Physics

Single correct answer type:

1. An unknown metal of mass heated to a temperature of was immersed into a

brass calorimeter of mass containing of water at a temperature of Calculate

the specific heat of the unknown metal if water temperature stabilizes at Specific heat

of brass is

(A)

(B)

(C)

(D)

Solution: (A)

Conservation of energy:

2. Two vectors and have equal magnitudes. The magnitude of ( ) is ‘ ’ times the

magnitude of ( ). The angle between and is:

(A) *

+

(B) *

+

(C) *

+

(D) *

+

Solution: (A)

Given | | | |

√ (√ )

Also | | | |

√ √

Squaring both sides:

i.e., *

+

3. A parallel plane capacitor having capacitance is charged by a battery to a potential

difference of between its plates. The charging battery is now disconnected and a porcelain

slab of dielectric constant is slipped between the plates. The work done by the capacitor on

the slab is:

(A) 560

(B)

(C)

(D)

Solution: (C)

( ) (

)

(

)

(

)

4. A closed organ pipe has a fundamental frequency of The number of overtones that

can be distinctly heard by a person with this organ pipe will be: Assume that the highest

frequency a person can hear is

(A)

(B)

(C)

(D)

Solution: (A)

Possible harmonics

No. of overtones

5. Consider a Young’s double slit experiment as shown in figure. What should be the slit

separation in terms of wavelength such that the first minima occurs directly in front of the slit

?

(A)

√

(B)

√

(C)

√

(D)

√

Solution: (C)

Path diff. in front of a slit on screen

√

(for minima)

For minima

(√ )

(√ )

6. The actual value of resistance , shown in the figure is This is measured in an

experiment as shown using the standard formula

, where and are the readings of the

voltmeter and ammeter, respectively. If the measured value of is less, then the internal

resistance of the voltmeter is:

(A)

(B)

(C)

(D)

Solution: (C)

Let the measured voltage be and

Let the measured current be and

Let the ammeter be ideal, thus

(

)

7. A hoop and a solid cylinder of same mass and radius are made of a permanent magnetic

material with their respective axes. But the magnetic moment of hoop is twice of solid cylinder.

They are placed in a uniform magnetic moments make a small angle with the field. If the

oscillation periods of hoop and cylinder are and respectively, then:

(A)

(B)

(C)

(D)

Solution: (B)

Time period of any magnetic body place in an external magnetic field is:-

√

√

……

√

……

From and

8. A particle executes simple harmonic motion with an amplitude of . When the particle is at

from the mean position, the magnitude of its velocity in units is equal to that of its

acceleration. Then, its periodic time in seconds is:

(A)

(B)

(C)

(D)

Solution: (A)

√

| | | |

9. The eye can be regarded as a single refracting surface. The radius of curvature of this

surface is equal to that of cornea . This surface separates two media of refractive

indices and . Calculate the distance from the refracting surface at which a parallel beam

of light will come to focus.

(A)

(B)

(C)

(D)

Solution: (A)

We know eye lens is convex refracting surface.

Use expression for refraction from curved surface

Refractive index of medium having incident ray

Refractive index of medium having reflected ray

10. Two forces and ,of magnitude and respectively, are at an angle with each

other. If the force is doubled, then their resultant also gets doubled. Then, the angle is:

(A)

(B)

(C)

(D)

Solution: (A)

| | | |

| | √

√ √

If is doubled

| | √ √

11. Charges and located at and , respectively, constitute an electric dipole. Distance

is the mid point of the dipole and is perpendicular to . A charge is placed at

where and The charge experiences an electrostatic force If is now

moved along the equatorial line to such that (

), the force on will be close to:

(

)

(A)

(B)

(C)

(D)

Solution: (A)

Electric field intensity at distance

Along the perpendicular bisector is:-

| |

So, force at distance on ‘

….

….

i.e.,

12. The self induced emf of a coil is volts. When the current in it is changed at uniform rate

from to in , the change in the energy of the inductance is:

(A)

(B)

(C)

(D)

Solution: (C)

| |

Energy of inductor

[ ]

13. A particle which is experiencing a force, given by , undergoes a displacement of

. If the particle had a kinetic energy of at the beginning of at the end of the

displacement?

(A)

(B)

(C)

(D)

Solution: (B)

Work done by force

By work energy theorem,

14. Two stars of masses each, and at distance rotate in a plane about

their common centre of mass Aa meteorite passes through moving perpendicular to the

star’s rotation plane. In order to escape from the gravitational field of this double star, the

minimum speed that meteorite should have at is: Take Gravitational constant

(A) ⁄

(B) ⁄

(C) ⁄

(D) ⁄

Solution: (C)

To escape out the total energy of small particle must be zero.

(

)

√

√

⁄

15. Two identical spherical balls of mass and radius each are stuck on two ends of a rod of

length and mass (see figure). The moment of inertia of the system about the axis passing

perpendicularly through the centre of the rod is:

(A)

(B)

(C)

(D)

Solution: (C)

[

]

16. Two kg of a monoatomic gas is at a pressure of ⁄ . The density of the gas is

⁄ . What is the order of energy of the gas due to its thermal motion?

(A)

(B)

(C)

(D)

Solution: (D)

Internal Energy of an g Ideal gas given by

(for n-moles)

[ for monoatomic gas]

17. A metal plate of area is illuminated by a radiation of intensity

. The work

function of the metal is The energy of the incident photons is and only of it

produces photo electrons per second and their maximum energy, respectively, will be: [

]

(A) and

(B) and

(C) and

(D) and

Solution: (C)

Maximum

Intensity ( )

(

)

and number of electrons per second. (

)

(

)

per sec.

18. A rigid massless rod of length has two masses attached at each end as shown in the

figure. The rod is pivoted at point P on the horizontal axis (see figure). When released from

initial horizontal position, its instantaneous angular acceleration will be:

(A)

(B)

(C)

(D)

Solution: (D)

Both the weight will provide torque in opposite direction:

Use: [where about axis of rotation

]

19. A cylindrical plastic bottle of negligible mass is filled with of water and left floating in

a pond with still water. If pressed downward slightly and released, it starts performing simple

harmonic motion at angular frequency If the radius of the bottle is then is close to:

density of water ⁄

(A) 5.00 rad

(B)

(C)

(D)

Solution: (C)

Let at equilibrium, length of bottle is immerged in water:

So,

….. [Equilibrium Condition]

Now it is further pushed by distance

[ ]

√

√

√

⁄

20. The modulation frequency of an radio station is which is of the carrier

wave. If another station approaches you for license what broadcast frequency will you allot?

(A)

(B)

(C)

(D)

Solution: (B)

Carrier wave frequency

Frequency used to

The allotted frequency showed be such that should not overlap with previous one.

Hence possible frequency

21. The Wheatstone bridge shown in figure here, gets balanced when the carbon resistor used

as has the colour code (Orange, Red, Brown). The resistors and are and ,

respectively. Assuming that the colour code for the carbon resistors gives their accurate values,

the colour code for the carbon resistor, used as , would be:

(A) Brown-Blue-Black

(B) Brown-Blue-Brown

(C) Grey-Black-Brown

(D) Red-Green-Brown

Solution: (B)

Condition for Balance wheat stone Bridge:-

For :- orange-Red-Brown

So,

Brown Blue Brown

22. Half mole of an ideal monoatomic gas is heated at constant pressure of from to

. Work done by gas is close to: Gas constant ⁄

(A)

(B)

(C)

(D)

Solution: (C)

23. The electric field of a plane polarized electromagnetic wave in free space at time is

given by an expression [ ]. The magnetic field is given by:

is the velocity of light

(A)

( ) [ ]

(B)

( ) [ ]

(C)

( ) [ ]

(D)

[ ]

Solution: (D)

At any time t;

Also,

|

|

( )

[ ]

Comparing coefficient both sides

| |

( )

So magnetic field

( )

24.

If binding energy per nucleon of is is and is then

what is value of the reaction

(A)

(B)

(C)

(D)

Solution: (B)

25. For the circuit shown below, the current through the Zener diode is:

(A) Zero

(B)

(C)

(D)

Solution: (C)

Current through

Current through

Current through Zennor diode

26. A particle starts form the origin at time and moves along the positive axis. The

graph of velocity with respect to time is shown in figure. What is the position of the particle at

time

(A) 10

(B) 9

(C)

(D)

Solution: (B)

Area enclosed by graph=displacement (As particle moving in same direction)

(

) Displacement

So, displacement

27. At some location one earth the horizontal component of earth’s magnetic field is

At this location, magnetic needle of length and pole strength is suspended from

its mid-point using a thread, it makes angles with horizontal in equilibrium. To keep this

needle horizontal, the vertical force that should be applied at one of its ends is:

(A)

(B)

(C)

(D)

Solution: (C)

at ,

28. Four equal point charges each are placed in the plane at and

. The work required to put a fifth charge at the origin of the coordinate system will be:

(A)

(B)

√

(C)

(

√ )

(D)

(

√ )

Solution: (C)

if (assumed)

*

√

√ +

[

√ ]

29. A current of was passed through an unknown resistor which dissipated a power of

Dissipated power when an ideal power supply of is connected across it is:

(A)

(B)

(C)

(D)

Solution: (C)

Power

When power supply is connected

30. The diameter and height of a cylinder are measured by a meter scale to be

and , respectively. What will be the value of its volume in appropriate significant

figures?

(A)

(B)

(C)

(D)

Solution: (C)

Converting to errors

(

) (

)

Chemistry

Single correct answer type:

1. The electron of an element with an atomic number of enters into the orbital:

(A)

(B)

(C)

(D)

Solution: (A)

, - (Last electron enters orbital)

2. Haemoglobin and gold sol are examples of:

(A) Positively charged sols

(B) Negatively and positively charged sols, respectively

(C). Negatively charged sols

(D) Positively and negatively charged sols, respectively

Solution: (D)

Gold sol is negatively charged, while the haemoglobin is positively charged.

3. Among the following reactions of hydrogen with halogens, the one that requires a catalyst is:

(A)

(B)

(C)

(D)

Solution: (B)

is least reactive halogen. So its reaction with required catalyst.

4. The number of - centre- -electron and -centre- -electron bonds in respectively, are:

(A)

(B)

(C)

(D)

Solution: (A)

has bridge bonds ( ) and terminal bonds ( )

5. The ground state energy of hydrogen atom is The energy second excited state of ion in

is:

(A)

(B)

(C)

(D)

Solution: (B)

.

/ .

/ (second exited state third shell, )

6. The amount of sugar( ) required to prepare of its aqueous solution is:

(A)

(B)

(C)

(D)

Solution: (C)

Then

( )

Moles of sucrose required

7. A reaction of cobalt( ) chloride and ethylenediamine in a mole ratio generates two isomeric

products (violet coloured) and (green coloured). A can show optical activity, but, is optically

inactive. What type of isomers does and represent?

(A) Coordination isomers

(B) Linkage isomers

(C) Ionisation isomers

(D) Geometrical isomers

Solution: (D)

, ( ) -

1 mole 2 mole

8. In the reaction of oxalate with permanganate in acidic medium, the number of electrons involved in

producing one molecule of is:

(A)

(B)

(C)

(D)

Solution: (C)

goes from oxidation state to oxidation state.

Total transfer for molecules of

Hence for molecule of only one is transferred.

9. A compound of formula has the lattice. which atom forms the lattice and what fraction of

tetrahedral voids is occupied by the other atoms:

(A). lattice -

Tetrahedral voids -

(B) lattice -

Tetrahedral voids - .

(C). lattice -

Tetrahedral voids -

(D). lattice -

Tetrahedral voids -

Solution: (B)

forms the lattice, number of atoms ( )

For close packing, no. of tetrahedral voids

If occupies one third of the tetrahedral voids, number of atoms

and formula will be

If forms , then

of the must occupied by to form

10. The pair that contains two bonds in each of the oxoacids is:

(A) and

(B) and

(C) and

(D) and

Solution: (D)

11. The process with negative entropy change is:

(A) Synthesis of ammonia from and

(B) Dissolution of iodine in water

(C) Dissociation of ( ) to ( ) and ( )

(D) Sublimation of dry ice

Solution: (A)

If a process leads to arrangement in a system, it will have negative entropy, whereas if disorder

increases, entropy will be positive. When number of moles of gas increases, or ice melts, the disorder

increases. Freezing of water, crystallization, combination reactions, etc, lead to arrangement.

( ) ( )

( ) ( ) ( )

Dissolution of

12. Elevation in the boiling point for molal solution of glucose is . The depression in the freezing point

for molal solution of glucose in the same solvent is . The relation between and is:

(A)

(B)

(C)

(D)

Solution: (A)

….(i)

…..(ii)

Devide .

/

13. Sodium metal on dissolution in liquid ammonia gives a deep blue solution due to the formation of:

(A) Sodamide

(B) Sodium-ammonia complex

(C) Sodium ion-ammonia complex

(D) Ammoniated electrons

Solution: (D)

( ) ( ( ) ) ( ( ) )

Blue colour Ammoniated electron

Reason of colour:

14. The electrolytes usually used in the electroplating of gold and silver, respectively, are:

(A) , ( ) - , ( ) -

(B) , ( ) - , -

(C) , ( ) - , ( ) -

(D) , ( ) - , ( ) -

Solution: (D)

Solution of dicyano complexes of Gold ( ) and Silver ( ) are used as electrolytes while performing

electroplating with them.

15. For an elementary chemical reaction, , the expression for , -

is:

(A) , - , -

(B) , - , -

(C) , - , -

(D) , - , -

Solution: (A)

, -

, - (formation of from ) , -

(A atoms combining to give )

Hence, , -

, - , -

16. is introduced in evacuated flask at . of the solid decomposed to

and as gases. The of the reaction at is

( )

(A)

(B)

(C)

(D)

Solution: (A)

( ) ( ) ( )

( ) ( ) ( )

No of moles

( )

( ) degree of dissociation

So no of moles at equilibrium.

( )

( )

At equilibrium

( ) ( )

(As ( ) is solid, its conc. remains constant

( )

( )

17. An ideal gas undergoes isothermal compression from to against a constant external

pressure of . Heat released in this process in used to increase the temperature of mole of

is the temperature of increases by:

(A)

(B)

(C)

(D)

Solution: (A)

Work done on isothermal irreversible for ideal gas

, -

Isothermal proceed to ideal gas,

Hence

Heat released is ( )

18. In the cell ( )| ( )| ( )| ( )| ( )| ( ) the cell potential is when a

molal

solution is used. The standard electrode potential of ( ⁄ ) electrode is:

{

}

(A)

(B)

(C)

(D)

Solution: (B)

( )

( )

( ) ( )

*, -, -+

( )

Volt

19. The difference in the number of unpaired electrons of a metal ion in its high-spin and low-spin

octahedral complexes is two. The metal ion is:

(A)

(B)

(C)

(D)

Solution: (B)

The electronic configuration for ( )

When it combines with weak ligands (high spin complex), it has unpaired electrons, whereas when it

bonds with strong ligands (low spin complex) then have unpaired electron

20. The major product obtained in the following reaction is:

(A)

(B)

(C)

(D)

Solution: (A)

Intramolecular aldol condensation

21. The major product of the following reaction is:

(A)

(B)

(C)

(D)

Solution: (D)

22. An aromatic compound having molecular formula on treating with aqueous ammonia and

heating

forms compound . The compound on reaction with molecular bromine and potassium hydroxide

provides

compound having molecular formula . The structure of is:

(A)

(B)

(C)

(D)

Solution: (D)

Hofmann rearrangement of amides

23. Which of the following tests cannot be used for identifying amino acids?

(A) Biuret test

(B) Xanthoproteic test

(C) Barfoed test

(D) Ninhydrin test

Solution: (C)

Barfoed’s test is a chemical test used for the detecting the presence of monosaccharides from reading

disaccharides.

24. The major product of the following reaction is:

(A)

(B)

(C)

(D)

Solution: (B)

25. The reaction that is involved in the ozone layer depletion mechanism in the stratosphere is:

(A)

(B) ( ) ( ) ( ) ( )

(C) ( ) → ( ) ( )

(D) ( ) → ( ) ( )

Solution: (A)

Onidation of methane is responsible for the majority of the ozone formation in troposphere.

26. What is the name of the following compound?

(A) -Bromo- -methylpent- -ene

(B) -Bromo- -methyl-1,2-dimethylprop -1-ene

(C) -Bromo-3-methylpent-3-ene

(D) -Bromo- -dimethylbut-1-ene

Solution: (A)

IUPAC refer

27.Which is the most suitable reagent for the following transformation?

(A) Tollen’s reagent

(B) ⁄

(C)

(D) ⁄

Solution: (B)

It is iodoform reaction.

28. The correct match between item and item is:

Item ‘I’ (Compound)

Item ‘II’ (Reagent)

( ) Lysine ( ) 1-naphthol

( ) Furfural ( ) ninhydrin

( ) Benzyl alcohol ( ) ( ) Styrene ( ) Ceric ammonium nitrate

(A) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

(B) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

(C) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

(D) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

Solution: (B)

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

Lysine ninhydrin [this is given by all the amino acid and Lysine is an amino acid)

Furfural 1-naphthol

Benzyl alcohol ceric ammonium nitrate

Styrene

Rapid furfural test is used to distinguish between glucose fructose. In this test sugar solution is added to 1-

naphthol and conc.

Cerric ammonium nitrate test is used for test of Alcohals

Styrene undergo oxidation when treated with

29. What will be the major product in the following mononitration reaction?

(A)

(B)

(C)

(D)

Solution: (A)

Addition of is an electrophilic addition. The increases the electron density on the left benzene ring

whereas the carbonyl group decreases the charge density of the right benzene group. As a result, is

added to the left benzene group due to higher charge density.

30. The major product of the following reaction is:

(A)

(B)

(C)

(D)

Solution: (C)

Mathematics

Single correct answer type:

1. Let .√

/

.√

/

If ( ) and ( ) respectively denote the real and imaginary parts

of then:

(A) ( ) (B) ( ) and ( )

(C) ( ) and ( ) (D) ( ) Solution: (A)

Let √

then ( )

Hence ( )

2. If ∑ *

+ ( )

then is equal to:

(A)

(B)

(C)

(D) ( ) Solution: (A)

LHS = ∑

∑

( )

( )

( ) ( )

∑

( ) ( ) ( )

∑

3. If the area of an equilateral triangle inscribed in the circle is

√ sq. units then is equal to: (A)

(B) (C)

(D) Solution: (A)

Given, area of equilateral triangle inscribed in the circle is √ , hence √

√ √ ……….(1)

Again,

Hence, radius of circle, √

4. A curve amongst the family of curves represented by the differential equation, ( ) which passes through ( ) is: (A) A circle with centre on the axis

(B) A circle with centre on the axis (C) A hyperbola with transverse axis along the axis

(D) An ellipse with major axis along the axis Solution: (A)

4

5

It passes through ( ) hence

5. With the usual notation, in if √ and √ then the ratio is:

(A) (B)

(C) (D) Solution: (A) Let angle B is , hence

√

( )

√

(Using sine rule)

√

√

( )

√

√

√

√

√

√

√

6. If mean and standard deviation of observations are and respectively,

then the variance of observations and is equal to: (A)

(B) (C)

(D) Solution: (B)

Mean

…(i)

Variance

( )

…(ii)

Now

New variance ∑

( )

7. The length of the chord of the parabola having equation √ √ is:

(A) √

(B) √

(C) √

(D) √ Solution: (A) Solving the equations of the given parabola and the line, we get

(√ √ )

| | √( ) √

Length of chord √( )

( ) √ ( )

( ) √ | |

√

8. If the probability of hitting a target by a shooter, in any shot, is

then the minimum number of

independent shots at the target required by him so that the probability of hitting the target at

least once is greater than

is:

(A) (B)

(C) (D) Solution: (B)

( )

Let no. of trials

( )

(not hit)

(

)

(

) (

)

So minimum number of trials

9. Two vertices of a triangle are ( ) and ( ) If its orthocenter is at the origin, then its third vertex lies in which quadrant? (A) Fourth (B) Second (C) Third (D) First Solution: (B)

Since is orthocenter of

Since the slope of is

hence the slope of will be

Equation of line is

( )

Similarly, since is parallel to y -axis hence will be parallel to x - axis and

Equation of line is

( )

On solving ( ) and ( )

point ( ) lies in quadrant

10. The value of (∑ ( ∑ )

) is:

(A)

(B)

(C)

(D)

Solution: (A)

∑ ( ( ∑ ))

∑ .

( )/

∑ .

( )/

∑ ( ( )

( ))

( )

( )

( )

11. A helicopter is flying along the curve given by

( ) A soldier positioned at the

point .

/ wants to shoot down the helicopter when it is nearest to him. Then this nearest

distance is:

(A)

√

(B)

(C)

√

(D) √

Solution: (A)

Let the point on curve be 4

5 and given point be (

)

For nearest point normal at P passes through

So slope of line Slope of normal at

| ( )

√

( )( )

( is not possible as )

Hence point is .

√ /

So √

√

12. The value of such that sum of the squares of the roots of the quadratic equation, ( ) has the least value is:

(A)

(B)

(C)

(D) Solution: (A)

( )

( )

( ) ( ) ( )

( ) *for minimum value of ( )+

13. Two sides of a parallelogram are along the lines, and If its

diagonals intersect at ( ) then one of its vertex is: (A) ( ) (B) ( ) (C) ( ) (D) ( ) Solution: (A)

Slope of

( ) and ( )

14. Let ( ) be a function defined by ( ) { | | √ } If be the set of all

points at which is not differentiable, then has exactly: (A) Two elements (B) One element (C) Three elements (D) Five elements Solution: (C)

From the graph it can be concluded than ( ) is non-differentiable at

√ in ( )

i.e., at points in ( )

15. The value of ∫

, - , -

⁄

⁄ where , - denotes the greatest integer less than or equal to

is:

(A)

( )

(B)

( )

(C)

( )

(D)

( )

Solution: (A)

∫

, - , -

∫

, -

∫

, -

∫

∫

∫

∫

, -

0

1

0

1

0

1

.

/ (

)

( )

16. If ∫

( ) where is a constant of integration, then ( ) is equal to:

(A)

(B)

(C)

(D) Solution: (A)

∫ (say)

Let

∫ (

) (

)

∫

[ ∫ ]

( )

( )

Hence ( )

17. Let be a differentiable function such that ( )

( )

( ) and ( ) Then

.

/

(A) Does not exist

(B) Exists and equals

(C) Exists and equals

(D) Exists and equals Solution: (B)

(where ( ))

This is a linear differential equation

Integrating factor ∫

So integrating the given differential equation using I.F.

∫

( )

( )

(

)

(

)

18. Let

2( )

3 where Then represents:

(A) An ellipse whose eccentricity is

√ when

(B) A hyperbola whose eccentricity is

√ when

(C) An ellipse whose eccentricity is √

when

(D) A hyperbola whose eccentricity is

√ when

Solution: (C)

If ( ) then this curve is a hyperbola whose

Eccentricity is √

√

If , then this curve is an ellipse whose

Eccentricity is √

√

19. Let [

] where Then the minimum value of ( )

is:

(A) √

(B) √

(C) √

(D) √ Solution: (A)

| | |

|

( ) ( ) ( )

( )

| |

4√ √

5

√

Hence minimum value of | |

is √

20. Let ( ) and ( ) be two given vectors where vectors and

are non-collinear. The value of for which vectors and are collinear, is:

(A) (B)

(C) (D) Solution: (A)

are collinear

21. The tangent to the curve, passing through the point ( ) also passes through the

point:

(A) .

/

(B) ( )

(C) .

/

(D) ( ) Solution: (A)

at ( )

Equaton of tangent is ( ) which passes through .

/

22. The plane which bisects the line segment joining the points ( ) and ( ) at right angles, passes through which one of the following points? (A) ( )

(B) ( ) (C) ( ) (D) ( ) Solution: (B)

The midpoint of line joining the given points is ( ) and it lies on the required plane.

Normal to the plane, the line joining the given points, has direction ratio

equation of plane is ( ) ( ) ( )

Hence plane passes through point ( ) 23. The number of values of ( ) for which the system of linear equations ( ) ( ) has a non-trivial solution, is: (A) Two (B) Three (C) Four (D) One Solution: (A) For non-trivial solutions

|

|

or

or

Since ( )

and

will satisfy the equation

Number of '

24. If ∫ ( )

∫ ( )

then .

/ is:

(A)

(B)

(C)

(D)

Solution: (B) Differentiating both side

( ) ( )

( )

( ) ( )

( )

( )

Hence .

/

25. On which of the following lines lies on the point of intersection of the line,

and the plane,

(A)

(B)

(C)

(D)

Solution: (B)

A general point on the given line

can we taken as ( )

If it also lies on the given plane then

( ) ( ) ( )

Point of intersection is ( ), which lies on

26. Let be in G.P. with for and be the set of pairs ( ) (the set of natural numbers) for which

|

|

Then the number of elements in is: (A) Infinitely many (B)

(C) (D) Solution: (A) (Let is common ratio of )

|

( ) (

)

( ) (

)

( ) (

)

| which is always true because are identical.

27. Consider the following three statements: is a prime number is a factor of

LCM of and is Then the truth value of which one of the following statements is true? (A) ( ) (B) ( ) ( ) (C) ( ) ( ) (D) ( ) ( ) Solution: (A)

is true

is False

is True

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( )

28. The value of

is:

(A)

(B)

(C)

(D)

Solution: (B)

.

/

29. The positive value of for which the co-efficient of in the expansion .√

/

is

is:

(A) √

(B)

(C)

(D) √

Solution: (C)

Coefficient of in (√

)

coefficient of in .√

/

A general term in the expansion .√

/

is (√ )

.

/ ( )

( )

coefficient of (for )

Now

30. Let be the set of natural numbers and two functions and be defined as

such that

( ) {

and ( ) ( ) Then is:

(A) Onto but not one-one (B) Both one-one and onto (C) One-one but not onto (D) Neither one-one nor onto Solution: (A)

( ) {

( ( )) ( )

( ( )) ( )

( ( )) is many one

( ( )) ( )

( ( )) ( )

( ( )) is onto