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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 = 8) = 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.

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