Time : 3 Hrs. MM : 100

INSTRUCTION :-

1. All question are compulsory.2. The question paper consist of 29 questions divided into three sections A, B and C. Section

A comprises of 10 questions of one mark each, section B comprises of 12 questions of fourmarks each and section C comprises of 07 questions of six marks each.

3. There is no overall choice. However, internal choice has been provided in 04 questions offour marks each and 02 questions of six marks each. You have to attempt only one of thealternatives in all such questions.

5. Use of calculators is not permitted. You may ask for logarithmic tables, if required.

1. If A {a,b,c}� and B {1,2,3}� and a function f : A B� is given by

f {(a,2),(b,3),(c,1)}� . Is f is a bijective function ?

2. Evaluate :1 1

sin / 3 sin2

�� �� �� � � � � �� �

3. Find the obtuse angle of inclination to the z-axis of a line that is inclined to x-axis at 45°

and to y-axis at 60°.

4. For what value of � are the vectors a 2i j k� � � �����

and b i 2j 3k� � �����

orthogonal ?

5. If a����

is a unit vector and (x a).(x a) 15� � �� �� �� �� �� �� �� �� �

then find | x |����

.

6. Form a 2 × 2 matrix ijA [a ]� where ija is given by ij

| 2i j |a

3i

�� .

7. If 2 3

A P Q1 0

� �� � �� �� �

where P is a symmetric matrix and Q is skew symmetric matrix

find the matrix P.

8. With out expanding, find the value of the determinant

0 q r r s

r q 0 p q

s r q p 0

� �

� �

� �

9. Evaluate the integral

1 8

2

(tan x)dx

1 x

�

��

10. Evaluate 2

1dx

9 25x��

11. Let A N N� � and � be the binary operation on A defined by

(a,b) (c,d) (a c,b d)� � � � Show that � is commutative and associative.

Find the identity element for � on A if any..

12. Solve the equation 1 11 x 1

tan tan x;(x 0)1 x 2

� ��� � � � � �

13. If x, y, z are different and

2 3

2 3

2 3

x x 1 x

y y 1 y 0

z z 1 z

�

� �

�; then show that xyz = –1

OR

Show that

1 a 1 11 1 1

1 1 b 1 abc 1a b c

1 1 1 c

�� �� � � � � �

�

14. Determine if funcion f(x) defined by

2 1x sin if x 0

f (x) x

0 if x 0

� ��� �� ��

is a continuous

function ?

15. If x 1 y y 1 x 0� � � � for 1 x 1� � � then prove that 2

dy 1

dx (1 x)� �

�

OR

If t

x a cos t log tan2

� �� � �

and y a sin t� then find dy

dx.

16. Find a point on the curve 2y (x 2)� � at which the tangent is parallel to the chord joining

the points (2, 0) and (4, 4).

OR

Show that the function f given by 1f (x) tan (sin x cos x);x 0�� � � is always an strictly

increasing function in (0, / 4)� .

17. Evaluate :

dx

cos(x a)cos(x b)� ��

OR

Evaluate : 2

dx

3x 13x 10� ��

18. Form the differential equation of the family of circles touching the x-axis at origin.

19. Solve the differential equation 2 2(x y )dx 2xydy 0� � �

20. A company has two plants to manufacturing scooters. Plant-I manufactures 70% of the

scooters and Plant-II manufactures 30%. At Plant-I, 30% of the scooters are rated of standard

quality and at plant-II, 90% of the scooters are rated of standard quality. A scooter is chosen

at random and is found to be of standard quality. Find the probability that it has come from

plant-II.

21. Let a i 4j 2k,b 3i 2j 7k� � � � � ���������

and c 2i j 4k� � �����

. Find a vector d����

which is

perpendicular to both a����

and b����

and c.d 15��������� .

22. Find the shortest distance between the lines l1 and l

2 whose vector equations are

r i j (2i j k)� � � � � �����

and r 2i j k (3i 5j 2k)� � � � � � �����

23. Obtain the inverse of the following matrix using elementary operations

0 1 2

A 1 2 3

3 1 1

� �� �� � �� �� �

24. If length of three sides of a trapezium other than base are equal to 10 cm. then find the area

of the trapezium when it is maximum.

OR

Show that the height of the cylinder of maximum volume that can be inscribed in a sphere

of radius R is 2R

3. Also find the maximum volume.

25. Evaluate :

/ 2

0

log sin x dx

�

�

OR

Evaluate : cot x tan x dx��

26. Find the area of the region enclosed between the two circles 2 2x y 4� � and

2 2(x 2) y 4� � � .

27. Find the distance of the point (–1, –5, –10) from the point of intersection of the line

r 2i j 2k (3i 4 j 2k)� � � � � � �����

and the plane r.(i j k) 5� � �����

.

28. Consider the experiment of tossing a coin. If the coin shows head, toss it again but if it

shows tail then throw a die. Find the conditional probability of the event that "the die

shows a number greater than 4'' given that "there is atleast one tail."

29. A dietician has to develop a special diet using two foods P and Q. Each packet (containing

30g) of food P contains 12 units of calcium, 4 units of iron, 6 units of cholesterol and 6

units of vitamin A. Each packet of the same quantity of food Q contains 3 units of calcium,

20 units of iron, 4 units of cholesterol and 3 units of vitamin A. The diet requires at least

240 units of calcium, atleast 460 units of iron and atmost 300 units of cholesterol. How

many packet of each food should be used to minimise the amount of vitamin A in the diet ?

What is the minimum amount of vitamin A ?

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