# Solved papers for CEE Kerala Engineering CEE Kerala Engineering Solved Paper-2006

### done CEE Kerala Engineering Solved Paper-2006

• question_answer1) In Youngs experiment, using red light $(\lambda =6600\overset{\text{o}}{\mathop{\text{A}}}\,),$ 60 fringes are seen in the field of view. How many fringes will be seen by using violet light $(\lambda =4400\overset{\text{o}}{\mathop{\text{A}}}\,)$ ?

A)  10

B)  20

C)  45

D)         90

E)  35

• question_answer2) At a temperature of$30{}^\circ C,$the susceptibility of a ferromagnetic material is found to be$\chi$. Its susceptibility at$333{}^\circ C$is:

A) $\chi$

B) $0.5\chi$

C) $2\chi$

D) $11.1\chi$

E) $0.09\chi$

• question_answer3) To cover a population of 20 lakh, a transmission tower should have a height ..... (Radius of earth = 6400 km, population per square km = 1000):

A) 25 m

B) 50 m

C) 75 m

D) 100 m

E) 39 m

• question_answer4) Two identical air core capacitors are connected in series to a voltage source of 15 V. If one of the capacitors is filled with a medium of dielectric constant 4, the new potential across this capacitor is:

A) 5V

B) 8V

C) 10 V

D) 12V

E) 3V

• question_answer5) The range of frequencies allotted for FM radio is:

A) 88 to 108 kHz

B) 88 to 108 MHz

C) 47 to 230 kHz

D) 47 to 230 MHz

E) 470 to 960 MHz

• question_answer6) A string of density$7.5\text{ }g\text{ }c{{m}^{-3}}$and area of cross-section$0.2\,m{{m}^{2}}$is stretched under a tension of 20 N. When it is plucked at the mid-point, the speed of the transverse wave on the wire is:

A) $116\,m{{s}^{-1}}$

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

C) $200\,m{{s}^{-1}}$

D) $80\,\,m{{s}^{-1}}$

E) $5900\,\,m{{s}^{-1}}$

• question_answer7) In co-axial cable the material used as spacer is:

A) teflon (or) polyethylene

B) glass or mica

C) a gaseous medium

D) glass

E) mica

• question_answer8) A work of$2\times {{10}^{-2}}J$is done on a wire of length 50 cm and area of cross-section$0.5\text{ }m{{m}^{2}}$. If the Youngs modulus of the material of the wire is$2\times {{10}^{10}}N{{m}^{-2}},$ then the wire must be:

A) elongated to 50.1414 cm

B) contracted by 2.0 mm

C) stretched by 0.707 mm

D) of length changed to 49.293 cm

E) of length changed to 50.2 cm

• question_answer9) Three blocks of masses${{m}_{1}},{{m}_{2}}$and${{m}_{3}}$are connected by massless string as shown kept on a frictionless table. They are pulled with a force${{T}_{3}}=40N.$If ${{m}_{1}}=10\,kg,{{m}_{2}}=6\,kg$and${{m}_{3}}=4\text{ }kg,$the tension${{T}_{2}}$will be:

A) 20 N

B) 40 N

C) 10 N

D) 32 N

E) 16 N

• question_answer10) Two identical cells whether connected in parallel or in series gives the same current when connected to an external resistance$1.5\,\Omega$. Find the value of internal resistance of each cell.

A) $1\,\Omega$

B) $0.5\,\Omega$

C) zero

D) $2\,\Omega$

E) $1.5\,\Omega$

• question_answer11) The binding energy per nucleon for deuteron and helium are 1.1 MeV and 7.0 MeV. The energy released when two deuterons ruse to form a helium nucleus is:

A) 23.6 MeV

B) 2.2 MeV

C) 30.2 MeV

D) 3.6 MeV

E) 28.0 MeV

• question_answer12) If the two vectors$\overrightarrow{A}=2\hat{i}+3\hat{j}+4\hat{k}$and$\overrightarrow{B}=\hat{i}+2\hat{j}-n\hat{k}$are perpendicular then the value of n is:

A) 1

B) 2

C) 3

D) 4

E) 5

• question_answer13) Force between two identical charges placed at a distance of r in vacuum is F. Now a slab of dielectric of dielectric constant 4 is inserted between these two charges. If the thickness of the slab is r/2, then the force between the charges will become:

A) $F$

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

C) $\frac{4}{9}F$

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

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

• question_answer14) A stone of mass m tied to a string of length$l$is rotating along a circular path with constant speed v. The torque on the stone is:

A) $mlv$

B) $\frac{mv}{l}$

C) $\frac{m{{v}^{2}}}{l}$

D) $m{{v}^{2}}l$

E) zero

• question_answer15) A copper disc of radius 0.1 m is rotated about its centre with 20 rev/s in a uniform magnetic field of 0.1 T with its plane perpendicular to the field. The emf induced across the radius of the disc is:

A) $\frac{\pi }{20}V$

B) $\frac{\pi }{10}V$

C) $20\pi \,mV$

D) $10\pi \,mV$

E) $2\pi \,mV$

• question_answer16) Water rises in a capillary tube to a height h. Choose false statement regarding capillary rise from the following:

A) On the surface of Jupiter, height will be less than h

B) In a lift moving up with constant acceleration height is less than h

C) On the surface of moon the height is more than h

D) In a lift moving down with constant acceleration height is less than h

E) At the poles, height is less than that at equator

• question_answer17) A magnetized wire of magnetic moment M and length L is bent in the form of a semicircle of radius r. The new magnetic moment is:

A) M

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

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

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

E) zero

• question_answer18) A proton, a deuteron and an alpha particle with the same kinetic energy enter a region of uniform magnetic field B at right angles to the field. The ratio of the radii of their circular paths is:

A) $1:1:1$

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

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

D) $\sqrt{2}:\sqrt{2}:1$

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

• question_answer19) LANDSAT series of satellites move in near polar orbits at an altitude of:

A) 3600km

B) 3000 km

C) 918 km

D) 512 km

E) 9200km

• question_answer20) A boat travels 50 km east, then 120 km North and finally it comes back to the starting point through the shortest distance. The total time of journey is 3 h. What is the average velocity, in$km\text{ }{{h}^{-1}},$over the entire trip?

A) zero

B) 100

C) 17

D) 33.33

E) 86.7

• question_answer21) The instantaneous displacement of a simple harmonic oscillator is given by $y=A\cos \left( \omega t+\frac{\pi }{4} \right)$Its speed will be maximum at the time:

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

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

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

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

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

• question_answer22) The surface area of a black body is$5\times {{10}^{-4}}{{m}^{2}}$and its temperature is$727{}^\circ C$. The energy radiated by it per minute is: $(\sigma =5.67\times {{10}^{-8}}J/{{m}^{2}}-s-{{k}^{4}})$

A) $1.7\times {{10}^{3}}J$

B) $2.5\times {{10}^{2}}J$

C) $8\times {{10}^{3}}J$

D) $3\times {{10}^{4}}J$

E) none of these

• question_answer23) Potential energy in a spring when stretched by 2 cm is U. Its potential energy, when stretched by 10 cm is:

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

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

C) $25U$

D) $5U$

E) none of these

• question_answer24) A toy cyclist completes one round of a square track of side 2 m in 40s. What will be the displacement at the end of 3 min?

A) 52 m

B) Zero

C) 16m

D) $2\sqrt{2}m$

E) $4\sqrt{2}m$

• question_answer25) The mass of a planet is six times that of the earth. The radius of the planet is twice that of the earth. If the escape velocity from the earth is v, then the escape velocity from the planet is:

A) $\sqrt{3}v$

B) $\sqrt{2}v$

C) $v$

D) $\sqrt{5}v$

E) $\sqrt{12}v$

• question_answer26) A particle of mass 5 g is executing simple harmonic motion with amplitude of 0.3 m and time period$(\pi /5)$s. The maximum value of the force acting on the particle is:

A) 5N

B) 4N

C) 0.5 N

D) 0.3 N

E) 0.15 N

• question_answer27) The inputs and outputs for different time intervals are given below the NAND gate

 Time Input A Input B Output Y ${{t}_{1}}\,to\,{{t}_{2}}$ 0 1 P ${{t}_{2}}\,to\,{{t}_{3}}$ 0 0 Q ${{t}_{3}}\,to\,{{t}_{4}}$ 1 0 R ${{t}_{4}}\,to\,{{t}_{5}}$ 1 1 S
The values taken by P, Q, R, S are respectively:

A) 1, 1, 1, 0

B) 0, 1, 0, 1

C) 0, 1, 0, 0

D) 1, 0, 1, 1

E) 1, 0, 1, 0

• question_answer28) A solenoid 600 mm long has 50 turns on it and is wound on an iron rod of 7.5 mm radius. Find the flux through the solenoid when the current in it is 3 A. The relative permeability of iron is 600:

A) 1.66 Wb

B) 1.66 nWb

C) 1.66 mWb

D) $1.66\mu Wb$

E) 1.66 pWb

• question_answer29) Two trains, each of length 200 m are running on parallel tracks. One overtakes the other in 20 s and one crosses the other in 10 s. The velocities of the two trains are:

A) 5 m/s, 10 m/s

B) 10 m/s, 30 m/s

C) 15 m/s, 5 m/s

D) 20 m/s, 21 m/s

E) 10 m/s, 5 m/s

• question_answer30) An electric dipole is placed at an angle of$30{}^\circ$with an electric field of intensity$2\times {{10}^{5}}N{{C}^{-1}}$. It experiences a torque equal to 4 Nm. Calculate the charge on the dipole if the dipole length is 2 cm.

A) $8\text{ }mC$

B) $4\text{ }mC$

C) $8\mu C$

D) $4\mu C$

E) $2mC$

• question_answer31) Two identical cells send the same current in$3\,\Omega$. resistance, whether connected in series or in parallel. The internal resistance of the cell should be:

A) $1\,\Omega$

B) $3\,\Omega$

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

D) $3.5\,\Omega$

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

• question_answer32) Which of the following statement is incorrect?

A) In LCR series AC circuit, as the frequency of the source increases, the impedance of the circuit first decreases and then increases

B) If the net reactance of an LCR series AC circuit is same as its resistance, then the current lags behind the voltage by$45{}^\circ$

C) At resonance, the impedance of an AC circuit becomes purely resistive

D) At resonance in LCR series AC circuit, the potential drops across inductor and capacitor are equal in magnitude but opposite in sign

E) Below resonance, voltage leads the current while above it, current leads the voltage

• question_answer33) A ray of light passes from vacuum into a medium of refractive index p, the angle of incidence is found to be twice the angle of refraction. Then the angle of incidence is:

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

B) $2{{\cos }^{-1}}\left( \frac{\mu }{2} \right)$

C) $2{{\sin }^{-1}}(\mu )$

D) $2{{\sin }^{-1}}\left( \frac{\mu }{2} \right)$

E) none of these

• question_answer34) The plot represents the flow of current through a wire at three different times. The ratio of charges flowing through the wire at different times is:

A) $2:1:2$

B) $1:3:3$

C) $1:1:1$

D) $2:3:4$

E) $2:3:3$

• question_answer35) In a gas, two waves of wavelengths 1 m and 1.01 m are superposed and produce 10 beats in s. The velocity of sound in the medium is:

A) 300 m/s

B) 336.7 m/s

C) 360.2 m/s

D) 270 m/s

E) 390 m/s

• question_answer36) The distance between the centres of carbon and oxygen atoms in the carbon monoxide molecule is $1.130\overset{\text{o}}{\mathop{\text{A}}}\,$ . Locate the centre of mass of Ac molecule relative to the carbon atom:

A)  $5.428\,\overset{\text{o}}{\mathop{\text{A}}}\,$

B)  $1.130\,\overset{\text{o}}{\mathop{\text{A}}}\,$

C)  $0.6457\,\overset{\text{o}}{\mathop{\text{A}}}\,$

D)                         $0.3260\,\overset{\text{o}}{\mathop{\text{A}}}\,$

E)  none of these

• question_answer37) The pressure inside two soap bubbles is 1.01 and 1.02 atmosphere respectively. The ratio of their respective volumes is:

A) 2

B) 4

C) 6

D) 8

E) 16

• question_answer38) If$\alpha$and$\beta$are the current gain in the CB and CE configurations respectively of the transistor circuit, then$\frac{\beta -\alpha }{\alpha \beta }=$

A) infinite

B) 1

C) 2

D) 0.5

E) zero

• question_answer39) Which of the following sets of quantities have same dimensional formula?

A) Frequency, angular frequency and angular momentum

B) Surface tension, stress and spring constant

C) Acceleration, momentum and retardation

D) Thermal capacity, specific heat and entropy

E) Work, energy and torque

• question_answer40) A varying magnetic flux linking a coil is given by$\phi -X{{t}^{2}}$. If at time$t=3\text{ }s,$the emf induced is 9 V, then the value of$X$is:

A) $0.66\text{ }Wb\text{ }{{s}^{-2}}$

B) $1.5\text{ }Wb\text{ }{{s}^{-2}}$

C) $-\text{ }0.66\text{ }Wb\text{ }{{s}^{-2}}$

D) $-1.5\text{ }Wb\text{ }{{s}^{-2}}$

E) $-\,0.33\,Wb{{s}^{-2}}$

• question_answer41) The apparent frequency of the whistle of an engine changes in the ratio$9:8$as the engine passes a stationary observer. If the velocity of the sound is$340\text{ }m{{s}^{-1}},$then the velocity of the engine is:

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

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

C) $340\text{ }m{{s}^{-1}}$

D) $\text{180 }m{{s}^{-1}}$

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

• question_answer42) The width of a single slit if the first minimum is observed at an angle$2{}^\circ$with a light of wavelength $6980\overset{\text{o}}{\mathop{\text{A}}}\,$:

A) 0.2mm

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

C) $2\times {{10}^{5}}mm$

D) 2 mm

E) 0.02mm

• question_answer43) A silicon specimen is made into a p-type semiconductor by doping, on an average, one indium atom per$5\times {{10}^{7}}$silicon atoms. If the number density of atoms in the silicon specimen is$5\times {{10}^{28}}atom/{{m}^{3}},$then the number of acceptor atoms in silicon per cubic centi metre will be:

A) $2.5\times {{10}^{30}}atom/c{{m}^{3}}$

B) $2.5\times {{10}^{35}}atom/c{{m}^{3}}$

C) $1\times {{10}^{13}}atom/c{{m}^{3}}$

D) $1\times {{10}^{15}}atom/c{{m}^{3}}$

E) none of the above

• question_answer44) In a photoelectric effect measurement, the stopping potential for a given metal is found to be${{V}_{0}}$volt when radiation of wavelength${{\lambda }_{0}}$is used. If radiation of wavelength$2{{\lambda }_{0}}$is used with the same metal then the stopping potential (in volt) will be:

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

B) $2{{V}_{0}}$

C) ${{V}_{0}}+\frac{hc}{2e{{\lambda }_{0}}}$

D) ${{V}_{0}}-\frac{hc}{2e{{\lambda }_{0}}}$

E) ${{V}_{0}}$

• question_answer45) A 20 kg ball moving with a velocity 6 m/s collides with a 30 kg ball initially at rest. If both of them coalesce, then the final velocity of the combined mass is:

A) 6 m/s

B) 5 m/s

C) 3.6 m/s

D) 2.4 m/s

E) 1.2 m/s

• question_answer46) A monkey climbs up and another monkey climbs down a rope hanging from a tree with same uniform acceleration separately. If the respective masses of monkeys are in the ratio $2:3,$the common acceleration must be:

A) g/5

B) 6g

C) g/2

D) g

E) g/3

• question_answer47) A running man has the same kinetic energy as that of a boy of half his mass. The man speeds up by$2\text{ }m{{s}^{-1}}$and the boy changes his speed by $x\text{ }m{{s}^{-1}},$so that the kinetic energies of the boy and the man are again equal. Then$x$in$m{{s}^{-1}}$is:

A) $-2\sqrt{2}$

B) $+2\sqrt{2}$

C) $\sqrt{2}$

D) 2

E) $1/\sqrt{2}$

• question_answer48) In artificial radioactivity,$1.414\times {{10}^{6}}$nuclei are disintegrated into${{10}^{6}}$nuclei in 10 min. The half-life in minutes must be:

A) 5

B) 20

C) 15

D) 30

E) 25

• question_answer49) In a certain region of space there are only molecules per cm3 on an average. The temperature there is 3 K. The pressure of this dilute gas is: $(k=1.38\times {{10}^{-~23}}J/K)$

A) $20.7\times {{10}^{-17}}N/{{m}^{2}}$

B) $15.3\times {{10}^{-15}}N/{{m}^{2}}$

C) $2.3\times {{10}^{-10}}N/{{m}^{2}}$

D) $5.3\times {{10}^{-5}}N/{{m}^{2}}$

E) $3.5\times {{10}^{-8}}N/{{m}^{2}}$

• question_answer50) A network of six identical capacitors, each of value C, is made as shown in the figure. The equivalent capacitance between the points A and B is:

A) $\frac{4C}{11}$

B) $3C/4$

C) $3C/2$

D) $3C$

E) $4C/3$

• question_answer51) Two cars A and B are travelling in the same direction with velocities${{v}_{1}}$and${{v}_{2}}({{v}_{1}}>{{v}_{2}})$. When the car A is at a distance d behind the car B, the driver of the car A applies the brake producing uniform retardation, a. There will be no collision when:

A) $d<\left( \frac{{{v}_{1}}-{{v}_{2}}}{2a} \right)$

B) $d>\frac{v_{1}^{2}-v_{2}^{2}}{2a}$

C) $d>\frac{{{({{v}_{1}}-{{v}_{2}})}^{2}}}{2a}$

D) $d<\frac{v_{1}^{2}-v_{2}^{2}}{2a}$

E) none of these

• question_answer52) A simple pendulum has a time period${{T}_{1}}$when on the earths surface and${{T}_{2}}$when taken to a height 2R above the earths surface where R is the radius of the earth. The value of$({{T}_{1}}/{{T}_{2}})$is:

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

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

C) $\sqrt{3}$

D) $9$

E) 3

• question_answer53) A symmetrical body is rotating about its axis of symmetry, its moment of inertia about the axis of rotation being$1\text{ }kg{{m}^{2}}$and its rate of rotation 2 rev/s. The angular momentum is:

A) $1.257\,kg\,{{m}^{2}}/s$

B) $12.57\,kg\,{{m}^{2}}/s$

C) $13.57\,kg\,{{m}^{2}}/s$

D) $20\,kg\,{{m}^{2}}/s$

E) $1.357\,kg\,{{m}^{2}}/s$

• question_answer54) The readings of a constant volume gas thermometer at$0{}^\circ C$and$100{}^\circ C$are 40 cm of mercury and 60 cm of mercury. If its reading at an unknown temperature is 100 cm of mercury column, then the temperature is:

A) $100{}^\circ C$

B) $50{}^\circ C$

C) $25{}^\circ C$

D) $300{}^\circ C$

E) none of these

• question_answer55) The amplitude of the sinusoidially oscillating electric field of a plane wave is 60 V/m. Then the amplitude of magnetic field is:

A) $2\times {{10}^{2}}T$

B) $6\times {{10}^{7}}T$

C) $6\times {{10}^{2}}T$

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

E) $3\times {{10}^{8}}T$

• question_answer56) A car of mass 1000 kg moves on a circular track of radius 20 m. If the coefficient of friction is 0.64, then the maximum velocity with which the car can move is:

A) 15 m/s

B) 11.2 m/s

C) 20 m/s

D) 18 m/s

E) 22.4 m/s

• question_answer57) If${{\varepsilon }_{0}}$and${{\mu }_{0}}$are the electric permittivity and magnetic permeability of free space and$\varepsilon$ and$\mu$are the corresponding quantities in the medium, the index of refraction of the medium in terms of above parameter is:

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

B) ${{\left( \frac{\varepsilon \mu }{{{\varepsilon }_{0}}{{\mu }_{0}}} \right)}^{1/2}}$

C) $\left( \frac{{{\varepsilon }_{0}}{{\mu }_{0}}}{\varepsilon \mu } \right)$

D) ${{\left( \frac{{{\varepsilon }_{0}}{{\mu }_{0}}}{\varepsilon \mu } \right)}^{1/2}}$

E) none of these

• question_answer58) Two planets have radii${{r}_{1}}$and${{r}_{2}}$and densities ${{d}_{1}}$and${{d}_{2}}$respectively. Then the ratio of acceleration due to gravity on them will be:

A) ${{r}_{1}}{{d}_{1}}:{{r}_{2}}{{d}_{2}}$

B) ${{r}_{1}}{{d}_{2}}:{{r}_{2}}{{d}_{1}}$

C) $r_{1}^{2}{{d}_{1}}:r_{2}^{2}{{d}_{2}}$

D) ${{r}_{1}}:{{r}_{2}}$

E) ${{r}_{1}}/\sqrt{{{d}_{1}}}:{{r}_{2}}\sqrt{{{d}_{2}}}$

• question_answer59) A physical quantity P is related to four measurable quantities a, b, c and d as follows $P=\frac{{{a}^{3}}{{b}^{2}}}{\sqrt{c}d}$ The percentage errors of measurement in a, b, c and d are 1%, 3%, 4% and 2%. The percentage error in the quantity P is:

A) 10%

B) 13%

C) 5%

D) 15%

E) 17%

• question_answer60) The ratio of the resistance of conductor at temperature$15{}^\circ C$to its resistance at temperature$37.5{}^\circ C$is$4:5$. The temperature coefficient of resistance the conductor is:

A) $\frac{1}{25}{}^\circ {{C}^{-1}}$

B) $\frac{1}{50}{}^\circ {{C}^{-1}}$

C) $\frac{1}{80}{}^\circ {{C}^{-1}}$

D) $\frac{1}{75}{}^\circ {{C}^{-1}}$

E) $\frac{1}{40}{{\,}^{\text{o}}}{{C}^{-1}}$

• question_answer61) A light ray of $5895\overset{\text{o}}{\mathop{\text{A}}}\,$ wavelength travelling in vacuum enters a medium of refractive index 1.5. The speed of light in the medium is:

A) $3\times {{10}^{8}}m/s$

B) $2\times {{10}^{8}}m/s$

C) $1.5\times {{10}^{8}}\text{ }m/s$

D) $6\times {{10}^{8}}m/s$

E) $5\times {{10}^{7}}m/s$

• question_answer62) The momentum of a body is increased by 25%. The kinetic energy is increased by about:

A) 25%

B) 5%

C) 56%

D) 38%

E) 65%

• question_answer63) Find the potential at the centre of a square of side$\sqrt{2}m$. Which carries at its four comers charges${{q}_{1}}=3\times {{10}^{-6}}C,{{q}_{2}}=-3\times {{10}^{-6}}C,$ ${{q}_{3}}=-4\times {{10}^{-6}}C,{{q}_{4}}=7\times {{10}^{-6}}C$

A) $2.7\times {{10}^{4}}V$

B) $1.5\times {{10}^{3}}V$

C) $3\times {{10}^{2}}V$

D) $5\times {{10}^{3}}V$

E) $3\times {{10}^{3}}V$

• question_answer64) The half-life of radon is 3.8 days. How many radon will be left out of 1024 mg after 38 days:

A) 1 mg

B) 2 mg

C) 3 mg

D) 4 mg

E) 7mg

• question_answer65) In a cubic unit cell of bcc structure, the lattic points (i.e., number of atoms) are:

A) 2

B) 6a

C) 8

D) 12

E) 5

• question_answer66) A 2 kg copper block is heated to$500{}^\circ C$and then it is placed on a large block of ice at$0{}^\circ C$. If the specific heat capacity of copper is $400J\,k{{g}^{-1}}{}^\circ {{C}^{-1}}$and latent heat of fusion of water is$3.5\times {{10}^{5}}J\text{ }k{{g}^{-1}},$the amount of ice that can melt is:

A) 7/8 kg

B) 7/5 kg

C) 8/7 kg

D) 5/7 kg

E) 7/3 kg.

• question_answer67) In a common-emitter amplifier, the load resistance of the output circuit is 1000 times the resistance of the input circuit. If$\alpha =0.98,$ then voltage gain is:

A) $49\times {{10}^{3}}$

B) $2.5\times {{10}^{2}}$

C) $1.5\times {{10}^{2}}$

D) 4.9

E) $3.5\times {{10}^{3}}$

• question_answer68) Two identical springs, each of spring constant K, are connected first series and then in parallel. A mass M is suspended from them. The ratio of the frequencies of vertical oscillations will be:

A) $2:1$

B) $1:1$

C) $1:4$

D) $4:1$

E) $1:2$

• question_answer69) A particle of mass$m=5$units is moving with a uniform speed $v=3\sqrt{2}$m in the$XOY$plane along the line$Y=X+4$. The magnitude of the angular momentum about origin is:

A) zero

B) 60 unit

C) 7.5 unit

D) $40\sqrt{2}$unit

E) 3.0 unit

• question_answer70) The plane faces of two identical plano-convex lenses each having a focal length of 50 cm are placed against each other to form a usual biconvex lens. The distance from this lens combination at which an object must be placed to obtain a real, inverted image which has the same size as the object is:

A) 50 cm

B) 25 cm

C) 100 cm

D) 40 cm

E) 125cm

• question_answer71) A solid sphere of volume V and density$\rho$floats at the interface of two immiscible liquids of densities${{\rho }_{1}}$and${{\rho }_{2}}$respectively. If${{\rho }_{1}}<\rho <{{\rho }_{2}},$then the ratio of volume of the parts of the sphere in upper and lower liquids is:

A) $\frac{\rho -{{\rho }_{2}}}{{{\rho }_{2}}-\rho }$

B) $\frac{{{\rho }_{2}}-\rho }{\rho -{{\rho }_{1}}}$

C) $\frac{\rho +{{\rho }_{1}}}{\rho +{{\rho }_{2}}}$

D) $\frac{\rho +{{\rho }_{2}}}{\rho +{{\rho }_{1}}}$

E) $\frac{\sqrt{{{\rho }_{1}}\,{{\rho }_{2}}}}{\rho }$

• question_answer72) A short solenoid of length 4 cm, radius 2 cm and 100 turns is placed inside and on the axis of a long solenoid of length 80 cm and 1500 turns. A current of 3 A flows through the short solenoid. The mutual inductance of two solenoids is:

A) $2.96\times {{10}^{-4}}H$

B) $5.3\times {{10}^{-5}}H$

C) $3.52\times {{10}^{-3}}H$

D) $8.3\times {{10}^{-5}}H$

E) $2.96\times {{10}^{-3}}H$

• question_answer73) Which one of the following has the highest molar conductivity?

A) Diaminedichloroplatinum (II)

B) Tetraaminedichlorocobalt (III) chloride

C) Potassium hexacyanoferrate (II)

D) Hexaaquochromium (III) bromide

E) Pentacarbonyl iron (0)

• question_answer74) One mole of magnesium in the vapour state absorbed$1200\text{ }kJ\text{ }mo{{l}^{-1}}$of energy. If the first and second ionization energies of Mg are 750 and$1450\text{ }kJ\text{ }mo{{l}^{-1}}$respectively, the final composition of the mixture is:

A) $31%\text{ }M{{g}^{+}}+69%\text{ }M{{g}^{2+}}$

B) $69%\text{ }M{{g}^{+}}+31%\text{ }M{{g}^{2+}}$

C) $86%\text{ }M{{g}^{+}}+14%\text{ }M{{g}^{2+}}$

D) $14%\text{ }M{{g}^{+}}+86%\text{ }M{{g}^{2+}}$

E) $13%\text{ }M{{g}^{+}}+87%\text{ }M{{g}^{2+}}$

• question_answer75) The first law of thermodynamic is expressed as:

A) $q-w=\Delta E$

B) $\Delta E=q-w$

C) $q=\Delta E-w$

D) $w=q+\Delta E$

E) none of the these

• question_answer76) Which of the following complexes are not correctly matched with hybridization of their central metal ion? 1. $[Ni{{(CO)}_{4}}]$ $s{{p}^{3}}$ 2. ${{[Ni{{(CN)}_{4}}]}^{2-}}$ $s{{p}^{3}}$ 3. ${{[Co{{F}_{6}}]}^{3-}}$ ${{d}^{2}}s{{p}^{3}}$ 4. ${{[Fe{{(CN)}_{6}}]}^{3-}}$ $s{{p}^{3}}{{d}^{2}}$ Select the correct answer using the codes given below:

A) 1 and 2

B) 1 and 3

C) 2 and 4

D) 1, 3 and 4

E) 2, 3 and 4

• question_answer77) During the transformation of$^{b}{{X}_{a}}{{\xrightarrow{{}}}^{d}}{{Y}_{c}}$ the number of $\beta$-particles emitted is:

A) $(b-d)/4$

B) $(c-a)+1/2(b-d)$

C) $(a-c)-1/2(b-d)$

D) $(b-d)+2(c-a)$

E) $(b-d)+1/2(c-a)$

• question_answer78) The Markownikoffs rule is the best applicable to the reaction between:

A) ${{C}_{2}}{{H}_{4}}+HCl$

B) ${{C}_{3}}{{H}_{6}}+B{{r}_{2}}$

C) ${{C}_{3}}{{H}_{6}}+HBr$

D) ${{C}_{3}}{{H}_{8}}+C{{l}_{2}}$

E) ${{C}_{2}}{{H}_{4}}+{{I}_{2}}$

• question_answer79) Phenol can be distinguished from ethanol by the following reagents except:

A) sodium

B) $NaOH/{{I}_{2}}$

C) neutral $FeC{{l}_{3}}$

D) $B{{r}_{2}}/{{H}_{2}}O$

E) phthalic anhydride/cone.${{H}_{2}}S{{O}_{4}}$and$NaOH$

• question_answer80) The enol form of acetone after treatment with${{D}_{2}}O,$gives:

A) ${{H}_{3}}C-\underset{\begin{smallmatrix} | \\ OD \end{smallmatrix}}{\mathop{C}}\,=C{{H}_{2}}$

B) ${{H}_{3}}C-\underset{\begin{smallmatrix} || \\ O \end{smallmatrix}}{\mathop{C}}\,=C{{D}_{3}}$

C) ${{H}_{2}}C=\underset{\begin{smallmatrix} | \\ OH \end{smallmatrix}}{\mathop{C}}\,=C{{H}_{2}}D$

D) ${{H}_{2}}C=\underset{\begin{smallmatrix} | \\ OH \end{smallmatrix}}{\mathop{C}}\,=CH{{D}_{2}}$

E) ${{D}_{2}}C=\underset{\begin{smallmatrix} | \\ OD \end{smallmatrix}}{\mathop{C}}\,-C{{D}_{3}}$

• question_answer81) Consider the following compounds:

 (i) chloroethene (ii) benzene (iii) 1, 3-butadiene (iv) 1, 3, 5-hexatriene
All the carbon atoms are$s{{p}^{2}}$hybridized in:

A) (i), (iii), (iv) only

B) (i), (ii) only

C) (ii), (iii), (iv) only

D) (iii), (iv) only

E) (i), (ii), (iii) and (iv)

• question_answer82) An alkene on reductive ozonolysis gives 2-molecules of$C{{H}_{2}}{{(CHO)}_{2}}$. The alkene is:

E) 1, 2-dimethyl cyclopropene

• question_answer83) Which of the following statements are correct concerning redox properties?

 1. A metal M for which$E{}^\circ$for the half reaction ${{M}^{n+}}+n{{e}^{-}}=M,$is very negative will be a good reducing agent 2. The oxidizing power of the halogens decreases from chlorine to iodine 3. The reducing power of hydrogen halides increases from hydrogen chloride to hydrogen iodide.

A) 1, 2 and 3

B) 1 and 2

C) 1 only

D) 2 and 3 only

E) 3 only

• question_answer84) Identify the compound Z, In this reaction sequence $C{{H}_{3}}C{{H}_{2}}COOH\xrightarrow[{}]{N{{H}_{3}}}X\xrightarrow[{}]{B{{r}_{2}}+KOH}$ $Y\xrightarrow[{}]{HN{{O}_{2}}}Z:$

A) $C{{H}_{3}}OH$

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

C) $C{{H}_{3}}C{{H}_{2}}OH$

D) $C{{H}_{3}}C{{H}_{2}}C{{H}_{2}}OH$

E) $C{{H}_{3}}N{{H}_{2}}$

• question_answer85) The following homogeneous gaseous reactions were experimentally found to be second order overall.

 1. $2NO\xrightarrow[{}]{{}}{{N}_{2}}+{{O}_{2}}$ 2. $3{{O}_{2}}\xrightarrow[{}]{{}}2{{O}_{3}}$ 3. ${{N}_{2}}{{O}_{3}}\xrightarrow[{}]{{}}NO+N{{O}_{2}}$ 4. ${{H}_{2}}+{{I}_{2}}\xrightarrow[{}]{{}}2HI$
Which of these are most likely to be elementary reactions that occur in one step?

A) 3 only

B) 1 and 3

C) 1 and 4

D) 3 and 4

E) 1, 2 and 3

• question_answer86) The successive ionization energy values for an element X are given below:

 (i) 1st ionization energy $=410\text{ }kJ\text{ }mo{{l}^{-1}}$ (ii) 2nd ionization energy$=820\text{ }kJ\text{ }mo{{l}^{-1}}$ (iii) 3rd ionization energy$=1100\text{ }kJ\text{ }mo{{l}^{-1}}$ (iv) 4th ionization energy$=1500\text{ }kJ\text{ }mo{{l}^{-1}}$ (v) 5th ionization energy$=3200\text{ }kJ\text{ }mo{{l}^{-1}}$
Find out the number of valence electron for the atom

A) 4

B) 3

C) 5

D) 2

E) 1

• question_answer87) On a humid day in summer, the mole fraction of gaseous${{H}_{2}}O$(water vapour) in the air at $25{}^\circ C$can be as high as 0.0287. Assuming a total pressure of 0.977 atm. What is the partial pressure of dry air?

A) 94.9 atm

B) 0.949 atm

C) 949 atm

D) 0.648 atm

E) 1.248 atm

• question_answer88) Match list I with list II and select the correct answer using the codes given below the lists.

 List I: Metal ions List II: Magnetic moment (BM) 1 $C{{r}^{3+}}$ A $\sqrt{35}$ 2 $F{{e}^{2+}}$ B $\sqrt{30}$ 3 $N{{i}^{2+}}$ C $\sqrt{24}$ 4 $M{{n}^{2+}}$ D $\sqrt{15}$ E $\sqrt{8}$
Codes:

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

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

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

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

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

• question_answer89) The number of optical isomers of $C{{H}_{3}}CH(OH)CH(OH)CHO$is:

A) zero

B) 2

C) 3

D) 4

E) 6

• question_answer90) For which of the following sparingly soluble salt, the solubility (s) and solubility product $({{K}_{sp}})$are related by the expression $s={{({{K}_{sp}}/4)}^{1/3}}$?

A) $BaS{{O}_{4}}$

B) $C{{a}_{3}}{{(P{{O}_{4}})}_{2}}$

C) $H{{g}_{2}}C{{l}_{2}}$

D) $A{{g}_{3}}P{{O}_{4}}$

E) $CuS$

• question_answer91) For the reaction$CO(g)+\frac{1}{2}{{O}_{2}}(g)\xrightarrow[{}]{{}}$$C{{O}_{2}}(g),\Delta H$and $\Delta S$ are$-283kJ$and$-87J{{K}^{-1}},$respectively. It was intended to carry out this reaction at 1000, 1500, 3000 and 3500 K. At which of these temperatures would this reaction be thermodynamically spontaneous?

A) 1500 and 3500 K

B) 3000 and 3500 K

C) 1000, 1500 and 3000 K

D) 1500, 3000 and 3500 K

E) At all these temperatures

• question_answer92) At certain temperature a 5.12% solution of cane sugar is isotonic with a 0.9% solution of an unknown solute. The molar mass of solute is:

A) 60

B) 46.17

C) 120

D) 90

E) 92.34

• question_answer93) Which of the following is true in respect of adsorption?

A) $\Delta G<0;\text{ }\Delta S>0;\text{ }\Delta H<0$

B) $\Delta G<0;\text{ }\Delta S<0;\text{ }\Delta H<0$

C) $\Delta G>0;\text{ }\Delta S>0;\text{ }\Delta H<0$

D) $\Delta G<0;\text{ }\Delta S<0;\text{ }\Delta H>0$

E) $\Delta G>0;\text{ }\Delta S>0;\text{ }\Delta H>0$

• question_answer94) Find the two third life$({{t}_{2/3}})$of a first order reaction in which$k=5.48\times {{10}^{-14}}$per second:

A) $201\times {{10}^{13}}s$

B) $2.01\times {{10}^{13}}s$

C) $201\times {{10}^{20}}s$

D) $0.201\times {{10}^{10}}s$

E) none of these

• question_answer95) If$(x/m)$is the mass of adsorbate adsorbed per unit mass of adsorbent. P is the pressure of the abdsorbate gas and a and b are constants, which of the following represents Langmuir adsorption isotherm?

A) $\log \left( \frac{x}{m} \right)=\log \left( \frac{a}{b} \right)+\frac{1}{a}\log p$

B) $\frac{x}{m}=\frac{b}{a}+\frac{1}{ap}$

C) $\frac{x}{m}=\frac{1+bp}{ap}$

D) $\frac{1}{(x/m)}=\frac{a}{b}+\frac{p}{a}$

E) $\frac{1}{(x/m)}=\frac{b}{a}+\frac{1}{ap}$

• question_answer96) The amine which will not liberate nitrogen on reaction with nitrous acid is:

A) Trimethyl amine

B) ethyl amine

C) sec-butyl amine

D) c-butyl amine

E) iso-propyl amine

• question_answer97) 5.6 g of an organic compound on burning with excess of oxygen gave 17.6 g of$C{{O}_{2}}$and 7.2 g of${{H}_{2}}O$. The organic compound is:

A) ${{C}_{6}}{{H}_{6}}$

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

C) ${{C}_{3}}{{H}_{8}}$

D) $C{{H}_{3}}COOH$

E) $C{{H}_{3}}CHO$

• question_answer98) One mole of acidified${{K}_{2}}C{{r}_{2}}{{O}_{7}}$on reaction with excess KI will liberate ....... mole(s) of ${{I}_{2}}$:

A) 6

B) 1

C) 7

D) 2

E) 3

• question_answer99) Which of the following exists as Zwitter ion?

A) p-aminophenol

B) Salicylic acid

C) Sulphanilic acid

D) Ethanolamine

E) p-amino acetophenone

• question_answer100) Match the lists I and II and pick the correct matching from the codes given below:

 List I List II A Thymine 1. Pyrimidine base B Thiamine 2. Enzyme C Insulin 3. Cell-wall component D Pepsin 4. Hormone E Phospholipids 5. $Vit\,{{B}_{1}}$
Codes:

A) A-4; B-3; C-1; D-5; E-2

B) A-5; B-3; C-4; D-1; E-2

C) A-3; B-2; C-1; D-5; E-4

D) A-2; B-4; C-1; D-3; E-5

E) A-1; B-5; C-4; D-2; E-3

• question_answer101) The halogen compound which most readily undergoes nucleophilic substitution is:

A) $C{{H}_{2}}=CHCl$

B) $C{{H}_{3}}CH=CHCl$

C) $C{{H}_{2}}=CHC(Cl)=C{{H}_{2}}$

D) $C{{H}_{2}}=CHC{{H}_{2}}Cl$

E) ${{C}_{6}}{{H}_{5}}Cl$

• question_answer102) Given the standard reduction potentials $Z{{n}^{2+}}/Zn=-0.74\,V,$ $C{{l}_{2}}/C{{l}^{-}}=1.36\,V,$ ${{H}^{+}}/1/2{{H}_{2}}=0\,V$and $F{{e}^{2+}}/F{{e}^{3+}}=0.77\,V$ The order of increasing strength as reducing agent is:

A) $C{{l}^{-}},Zn,{{H}_{2}},F{{e}^{2+}}$

B) ${{H}_{2}},Zn,F{{e}^{2+}},Cl$

C) $C{{l}^{-}}F{{e}^{2+}},Zn,{{H}_{2}}$

D) ${{H}_{2}}F{{e}^{2+}},C{{l}^{-}},Zn$

E) $C{{l}^{-}},F{{e}^{2+}},{{H}_{2}},Zn$

• question_answer103) The reaction$2A+B+C\xrightarrow{{}}D+E$is found to be first order in A, second in B and zero order in C. What is the effect on the rate of increasing concentration of A, B and C two times?

A) 72 times

B) 8 times

C) 24 times

D) 36 times

E) none of the these

• question_answer104) Match list I with list II and select the correct answer using the codes given below the lists

 List-I Type of ore List-II Example 1. Oxide ore A. Feldspar 2. Sulphide ore B. barytes 3. Sulphate ore C. Fluorspar 4. Halide ore D. Galena E. Corundum
Codes:

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

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

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

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

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

• question_answer105) Consider the following halogen containing compounds:

 (A) $CHC{{l}_{3}}$ (B) $CC{{l}_{4}}$ (C) $C{{H}_{2}}C{{l}_{2}}$ (D) $C{{H}_{3}}Cl$ (E)
The compounds with a net zero dipole moment are:

A) B and E only

B) C only

C) C and D only

D) A and D only

E) B only

• question_answer106) For the reaction${{N}_{2}}(g)+3{{H}_{2}}(g)$ $2N{{H}_{3}}(g);$ $\Delta H=-93.6\,kJ\,mo{{l}^{-1}},$the concentration of $N{{H}_{3}}$ at equilibrium can be increased by:

 (1) lowering the temperature (2) low pressure (3) excess of ${{N}_{2}}$ (4) excess of ${{H}_{2}}$

A) (2) and (4) are correct

B) (2) only is correct

C) (1), (2) and (3) are correct

D) (3) and (4) are correct

E) (1), (3) and (4) are correct

• question_answer107) Which of the following is bacteriostatic?

A) Penicillin

B) Erythromycin

C) Amino glycodine

D) Ofloxacin

E) Bithional

• question_answer108) Which one of the following set of quantum numbers is not possible for electron in the ground state of an atom with atomic number 19?

A) $n=2,l=0,\text{ }m=0$

B) $n=2,l=1,m=0$

C) $n=3,l=1,m=-1$

D) $n=3,l=2,\text{ }m=+2$

E) $n=4,\text{ }l=0,\text{ }m=0$

• question_answer109) Match list I and list II and pick out correct matching codes from the given choices:

 List-I Compound List-II Structure A.$Cl{{F}_{3}}$ 1. Square planar B.$PC{{l}_{5}}$ 2. Tetrahedral. C. $I{{F}_{5}}$ 3. Trigonal bipyramidal D. $CC{{l}_{4}}$ 4. Square pyramidal E. $Xe{{F}_{4}}$ 5. T-shaped
Codes:

A) A-5, B-4, C-3, D-2, E-1

B) A-5, B-3, C-4, D-2, E-1

C) A-5, B-3, C-4, D-1, E-2

D) A-4, B-3, C-5, D-2, E-1

E) A-3, B-4, C-1, D-5, E-2

• question_answer110) The sequence that correctly describes the relative bond strength pertaining to oxygen molecule and its cation or anions is:

A) $O_{2}^{2-}>O_{2}^{-}>{{O}_{2}}>O_{2}^{+}$

B) ${{O}_{2}}>O_{2}^{+}>O_{2}^{-}>O_{2}^{2-}$

C) $O_{2}^{+}>{{O}_{2}}>O_{2}^{2-}>O_{2}^{-}$

D) $O_{2}^{+}>{{O}_{2}}>O_{2}^{-}>O_{2}^{2-}$

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

• question_answer111) The hybrid rocket propellant consists of:

A) acrylic rubber and liquid nitrogen tetraoxide

B) polyurethane and ammonium perchlorate

C) nitroglycerine and nitrocellulose

D) liquid hydrogen and liquid oxygen

E) hydrogen peroxide

• question_answer112) The orbital angular momentum of an electron revolving in a p-orbital is:

A) zero

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

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

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

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

• question_answer113) Boric acid is used in carom boards for smooth gliding of pawns because:

A) ${{H}_{3}}B{{O}_{3}}$molecules are loosely chemically bonded and hence soft

B) its low density makes it fluffy

C) it can be powdered to a very small grain size

D) it is chemically inert with the plywood

E) H-bonding in ${{H}_{3}}B{{O}_{3}}$ gives it a layered structure

• question_answer114) Consider the following molecules or ions:

 (i) ${{H}_{2}}O$ (ii) $NH_{4}^{+}$ (iii) $SO_{4}^{2-}$ (vi) $ClO_{4}^{-}$ (v) $N{{H}_{3}}$
$s{{p}^{3}}$hybridization is involved in the formation of:

A) (i), (ii), (v) only

B) (i), (ii) only

C) (ii) only

D) (i), (ii), (iii), (iv) only

E) (i), (ii), (iii), (iv) and (v)

• question_answer115) Gases X, Y, Z, P and Q have the van der Waals constants a and b (in CGS units) as shown below:

 X Y Z P Q a 6 6 20 0.05 30 b 0.025 0.15 0.1 0.02 0.2
The gas with the highest critical temperature is:

A) P

B) Q

C) Y

D) Z

E) X

• question_answer116) How many optically active stereoisomers are possible for butane-2, 3-diol?

A) 0

B) 1

C) 2

D) 3

E) 4

• question_answer117) Calculate the equilibrium constant for the reaction, at$25{}^\circ C$ $Cu(s)+2A{{g}^{+}}(aq)\xrightarrow[{}]{{}}C{{u}^{2+}}(aq)+2Ag(s)$ at$25{}^\circ C,E_{cell}^{o}=0.47V,R=8.314\,J{{K}^{-1}}$ $F=96500\,C$:

A) $1.8\times {{10}^{15}}$

B) $8.5\times {{10}^{15}}$

C) $1.8\times {{10}^{10}}$

D) $85\times {{10}^{15}}$

• question_answer118) Which of the following statements regarding the${{S}_{N}}1$reaction shown by alkyl halide is not correct?

A) The added nucleophile plays no kinetic role in${{S}_{N}}1$reaction

B) The${{S}_{N}}1$reaction involves the inversion of configuration of the optically active substrate

C) The${{S}_{N}}1$reaction on the chiral starting material ends up with racemization of the product

D) The more stable the carbocation intermediate the faster the${{S}_{N}}1$reaction

E) Polar protic solvent increases the rate of ${{S}_{N}}1$reaction

• question_answer119) Pick out the statement (s) which is (are) not true about the diagonal relationship of$Li$and Mg:

 (i) polarising powers of$L{{i}^{+}}$and$M{{g}^{2+}}$are almost same (ii) like$Li,$$Mg$decomposes water very fast (iii)$LiCl$and$MgC{{l}^{2}}$are deliquescent (iv) like$Li,Mg$do not form solid bicarbonates

A) (i) and (ii)

B) (ii) and (iii)

C) only (ii)

D) only (i)

E) (ii) and (iv)

• question_answer120) Which of the following concentration terms is/are independent of temperature?

A) Molarity

B) Molarity and mole fraction

C) Mole fraction and molality

D) Molality and normality

E) Only molality

• question_answer121) For an equilateral triangle the centre is the origin and the length of altitude is a. Then, the equation of the circumcircle is:

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

B) $3{{x}^{2}}+3{{y}^{2}}=2{{a}^{2}}$

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

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

E) $9{{x}^{2}}+9{{y}^{2}}=4{{a}^{2}}$

• question_answer122) If the two pair of lines${{x}^{2}}-2mxy-{{y}^{2}}=0$and ${{x}^{2}}-2nxy-{{y}^{2}}=0$are such that one of them represents the bisector of the angles between the other, then:

A) $mn+1=0$

B) $mn-1=0$

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

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

E) none of these

• question_answer123) Let$f$be a function such that$f(1)=10$and $f(x)\ge 2$for$1\le x<4$. How small can$f(4)$possibly be?

A) 8

B) 12

C) 16

D) 2

E) 10

• question_answer124) The solution of the differential equation $x\frac{dy}{dx}+2y={{x}^{2}}$is:

A) $y=\frac{{{x}^{2}}+c}{4{{x}^{2}}}$

B) $y=\frac{{{x}^{2}}}{4}+c$

C) $y=\frac{{{x}^{4}}+c}{{{x}^{2}}}$

D) $y=\frac{{{x}^{4}}+c}{4{{x}^{2}}}$

E) $y=\frac{{{x}^{3}}}{4}+\frac{c}{{{x}^{2}}}$

• question_answer125) $\frac{1}{\cos 80{}^\circ }-\frac{\sqrt{3}}{\sin 80{}^\circ }$is equal to:

A) $\sqrt{2}$

B) $\sqrt{3}$

C) 2

D) 4

E) $\sqrt{5}$

• question_answer126) $y=-A\text{ }cos\text{ }5x+B\text{ }sin\text{ }5x$satisfies the differential equation:

A) $\frac{{{d}^{2}}y}{d{{x}^{2}}}+10\frac{dy}{dx}+25y=0$

B) $\frac{{{d}^{2}}y}{d{{x}^{2}}}-10\frac{dy}{dx}+25y=0$

C) $\frac{{{d}^{2}}y}{d{{x}^{2}}}-25y=0$

D) $\frac{{{d}^{2}}y}{d{{x}^{2}}}+25y=0$

E) none of the above

• question_answer127) If $\int{\frac{\sqrt{x}}{x+1}}dx=A\sqrt{x}+B{{\tan }^{-1}}\sqrt{x}+c,$then:

A) $A=1,\text{ }B=1$

B) $A=1,\text{ }B=2$

C) $A=2,B=2$

D) $A=2,B=-2$

E) $A=-2,B=-2$

• question_answer128) The equation of the tangent to the curve $y={{(1+x)}^{y}}+{{\sin }^{-1}}({{\sin }^{2}}x)$at$x=0$is:

A) $x-y+1=0$

B) $x+y+1=0$

C) $2x-y+1=0$

D) $x+2y+2=0$

E) $2x+y-1=0$

• question_answer129) $\int{\frac{{{x}^{3}}\sin [{{\tan }^{-1}}{{(x)}^{4}}]}{1+{{x}^{8}}}}dx$is equal to:

A) $\frac{1}{4}\cos [{{\tan }^{-1}}({{x}^{4}})]+c$

B) $\frac{1}{4}\sin [{{\tan }^{-1}}({{x}^{4}})]+c$

C) $-\frac{1}{4}\cos [{{\tan }^{-1}}({{x}^{4}})]+c$

D) $\frac{1}{4}{{\sec }^{-1}}[{{\tan }^{-1}}({{x}^{4}})]+c$

E) $-\frac{1}{4}{{\cos }^{-1}}[{{\tan }^{-1}}({{x}^{4}})]+c$

• question_answer130) An anti-aircraft gun can take a maximum of four shots at any plane moving away from it. The probabilities of hitting the plane at the 1 st, 2 nd, 3 rd and 4 th shots are 0.4, 0.3, 0.2 and 0.1 respectively. What is the probability that at least one shot hits the plane?

A) 0.6976

B) 0.3024

C) 0.72

D) 0.6431

E) 0.7391

• question_answer131) The statement$\tilde{\ }(p\to q)$is equivalent to:

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

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

C) $p\wedge q$

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

E) $p\vee q$

• question_answer132) ${{I}_{n}}=\int_{0}^{\pi /4}{{{\tan }^{n}}x}dx,$then $\underset{n\to \infty }{\mathop{\lim }}\,n[{{I}_{n}}+{{I}_{n+2}}]$equal is:

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

B) 1

C) $\infty$

D) zero

E) none of these

• question_answer133) A bag contains 3 black, 3 white and 2 red balls. One by one, three balls are drawn without replacement. The probability that the third ball is red is equal to:

A) $\frac{2}{56}$

B) $\frac{3}{56}$

C) $\frac{1}{56}$

D) $\frac{12}{56}$

E) $\frac{14}{56}$

• question_answer134) The angle between the line $\frac{x-3}{2}=\frac{y-1}{1}=\frac{z+4}{-2}$and the plane, $x+y+z+5=0$is:

A) ${{\sin }^{-1}}\left( \frac{2}{\sqrt{3}} \right)$

B) ${{\sin }^{-1}}\left( \frac{1}{\sqrt{3}} \right)$

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

D) ${{\sin }^{-1}}\left( \frac{1}{3\sqrt{3}} \right)$

E) ${{\sin }^{-1}}(2)$

• question_answer135) A vector perpendicular to$2\hat{i}+\hat{j}+\hat{k}$and coplanar with$\hat{i}+2\hat{j}+\hat{k}$ and$\hat{i}+\hat{j}+2\hat{k}$is:

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

B) $\hat{i}+7\hat{j}-\hat{k}$

C) $5(\hat{j}+\hat{k})$

D) $2\hat{i}-7\hat{j}-\hat{k}$

E) $5(\hat{i}+\hat{k})$

• question_answer136) If$\left[ \begin{matrix} 2 & 1 \\ 3 & 2 \\ \end{matrix} \right]A\left[ \begin{matrix} -3 & 2 \\ 5 & -3 \\ \end{matrix} \right]=\left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right],$then A is equal to:

A) $-\left[ \begin{matrix} 1 & 1 \\ 1 & 0 \\ \end{matrix} \right]$

B) $\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \\ \end{matrix} \right]$

C) $\left[ \begin{matrix} 1 & 0 \\ 1 & 1 \\ \end{matrix} \right]$

D) $\left[ \begin{matrix} 0 & 1 \\ 1 & 1 \\ \end{matrix} \right]$

E) $\left[ \begin{matrix} 2 & 1 \\ 0 & 1 \\ \end{matrix} \right]$

• question_answer137) The point on the curve$y=2{{x}^{2}}-6x-4$at which the tangent is parallel to the$x-$axis, is:

A) $\left( \frac{3}{2},\frac{13}{2} \right)$

B) $\left( -\frac{5}{2},-\frac{17}{2} \right)$

C) $\left( \frac{3}{2},\frac{17}{2} \right)$

D) $(0,-4)$

E) $\left( \frac{3}{2},-\frac{17}{2} \right)$

• question_answer138) If $\overrightarrow{a}=2\hat{i}-3\hat{j}+p\hat{k}$and$\overrightarrow{a}\times \overrightarrow{b}=4\hat{i}+2\hat{j}-2\hat{k},$then p is:

A) 0

B) $-1$

C) 1

D) 2

E) $-2$

• question_answer139) Let$\overrightarrow{a}=\hat{i}-\hat{j},\overrightarrow{b}=\hat{j}-\hat{k},\overrightarrow{c}=\hat{k}-\hat{i}$. If$\overrightarrow{d}$is a unit vector such that$\overrightarrow{a}.\overrightarrow{d}=0=[\overrightarrow{b}\overrightarrow{c}\overrightarrow{d}],$then d is (are):

A) $\pm \frac{\hat{i}+\hat{j}-\hat{k}}{\sqrt{3}}$

B) $\pm \frac{\hat{i}+\hat{j}-2\hat{k}}{\sqrt{6}}$

C) $\pm \frac{\hat{i}+\hat{j}+\hat{k}}{\sqrt{3}}$

D) $\pm \hat{k}$

E) $\hat{i}+\hat{j}$

• question_answer140) A tower subtends an angle$\alpha$ at a point A in the plane of its base and the angle of depression of the foot of the tower at a point b ft just above A is $\beta$. Then, the height of the tower is:

A) $b\text{ }tan\text{ }\alpha \text{ }cot\text{ }\beta$

B) $b\text{ }\cot \,\alpha \text{ tan}\,\beta$

C) $b\text{ cot}\,\alpha \text{ }cot\,\beta$

D) $b\text{ }\tan \,\alpha \text{ tan}\,\beta$

E) $b\text{ ta}{{\text{n}}^{2}}\,\alpha \text{ }cot\,\beta$

• question_answer141) If$\alpha$and$\beta$are the roots of the equation ${{x}^{2}}-7x+1=0,$then the value of$\frac{1}{{{(\alpha -7)}^{2}}}+\frac{1}{{{(\beta -7)}^{2}}}$is:

A) 45

B) 47

C) 49

D) 50

E) 51

• question_answer142) The equation of the sphere whose centre is $(6,-1,\text{ }2)$and which touches the plane $2x-y+2z-2=0,$is:

A) ${{x}^{2}}+{{y}^{2}}+{{z}^{2}}-12x+2y-4z-16=0$

B) ${{x}^{2}}+{{y}^{2}}+{{z}^{2}}-12x+2y-4z=0$

C) ${{x}^{2}}+{{y}^{2}}+{{z}^{2}}-12x+2y-4z+16=0$

D) ${{x}^{2}}+{{y}^{2}}+{{z}^{2}}-12x+2y-4z+6=0$

E) ${{x}^{2}}+{{y}^{2}}+{{z}^{2}}-12x+2y-4z-5=0$

• question_answer143) Let X and Y be the sets of all positive divisors of 400 and 1000 respectively (including 1 and the number). Then$n(X\cap Y)$is equal to:

A) 4

B) 6

C) 8

D) 10

E) 12

• question_answer144) The value of$lo{{g}_{2}}lo{{g}_{2}}lo{{g}_{4}}256+2lo{{g}_{\sqrt{2}}}2$is:

A) 1

B) 2

C) 3

D) 4

E) 5

• question_answer145) The radius of a sphere is measured as 5 cm with an error possibly as large as 0.02 cm. The error and percentage error in computing the surface area of the sphere are:

A) $0.87\pi \,and\,0.2%$

B) $0.8\pi \,and\,0.8%$

C) $0.4\pi \,and\,0.4%$

D) $\pi \,and\,1%$

E) $0.6\pi \,and\,0.6%$

• question_answer146) If$\int{x}\,f(x)dx=\frac{f(x)}{2},$then$f(x)$is equal to:

A) ${{e}^{x}}$

B) ${{e}^{-x}}$

C) $\log x$

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

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

• question_answer147) The value of $\left| \begin{matrix} \cos (x-a) & \cos (x+a) & \cos x \\ \sin (x+a) & \sin (x-a) & \sin x \\ \cos a\tan x & \cos a\cot x & \cos ec2x \\ \end{matrix} \right|$is equal to:

A) 1

B) $sin\text{ }a\text{ }cos\text{ }a$

C) 0

D) $sin\text{ }x\text{ }cos\text{ }x$

E) $cosec\text{ }2x$

• question_answer148) The eccentricity of the hyperbola in the standard form$\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1$passing through (3,0) and$(3\sqrt{2},2)$is:

A) $\frac{13}{3}$

B) $\sqrt{13}$

C) $\sqrt{3}$

D) $\frac{\sqrt{13}}{3}$

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

• question_answer149) If a, b and c are in geometric progression and the roots of the equations$a{{x}^{2}}+2bx+c=0$are $\alpha$ and $\beta$ and those of$e{{x}^{2}}+2\text{ }bx+a=0$are $\gamma$ and $\delta ,$ then:

A) $\alpha \ne \beta \ne \gamma \ne \delta$

B) $\alpha \ne \beta and\,\gamma \ne \delta$

C) $a\alpha =a\beta =c\gamma =c\delta$

D) $\alpha =\beta \,and\,\gamma \ne \delta$

E) $\alpha \ne \beta \,and\,\gamma =\delta$

• question_answer150) If in the triangle$ABC,B=45{}^\circ ,$then ${{a}^{4}}+{{b}^{4}}+{{c}^{4}}$is equal to:

A) $2\text{ }{{a}^{2}}({{b}^{2}}+{{c}^{2}})$

B) $2{{c}^{2}}({{a}^{2}}+{{b}^{2}})$

C) $2{{b}^{2}}({{a}^{2}}+{{c}^{2}})$

D) $2({{a}^{2}}{{b}^{2}}+{{b}^{2}}{{c}^{2}}+{{c}^{2}}{{a}^{2}})$

E) $2{{a}^{2}}{{b}^{2}}+2{{b}^{2}}{{c}^{2}}+3{{a}^{2}}{{c}^{2}}$

• question_answer151) Let$f$be twice differentiable function such that$f(x)=-f(x)$and$f(x)=g(x),$$h(x)=\{f{{(x)}^{2}}\}+{{\{g(x)\}}^{2}}.$then$h(5)=11,$is equal to:

A) 22

B) 11

C) 0

D) 20

E) none of these

• question_answer152) Suppose A is a matrix of order 3 and$B=|A|{{A}^{-1}}.$If $|A|=-5,$then$|B|$is equal to:

A) 1

B) $-5$

C) $-1$

D) $25$

E) $-125$

• question_answer153) A differentiable function$f(x)$is defined for all $x>0$and satisfies$f({{x}^{3}})=4{{x}^{4}}$for all$x>0$. The value of$f(8)$is:

A) $\frac{16}{3}$

B) $\frac{32}{3}$

C) $\frac{16\sqrt{2}}{3}$

D) $\frac{32\sqrt{2}}{3}$

E) $\frac{32{{(2)}^{1/3}}}{3}$

• question_answer154) The equation of the hyperbola whose vertices are at (5, 0) and$(-\text{ }5,0)$and one of the directrices is$x=\frac{25}{7},$is:

A) $\frac{{{x}^{2}}}{25}-\frac{{{y}^{2}}}{24}=1$

B) $\frac{{{x}^{2}}}{24}-\frac{{{y}^{2}}}{25}=1$

C) $\frac{{{x}^{2}}}{16}-\frac{{{y}^{2}}}{25}=1$

D) $\frac{{{x}^{2}}}{25}-\frac{{{y}^{2}}}{16}=1$

E) $\frac{{{x}^{2}}}{25}-\frac{{{y}^{2}}}{24}=-1$

• question_answer155) If a and P are different complex numbers with $|\beta |=1,$then$\left| \frac{\beta -\alpha }{1-\alpha \beta } \right|$is:

A) 0

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

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

D) 1

E) 2

• question_answer156) $\int_{0}^{2}{[{{x}^{2}}]}dx$is:

A) $2-\sqrt{2}$

B) $2+\sqrt{2}$

C) $\sqrt{2}-1$

D) $-\sqrt{2}-\sqrt{3}+5$

E) none of these

• question_answer157) The point (4, 1) undergoes the following three transformations successively: (i) reflection about the line$y=x$ (ii) translation through a distance of 2 unit along the positive direction of$x-$axis (iii) rotation through an angle of$\frac{\pi }{4}$about the origin in the anticlockwise direction The final position of the point is:

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

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

C) $(-\sqrt{2},7\sqrt{2})$

D) $(\sqrt{2},7\sqrt{2})$

E) $(\sqrt{2},-7\sqrt{2})$

• question_answer158) Suppose a circle passes through (2, 2) and (9, 9) and touches the$x-$axis at P. If O is the origin, then OP is equal to:

A) 4

B) 5

C) 6

D) 9

E) 11

• question_answer159) If ${{e}^{{{e}^{x}}}}={{a}_{0}}+{{a}_{1}}x+{{a}_{2}}{{x}^{2}}+......,$then:

A) ${{a}_{0}}=1$

B) ${{a}_{0}}=e$

C) ${{a}_{0}}={{e}^{e}}$

D) ${{a}_{0}}={{e}^{2}}$

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

• question_answer160) The period of the function $f(x)=|\sin x|+|\cos x|$is:

A) $2\pi$

B) $3\pi$

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

D) $\pi$

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

• question_answer161) The equation of the line passing through the origin and the point of intersection of the lines$\frac{x}{a}+\frac{y}{b}=1$and$\frac{x}{b}+\frac{y}{a}=1$is:

A) $bx-ay=0$

B) $x+y=0$

C) $ax-by=0$

D) $x-y=0$

E) $ax+by=0$

• question_answer162) If$\overrightarrow{a}$and$\overrightarrow{b}$are unit vectors such that $[\overrightarrow{a}\,\overrightarrow{b}\,\overrightarrow{a}\times \overrightarrow{b}]=\frac{1}{4},$then angle between$\overrightarrow{a}$and$\overrightarrow{b}$is:

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

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

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

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

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

• question_answer163) If the eccentricities of the ellipse $\frac{{{x}^{2}}}{4}+\frac{{{y}^{2}}}{3}=1$ and the hyperbola$\frac{{{x}^{2}}}{64}-\frac{{{y}^{2}}}{{{b}^{2}}}=1$are reciprocals of each other, then${{b}^{2}}$is equal to:

A) 192

B) 64

C) 16

D) 32

E) 128

• question_answer164) The order and degree of the differential equation$\sqrt{\sin x}(dx+dy)=\sqrt{\cos x}(dx-dy)$is:

A) (1, 2)

B) (2, 2)

C) (1, 1)

D) (2, 1)

E) (0, 1)

• question_answer165) The domain of the real valued function$f(x)=\sqrt{5-4x-{{x}^{2}}}+{{x}^{2}}\log (x+4)$is:

A) $-5\le x\le 1$

B) $-5\le x$ and$x>1$

C) $-4<x\le 1$

D) $\phi$

E) $0\le x\le 1$

• question_answer166) If${{a}_{1}},{{a}_{2}},.....,{{a}_{50}}$are in GP, then $\frac{{{a}_{1}}-{{a}_{3}}+{{a}_{5}}-....+{{a}_{49}}}{{{a}_{2}}-{{a}_{4}}+{{a}_{6}}-....+{{a}_{50}}}$is equal to:

A) 0

B) $1$

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

D) $\frac{{{a}_{25}}}{{{a}_{24}}}$

E) $\frac{2{{a}_{1}}}{3{{a}_{2}}}$

• question_answer167) Suppose the number of elements in set A is p, the number of elements in set B is q and the number of elements in$A\times B$is 7. Then ${{p}^{2}}+{{q}^{2}}$is equal to:

A) 42

B) 49

C) 50

D) 51

E) 55

• question_answer168) The value of$\cos [2{{\tan }^{-1}}(-7)]$is:

A) $\frac{49}{50}$

B) $-\frac{49}{50}$

C) $\frac{24}{25}$

D) $-\frac{24}{25}$

E) $\frac{48}{49}$

• question_answer169) $\int\limits_{0}^{\pi }{|\cos x|}dx$is equal to:

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

B) $-2$

C) $1$

D) $-1$

E) $2$

• question_answer170) $\int{\left( \frac{\sin 2x}{\sin 3x\sin 5x} \right)}dx$is equal to:

A) $\frac{1}{5}{{\log }_{e}}|\sin 5x|-\frac{1}{3}{{\log }_{e}}|\sin 3x|+c$

B) $\frac{1}{3}{{\log }_{e}}|\sin 3x|-\frac{1}{5}{{\log }_{e}}|\sin 5x|+c$

C) $\frac{1}{3}{{\log }_{e}}|\sin 3x|+\frac{1}{5}{{\log }_{e}}|\sin 5x|+c$

D) $-\frac{1}{2}\cos 2x+\frac{1}{3}{{\log }_{e}}|\sin 3x|$ $+\frac{1}{5}{{\log }_{e}}|sin5x|+c$

E) $-\frac{1}{2}\cos 2x-\frac{1}{3}{{\log }_{e}}|\sin 3x|$$-\frac{1}{5}{{\log }_{e}}|\sin 5x|+c$

• question_answer171) If$\left| \begin{matrix} 2i & -3i & 1 \\ 3 & 3i & -1 \\ 4 & 3 & i \\ \end{matrix} \right|=x+iy,$then:

A) $x=3,\text{ }y=1$

B) $x=2,\text{ }y=5$

C) $x=0,y=0$

D) $x=1,\text{ }y=1$

E) $x=0,\text{ }y=5$

• question_answer172) If $y={{\log }_{a}}x,x>0,$then,$\frac{dy}{dx}$is equal to:

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

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

C) $\frac{1}{x}{{\log }_{x}}a$

D) $\frac{1}{a}{{\log }_{e}}x$

E) $\frac{1}{x}{{\log }_{a}}e$

• question_answer173) The value of$\left| \begin{matrix} ^{10}{{C}_{4}} & ^{10}{{C}_{5}} & ^{11}{{C}_{m}} \\ ^{11}{{C}_{6}} & ^{11}{{C}_{7}} & ^{12}{{C}_{m+2}} \\ ^{12}{{C}_{8}} & ^{12}{{C}_{9}} & ^{13}{{C}_{m+4}} \\ \end{matrix} \right|=0$when m is equal to:

A) 6

B) 5

C) 4

D) 1

E) 2

• question_answer174) The minimum value of$2\cos \theta +\frac{1}{\sin \theta }$ $+\sqrt{2}\tan \theta$the interval$\left( 0,\frac{\pi }{2} \right)$is:

A) $2+\sqrt{2}$

B) $3\sqrt{2}$

C) $2\sqrt{3}$

D) $3+\sqrt{2}$

E) 7

• question_answer175) If$\frac{1}{^{4}{{C}_{n}}}=\frac{1}{^{5}{{C}_{n}}}+\frac{1}{^{6}{{C}_{n}}},$then n is equal to:

A) 3

B) 2

C) 1

D) 0

E) 4

• question_answer176) Let${{x}_{1}}$and${{x}_{2}}$be solutions of the equation${{\sin }^{-1}}\left( {{x}^{2}}-3x+\frac{5}{2} \right)=\frac{\pi }{6}$.Then, the value of $x_{1}^{2}+x_{2}^{2}$is:

A) 4

B) 5

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

D) 6

E) $\frac{15}{2}$

• question_answer177) If the points (- 1, 2, - 3), (4, a, 1) and (b, 8, 5) are collinear, then a and b are respectively equal to:

A) 5 and 5

B) 9 and 5

C) 5 and 9

D) $-5$and 9

E) 5 and$-9$

• question_answer178) The locus of the point$(l,m)$so that$lx+my=1$ touches the circle${{x}^{2}}+{{y}^{2}}={{a}^{2}}$is:

A) ${{x}^{2}}+{{y}^{2}}-ax=0$

B) ${{x}^{2}}+{{y}^{2}}=\frac{1}{{{a}^{2}}}$

C) ${{y}^{2}}=4ax$

D) ${{x}^{2}}+{{y}^{2}}-ax-ay+{{a}^{2}}=0$

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

• question_answer179) If the derivative of the function$f(x)$is every where continuous and is given by $f(x)=\left\{ \begin{matrix} b{{x}^{2}}+ax+4; & x\ge -1 \\ a{{x}^{2}}+b; & x<-1 \\ \end{matrix}, \right.$then:

A) $a=2,b=-3$

B) $a=3,b=2$

C) $a=-2,b=-3$

D) $a=-3,b=-2$

E) $a=-1,b=-2$

• question_answer180) Let$a={{e}^{i\frac{2\pi }{3}}}$. Then the equation whose roots are$a+{{a}^{-2}}$and${{a}^{2}}+{{a}^{-4}}$is:

A) ${{x}^{2}}-2x+4=0$

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

C) ${{x}^{2}}+x+4=0$

D) ${{x}^{2}}+2x-4=0$

E) ${{x}^{2}}+2x+4=0$

• question_answer181) Let$n=2006!.$Then $\frac{1}{{{\log }_{2}}n}+\frac{1}{{{\log }_{3}}n}+...+\frac{1}{{{\log }_{2006}}n}$is equal to:

A) 2006

B) 2005

C) 2005!

D) 1

E) 0

• question_answer182) $\int{{{e}^{x}}}\{log\text{ }sin\text{ }x+cot\text{ }x\}dx$ is equal to:

A) ${{\text{e}}^{x}}cot\text{ }x+c$

B) ${{e}^{x}}\,log\text{ }sin\text{ }x+c$

C) ${{e}^{x}}log\text{ }sin\text{ }x+tan\text{ }x+c$

D) ${{e}^{x}}+sin\text{ }x+c$

E) $log(sin\,x+cos\,x)+{{e}^{x}}+c$

• question_answer183) $\int\limits_{-10}^{10}{\log \left( \frac{a+x}{a-x} \right)}dx$is equal to:

A) 0

B) $-2\log (a+10)$

C) $2\log \left( \frac{a+10}{a-10} \right)$

D) $2\log (a+10)$

E) $2$

• question_answer184) If$f(x+y)=f(x)f(y)$for all real$x$and$y,$ $f(6)=3$and$f(0)=10,$then$f(6)$is:

A) 30

B) 13

C) 10

D) 0

E) 6

• question_answer185) The value of the determinant, $\left| \begin{matrix} \sqrt{13}+\sqrt{3} & 2\sqrt{5} & \sqrt{5} \\ 15+\sqrt{26} & 5 & \sqrt{10} \\ 3+\sqrt{65} & \sqrt{15} & 5 \\ \end{matrix} \right|$is:

A) $5(\sqrt{6}-5)$

B) $5\sqrt{3}(\sqrt{6}-5)$

C) $\sqrt{5}(\sqrt{6}-\sqrt{3})$

D) $\sqrt{2}(\sqrt{7}-\sqrt{5})$

E) $3(\sqrt{5}-\sqrt{2})$

• question_answer186) If${{\alpha }_{1}},{{\alpha }_{2}},{{\alpha }_{3}},{{\alpha }_{4}}$are the roots of the equation${{x}^{4}}+(2-\sqrt{3}){{x}^{2}}+2+\sqrt{3}=0,$then the value of$(1-{{\alpha }_{1}})(1-{{\alpha }_{2}})(1-{{\alpha }_{3}})(1-{{\alpha }_{4}})$is:

A) 1

B) 4

C) $2+\sqrt{3}$

D) 5

E) 0

• question_answer187) The values of$\lambda$ so that the line$3x-4y=\lambda$ touches${{x}^{2}}+{{y}^{2}}-4x-8y-5=0$are:

A) $-35,15$

B) $3,-5$

C) $35,-15$

D) $-3,5$

E) 20, 15

• question_answer188) Suppose$0<t<\frac{\pi }{2}$and$\sin t+\cos t=\frac{1}{5}$ then $\tan \frac{t}{2}$ is equal to:

A) 2

B) 3

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

D) 5

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

• question_answer189) The point of intersection of the line$\overrightarrow{r}=7\hat{i}+10\hat{j}+13\hat{k}+s(2\hat{i}+3\hat{j}+4\hat{k})$and$\overrightarrow{r}=3\hat{i}+5\hat{j}+7\hat{k}+t(\hat{i}+2\hat{j}+3\hat{k})$is:

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

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

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

D) $\hat{i}-\hat{j}-\hat{k}$

E) $\hat{i}+\hat{j}+\hat{k}$

• question_answer190) The value of $cos\text{ }480{}^\circ -sin\text{ }150{}^\circ +sin\text{ }600{}^\circ .cos\text{ }390{}^\circ$is equal to:

A) 0

B) 1

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

D) $-1$

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

• question_answer191) Define$f(x)=\int\limits_{0}^{x}{\sin t}\,dt,\,x\ge 0$Then:

A) $f$is increasing only in the interval $\left[ 0,\frac{\pi }{2} \right]$

B) $f$is decreasing in the interval $[0,\pi ]$

C) $f$attains maximum at $x=\frac{\pi }{2}$

D) $f$attains minimum at$x=\pi$

E) $f$attains maximum at$x=\pi$

• question_answer192) Let $f(x)=\frac{{{\sin }^{2}}\pi x}{1+{{\pi }^{x}}}$ Then, $\int{(f(x)+f(-x))}dx$is equal to:

A) $0$

B) $x+c$

C) $\frac{x}{2}-\frac{\cos \pi x}{2\pi }+c$

D) $\frac{1}{1+{{\pi }^{x}}}\frac{{{\cos }^{2}}\pi x}{2\pi }+c$

E) $\frac{x}{2}-\frac{\sin 2\pi x}{4\pi }+c$

• question_answer193) For the arithmetic progression $a,(a+d),(a+2d),(a+3d),....,(a+2nd),$ the mean deviation from mean is:

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

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

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

D) $\frac{(n+1)d}{2}$

E) $\frac{n(n-1)d}{2n-1}$

• question_answer194) If the angles of a triangle are in the ratio$3:4:5,$then the ratio of the largest side to the smallest side of the triangle is:

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

B) $\frac{\sqrt{3}+1}{2\sqrt{2}}$

C) $\frac{\sqrt{3}+1}{2}$

D) $\frac{\sqrt{3}+1}{\sqrt{2}}$

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

• question_answer195) If$\frac{\sin (x+y)}{\sin (x-y)}=\frac{a+b}{a-b},$then$\frac{\tan x}{\tan y}$is equal to:

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

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

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

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

E) $\frac{{{a}^{2}}-{{b}^{2}}}{{{a}^{2}}+{{b}^{2}}}$

• question_answer196) Let R be a relation on the set of integers given by$aRb\Leftrightarrow a={{2}^{k}}.b$for some integer k. Then R is:

A) an equivalence relation

B) reflexive but not symmetric

C) reflexive and transitive but not symmetric

D) reflexive and symmetric but not transitive

E) symmetric and transitive but not reflexive

• question_answer197) . The position of reflection of the point (4, 1) about the line$y=x-1$is:

A) (1, 2)

B) (3, 4)

C) $(-\text{ }1,0)$

D) $(-2,-1)$

E) (2, 3)

• question_answer198) Derivative of${{\sec }^{-1}}\left( \frac{1}{1-2{{x}^{2}}} \right)$w.r.t. ${{\sin }^{-1}}(3x-4{{x}^{3}})$is:

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

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

C) 1

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

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

• question_answer199) Let ABCD be a parallelogram and let E be the midpoint of side AB. If EC is perpendicular to ED, then:

A) $ED=EC$

B) $EB=EC$

C) $EA=ED$

D) $EC+ED=2BC$

E) $EC+ED=2DC$

• question_answer200) The radius of a circle is increasing at the rate of 0.1 cm/s. When the radius of the circle is 5 cm, the rate of change of its area, is:

A) $-\pi \,c{{m}^{2}}/s$

B) $10\pi \,c{{m}^{2}}/s$

C) $0.1\pi \,c{{m}^{2}}/s$

D) $5\pi \,c{{m}^{2}}/s$

E) $\pi \,c{{m}^{2}}/s$

• question_answer201) Let$D=\{1,2,3,5,6,10,15,30\}$. Define the operations +, .and on D as follows $a+b=LCM(a,b),$$a.b=GCD(a,b)$and $a=\frac{30}{a}.$Then$(15+6).10$is equal to:

A) 1

B) 2

C) 3

D) 5

E) 10

• question_answer202) If$A+B=\frac{\pi }{4},$then$(tan\text{ }A+1)\text{ }(tan\text{ }B+1)$equals:

A) 1

B) $\sqrt{3}$

C) 2

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

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

• question_answer203) ${{\log }_{e}}3-\frac{{{\log }_{e}}9}{{{2}^{2}}}+\frac{{{\log }_{e}}27}{{{3}^{2}}}-\frac{{{\log }_{e}}81}{{{4}^{2}}}+...$is:

A) $({{\log }_{e}}3)({{\log }_{e}}2)$

B) ${{\log }_{e}}3$

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

D) $\frac{{{\log }_{e}}5}{{{\log }_{e}}3}$

E) $\frac{{{\log }_{e}}3}{{{\log }_{e}}2}$

• question_answer204) The roots of the equation $(q-r){{x}^{2}}+(r-p)x+(p-q)=0$are:

A) $\frac{r-p}{q-r},1$

B) $\frac{p-q}{q-r},1$

C) $\frac{p-r}{q-r},2$

D) $\frac{q-r}{p-q},2$

E) $\frac{r-p}{p-q},1$

• question_answer205) The interior angles of a polygon are in AP. The smallest angle is$120{}^\circ$and the common difference is$5{}^\circ$. The number of sides of the polygon is:

A) 9

B) 10

C) 16

D) 5

E) 8

• question_answer206) The direction cosines of the line$4x-4=1-3y=2z-1$are:

A) $\frac{3}{\sqrt{56}},\frac{-4}{\sqrt{56}},\frac{6}{\sqrt{56}}$

B) $\frac{3}{\sqrt{29}},\frac{-4}{\sqrt{29}},\frac{6}{\sqrt{29}}$

C) $\frac{3}{\sqrt{61}},\frac{-4}{\sqrt{61}},\frac{6}{\sqrt{61}}$

D) $4,-3,2$

E) $\frac{4}{\sqrt{29}},\frac{-3}{\sqrt{29}},\frac{2}{\sqrt{29}}$

• question_answer207) The digit at the unit place in the number${{19}^{2005}}+{{11}^{2005}}-{{9}^{2005}}$is:

A) 2

B) 1

C) 0

D) 8

E) 9

• question_answer208) If$x{{e}^{xy}}=y+si{{n}^{2}}x,$then$\frac{dy}{dx}$at$x=0$is:

A) $-1$

B) 0

C) 1

D) 2

E) $-2$

• question_answer209) The number of triangles which can be formed by using the vertices of a regular polygon of $(n+3)$sides is 220. Then n is equal to:

A) 8

B) 9

C) 10

D) 11

E) 12

• question_answer210) lf$[x]$denotes the greatest integer$\le x,$then$\left[ \frac{2}{3} \right]+\left[ \frac{2}{3}+\frac{1}{99} \right]+\left[ \frac{2}{3}+\frac{2}{99} \right]+....+\left[ \frac{2}{3}+\frac{98}{99} \right]$is equal to:

A) 99

B) 98

C) 66

D) 65

E) 33

• question_answer211) The centre of the sphere passing through the origin and through the intersection points of the plane$\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$with axes is:

A) $\left( \frac{a}{2},0,0 \right)$

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

C) $\left( 0,0,\frac{a}{2} \right)$

D) $\left( \frac{a}{2},\,\frac{b}{2},\,0 \right)$

E) $\left( \frac{a}{2},\frac{a}{2},\frac{c}{2} \right)$

• question_answer212) In an arithmetic progression, the 24 th term is 100. Then the sum of the first 47 terms of the arithmetic progression is:

A) 2300

B) 2350

C) 2400

D) 4600

E) 4700

• question_answer213) If the vertex of the parabola$y={{x}^{2}}-16x+k$lies on$x-$axis, then the value of k is:

A) 16

B) 8

C) 64

D) $-64$

E) $-8$

• question_answer214) Let co be an imaginary root of${{x}^{n}}=1$. Then $(5-\omega ){{(5-\omega )}^{2}}....(5-{{\omega }^{n-1}})$is:

A) 1

B) $\frac{{{5}^{n}}+1}{4}$

C) ${{4}^{n-1}}$

D) $\frac{{{5}^{n}}-1}{4}$

E) ${{5}^{n-1}}$

• question_answer215) An integrating factor of the differential equation, $(1+y+{{x}^{2}}y)dx+(x+{{x}^{3}})dy=0$is:

A) $\log x$

B) $x$

C) ${{e}^{x}}$

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

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

• question_answer216) The line$2x-y=1$bisects angle between two lines. If equation of one line is$y=x,$then the equation of the other line is:

A) $7x-y-6=0$

B) $x-2y+1=0$

C) $3x-2y-1=0$

D) $x-7y+6=0$

E) $2x-3y+1=0$

• question_answer217) If$f(x)=\tan x-{{\tan }^{3}}x+{{\tan }^{5}}x-.....$to$\infty$with $0<x<\frac{\pi }{4},$ then$\int_{0}^{\frac{\pi }{4}}{f(x)}dx$is equal to:

A) 1

B) 0

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

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

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

• question_answer218) The standard deviation of n observations ${{x}_{1}},{{x}_{2}},.....,{{x}_{n}}$is 2. If$\sum\limits_{i=1}^{n}{{{x}_{i}}}=20$and$\sum\limits_{i=1}^{n}{x_{i}^{2}}=100,$then n is:

A) 10 or 20

B) 5 or 10

C) 5 or 20

D) 5 or 15

E) 25

• question_answer219) Area (in sq unit) enclosed by $y=1,\,2x+y=2$ and$x+y=2$is:

A) $\frac{1}{2}sq\,unit$

B) $\frac{1}{4}sq\,unit$

C) $1sq\,unit$

D) $2\,sq\,unit$

E) 4 sq unit

• question_answer220) The ratio in which the line$x+y=4$divides the line joining the points$(1,-1)$and (5, 7) is:

A) $1:2$

B) $2:1$

C) $1:3$

D) $3:1$

E) $3:2$

• question_answer221) The set of all$x$satisfying the inequality$\frac{4x-1}{3x+1}\ge 1$is:

A) $\left( -\infty ,-\frac{1}{3} \right)\cup \left[ \frac{1}{4},\infty \right)$

B) $\left( -\infty ,-\frac{2}{3} \right)\cup \left[ \frac{1}{5},\infty \right)$

C) $\left( -\infty ,-\frac{1}{3} \right)\cup \left[ 2,\infty \right)$

D) $\left( -\infty ,-\frac{2}{3} \right)\cup \left[ 4,\infty \right)$

E) $\left( -\infty ,-\frac{1}{3} \right)\cup \left[ \frac{1}{2},\infty \right)$

• question_answer222) $\left| \begin{matrix} a+x & b & c \\ a & b+y & c \\ a & b & c+z \\ \end{matrix} \right|$is equal to:

A) $abc\left( 1+\frac{x}{a}+\frac{y}{b}+\frac{z}{c} \right)$

B) $abc\left( 1+\frac{a}{x}+\frac{b}{y}+\frac{c}{z} \right)$

C) $xyz\left( 1+\frac{a}{x}+\frac{b}{y}+\frac{c}{z} \right)$

D) $xyz\left( 1+\frac{x}{a}+\frac{y}{b}+\frac{z}{c} \right)$

E) $xyz(a+b+c+1)$

• question_answer223) If$^{18}{{C}_{15}}+c{{(}^{18}}{{C}_{16}}){{+}^{17}}{{C}_{16}}+1{{=}^{n}}{{C}_{3}},$then n is equal to:

A) 19

B) 20

C) 18

D) 24

E) 21

• question_answer224) $\underset{n\to \infty }{\mathop{\lim }}\,\left( \frac{{{1}^{2}}}{1-{{n}^{3}}}+\frac{{{2}^{2}}}{1-{{n}^{3}}}+....+\frac{{{n}^{2}}}{1-{{n}^{3}}} \right)$is equal to:

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

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

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

D) $-\frac{1}{6}$

E) 0

• question_answer225) If $\overrightarrow{A}=\hat{i}+2\hat{j}+3\hat{k},\overrightarrow{B}=-\hat{i}+2\hat{j}+\hat{k}$and $\overrightarrow{C}=3\hat{i}+\hat{j},$then$\overrightarrow{A}+t\overrightarrow{B}$ is perpendicular to $\overrightarrow{C},$ if t is equal to:

A) $-5$

B) 4

C) 5

D) $-4$

E) $-7$

• question_answer226) The number of ways in which one can select three distinct integers between 1 and 30, both inclusive, whose sum is even, is:

A) 455

B) 1575

C) 1120

D) 2030

E) 1930

• question_answer227) The set of values of$x$satisfying$2\le \,|x-3|<4$ is:

A) $(-1,1]\cup [5,7)$

B) $-4\le x\le 2$

C) $-1<x<7or\,x\ge 5$

D) $x<7or\,x\ge 5$

E) $-\infty \,<x\le 1\,or\,\,5\le x<8$

• question_answer228) If$i=\sqrt{-1},$then $4+5{{\left( -\frac{1}{2}+\frac{i\sqrt{3}}{2} \right)}^{334}}+3{{\left( -\frac{1}{2}+\frac{i\sqrt{3}}{2} \right)}^{365}}$is equal to:

A) $1-i\sqrt{3}$

B) $-1+i\sqrt{3}$

C) $i\sqrt{3}$

D) $-i\sqrt{3}$

E) $1+i\sqrt{3}$

• question_answer229) The angle between the lines$\sqrt{3}x-y-2=0$ and$x-\sqrt{3}y+1=0$is:

A) $90{}^\circ$

B) ${{60}^{\text{o}}}$

C) $45{}^\circ$

D) $15{}^\circ$

E) $30{}^\circ$

• question_answer230) Let$f(x)=x-[x],$for every real$x,$where$[x]$is the greatest integer less than or equal to$x$. Then,$\int\limits_{-1}^{1}{f(x)}dx$is:

A) 1

B) 2

C) 3

D) 0

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

• question_answer231) If$\int\limits_{0}^{{{x}^{2}}}{f(t)}dt=x\cos \pi x,$then the value of$f(4)$is:

A) $1$

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

C) $-1$

D) $\frac{-1}{4}$

E) $-4$

• question_answer232) If$y=\left( 1+\frac{1}{x} \right)\left( 1+\frac{2}{x} \right)\left( 1+\frac{3}{x} \right)......\left( 1+\frac{n}{x} \right)$ and$x\ne 0$then$\frac{dy}{dx}$when$x=-1$is:

A) $n!$

B) $(n-1)!$

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

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

E) $(n+1)!$

• question_answer233) If in a triangle ABC,$a=15,\text{ }b=36,\text{ }c=39,$then sin c is equal to:

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

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

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

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

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

• question_answer234) Let A = {1, 2, 3, 4}, B = {2, 4, 6}. Then the number of sets C such that$A\cap B\subseteq C\subseteq A\cup B$is:

A) $6$

B) $9$

C) $8$

D) $10$

E) $12$

• question_answer235) A force of magnitude $\sqrt{6}$ acting along line joining the points$A(2,-1,1)$and B (3, 1, 2) displaces a particle from A to B. The work done by ?e force is:

A) 6

B) $6\sqrt{6}$

C) $\sqrt{6}$

D) 12

E) $2\sqrt{6}$

• question_answer236) If z is a complex number such that $\operatorname{Re}(z)=\operatorname{Im}(z),$then:

A) $\operatorname{Re}({{z}^{2}})=0$

B) $\operatorname{Im}({{z}^{2}})=0$

C) $\operatorname{Re}({{z}^{2}})=\operatorname{Im}({{z}^{2}})$

D) $\operatorname{Re}({{x}^{2}})=-\operatorname{Im}({{z}^{2}})$

E) ${{z}^{2}}=0$

• question_answer237) Equation of the plane parallel to the planes $c+2y+3z-5=0,\text{ }x+2y+3z-7=0$and equidistant from them is:

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

B) $x+2y+3z-1=0$

C) $x+2y+3z-8=0$

D) $x+2y+\text{ }3z-3=0$

E) $x+2y+3z-10=0$

• question_answer238) The number of binary operations that can be defined on the set$A=\{a,\text{ }b,\text{ }c\}$ is:

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

B) ${{3}^{4}}$

C) ${{3}^{9}}$

D) ${{9}^{3}}$

E) 3

• question_answer239) If${{(2{{x}^{2}}-x-1)}^{5}}={{a}_{0}}+{{a}_{1}}x+{{a}_{2}}{{x}^{2}}+...$ $+{{a}_{10}}{{x}^{10}},$ then${{a}_{2}}+{{a}_{4}}+{{a}_{6}}+{{a}_{8}}+{{a}_{10}}$is equal to:

A) 15

B) 30

C) 16

D) 32

E) 17

• question_answer240) If the plane$2x-y+z=0$is parallel to the line $\frac{2x-1}{2}=\frac{2-y}{2}=\frac{z+1}{a},$then the value of a is:

A) 4

B) $-4$

C) 2

D) $-2$

E) 0