## Maths-1998-Set II

**Q2) Express the matrix
A = 3 1
-4 -1
as the sum of a symmetric and a skew symmetric matrix.**

Ans2) We know for a square matrix A. 1/2(A + A’ ) is sym and 1/2(A – A’ ) is skew-symmetric. Also

A = 1/2 (A + A’ ) + 1/2 (A – A’ )

Now

A =

3 1

-4 -1

=>A’ =

3 -4

1 -1

… 1/2 (A + A’ ) = 1/2

6 -3

-3 -2

1/2(A – A’ ) = 1/2

0 5

-5 0

Thus A =

0 -3/2

-3/2 -1

+

0 5/2

-5/2 0

where first one is symmetric and the second skew symmetric

**Q3) Evaluate
lim
x->/2 (2x – )/cos x**

Ans3)

lim

x->/2 (2x – )/cos x putting /2 – x = , as x -> /2 , -> 0

= lim

->0 (2(/2 – ) – )/cos (/2 – )

= lim

->0

-2/sin = -2 (as lim /Sin = 1)

->0

## Maths-1999-Set I

(i) All Questions are compulsory

(ii) Question number 1 to 15 are of 2 marks each

(iii) Question number 16 to 25 are of 4 marks each

(iv) Question number 26 to 30 are of 6 marks each

**Q1) A coin is tossed 12 times. What is the Probability of getting exactly 8 tails? (Marks 2)**

Ans1) When a coin is tossed, we have S = {H, T}

p = P (getting a tail) = 1/2

… q = 1 – p = 1/2

Here n = 12

Now the Probability Distribution

B(n, p) = B(12, 1/2)

i.e. (q + p)12 where q = p = 1/2

… P (X = = 12C8 . q12 – 8 . p8

= 12C4 . q4 p8 = 12×11x10×9 x (1/2)4 + 8

4

= 498 = 495

212 4096

**Q2) Three coins are tossed simultaneously. List the sample space for the event. (Marks 2)**

Ans2) Sample space S, of tossing 3 coins simultaneously is given by:-

S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

**Q3) Two cards are drawn without replacement from a well shuffled pack of 52 cards. What is the probability that one is red queen and the other is a king of black colour? (Marks 2)**

Ans3) We know that there are 2 red queens and two kings of black colour in a pack of 52 cards.

P (drawing a red queen) = 2/52 = 1/26

Since the card is not replaced, after the firs card, we have 51 cards left.

… P (drawing a black king) = 2/51

But these two draws are interchangeable.

… The reqd. probability = 2 x 1/26 x 2/51 = 2/(13 x 51) = 2/663

**Q4) Find a unit vector perpendicular to both = 3 + – 2 and = 2 + 3 – (Marks 2)**

Ans4) Here, = 3 + – 2 and = 2 + 3 -

=> x =

3 1 -2

2 3 -1

= 5 – + 7

… = Unit vector | to and

= ( x )/[ x ] = (5 – + 7)/(52 + (-1)2 + 72)

= (5 – + 7)/(75) = 1/(53) (5 – + 7)

## Maths-1999-Set II

**Q1) If y = cot x, show that d2y/dx2 + 2y dy/dx = 0 (Marks 2)**

Ans1) Here y = cot x

differentiating both sides w.r.t. x:-

dy/dx = -cosec2x

differentiating again w.r.t. x:-

d2y/dx2 = -d/dx (cosec2x)

d2y/dx2 = -2cosecx(-cosec x . cot x ) = +2cosec2x . cot x

or d2y/dx2 = -2 dy/dx . y

or d2y/dx2 + 2y dy/dx = 0.

**Q3) If
A = 2 2
-3 1
4 0 B = 6 2
1 3
0 4 **

**find the matrix C such that A + B + C is a zero matrix. (Marks 2)**

Ans3) Here

A = 2 2

-3 1

4 0 B = 6 2

1 3

0 4

we want to find C such that A + B + C = 0

< => C = 0 – A – B = -A – B

= – 2 2

-3 1

4 0 – 6 2

1 3

0 4

= -2 -2

3 -1

-4 0 + -6 -2

-1 -3

0 -4

= -8 -4

2 -4

-4 -4

**Q5) Construct a 3 x 2 matrix whose elements in the ith row and the jth coloumn are given by:-
aij = (3i + j)/2 (Marks 2)**

Ans5) We want a 3 x 2 matrix

=> 1 i 3 and 1 j 2

where aij = (3i + j)/2

… a11 = (3 + 1)/2 = 2; a12 = (3 + 2)/2 = 5/2;

a21 = (6 + 1)/2 = 7/2; a22 = (6 + 2)/2 = 4;

a31 = (9 + 1)/2 = 5; a32 = (9 + 2)/2 = 11/2

… The reqd matrix is :-

a11 a12

a21 a22

a31 a32

=

2 5/2

7/2 4

5 11/2

## Maths-2000-Set I

**Q1) Evaluate
{(1 – Cos x)/x2}**

Ans1)

{(1 – Cos x)/x2} = (2Sin2x/2)/x2

= 2/4 (Sin(x/2)/(x/2))2 =2/4 . 12 = 1/2 (as Sinx/x=1)

**Q2) A particle moves along a straight line so that S = t . Show that the acceleration is negative and is proportional to the cube of velocity.**

Ans2) Here the equation of motion is

S = t

differentiating both sides w.r.t. ‘t’

v = dS/dt = 1/2t = 1/2S ( S = t )

or v = 1/2S

differentiating again w.r.t. ‘t’

accn = dv/dt = -1/2s2 ds/dt

= -2/4s2 ds/dt

= -(2/(1/v2)). v = -2v3

accn is negative and is proportional to cube of velocity.

**Q3) Evaluate
Cos4xdx**

Ans3) let

I = Cos4xdx = (Cos2x)2dx

= ((1 + Cos 2x)/2)2dx = 1/4 (1 + Cos22x + 2Cos 2x)dx

=1/4 [1 + (1 + Cos 4x)/2 + 2Cos 2x]dx

= 1/4[x + x/2 + (Sin 4x)/8 + (2Sin 2x)/2] + C

= 3x/8 + (Sin 4x)/32 + (1/4)Sin 2x + C

**Q4) Evaluate
(1 + tan x)/(x + log sec x) dx**

Ans4) Let

I = (1 + tan x)/(x + log sec x) dx

= log | x + log sec x | + C

[ as f'(x)/f(x) dx = log | f(x) | + C ]

## Maths-2000-Set II

**Q2) Find a matrix X such that A + 2B + X = 0, where
A= 2 -1
3 5
B=-1 1
0 2**

Ans2) Here

A= 2 -1

3 5

B= -1 1

0 2

A + 2B + X = 0

X = -A – 2B

= – 2 -1 – -1 1

3 5 2 0 2

= 0 -1

-3 -9

**Q4) Calculate Spearman’s rank correlation from the following Data:
x 1 2 3 4 5 6
y 2 3 1 6 5 4**

Ans4) Here :-

X Y RX RY di=RX – RY di2

1 2 6 5 1 1

2 3 5 4 1 1

3 1 4 6 -2 4

4 6 3 1 2 4

5 5 2 2 0 0

6 4 1 3 -2 4

di2 = 14

R, the rank Correlation

= 1 – (6di2/n(n2 – 1))

= 1 – (6 x 14)/6(35)

= 1 – 14/35 = 21/35 = 0.6

**Q7) Evaluate
1+ Cot x dx
x + log Sin x**

Ans7) Let

I= 1 + Cot x dx

x + log Sin x

= log | x + log Sin x | + C [ f'(x)/f(x) dx = log|f(x)| + c and d/dx(x + logSin x) = 1+Cot x ]

## Physics-1996-Set I

**Q 1. What is the work done in moving a charge of 10 nC between two points on an equipotential surface ? (1 mark)**

Ans 1. dw = qdv

on equipotential surface dv = 0

dw = 0 i.e,no work is done in moving a change on an equipotential surface.

**Q 2. Name the device used for measuring the internal resistance of a secondary cell ? (1 mark)**

Ans 2. Internal resistance of a secondary cell can be measured by a potentiometer.

**Q 3. What is the nature of magnetic field in a moving coil galvanometer ? (1 mark)**

Ans 3. A radial magnetic fields exists in a moving coil galvanometer.

**Q 4.If a rate of change of current of 2 A/s induces an e.m.f.of 10mV in a solenoid, what is the self -inductance of the solenoid ? (1 mark)**

Ans 4. Given dI = 2 A/S , 10mv = 10-3V

dt

| | = L dI

dt

or 10 x 10

-3 = L x 2

or L = 10-2 = 5 x 10-3 H = 5mH

2

**Q 5. What type of magnetic material is used in making permanent magnets ? (1 mark)**

Ans 5. A substance like steel which as a large retentivity and a large coercivity is used in making permanent magnets.

**Q 6. Two metals A, B have work-functions 2eV, 4eV respectively. Which metal has lower threshold wavelength for photoelectric effect ? (1 mark)**

Ans 6. Work function W = h v

0 = hc where is threshold wavelength.

x 1

W

Metal B has higher value of W & hence lower threshold wavelength.

## Physics-1996-Set II

**[Only those questions and answers which are different from set 1 have been given here.]**

**Q.1.If a rate of change of current of 4 A/s induces an e.m.f. of 20 mV in a solenoid, what is the self-inductance of the solenoid ? (1 mark)**

Ans 1.

…………….. dI = 4A/S

…………… …….dt

…………… ……. = 20mv = 20 x 10-3 V

…………… ……. | | = LdI

…………… ……………….dt

20 x 10-3 = L x 4 or L = 20 x 10-3 = 5 x10-3 H

…………… …………………. …………… 4

**Q 4. What is the work done in moving a charge of 20 nC between two points on an equipotential surface ? (1 mark)**

Ans 4. The work done in moving a change zone between two points on equipotential service is zero work done = q dv

…………… …dv = 0

…….work done = 0

**Q 5. Two stars A and B have magnitudes +1 and -1 respectively. Which of these two appears brighter? (1 mark)**

Ans 5. More negative the magnitude,brighter is the star. Hence B is brighter than A.

**Q 6. Define transconductance of a transistor. (1mark)**

Ans 6. Transconductance of a transistor is the ratio of change in collector current to the corresponding change in base-emitter voltage.

**Q 7. Which of the following has the shortest wavelength : microwaves,ultra-violet rays X-rays ? (1 mark)**

Ans 7. X-rays have the shortest wave-length among the three given wavelengths.

## Physics-1996-Set III

**[Only those questions and answers which are different from Set I have been given here.] **

**Q 2. If a rate of change of current of 3 A/s induces an e.m.f. of 30 mV in a solenoid,what is self-inductance of the solenoid ?(1 mark)**

Ans 2. dI = 3A/S

dt

= 30 mv = 30 x 10-3

| | = L dI

dt

30 x 10-3 = L x 3 or L = 10-2 H.

**Q 4. Name the device used for measuring the e.m.f. of a cell.
(1 mark)**

Ans 4 Potentiometer is used for measuring the emf of a cell.

**Q 5. What is the work done in moving a charge 40nC between two points on an equipotential surface ?(1 mark)**

Ans 5 dw = q dv

on an equipotential surface dv = 0

dw = 0

**Q 6. Two stars A and B have magnitudes +1 and -1 respectively.Which of these two appear less bright? (1mark)**

Ans 6. Star with positive magnitude i.e,+1 will appear less bright

## Physics-1997-Set I

**Q1) Horizontal component of Earth’s magnetic field at a place is 3 times the vertical component. What is the value of angle of dip at this place? (Marks 1)**

Ans1) tan = BV/BH = 1/3

or = 30o

**Q2) Force between two point electric charges kept at a distance d apart in air is F. If these charges are kept at the same distance in water, how does the force between them change? (Marks 1)**

Ans2) F = (1/4or) q1q2/r2

where r is the relative permittivity.

r = 1 for air & 81 for water.

Fa/Fw = 81/1

or Fw = 81 Fa

**Q3) Give any two factors on which thermo-electric emf produced in a thermo-couple depends. (Marks 1)**

Ans3) Thermoemf depends on

(i) Temp difference between the junctions

(ii) Nature of metals forming thermocouple

**Q4) The electric current in a wire in the direction from B to A is decreasing. What is the direction of induced current in the metallic loop kept above the wire as shown in the figure? (Marks 1)**

Ans4)

Induced current will be clockwise

**Q5) Name the electromagnetic radiations used for viewing objects through haze and fog. (Marks 1)**

Ans5) Infrared rays.

**Q6) Give the ratio of the number of holes and the number of conduction electrons in an intrinsic semiconductor. (Marks 1)**

Ans6) One

**Q7) In the given diagram, is the diode D forward or reversed biased? (Marks 1)**

Ans7) The diode is reverse biased since n-type is at a potential higher than p-type.

**Q8) Name the planet which has maximum value of albedo**.

(Marks 1)

Ans8) Venus

**Q9) Two point electric charges of unknown magnitude and sign are placed a distance ‘d’ apart. The electric field intensity is zero at a point, not between the charges but on the line joining them. Write two essential conditions for this to happen. (Marks 2)**

Ans9) (i) They should be of opposite signs

(ii) They must be of unequal magnitude.