12th IIT 21st May Test Paper - Solution - YLCeducation.ylc.co.in/.../08/12th-IIT-21st-May-Test-Paper-Solution.pdf · M g a Q4 A block of mass 1 kg is kept over another block of plane

12th IIT 21st May Test Solution

101, Blue Diamond Complex, Fatehgunj, Vadodara-02 .Tel. No. 2787268 / 2750018 / 2782232

2

Subject-PHYSICS Part : 1

SECTION-A

(NEXT EIGHT QUESTIONS: SINGLE CHOICE – ONLY ONE CORRECT ANSWER)

Q1 The electric field in a region (relative permittivity = 1) varies with distance “r” from origin according to

relation ,ˆ2rArE where A is a positive constant. The volume charge density in the region is proportional to

(a.) r (b.) 2r (c.) r/1 (d.) 2/1 r

Sol. (a)

dxxxrKQ

rKArE

r2

022

2 4)( [where Q is total charge inside sphere of radius ].r

dxxrr

0

24 x

Q2 A solid hemispherical uniform charged body having charge Q is kept symmetrically along the y-axis as shown

in figure. The electric potential at a distance d from the origin along the x -axis at point P will be

(a.)

dQ

041

(b.)

Less than dQ

041

(c.) More than

dQ

041

and less than dQ

042

(d.)

More thandQ

042

Sol. (a)

By symmetry, potential at P is same due to upper & lower half

Q3 A steel block of mass M is being lifted by a helicopter, which is accelerating upward with an acceleration of

magnitude a. The block is attached to a horizontal bar by two cables, which make angles of θ1 and θ2 w.r.t.

horizontal as shown in the figure. The tension in the cable 1 in terms of M, θ1, θ2, g and a is



3

(a.) 21

1

sincos

agM

(b.) 21

1

cossin

agM

(c.) 21

2

cossin

agM

(d.)

21

2

sincos

agM

Sol. (d)

T1cos1 = T2cos2

T1sin1 + T2sin2 – Mg = Ma

T1sin1 + 2

211

cossincos

T– Mg = Ma

T1 = 21

2

sincos

agM

Q4 A block of mass 1 kg is kept over another block of plane. Block of mass 1 kg is being moved continuously with

velocity 1 m/s towards positive X-axis and lower block is moved continuously with velocity 1 m/s towards

positive Z-axis as shown in figure. If coefficient of friction between two blocks is 0.1, the rate at which heat is

generated will be )/10( 2smg

(a.) J/sec1 (b.) J/sec2 (c.) J/sec2 (d.)

J/sec2

1

Sol. (c) Rate of heat generation = power developed by friction

.22)1( RR VmgVf

Q5 The following set of figure show two cases of collision between two balls. Let

1u and

2u be the velocities

before collision and

1V ,

2V be the velocities after collision.

The coefficient of restitution,

e =

12

12

uu

VV

Student-A: Equation holds for both cases as e is property of material of the colliding bodies. Student-B: Equation holds for case II only.

(a.) Student-A is incorrect, Student-B is correct. (b.) Student-A is correct, Student-B is incorrect.



4

(c.) Both are correct (d.) Both are incorrect

Sol. (a)

We take velocity of approach along line of collision. In first case, 21app uuv

Q6 A gas is enclosed in a cylinder with a movable frictionless piston. Its initial thermodynamic state at pressure

Pa10P 5i and volume 33

i m10V changes to a final state at Pa10321P 5

f

and 33

f m108V

in an adiabatic quasi-static process, such that 53VP constant. Consider another thermodynamic process that

brings the system from the same initial stat to the same final state in two steps: an isobaric expansion at iP

followed by an isochoric (isovolumetric) process at volumes .Vf The amount of heat supplied to the system in

the two step process is approximately (a.) J112 (b.) J294 (c.) J588 (d.) J813

Sol. (c)

Q7 A motor boat is to reach at a point 30 upstream on the other side of a river flowing with velocity .5 1ms The

velocity of motor boat with respect to water is .35 1ms The driver should steer the boat at an angle in

degrees w.r.t the line of destination from the starting point (a.) 30 (b.)

31sin 1

(c.)

31sin 1

(d.)

32sin 1

Sol. (a)

The velocity of motor boat is given as



5

wmwm vvv

302/1sin

120sin35

sin5

Q8 A water cooler of storage capacity 120 litres can cool water at constant rate of P watts. In a closed circulation

system (as shown schematically in the figure), the water from the cooler is used to cool an external device that

generates constantly kW3 of heat (thermal load). The temperature of water fed into the device cannot exceed

C30 and the entire stored 120 litres of water is initially cooled to C.10 The entire system is thermally

insulated. The minimum value of P (in watts) for which the device can be operated for 3 hours is:

(Specific heat of water is 11KkgkJ4.2 and the density of water is 3mkg1000 )

(a.) 1600 (b.) 2067 (c.) 2533 (d.) 3933

Sol. (b)

SECTION-B

(MULTIPLE CHOICE – ONE OR MORE THAN ONE CORRECT ANSWER)

Q1 In the presence of stationary charges in the space, line integral of the electric field

dE for a closed path as

shown, is carried out:

(a.) The value of the integral is independent of the path (b.) The value of the integral is independent of where charges may be located relative to the path (c.) The value of the integral is always is constant (d.) The value of the integrals is always negative



6

Sol. (a, b, c)

In an electric field dV=

dE for a closed loop, dV = 0

Q2 The figure shows, two point charges )0(2 Q and .Q The charges divide the line joining them in three

parts I, II and III

(a.) Region III has a local maxima of electric field (b.) Region I has a local minima of electric field

(c.) Equilibrium position for a test charge lies in region II (d.) Region II has a local minima of electric field

Sol. (a, d)

Q3 A particle A moves with a velocity 1v and a particle B moves with a velocity ,2v w.r.t. ground

and r is the length of the vector, ,AB then

(a.) 0

dtdr

if |||| 12 vv

(b.) 0

dtdr

if |||| 12 vv

(c.) 0

dtdr

if ABvABv .. 12

(d.)

0dtdr

if 0.)( 21 BAvv

Sol. (c, d)

ABuvvdtdr .)( 12 where ABu is a unit vector along AB

Q4 Two blocks of masses m2 and m are connected as shown in the figure. Now the string between the blocks is

suddenly broken.

(a.) gaA (b.)

2gaA

(c.) gaB (d.)

2gaB

Sol. (b, c) ‘B’ is in free fall, as string is cut, hence gaB tension in string, before it is cut, is Mg. hence net force on A

after cutting string is Mg 2gaA

Q5 A ball tied to the end of a string swings in a vertical circle under the influence of gravity (a.) When the string makes an angle 90 with the vertical the tangential acceleration is zero and radial acceleration



7

is somewhere between maximum and minimum (b.) When the string makes an angle 90 with the vertical the tangential acceleration is maximum and radial

acceleration is somewhere between maximum and minimum (c.) At no place in the circular motion, tangential acceleration is equal to radial acceleration in magnitude (d.) Throughout the path whenever radial acceleration has its extreme value, the tangential acceleration is zero

Sol. (b, d) In the shown diagram at position A and C radial acceleration has its maximum values and there

is no component of acceleration in tangential direction.

At position B and D tangential acceleration is maximum )(g and radial acceleration in some

where between maximum and minimum. Q6 Bodies of mass 0.5 kg resting on a horizontal frictionless tabletop are connected with an unstretched spring of

length L = 20 cm and of spring constant 16 N/m. The mass of the spring is negligible. At a certain moment the

bodies are given an initial speed of 0v = 0.36 m/s towards the wall on the right. The body at the right collides

with the wall totally elastically

(a.) There will be 2 collisions with the wall (b.) After 1st collision centre of mass comes to rest (c.) After 2nd collision centre of mass moves to left with speed v0 (d.) After all collisions are over, the system oscillates about the centre of mass

Sol. (a, b, c)

After 1st collision two are moving with speeds 0v in opposite direction. Hence centre of mass is at rest & two

start oscillating with initial position as mean position. So mass on right again collides with wall when at mean

position & both masses have same speed towards left after collision.

SECTION-D

Integer Answer Type

This section Contains 6 questions. The answer to each of the questions is a single digit integer, ranging from 00 to

99. The appropriate bubbles below the respective question numbers in the OMR have to be darkened.

Q1 A tank is filled upto a height 10 m with a liquid and is placed on a platform of height 2m from the ground. The

distance h from the ground where a small hole should be punched so that the liquid stream coming out of the

hole covers maximum horizontal distance in meters is



8

Sol. (6)

For maximum horizontal range, hole should be mid point or as close as possible to mid point between ground & liquid surface.

Q2 A point mass is subjected to two simultaneous sinusoidal displacements in x -direction, tAtx sin)(1 and

.3

2sin)(2

tAtx A third sinusoidal displacement

34sin)(3

tBtx is added such that its

average kinetic energy of SHM due to combination of the three displacements is four time its average kinetic

energy due to )(1 tx only. If values of B is ,nA write value of :n

Sol. (3) Q3 J600 of heat is added to a monatomic gas in a process in which the gas performs a work of .150 J If molar

heat capacity for the process is ,nR write value of :n

Sol. (2)

Q4 A wave is represented by the equation )02.0100(2sin10)10002.0(2sin10 xttxy on string

of length 1 m, where x and y are in cm and t is time in second. If String vibrates with n loops, write value

of n

Sol. (4) Q5 Velocity versus displacement curve of a particle moving in straight line is shown in the figure. From a point P,

a line is drawn perpendicular to displacement axis and line PR is drawn normal to the curve at P. The

instantaneous acceleration of the particle at point P is



9

Sol. (1)

21212

PRofslope12 ms

dxdvva

Q6 n moles of hydrogen gas is mixed with 2 moles in helium gas. The speed of sound in mixture is 12

times the

RMS speed of gas molecules in the mixture. Find .n

Sol. (2)

4233

21

00

frMRT

MrRT

242

)3(2)5(

nn

nf



10

Subject-CHEMISTRY Part : 2

SECTION-A


Q1 In the precipitation titration of KCl against AgNO3, K2CrO4 is used as an indicator. End point is detected by

appearance of deep colored precipitate of Ag2CrO4. Determine the concentration of chromate ion required for

detection of end point. Ksp of AgCl = 10–10 and Ksp of Ag2CrO4 = 10–12 (a.) 0.1 M (b.) 0.01M (c.) 0.001M (d.) 0.0001M

Sol. (b)

At end point [Ag+] = SPK =10–5

[CrO42–] =

25

12

10

10

= 0.01

Q2 Normility of “20 vol” H2O2 is (a.) 20 (b.) 10 (c.) 58.3 (d.) 79.1

Q3 The first four ionization energy values of an element are 190, 580, 870 and 5960 kcal. The number of valence

electrons in the element is (a.) 1 (b.) 2 (c.) 3 (d.) 4

Sol. (c)

Differences in first three ionization energies are less but fourth one is very high. This indicates that there are 3

valence electrons

Q4 The correct order of increasing s character of the orbital of B which overlaps with the orbital of F to form B-F

bond in 2BF , BF3 and BF4 is

(a.) 432 BFBFBF (b.) 234 BFBFBF (c.) 423 BFBFBF (d.)

342 BFBFBF

Sol. (b)

%)25(%),33.33(%),50( 34

232 spBFspBFspBF

Q5 Which of the given statement is/are correct for the given reaction knowing the fact that higher the number of

hyperconjugative structures higher the stability of carbocation

CH)(CHC)(CD 3333 CH)(CDC)(CH 3333

(a.) The equilibrium constant should be higher than one for the above reaction



11

(b.) The equilibrium constant should be lower than one for the above reaction (c.) C)(CD 33 will be more stable than C)(CH 33 due to higher electron releasing nature of deuterium than

hydrogen (d.) C)(CD 33 will be more stable than C)(CH 33 due to presence of stronger DC bond than HC bond

Sol. (a) cCH 33 )( is more stable than CCD

3)3( due to weaker bond strength of C-H bond

Q6 If the speed of electron in first Bohr orbit of hydrogen be ‘x’, then speed of the electron in second orbit of He+

is: (a.) x / 2 (b.) 2x (c.) x (d.) 4x

Sol. nZV

nZVH

1x

22

HeV

1HeV

1HeV

x xVHe

Q7 The molar entropy of vaporization of acetic acid is 14.4 cal K–1 mol–1 at its boiling point C.118 The latent

heat of vaporization of acetic acid is

(a.) 49 cal g–1 (b.) 64 cal g–1 (c.) 94 cal g–1 (d.) 84 cal g–1

Q8 The crystal system of a compound with unit cell dimensions nm504.0,387.0,387.0 cba and

90 and 120 is:

(a.) Cubic (b.) Hexagonal (c.) Orthorhombic (d.) Rhombohedral

SECTION-B


Q1 Three metals of alkaline earth metal group (A, B and C) when reacted with a fixed volume of liquid bromine separately gave product (metal bromide) whose mass is plotted against mass of metals taken as shown in the figure:



12

From the plot, predict what relation can be concluded between the atomic weights of A, B and C?

(a.) C > B (b.) B > A

(c.) C < A < B (d.) Data is inadequate to predict

Sol. (a, b)

Since mass of bromine taken is constant, same moles of 2MBr will be produced in each case.

Since, moles = (mass) / molecular weight.

Heaviest metal will produce maximum mass of product. Hence, correct order of atomic mass of A, B, C is

CBA

Q2 Two liquids A and B form an ideal solution. The solution has a vapor pressure of 700 Torr at .C80 It is

distilled till 2/3rd of the solution is collected as condensate. The composition of the condensate is 75.0' Ax

and that of the residue is .30.0" Ax If the vapor pressure of the residue at C80 is 600 Torr, which of the

following is/are true?

(a.) The composition of the original liquid was 6.0Ax (b.) The composition of the original liquid was 4.0Ax

(c.) Torr3/25000 AP (d.) Torr5000 BP

Sol. (a, c, d)

700 BBAA PxPx … (i)

6007.030.0"" BABBAA PPPxPx … (ii)

If moles of A & B initially are x & y then

)(31

3.0)(32

75.0 yxyxx

yxxxA

or yx

yxB

Solving given

6.0Ax 4.0Bx

torr500&torr3

2500 BA PP



13

Q3 A open ended mercury manometer is used to measure the pressure exerted by a trapped gas as shown in the

figure. Initially manometer shows no difference in mercury level in both columns as shown in diagram.

After sparking '' A dissociates according to following reaction

)(3)()( gCgBgA

If pressure of Gas "" A decreases to 0.9 atm. Then (Assume temperature to be constant and is 300 K)

(a.) Total pressure increased to 1.3 atm (b.) Total pressure increased by 0.3 atm

(c.) Total pressure increased by 22.3 cm to Hg (d.) Difference in mercury level is 228 mm

Sol. (a, b, d)

atm.3.11.031)31( TP

atm2.0P or cm3.076 of Hg or mm3.0760 of Hg

Q4 According to Maxwell’s distribution of molecular speeds: (a.)

mpavrms vvv

(b.) Only a small fraction of molecules are having either very high speeds or very low speeds.

(c.) As temperature increases, the fraction of molecules having velocity equal to mpv decreases.

(d.) For two gas samples at same temperature, the gas having higher molar mass will have larger fraction of

molecules possessing velocity equal to .vmp

Q5 Which of the following molecules posses zero dipole moment (a.) BF3 (b.) NF3 (c.) ClF3 (d.) CF4

Sol. (a, d)

Q6 Which of the following is(are) deactivating group(s) by mesomeric effect for electrophilic substitution reaction

when attached to benzene ring? (a.)

3RN

(b.) COOH (c.) RCHCH (d.) CN



14

Sol. (b, d)

(a) is electron withdrawing group by inductive effect.

(c) is electron releasing group when attached to benzene ring.

SECTION-D

Integer Answer Type



Q1 4.48 L of an ideal gas at STP requires 12 cal to raise the temperature by C15 at constant volume. The pC of

the gas is ______ cal.

Sol. (6)

4.48 L required 12 cal

22.4 L required cal6048.4

124.22

For C15 60 cal is required

for C1 cal415

601

is required

cal.4vC RCC vp cal6pC

Q2 One mole of van der waal’s gas, having compressibility factor (Z) = 0.8 exerts 2 atm pressure in a 10 L vessel.

Find the intercept along Y-axis when a graph plotted between mPV vs. P where P = pressure and mV Molar

volume of the real gas. (Assume at this condition pressure of ideal gas is 2.02 bar).

Sol. (25)

2.02 bar 2 atm

Based on given information force at attraction in real gas is negligible because gasidealgasreal PP

RTb)(VnaP 2

is RTb)(VP RTPbPV

RTPbPV

For PPV slope is b

Value of intercept is RT

RTPVZ

RT1028.0

250.8

102RT



15

Q3 Mixture of equal molecules of 2N and 5PCl is kept at 6 atm. Pressure. At equilibrium total pressure of the

vessel is 7.5 atm. Find the % degree of dissociation of .PCl5 Assume constant temperature.

Sol. (50)

5PCl 23 ClPCl

P3 P P

Total pressure 7.53P3

1.5P

degree of dissociation is 50% Q4 62HB excess of 3NH at high temperature give Product .x Hybridisation of B in P is .ysp Find the value

of .x

Sol. (2)

xtemphighNHofexcess

62 (BN)HB 3 (Boron nitride)

Hybridization is 2sp Ans. 2

Q5 Find the total number of isomeric alkynes possible of FClBrI.C3

Sol. (8)

F C C

Cl

Br

IC 2 Br C C

Cl

IC 2

F

F

C CCl Br

I

C 2

F

C C Cl

Br

I C 2

Q6 1 M 1000 ml aqueous solution of HCl is diluted to 2 litre. 1 litre of this solution is again diluted to 2 litre this

process is repeated for 3 more times. 20 ml of final solution is titrated with strong base .(OH)X 2 5 ml of

2(OH)X is required. Then normality of strong base is .YN

Find the value of .Y

Sol. (8)

Molarity of HCl is changed by

3216842MMMMMM

2211 VNVN



16

NNN 220

32

82NN



17

Subject-Mathematics Part : 3

SECTION-A


Q1 The equation ||)3(|1| )23(log 2)1( xxx xxx has

(a.) Unique solution (b.) Two solutions (c.) No solution (d.) More than two solutions

Sol. (c)

Q2 The coefficient of 49x in the product )99(...)3()1( xxx is

(a.) 299 (b.) 2500 (c.) 1 (d) 5000

Sol. (b)

Here, number of factors 50

The coefficient of 99...53149 x

)991(2

50

2500

Q3 If the number of ways of selecting n cards out of unlimited number of cards bearing the number .3,9,0 So

that they cannot be used to write the number 903 is ,93 then n is equal to

(a.) 3 (b.) 4 (c.) 5 (d.) 6

Sol. (c)

We can not write 903

If in the section of n cards, we get either 9( or 9(),3 or 0(),0 or ),3 (only ),0 (only )3 or (only )9

Can be selected n....222 times n2



18

Similarly, 9( or )0 or 3( or )0 can be selected n2

In the above selection (only )0 or (only )3 or (only )9 is repeated twice

Total ways 933222 nnn

962.3 n

52322 n

5n

Q4 The number of distinct values of a 22 determinant whose entries are from the set }1,0,1{ is

(a.) 3 (b.) 4 (c.) 5 (d.) 6

Sol. (c)

Possible values are 2,1,0,1,2

i.e., 111

10,0

0011

,11101

21111

,21111

Q5

If is a non real cube root of ,2 then the value of 122

31121

22

is equal to

(a.) 11 (b.) 12 (c.) 13 (d.) 0

Sol. (c)

12231121

22

applying 133 RRR

Then 131)2(616122

31121

322

)2,)2(( 33/1

Q6 If

233033

2 yx and ,441

5142

xy then

(a.)

230103

yx (b.)

021112

x

(c.)

212321

yx (d.)

211211

y

Sol. (b)



19

Q7

If ,122212221

A then Aadj is equal to

(a.) A (b.) TA (c.) A3 (d.) TA3

Sol. (d)

122212221

A

366636663

333231

232221

131211

CCCCCCCCC

Ac

.3122212

2213

366636

663')(adj Tc AAA

Q8 The decimal parts of the logarithms of two numbers taken at random are found to six places. Probability that

second can be subtracted from one first without borrowing is (a.) 6

21

(b.) 6

209

(c.) 6

2011

(d.) 6

203

Sol. (c)



20

Let

6

16543211

ii xxxxxxxD and

6

16543212

ii yyyyyyyD

Digit after decimal can be .9,...,3,2,1,0

Now, 2D subtract from 1D without borrowing.

ii yx

Let )90( iy

9,...,1, ix

After choosing ii xy , can be choosen in )10( ways.

Favourable ways

9

0

)10(

9

0

9

0

10

5521091010

And total number of ways 1001010

Required probability (for six digits) 66

2011

10055

SECTION-B


Q1 A bag contains four tickets marked with ,222,211,121,112 one ticket is drawn at random from the bag. Let

)3,2,1( iEi denote the event that thi digit on the ticket is .2 Then

(a.) 1E and 2E are independent (b.)

2E and 3E are independent

(c.) 3E and 1E are independent (d.)

21 , EE and 3E are independent

Sol. (a, b, c)

We have, 21

42)( iEP for ,3,2,1i

Also, for

).()(41)(, jiji EPEPEEPji



21

Therefore, iE and jE are independent for ,ji

Also, )()()(41)( 321321 EPEPEPEEEP

321 ,, EEE are not independent.

Q2

The determinant 22

22

22

xcbcacbcxbabacabxa

is divisible by

(a.) x (b.) 2x (c.) 3x (d.) 4x

Sol. (a, b, c, d)

Taking cba ,, commons from 321 ,, CCC respectively and then multiplying from cba ,, in 321 ,, RRR

respectively.

2222

2222

2222

xcccbxbbaaxa

Applying ,3211 RRRR then

2222

2222

222222222222

xcccbxbb

cbaxcbaxcbax

Applying 122 CCC and ,133 CCC then

)(0

000

22224

22

22

0222

cbaxxxc

xbcbax

Q3

Let ,122212221

A then

(a.) OIAA 32 54 (b.)

)4(51

31 IAA

(c.) 3A is not invertible (d.) 2A is invertible

Sol. (a, b, d)



22

Q4 Two roads are represented by the equations 6 xy and .8 yx An inspection bunglow has to be so

constructed that it is at a distance of 100 from each of the roads. Possible location of the bunglow is given by

(a.) )7,12100( (b.) )7,21001( (c.) )21007,1( (d.) )21007,1(

Sol. (a, b, c, d)

Q5 From the point )3,0(A on the circle ,0)3(4 22 yxx a chord AB is drawn and extended to a point

,M such that .2ABAM An equation of the locus of M is



23

(a.) 0)2(6 22 yxx (b.) 0)3(8 22 yxx

(c.) 096822 yxyx (d.) 044622 yxyx

Sol. (b, c)

Q6 The normals to the parabola axy 42 from the point )2,5( aa are

(a.) axy 3 (b.) axy 122 (c.) axy 333 (d.) axy 3

Sol. (a, b)

Any normal to parabola axy 42 may be taken as

)(2 2atxtaty or 02 3 atattxy … (i)

If Eq. (i) passes through ),2,5( aa then

0252 3 atatata

or 0233 tt

or 0)2()1( 2 ttt

1,2,1 t

From Eq. (i) normals are 03 axy and 0122 axy

SECTION-D

Integer Answer Type





24

Q1 If 7

2 i

e

and ,)(20

10

k

kk xAAxf then the value of

6

0

220 )()(

r

nn

nn

r xAxAAnxf then n

must be equal to

Sol. (7)

Q2 Let CBA ,, be three events. If the probability of occurring exactly one event out of A and B is ,1 a out

of B and C is ,21 a out of C and A is a1 and that of occurring three events simultaneously is .2a If

the probability that atleast one out of CBA ,, will occur is greater than then the value of 10 must be

Sol. (5)



25

Q3 Suppose a matrix A satisfies .752 OIAA If ,5 bIaAA then the value of

59

ab must be

Sol. (4)

Q4

The value of 4log4log3log3log

9log8log3log512log

33

82

43

43 is 2 then must be

Sol. (5)

4log4log3log3log

9log8log3log512log

33

82

43

43

23

23

222

23

3

29

3

2log2log3log3log

3log2log3log2log 3

2

2

2log22log2

3log313log

3log222log3

3log212log9

33

22

23

23

322

2319

1034

415

Q5 The trigonometric equation xxx 4cosec2coseccosec has a solution nn (,

7)12(

integer), then n

should not be an integer of the type pp (),60(,7 integer). The numerical quantity should be

Sol. (3)

The given equation is not defined for



26

Q6

If ,2

2tan50

142

1

r rr

rS then the value of

Stan

12752552

must be

Sol. (2)

Documents

12th IIT 21st May Test Paper - Solution - YLCeducation.ylc.co.in/.../08/12th-IIT-21st-May-Test-Paper-Solution.pdf · M g a Q4 A block of mass 1 kg is kept over another block of plane