CHRIST JUNIOR COLLEGEI PREPARATORY EXAMINATION – JANUARY 2015

II PUC – MATHEMATICS (35)

Time: 3 Hrs. 15min. Max. Marks: 100

PART - A

Answer all the questions 10x1=101. Let ¿ be the binary operation on N given by a ¿ b = L.C.M of a and b , find 20¿ 16

2. Find the value of sin−1(sin( 3π5 ))3. Find |4 A| if A=[4 −1

3 2 ] 4. Construct a 2x2 matrix A = [a ij] whose element are given by a ij=¿ 2i+ j ¿ .

5. The function f ( x )= 1x−3 is not continuous at x =3 justify the statement.

6. Evaluate ∫ tan22x .dx 7. Find the direction ratios of the vector , joining the points P (3,4,0) and Q (-2,-3,-4) ,

directed from P to Q 8. Find the equation of the plane with the intercept 3,4 and 5 on x, y and z axes

respectively.9. Define the term ‘constraints’ in the LPP.10. If P(E) = 0.6 and P( E∩F) = 0.2 then find P(F/E).

PART –B

Answer any ten questions : 10x2=2011. Show that the relation R in the set of integers given by R = {(a ,b );5∣(a−b)}

is symmetric and transitive.12. Find the value of tan−1 (√3 )−sec−1 (−2 )

13. Write cot−1( 1

√x2−1 ) , x>1 in the simplest form.

14. Find the area of triangle whose vertices are (2,0) (-1,0) and (0,3) by using determinants.

15. Find dydx, if sin2 x+cos2 y=k where k is a constant.

16. If y=log8 (logx ) , find dydx

17. Find the approximate change in the volume V of a cube of a side x meters caused by increasing by 5 %

18. Evaluate ∫ cos2 x+2sin2 x

cos2 x.dx

19. Evaluate ∫ dx

x2+4 x+9

20. Find the order and degree of the differential

equation xyd2 yd x2

+x ( dydx )2

− ydydx

=sin ( dydx ) 21. Find the area of parallelogram whose adjacent sides are given by the vectors

a⃗=i− j+k and b⃗=3 i+ j+4 k

22. If a⃗ is a unit vector and ( x⃗−a⃗ ) . ( x⃗+a⃗ )=24 then find|x⃗| 23. Find the angle between the pairs of lines r⃗=3i+5 j−k+λ (i+ j+k )

and r⃗=7 i++4k+μ (2 i+2 j+2k ).24. Two cards drawn at random and without replacement from a pack of 52 playing cards.

Find the probability that both the cards are black.

PART-C

Answer any Ten questions 10x3=3025. Show that the relation R in the set of real numbers R defined as R={(a ,b );a≤b2 }

is neither reflexive nor symmetric nor transitive.

26. Find the value of x , if tan−1( x−1x−2 )+ tan−1( x+1x+2 )=π4

27. If A =[1 54 7] , find A−1 by elementary operations .

28. Differentiate √ ( x+1 ) ( x+2 )( x+3 ) ( x+4 )

w.r.t x

29. Verify mean value theorem for the function f(x) = x2-4x-3 in the interval [1 ,4]. 30. Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.

31. Find ∫ 11−tanx

.dx

32. Evaluate ∫ ex . cosx .dx

33. Find the area of the region bounded by y2=9x , x=2 , x=4∧the x−axis¿ the first quadrant

34. Form the differential equation of the family of circles touching the y –axis at origin.35. Prove that for any two vectors a⃗∧b⃗ ,|a⃗+ b⃗|≤|⃗a|+|⃗b| 36. Find x such that the four points

A (3,2,1 )B (4 , x ,5 )C (4,2 ,−2 )∧D (6,5 ,−1 )are coplanar 37. Find the equation of the plane through the line of intersection of the planes

x+y+z=1 and2x+3y+4z=5 which is perpendicular to the plane x-y+z=038. Consider the experiment of tossing two fair coins simultaneously , find the probability

that both are head given that at least one of them is a head.

PART-D

Answer any six questions 6x5=3039. Let R+ be the set of all non – negative real numbers . show that the function

f :R+→¿ defined by f(x) = x2+4 is invertible . Also write the inverse of f(x).

40. If A=[−1 22 3 ] ,B=[ 1 −3

−3 4 ] verify that AB – BA is a skew symmetric matrix and

AB +BA is a symmetric matrix.41. Solve the following by using matrix method 2x+y+z =1 , x-2y-z =3/2 , 3y – 5z =9 .

42. If x=a ( cost+ tsint ) and y=a(sint−tcost ) find d2 yd x2

43. Sand is pouring from a pipe at the rate of 12cm3/sec . The falling sand forms a cone on the ground in such a way that the height of the cone is always one- sixth of the radius of the base . How fast is the height of the sand cone increasing when the height is 4 cm ?

44. Find the integral of √a2−x2 w.r.t x and hence evaluate √1+4 x− x2

45. Using integration find the area of the region bounded by the triangle whose vertices are (-1,0) (1,3) and (3,2)

46. Solve the differential equation xdydx

+2 y=x2logx

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

48. Find the probability of getting at most two sixes in six throws of a single die .

PART-E

Answer any one question 1x10=10

49. a. Prove that ∫a

b

f ( x ) . dx=∫a

b

f (a+b−x ) . dx and hence evaluate ∫π6

π3

( 11+√tanx ) .dx

b. Prove that |1 a a2

1 b b2

1 c c2|= (a−b ) (b−c ) (c−a )

50. a. A diet is to contain at least 80 units of vitamin A and 100 units of minerals .Two foods F1 and F2 are available . Food F1 costs Rs.4 per unit food and F2 costs Rs. 6 per unit . One unit of food F1 contains 3 units of vitamin A and 4 units of minerals . One unit of food F2

contains 6 units of vitamin A and 3 units of minerals. Formulate this as a linear programming problem. Find the minimum cost for diet that consists of mixture of these two foods and also meets the minimal nutritional requirements.

b. Discuss the continuity of the function f ( x )={ |x|+3 ,if x ≤−3−2 x ,if−3<x<36 x+2 , if x≥3

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