and n-pentane is

A) n-pentane > 2, 2-dimethylpropane > 2-methylbutane

B) n-pentane > 2-methylbutane > 2, 2-dimethylpropane

C) 2, 2-dimethylpropane > 2-methylbutane > n-pentane

D) 2-methylbutane > n-pentane > 2, 2-dimethylpropane

E) 2-methylbutane > 2, 2-dimethylpropane > n-pentane

• question_answer112) For preparing an alkane, a saturated solution of sodium or potassium salt of a carboxylic acid is subjected to

A) hydrolysis

B) oxidation

C) hydrogenation

D) hydration

E) electrolysis

• question_answer113) The stablest radical among the following is

A) ${{C}_{6}}{{H}_{5}}-C{{H}_{2}}-\overset{\bullet }{\mathop{C}}\,{{H}_{2}}$

B) $C{{H}_{3}}\overset{\bullet }{\mathop{C}}\,{{H}_{2}}$

C) ${{C}_{6}}{{H}_{5}}-\overset{\bullet }{\mathop{C}}\,H-C{{H}_{3}}$

D) $C{{H}_{3}}-\overset{\bullet }{\mathop{C}}\,H-C{{H}_{3}}$

E) $C{{H}_{3}}-\overset{\bullet }{\mathop{C}}\,{{H}_{2}}-\overset{\bullet }{\mathop{C}}\,{{H}_{2}}$

• question_answer114) The temporary effect in which there is complete transfer of a shared pair of pi-electrons to one of the atoms joined by a multiple bond on the demand of an attacking reagent is called

A) inductive effect

B) positive resonance effect

C) negative resonance effect

D) hyperconjugation

E) electromeric effect

• question_answer115) Among the following pairs, the pair that illustrates stereoisomerism is

A) 1-butanol and 2-butanol

B) cis-2-butene and trans-2-butene

C) dimethyl ether and ethanol

D) acetone and propanal

E) ethanol and ethanol

• question_answer116) The compound$CHCl=CHCHOHCOOH$ with molecular formula${{C}_{4}}{{H}_{5}}{{O}_{3}}Cl$ can exhibit

A) geometric, optical, position and functional isomerism

B) geometric, optical and functional isomerism

C) position and functional isomerism only

D) geometric and optical isomerism only

E) geometric isomerism only

• question_answer117) Which of the following is the correct method of preparation of methyl fluoride?

A) $C{{H}_{4}}+HF\xrightarrow{{}}$

B) $C{{H}_{3}}OH+HF\xrightarrow{{}}$

C) $C{{H}_{4}}+{{F}_{2}}\xrightarrow{{}}$

D) $C{{H}_{3}}Br+AgF\xrightarrow{{}}$

E) $C{{H}_{3}}N{{H}_{2}}+HF\xrightarrow{{}}$

• question_answer118) When 3-phenylpropene reacts with$HBr$in the presence of peroxide, the major product formed is

A) 2-bromo 1-phenylpropane

B) 1, 2-dibromo 3-phenylpropane

C) 3 - (o-bromophenyl) propene

D) 1-bromo 3-phenylpropane

E) 3 (p-bromophenyl) propene

• question_answer119) Reaction of butanone with methyl magnesium bromide following by hydrolysis gives

A) 2-methyl-2-butanol

B) 2-butanol

C) 3-methyl-2-butanol

D) 2, 2-dimethyl-1- butanol

E) 2-pentanol

• question_answer120) The hydroxyl compound that gives a precipitate immediately when treated with concentrated hydrochloric acid and anhydrous zinc chloride is

A) 3-methyl-2-butanol

B) 3-methyl-l-butanol

C) 1-butanol

D) 2-methyl-2-butanol

E) 2, 3-dimethyl-l-butanol

• question_answer121) If the standard deviation of 3, 8, 6, 10, 12, 9, 11, 10, 12, 7 is 2.71, then the standard deviation of 30, 80, 60, 100, 120, 90, 110, 100, 120, 70 is

A) 2.71

B) $27.1$

C) $(2.71)\sqrt{10}$

D) $(27.1)\sqrt{2}$

E) 0.271

• question_answer122) The domain of the function${{\cos }^{-1}}({{\log }_{2}}({{x}^{2}}+5x+8))$is

A) $[2,\text{ }3]$

B) $[-2,\text{ }2]$

C) $[3,\text{ }1]$

D) $(-2,-2)$

E) $[-3,-2]$

• question_answer123) $\underset{x\to 0}{\mathop{\lim }}\,\frac{{{(1+2x)}^{10}}-1}{x}$is equal to

A) 5

B) 10

C) 15

D) 20

E) 0

• question_answer124) The range of the function $f(x)={{\log }_{e}}(3{{x}^{2}}+4)$is equal to

A) $[{{\log }_{e}}2,\infty )$

B) $[{{\log }_{e}}3,\infty )$

C) $[2{{\log }_{e}}3,\infty )$

D) $[0,\infty )$

E) $[2{{\log }_{e}}2,\infty )$

• question_answer125) $\underset{x\to 2}{\mathop{\lim }}\,\frac{{{x}^{100}}-{{2}^{100}}}{{{x}^{77}}-{{2}^{77}}}$is equal to

A) $\frac{100}{77}$

B) $\frac{100}{77}({{2}^{22}})$

C) $\frac{100}{77}({{2}^{21}})$

D) $\frac{100}{77}({{2}^{23}})$

E) $\frac{100}{77}({{2}^{24}})$

• question_answer126) $\underset{k\to \infty }{\mathop{\lim }}\,\left( \frac{{{1}^{3}}+{{2}^{3}}+{{3}^{3}}+......{{k}^{3}}}{{{k}^{4}}} \right)$is equal to

A) $0$

B) $2$

C) $\frac{1}{3}$

D) $\infty$

E) $\frac{1}{4}$

• question_answer127) If$y={{\sin }^{2}}{{\cot }^{-1}}\sqrt{\frac{1+x}{1-x}},$then$\frac{dy}{dx}$is equal to

A) $2\sin 2x$

B) $\sin 2x$

C) $\frac{1}{2}$

D) $-\frac{1}{2}$

E) $\cos 2x$

• question_answer128) If$x={{\sin }^{-1}}(3t-4{{t}^{3}})$and$y={{\cos }^{-1}}(\sqrt{1-{{t}^{2}}}),$then $\frac{dy}{dx}$is equal to

A) $\frac{1}{2}$

B) $\frac{2}{3}$

C) $\frac{1}{3}$

D) $\frac{2}{5}$

E) $\frac{1}{5}$

• question_answer129) If$y=(x+1)(x+2)(x+3)(x+4)(x+5),$then the value of $\frac{dy}{dx}$at$x=0$is equal to

A) 374

B) 742

C) 472

D) 247

E) 274

• question_answer130) If$y={{\cot }^{-1}}\left( \tan \frac{x}{2} \right),$then$\frac{dy}{dx}$ is equal to

A) $\frac{1}{2}$

B) $0$

C) $\frac{x}{2}$

D) $-\frac{1}{2}$

E) $-\frac{x}{2}$

• question_answer131) If$y={{({{\sin }^{-1}}x)}^{2}},$then$(1-{{x}^{2}})\frac{{{d}^{2}}y}{d{{x}^{2}}}-x\frac{dy}{dx}$is equal to

A) $0$

B) $-1$

C) $-2$

D) $1$

E) $2$

• question_answer132) If${{x}^{y}}.{{y}^{x}}=16,$then$\frac{dy}{dx}$at (2, 2) is

A) 1

B) 2

C) $-1$

D) $-2$

E) 0

• question_answer133) If$2y=si{{n}^{-1}}(x+5y),$then$\frac{dx}{dy}$is equal to

A) $cos\text{ }2y-5$

B) $2cosy+5$

C) $cos\text{ }2y+5$

D) $2cos2y+5$

E) $2cos2y-5$

• question_answer134) The total revenue in rupees received from the sale of x units of a product is given by$R(x)=13{{x}^{2}}+26x+15$. Then, the marginal revolution rupees, when$x=15$is

A) 116

B) 126

C) 136

D) 416

E) 146

• question_answer135) The function$f(x)={{(x(x-2))}^{2}}$is increasing in the set

A) $(-\infty ,0)\cup (2,\infty )$

B) $(-\infty ,1)$

C) $(0,1)\cup (2,\infty )$

D) $(1,2)$

E) $(0,2)$

• question_answer136) If a tangent of the curve$y=2+\sqrt{4x+1}$has slope$\frac{2}{5}$at a point, then the point is

A) (0, 2)

B) $\left( \frac{3}{4},4 \right)$

C) (2, 5)

D) (7, 6)

E) (6, 7)

• question_answer137) The equation of the line parallel to x-axis and tangent to the curve$y=\frac{1}{{{x}^{2}}+2x+5}$is

A) $y=\frac{1}{4}$

B) $y=4$

C) $y=\frac{1}{2}$

D) $y=0$

E) $y=2$

• question_answer138) The equation of the tangent to the curve$x=\frac{t-1}{t+1},y\frac{t+1}{t-1}$is$t=2$

A) $x+9y-6=0$

B) $9x-y-6=0$

C) $9x+y+6=0$

D) $x+9y+6=0$

E) $9x+y-6=0$

• question_answer139) The point on the hyperbola $3{{x}^{2}}-4{{y}^{2}}=72$ which is nearest to the line$3x+2y+1=0$is

A) $(-\text{ }6,3)$

B) (6, 3)

C) $(-\text{ }6,-3)$

D) $(\text{ }6,-3)$

E) $(\sqrt{24},0)$

• question_answer140) The value of$x$in the interval [4, 9] at which the function$f(x)=\sqrt{x}$ satisfies the mean value theorem is

A) $\frac{13}{4}$

B) $\frac{17}{4}$

C) $\frac{21}{4}$

D) $\frac{23}{4}$

E) $\frac{25}{4}$

• question_answer141) $\int{\frac{dx}{(x+1)\sqrt{x}}}$is equal to

A) ${{\tan }^{-1}}\sqrt{x}+C$

B) $2{{\tan }^{-1}}x+C$

C) $2{{\tan }^{-1}}(\sqrt{x})+C$

D) ${{\tan }^{-1}}\left( {{x}^{\frac{3}{2}}} \right)+C$

E) $2{{\tan }^{-1}}\left( {{x}^{\frac{3}{2}}} \right)+C$

• question_answer142) $\int{\frac{\log x}{{{x}^{2}}}}dx$is equal to

A) $\frac{\log x}{x}+\frac{1}{{{x}^{2}}}+C$

B) $-\frac{\log x}{x}+\frac{2}{x}+C$

C) $-\frac{\log x}{x}-\frac{1}{2x}+C$

D) $x\log x+\frac{1}{{{x}^{2}}}+C$

E) $-\frac{\log x}{x}-\frac{1}{x}+C$

• question_answer143) If$\int{\frac{x{{\sin }^{-1}}x}{\log \cos x}dx=-\log (\log \cos x)+C,}$is equal to

A) $\tan x$

B) $-sin\,x$

C) $-\cos x$

D) $-\tan \,x$

E) $sin\,x$

• question_answer144) $\int{\frac{x{{\sin }^{-1}}x}{\sqrt{1-{{x}^{2}}}}}dx$is equal to

A) $x-{{\sin }^{-1}}x+C$

B) $x-\sqrt{1-{{x}^{2}}}{{\sin }^{-1}}x+C$

C) $x+{{\sin }^{-1}}x+C$

D) $x+\sqrt{1-{{x}^{2}}}{{\sin }^{-1}}x+C$

E) $x{{\sin }^{-1}}x+\sqrt{1-{{x}^{2}}}+C$

• question_answer145) $\int{\frac{4{{e}^{x}}+6{{e}^{-x}}}{9{{e}^{x}}-4{{e}^{-x}}}}dx$is equal to

A) $\frac{3}{2}x+\frac{35}{36}\log |9{{e}^{2x}}-4|+C$

B) $\frac{3}{2}x-\frac{35}{36}\log |9{{e}^{2x}}-4|+C$

C) $-\frac{3}{2}x+\frac{35}{36}\log |9{{e}^{2x}}-4|+C$

D) $-\frac{5}{2}x+\frac{35}{36}\log |9{{e}^{2x}}-4|+C$

E) $\frac{5}{2}x+\frac{35}{36}\log |9{{e}^{2x}}-4|+C$

• question_answer146) $\int{\sqrt{\frac{1-x}{1+x}}}dx$is equal to

A) ${{\sin }^{-1}}x+\sqrt{1-{{x}^{2}}}+C$

B) ${{\sin }^{-1}}x-2\sqrt{1-{{x}^{2}}}+C$

C) $2{{\sin }^{-1}}x-\sqrt{1-{{x}^{2}}}+C$

D) ${{\sin }^{-1}}x-\sqrt{1-{{x}^{2}}}+C$

E) $-{{\cos }^{-1}}x-\sqrt{1-{{x}^{2}}}+C$

• question_answer147) $\int{\frac{dx}{1+\tan x}}$is equal to

A) $\frac{1}{2}+\frac{1}{2}\log |\cos x+\sin x|+C$

B) $\frac{x}{2}+\frac{1}{2}\log |\cos x-\sin x|+C$

C) $\frac{1}{2}+\frac{1}{2}\log |\cos x-\sin x|+C$

D) $\frac{x}{2}+\frac{1}{2}\log |\cos x+\sin x|+C$

E) $\frac{1}{2}+\frac{1}{2}\log |\cos x+\sin x|+C$

• question_answer148) If$\int_{a}^{0}{\frac{{{x}^{2}}-1}{1-x}dx=-\frac{1}{2},}$then the value of a is equal to

A) $-1$

B) $1$

C) $2$

D) $-2$

E) $-\frac{1}{2}$

• question_answer149) The value of the integral$\int_{0}^{1}{x{{(1-x)}^{5}}dx}$is equal to

A) $\frac{1}{6}$

B) $\frac{1}{7}$

C) $\frac{6}{7}$

D) $\frac{5}{6}$

E) $\frac{1}{42}$

• question_answer150) If$[x]$denotes the greatest integer less than or equal to x, then the value of$\int_{0}^{2}{(|x-2|+[x])}dx$is equal to

A) 2

B) 3

C) 1

D) 4

E) $\frac{3}{2}$

• question_answer151) $\int_{0}^{1}{x{{e}^{-5x}}dx}$is equal to

A) $\frac{1}{25}-\frac{6{{e}^{-5}}}{25}$

B) $\frac{1}{25}+\frac{6{{e}^{-5}}}{25}$

C) $-\frac{1}{25}-\frac{6{{e}^{-5}}}{25}$

D) $\frac{1}{25}-\frac{1}{5}{{e}^{-5}}$

E) $\frac{1}{25}+\frac{1}{5}{{e}^{-5}}$

• question_answer152) The area bounded by the curve$y=sin\text{ }x$ between$x=0$and$x=2\pi$is (in square units)

A) 1

B) 2

C) 0

D) 4

E) $2\pi$

• question_answer153) The differential equation representing the family of curves${{y}^{2}}=2c(x+\sqrt{c}),$where c is a positive parameter, is of

A) order 1, degree 2

B) order 1, degree 3

C) order 2, degree 3

D) order 2, degree 2

E) order 1, degree 1

• question_answer154) An integrating factor of the differential equation${{(1+x)}^{2}}\frac{dy}{dx}+xy=x$is

A) $\frac{x}{1+{{x}^{2}}}$

B) $\frac{1}{2}\log (1+{{x}^{2}})$

C) $\sqrt{1+{{x}^{2}}}$

D) $x$

E) $\frac{1}{1+{{x}^{2}}}$

• question_answer155) The solution of the differential equation$x\frac{dy}{dx}+y=\frac{1}{{{x}^{2}}}$at (1, 2) is

A) ${{x}^{2}}y+1=3x$

B) ${{x}^{2}}y+1=0$

C) $xy+1=3x$

D) ${{x}^{2}}(y+1)=3x$

E) ${{x}^{2}}y=3x+1$

• question_answer156) The general solution of the differential equation$\frac{dy}{dx}={{e}^{y}}({{e}^{x}}+{{e}^{-x}}+2x)$is

A) ${{e}^{-y}}={{e}^{x}}-{{e}^{-x}}+{{x}^{2}}+C$

B) ${{e}^{-y}}={{e}^{-x}}-{{e}^{x}}-{{x}^{2}}+C$

C) ${{e}^{-y}}=-{{e}^{-x}}-{{e}^{x}}-{{x}^{2}}+C$

D) ${{e}^{y}}={{e}^{-x}}+{{e}^{x}}+{{x}^{2}}+C$

E) ${{e}^{y}}={{e}^{-x}}+{{e}^{x}}+C$

• question_answer157) If the function$f:[1,\infty )\to [1,\infty )$is defined by $f(x)={{2}^{x(x-1)}},$then ${{f}^{-1}}(x)$is

A) ${{\left( \frac{1}{2} \right)}^{x(x-1)}}$

B) $\frac{1}{2}(1-\sqrt{1+4{{\log }_{2}}x})$

C) $\frac{1}{2}\sqrt{1+4{{\log }_{2}}x}$

D) $\frac{1}{2}[1+\sqrt{1+4{{\log }_{2}}x}]$

E) not defined

• question_answer158) If$n(A)=8$and$n(A\cap B)=2,$ then$n((A\cap B)\cap A)$is equal to

A) 2

B) 4

C) 6

D) 8

E) 10

• question_answer159) If$f(x)=\sin x+\cos x,x\in (-\infty ,\infty )$and$g(x)={{x}^{2}},x\in (-\infty ,\infty ),$then$(fog)(x)$is equal to

A) 1

B) 0

C) ${{\sin }^{2}}(x)+\cos ({{x}^{2}})$

D) $\sin ({{x}^{2}})+{{\cos }^{2}}(x)$

E) $\sin ({{x}^{2}})+\cos ({{x}^{2}})$

• question_answer160) If the set A contains 5 elements, then the number of elements in the power set $P(A)$ is equal to

A) 32

B) 25

C) 16

D) 8

E) 10

• question_answer161) The domain of the function$f(x)=\frac{1}{\sqrt{9-{{x}^{2}}}}$is

A) $-3\le x\le 3$

B) $-3<x<3$

C) $-9\le x\le 9$

D) $-9<x<9$

E) $-\infty <x<\infty$

• question_answer162) The period of the function$f(x)=|\sin 2x|+\cos 8x|$is

A) $2\pi$

B) $\pi$

C) $\frac{2\pi }{3}$

D) $\frac{\pi }{2}$

E) $\frac{\pi }{4}$

• question_answer163) The value of$i-{{i}^{2}}+{{i}^{3}}-{{i}^{4}}+....-{{i}^{100}}$is equal to

A) $i$

B) $-i$

C) $1-i$

D) $1+i$

E) 0

• question_answer164) If the imaginary part of$\frac{2+i}{ai-1}$is zero, where a is a real number, then the value of a is equal to

A) $\frac{1}{2}$

B) $2$

C) $-\frac{1}{2}$

D) $-2$

E) $\frac{3}{2}$

• question_answer165) The argument of the complex number$\left( \frac{i}{2}-\frac{2}{i} \right)$is equal to

A) $\frac{\pi }{4}$

B) $\frac{3\pi }{4}$

C) $\frac{\pi }{12}$

D) $\frac{\pi }{2}$

E) $\frac{3\pi }{2}$

• question_answer166) Let${{z}_{1}}=3+4i$and${{z}_{2}}=-1+2i$. Then, $|{{z}_{1}}+{{z}_{2}}{{|}^{2}}-2(|{{z}_{1}}{{|}^{2}}+|{{z}_{2}}{{|}^{2}})$is equal to

A) $|{{z}_{1}}-{{z}_{2}}{{|}^{2}}$

B) $-|{{z}_{1}}-{{z}_{2}}{{|}^{2}}$

C) $|{{z}_{1}}{{|}^{2}}+|{{z}_{2}}{{|}^{2}}$

D) $|{{z}_{1}}{{|}^{2}}-|{{z}_{2}}{{|}^{2}}$

E) $|{{z}_{1}}{{|}^{2}}+|{{z}_{2}}{{|}^{2}}-2|{{z}_{1}}||{{z}_{2}}|$

• question_answer167) If${{z}_{1}}$and${{z}_{2}}$are two non-zero complex numbers such that$|{{z}_{1}}+{{z}_{2}}|=|{{z}_{1}}|+|{{z}_{2}}|,$then arg$\left( \frac{{{z}_{1}}}{{{z}_{2}}} \right)$is equal to

A) $0$

B) $-\pi$

C) $-\frac{\pi }{2}$

D) $\frac{\pi }{2}$

E) $\pi$

• question_answer168) If the equation${{x}^{2}}-(2+m)x+({{m}^{2}}-4m+4)=0$in$x$has equal roots, then the values of m are

A) $\frac{2}{3},1$

B) $\frac{2}{3},6$

C) $0,1$

D) $0,2$

E) $\frac{2}{3},0$

• question_answer169) The number of integral values of b, for which the equation${{x}^{2}}+bx-16=0$has integral roots, is

A) 2

B) 3

C) 4

D) 5

E) 6

• question_answer170) If$(1+i)$is a root of the equation${{x}^{2}}-x(1-i)=0,$then the other root is

A) $1-i$

B) $i$

C) $-i$

D) $2i$

E) $-2i$

• question_answer171) If the roots of the quadratic equation $3{{x}^{2}}+2x+{{a}^{2}}-a=0$in$x$are of opposite signs, then a lies in the interval

A) B) $(-\infty ,0)$

C) $(-1,0)$

D) $(0,1)$

E) $(1,3)$

• question_answer172) The number of real roots of the equation$|x{{|}^{2}}-3|x|+2=0$is

A) 1

B) 2

C) 3

D) 6

E) 4

• question_answer173) Let a, b, c be positive real numbers. If $\frac{{{x}^{2}}-bx}{ax-c}=\frac{m-1}{m+1}$has two roots which are numerically equal but opposite in sign, then the value of m is

A) $c$

B) $\frac{1}{c}$

C) $\frac{a+b}{a-b}$

D) $1$

E) $\frac{a-b}{a+b}$

• question_answer174) If the 9th term of an AP is zero, then the ratio of 29th term to 19th term is

A) 1 : 2

B) 1 : 3

C) 2 : 1

D) 3 : 1

E) 9 : 1

• question_answer175) Let${{S}_{1}},{{S}_{2}},.........{{S}_{101}}$be consecutive terms of an AP. If$\frac{1}{{{S}_{1}}{{S}_{2}}}+\frac{1}{{{S}_{2}}{{S}_{3}}}+......+\frac{1}{{{S}_{100}}{{S}_{101}}}=\frac{1}{6}$and ${{S}_{1}}+{{S}_{101}}=50,$then$|{{S}_{1}}-{{S}_{101}}|$is equal to

A) 10

B) 20

C) 30

D) 40

E) 50

• question_answer176) If${{a}_{1}},{{a}_{2}},{{a}_{3}},.......,{{a}_{n}}$are in AP and${{a}_{1}}=0,$then the value of $\left( \frac{{{a}_{3}}}{{{a}_{2}}}+\frac{{{a}_{4}}}{{{a}_{3}}}+....\frac{{{a}_{n}}}{{{a}_{n-1}}} \right)-{{a}_{2}}\left( \frac{1}{{{a}_{2}}}+\frac{1}{{{a}_{3}}}+.....+\frac{1}{{{a}_{n-2}}} \right)$ is equal to

A) $(n-2)+\frac{1}{(n-2)}$

B) $\frac{1}{(n-2)}$

C) $(n-2)$

D) $(n-1)$

E) $(n+2)$

• question_answer177) The value of${{1}^{2}}-{{2}^{2}}+{{3}^{2}}-{{4}^{2}}+...\text{ 1}{{\text{1}}^{2}}$is equal to

A) 55

B) 66

C) 77

D) 88

E) 99

• question_answer178) Let${{S}_{n}}$denote the sum of first n terms of an AP and${{S}_{2n}}=3{{S}_{n}}.$If${{S}_{3n}}=k{{S}_{n}},$then the value of k is equal to

A) 4

B) 5

C) 6

D) 7

E) 8

• question_answer179) The first four terms of an AP are $a,9,3a-b,3a$$+b$. The 2011th term of the AP is

A) 2015

B) 4025

C) 5030

D) 6035

E) 8045

• question_answer180) If$(n+2)!=2550\times n!,$ then the value of n is equal to

A) 8

B) 49

C) 50

D) 51

E) 52

• question_answer181) If$^{n}{{C}_{r-1}}=28{{,}^{n}}{{C}_{r}}=56$and$^{n}{{C}_{r+1}}=70,$then the value of r is equal to

A) 6

B) 2

C) 3

D) 4

E) 5

• question_answer182) The number of integers greater than 6000 that can be formed with 3, 5, 6, 7 and 8, where no digit is repeated, is

A) 120

B) 192

C) 216

D) 72

E) 202

• question_answer183) The sum of the coefficients in the expansion of ${{\left( {{x}^{2}}-\frac{1}{3} \right)}^{199}}\times {{\left( {{x}^{3}}+\frac{1}{2} \right)}^{200}}$is

A) $\frac{1}{3}$

B) $-\frac{1}{3}$

C) $\frac{2}{3}$

D) $\frac{3}{2}$

E) $0$

• question_answer184) If${{(1+ax)}^{n}}=1+6x+\frac{27}{2}{{x}^{2}}+....+{{a}^{n}}{{x}^{n}},$then the values of a and n are respectively

A) $2,3$

B) $3,2$

C) $\frac{3}{2},4$

D) $1,6$

E) $\frac{3}{2},6$

• question_answer185) If${{(1-x)}^{n}}={{c}_{0}}-{{c}_{1}}x+{{c}_{2}}{{x}^{2}}-{{c}_{3}}{{x}^{3}}$ $+....+{{(-1)}^{n}}{{c}_{n}}{{x}^{n}},$ then$\frac{{{c}_{0}}}{2}-\frac{{{c}_{1}}}{3}+\frac{{{c}_{2}}}{4}-\frac{{{c}_{3}}}{5}+.....+{{(-1)}^{n}}\frac{{{c}_{n}}}{n+2}$is

A) $\frac{1}{n(n+1)}$

B) $\frac{1}{(n+1)(n+2)}$

C) $\frac{1}{(n+1)(n+3)}$

D) $\frac{1}{n(n+3)}$

E) $\frac{1}{(n+2)(n+3)}$

• question_answer186) If$A=\left[ \begin{matrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b \\ \end{matrix} \right]$is a matrix satisfying$A{{A}^{T}}=9{{I}_{3}},$then the values of a and b are respectively

A) $1,2$

B) $-1,\text{ }2$

C) $-1,-2$

D) $2,1$

E) $-2,-1$

• question_answer187) If$A=\left[ \begin{matrix} 2 & 1 \\ 0 & x \\ \end{matrix} \right]$and${{A}^{-1}}=\left[ \begin{matrix} \frac{1}{2} & \frac{1}{6} \\ 0, & \frac{1}{x} \\ \end{matrix} \right]$,then the value of$x$is equal to

A) $-3$

B) $3$

C) $-2$

D) $6$

E) $-6$

• question_answer188) If$A=\left( \begin{matrix} \cos \alpha & \sin \alpha \\ -{{\sin }^{10}}\alpha & \cos \alpha \\ \end{matrix} \right),$then${{A}^{10}}$is equal to

A) $\left( \begin{matrix} {{\cos }^{10}}\alpha & {{\sin }^{10}}\alpha \\ -{{\sin }^{10}}\alpha & {{\cos }^{10}}\alpha \\ \end{matrix} \right)$

B) $\left( \begin{matrix} {{\cos }^{10}}\alpha & -{{\sin }^{10}}\alpha \\ {{\sin }^{10}}\alpha & {{\cos }^{10}}\alpha \\ \end{matrix} \right)$

C) $\left( \begin{matrix} {{\cos }^{10}}\alpha & {{\sin }^{10}}\alpha \\ -{{\sin }^{10}}\alpha & -{{\cos }^{10}}\alpha \\ \end{matrix} \right)$

D) $\left( \begin{matrix} \cos 10\alpha & \sin 10\alpha \\ -\sin 10\alpha & \cos 10\alpha \\ \end{matrix} \right)$

E) $\left( \begin{matrix} \cos 10\alpha & -\sin 10\alpha \\ -\sin 10\alpha & -\cos 10\alpha \\ \end{matrix} \right)$

• question_answer189) If$A=\left[ \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & -1 \\ \end{matrix} \right]$and$I$is the unit matrix of order 3, then${{A}^{2}}+2{{A}^{4}}+4{{A}^{6}}$is equal to

A) $7{{A}^{8}}$

B) $7{{A}^{7}}$

C) $8I$

D) $6I$

E) $I$

• question_answer190) If$A=\left[ \begin{matrix} x & 1 \\ 1 & 0 \\ \end{matrix} \right]$and${{A}^{2}}$is the unit matrix, then the value of${{x}^{3}}+x-2$is equal to

A) $-8$

B) $-2$

C) 0

D) $1$

E) $8$

• question_answer191) If$(b+c)(y+z)-ax=b-c$, $(c+a)(z+x)-by=c-a$ and$(a+b)(x+y)-cz=a-b,$where $a+b+c\ne 0,$then$x$is equal to

A) $\frac{c+b}{a+b+c}$

B) $\frac{c-b}{a+b+c}$

C) $\frac{a-b}{a+b+c}$

D) $\frac{a+b}{a+b+c}$

E) $\frac{b-c}{a+b+c}$

• question_answer192) If$|2x-3|<|x+5|,$then$x$lies in the interval

A) $(-3,5)$

B) $(5,9)$

C) $\left( -\frac{2}{3},8 \right)$

D) $\left( -8,\frac{2}{3} \right)$

E) $\left( -5,\frac{2}{3} \right)$

• question_answer193) The solution set of$\frac{x+3}{x-2}\le 2$is

A) $(-\infty ,\infty )$

B) $(-\infty ,2]\cup [7,\infty )$

C) $(-\infty ,2)\cup [7,\infty )$

D) $[7,\infty )$

E) $(-\infty ,2)$

• question_answer194) Let p: roses are red and q : the sun is a star. Then, the verbal translation of$(-\text{ }p)/q$is

A) roses are not red and the sun is not a star

B) it is not true that roses are red or the sun is not a star

C) it is not true that roses are red and the sun is not a star

D) roses are not red or the sun is a star

E) it is not true that roses are red and the sun is a star

• question_answer195) The statement$p\to (\tilde{\ }q)$ is equivalent to

A) $q\to p$

B) $\tilde{\ }q\vee \tilde{\ }p$

C) $p\wedge \tilde{\ }q$

D) $\tilde{\ }q\to p$

E) $\tilde{\ }p\vee q$

• question_answer196) The negation of$(p\vee \tilde{\ }q)\wedge q$is

A) $(p\vee q)\wedge \tilde{\ }q$

B) $(p\wedge \tilde{\ }q)\vee q$

C) $(\tilde{\ }p\wedge q)\vee \tilde{\ }q$

D) $(p\wedge \tilde{\ }q)\vee \tilde{\ }q$

E) $(\tilde{\ }p\vee \tilde{\ }q)\wedge \tilde{\ }q$

• question_answer197) The value of$cos\text{ }20{}^\circ +cos\text{ }100{}^\circ +cos\text{ }140{}^\circ$is equal to

A) $\frac{1}{2}$

B) $\frac{1}{\sqrt{3}}$

C) $\sqrt{3}$

D) $0$

E) $1$

• question_answer198) If $\frac{-\pi }{2}<\theta <\frac{\pi }{2}$and$\theta \ne \pm \frac{\pi }{4},$then the value of$\cot \left( \frac{\pi }{4}+\theta \right)\cot \left( \frac{\pi }{4}-\theta \right)$is

A) 0

B) $-1$

C) 1

D) $-2$

E) 2

• question_answer199) If$\sin \theta =3\sin (\theta +2\alpha ),$then the value of$\tan (\theta +\alpha )+2\tan \alpha$is

A) $3$

B) $2$

C) $-1$

D) $0$

E) $1$

• question_answer200) If$\alpha ,\beta ,\gamma \in [0,\pi ]$and if$\alpha ,\beta ,\gamma$are in AP, then $\frac{sin\alpha -\sin \gamma }{\cos \gamma -\cos \alpha }$is equal to

A) $\sin \beta$

B) $\cos \beta$

C) $\cot \beta$

D) $2\cos \beta$

E) $\cos ec\beta$

• question_answer201) If$2{{\sin }^{-1}}x-{{\cos }^{-1}}x=\frac{\pi }{2},$then$x$is equal to

A) $\frac{1}{\sqrt{2}}$

B) $-\frac{1}{\sqrt{2}}$

C) $\frac{-\sqrt{3}}{2}$

D) $\frac{\sqrt{3}}{2}$

E) $\frac{1}{2}$

• question_answer202) The value of$\frac{1}{8}(3-4\text{ }cos\text{ }2\theta +cos\text{ }4\theta )$is

A) $\cos 4\theta$

B) $\sin 4\theta$

C) ${{\sin }^{4}}\theta$

D) ${{\cos }^{4}}\theta$

E) ${{\sin }^{4}}(\theta /2)$

• question_answer203) If$8\text{ }cos\text{ }2\theta +8\text{ }sec\text{ }2\theta =65,\text{ }0<\theta <\frac{\pi }{2},$then the value of$4\text{ }cos\text{ }4\theta$is equal to

A) $\frac{-23}{8}$

B) $\frac{-31}{8}$

C) $\frac{-31}{32}$

D) $\frac{-33}{32}$

E) $\frac{-32}{4}$

• question_answer204) The value of${{\tan }^{-1}}(2)+{{\tan }^{-1}}(3)$is equal to

A) $\frac{3\pi }{4}$

B) $\frac{\pi }{4}$

C) $\frac{\pi }{3}$

D) ${{\tan }^{-1}}(6)$

E) ${{\tan }^{-1}}(5)$

• question_answer205) The equation$k\text{ }sin\text{ }x+cos\text{ }2x=2k-7$has a solution, if

A) $k>6$

B) $2\le k\le 6$

C) $k<2$

D) $-6\le k\le -2$

E) $k\le -6$

• question_answer206) The distance between the points $(a\cos \alpha ,a\sin \alpha )$and$a\cos \beta ,a\sin \beta$is

A) $2\left| \sin \left( \frac{\alpha -\beta }{2} \right) \right|$

B) $2\left| a\sin \left( \frac{\alpha -\beta }{2} \right) \right|$

C) $2\left| a\cos \left( \frac{\alpha -\beta }{2} \right) \right|$

D) $\left| a\cos \left( \frac{\alpha -\beta }{2} \right) \right|$

E) $2|a(1-\cos (\alpha -\beta ))|$

• question_answer207) The vertices of the rectangle ABCD are$A(-1,$$0),$$B(2,\text{ }0),C(a,\text{ }b)$and$D(-1,\text{ }4)$Then, the length of the diagonal AC is

A) 2

B) 3

C) 4

D) 5

E) 6

• question_answer208) If a straight line passes through the points $\left( \frac{-1}{2},1 \right)$and (1, 2), then its x-intercept is

A) $-2$

B) $-1$

C) $2$

D) $1$

E) $0$

• question_answer209) The line parallel to the x-axis and passing through the point of intersection of the lines $ax+2by+3b=0$and $bx-2ay-3a=0,$where $(a,b)\ne (0,0)$is

A) above the x-axis at a distance of$\frac{3}{2}$

B) above the x-axis at a distance of$\frac{2}{3}$

C) below the x-axis at a distance of$\frac{2}{3}$

D) below the x-axis at a distance of$\frac{3}{2}$

E) below the x-axis at a distance of 3

• question_answer210) The line L has intercepts a and b on the coordinate axes. Keeping the origin fixed, the coordinate axes are rotated through a fixed angle. If the line L has intercepts p and q on the rotated axes, then$\frac{1}{{{a}^{2}}}+\frac{1}{{{b}^{2}}}$is equal to

A) ${{p}^{2}}+{{q}^{2}}$

B) ${{p}^{2}}-{{q}^{2}}$

C) $\frac{1}{{{p}^{2}}}+\frac{1}{{{q}^{2}}}$

D) $\frac{1}{{{p}^{2}}}-\frac{1}{{{q}^{2}}}$

E) $\frac{1}{{{q}^{2}}}-\frac{1}{{{p}^{2}}}$

• question_answer211) The equation of the perpendicular bisector of the line segment joining$A(-2,\text{ }3)$and$B(6,-5)$is

A) $x-y=-1$

B) $x-y=3$

C) $x+y=3$

D) $x+y=1$

E) $x+y=-1$

• question_answer212) The vertices of the $\Delta PQR$ are P (0, b), Q (0, 0) and R (a, 0). If the medians PM and QN of PQR are perpendicular, then

A) ${{b}^{2}}=2{{a}^{2}}$

B) $b={{a}^{2}}$

C) ${{a}^{2}}=2{{b}^{2}}$

D) $a=b$

E) $a=-b$

• question_answer213) The slope of the straight line which does not intersect$x-$axis is equal to

A) $\frac{1}{2}$

B) $\frac{1}{\sqrt{2}}$

C) $\sqrt{3}$

D) $1$

E) $0$

• question_answer214) The length of the tangent drawn from any point on the circle${{x}^{2}}+{{y}^{2}}+2fy+\lambda =0$to the circle${{x}^{2}}+{{y}^{2}}+2fy+\mu =0,$where$\mu >\lambda >0$,is

A) $\sqrt{\mu -\lambda }$

B) $\sqrt{\mu +\lambda }$

C) $\sqrt{{{\mu }^{2}}-{{\lambda }^{2}}}$

D) $\mu +\lambda$

E) $\mu -\lambda$

• question_answer215) The sum of the minimum distance and the maximum distance from the point$(4,-3)$to the circle${{x}^{2}}+{{y}^{2}}+4x-10y-7=0$is

A) 20

B) 12

C) 10

D) 16

E) 22

• question_answer216) The equation of one of the diameters of the circle${{x}^{2}}-{{y}^{2}}-6x+2y=0$is

A) $x-3y=0$

B) $x+3y=0$

C) $3x+y=0$

D) $3x-y=0$

E) $x+2y=0$

• question_answer217) The parametric equations of the circle${{x}^{2}}+{{y}^{2}}+x+\sqrt{3}y=0$are

A) $x=1+\cos \theta ,y=\frac{\sqrt{3}}{2}+\sin \theta$

B) $x=-\frac{1}{2}+\cos \theta ,y=-\frac{\sqrt{3}}{2}+\sin \theta$

C) $x=\frac{1}{2}+\cos \theta ,y=-\frac{\sqrt{3}}{2}+\sin \theta$

D) $x=\frac{1}{2}+\frac{1}{2}+\cos \theta ,y=\frac{\sqrt{3}}{2}+\frac{1}{2}+\sin \theta$

E) $x=\cos \theta -1,y=\frac{\sqrt{3}}{2}+\sin \theta$

• question_answer218) An equilateral triangle is inscribed in the parabola${{y}^{2}}=4x$. If a vertex of the triangle is at the vertex of the parabola, then the length of side of the triangle is

A) $\sqrt{3}$

B) $8\sqrt{3}$

C) $4\sqrt{3}$

D) $3\sqrt{3}$

E) $2\sqrt{3}$

• question_answer219) The equation of the latusrectum of the conic${{y}^{2}}=\frac{5}{2}x$is

A) $8x-5=0$

B) $8x+5=0$

C) $5x+8=0$

D) $x-5=0$

E) $x-8=0$

• question_answer220) For each point$(x,\text{ }y)$on an ellipse, the sum of the distances from$(x,\text{ }y)$to the points$(2,0)$and$(-2,0)$is 8. Then, the positive value of$x$so that$(x,3)$lies on the ellipse is

A) $2$

B) $2\sqrt{3}$

C) $\frac{1}{\sqrt{3}}$

D) $4$

E) $0$

• question_answer221) The focus of the parabola${{y}^{2}}+6x-2y+13=0$ is at the point

A) $\left( \frac{7}{2},1 \right)$

B) $\left( \frac{-1}{2},1 \right)$

C) $\left( -2,\frac{1}{2} \right)$

D) $\left( -\frac{7}{2},1 \right)$

E) $\left( -\frac{1}{2},-1 \right)$

• question_answer222) The distance between the vertex of the parabola$y={{x}^{2}}-4x+3$and the centre of the circle${{x}^{2}}=9-{{(y-3)}^{2}}$is

A) $2\sqrt{3}$

B) $3\sqrt{2}$

C) $2\sqrt{2}$

D) $\sqrt{2}$

E) $2\sqrt{5}$

• question_answer223) If a is perpendicular to b, then the vector$a\times \{a\times \{a\times (a\times b)\}\}$is equal to

A) $|a{{|}^{2}}b$

B) $|a|b$

C) $|a{{|}^{3}}b$

D) $|a{{|}^{4}}b$

E) 0

• question_answer224) If the vector$8i+aj$of magnitude 10 is in the direction of the vector$4i+3j,$then the value of a is equal to

A) 6

B) 3

C) $-3$

D) 5

E) $-6$

• question_answer225) If$a=2i-7j+k$and$b=i+3j-5k$and $a.mb=120,$then the value of m is equal to

A) 5

B) $-24$

C) $-5$

D) $120$

E) $24$

• question_answer226) If the angle between a and c is$25{}^\circ ,$the angle between b and c is$65{}^\circ ,$and$a+b=c,$then the angle between a and b is

A) $40{}^\circ$

B) $115{}^\circ$

C) $25{}^\circ$

D) $65{}^\circ$

E) $90{}^\circ$

• question_answer227) The position vector of the centroid of the$\Delta ABC$is$2i+4j+2k$. If the position vector of the vertex A is$2i+6j+4k,$then the position vector of midpoint of BC is

A) $2i+3j+k$

B) $2i+3j-k$

C) $2i-3j-k$

D) $-2i-3j-k$

E) $2i-3j+k$

• question_answer228) The projection of the vector$2i+a\text{ }j-k$on the vector$i-2j+k$ is$\frac{-5}{\sqrt{6}}$. Then, the value of a is equal to

A) 1

B) 2

C) $-2$

D) 3

E) $-3$

• question_answer229) A unit vector in the$XOY-$plane that makes an angle$30{}^\circ$with the vector$i+j$and makes an angle$60{}^\circ$with$i-j$is

A) $\frac{1}{4}[(\sqrt{6}+\sqrt{2})i-(\sqrt{6}-\sqrt{2})j]$

B) $\frac{1}{4}[(\sqrt{6}-\sqrt{2})i+(\sqrt{6}+\sqrt{2})j]$

C) $\frac{1}{4}[(\sqrt{6}-\sqrt{2})i+(\sqrt{6}+\sqrt{2})j]$

D) $\frac{1}{3}[(\sqrt{6}+\sqrt{2})i+(\sqrt{2}-\sqrt{6})j]$

E) $\frac{1}{4}[(\sqrt{6}+\sqrt{2})i+(\sqrt{6}-\sqrt{2})j]$

• question_answer230) The angle between the line$r=(i+2j+3k)+\lambda (3i+3j+4k)$and the plane $r.(i+j-2k)=0$is

A) $0{}^\circ$

B) $60{}^\circ$

C) $30{}^\circ$

D) $90{}^\circ$

E) $45{}^\circ$

• question_answer231) The lines$r=i+j-k+\lambda (3i-j)$and$r=4i-k+\mu (2i+3k)$intersect at the point

A) (0, 0, 0)

B) (0, 0, 1)

C) (0, - 4, -1)

D) (4, 0, -1)

E) (4, 1, -1)

• question_answer232) An equation of the plane through the points (1, 0, 0) and (0, 2, 0) and at a distance$\frac{6}{7}$units from the origin is

A) $6x+3y+z-6=0$

B) $6x+3y+2z-6=0$

C) $6x+3y+z+6=0$

D) $6x+3y+2z+6=0$

E) $6x+2y+3z+6=0$

• question_answer233) The projection of a line segment on the axes are 9,12 and 8. Then, the length of the line segment is

A) 15

B) 16

C) 17

D) 18

E) 21

• question_answer234) The straight line passing through the point $(1,0,-2)$ and perpendicular to the plane $x-2y+5z-7=0$is

A) $\frac{x-1}{1}=\frac{y}{0}=\frac{z-5}{-2}$

B) $\frac{x-1}{5}=\frac{y}{-2}=\frac{z+2}{1}$

C) $\frac{x-5}{-2}=\frac{y-1}{-5}=\frac{z}{1}$

D) $\frac{x-1}{-1}=\frac{y}{-2}=\frac{z-2}{5}$

E) $\frac{x-1}{1}=\frac{y}{-2}=\frac{z+2}{5}$

• question_answer235) The equation of the plane passing through $(1,2,3)$and parallel to$3x-2y+4z=5$is

A) $3x-2y+4=11$

B) $3x-2y+4z=0$

C) $3x-2y+4z=10$

D) $3(x-1)-2(y-2)+4(z-3)=5$

E) $3(x-1)-2(y-2)+4(z-3)=11$

• question_answer236) If the straight lines$\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{0}$and $\frac{x-1}{k}=\frac{y-4}{2}=\frac{z-5}{1}$are coplanar, then the value of$k$is

A) $-3$

B) $0$

C) $1$

D) $-2$

E) $6$

• question_answer237) The line$\frac{x-{{x}_{1}}}{0}=\frac{y-{{y}_{1}}}{1}=\frac{z-{{z}_{1}}}{2}$is

A) perpendicular to the$x-$axis

B) perpendicular to the$yz-$plane

C) parallel to the$y-$axis

D) parallel to the $xz-$plane

E) perpendicular to the $z-$axis

• question_answer238) The AM of 9 terms is 15. If one more term is added to this series, then the AM becomes 16. The value of the added term is

A) 30

B) 27

C) 25

D) 23

E) 20

• question_answer239) If the average of the numbers$1,2,3...,98,99,x$ is$100x,$then the value of$x$is

A) $\frac{51}{100}$

B) $\frac{50}{99}$

C) $\frac{1}{2}$

D) $\frac{51}{99}$

E) $\frac{50}{101}$

• question_answer240) If the median of$\frac{x}{5},x,\frac{x}{4},\frac{x}{2},\frac{x}{3}(x>0)$is 8, then the value of$x$is

A) 24

B) 32

C) 8

D) 16

E) 40

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