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65(B) 1 C/1 amo b Z§. Roll No. ( ) MATHEMATICS (FOR BLIND CANDIDATES ONLY) : 3 : 100 Time allowed : 3 hours Maximum Marks : 100 Series : SGN/C H¥ n¶m Om±M H a b| {H Bg àíZ-nÌ ‘| ‘w {ÐV n¥ð 15 h¢ & àíZ-nÌ ‘| Xm{hZo hmW H s Amoa {XE JE H moS Zå~a H mo NmÌ CÎma-nwpñVH m Ho ‘w I-n¥ ð na {bI| & H¥ n¶m Om±M H a b| {H Bg àíZ-nÌ ‘| 29 àíZ h¢ & H¥ n¶m àíZ H m CÎma {bIZm ewê H aZo go nhbo , àíZ H m H« ‘m§H Adí¶ {bI| & Bg àíZ-nÌ H mo n‹ TZo Ho {bE 15 {‘ZQ H m g‘¶ {X¶m J¶m h¡ & àíZ-nÌ H m {dVaU ny dm©• ‘| 10.15 ~Oo {H ¶m OmEJm & 10.15 ~Oo go 10.30 ~Oo VH NmÌ Ho db àíZ-nÌ H mo n‹ T| Jo Am¡a Bg Ad{Y Ho Xm¡amZ do CÎma-nw pñVH m na H moB© CÎma Zht {bI|Jo & Please check that this question paper contains 15 printed pages. Code number given on the right hand side of the question paper should be written on the title page of the answer-book by the candidate. Please check that this question paper contains 29 questions. Please write down the Serial Number of the question before attempting it. 15 minute time has been allotted to read this question paper. The question paper will be distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the students will read the question paper only and will not write any answer on the answer-book during this period. SET – 4 narjmWu H moS H mo CÎma-nwpñVH m Ho ‘wI- ð na Adí¶ {bI| & Candidates must write the Code on the title page of the answer- book. HmoS Z§. Code No. 65(B)

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65(B) 1 C/1

amob Z§. Roll No.

( )

MATHEMATICS (FOR BLIND CANDIDATES ONLY)

: 3 : 100 Time allowed : 3 hours Maximum Marks : 100

Series : SGN/C

H¥$n¶m Om±M H$a b| {H$ Bg àíZ-nÌ ‘| ‘w{ÐV n¥ð 15 h¢ & àíZ-nÌ ‘| Xm{hZo hmW H$s Amoa {XE JE H$moS> Zå~a H$mo N>mÌ CÎma-nwpñVH$m Ho$

‘wI-n¥ð> na {bI| & H¥$n¶m Om±M H$a b| {H$ Bg àíZ-nÌ ‘| 29 àíZ h¢ & H¥$n¶m àíZ H$m CÎma {bIZm ewê$ H$aZo go nhbo, àíZ H$m H«$‘m§H$ Adí¶ {bI| & Bg àíZ-nÌ H$mo n‹T>Zo Ho$ {bE 15 {‘ZQ> H$m g‘¶ {X¶m J¶m h¡ & àíZ-nÌ H$m

{dVaU nydm©• ‘| 10.15 ~Oo {H$¶m OmEJm & 10.15 ~Oo go 10.30 ~Oo VH$ N>mÌ Ho$db àíZ-nÌ H$mo n‹T>|Jo Am¡a Bg Ad{Y Ho$ Xm¡amZ do CÎma-nwpñVH$m na H$moB© CÎma Zht {bI|Jo &

Please check that this question paper contains 15 printed pages. Code number given on the right hand side of the question paper

should be written on the title page of the answer-book by the candidate.

Please check that this question paper contains 29 questions. Please write down the Serial Number of the question before

attempting it. 15 minute time has been allotted to read this question paper. The

question paper will be distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the students will read the question paper only and will not write any answer on the answer-book during this period.

SET – 4

narjmWu H$moS> H$mo CÎma-nwpñVH$m Ho$ ‘wI-n¥ð> na Adí¶ {bI| & Candidates must write the Code on the title page of the answer-book.

H$moS> Z§. Code No.

65(B)

65(B) 2 C/1

:

(i)

(ii) - 29

(iii) - . 1-4 - , 1

(iv) - . 5-12 - 2

(v) - . 13-23 - I 4

(vi) - . 24-29 - II 6

General Instructions :

(i) All questions are compulsory.

(ii) This question paper contains 29 questions.

(iii) Questions No. 1-4 in Section A are very short-answer type questions carrying 1 mark each.

(iv) Questions No. 5-12 in Section B are short-answer type questions carrying 2 marks each.

(v) Questions No. 13-23 in Section C are long-answer – I type questions carrying 4 marks each.

(vi) Questions No. 24-29 in Section D are long-answer – II type questions carrying 6 marks each.

65(B) 3 C/1

SECTION – A

1 4 1

Question numbers 1 to 4 carry 1 mark each.

1. A = 3 B A B 3 , | B | = 2 ,

| A |

If A = 3B, where A and B are square matrices of order 3 and | B |

= 2, then find | A |.

2. : .

.

sin–1 x1 – x2 dx.

Find : .

.

sin–1 x1 – x2 dx.

3. tan x sec x

Write the derivative of tan x with respect to sec x.

4. a ^b | a – ^b | = 1 , a ^b

If a and ^b are two unit vectors and | a – ^b | = 1, then find the acute

angle between a and ^b.

65(B) 4 C/1

SECTION – B

5 12 2

Question numbers 5 to 12 carry 2 marks each.

5. : cot –1 (–x) = – cot –1 x, x .

Prove that : cot –1 (–x) = – cot –1 x, x .

6. sin

sin–1 27 + cos–1 2x = 1 , x

If sin

sin–1 27 + cos–1 2x = 1, then find the value of x.

7. X Y =

3 2

1 4 2X + 3Y =

7 4

–1 10

Find the matrix X if Y =

3 2

1 4 and 2X + 3Y =

7 4

–1 10

8. f(x) = x3 – 5x2 – 3x [1, 3]

Verify Mean value theorem, for the function f(x) = x3 – 5x2 – 3x in

the interval [1, 3].

65(B) 5 C/1

9. , 6y = x3 + 2

x- y – 2

A particle moves along the curve 6y = x3 + 2. Find the points on the

curve at which the y – co-ordinate is changing 2 times as fast as the

x – co-ordinate.

10. : .

. xx

xd

)a(sin

a) ( sin

.

Find : .

. xx

xd

)a(sin

a) ( sin

11. a = i – j + 3k b = 2 i – 7 j

+ k

The adjacent sides of a parallelogram are determined by the

vectors a = i – j + 3k and b = 2 i – 7 j + k. Find the vectors

determining its diagonals and hence find the area of the

parallelogram.

12.

A die is tossed thrice. Find the probability of getting an odd

number at least once.

65(B) 6 C/1

SECTION – C

13 23 4

Question numbers 13 to 23 carry 4 marks each.

13. :

xy xz x2 + 1

y2 + 1 yz xy yz z2 + 1 xz

= 1 + x2 + y2 + z2

A =

3 2 0

1 4 0 0 0 5

, A2 – 7A + 10 I3 = O A–1

Using properties of determinants, prove that :

xy xz x2 + 1

y2 + 1 yz xy yz z2 + 1 xz

= 1 + x2 + y2 + z2

OR

If A =

3 2 0

1 4 0 0 0 5

, show that A2 – 7A + 10 I3 = O, hence find A–1.

65(B) 7 C/1

14. f (x) =

(3sin x –1)2

x log (1 + 5x) , x 0

k , x = 0

x = 0 , k

y = 1 – x1 + x , (1 – x2)

dydx + y = 0.

If the function f (x) =

(3sin x –1)2

x log (1 + 5x) , x 0

k , x = 0

is continuous at x = 0, find k.

OR

If y = 1 – x1 + x , prove that (1 – x2)

dydx + y = 0

15. (sin x)y = x + y ,

dydx =

1 – (x + y) y cot x(x + y) log sin x – 1

If (sin x)y = x + y, then show that

dydx =

1 – (x + y) y cot x(x + y) log sin x – 1

65(B) 8 C/1

16. :

..

x2 + xx3 – x2 + x – 1

dx

: .

.e2x.cos (3x + 1) dx

Find :

..

x2 + xx3 – x2 + x – 1

dx

OR

Find : .

.e2x.cos (3x + 1) dx

17. :

0

2

d cossin

sinx

xx

x

Evaluate :

0

2

d cossin

sinx

xx

x

18.

dydx – 3 cot x . y = sin 2x; y = 2 x =

2

Find the particular solution of the differential equation

dydx – 3 cot x . y = sin 2x ; given that y = 2 when x =

2

65(B) 9 C/1

19.

x –

Form the differential equation representing the family of ellipses

having foci on x – axis and centre at origin.

20. A(3, 2, 1), B(4, x, 5), C(4, 2, –2) D(6, 5, –1)

, x

If the four points A(3, 2, 1), B(4, x, 5), C(4, 2, –2) and

D(6, 5, –1) are coplanar, then find the value of x.

21. , ,

:

r = ( i + 2 j + 3k) + ( i – 3 j + 2k)

r = (4 i + 5 j + 6k) + (2 i + 3 j + k).

Find the shortest distance between the lines whose vector

equations are given below :

r = ( i + 2 j + 3k) + ( i – 3 j + 2k) and

r = (4 i + 5 j + 6k) + (2 i + 3 j + k).

65(B) 10 C/1

22. A 4 4 , B 3 5

C 5 3

,

B

From three bags, bag A contains 4 red and 4 black balls, bag B

contains 3 red and 5 black balls and bag C contains 5 red and 3

black balls. A bag out of the three is selected at random and a ball

is drawn from it randomly. If the drawn ball is found to be red,

find the probability that it was drawn from bag B.

23. 52

From a well shuffled pack of 52 playing cards, two cards are

drawn at random without replacement. Write the probability

distribution of number of kings. Hence find the mean of the

distribution.

65(B) 11 C/1

SECTION – D

24 29 6 Question numbers 24 to 29 carry 6 marks each.

24. f : [–1, 1] → , f(x) = x

x + 2 ,

f : [–1, 1] → (f ), f–1

–1

3

f–1

1

5

a * b = (a + b)

2 , a, b R

(i)

(ii)

*

Show that f : [–1, 1] → given by f(x) = x

x + 2 is one – one. Find

the inverse of the function f : [–1, 1] → (Range of f) and hence

find f–1

–1

3 and f–1

1

5 .

OR

65(B) 12 C/1

Find whether the binary operation :

a * b = (a + b)

2 , a, b R is

(i) Commutative

(ii) Associative

Hence, find if the operation * has identity or not.

25. :

x – y + 2z = 7 ; 3x + 4y – 5z = – 5 ; 2x – y + 3z = 12.

Solve the following system of equations using matrix method :

x – y + 2z = 7 ; 3x + 4y – 5z = – 5 ; 2x – y + 3z = 12.

26. y2 = 4x

(0, 3)

28

?

65(B) 13 C/1

Find the equation of the normal to the curve y2 = 4x, which passes

through the point (0, 3).

OR

A wire of length 28 cm is cut into two pieces. One made into a

square and the other into a circle. Find the lengths of the two

pieces so that the combined area of square and the circle is

minimum.

27. y2 = 9x, x = 2, x = 4 x –

2

.. (2x2 + 3x + 1) dx.

Using integration, find the area of the region bounded by y2 = 9x,

x = 2, x = 4 and the x – axis in the first quadrant.

OR

By the method of limit of sum, find the value of the following

definite integral.

2

.. (2x2 + 3x + 1) dx.

65(B) 14 C/1

28. (–1, –5, –10) r = 2 i – j + 2k + (3 i + 4 j + 2k)

r . ( i – j + k) = 5

Find the distance of the point (–1, –5, –10) from the point of

intersection of the line r = 2 i – j + 2k + (3 i + 4 j + 2k) and the

plane r . ( i – j + k) = 5

29. X Y

A 10 , B 12

C 8 1 ..

:

A B C

X 1 2 3

Y 2 2 1

X 1 ` 16 Y 1 ` 20 ,

(i)

(L.P.P.)

(ii) ?

65(B) 15 C/1

A dietician wishes to mix two kinds of food X and Y in such a way

that the mixture contains at least 10 units of Vitamin A, 12 units of

vitamin B and 8 units of vitamin C. The vitamin contents of one kg

food is given below :

Food Vitamin A Vitamin B Vitamin C

X 1 2 3

Y 2 2 1

One kg of food X costs ` 16 and one kg of food Y costs ` 20. Then

(i) for finding the least cost of the mixture which will produce the

required diet, write an L.P.P.

(ii) What is the importance of balanced diet in a person’s life ?

__________

65(B) 16 C/1