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F.M 100 Tim e 3 Hrs HALF-YEARLY EXAMINATION 2018-2019 SUBJECT: MATHEMATICS CLASS- XII (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 question from Section A and all question EITHER from Section B OR Section C Section A: Internal choice has been provided in three questions of 4 marks each and two questions of 6 marks each. Section B: Internal choice has been provided in two questions of 4 marks each. Section C: Internal choice has been provided in two questions of 4 marks each. All working, including rough work, should be done on the same sheet as, and adjacent to the rest of the answer. The intended marks for questions or parts of questions are given in brackets [ ]. Mathematical tables and graph papers are provided. SECTION A(80 Marks) Question 1 (i) The binary operation *: R X R R is defined as a * b = 2a + b. Find ( 2 * 3 ) * 4. 5 a (ii) If A = and A is symmetric matrix, show that a = b. b 0 (iii) Solve 3 tan 1 x + cot 1 x = (iv) Without expanding at any stage, find the value of: 2 x y+z 2 y z+ x 2 z x+ y (v) Find the value of k so that f(x) defined as: [10 2] [ ] | | { (vi) (vii) (viii) (ix) x 2 2 x 3 f(x) = is continuous at x = -1 , x 1 x+1 k , x= 1 Find the approximate change in the volume V of a cube of side x metres caused by decreasing the side by 1%. 1 1 1 If A and B are events such that P(A) = , P(B) = and P(A B) = , then find: 2 3 4 P(A|B) and P(B|A) 1 1 In a race, the probabilities of A and B winning a race are 3 respectively. Find the 6 probability of neither of them winning the race. Using L Hospital s Rule, evaluate: 1 lim x tan x x 2 sec x 4 Find the intervals in which the function f(x ) is strictly increasing where, f(x)=10-6x-2x 2. (x) Question 2 [4] Let R+ be the set of all positive real numbers and f: R+ [4, ): f(x) = x2+ 4. Show that inverse of f exists and find f-1 Question 3 [4] Solve the equation for x: sin 1 x +sin 1(1 x)=cos 1 x , x 0 Question 4 If x = tan Question 5 [4] y d2 y ( dy 1 + 2 x a ) =0 log , prove that (1 + x2) 2 dx dx a { [4] 2 x , x 1 (a) Show that the function f(x) = f ( x )= 1 is continuous at x=1 but not differentiable. , x >1 x OR x (b) Verify Rolle s theorem for f(x) = e sin x on [0, ]. Question 6 [4] x sin 1 x dy y 2 (a) If y = , prove that ( 1 x ) dx =x + x 2 1 x OR x dy x y = (b) If x = e y , prove that dx x log x Question 7 [4] (a) Using matrices, solve the following system of equations: 2 x 3 y+5 z=11 3 x+2 y 4 z= 5 x+ y 2 z= 3 OR (b) Using elementary transformations, find the inverse of the matrix: 0 1 2 1 2 3 3 1 1 Question 8 [4] (a) Find the points on the curve y = 4 x 3 3 x+5 at which the equation of the tangent is parallel to the x-axis. OR (b) Water is dripping out from a conical funnel of semi-verticle angle at the uniform rate of 4 2cm2/sec in the surface, through a tiny hole at the vertex of the bottom. When the slant height of the water level is 4cm, find the rate of decrease of the slant height of the water. [ ] Question 9 [4] (a) Find the equations of the normals to the curve y = x3+2x+6 which are parallel to the line x+14y+4=0. 2 OR (b) A circular disc of radius 3 cm. is heated. Due to expansion its radius increases at the rate of 0 05 cm/s. Find the rate at which its area increases when the radius is 3 2 cm. Question 10 [4] An urn contains 10 white and 3 black balls while another urn contains 3 white and 5 black balls. Two balls are drawn from the first urn and put into the second urn and then a ball is drawn from the second urn. Find the probability that the ball drawn from the second urn is a white ball. Question 11 A cone is inscribed in a sphere of radius 12cm. If the volume of the cone is maximum, find its height. [6] Question 12 [6] (a) Show that the altitude of a right circular cone of maximum volume that can be inscribed in a sphere of 4r radius r is . 3 OR (b) An open topped box is to be made by removing equal squares from each corner of a 3 m by 8 m rectangle sheet of aluminium and by folding up the sides. Find the volume of the largest such box. Question 13 [6] A speaks truth in 60% of the cases, while B in 40% of the cases. In what percent of cases are they likely to contradict each other in stating the same fact? OR Box I contains 2 white and 3 black balls. Box II contains 4 white and 1 black ball and box III contains 3 white and 4 black balls. A dice having 3 red, 2 yellow and 1 green face, is thrown to select the box. If red face turns up, we pick Box I, if a yellow face turns up, we pick Box II, otherwise we pick Box III. Then we draw a ball from the selected box. If the ball drawn is white, what is the probability that the dice had turned up with a red face? Question 14 [6] (a) 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: (i) The probability distribution of X (ii) Mean of X (iii) Variance of X OR (b) The difference between mean and variance of a binomial distribution is 1 and the difference of their squares is 11. Find the distribution. SECTION B (20 Marks) Question 15 [6] (a) Find the Cartesian equation of the line passing through the points ( -1, 0, 2) and (3, 4, 6). x 11 y+ 2 z +8 = = (b) Find the image of the point (2, -1, 5) in the line . Also, find the length of the 10 4 11 perpendicular from the point (2, -1, 5) to the line. Question 16 [4] 3 Find the equation of a line passing through the points P(-1,3,2) and Q(-4,2,-2). Also, if the point R(5,5,k) is collinear with the points P and Q, find k. Question 17 (a) Find the shortest distance between the following lines: [4] r =( 2i+ 4 j+5 k ) + ( 4 i+6 j+8 k) r =( i+2 j+3 k )+ k (2 i+ 3 j+4 k ) OR (b) Find the equation of the plane passing through (2,-3,1) and (-1,1,-7) and is perpendicular to the plane x 2y + 5z + 1 = 0. Question 18 [6] (a) Find the equation of the plane passing through the intersection of the planes x + y + z + 1 = 0 and 2x 3y + 5z 2 = 0 and the point (-1, 2, 1). OR (b) Find the equations of planes parallel to the plane 2x 4y + 4z = 7 and which are at a distance of 5 units from ( 3,-1,2). SECTION C (20 Marks) Question 19 (a) Find the coefficient of correlation from the regression lines: x 2y + 3 = 0 and 4x 5y + 1 = 0 (b) Find the line of regression of y on x from the following table: x 1 2 3 Y 7 6 5 Hence, estimate the value of y, when x = 6. Question 20 (a) From the given data: Variable Mean Standard Deviation [6] 4 4 5 3 [4] X 6 4 y 8 6 2 . Find: 3 (i) Regression coefficients byx and bxy. (ii) Regression line x on y. (iii) Most likely value of x when y =14. OR (b) Given that the observations are (9, -4), (10, -3), (11, -1), (13, 1), (14, 3), (15, 5), (16, 8), find the two lines of regression. Estimate the value of y, when x = 13.5. and correlation coefficient: Question 21 [6] (a) A manufacturing company makes two types of teaching aids A and b of Mathematics for Class X. Each type of A requires 9 labour hours for fabricating and 1 labour hour for finishing. Each type of B requires 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available per week are 180 and 30 respectively. The company makes a profit of `80 on each piece of type A and `120 on each piece of type B. How many pieces of type A and type B should be manufactured per week to get a maximum profit? Formulate this as Linear Programming Problem and solve it. Identify the feasible region from the rough sketch. 4 OR (b) A toy company manufactures two types of dolls A and B. Market test and available resources have indicated that the combined production level should not exceed 1200 dolls per week and the demands for the dolls of type B is atmost half of that for dolls of type A. Further, the production level of type A can exceed three times the production of dolls of other type by at most 600 units. If the company makes profit of `12 and `16 per doll respectively on dolls A and B, how many of each type of dolls should be produced weekly, in order to maximise the profit? Question 22 [4] Find the regression coefficient b yx and bxy and the two lines of regression for the following data. Also, compute the correlation coefficient. X 2 6 4 7 5 Y 8 8 5 6 2 5
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