# Solved papers for CEE Kerala Engineering CEE Kerala Engineering Solved Paper-2005

### done CEE Kerala Engineering Solved Paper-2005

• question_answer1) The frequency of X-rays, $\gamma$-rays and ultraviolet rays are respectively a, b and c then:

A) $a<b,\text{ }b>c$

B) $a>b,\text{ }b>c$

C) $a>b,\text{ }b<c$

D) $a<b,\text{ }b<c$

E) $a=b=c$

• question_answer2) If c is the speed of electromagnetic waves in vacuum, its speed in a medium of dielectric constant K and relative permeability${{\mu }_{r}}$is:

A) $v=\frac{1}{\sqrt{{{\mu }_{r}}K}}$

B) $v=c\sqrt{{{\mu }_{r}}K}$

C) $v=\frac{c}{\sqrt{{{\mu }_{r}}K}}$

D) $v=\frac{K}{\sqrt{{{\mu }_{r}}c}}$

E) $v=\frac{{{\mu }_{r}}}{\sqrt{cK}}$

• question_answer3) A red coloured object illuminated by mercury vapour lamp, when seen through a green filter, will appear:

A) red

B) blue

C) violet

D) white

E) black

• question_answer4) Time taken by sunlight to pass through a window of thickness 4 mm whose refractive index is$\frac{3}{2}$is:

A) $2\times {{10}^{-4}}s$

B) $2\times {{10}^{4}}s$

C) $2\times {{10}^{-11}}s$

D) $2\times {{10}^{11}}s$

E) $2\times {{10}^{8}}s$

• question_answer5) Two thin lenses of focal length 20 cm and 25 cm are in contact. The effective power of the combination is:

A) 4.5 D

B) 18 D

C) 45 D

D) 2.5 D

E) 9 D

• question_answer6) The magnification of the image when an object is placed at a distance$x$from the principal focus of a mirror of focal length$f$is:

A) $\frac{x}{f}$

B) $1+\frac{f}{x}$

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

D) $1-\frac{f}{x}$

E) $\frac{1+f}{x}$

• question_answer7) In the Youngs double slit experiment, the central maxima is observed to be${{I}_{0}}$. If one of the slits is covered, then the intensity at the central maxima will become:

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

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

C) $\frac{{{I}_{0}}}{4}$

D) ${{I}_{0}}$

E) $I_{0}^{2}$

• question_answer8) The ratio of the de-Broglie wavelength of an $\alpha -$particle and a proton of same kinetic energy is:

A) $1:2$

B) $1:1$

C) $1:\sqrt{2}$

D) $4:1$

E) $\sqrt{2}:1$

• question_answer9) Which of the following is not conserved in nuclear reaction?

A) Total energy

B) Mass number

C) Charge number

D) Nucleon number

E) Number of fundamental particles

• question_answer10) The number of $\alpha$-particles and p-particles respectively emitted in the reaction$_{88}{{A}^{196}}{{\to }_{78}}{{B}^{164}}$are:

A) 8 and 8

B) 8 and 6

C) 6 and 8

D) 6 and 6

E) 8 and 4

• question_answer11) The counting rate observed from a radioactive source ate = 0 s was 1600 count/s and at$t=8$s it was 100 counts/s. The counting rate observed as counts per second at$t=6\text{ }s,$will be:

A) 400

B) 300

C) 250

D) 200

E) 150

• question_answer12) If${{D}_{e}},\text{ }{{D}_{b}}$and${{D}_{c}}$are the doping levels of emitter, base and collector respectively of a transistor, then:

A) ${{D}_{e}}={{D}_{b}}={{D}_{c}}$

B) ${{D}_{e}}<{{D}_{b}}={{D}_{c}}$

C) ${{D}_{e}}>{{D}_{b}}>{{D}_{c}}$

D) ${{D}_{e}}<{{D}_{b}}<{{D}_{c}}$

E) ${{D}_{e}}>{{D}_{c}}>{{D}_{b}}$

• question_answer13) The relation between a and p parameters of a transistor is:

A) $\alpha =\frac{1+\beta }{\beta }$

B) $\alpha =\frac{1-\beta }{\beta }$

C) $\alpha =\frac{\beta }{1+\beta }$

D) $\alpha =\frac{\beta }{1-\beta }$

E) $\alpha =\beta$

• question_answer14) A p-n junction in series with a resistance of$5\,k\Omega$is connected across a 50 V DC source. If the forward bias resistance of the junction is$50\,\Omega ,$, the forward bias current is:

A) 8.8 mA

B) 1 mA

C) 2 mA

D) 20 mA

E) 9.9 mA

• question_answer15) A transistor connected at common-emitter mode contains load resistance of$5\,k\,\Omega ,$and an input resistance of$1\,k\,\Omega$. If the input peak voltage is 5 mV and the current gain is 50, find the voltage gain:

A) 250

B) 500

C) 125

D) 50

E) 75

• question_answer16) If${{n}_{1}}$and${{n}_{2}}$are the refractive indices of the core and the cladding respectively of an optic fibre, then:

A) ${{n}_{1}}={{n}_{2}}$

B) ${{n}_{1}}<{{n}_{2}}$

C) ${{n}_{2}}<{{n}_{1}}$

D) ${{n}_{2}}=2{{n}_{1}}$

E) ${{n}_{2}}=\sqrt{2{{n}_{1}}}$

• question_answer17) If a radio receiver amplifies all the signal frequencies equally well, it is said to have high:

A) fidelity

B) distortion

C) sensibility

D) sensitivity

E) selectivity

• question_answer18) The waves relevant to telecommunications are:

A) visibe light

B) infrared

C) ultraviolet

D) microwave

E) none of the above

• question_answer19) A TV tower has a height of 100 m. What is the maximum distance up to which the TV transmission can be received$(R=8\times {{10}^{6}}m)$?

A) 34.77 km

B) 32.70 km

C) 40 km

D) 40.70 km

E) 42.75 km

• question_answer20) The dimensional formula of magnetic flux is:

A) $[{{M}^{1}}{{L}^{0}}{{T}^{-2}}{{A}^{-1}}]$

B) $[{{M}^{1}}{{L}^{2}}{{T}^{-1}}{{A}^{-1}}]$

C) $[{{M}^{1}}{{L}^{2}}{{T}^{-1}}{{A}^{-2}}]$

D) $[{{M}^{1}}{{L}^{2}}{{T}^{0}}{{A}^{-1}}]$

E) $[{{M}^{1}}{{L}^{2}}{{T}^{-2}}{{A}^{-1}}]$

• question_answer21) A physical quantity A is related to four observables a, b, c and d as follows: $A=\frac{{{a}^{2}}{{b}^{3}}}{c\sqrt{d}}$ The percentage errors of measurement in a, b, c and d are 1%, 3%, 2% and 2% respectively. What is the percentage error in the quantity A?

A) 12%

B) 7%

C) 5%

D) 16%

E) 14%

• question_answer22) A body starting from rest moves with constant acceleration. The ratio of distance covered by the body during the 5th second to that covered in 5 s is:

A) $\frac{9}{25}$

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

C) $\frac{25}{5}$

D) $\frac{1}{25}$

E) $25$

• question_answer23) The area under acceleration-time graph gives:

A) distance travelled

B) change in acceleration

C) force acting

D) change in velocity

E) work done

• question_answer24) A particle is displaced from a position$(2\hat{i}-\hat{j}+\hat{k})$to another position$(3\hat{i}+2\hat{j}-2\hat{k})$ under the action of the force of$(2\hat{i}+\hat{j}-\hat{k})$. The work done by the force in an arbitrary unit is:

A) 8

B) 10

C) 12

D) 16

E) 20

• question_answer25) From the top of tower, a stone is thrown up. It reaches the ground in${{t}_{1}}$second. A second stone thrown down with the same speed reaches the ground in${{t}_{2}}$second. A third stone released from rest reaches the ground in${{t}_{3}}$second. Then:

A) ${{t}_{3}}=\frac{({{t}_{1}}+{{t}_{2}})}{2}$

B) ${{t}_{3}}=\sqrt{{{t}_{1}}{{t}_{2}}}$

C) $\frac{1}{{{t}_{3}}}=\frac{1}{{{t}_{1}}}-\frac{1}{{{t}_{2}}}$

D) $t_{3}^{2}=t_{2}^{2}-t_{1}^{2}$

E) ${{t}_{3}}=\frac{({{t}_{1}}-{{t}_{2}})}{2}$

• question_answer26) An object is projected at an angle of$45{}^\circ$with the horizontal. The horizontal range and maximum height reached will be in the ratio:

A) $1:2$

B) $2:1$

C) $1:4$

D) $4:1$

E) $4:\sqrt{2}$

• question_answer27) If the length of the seconds hand in a stop-clock is 3 cm, the angular velocity and linear velocity of the tip is:

A) $0.2047\text{ }rad/s,\text{ }0.0314\text{ }m{{s}^{-1}}$

B) $0.2547\text{ }rad/s,\text{ }0.314\text{ }m{{s}^{-1}}$

C) $0.1472\text{ }rad/s,\text{ }0.06314\text{ }m{{s}^{-1}}$

D) $0.1047\text{ }rad/s,\text{ }0.00314\text{ }m{{s}^{-1}}$

E) $0.347\text{ }rad/s,\text{ }0.314\text{ }m{{s}^{-1}}$

• question_answer28) A player caught a cricket ball of mass 150 g moving at the rate of$20\text{ }m{{s}^{-1}}$. If the catching process be completed in 0.1 s, the force of the blow exerted by the ball on the hands of the player is:

A) 0.3 N

B) 30 N

C) 300 N

D) 3000 N

E) 3N

• question_answer29) A uniform metal chain is placed on a rough table such that one end of it hangs down over the edge of the table. When one-third of its length hangs over the edge, the chain starts sliding. Then, the coefficient of static friction is:

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

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

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

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

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

• question_answer30) Two masses M and M/2 are joined together by means of light inextensible string passed over a frictionless pulley as shown in the figure. When the bigger mass is released, the small one will ascend with an acceleration of:

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

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

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

D) g

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

A) both momentum and kinetic energies are conserved

B) both momentum and kinetic energies are not conserved

C) only energy is conserved

D) only mechanical energy is conserved

E) only momentum is conserved

• question_answer32) A ball is released from the top of a tower. The ratio of work done by force of gravity in first, second and third second of the motion of the ball is:

A) $1:2:3$

B) $1:4:9$

C) $1:3:5$

D) $1:5:3$

E) $1:3:2$

• question_answer33) When the kinetic energy of a body is doubled, its momentum increases by ...... times.

A) $\sqrt{2}$

B) $2$

C) $4$

D) $2\sqrt{2}$

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

• question_answer34) Three identical spheres, each of mass 1 kg are kept as shown in figure below, touching each other, with their centres on a straight line. If their centres are marked P, Q, R respectively, the distance of centre of mass of the system from P is:

A) $\frac{PQ+PR+QR}{3}$

B) $\frac{PQ+PR}{3}$

C) $\frac{PQ+QR}{3}$

D) $\frac{PR+QR}{3}$

E) $\frac{PQ+QR+PR}{6}$

• question_answer35) The moment of inertia of a thin rod of mass M and length L, about an axis perpendicular to the rod at a distance $\frac{L}{4}$ from one end is:

A) $\frac{M{{L}^{2}}}{6}$

B) $\frac{M{{L}^{2}}}{12}$

C) $\frac{7M{{L}^{2}}}{24}$

D) $\frac{7M{{L}^{2}}}{12}$

E) $\frac{7M{{L}^{2}}}{48}$

• question_answer36) A body rolls down an inclined plane. If its kinetic energy of rotation is 40% of its kinetic energy of translation, then the body is:

A) solid cylinder

B) solid sphere

C) disc

D) ring

E) hollow cylinder

• question_answer37) Which of the following statements about the gravitational constant is true?

A) It is a force

B) It has no unit

C) It has same value in all systems of units

D) It depends on the value of the masses

E) It does not depend on the nature of the medium in which the bodies are kept

• question_answer38) Four particles each of mass M, are located at the vertices of a square with side L. The gravitational potential due to this at the centre of the square is:

A) $-\sqrt{32}\frac{GM}{L}$

B) $-\sqrt{64}\frac{GM}{{{L}^{2}}}$

C) zero

D) $\sqrt{32}\frac{GM}{L}$

E) $8\frac{GM}{{{L}^{2}}}$

• question_answer39) Two identical solid copper spheres of radius R are placed in contact with each other. The gravitational attraction between them is proportional to:

A) ${{R}^{2}}$

B) ${{R}^{-2}}$

C) ${{R}^{4}}$

D) ${{R}^{-4}}$

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

• question_answer40) The modulus of elasticity is dimensionally equivalent to:

A) strain

B) force

C) stress

D) coefficient of viscosity

E) surface tension

• question_answer41) Radius of an air bubble at the bottom of the lake is r and it becomes 2 r when the air bubble rises to the top surface of the lake. If P cm of, water be the atmospheric pressure, then the depth of the lake is:

A) 2P

B) 8P

C) 4P

D) 7P

E) 5P

• question_answer42) A manometer connected to a closed tap reads$4.5\times {{10}^{5}}Pa$. When the tap is opened the reading of the manometer falls to$4\times {{10}^{5}}Pa$. Then the velocity of flow of water is:

A) $7\text{ }m{{s}^{-1}}$

B) $8\text{ }m{{s}^{-1}}$

C) $9m{{s}^{-1}}$

D) $12m{{s}^{-1}}$

E) $10\text{ }m{{s}^{-1}}$

• question_answer43) What is the velocity v of a metallic ball of radius r falling in a tank of liquid at the instant when its acceleration is one-half that of a freely falling body? (The densities of metal and of liquid are$\rho$and$\sigma$respectively, and the viscosity. of the liquid is$\eta$):

A) $\frac{{{r}^{2}}g}{9\eta }(\rho -2\sigma )$

B) $\frac{{{r}^{2}}g}{9\eta }(2\rho -\sigma )$

C) $\frac{{{r}^{2}}g}{9\eta }(\rho -\sigma )$

D) $\frac{2{{r}^{2}}g}{9\eta }(\rho -\sigma )$

E) $\frac{{{r}^{2}}g}{18\eta }(\rho -2\sigma )$

• question_answer44) A black body has maximum wavelength${{\lambda }_{m}}$at 2000 K. Its corresponding wavelength at 3000 K will be:

A) $\frac{3}{2}{{\lambda }_{m}}$

B) $\frac{2}{3}{{\lambda }_{m}}$

C) $\frac{16}{81}{{\lambda }_{m}}$

D) $\frac{81}{16}{{\lambda }_{m}}$

E) $\frac{4}{3}{{\lambda }_{m}}$

• question_answer45) The value of$\frac{PV}{T}$for one mole of an ideal gas is nearly equal to:

A) $2\,J\,mo{{l}^{-1}}{{K}^{-1}}$

B) $8.3\,mo{{l}^{-1}}{{K}^{-1}}$

C) $4.2\,J\,mo{{l}^{-1}}{{K}^{-1}}$

D) $2cal\,mo{{l}^{-1}}{{K}^{-1}}$

E) $4\,cal\,mo{{l}^{-1}}{{K}^{-1}}$

• question_answer46) The volume of a metal sphere increases by 0.24% when its temperature is raised by$40{}^\circ$ C. The coefficient of linear expansion of the metal is...$/{}^\circ C$.

A) $2\times {{10}^{-5}}$

B) $6\times {{10}^{-5}}$

C) $18\times {{10}^{-5}}$

D) $1.2\times {{10}^{-5}}$

E) $2.1\times {{10}^{-5}}$

• question_answer47) The temperature of equal masses of three different liquids A, B and C are$12{}^\circ C,\text{ }19{}^\circ C$and $28{}^\circ C$respectively. The temperature when A and B are mixed is$16{}^\circ C$and when B and C are mixed is$23{}^\circ C$. The temperature when A and C are mixed is:

A) $18.2{}^\circ C$

B) $22{}^\circ C$

C) $20.2{}^\circ C$

D) $24.2{}^\circ C$

E) $20.8{}^\circ C$

• question_answer48) The time period of the seconds hand of a watch is:

A) $1\,h$

B) $1\,s$

C) $12\,h$

D) $1\text{ }min$

E) $0.1\,h$

• question_answer49) A particle starts SHM from the mean position. Its amplitude is a and total energy E. At one instant its kinetic energy is$3\frac{E}{4}$. Its displacement at that instant is:

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

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

C) $\frac{a}{\sqrt{\left( \frac{3}{2} \right)}}$

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

E) $a$

• question_answer50) A particle executes linear simple harmonic motion with an amplitude of 2 cm. When the particle is at 1 cm from the mean position the magnitude of its velocity is equal to that of its acceleration. Then its time period in second is:

A) $\frac{1}{2\pi \sqrt{3}}$

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

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

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

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

• question_answer51) A closed organ pipe and an open organ pipe are tuned to the same fundamental frequency. The ratio of their lengths is:

A) $1:1$

B) $2:1$

C) $1:4$

D) $1:2$

E) $4:1$

• question_answer52) An observer standing near the sea shore/min. If the wavelength of the water wave is 10 m then the velocity of water wave is:

A) $540\,m{{s}^{-1}}$

B) $5.4m{{s}^{-1}}$

C) $0.184\text{ }m{{s}^{-1}}$

D) $9\text{ }m{{s}^{-1}}$

E) $48.6\text{ }m{{s}^{-1}}$

• question_answer53) A set of 24 tuning forks are so arranged that each gives 6 beats/s with the previous one. If the frequency of the last tuning fork is double that of the first, frequency of the second tuning fork is:

A) 138 Hz

B) 132 Hz

C) 144 Hz

D) 276 Hz

E) 270 Hz

• question_answer54) The electrostatic field due to a charged conductor just outside the conductor is:

A) zero and parallel to the surface at every point inside the conductor

B) zero and is normal to the surface at every point inside the conductor

C) parallel to the surface at every point and zero inside the conductor

D) normal to the surface at every point and zero inside the conductor

E) normal to the surface at every point and non-zero inside the conductor

• question_answer55) A point charge + q is placed at the midpoint of a cube of side a. The electric flux emerging from the cube is:

A) zero

B) $\frac{3q{{a}^{2}}}{{{\varepsilon }_{0}}}$

C) $\frac{q}{{{\varepsilon }_{0}}}$

D) $\frac{{{\varepsilon }_{0}}}{4q{{a}^{2}}}$

E) $\frac{{{\varepsilon }_{0}}}{q}$

• question_answer56) Figure below shows four plates each of area A and separated from one another by a distance d. What is the capacitance between P and Q?

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

B) $\frac{2{{\varepsilon }_{0}}A}{d}$

C) $\frac{3{{\varepsilon }_{0}}A}{d}$

D) $\frac{4{{\varepsilon }_{0}}A}{d}$

E) Zero

• question_answer57) A soap bubble is charged to a potential of 16 V. Its radius is, then doubled. The potential of the bubble now will be:

A) 16V

B) 8V

C) 4V

D) 2V

E) zero

• question_answer58) A parallel plate capacitor of capacitance$10\mu F$is charged to$1\mu C$. The charging battery is removed and then the separation between the plates is doubled. Work done during the process is:

A) $5\text{ }m\text{ }J$

B) $0.05\text{ }m\text{ }J$

C) $\text{1 }m\text{ }J$

D) $10\text{ }m\text{ }J$

E) $\text{50 }m\text{ }J$

• question_answer59) A 10 0 electric heater operates on a 110 V line. The rate at which heat is developed in watts is:

A) 1310 W

B) 670 W

C) 810 W

D) 1210 W

E) 1100 W

• question_answer60) For a certain thermocouple, if the temperature of the cold junction is$0{}^\circ C,$the neutral temperature and inversion temperatures are$285{}^\circ C$and$570{}^\circ C$ respectively. If the cold junction is brought to $10{}^\circ C,$then the new neutral and inversion temperatures are respectively:

A) $285{}^\circ C$ and $560{}^\circ C$

B) $285{}^\circ C$ and$570{}^\circ C$

C) $295{}^\circ C$ and$560{}^\circ C$

D) $275{}^\circ C$ and$560{}^\circ C$

E) $275{}^\circ C$ and$570{}^\circ C$

• question_answer61) In which of the following substances does resistance decrease with increase in temperature?

A) Copper

B) Carbon

C) Constantan

D) Silver

E) Sodium

• question_answer62) Resistors P and Q are connected in the gaps of the meter bridge. The balancing point is obtained$\frac{1}{3}$m from the zero end. If a$6\,\Omega$resistance is connected in series with P the balance point shifts to$\frac{2}{3}$m from the same end. P and Q are:

A) 4, 2

B) 2, 4

C) both (a) and (b)

D) neither (a) nor (b)

E) unpredictable

• question_answer63) The currents ii and 13 through the resistors${{R}_{1}}(=10\,\Omega )$and${{R}_{2}}(=30\,\Omega )$in the circuit -diagram with${{E}_{1}}=3V\,,{{E}_{2}}=3V$and${{E}_{3}}=2V$are respectively:

A) 0.2 A, 0.1 A

B) 0.4 A, 0.2 A

C) 0.1 A, 0.2 A

D) 0.2 A, 0.4 A

E) 0.4 A, 0.1 A

• question_answer64) An$\alpha -$particle with a specific charge of $2.5\times {{10}^{7}}C\,k{{g}^{-1}}$ moves with a speed of$2\times {{10}^{5}}$ $m{{s}^{-1}}$in a perpendicular magnetic field of 0.05 T. Then the radius of the circular path described by it is:

A) 8 cm

B) 4 cm

C) 16cm

D) 2cm

E) 32 cm

• question_answer65) A cyclotron can be used to accelerate:

A) $\alpha -$particles

B) $\beta -$particles

C) neutrons

D) neutrino

E) positron

• question_answer66) The magnitude of the earths magnetic field at a place is${{\beta }_{0}}$and the angle of dip is$\delta$. A horizontal conductor of length$l$lying magnetic north-south moves eastwards with a velocity v. The emf induced across the conductor is:

A) zero

B) ${{B}_{0}}lv\,\sin \delta$

C) ${{B}_{0}}\,lv$

D) ${{B}_{0}}lv\,\cos \delta$

E) ${{B}_{0}}\sin \delta$

• question_answer67) A miiliammeter of range 0 - 30 mA has internal resistance of$20\,\Omega$. The resistance to be connected in series to convert it into a voltmeter of maximum reading 3V is:

A) $49\,\Omega$

B) $80\,\Omega$

C) $40\,\Omega$

D) $30\,\Omega$

E) $50\,\Omega$

• question_answer68) A straight conductor of length I carrying a current 7, is bent in the form of a semicircle. The magnetic field (in tesla) at the centre of the semicircle is:

A) $\frac{{{\pi }^{2}}I}{l}\times {{10}^{-7}}$

B) $\frac{\pi I}{l}\times {{10}^{-7}}$

C) $\frac{\pi I}{{{l}^{2}}}\times {{10}^{-7}}$

D) $\frac{\pi {{I}^{2}}}{l}\times {{10}^{-7}}$

E) $\frac{\pi {{I}^{2}}}{{{l}^{2}}}\times {{10}^{-7}}$

• question_answer69) A coil having an inductance of 0.5 H carries a current which is uniformly varying from 0 to 10 A in 2 s. The emf (in volts) generated in the coil is:

A) 10

B) 5

C) 2.5

D) 1.25

E) 0.25

• question_answer70) If an alternating voltage is represented as E = 141 sin (628 t), then the rms value of the voltage and the frequency are respectively:

A) 141 V, 628 Hz

B) 100 V, 50 Hz

C) 100 V, 100 Hz

D) 141 V, 100 Hz

E) 100 V, 314 Hz

• question_answer71) A step-down transformer is used on a 1000 V line to deliver 20 A at 120 V at the secondary coil. If the efficiency of the transformer is 80%, the current drawn from the line is:

A) 3 A

B) 30 A

C) 0.3 A

D) 2.4 A

E) 24 A

• question_answer72) For the series LCR circuit shown in the figure, what is the resonance frequency and the amplitude of the current at the resonating frequency?

A) $2500\,rad-{{s}^{-1}}and\,5\sqrt{2}A$

B) $2500\text{ }rad-{{s}^{-1}}and\text{ }5\text{ }A$

C) $2500\text{ }rad-{{s}^{-1}}and\,\frac{5}{\sqrt{2}}\text{ }A$

D) $250\,rad-{{s}^{-1}}and\,5\sqrt{2}A$

E) $25\,rad-{{s}^{-1}}and\,5\sqrt{2}A$

• question_answer73) What would be the heat released when an aqueous solution containing 0.5 mole of$HN{{O}_{3}}$is mixed with 0.3 mole of$O{{H}^{-}}$(enthalpy of neutralization is$-57.1\text{ }kJ$)?

A) 28.5 kJ

B) 17.1 kJ

C) 45.7 kJ

D) 1.7 kJ

E) 2.85 kJ

• question_answer74) $A(g)+3B(g)4C(g)$ Initially concentration of A is equal to that of B. The equilibrium concentrations of A and C are equal.${{K}_{c}}$is:

A) 0.08

B) 0.8

C) 8

D) 80

E) $\frac{1}{8}$

• question_answer75) Two moles of$PC{{l}_{5}}$is heated in a closed vessel of 2L capacity. When the equilibrium is attained 40% of it has been found to be dissociated. What is the${{K}_{c}}$in$mol/d{{m}^{3}}$?

A) 0.532

B) 0.266

C) 0.133

D) 0.174

E) 0.25

• question_answer76) Dry air is passed through a solution containing 10 g of a solute in 90 g of water and then through pure water. The loss in weight of solution is 2.5 g and that of pure solvent is 0.05 g. Calculate the molecular weight of the solute.

A) 50

B) 180

C) 102

D) 25

E) 51

• question_answer77) The vant Hoff factor of$BaC{{l}_{2}}$at 0.01 M concentration is 1.98. The percentage of dissociation of$BaC{{l}_{2}}$at this concentration is:

A) 49

B) 69

C) 89

D) 98

E) 100

• question_answer78) The standard electrode potentials of$A{{g}^{+}}/Ag$is$+0.80\text{ }V$and$C{{u}^{+}}/Cu$is$+0.34\text{ }V$. These electrodes are connected through a salt bridge and if:

A) copper electrode acts as a cathode then$E{}^\circ$cell is$+0.46\text{ }V$

B) silver electrode acts as anode then$E{}^\circ$cell is$-0.34\text{ }V$

C) copper electrode acts as anode then$E{}^\circ$cell is$+0.46V$

D) silver electrode acts as a cathode then$E{}^\circ$cell is$-0.34\text{ }V$

E) silver electrode acts as anode and$E{}^\circ$cell is $+1.14V$

• question_answer79) In alkaline medium$Cl{{O}_{2}}$oxidizes${{H}_{2}}{{O}_{2}}$to${{O}_{2}}$ and itself gets reduced to$C{{l}^{-}}$. How many moles ofH^02 are oxidized by 1 mole of$Cl{{O}_{2}}$?

A) 1.0

B) 1.5

C) 2.5

D) 3.5

E) 5.0

• question_answer80) For the reaction $2{{N}_{2}}{{O}_{5}}(g)\xrightarrow[{}]{{}}4N{{O}_{2}}(g)+{{O}_{2}}(g)$ if the concentration of$N{{O}_{2}}$increases by $5.2\times {{10}^{-3}}$M in 100 s then the rate of the reaction is:

A) $1.3\times {{10}^{-5}}M{{s}^{-1}}$

B) $0.5\times {{10}^{-4}}M{{s}^{-1}}$

C) $7.6\times {{10}^{-4}}M{{s}^{-1}}$

D) $2\times {{10}^{-3}}M{{s}^{-1}}$

E) $2.5\times {{10}^{-5}}M{{s}^{-1}}$

• question_answer81) A first order reaction is 10% complete in 20 min. The time taken for 19% completion is:

A) 30 min

B) 40 min

C) 50 min

D) 38 min

E) 45 min

• question_answer82) Lyophilic sols are more stable than lyophobic sols because the particles:

A) are positively charged

B) are negatively charged

C) are solvated

D) repel each other

E) are heavy

• question_answer83) Potassium stearate is obtained by the saponification of an oil or fat. It has the formula$C{{H}_{3}}-{{(C{{H}_{2}})}_{16}}-CO{{O}^{-}}{{K}^{+}}$. The molecule has a lyophobic end$[C{{H}_{3}}]$and a lyophilic end$CO{{O}^{-}}{{K}^{+}}$Potassium stearate is an example for:

A) lyophobic colloid

B) lyophilic colloid

C) multimolecular colloid

D) macromolecular colloid

E) associated colloid or micelle

• question_answer84) (A) ${{K}_{4}}[Fe{{(CN)}_{6}}]$ (B) ${{K}_{3}}[Cr{{(CN)}_{6}}]$ (C) ${{K}_{3}}[Co{{(CN)}_{6}}]$ (D) ${{K}_{2}}[Ni{{(CN)}_{4}}]$ Select the complexes which are diamagnetic:

A) (A), (B) and (C)

B) (B), (C) and (D)

C) (A), (C) and (D)

D) (A), (B) and (D)

E) (B) and (D)

• question_answer85) Which is not true of the co-ordination compound$[Co{{(en)}_{2}}C{{l}_{2}}]Cl$?

A) Exhibits geometrical isomerism

B) Exhibits optical isomerism

C) Exhibits ionization isomerism

D) Is an octahedral complex

E) Is a cationic complex

• question_answer86) The IUPAC name of the compound is: $HOOC-C{{H}_{2}}-\underset{\begin{smallmatrix} | \\ COOH \end{smallmatrix}}{\mathop{CH}}\,-C{{H}_{2}}-C{{H}_{2}}-COOH$

A) 2(carboxymethyl)-pentane-1, 5-dioic acid

B) 3-carboxyhexane-1, 6-dioic acid

C) butane-1, 2, 4-tricarboxylic acid

D) 4-carboxyhexane-1, 6-dioic acid

E) 1, 2-dicarboxy pentanoic acid

• question_answer87) How much of sulphur is present in an organic compound, if 0.53 g of the compound gave 1.158 g of$BaS{{O}_{4}}$on analysis?

A) 10%

B) 15%

C) 20%

D) 25%

E) 30%

• question_answer88) An alkene having the molecular formula ${{C}_{9}}{{H}_{18}}$on ozonolysis gives 2, 2-dimethyl propanal and 2-butanone. The alkene is:

A) 2, 2, 2-trimethyl-3-hexene

B) 2, 2, 6-trimethyl-3-hexane

C) 2, 3, 4-trimethyl-2-hexene

D) 2, 2, 4-trimethyl-3-hexene

E) 2, 2,-dimethyl 1-3 heptene

• question_answer89) Observe the following reactions and predict the nature of A and B:

A) A and B both are

B) A and B both are

C)

D)

E)

• question_answer90) Nitration of aniline in strongly acidic medium, result in the formation of m-nitroaniline also. This is because:

A) amino group is meta orienting during electrophilic substitution reaction

B) nitro group goes always to the meta position irrespective of the substituents

C) nitration of aniline is a nucleophilic substitution reaction in strongly acidic medium

D) in strongly acidic conditions aniline is present as anilinium ion

E) strong acids generate nitrite anion which can attack only the meta position

• question_answer91) How many$\sigma$and$\pi$bonds are present in toluene?

A) $3\pi +\text{ }8\sigma$

B) $3\pi +10\sigma$

C) $3\pi +15\sigma$

D) $6\pi +3\sigma$

E) $6\pi +6\sigma$

• question_answer92) Which of the following Fischers projection formula is identical to D-glyceraldehyde?

A)

B)

C)

D)

E)

• question_answer93) The name of the compound fig. is:

• question_answer94) When 32.25 g of ethyl chloride is subjected to dehydrohalogenation reaction the yield of the alkene formed is 50%. The mass of the product formed is: (atomic mass of chlorine is 35.5)

A) 14 g

B) 28 g

C) 64.5 g

D) 56 g

E) 7g

• question_answer95) Chlorination of toluene in presence of light and heat followed by treatment with aqueous $NaOH$gives:

A) o-cresol

B) p-cresol

C) mixture of o-cresol and p-cresol

D) benzoic acid

E) 1, 3, 5-trihydroxy toluene

• question_answer96) $C{{H}_{3}}-CHO-HCN\xrightarrow{{}}A$. Compound A on hydrolysis gives:

A) $C{{H}_{3}}C{{H}_{2}}COOH$

B) $C{{H}_{3}}\text{ }C{{H}_{2}}C{{H}_{2}}N{{H}_{2}}$

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

D) $C{{H}_{3}}.COCH=NOH$

E) $C{{H}_{3}}\underset{\begin{smallmatrix} | \\ OH \end{smallmatrix}}{\mathop{CH}}\,COOH$

• question_answer97) Which of the following does not undergo Cannizaros reaction?

A) Benzaldehyde

B) 2-methylpropanal

C) p-methoxybenzaldehyde

D) 2, 2-dimethylpropanal

E) Formaldehyde

• question_answer98) Identify the product in the following sequence 3, 4, 5-tribromoanilin$\xrightarrow[(2)\,{{H}_{3}}P{{O}_{2}}]{(1)\,diazotization}$?

A) 3, 4, 5-tribromobenzene

B) 1, 2, 3-tribromobenzene

C) 2, 4, 6-tribromobenzene

D) 3, 4, 5-tribromonitrobenzene

E) 3, 4, 5-tribromophenol

• question_answer99) Among the amines$(A){{C}_{6}}{{H}_{5}}N{{H}_{2}}$$(B)C{{H}_{3}}N{{H}_{2}}$$(C){{(C{{H}_{3}})}_{2}}NH$$(D){{(C{{H}_{3}})}_{3}}N,$the order of basicity is:

A) $A<B<D<C$

B) $D<C<B<A$

C) $A>B>C>D$

D) $B<C<D<A$

E) $D<C<B<A$

• question_answer100) The number average molecular mass and mass average molecular mass of a polymer are respectively 30,000 and 40, 000. The poly dispersity index of the polymer is:

A) < 1

B) > 1

C) 1

D) 0

E) $-1$

• question_answer101) In biological systems, the RNA molecules direct the synthesis of specific proteins which are characteristic of each kind of organism. This process is known is:

A) transcription

B) mutation

C) replication

D) translation

E) flocculation

• question_answer102) Pick up the correct statement:

A) CO which is major pollutant resulting from the combustion of fuels in automobiles plays a major role in photochemical smog

B) Classical smog has an oxidizing character while the photochemical smog is reducing in character

C) Photochemical smog occurs in day time whereas the classical smog occurs in early morning hours

D) During formation of smog the level of ozone in the atmosphere goes down

E) Classical smog is good for health but not photochemical smog

• question_answer103) In Antarctica ozone depletion is due to the formation of following compound :

A) acrolein

B) peroxyacetyl nitrate

C) $S{{O}_{2}}$and$S{{O}_{3}}$

D) chlorine nitrate

E) formaldehyde

• question_answer104) $100\text{ }g\text{ }CaC{{O}_{3}}$is treated with 1 L of $1N\text{ }HCl$. What would be the weight of$C{{O}_{2}}$liberated after the completion of the reaction?

A) 55 g

B) 11 g

C) 22 g

D) 33 g

E) 44 g

• question_answer105) The relationship between the energy${{E}_{1}}$of the radiation with a wavelength $8000\overset{\text{o}}{\mathop{\text{A}}}\,$ and the energy${{E}_{2}}$of the radiation with a wavelength $16000\overset{\text{o}}{\mathop{\text{A}}}\,$ is:

A) ${{E}_{1}}=6{{E}_{2}}$

B) ${{E}_{1}}=2{{E}_{2}}$

C) ${{E}_{1}}=4{{E}_{2}}$

D) ${{E}_{1}}=1/2{{E}_{2}}$

E) ${{E}_{1}}={{E}_{2}}$

• question_answer106) If the molecule of$HCl$were totally polar, the expected value of dipole moment is 6.12 D (debye), but the experimental value of dipole moment was 1.03 D. Calculate the percentage ionic character:

A) 17

B) 83

C) 50

D) zero

E) 90

• question_answer107) Which one of the following molecules has the smallest bond angle?

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

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

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

D) ${{H}_{2}}Se$

E) ${{H}_{2}}S$

• question_answer108) If the absolute temperature of a gas is doubled and the pressure is reduced to one half, the volume of the gas will:

A) remain unchanged

B) be doubled

C) increase four fold

D) be halved

E) be reduced to one-fourth

• question_answer109) To what temperature must a neon gas sample be heated to double its pressure, if the initial volume of gas at$75{}^\circ C$is decreased by 15.0%?

A) $319{}^\circ C$

B) $592{}^\circ C$

C) $128{}^\circ C$

D) $60{}^\circ C$

E) $90{}^\circ C$

• question_answer110) When electric current is passed through an ionic hydride in molten state:

A) hydrogen is liberated at anode

B) hydrogen is liberated at cathode

C) no change takes place

D) hydride for migrates towards cathode

E) hydride ion remains in solution

• question_answer111) The order of first ionization energies of the elements$Li,Be,B,Na$is:

A) $Li>Be>B>Na$

B) $Be>B>Li>Na$

C) $Na>Li>B>Be$

D) $Be>Li>B>Na$

E) $B>Be>Li>Na$

• question_answer112) The froth-floatation process is based upon:

A) the difference in the specific gravity of ore and gangue particles

B) the magnetic properties of gangue and ore

C) preferential wetting of gangue particles by oil

D) preferential wetting of ore particles by oil

E) the solubility of ore particles in suitable reagent

• question_answer113) Which one of the following statements is true for all the alkali metals?

A) Their nitrates decompose on heating to give$N{{O}_{2}}$and${{O}_{2}}$

B) Their carbonates decompose on heating to give$C{{O}_{2}}$and the metal oxide

C) They react with oxygen to give mainly the oxide${{M}_{2}}O$

D) They react with halogens to give the halides MX

E) They react with nitrogen to give nitrides

• question_answer114) The order of acidic strength of boron trihalides is:

A) $B{{F}_{3}}<BC{{l}_{3}}<BB{{r}_{3}}<B{{I}_{3}}$

B) $B{{l}_{3}}<BB{{r}_{3}}<BC{{l}_{3}}<B{{F}_{3}}$

C) $BC{{l}_{3}}<BB{{r}_{3}}<B{{l}_{3}}<B{{F}_{3}}$

D) $BB{{r}_{3}}<BC{{l}_{3}}<B{{F}_{3}}<B{{I}_{3}}$

E) $B{{F}_{3}}<B{{I}_{3}}<BC{{I}_{3}}<BB{{r}_{3}}$

• question_answer115) The carbide which reacts with water to form ethyne is:

A) $Ca{{C}_{2}}$

B) $SiC$

C) $M{{g}_{2}}{{C}_{3}}$

D) $A{{l}_{4}}{{C}_{3}}$

E) $B{{e}_{2}}C$

• question_answer116) Effective magnetic moment of$S{{c}^{\text{3}+}}$ion is:

A) 1.73

B) 0

C) 5.92

D) 2.83

E) 3.87

• question_answer117) Potassium permanganate acts as an oxidant in alkaline and acidic media. The final products formed from $KMn{{O}_{4}}$ in the two conditions are respectively:

A) $Mn{{O}^{2-}}$and $M{{n}^{3+}}$

B) $M{{n}^{3+}}$and $M{{n}^{2+}}$

C) $M{{n}^{2+}}$and $M{{n}^{3+}}$

D) $Mn{{O}_{2}}$and $M{{n}^{2+}}$

E) $M{{n}^{2+}},Mn{{O}_{2}}$

• question_answer118) Calculate the mass loss in the following. $_{1}^{2}H+_{1}^{3}H\xrightarrow[{}]{{}}_{2}^{4}He+_{0}^{1}n$ [Given the masses: $^{2}H=2.014{{;}^{3}}H=3.016;$ $He=4.004;\text{ }n=1.008amu$]

A) 0.018 amu

B) 0.18 amu

C) 0.0018 amu

D) 1.8 amu

E) 18 amu

• question_answer119) Match list I and list II and select the correct answer using the code given below the lists:

 List-I Nuclear Reactor components List-II Substance used 1. Moderator A. Uranium 2. Control rods B. Graphite 3. Fuel rods C. Boron 4. Coolant D. Lead E. Sodium
Codes:

A) 1-B, 2-A, 3-C, 4-E

B) 1-B, 2-C, 3-A, 4-E

C) 1-E, 2-B, 3-A, 4-C

D) 1-C, 2-D, 3-A, 4-B

E) 1-D, 2-C, 3-B, 4-A

• question_answer120) $\Delta H$and$\Delta S$for a reaction are$+30.558\text{ }kJ$ $mo{{l}^{-1}}$and$0.066\,kJ\,{{K}^{-1}}mo{{l}^{-1}}$at 1 aim pressure. The temperature at which free energy change is equal to zero and the nature of the reaction below this temperature are:

A) 483 K, spontaneous

B) 443 K, non-spontaneous

C) 443 K, spontaneous

D) 463 K, non-spontaneous

E) 463 K, spontaneous

• question_answer121) $\sum\limits_{r=0}^{m}{^{n+r}{{C}_{n}}}$is equal to:

A) $^{n+m+1}{{C}_{n+1}}$

B) $^{n+m+2}{{C}_{n}}$

C) $^{n+m+3}{{C}_{n-1}}$

D) 0

E) none of these

• question_answer122) If$P(n,r)=1680$and$C(n,r)=70,$then $69n+r!$is equal to:

A) 128

B) 576

C) 256

D) 625

E) 1152

• question_answer123) $\left( 1+\frac{{{C}_{1}}}{{{C}_{0}}} \right)\left( 1+\frac{{{C}_{2}}}{{{C}_{1}}} \right)\left( 1+\frac{{{C}_{3}}}{{{C}_{2}}} \right).....\left( 1+\frac{{{C}_{n}}}{{{C}_{n-1}}} \right)$is equal to:

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

B) $\frac{{{(n+1)}^{n}}}{(n-1)!}$

C) $\frac{{{(n-1)}^{n}}}{n!}$

D) $\frac{{{(n+1)}^{n}}}{n!}$

E) $\frac{n-1}{n!}$

• question_answer124) In the expansion of${{(1+3x+2{{x}^{2}})}^{6}}$the coefficient of ${{x}^{11}}$ is:

A) 144

B) 288

C) 216

D) 576

E) $({{2}^{11}})(3)$

• question_answer125) ${{10}^{n}}+3({{4}^{n+2}})+5$is divisible by$(n\in N)$:

A) 7

B) 5

C) 9

D) 17

E) 13

• question_answer126) $f(x)=$ $\left| \begin{matrix} 1 & x & x+1 \\ 2x & x(x-1) & (x+1)x \\ 3x(x-1) & x(x-1)(x-2) & (x+1)x(x-1) \\ \end{matrix} \right|$ then$f(100)$is equal to:

A) 0

B) 1

C) 100

D) $-100$

E) $-1$

• question_answer127) If$p{{\lambda }^{4}}+q{{\lambda }^{3}}+r{{\lambda }^{2}}+s\lambda +t=$ $\left| \begin{matrix} {{\lambda }^{2}}+3\lambda & \lambda -1 & \lambda +3 \\ \lambda +1 & 2-\lambda & \lambda -3 \\ \lambda -3 & \lambda +4 & 3\lambda \\ \end{matrix} \right|$ then$t$is equal to:

A) 33

B) 22

C) 21

D) $-33$

E) 54

• question_answer128) Let A be a square matrix of order 3.If $|A|=-2,$then the value of determinant of $|A|adj\,A$is:

A) 8

B) $-8$

C) $-1$

D) 32

E) $-32$

• question_answer129) If$f(x)=\left| \begin{matrix} x-3 & 2{{x}^{2}}-18 & 3{{x}^{3}}-81 \\ x-5 & 2{{x}^{2}}-50 & 4{{x}^{3}}-500 \\ 1 & 2 & 3 \\ \end{matrix} \right|,$then$f(1).f(3)+f(3).f(5)+f(5).f(1)$is equal to:

A) $f(1)$

B) $f(3)$

C) $f(1)+f(3)$

D) $f(1)+f(5)$

E) $f(1)+f(3)+f(5)$

• question_answer130) If$x=(\beta -\gamma )(\alpha -\delta ),y=(\gamma -\alpha )(\beta -\delta ),$ $z=(\alpha -\beta )(\gamma -\delta ),$then the value of${{x}^{3}}+{{y}^{3}}+{{z}^{3}}-3xyz$is:

A) 0

B) ${{\alpha }^{6}}+{{\beta }^{6}}+{{\gamma }^{6}}+{{\delta }^{6}}$

C) ${{\alpha }^{6}}{{\beta }^{6}}{{\gamma }^{6}}{{\delta }^{6}}$

D) 1

E) none of these

• question_answer131) If $A=\left[ \begin{matrix} 1 & -1 & 1 \\ 0 & 2 & -3 \\ 2 & 1 & 0 \\ \end{matrix} \right]$and$B=(adj\text{ }A),$and $C=5A,$then $\frac{|adj\,B|}{|C|}$is equal to:

A) 5

B) 25

C) $-1$

D) 1

E) 125

• question_answer132) The output s as a Boolean expression in the inputs${{x}_{1}},{{x}_{2}}$and${{x}_{3}}$for the logic circuit in the following figure is:

A) ${{x}_{1}},x_{2}^{}+x_{2}^{}+{{x}_{3}}$

B) ${{x}_{1}}+x_{2}^{}{{x}_{3}}+{{x}_{3}}$

C) $({{x}_{1}}{{x}_{2}})+{{x}_{1}}x_{2}^{}{{x}_{3}}$

D) ${{x}_{1}}+x{{}_{2}}+{{x}_{3}}$

E) ${{x}_{1}}x{{}_{2}}+x{{}_{2}}{{x}_{3}}$

• question_answer133) The. feasible region for the following constraints${{L}_{1}}\ge 0,{{L}_{2}}\ge 0,{{L}_{3}}=0,x\ge 0,y\ge 0$in the diagram shown is:

A) area DHF

B) area AHC

C) line segment EG

D) line segment GI

E) line segment 1C

• question_answer134) Let${{D}_{70}}=\{1,2,5,7,10,14,35,70\}$. Define$+,$$.,$ and$,,$by$a+b=1\,cm(a,b),a.b-\gcd (a,b)$and $a=\frac{70}{a}$for all$a,b\in {{D}_{70}}$.The value of$(2+7)$$(14.10)$is:

A) 7

B) 14

C) 35

D) 5

E) 1

• question_answer135) Let a be any element in a Boolean Algebra B. If$a+x=1$and$ax=0,$then:

A) $x=1$

B) $x=0$

C) $x=a$

D) $x=a$

E) $x=a+a$

• question_answer136) If the sides of the triangle are $p,q,\sqrt{{{p}^{2}}+{{q}^{2}}+pq},$then the greatest angle is:

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

B) $\frac{5\pi }{4}$

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

D) $\frac{7\pi }{4}$

E) $\frac{5\pi }{3}$

• question_answer137) then 5 is equal to:

A) $x.(y+z)$

B) $x.(y+z)$

C) $x.(y+z)$

D) $(x+y).z$

E) $x.y+z$

• question_answer138) If$\sin \left( {{\sin }^{-1}}\frac{1}{5}+{{\cos }^{-1}}x \right)=1,$then the value of$x$is:

A) $-1$

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

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

D) $1$

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

• question_answer139) $\sin \left[ 3{{\sin }^{-1}}\left( \frac{1}{5} \right) \right]$is equal to:

A) $\frac{71}{125}$

B) $\frac{74}{125}$

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

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

E) $\frac{-3}{5}$

• question_answer140) $\frac{\cos 9{}^\circ +\sin 9{}^\circ }{\cos 9{}^\circ -\sin 9{}^\circ }$ is equal to:

A) $tan\text{ }26{}^\circ$

B) $tan\text{ 81}{}^\circ$

C) $tan\text{ 51}{}^\circ$

D) $tan\text{ 54}{}^\circ$

E) $tan\text{ 46}{}^\circ$

• question_answer141) ${{\cos }^{-1}}\left( \frac{3+5\cos x}{5+3\cos x} \right)$is equal to:

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

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

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

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

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

• question_answer142) If$\Delta ={{a}^{2}}-{{(b-c)}^{2}}$where$\Delta$is the area of triangle ABC, then$tan\Delta$is equal to:

A) $\frac{15}{16}$

B) $\frac{8}{17}$

C) $\frac{8}{15}$

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

E) $\frac{11}{15}$

• question_answer143) If$0<\phi <\frac{\pi }{2},x=\sum\limits_{n=0}^{\infty }{{{\cos }^{2n}}\phi },y=\sum\limits_{n=0}^{\infty }{{{\sin }^{2n}}\phi }$and $z=\sum\limits_{n=0}^{\infty }{{{\cos }^{2n}}\phi }{{\sin }^{2}}n\phi ,$then:

A) $xyz=xz+y$

B) $xyz=xy+z$

C) $xyz=x+y+z$

D) $xyz=yz+x$

E) $xyz=x+yz$

• question_answer144) If in a triangle $ABC,a=5,b=4,A=\frac{\pi }{2}+B,$ then C:

A) is${{\tan }^{-1}}\left( \frac{1}{9} \right)$

B) is ${{\tan }^{-1}}\left( \frac{9}{40} \right)$

C) cannot be evaluated

D) is$2{{\tan }^{-1}}\left( \frac{1}{9} \right)$

E) is$2{{\tan }^{-1}}\left( \frac{1}{40} \right)$

• question_answer145) ABC is a right angled isosceles triangle with$\angle B=90{}^\circ$. If D is a point on AB so that$\angle CDB=15{}^\circ$and, if$AD=35\text{ }cm,$then CD is equal to:

A) $35\sqrt{2}cm$

B) $70\sqrt{2}cm$

C) $\frac{35\sqrt{3}}{2}cm$

D) $35\sqrt{6}cm$

E) $\frac{35\sqrt{2}}{2}cm$

• question_answer146) If$\angle A=90{}^\circ$in the triangle ABC, then ${{\tan }^{-1}}\left( \frac{c}{a+b} \right)+{{\tan }^{-1}}\left( \frac{b}{a+c} \right)$is equal to:

A) 0

B) 1

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

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

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

• question_answer147) The shadow of a tower is found to be 60 m shorter when the suns altitude changes from $30{}^\circ$to$60{}^\circ$. The height of the tower from the ground is approximately equal to:

A) 62 m

B) 301 m

C) 101 m

D) 75 m

E) 52 m

• question_answer148) ABCD is a rectangular field. A vertical lamp post of height 12m stands at the corner A. If the angle of elevation of its top from B is$60{}^\circ$ and from C is$45{}^\circ$, then the area of the field is:

A) $48\sqrt{2}\,sq\,m$

B) $48\sqrt{3}\,sq\,m$

C) $48\,sq\,m$

D) $12\sqrt{2}\,sq\,m$

E) $12\sqrt{3}\,sq\,m$

• question_answer149) If the points$(k,3),(2,k),(-k,3)$are collinear, then the values of k are:

A) 2, 3

B) 1, 0

C) 1, 2

D) $1,-1/2$

E) 0, 3

• question_answer150) If$A(3,5),\text{ }B(-5,-4),C(7,10)$are the vertices of a parallelogram, taken in the order, then the co-ordinates of the fourth vertex are:

A) (10, 19)

B) (15, 10)

C) (19, 10)

D) (19, 15)

E) (15, 19)

• question_answer151) ABC is a triangle with vertices$A(-1,4),$ $B(6,-2)$and$C(-2,4)$. D, E and F are the points which divide each AB, BC and CA respectively in the ratio$3:1$internally. Then, the centroid of me triangle DEF is:

A) (3, 6)

B) (1, 2)

C) (4, 8)

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

E) $(-1,2)$

• question_answer152) If the pairs of lines${{x}^{2}}-2nxy-{{y}^{2}}=0$and ${{x}^{2}}-2mxy-{{y}^{2}}=0$are such that one of them represents the bisectors of the angles between the other, then:

A) $\frac{1}{n}+\frac{1}{m}=0$

B) $\frac{1}{n}-\frac{1}{m}=0$

C) $nm-1=0$

D) $nm+1=0$

E) $\frac{1}{m}-\frac{1}{n}=0$

• question_answer153) The angle between the pair of straight lines${{y}^{2}}{{\sin }^{2}}\theta -xy{{\sin }^{2}}\theta +{{x}^{2}}({{\cos }^{2}}\theta -1)=0$is:

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

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

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

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

E) $\pi$

• question_answer154) If the equation of base of an equilateral triangle is $2x-y=1$and the vertex is$(-1,2),$then the length of the side of the triangle is:

A) $\frac{2}{\sqrt{15}}$

B) $\sqrt{\frac{20}{3}}$

C) $\sqrt{\frac{8}{15}}$

D) $\sqrt{\frac{15}{2}}$

E) $\sqrt{2}$

• question_answer155) The image of the origin with reference to the line$4x+3y-25=0$is:

A) $(-8,6)$

B) (8, 6)

C) $(-3,4)$

D) $(8,-6)$

E) $(-4,-3)$

• question_answer156) The lines$2x-3y=5$and$3x-4y=7$are diameters of a circle having area as 154 sq unit. Then, the equation of the circle is:

A) ${{x}^{2}}+{{y}^{2}}+2x-2y=62$

B) ${{x}^{2}}+{{y}^{2}}+2x-2y=47$

C) ${{x}^{2}}+{{y}^{2}}-2x+2y=47$

D) ${{x}^{2}}+{{y}^{2}}-2x+2y=62$

E) ${{x}^{2}}+{{y}^{2}}-2x-2y=47$

• question_answer157) A circle is drawn to cut a chord of length 2a unit along x-axis and to touch they-axis. The locus of the centre of the circle is:

A) ${{x}^{2}}+{{y}^{2}}={{a}^{2}}$

B) ${{x}^{2}}-{{y}^{2}}={{a}^{2}}$

C) $x+y={{a}^{2}}$

D) ${{x}^{2}}-{{y}^{2}}=4{{a}^{2}}$

E) ${{x}^{2}}+{{y}^{2}}=4{{a}^{2}}$

• question_answer158) If the equation of the tangent to the circle ${{x}^{2}}+{{y}^{2}}-2x+6y-6=0$parallel to $3x-4y+7=0$is$3x-4y+k=0,$then the values of k are:

A) $5,-35$

B) $-5,35$

C) $7,-32$

D) $-7,32$

E) $3,-13$

• question_answer159) The locus of a point which moves so that the ratio of the length of the tangents to the circles${{x}^{2}}+{{y}^{2}}+4x+3=0$and${{x}^{2}}+{{y}^{2}}-6x+5=0$is$2:3,$is:

A) $5{{x}^{2}}+5{{y}^{2}}-60x+7=0$

B) $5{{x}^{2}}+5{{y}^{2}}+60x-7=0$

C) $5{{x}^{2}}+5{{y}^{2}}-60x-7=0$

D) $5{{x}^{2}}+5{{y}^{2}}+60x+7=0$

E) $5{{x}^{2}}+5{{y}^{2}}+60x+12=0$

• question_answer160) The foci of Ac ellipse$\frac{{{x}^{2}}}{16}+\frac{{{y}^{2}}}{{{b}^{2}}}=1$and the hyperbola$\frac{{{x}^{2}}}{144}-\frac{{{y}^{2}}}{81}=\frac{1}{25}$coincide. Then, the value of${{b}^{2}}$is:

A) 1

B) 5

C) 7

D) 9

E) 36

• question_answer161) The eccentricity of the ellipse$25{{x}^{2}}+16{{y}^{2}}-150x-175=0$is:

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

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

C) $\frac{4}{5}$

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

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

• question_answer162) Suppose S and S are foci of the ellipse$\frac{{{x}^{2}}}{25}+\frac{{{y}^{2}}}{16}=1.$If p is a variable point on the ellipse and if $\Delta$ is area of the triangle PSS then the maximum value of$\Delta$is:

A) 8

B) 12

C) 16

D) 20

E) 24

• question_answer163) The equation of the hyperbola in the standard from (whit transverse axis along the$x-$axis) having the length of the latus rectum = 9 unit and eccentricity$=\frac{5}{4}$is:

A) $\frac{{{x}^{2}}}{16}-\frac{{{y}^{2}}}{18}=1$

B) $\frac{{{x}^{2}}}{36}-\frac{{{y}^{2}}}{27}=1$

C) $\frac{{{x}^{2}}}{64}-\frac{{{y}^{2}}}{36}=1$

D) $\frac{{{x}^{2}}}{36}-\frac{{{y}^{2}}}{64}=1$

E) $\frac{{{x}^{2}}}{16}-\frac{{{y}^{2}}}{9}=1$

• question_answer164) If $|\overrightarrow{a}|=|\overrightarrow{b}|=1$and$|\overrightarrow{a}+\overrightarrow{b}|=\sqrt{3},$then the value of $(3\overrightarrow{a}-4\overrightarrow{b}).(2\overrightarrow{a}+5\overrightarrow{b})$is:

A) $-21$

B) $-\frac{21}{2}$

C) 21

D) $\frac{21}{2}$

E) $\frac{59}{2}$

• question_answer165) If$|\overrightarrow{a}|=3,|\overrightarrow{b}|=4,|\overrightarrow{c}|=5$and$\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$are such that each is perpendicular to the sum of other two, then$|\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}|$is:

A) $5\sqrt{2}$

B) $\frac{5}{\sqrt{2}}$

C) $10\sqrt{2}$

D) $10\sqrt{3}$

E) $5\sqrt{3}$

• question_answer166) A unit vector in the plane of$\hat{i}+2\hat{j}+\hat{k}$and$\hat{i}+\hat{j}+2\hat{k}$hand perpendicular to$2\hat{i}+\hat{j}+\hat{k}$is:

A) $\hat{j}-\hat{k}$

B) $\frac{\hat{i}+\hat{j}}{\sqrt{2}}$

C) $\frac{\hat{j}+\hat{k}}{\sqrt{2}}$

D) $\frac{\hat{j}-\hat{k}}{\sqrt{2}}$

E) $5(\hat{j}-\hat{k})$

• question_answer167) If$\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$are unit coplanar vectors, then$[2\overrightarrow{a}-\overrightarrow{b},2\overrightarrow{b}-\overrightarrow{c},2\overrightarrow{c}-\overrightarrow{a}]$is equal to

A) $1$

B) $0$

C) $-\sqrt{3}$

D) $\sqrt{3}$

E) $6$

• question_answer168) If$\overrightarrow{u},\overrightarrow{v},\overrightarrow{w}$be vectors such that$\overrightarrow{u}+\overrightarrow{v}+\overrightarrow{w}=\overrightarrow{0},$ and$|\overrightarrow{u}|=3,|\overrightarrow{v}|=4,|\overrightarrow{w}|=5,$then $\overrightarrow{u}.\overrightarrow{v}+\overrightarrow{v}.\overrightarrow{w}+\overrightarrow{w}.\overrightarrow{u}$equal to:

A) 47

B) $-47$

C) 0

D) 25

E) $-25$

• question_answer169) If $\vec{a}$ is perpendicular to$\overrightarrow{b}$and$\overrightarrow{c},|\overrightarrow{a}|=2,$$|\overrightarrow{b}|=3|\overrightarrow{c}|=4$and the angle between$\overrightarrow{b}$and$\overrightarrow{c}$is$\frac{2\pi }{3},$then$[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}]$is equal to:

A) $4\sqrt{3}$

B) $6\sqrt{3}$

C) $12\sqrt{3}$

D) $18\sqrt{3}$

E) $8\sqrt{3}$

• question_answer170) If$\overrightarrow{a},\overrightarrow{b}$and$\overrightarrow{c}$are perpendicular to$\overrightarrow{b}+\overrightarrow{c},\overrightarrow{c}+\overrightarrow{a}$ and$\overrightarrow{a}+\overrightarrow{b}$respectively and, if $|\overrightarrow{a}+\overrightarrow{b}|=6,|\overrightarrow{b}+\overrightarrow{c}|=8$and$|\overrightarrow{c}+\overrightarrow{a}|=10,$then $|\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}|$is equal to:

A) $5\sqrt{2}$

B) 50

C) $10\sqrt{2}$

D) 10

E) 20

• question_answer171) If (2, 3, 5) is one end of a diameter of the sphere${{x}^{2}}+{{y}^{2}}+{{z}^{2}}-6x-12y-2z+20=0,$then co-ordinates of the other end of the diameter are:

A) (4, 3, 5)

B) $(4,9,-3)$

C) (4, 9, 3)

D) $(4,3,-3)$

E) (4, 9, 5)

• question_answer172) The equation of the plane through the point $(2,-1,-3)$and parallel to the lines$\frac{x-1}{3}=\frac{y+2}{2}=\frac{z}{-4}$and$\frac{x}{2}=\frac{y-1}{-3}=\frac{z-2}{2}$is:

A) $8x+14y+13z+37=0$

B) $8x-14y+13z+37=0$

C) $8x+14y-13z+37=0$

D) $8x+14y+13z-37=0$

E) $8x-14y-13z-37=0$

• question_answer173) If a line makes angles$\alpha ,\beta ,\gamma$with the co-ordinate axes, then$cos\text{ }2\alpha +cos\text{ }2\beta +cos\text{ }2\gamma$is:

A) 3

B) $-2$

C) 2

D) $-3$

E) $-1$

• question_answer174) If for a plane, the intercepts on the co-ordinate axes are 8,4,4, then the length of the perpendicular from the origin on to the plane is:

A) $\frac{8}{3}$

B) $\frac{3}{8}$

C) $3$

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

E) $\frac{4}{5}$

• question_answer175) The equation of the sphere concentric with the sphere$2{{x}^{2}}+2{{y}^{2}}+2{{z}^{2}}-6x+2y-4z=1$ and double its radius is:

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

B) ${{x}^{2}}+{{y}^{2}}+{{z}^{2}}-6x+2y-4z=1$

C) $2{{x}^{2}}+2{{y}^{2}}+2{{z}^{2}}-6x+2y-4z-15=0$

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

E) $2{{x}^{2}}+2{{y}^{2}}+2{{z}^{2}}-6x+2y-4z-25=0$

• question_answer176) If a plane meets the co-ordinate axes at A, B and C such that the centroid of the triangle is (1, 2, 4), then the equation of the plane is:

A) $x+2y+4z=12$

B) $4x+2y+z=12$

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

D) $4x+2y+z=3$

E) $x+y+z=12$

• question_answer177) The position vector of the point where the line$\overrightarrow{r}=\hat{i}-\hat{j}+\hat{k}+t(\hat{i}+\hat{j}-\hat{k})$meets the plane $\overrightarrow{r}.(\hat{i}+\hat{j}+\hat{k})=5$is:

A) $5\hat{i}+\hat{j}-\hat{k}$

B) $5\hat{i}+3\hat{j}-3\hat{k}$

C) $2\hat{i}+\hat{j}+2\hat{k}$

D) $5\hat{i}+\hat{j}+\hat{k}$

E) $4\hat{i}+2\hat{j}-2\hat{k}$

• question_answer178) If the distance of the point (1, 1, 1) from the origin is half its distance from the plane $x+y+z+k=0,$then k is equal to:

A) $\pm \,3$

B) $\pm 6$

C) $-3,9$

D) $3,-9$

E) 3, 9

• question_answer179) Two persons A and B throw a die alternately till one of them gets a 3 and wins the game, the respective probabilities of winning, if A begins, are:

A) $\frac{7}{11},\frac{4}{11}$

B) $\frac{6}{11},\frac{5}{11}$

C) $\frac{5}{6},\frac{1}{6}$

D) $\frac{4}{7},\frac{3}{7}$

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

• question_answer180) If three natural numbers from 1 to 100 are selected randomly, then probability that all are divisible by both 2 and 3, is:

A) $\frac{4}{105}$

B) $\frac{4}{33}$

C) $\frac{4}{35}$

D) $\frac{4}{1155}$

E) $\frac{3}{1155}$

• question_answer181) The average monthly salary of workers in a factory is Rs. 206. If the average monthly salary of males and females are Rs. 210 and Rs. 190 respectively, the percentage of female employed in the factory is:

A) 10

B) 50

C) 30

D) 40

E) 20

• question_answer182) 5 boys and 5 girls are sitting in a row randomly. The probability that boys and girls sit alternatively is:

A) $\frac{5}{126}$

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

C) $\frac{4}{126}$

D) $\frac{6}{126}$

E) $\frac{1}{63}$

• question_answer183) The function$f$satisfies the functional equation$3f(x)+2f\left( \frac{x+59}{x-1} \right)=10x+30$for all real$x\ne 1,$The value of$f(7)$is:

A) 8

B) 4

C) $-8$

D) 11

E) 44

• question_answer184) The value of$f$at$x=0$so that function$f(x)=\frac{{{2}^{x}}-{{2}^{-x}}}{x},x\ne 0,$is continuous at$x=0,$is:

A) 0

B) log 2

C) 4

D) ${{e}^{4}}$

E) log 4

• question_answer185) The domain of ${{\sin }^{-1}}\left( {{\log }_{3}}x \right)$is:

A) $[-1,1]$

B) $[0,1]$

C) $[0,\infty ]$

D) $R$

E) $\left[ \frac{1}{3},3 \right]$

• question_answer186) $\underset{x\to 0}{\mathop{\lim }}\,{{\left\{ \frac{1+\tan x}{1+\sin x} \right\}}^{\cos ecx}}$is equal to:

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

B) 1

C) e

D) ${{e}^{2}}$

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

• question_answer187) If $y={{\sec }^{-1}}\frac{x+1}{x-1}+{{\sin }^{-1}}\frac{x-1}{x+1}$,then$\frac{dy}{dx}$is:

A) $1$

B) $0$

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

D) $\frac{x+1}{x-1}$

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

• question_answer188) If$y={{a}^{x}}.{{b}^{2x-1}},$then$\frac{{{d}^{2}}y}{d{{x}^{2}}}$is:

A) ${{y}^{2}}.\log a{{b}^{2}}$

B) $y.\log a{{b}^{2}}$

C) ${{y}^{2}}$

D) $y.{{(\log {{a}^{2}}b)}^{2}}$

E) $y.{{(\log a{{b}^{2}})}^{2}}$

• question_answer189) The value of$\frac{d}{dx}\left[ {{\tan }^{-1}}\left( \frac{\sqrt{x}(3-x)}{1-3x} \right) \right]$

A) $\frac{1}{2(1+x)\sqrt{x}}$

B) $\frac{3}{(1+x)\sqrt{x}}$

C) $\frac{2}{(1+x)\sqrt{x}}$

D) $\frac{3}{2(1-x)\sqrt{x}}$

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

• question_answer190) The derivative of$y=(1-x)(2-x)...(n-x)$at $x=1$is equal to:

A) $0$

B) $(-1)(n-1)!$

C) $n!-1$

D) ${{(-1)}^{n-1}}(n-1)!$

E) ${{(-1)}^{n}}(n-1)!$

• question_answer191) Let$f(x+y)=f(x)f(y)$and$f(x)=1+\sin (3x)g(x),$where$g(x)$is continuous, then$f(x)$is:

A) $f(x)g(0)$

B) $3g(0)$

C) $f(x)\cos 3x$

D) $3f(x)g(0)$

E) $3f(x)g(x)$

• question_answer192) If$sin\text{ }y=x\text{ }sin(a+y),$then$\frac{dy}{dx}$is:

A) $\sin (a+y)$

B) ${{\sin }^{2}}(a+y)$

C) $\frac{{{\sin }^{2}}(a+y)}{\sin a}$

D) $\frac{\sin (a+y)}{\sin a}$

E) $\cos (a+y)$

• question_answer193) If$y={{\tan }^{-1}}\left( \frac{a\cos x-b\sin x}{b\cos x+a\sin x} \right),$then$\frac{dy}{dx}$is equal to:

A) 2

B) $-1$

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

D) 0

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

• question_answer194) Let$f$be continuous on [1, 5] and differentiable in (1, 5). If$f(1)=-3$and$f(x)\ge 9$for all$x\in (1,5)$then:

A) $f(5)\ge 33$

B) $f(5)\ge 36$

C) $f(5)\le 36$

D) $f(5)\ge 9$

E) $f(5)\le 9$

• question_answer195) If$4{{x}^{2}}+p{{y}^{2}}=45$and${{x}^{2}}-4{{y}^{2}}=5$cut orthogonally, then the value of p is:

A) $\frac{1}{9}$

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

C) 3

D) 18

E) 9

• question_answer196) If a particle moves such that the displacement is proportional to the square of the velocity acquired, then its acceleration is:

A) proportional to${{s}^{2}}$

B) proportional to $\frac{1}{{{s}^{2}}}$.

C) proportional to $s$.

D) proportional to$\frac{1}{s}$.

E) a constant

• question_answer197) The maximum value of$xy$when$x+2y=8$is:

A) 20

B) 16

C) 24

D) 8

E) 4

• question_answer198) The function$f(x)={{\tan }^{-1}}(\sin x+\cos x),$$x>0$is always an increasing function on the interval:

A) $(0,\pi )$

B) $\left( 0,\frac{\pi }{2} \right)$

C) $\left( 0,\frac{\pi }{4} \right)$

D) $\left( 0,\frac{3\pi }{4} \right)$

E) $\left( 0,\frac{5\pi }{4} \right)$

• question_answer199) The radius of a cylinder is increasing at the rate of 3 m/s and its altitude is decreasing at the rate of 4 m/s. The rate of change of volume when radius is 4 m and altitude is 6 m, is:

A) $80\text{ }\pi \text{ }cu\text{ }m/s$

B) $\text{144 }\pi \text{ }cu\text{ }m/s$

C) 80 cu m/s

D) 64 cu m/s

E) $-\text{ }80\text{ }\pi \text{ }cu\text{ }m/s$

• question_answer200) A ladder 10 m long rests against a vertical wall with the lower end on the horizontal ground. The lower end of the ladder is pulled along the ground away from the wall at the rate of 3 cm/s. The height of the upper end while it is descending at the rate of 4 cm/s, is:

A) $4\sqrt{3}m$

B) $5\sqrt{3}m$

C) $5\sqrt{2}m$

D) $8\,m$

E) $6\,m$

• question_answer201) $\int{\frac{dx}{\sin (x-a)\sin (x-b)}}$is:

A) $\frac{1}{\sin (a-b)}\log \left| \frac{\sin (x-a)}{\sin (x-b)} \right|+c$

B) $\frac{-1}{\sin (a-b)}\log \left| \frac{\sin (x-a)}{\sin (x-b)} \right|+c$

C) $\log \sin (x-a)\sin (x-b)+c$

D) $\log \left| \frac{\sin (x-a)}{\sin (x-b)} \right|$

E) $\frac{1}{\sin (x-a)}\log \sin (x-a)\sin (x-b)+c$

• question_answer202) If an antiderivative of$f(x)$is${{e}^{x}}$and that of$g(x)$is $\cos x,$then$\int{f(x)}\cos x\,dx+$$\int{g(x)}\,{{e}^{x}}dx$is equal to:

A) $f(x)g(x)+c$

B) $f(x)+g(x)+c$

C) ${{e}^{x}}\cos x+c$

D) $f(x)-g(x)+c$

E) ${{e}^{x}}\cos x+f(x)g(x)+c$

• question_answer203) $\int{{{e}^{x\log a}}{{e}^{x}}}dx$is equal to:

A) $\frac{{{a}^{x}}}{\log \,\,ae}+c$

B) $\frac{{{e}^{x}}}{1+{{\log }_{e}}a}+c$

C) ${{(ae)}^{x}}+c$

D) $\frac{{{(ae)}^{x}}}{{{\log }_{e}}ae}+c$

E) $\frac{{{a}^{x}}{{e}^{x}}}{{{\log }_{x}}a}+c$

• question_answer204) $\int{\sqrt{{{e}^{x}}-1}}dx$is equal to

A) $2[\sqrt{{{e}^{x}}-1}-{{\tan }^{-1}}\sqrt{{{e}^{x}}-1}]+c$

B) $\sqrt{{{e}^{x}}-1}-{{\tan }^{-1}}\sqrt{{{e}^{x}}-1}+c$

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

D) $2[\sqrt{{{e}^{x}}-1}+{{\tan }^{-1}}\sqrt{{{e}^{x}}-1}]+c$

E) $2[\sqrt{{{e}^{x}}-1}-{{\tan }^{-1}}\sqrt{{{e}^{x}}+1}]+c$

• question_answer205) If ${{I}_{1}}=\int{{{\sin }^{-1}}}x\,dx$and${{I}_{2}}=\int{{{\sin }^{-1}}}\sqrt{1-{{x}^{2}}}\,dx$then:

A) ${{I}_{1}}={{I}_{2}}$

B) ${{I}_{2}}=\frac{\pi }{2}{{I}_{1}}$

C) ${{I}_{1}}+{{I}_{2}}=\frac{\pi }{2}x$

D) ${{I}_{1}}+{{I}_{2}}=\frac{\pi }{2}$

E) ${{I}_{1}}-{{I}_{2}}=\frac{\pi }{2}x$

• question_answer206) $\int{{{\cos }^{-3/7}}x{{\sin }^{-11/7}}}x\,dx$is equal to:

A) $\log |{{\sin }^{4/7}}x|+c$

B) $\frac{4}{7}{{\tan }^{4/7}}x+c$

C) $\frac{-7}{4}{{\tan }^{-4/7}}x+c$

D) $\log |{{\cos }^{3/7}}x|+c$

E) $\frac{7}{4}{{\tan }^{-4/7}}x+c$

• question_answer207) $\int{\frac{(\sin \theta +\cos \theta )}{\sqrt{\sin 2\theta }}}d\theta$is equal to:

A) $\log |\cos \theta -\sin \theta +\sqrt{\sin 2\theta }|+c$

B) $\log |\sin \theta -\cos \theta +\sqrt{\sin 2\theta }|+c$

C) ${{\sin }^{-1}}(\sin \theta -\cos \theta )+c$

D) ${{\sin }^{-1}}(\sin \theta +\cos \theta )+c$

E) ${{\sin }^{-1}}(\cos \theta -\sin \theta )+c$

• question_answer208) $\int_{\pi /6}^{\pi /3}{\frac{dx}{1+\sqrt{\tan x}}}$is equal to:

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

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

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

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

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

• question_answer209) $\int_{-\pi }^{\pi }{\frac{{{\sin }^{4}}x}{{{\sin }^{4}}x+{{\cos }^{4}}x}}dx$is equal to:

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

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

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

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

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

• question_answer210) The value of$\int_{0}^{\frac{\pi }{2}}{\frac{{{2}^{\sin x}}}{{{2}^{\sin x}}+{{2}^{\cos x}}}}dx$ is:

A) $2$

B) $\pi$

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

D) $2\pi$

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

• question_answer211) If$f$is continuous function, then:

A) $\int_{-2}^{2}{f(x)}dx=\int_{0}^{2}{[f(x)-f(-x)]}\,dx$

B) $\int_{-3}^{5}{2f(x)}dx=\int_{-6}^{10}{f(x-1)}\,dx$

C) $\int_{-3}^{5}{f(x)\,}dx=\int_{-4}^{4}{f(x-1)}\,dx$

D) $\int_{-3}^{5}{f(x)\,}dx=\int_{-2}^{6}{f(x-1)}\,dx$

E) $\int_{-3}^{5}{f(x)\,}dx=\int_{-6}^{10}{f\left( \frac{x}{2} \right)}\,dx$

• question_answer212) The area of the region bounded by${{y}^{2}}=4ax$and${{x}^{2}}=4ay,\text{ }a>0$in sq unit, is:

A) $16\frac{{{a}^{2}}}{3}$

B) $14\frac{{{a}^{2}}}{3}$

C) $13\frac{{{a}^{2}}}{3}$

D) $16{{a}^{2}}$

E) $4{{a}^{2}}$

• question_answer213) An integrating factor of the differential equation $x\frac{dy}{dx}+y\log x=x{{e}^{x}}{{x}^{-\frac{1}{2}\log x}}$$(x>0)$is:

A) ${{x}^{\log x}}$

B) ${{(\sqrt{x})}^{\log x}}$

C) ${{(\sqrt{e})}^{{{(\log x)}^{2}}}}$

D) ${{e}^{{{x}^{2}}}}$

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

• question_answer214) The solution of ${{e}^{dy/dx}}=(x+1),y(0)=3$is:

A) $y=x\log x-x+2$

B) $y=(x+1)\log |x+1|-x+3$

C) $y=(x+1)\log |x+1|+x+3$

D) $y=x\log x+x+3$

E) $y=-(x+1)\log |x+1|+x+3$

• question_answer215) Solution of the differential equation $\frac{dy}{dx}\tan y=\sin (x+y)+\sin (x-y)$is:

A) $sec\text{ }y+2\text{ }cos\text{ }x=c$

B) $sec\text{ }y-2\text{ }cos\text{ }x=c$

C) $cos\text{ }y-2\text{ }sin\text{ }x=c$

D) $tan\text{ }y-2\text{ }sec\text{ }y=c$

E) $sec\text{ }y+2\text{ }sin\text{ }x=c$

• question_answer216) Solution of the differential equation $\frac{dy}{dx}+\frac{y}{x}=\sin x$is:

A) $x(y+\cos x)=\sin x+c$

B) $x(y-\cos x)=\sin x+c$

C) $x(y\cos x)=\sin x+c$

D) $x(y-\cos x)=\cos x+c$

E) $x(y+\cos x)=\cos x+c$

• question_answer217) If ${{N}_{a}}=\{an:n\in N\},$then${{N}_{5}}\cap {{N}_{7}}$is equal to:

A) ${{N}_{7}}$

B) $N$

C) ${{N}_{35}}$

D) ${{N}_{5}}$

E) ${{N}_{12}}$

• question_answer218) If$f(x)=\frac{\alpha \,\,x}{x+1},x\ne 1,$for what value of $\alpha$ is$f[f(x)]=x?$

A) $\sqrt{2}$

B) $-\sqrt{2}$

C) 1

D) 2

E) $-1$

• question_answer219) If$n(A)=4,n(B)=3,n(A\times B\times C)=24,$then$n(C)$is equal to:

A) 288

B) 1

C) 12

D) 17

E) 2

• question_answer220) Two finite sets have m and n elements respectively. The total number of subsets of first set is 56 more than the total number of subsets of the second set. The values of m and n respectively are:

A) 7, 6

B) 6, 3

C) 5, 1

D) 7, 8

E) 3, 6

• question_answer221) The number of elements in the set$\{(a,b):$$2{{a}^{2}}+3{{b}^{2}}=35,\text{ }a,b\in Z\},$where Z is the set of all integers, is:

A) 2

B) 4

C) 8

D) 12

E) 16

• question_answer222) If$z=r{{e}^{i\theta }},$then$|{{e}^{iz}}|$is equal to:

A) $1$

B) ${{e}^{2r\sin \theta }}$

C) ${{e}^{r\sin \theta }}$

D) $r{{e}^{\sin \theta }}$

E) ${{e}^{-r\sin \theta }}$

• question_answer223) ${{i}^{2}}+{{i}^{4}}+{{i}^{6}}+...$upto$(2k+1)$terms,$k\in N$is:

A) 0

B) 1

C) $-1$

D) k

E) $k+1$

• question_answer224) If$z=\sqrt{2}-i\sqrt{2}$is rotated through an angle$45{}^\circ$in the anticlockwise direction about the origin, then the co-ordinates of its new position are:

A) $(2,0)$

B) $(\sqrt{2},\sqrt{2})$

C) $(\sqrt{2},-\sqrt{2})$

D) $(\sqrt{2},0)$

E) $(4,0)$

• question_answer225) If$z=\frac{7-i}{3-4i},$then${{z}^{14}}$is equal to:

A) ${{2}^{7}}$

B) ${{2}^{7}}i$

C) ${{2}^{14}}i$

D) $-{{2}^{7}}i$

E) $-{{2}^{14}}$

• question_answer226) If${{(\sqrt{8}+i)}^{50}}={{3}^{49}}(a+ib),$then${{a}^{2}}+{{b}^{2}}$is:

A) 3

B) 8

C) 9

D) $\sqrt{8}$

E) 4

• question_answer227) If$3{{p}^{2}}=5p+2$and$3{{q}^{2}}=5q+2$where$p\ne q,$then the equation whose roots are$3p-2q$ and$3q-2p$is:

A) $3{{x}^{2}}-5x-100=0$

B) $5{{x}^{2}}+3x+100=0$

C) $3{{x}^{2}}-5x+100=0$

D) $3{{x}^{2}}+5x-100=0$

E) $5{{x}^{2}}-3x-100=0$

• question_answer228) If$x=8+3\sqrt{7}$and$xy=1,$then the value of$\frac{1}{{{x}^{2}}}+\frac{1}{{{y}^{2}}}$is:

A) 254

B) 192

C) 292

D) 66

E) 62

• question_answer229) If$\alpha$and$\beta$are the roots of the equation${{x}^{2}}-6x+a=0$and satisfy the relation $3\alpha +2\beta =16,$then the value of a is:

A) $-8$

B) 8

C) $-16$

D) 9

E) none of these

• question_answer230) The solution set of the equation$pq{{x}^{2}}-{{(p+q)}^{2}}x+{{(p+q)}^{2}}=0$is:

A) $\left\{ \frac{p}{q},\frac{q}{p} \right\}$

B) $\left\{ pq,\frac{p}{q} \right\}$

C) $\left\{ \frac{q}{p},pq \right\}$

D) $\left\{ \frac{p+q}{p},\frac{p+q}{q} \right\}$

E) $\left\{ \frac{p-q}{p},\frac{p-q}{q} \right\}$

• question_answer231) If the roots a, Rot the equation $\frac{{{x}^{2}}-bx}{ax-c}=\frac{\lambda -1}{\lambda +1}$are such that$\alpha +\beta =0,$then the value of$\lambda$is:

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

B) $c$

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

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

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

• question_answer232) If$x,y,z$are in AP, then $\frac{1}{\sqrt{x}+\sqrt{y}},\frac{1}{\sqrt{z}+\sqrt{x}},$ $\frac{1}{\sqrt{y}+\sqrt{z}}$ are in:

A) AP

B) GP

C) HP

D) AP and HP

E) AP and GP

• question_answer233) If${{A}_{i}}\left[ \begin{matrix} {{a}^{i}} & {{b}^{i}} \\ {{b}^{i}} & {{a}^{i}} \\ \end{matrix} \right]$and, if$|a|<1,|b|<1,$then $\sum\limits_{i=1}^{\infty }{det({{A}_{i}})}$ is equal to:

A) $\frac{{{a}^{2}}}{{{(1-a)}^{2}}}-\frac{{{b}^{2}}}{{{(1-b)}^{2}}}$

B) $\frac{{{a}^{2}}-{{b}^{2}}}{(1-{{a}^{2}})(1-{{b}^{2}})}$

C) $\frac{{{a}^{2}}}{{{(1-a)}^{2}}}+\frac{{{b}^{2}}}{{{(1-b)}^{2}}}$

D) $\frac{{{a}^{2}}}{{{(1+a)}^{2}}}-\frac{{{b}^{2}}}{{{(1+b)}^{2}}}$

E) $\frac{a}{1+a}-\frac{b}{1+b}$

• question_answer234) The product$(32){{(32)}^{1/6}}{{(32)}^{1/36}}.....$to$\infty$is:

A) 16

B) 32

C) 64

D) 0

E) 62

• question_answer235) If AM and GM of$x$and y are in the ratio$p:q,$then$x:y$is:

A) $p-\sqrt{{{p}^{2}}+{{q}^{2}}}:p+\sqrt{{{p}^{2}}+{{q}^{2}}}$

B) $p+\sqrt{{{p}^{2}}-{{q}^{2}}}:p-\sqrt{{{p}^{2}}-{{q}^{2}}}$

C) $p:q$

D) $p+\sqrt{{{p}^{2}}+{{q}^{2}}}:p-\sqrt{{{p}^{2}}+{{q}^{2}}}$

E) $q+\sqrt{{{p}^{2}}-{{q}^{2}}}:q-\sqrt{{{p}^{2}}-{{q}^{2}}}$

• question_answer236) If$x,y,z$are in AP and${{\tan }^{-1}}x,{{\tan }^{-1}}y$and ${{\tan }^{-1}}z$ are also in AP, then:

A) $x=y=z$

B) $x=y=-z$

C) $x=1,y=2,z=3$

D) $x=2,y=4,z=6$

E) $x=2y=3z$

• question_answer237) The coefficient of${{x}^{3}}$the expansion of${{3}^{x}}$is:

A) $\frac{{{3}^{3}}}{6}$

B) $\frac{{{(\log 3)}^{3}}}{3}$

C) $\frac{\log ({{3}^{3}})}{6}$

D) $\frac{{{(\log 3)}^{3}}}{6}$

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

• question_answer238) The sum of the series$1+\frac{3}{2!}+\frac{7}{3!}+\frac{15}{4!}+....$to$\infty$is:

A) $e(e+1)$

B) $e(1-e)$

C) $3e-1$

D) $3e$

E) $e(e-1)$

• question_answer239) If${{\log }_{0.3}}(x-1)<{{\log }_{0.09}}(x-1),$then$x\ne 1$lies in:

A) (1, 2)

B) $(0,\text{ }1)$

C) $(1,\infty )$

D) $(2,\infty )$

E) $(0.09,0.3)$

• question_answer240) A student is to answer 10 out of 13 questions in an examination such that he must choose at least 4 from the first five questions. The number of choices available to him is:

A) 140

B) 196

C) 280

D) 346

E) 265