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PRE-BOARD EXAMINATION 2019 20 (Maximum Marks: 100) Class XII Time: 3 Hours MATHEMATICS Candidates are allowed additional 15 minutes for only reading the paper. They must NOT start writing during this time. The question paper consists of three sections A, B and C. Candidates are required to attempt all questions from Section A and all questions EITHER from Section B OR Section C. Section A: Internal choice has been provided in three questions of four marks each and two questions of six marks each. Section B: Internal choice has been provided in two questions of four marks each. Section C: Internal choice has been provided in two questions of four marks each. All working, including rough work, should be done on the same sheet as, and adjacent to the rest of the answer. SECTION A (80 Marks) Question 1. Each question carry 2 marks [10x2 =20 marks] 1. Determine whether the binary operation * defined on Q defined as a * b ab 1 , is commutative and associative. 2. Find the intervals in which the function f ( x) sin x cos x , 0 x 2 , is strictly increasing. 3. Two integers are selected at random from the integers 1 to 11. If the sum is even, find the probability that both the numbers are odd. 4. Using LHospital s Rule, evaluate: Lt x x . x 0 5. Evaluate: 1 sin x cos 2 x dx 2 6. Using properties of definite integrals, evaluate: sin 2 x a b cos xdx 0 dy sin x y sin x y . 7. Solve the differential equation: dx 3 1 8. Evaluate: sin 1 2 tan 1 . 3 2 9. Find a point on the curve x 2 y 2 2 x 4 y 1 0 at which the tangent is parallel to the chord joining (2, 0) and (4, 4). 10. Using matrix method, check whether the following system of equations is consistent or not: 2 x 5 y 7; 6 x 15 y 13. -2Question 2. If the function f ( x) 2 x 3 is invertible, then find its inverse. Hence prove fof 1 x x. [4] Question 3. 1 x2 1 x2 Prove that tan 1 x2 1 x2 1 1 cos 1 x 2 . 4 2 [4] Question 4. 1 5 2 Using elementary transformations, find the inverse of 1 1 7 0 3 4 [4] Question 5. Prove that the function f ( x) x 5 , x R is continuous at x 5 but not differentiable. OR a 2 b2 Using properties of determinants: a 2 b2 c2 b2 c2 b2 c2 a 2 4a 2b 2 c 2 c 2 a2 [4] Question 6. 2 x 3 , then prove that x y2 xy1 y . a bx If y x log [4] Question 7. 1 Evaluate 3x 2 2 x 1 dx expressing as the limit of a sum. [4] 0 Question 8. The mean and variance of a binomial distribution are 4 and 2 respectively. Find the probability of atleast 6 successes. OR Show that the relation R in the set A {x / x W , 0 x 12} given by R { a, b : a b is a multiple of 4} is an equivalence relation. Also find the set of all elements related to 2. [4] -3Question 9. Solve the following differential equation: dy y sin x cos x . dx x x [4] OR If Rolle s theorem holds for the function f ( x) x3 bx 2 ax 5 , in 1,3 at c 2 1 , 3 then find the values of a & b. Question 10. 11 5 14 1 2 3 Given two matrices A and B: A 1 4 1 and B 1 1 2 . Find AB and use it to 7 1 1 3 2 6 solve the following system of linear equations: x 2 y 3z 6, x 4 y z 12, x 3 y 2 z 1. OR Water is leaking out from a conical funnel at the rate of 5 cm3 / sec . If the radius of the base of the funnel is 10 cm and its height is 20 cm, find the rate at which the water level is dropping when it is 5 cm from the top. [4] Question 11. Given 2 bags A and B as follows: Bag A contains 3 red and 2 white balls; bag B contains 2 red and 5 white balls. A bag is selected at random, a ball is drawn and put into the other bag; then a ball is drawn from the second bag. Find the probability that both balls drawn are of the same colour. [6] Question 12. A window is in the shape of a rectangle surmounted by a semi-circle. If the perimeter of the window is 30 m, find the dimensions of the window so that the maximum possible light is admitted. [6] Question 13. cos 2 x 0 1 sin x cos xdx 2 Evaluate: OR Evaluate: sin x cos x dx. 9 16sin 2 x [6] -4Question 14. From a lot of 6 items containing 2 defective items, a sample of 4 items are drawn at random. Let the random variable X denote the number of defective items in the sample. If the sample is drawn without replacement, find: (a) the probability distribution of X (b) mean of X (c) variance of X [6] SECTION B ( 20 marks) Question 15. with y axis and 4 2 with z axis. [2] (b) Find the volume of the parallelepiped whose three coterminous edges are 2i 3 j 4k , (a) Find a vector r of magnitude 3 2 units which makes an angle of i 2 j k and 3i j 2k . [2] (c) Find the angle between the line r i 2 j k i j k and the plane r 2i j k 4 [2] Question 16. Find the equation of the planes parallel to the plane 2 x 4 y 4 z 7 and which are at a distance of 5 units from the point P 3, 1, 2 . OR Find the image of the point A 2, 1,5 in the line Question 17. Show that the lines [4] x 11 y 2 z 8 . 10 4 11 x 1 y 1 z 10 x 4 y 3 z 1 and intersect. 2 3 8 1 4 7 Also find the coordinates of the point of intersection. OR Find the equation of the plane passing through the point 1, 2,1 and perpendicular to the line joining the points A 3, 2,1 and B 1, 4, 2 . Question 18. Find the area enclosed by the curves y x 2 and y x 2 2 x and the lines x 1 and x 3. [4] -5SECTION C [20 marks] Question 19. (a) A furniture dealer deals in only two items tables and chairs. He has Rs 20000 to invest and a space to store at most 80 pieces. A table costs him Rs 800 and a chair costs him Rs 200. He can sell a table for Rs 950 and a chair for Rs 280. Assume that he can sell all the items that he buys. Formulate this problem as an L.P.P. in order to maximize his profit. [2] (b) The demand function for a certain product is given by the function p 200 20 x x 2 where x is the number of units demanded and p is the price per unit. Find the marginal revenue. [2] (c) Find coefficient of correlation for the regression lines: x 2 y 3 0 & 4 x 5 y 1 0. [2] Question 20. Find the line of regression of x on y from the following table: [4] Physics 15 12 8 8 7 7 7 6 5 3 History 10 25 17 11 13 17 20 13 9 15 Using this data: (i) Find the line of regression in which Physics is taken as the independent variable. (ii) A candidate had scored 10 marks in physics but he was absent from history test. Estimate his probable score for the latter test. OR You are given the following data: Variable x y Mean 36 85 Standard deviation 11 8 And correlation coefficient = 0.66. Find: (i) The two regression coefficients bxy and byx (ii) The two egression equations (iii) Most likely value of y when x 10. -6Question 21. A product can be manufactured at a total cost C ( x) x2 100 x 40, where x is the number 100 x of units produced. The price at which each unit can be sold is given by P 200 . 400 Determine the production level at which the profit is maximum. What is the price per unit and the total profit at the level of production? [4] OR The average cost function associated with producing and marketing x units of an item is given by AC= 2 x 11 50 . Find (i) the total cost function and the marginal cost function x (ii)the range of values of output x for which AC is increasing. Question 22. To maintain one s health, a person must fulfil certain minimum daily requirements for the following three nutrients calcium, protein and calories. His diet consists of only food items I and II whose nutrient contents are shown below: Food I Food II Minimum daily requirement Calcium 10 4 20 Protein 5 5 20 calories 2 6 12 If food I costs Rs 6 per unit and food II costs Rs 10 per unit, find the combination of food items so that the cost may be minimum. [6]
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