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Class 12 SHARADA MANDIR SCHOOL First Preliminary Examination November 2016 MATHEMATICS PAPER (3 HOURS) Max Marks -100 Section A Answer Question 1 (compulsory) and Five other questions. Section C Answer two questions from either Section C. 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 [ ]. .. SECTION A Question 1. [10 3 = 30] i. Find the value of k if M = ii. Find the equation of tangent to the ellipse axis iii. Evaluate: iv. Solve v. Evaluate the integral vi. Evaluate vii. Find the regression coefficient byx and bxy ,if , , , and n=6. Also find the correlation coefficient r A machine manufactured by a firm consists of two parts A and B. But out of 100 A s manufactured 9 are likely to be defective and out of 100 B s manufactured, 5 are likely to be defective. Then find the probability that a machine manufactured by the firm is free from any defect viii. = ,then find the value of x which makes an angle of 30 with the x- ix. Express the complex number in the polar form x. Find the equation of hyperbola with centre at the origin, the length of transverse axis is 6 and one focus at (0,4) Question 2. (a) Show that = (a+b+c) ( [5] (b) Using matrices solve the following system of equations: 3x-2y+z=2, 2x+y-3z = -5, -x+2y+z=6 [5] Question 3. (a) Draw the switching circuit for the given expression P.R+Q.(Q +R).(P+QR).Simplify the expression and construct the switching circuit for the simplified expression [5] (b) Prove that [5] Question 4. (a) Verify Lagrange s mean value theorem for the function f(x) = x 2sinx on [(b) Find the equation of the parabola with latus rectum joining points (4,6) and (4,-2) [5] [5] Question 5: (a) If , then prove that [5] (b) A closed rectangular box is made up of aluminum sheet of negligible thickness and the length of the box is twice its breadth .If the capacity of the box is 243c then compute the dimension of the box of least surface area [5] Question 6: (a) Evaluate [5] (b) Evaluate [5] Question 7: (a) Find the coefficient of correlation between X and Y from the following data by Karl Pearson s method. [5] X: Y: (b) 12 14 13 17 14 18 15 19 16 20 17 24 18 28 Given observations (10,-5),(10,-3) ,(11,-2) , (11,0) , (12,1) , (15,6) , (16,4) , (11,-2) (i) Find byx and bxy (ii) predict the value of y when x=14 (iii) Predict the value of x when y=3 [5] Question 8: (a) The odds against a husband who is 45 yrs old, living till he is 70yr are 7:5 and the odds against his wife who is now 36,living till she is 61 yrs are 5:3.Find the probability that the couple will be alive 25 yrs hence and also that at least one of them will be alive 25 yrs hence [5] (b) Two balls are drawn from an urn containing 2 white balls 3 red balls and 4 black balls one by one without replacement what is the probability that i) Both the balls are of same colour ii) At least one ball is red iii) Neither of 2 balls is white [5] Question 9: (a) If z=x+iy, (b) Evaluate .Find the locus of z [5] [5] Pg. 3 MATHEMATICS Std. 12 SECTION C Question 10: (a) A machine costing Rs 2 lakhs has effective life of 7 yrs and its scrap value is Rs 30000. What amount should the company put into a sinking fund earning 55 per annum, so that it can replace the machine after its useful life. Assume that a new machine will cost Rs 3 lakhs after 7 yrs. (b) The average cost function AC for a commodity is given by AC in terms of output x .Then find i) The marginal cost function of x ii) The output for which AC increases [5] [5] Question 11: (a) If the bankers gain on a bill will be th of the bankers discount, the rate of interest being 10% per annum, then find the unexpired period of the bill [5] (b) The fixed cost of a product is Rs30000 and the variable cost per unit is Rs 800. If the demand function is p(x)= 4500-100x. i) Find revenue function R(X) ii)Find break even point iii)Find cost function of x [5] Question 12: (a) Calculate the index number for the year 2014 with 2010 as the base year by the simple aggregate method [5] Commodity Price 2010 2014 A 20 25 B 30 30 C 10 15 D 25 35 E 40 45 F 50 55 (b) The following table shows the average monthly production of coal for the year 19871996.Calculate the four quarterly moving averages and show this moving average on a graph. [5] Year 1987 Monthly 50 production 1988 36.5 1989 43 1990 44.5 1991 38.9 1992 38.1 1993 32.6 1994 41.7 1995 41.1 1996 33.8
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