Physics

Single correct answer type:

1. Two guns and can fire bullets at speeds ⁄ and ⁄ respectively. From a point on a

horizontal ground, they are fired in all possible directions. The ratio of maximum areas covered by the

bullets fired by the two guns, on the ground is:

(A)

(B)

(C)

(D)

Solution: (A)

( )

( )

*

+

2. To mop-clean a floor, a cleaning machine presses a circular mop of radius vertically down with a

total force and rotates it with a constant angular speed about its axis. If the force is distributed

uniformly over the mop and if coefficient of friction between the mop and the floor is the torque,

applied by the machine on the mop is:

(A)

(B) ⁄

(C) ⁄

(D) ⁄

Solution: (A)

Net torque ∑

∫ ∫ (

)

3. A piece of wood of mass is dropped from the top of a height building. At the same

time, a bullet of mass is fired vertically upward, with a velocity from the ground.

The bullet gets embedded in the wood. Then the maximum height to which the combined system

reaches above the top of the building before falling below is:

( )

(A)

(B)

(C)

(D)

Solution: (A)

Using relative velocity

Time for collision,

⁄

⁄

⁄

So from top of building

4. A heat source at is connected to another heat reservoir at by a copper slab

which is thick. Given that the thermal conductivity of copper is the energy flux

through it in the steady state is:

(A)

(B)

(C)

(D)

Solution: (C)

⁄

5. A magnet of total magnetic moment is placed in a time varying magnetic field,

( ) where Tesla and ⁄ The work done for reversing the direction of the

magnetic moment at second, is:

(A)

(B)

(C)

(D)

Solution: (B)

at

( )

6. Two electric dipoles, with respective dipole moments and are

placed on the -axis with a separation as shown in the figure

The distance from at which both of them produce the same potential is:

(A)

√

(B) √

√

(C) √

√

(D)

√

Solution: (B)

( )

( )

(√

√ )

7. The density of a material in units is In certain units in which the unit of length is

and the unit of mass is the numerical value of density of the material is:

(A)

(B)

(C)

(D)

Solution: (C)

[

] [ ]

[ ] [ ]

[

] [

]

8. Three Carnot engines operate in series between a heat source at a temperature and a heat sink at

temperature (see figure). There are two other reservoirs at temperature and as shown, with

The three engines are equally efficient if:

(A) ( )

( )

(B) ( )

( )

(C) ( )

( )

(D) ( )

( )

Solution: (C)

Given

….(1)

….(2)

From (1) and (2)

( ) ⁄ (

) ⁄

9. A homogeneous solid cylindrical roller of radius and mass is pulled on a cricket pitch by a

horizontal force. Assuming rolling without slipping, angular acceleration of the cylinder is:

(A)

(B)

(C)

(D)

Solution: (C)

( )

( )

For pure rolling

( )

From ( ) ( ) and ( )

10. A block of mass is kept on a plateform which starts from rest with constant acceleration ⁄

upward, as shown in fig. Work done by normal reaction on block in time is:

(A)

(B)

(C)

(D)

Solution: (D)

11. To get output at for the given logic gate circuit the input values must be:

(A)

(B)

(C)

(D)

Solution: (D)

( ) ( )

( ) ( ) ( )

12. A train moves towards a stationary observer with speed ⁄ The train sounds a whistle and its

frequency registered by the observer is If the speed of the train is reduced to ⁄ the frequency

registered is If speed of sound is ⁄ then the ratio ⁄ is:

(A) ⁄

(B) ⁄

(C) ⁄

(D) ⁄

Solution: (C)

(

)

(

)

(

)

13. A TV transmission tower has a height of and the height of the receiving antenna is

What is the maximum distance upto which signals can be broadcasted from this tower in LOS(Line of

Sight) mode? (Given : radius of earth )

(A)

(B)

(C)

(D)

Solution: (D)

√ √

√ √

[√ √ ]

[ √ √ ]

[ √ √ ]

[ ]

14. A potentiometer wire having length and resistance is joined to a cell of emf and

internal resistance cell having ⁄ and internal resistance is connected. The length at

which the galvanometer as shown fig. shows no deflection is:

(A)

(B)

(C)

(D)

Solution: (D)

Potential difference across .resistance

(

)

For no current in galvanometer

potential difference across

15. Water flows into a large tank with flat bottom at the rate of Water is also leaking out of

a hole of area at its bottom. If the height of the water in the tank remains steady then this height

is:

(A)

(B)

(C)

(D)

Solution: (A)

Amount of water going out = amount of water coming in.

√

( )

( )

16. In an electron microscope, the resolution that can be achieved is of the order of the wavelength of

electrons used. To resolve a width of the minimum electron energy required is close to:

(A)

(B)

(C)

(D)

Solution: (B)

( ) √

√

17. In the given circuit the cells have zero internal resistance. The currents (in Amperes) passing through

resistance and respectively, are:

(A)

(B)

(C)

(D)

Solution: (C)

Assume as

as

Current in is zero

from to

18. A charge is distributed over three concentric spherical shells of radii ( ) such that

their surface charge densities are equal to one another.

The total potential at a point at distance from their common centre, where would be:

(A)

( )

(B) ( )

( )

(C)

(D) ( )

( )

Solution: (D)

( )

( )

( )

[ ] (

)

19. In a Young’s double slit experiment slit separation on observes a bright fringe at angle

by using light of wavelength When the light of wavelength is used a bright fringe is seen at

the same angle in the same set up. Given that and are in visible range ( ) their

values are:

(A)

(B)

(C)

(D)

Solution: (C)

( )

Putting

20. A string of length and mass is fixed at both ends. The tension in the string is The

string is set into vibration using an external vibrator of frequency The separation between

successive nodes on the string is close to:

(A)

(B)

(C)

(D)

Solution: (A)

√

√

21. In the cube of side shown in the figure, the vector from the central point of the face to

the central point of the face will be:

(A)

( )

(B)

( )

(C)

( )

(D)

( )

Solution: (C)

22. If the magnetic field of a plane electromagnetic wave is given by (The speed of light ⁄

* (

)+ then the maximum electric field associated with it is:

(A) ⁄

(B) ⁄

(C) ⁄

(D) ⁄

Solution: (A)

⁄ ⁄

23. A parallel plate capacitor is of area and a separation The gap is filled with three

dielectric materials of equal thickness (see figure) with dielectric constant and

The dielectric constant of a material which when fully inserted in above capacitor, gives same

capacitance would be:

(A)

(B)

(C)

(D)

Solution: (D)

( )

24. A solid metal cube of edge length is moving in a positive -direction at a constant speed of

⁄ There is a uniform magnetic field of in the positive -direction. The potential difference

between the two faces of the cube perpendicular to the -axis, is:

(A)

(B)

(C)

(D)

Solution: (A)

( )

25. A uniform metallic wire has a resistance of and is bent into an equilateral triangle. Then, the

resistance between any two vertices of the triangle is:

(A)

(B)

(C)

(D)

Solution: (D)

Req

26. A satellite is moving with a constant speed in circular orbit around the earth. An object of mass

is ejected from the satellite such that it just escapes from the gravitational pull of the earth. At the time

of ejection, the kinetic energy of the object is:

(A)

(B)

(C)

(D)

Solution: (B)

√

(√ )

27. A carbon resistor is color coded with green, black, red and brown respectively. The maximum

current which can be passed through this resistor is:

(A)

(B)

(C)

(D)

Solution: (C)

Green Black Red Silver tolerance

28. An insulating thin rod of length has a linear charge density ( )

on it. The rod is rotated

about an axis passing through the origin ( ) and perpendicular to the rod. If the rod makes

rotations per second, then the time averaged magnetic moment of the rod is:

(A)

(B)

(C)

(D)

Solution: (A)

⁄

Current due to elementary charge

Magnetic dipole moment

∫ ∫

29. A plano convex lens of refractive index and focal length is kept in contact with another plano

concave lens of refractive index and focal length If the radius of curvature of their spherical faces

is each and the and are related as:

(A)

(B)

(C)

(D)

Solution: (D)

( ) (

)

( ) (

)

30. Using a nuclear counter the count rate of emitted particles from a radioactive source is measured. At

it was counts per second and seconds it was counts per second. The count rate

observed, as counts per second, at seconds is close to:

(A)

(B)

(C)

(D)

Solution: (B)

After seconds count rate changes from per second to per second.

in half life half life is half life. Count rate after second

Chemistry

Single correct answer type:

1. The increasing order of the values of the following compounds is:

(A)

(B)

(C)

(D)

Solution: (B)

Acidic strength of phenol is dependent on the substituents attached to it. Presence of electron

withdrawing groups increase the tendency to attract electrons (Acidity), thereby decreasing the

value. is a strong withdrawing group, is a weak electron withdrawing group as it has

effects whereas is an electron donating group. Hence, the correct order of values

is:

order of values:

2. Water filled in two glasses A and B gave BOD values of and respectively. The correct

statement regrading them, is

(A) A is suitable for drinking, whereas B is not

(B) B is more polluted than A.

(C) Both A and B are suitable for drinking

(D) A is ore polluted than B.

Solution: (B)

A water supply with a BOD level of 3-5 ppm is considered moderately clean. In water with a

BOD level of 6-9 ppm, the water is considered somewhat polluted because there is usually

organic matter present and bacteria are decomposing this waste. Hence, B with a higher value

of BOD level is more polluted.

3. The major product of the following reaction is:

(A)

(B)

(C)

(D)

Solution: (A)

Addition of alc. to haloalkanes follows the Saytzeff rule which points towards greater

stability. Saytzeff’s rule implies that elimination reactions favour formation of alkene with greater

number of substituents.

Not only that, but it also gives the molecule an extended conjugate system which further

enhances its stability.

4. Which of the graphs shown below does not represent the relationship between incident light

and the electron ejected from metal surface?

(A)

(B)

(C)

(D)

Solution: (D)

………. Photoelectric effect.

Intercept must be

5. Which primitive unit cell has unequal edge lengths ( ) and all axial angles different

from

(A) Hexagonal

(B) Monoclinic

(C) Triclinic

(D) Tetragonal

Solution: (D)

The most inorganized system has and is called the triclinic system.

6. The type of hybridization and no. of lone pair(s) of electron of in , respectively, are:

(A) and

(B) and

(C) and

(D) and

Solution: (C)

Steric no. hybridisation

Geometry octahedral

No. of lone pairs on

7. The total number of isomers for a square planar complex: [ ( )( )( )] is:

(A)

(B)

(C)

(D)

Solution: (B)

Possible Structural isomers No of geometrical isomer by each

[ ( )( )( )]

[ ( )( )( )]

[ ( )( )( )]

[ ( )( )( )]

Total number of isomers will be .

8. Hall Heroult’s process is given by:

(A)

(B)

(C) ( ) ( ) ( ) ( )

(D) →

Solution: (C)

Hall Heroult’s process:

At cathode:

At cathode:

Overall reaction: 2

9. The chemical nature of hydrogen peroxide is:

(A) Oxidizing and reducing agent in both acidic and basic medium.

(B) Oxidizing agent in acidic medium, but not in basic medium

(C) Oxidizing and reducing agent in acidic medium, but not in basic medium

(D) Reducing agent in basic medium, but not in acidic medium.

Solution: (A)

acts as an oxidizing agent, as well as a reducing agent in both acidic as well as basic

mediums.

10. Two pi and half sigma bonds are present in:

(A)

(B)

(C)

(D)

Solution: (C)

According to ; electronic configuration of is:

( )

i.e., bonds and bonds.

11. Wilkinson catalyst is:

(A) [( ) ]

(B) [( ) ]( )

(C) [( ) ]

(D) [( ) ]

Solution: (A)

Chlorotris(triphenylphosphine)rhodium(I), is known as Wilkinson’s catalyst. It is used as a

homogeneous hydrogenation catalyst. It is a square planar 16-electron complex. The oxidation

state of Rhodium in it is +1.

12. The values of

for the following reactions at are, respectively:

(At )

( ) ( ) ( )

( ) ( )

( ) ( ) ( )

the value of ⁄ are respectively:

(A)

(B)

(C)

(D)

Solution: (B)

( )

( )

Equations 1

Equations 2:

Equations 3:

( )

13. Liquids and form an ideal solution. In the entire composition range. At the

vapor pressure of pure A and pure B are and , respectively. The

composition of the vapor in equilibrium with a solution containing 40 mole percent of A at

this temperature is:

(A)

(B)

(C)

(D)

Solution: (C)

Since,

and

According to Raoult’s Law:

( )( ) ( )( )

Method 1

Now,

Method 2

We have the total pressure and the individual partial vapour pressures of the two liquids,

,

and . Each gas exerts a partial

pressure proportional to its mole fraction. Hence, mole fractions of A and B in vapour

phases are 0.28 and 0.72 respectively.

14. Which of the following is not an example of heterogeneous catalyst reaction?

(A) Combustion of Coal

(B) Hydrogenation of Vegetable oils

(C) Ostwald’s process

(D) Haber’s process

Solution: (A)

( ) Since Combustion of Coal is a fast and spontaneous process, no catalyst is required.

( ) Catalyst used is ( )

( ) Catalyst used is ( ) ( ) and ( )

( ) Catalyst used is ( )

15. A mixture of mol of ( ) and of sodium sulphate was dissolved in water and

the volume was made up to The mass of calcium sulphate formed and the

concentration of in resulting solution, respectively, are : (Molar mass of ( )

and are and respectively; of ( ) )

(A)

(B)

(C)

(D)

Solution: (B)

( )

[ ]

Remaining is

The solution will have ions coming from both ( ) and . The concentration of

is 0.007, which is negligible as compared to 0.1 moles of ( ) .

[ ]

16. The total number of isotopes of hydrogen and number of radioactive isotopes among them,

respectively, are

(A) and

(B) and

(C) and

(D) and

Solution: (A)

Isotopes of Hydrogen:

Among all three isotopes only Tritium is radioactive.

17. If dichloromethane (DCM) and water are used for differential extraction, which one of

the following statements is correct?

(A) DCM and would stay as upper and lower layer respectively in the separating funnel

(S.F.)

(B) DCM and will be miscible clearly

(C) DCM and would stay as lower and upper layer respectively in the S.F

(D) DCM and will make turbid/colloidal mixture

Solution: (C)

Dichloromethane and water are immiscible in nature. Also, Dichloromethane has higher density

than water so forms bottom layer (Layer ) in the separating funnel.

18. Consider the following reduction processes:

( )

( )

( )

( )

The educing power of the metals increases in the order:

(A)

(B)

(C)

(D)

Solution: (D)

Higher is the reduction potential, higher will be its tendency to get oxidized and thus higher will

be its reducing property. Consequently, its reduction ability will be low.

19. The electronegativity of aluminum is similar to:

(A) Beryllium

(B) Boron

(C) Carbon

(D) Lithium

Solution: (A)

of

of

of

of

Due to diagonal relationship between and they exhibit similar properties and thus, have

similar electronegativity values.

20. The major product of the following reaction is:

(A)

(B)

(C)

(D)

Solution: (B)

21. The major product ‘X’ formed in the following reaction is:

(A)

(B)

(C)

(D)

Solution: (A)

doesn’t reduce esters and alkenes.

22. The decreasing order of ease of alkaline hydrolysis for the following esters is

(A)

(B)

(C)

(D)

Solution: (B)

The speed of hydrolysis will depend on how likely a molecule is to accepting ions.

More acidic a molecule, higher will be the speed of reaction. is a strong EWG

(Electron withdrawing group), is a weak EWG and is an EDG.

Hence, the increasing order of rate of hydrolysis will be:

23. Which dicarboxylic acid in presence of a dehyrating gent is least reactive to give

ananhydride?

(A)

(B)

(C)

(D)

Solution: (D)

Cyclic anhydrides in membered ring are less stable.

24. Which hydrogen in compound ( ) is easily replaceable during bromination reaction in

presence of light ?

(A) - hydrogen

(B) - hydrogen

(C) -hydrogen

(D) - hydrogen

Solution: (C)

Allylic radial is more stable due to resonance.

Rate of reaction Stability of free radical

25. The major product formed in the reaction given below will be :

(A)

(B)

(C)

(D)

Solution: (C)

26. Consider the given plots for a reaction obeying Arrhenius equation ( ) (k

and are rate constant and activetion energy, respectively )

(A) I is wrong but II is right

(B) Both I and II are wrong

(C) I is right but II is wrong

(D) Both I and II are correct

Solution: (C)

Arhaneous Solution: ⁄

K decreases exponentially with

The second graph should be an asymptotic curve tangetial to positive x axis.

graph is correct and second is incorrect.

27. The metal used for making X-ray tube window is:

(A)

(B)

(C)

(D)

Solution: (C)

Beryllium is used in making windows in X-ray tubes.

28. A process has and Out of the values given below,

choose the minimum given below, choose the minimum temperature above which the process

will be spontaneous:

(A)

(B)

(C)

(D)

Solution: (A)

……. (Gibbs Helmholtz Equation)

For a reaction to be spontaneous,

So,

i.e. Minimum temp. should be

29. The effect of lanthanoid contraction in the lanthanoid series of elements by and large

means:

(A) Increase in atomic radii and decrease in ionic radii

(B) decrease in atomic radii and increases in ionic radii

(C) increase in both atomic and ionic radii

(D) decrease in both atomic and ionic

Solution: (B)

30. The correct structure of product in the following reaction is:

( ) ( )

(A)

(B)

(C)

(D)

Solution: (A)

Mathematics

Single correct answer type:

1. If the area enclosed between the curves and ( ), is square unit. Then is

(A) √

(B)

√

(C) √

(D)

√

Solution: (B) Point of intersection

( )

(

)

Point of intersection .

/

Area ∫ (√

) .

√

⁄

⁄

/

⁄

√

2. In a class of students numbered to 140, all even numbered students opted Mathematics course, those whose number is divisible by 3 opted Physics course and those whose number is divisible by 5 opted Chemistry course. Then the number of students who did not opt for any of the three courses is: (A) (B)

(C) (D) Solution: (C)

[

]

[

]

[

]

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

[

] [

] [

] [

]

Hence, number of students who did not opt for any of the three courses

3. Let ∫ ( )

. If is minimum then the ordered pair ( ) is

(A) ( √ )

(B) (√ √ )

(C) ( √ )

(D)( √ √ )

Solution: (D)

∫ ( )

From figure minimum area is when ordered pair ( ) ( √ √ )

4. If

, (

) and .

/

then .

/ equals:

(A)

(B)

(C)

(D)

Solution: (C)

( )

This is linear differential equation

Integrating factor ∫

Hence ∫

Given .

/

Hence .

/

5. An unbiased coin is tossed. If the outcome is a head then a pair of unbiased dice is rolled and the sum of the numbers obtained on them is noted. If the toss of the coin results in tail then a card from a well-shuffled pack of nine cards numbered is randomly picked and the

number on the card is noted. The probability that the noted number is either or is

(A)

(B)

(C)

(D)

Solution: (B) ( ) ( ) ( ) ( ) ( )

6. If ∑ (

)

then k equals

(A) (B)

(C) (D) Solution: (B)

∑(

)

Now

Let given sum be so

∑( )

( ) (

)

Given

7. If the third term in the binomial expansion of ( ) equals then a possible value

of is:

(A) √

(B)

(C) √

(D)

Solution: (D)

In the expansion of ( )

third term say ( )

( )

Taking logarithm to the base on both sides

( ) ( )

Here

8. If are the lengths of the sides of a triangle, then cannot be equal to:

(A)

(B)

(C)

(D)

Solution: (D)

sides of triangle

….(i)

….(ii)

…..(iii)

From (i)

* ( √

)+ * (

√

) +

. √

√

/ …..(iv)

from (ii)

…..(v) from (iii)

( √

)(

√

)

. √

/ .

√

/ …..(vi)

from (iv), (v), (vi)

( √

√

)

now check options

9. Consider a triangular plot ABC with sides and . A vertical lamp-post at the mid point D of AC subtends an angle at B. The height (in m) of the lamp-post is:

(A) √

(B)

√

(C)

√

(D) √ Solution: (B)

Length of median

√ ( ) ( )

√ ( )

√ √

√ √

Let be height of tower, given

√ √

√

√

10.The equation of a tangent to the hyperbola, , parallel to the line , is

(A) (B)

(C) (D) Solution: (C)

Hyperbola is

Equation of its tangent in slope from is √ Hence tangent with slope is

11. The shortest distance between the point .

/ and the curve √ ( ), is

(A) √

(B)

(C)

(D) √

Solution: (D)

Let be the nearest to .

/ then normal at will pass through .

/ Let Co-ordinates of be

.

/

Hence equation of normal is

This line passes through .

/

*( ) +

hence nearest point is ( )

distance √.

/

( )

√

12. The mean of five observations is and their variance is If three of the given five

observations are and then a ratio of other two observations is

(A)

(B)

(C)

(D)

Solution: (B)

…(i)

( )

…(ii) ( )

So ratio is

or

13. Consider the quadratic equation ( ) ( ) Let S be the set of all

integral values of c for which one root of the equation lies in the interval ( ) and its other root

lies in the interval ( ). Then the number of elements in S is

(A)

(B)

(C)

(D)

Solution: (A)

Case …(i) ( )

…(ii) ( )

( )

…(iii) ( )

( )

…(iv)

Here ( ) ( ) ( ) ( )

(

)

Case …(i) ( )

…(ii) ( ) …(iii)

( )

…(iv)

( ) ( ) ( ) ( )

Hence, .

/

Hence, total number of element in

14. If the system of equations has inifinitely

many solutions, then equals

(A)

(B)

(C)

(D)

Solution: (A)

………(i) ………(ii)

……..(iii)

|

| ( ) ( ) ( )

for From (ii) (i),

From (iii) – (ii),

Hence for infinite solution,

Hence for infinite solution Hence,

15. Let and be any two non-zero complex numbers such that If

then maximum value of is

Note: In actual paper value of was asked. Hence none of the options given were correct. So

we have modified the question as well as options.

(A)

(B)

(C)

(D)

√

Solution: (C)

Let .

/

(

)

|

| ( )

|

| ( )

( )

( )

(

)

√

√

√

16. The plane passing through the point ( ) and parallel to the lines

and

also passes through the point:

(A) ( )

(B) ( )

(C) ( )

(D) ( )

Solution: (D)

|

|

Equation of plane is ( ) ( ) ( )

Hence point ( ) satisfy the equation of plane.

17. If the line intersects the x-axis at the point A and the y-axis at the point B,

then the incentre of the triangle OAB, where O is the origin, is:

(A) ( )

(B) ( )

(C) ( )

(D) (2, 2)

Solution: (D)

(

)

( ( ) ( ) ( )

( ) ( ) ( )

) ( )

18.The sum of all values of (

) satisfying

is

(A)

(B)

(C)

(D)

Solution: (A)

Let

.

/

( )

( )

(

)

Sum of values of is

19. Let be a natural number and

Then ∫

( )

is equal to (where

is a constant of integration)

(A)

.

/

(B)

.

/

(C)

.

/

(D)

.

/

Solution: (C)

∫( )

∫( )

( )

∫ .

/

∫.

/

Put

( )

∫

.

/ ( )

( )

(

)

20. For each let , - be the greatest integer less than or equal to t.

Then,

( ) ., -

/

, -

(A) equals

(B) equals

(C) does not exist

(D) equal 1

Solution: (A)

( ) ., - /

, -

( ( )) . /

( )( )

( ) ( )

( )

21. Let ( ) { ( )

Let S be the set of points in the interval ( ) at

which is not differentiable. Then S:

(A) equals * +

(B) equals * +

(C) is an empty set

(D) equal * +

Solution: (A)

From the graph we can easily conclude that ( ) is non-derivable at

22. If the parabolas ( ) and have a common normal, then which one of

the following is a valid choice for the ordered triad ( )

(A) ( )

(B) .

/

(C) .

/

(D) All of above

Solution: (D)

Note: In actual paper all four options were correct. Hence we have changed some options to

make this question appropriate.

Axis of both parabola is x-axis.

As we know, axis of parabola is always a normal to the parabola.

Hence for all values of ( ) a common normal is present.

23. If a circle passing through the point ( ) touches the circle

externally at the point ( ) then the radius of C is:

(A) 4

(B) 5

(C) √

(D) √

Solution: (B)

Tangent at ( ) is

( ) ( ) ( ) ( )

Required circle is

( ) ( ) ( )

It pass through ( )

( )

radius √

24. Consider the statement: ( ) is prime” Then which one of the following is true?

(A) ( ) is false but ( ) is true

(B) Both ( ) and ( ) are false

(C) Both ( ) and ( ) are true

(D) ( ) is false but ( ) is true

Solution: (C)

( )

( )

( )

Hence ( ) and ( ) both are prime. Hence both are true.

25. The sum of all two digit positive numbers which when divided by yield 2 or as remainder

is

(A)

(B)

(C)

(D)

Solution: (A)

Two digit numbers of the form are

Two digit numbers of the form are

Sum of all the above numbers equals to

( )

( )

26. Iet be a function such that ( ) ( ) ( ) ( ) . Then ( )

equals

(A)

(B)

(C)

(D)

Solution: (D)

( )

( )

( )

( )

( )

( )

( ) and

( )

( )

27. A point P moves on the line If ( ) ( ) are fixed points, then the

locus of the centroid of is a line:

(A) with slope

(B) with slope

(C) Parallel to y-axis

(D) parallel to x-axis

Solution: (A)

Let coordinate of centroid is ( ) and coordinate of point is ( )

Hence,

and

Since ( ) lies on line

Hence, ( ) ( )

Hence locus of centroid is which has slope

28. Let and [ ( ) ( ) ( ) ( )

], , -. If the minimum value

of ( ) is 8, then a value of is :

(A) (√ )

(B) (√ )

(C)

(D)

Solution: (C)

| ( ) ( ) ( ) ( )

|

| ( )

( )

|

( ) ( )

Because minimum value of

( )

29. Let A be a point on the line ( ) ( ) ( ) and ( ) be a point in

the space. Then the value of for which the vector is parallel to the plane is

:

(A)

(B)

(C)

(D)

Solution: (B)

Let is ( )

( ) ( ) ( ) which is parallel to plane

( ) ( ) ( )

30. Let ( ) and ( ) be three vectors

such that and is perpendicular to Then a possible value of ( ) is

(A) .

/

(B) ( )

(C) .

/

(D) ( )

Solution: (A)

Because so ………..(i)

Because is perpendicular to so ( ) ………..(ii)

( ) ( ) where

.

/ satisfied the above triplet.