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12th Maths Practice paper-04

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MATHEMATICS Class II PreUni Plus 2016-2017 Practice Paper - 04 Take Home Max. Marks Duration 100 3 hours 15 minutes Instructions 1. Do not overwrite. 2. Draw a line using pen at the end of each answer. 3. Steps adopted in solving problems should be clearly shown. 4. In case of violation of the above conditions, the answer scripts will not be considered for valuation. PART - A I. Answer ALL the following (ONE mark each) [10 1 = 10] 1. Give an example of a relation which is transitive but neither reflexive nor symmetric. 3 2. Find the value of sin 1 sin . 5 3. Construct a 2 2 matrix A = [aij] whose elements are given by a ij i j 2 2 . 3 2 1 0 4. Find X if Y = and 2X Y 1 4 3 2 5. If y = sec tan x find dy . dx 6. Find the integral of tan2(2x 3) w.r.t. x. 7. Find the direction cosines of the vector i 2 j 3k . 8. Find the distance of the plane 2x 3y + 4z 6 = 0 from the origin. 9. Define feasible region of a linear programming problem. 3 1 1 10. If P A , P B find P(A B). If A and B are independent events such that P(A) , then find 5 5 5 P(A B). PART - B II. Answer any TEN of the following (TWO marks each) 11. Show that f : R0 R0 defined by f(x) = 1 is one-one and onto where R0 is the set of all non-zero real x numbers. 3 cos x 12. Express tan 1 x in the simplest form. , 2 1 sin x 2 3 3 13. Evaluate tan sin 1 cot 1 . 5 2 2P1617MPP4 [10 2 = 20] 1 14. Find the area of the triangle whose vertices are (3, 8), ( 4, 2) and (5, 1) using determinants. 15. If f is differentiable at a point c, show that it is also continuous at that point. 16. Find the derivative of log x cos x w.r.t. x 17. If the radius of the sphere is measured 7 m with an error of 0.02 m then find the approximate error in calculating its volume. 18. Evaluate cos x sin x dx . 1 sin 2x 19. Evaluate sin 1 x dx . 1 dy 5 3 d 2 y 20. Find the order and degree of the differential equation 1 2 dx dx 21. Show that the points A(2, 3, 4), B( 1, 2, 1), C(5, 8, 7) are collinear. 22. Find and if 2i 2 j 27k i j k 0 . 23. Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector 3i 5j 6k . 24. In a school there are 1000 students, out of which 430 are girls. It is known that out of 430, 10% of the girls study in class XII. What is the probability that a student chosen randomly studies in class XII given that the chosen student is a girl. PART - C III. Answer any TEN of the following (THREE marks each) [10 3 = 30] 25. Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b) |a b| is even} is an equivalence relation. 1 x 1 1 26. Solve tan 1 tan x . 1 x 2 27. If A and B are square matrices of the same order then show that (AB) 1 = B 1 A 1. 28. Verify Mean value theorem for f(x) = x3 5x2 3x in the closed interval [1, 3] 29. Find dy , if x a cos sin , y a sin cos . dx 30. Find the intervals in which the function f(x) = sin x + cos x, 0 x 2 is strictly increasing or strictly decreasing. 5 31. Find x 1 dx as the limit of a sum. 0 32. Evaluate 2P1617MPP4 x 1 x log x 2 x dx 2 33. Form the differential equation of the family of circles touching the x-axis at the origin. 34. Find the area of the region bounded by the line y = 3x + 2, the x-axis and the ordinates at x = 1 and x = 1. 35. Show that the vectors a, b and c are co-planar if a b, b c and c a are co-planar. 36. Two adjacent sides of a parallelogram are 2i 4 j 5k and i 2 j 3k . Find the unit vector parallel to its diagonal. Also find its area. x 3 y 1 z 3 x 1 y 4 z 5 and 3 5 4 1 1 2 37. Find the angle between the pair of lines 38. Find the probability distribution of number of doublets in the three throws of a pair of dice. PART - D IV. Answer any SIX of the following (FIVE marks each) [6 5 = 30] 39. f : N R defined by f(x) = 4x2 + 12x + 15. Show that f : N S, S is the range of f is invertible. Find the inverse of f. 1 4 40. By using elementary row operations find the inverse of the following matrix A . 4 2 1 1 2 2 0 1 41. Use product 0 2 3 9 2 3 to solve the system of equations x y + 2z = 1, 2y 3z = 1 and 3 2 4 6 1 2 3x 2y + 4z = 2. 42. If y = (tan 1 x)2 show that (x2 + 1)y2 + 2x (x2 + 1)y1 = 2. 43. Sand is pouring from a pipe at the rate of 12 cm3 / s. 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. Evaluate x 3 5 4x x 2 dx . 45. Find the area of the region enclosed between the two circles x2 + y2 = 4 and (x 2)2 + y2 = 4. x x 46. Show that the differential equation 2ye y dx y 2xe y dy 0 is homogeneous and solve it. 47. Derive the equation of the plane in the normal form both in vector and Cartesian form. 48. An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of an accidents are 0.01, 0.03 and 0.15 respectively. One of the insured person met with an accident. What is the probability that he is a scooter driver? 2P1617MPP4 3 PART - E V. Answer any ONE of the following (TEN marks) a [1 10 = 10] 2 a 49. (a) Prove that f x dx f a x dx and hence evaluate log sin xdx . 0 [6] 0 0 k cos x 2x if (b) Find k, so that f is continuous at x . f x 2 3 if 2 x 2 x [4] 50. (a) Two godowns A and B have grain capacity of 100 quintals and 50 quintals respectively. They supply to 3 ration shops D, E and F whose requirements are 60, 50 and 40 quintals respectively The cost of transportation per quintal from the godowns to the shops are given in the following table. Transportation cost / qunintal (in Rs.) From / To A B D E 6 3 4 2 F 2.50 3 [6] How should the supplies be transported in order that the transportation cost is minimum? What is the minimum cost? x y 2z x (b) Evaluate z y z 2x z x y y [4] z x 2y *** 2P1617MPP4 4

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