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II P.U.C.MATHEMATICS- ANNUAL EXAM-2014 Max. Marks:100

PART-A

ANSWER ALL THE QUESTIONS 10X1=10

1. Define equivalence relation.

2. Find the principal value of 1sec ( 2)− − .

3. If [ ]2 3 4A = ,210

B =

find AB.

4. without using direct expansion 102 18 36

1 3 417 3 6

Evaluate

5. If 0sin( )y x= , the find dydx

6. Evaluate xxe dx−∫

7. Find the direction cosines of the vector ˆˆ ˆ 2i j k+ −

8. Find the vector equation of a line joining passing through the points (-1,0,2) and (3,4,6)

9. Define feasible region?

10. If 7( )13

P A = . 9( )13

P B = and . 4( )13

P A B = . Evaluate P(A/B).

PART-B

Answer any Ten question 10x2=20

11.Show that the relation R in R defined as R ( ) }{ , : ,a b a b= ≤ is reflexive and transitive but not symmetric.

12. Show that 1 2 3tan tan tan2 11 4

− − −+ =

13. write2

1 1 1tan xx

− + −

, 0x ≠ ,in the simplest form.

14. If each element of any row (or column ) is multiple of K, then show that the value of whole det is multiplied by K.

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15. Show that the function f given by ( )3 3,1,

xf x

+=

0 0

if xif x

≠=

is not continuous at 0x = .

16. Differentiate sin xx . . . .w r t x

17. Find the point on the curve 2 2 25x y+ = where the tangent is parallel to x-axis.

18. Evaluate 1 dx

x x+∫

19.Evaluate 2

1

0

sin x x dx

π

−∫

20. Form the D.E of the family of ellipses having foci on y-axis and centre at origin .

21. Show that the points ( ) ( ) ( )ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ2 , 3 5 , 3 4 4A i j k B i j k C i j k− + − − − − are the vertices of a right angled

triangle.

22. Find 2

,a b−

if two vectors a

and b

are such that 2, 3a b= =

and a

. 4.b =

23.Find the angle between the pair of lines given by ˆ ˆˆ ˆ ˆ ˆ3 2 4 ( 2 2 )r i j k i j kλ= + − + + +

and ˆˆ ˆ ˆ ˆ5 2 (3 2 6 )r i j i j kµ= − + + +

24.If A and B are two independent events, then the probability of occurrence of at least one of A and B is given by ( )1 ( )P A P B′ ′−

PART – C

Answer any Ten questions. 10x3=30

25. On Z, the binary operation * is defined by a*b =a-b, ∀ a, bε z, verify whether * is commutative or Associate.

26. Prove that 1 1 13 8 84sin sin cos5 17 85

− − − − =

.

27. solve the equation for , ,x y z and t ,if 1 1 3 5

2 3 30 2 4 6

x zy t

− + =

28. Prove that every differentiable function is continuous.

29. If sec , tan ,x a y bθ θ= = Find dydx

also express dydx

in term of x & y.

30. Find the intervals in which the function given by ( ) sin 3 ,f x x= 0,2

x π ∈ is (a) increasing (b)

Decreasing

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31. Find ( )( )

2

2 21 4x dx

x x+ +∫

32. Find 1 4 5

15 1x x

−+∫ dx

33. Find the Area of the region bounded by the curve 2y x= & the lines 1x = , 4x = and x axis−

34. Find the General solution of the D.E ( )1 , 22

dy x ydx y

+= ≠

−

35. Three Vectors , a b and c

satisfy the condition 0a b c+ + =

.Evaluate the quantity

. . . ,a b b c c aµ = + +

if 1, 1 2a b and c= = =

.

36. Find a unit vector perpendicular to each of the vectors ( )a b+

and ( )a b−

,Where ˆˆ ˆa i j k= + + ,

ˆˆ ˆ2 3b i j k= + +

37. Find the vector equation of the plane passing through the intersection of the planes ˆˆ ˆ.( ) 6r i j k+ + =

and ˆˆ ˆ.(2 3 4 ) 5r i j k+ + = −

.

38. If a fair coin is tossed 10times,find the probability of (i) exactly six heads (ii) at most six heads.

PART - D 6 5 30× =

Answer any SIX questions

39. Let :f N R→ be a function defined as 2( ) 4 12 15f x x x= + + .Show that :f N S→ ,where ,S is the range of f ,is invertible.Find the inverse of f

40. If 0 6 7 0 1 1 26 0 8 , 1 0 2 , 2

7 8 0 1 2 0 3A B C

= − = = − −

Verify that ( )A B C AC BC+ = +

41.Solve by using Matrix method 3 2 3 8,2 1x y z x y z− + = + − = , 4 3 2 4x y z− + =

42.If ( )21tany x−= Show that ( )22 22 11 2 ( 1) 2x y x x y+ + + = .

43. A ladder 5m long is leaning against a wall.The bottom of the ladder is pulled along the ground, away from the wall,at the rate of 2cm/s.How fast is its height on the wall decreasing when the foot of the ladder is 4m away from the wall?

44. Prove that 2

2 2 2 2 1sin2 2x a xa x d x a x C

a− − = − + + ∫ ,hence evaluate 21 4x x dx+ −∫ .

45. Find the Area bounded by the curve 2 4x y= and the line 4 2x y= − .

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46.Find the General solution of the Differential equation cosdy y xdx

− = .

47. Derive the equation of a line in space passing through two given points both in the vector and Cartesian form.

48. If the Sum of the mean and variance of a binomial distribution for 5 trials be 1.8, find the distribution.

Part- E

Answer any One question 1x10=10

49. (a) Prove that 0 0

( ) ( )a a

f x f a x dx= −∫ ∫ ,hence evaluate ( )( )2

0

11 1

dxx x

∞

+ +∫ .

(b) Find all the points of discontinuity of the function f defined by 2, 1

( ) 0, 12, 1

x xf x x

x x

+ <= = − >

50. (a) Solve the following problem graphically: Minimise and Maximise 3 9Z x y= +

Subject to the constraints: 3 60; 10; 0, 0x y x y x y and x y+ ≤ + ≥ ≤ ≥ ≥

(b) If , ,x y z are different and

2 3

2 3

2 3

11 01

x x xy y yz z z

+∆ = + =

+,then show that1 0xyz+ = .

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