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SCOTTISH HIGH INTERNATIONAL SCHOOL PRE BOARD 1 EXAMINATION (2020 21) MATHEMATICS, XII ISC M.M. 80 Time : 3 Hrs Instructions SECTION A (65 Marks) Q1. [15 X 1 = 15] (i) A and B are invertible matrices of the same order such that |(AB)-1| = 8, If |A| = 2, then |B| is (A) 16 (B) 4 (C) 6 (D) 1/ 16 (ii) Determine the point on the curve y = x2 4x + 3 at which the normal is parallel to a line whose slope is 1/2. (A) (1,0) (B) (0, 1) (C) (9/2, 21/4) (D) (9/2, 0) (iii) If P(A) = 2/5, P(B) = and P(A U B) = 9/10, then A & B are (A) Mutually exclusive (B) Independent (C) Exhaustive (D) None of these (iv) If a function f : R+ R is defined by f(x) = 3 + 2x2, the pre image(s) of 35 and 75 are (A) {-4, 4}; {-6, 6} (B) -4, -6 (C) {6, -6}; {4, -4} (D) 4, 6 (v) If f(x) = log x, then the derivative of f (2log x) is (A) logx/ x (B) x/ logx (C) x logx (D) 1/ xlogx (vi) For which value of x is the following matrix not invertible 1 1 4 [ 2 1 3] 0 7 (A) - 1 (B) 0 (C) 1 (D) R {1} (vii) The order and degree of the differential equation y 3 (y )4 + 5y 7 = 0 is (A) 2, 4 (B) 3, 4 (C) 3, 1 (D) 1, 1 (viii) If A and B are two events such that P(A) = 1/5, P(B) = and P(A/ B) = 2/5, then P(A B ) is (A) 13/ 20 (B) 7/10 (C) 7/20 (D) 5/7 (ix) In which interval is the function f(x) = x + (A) (-1, 1) (x) 1 (B) R (-1, 1) The function given by f(x) = 1 + x2 is (A) Injection (B) surjection increasing ? (C) ( - , - 1) (C) bijection (D) (1, ) (D) None of these (xi) For the given determinant, evaluate a11A31 + a12A32 + a13A33 , where aij denotes element at (i.j) position and Aij denotes co-factor of the corresponding element : 117 324 268 |513 189 712| 822 73 196 (xii) If A and B are events such that P(A) = 0.5, P(B) = 0.3 and P(A U B) = 0.6, find P(B A). (xiii) Evaluate cos 1 (cos 9 /8) (xiv) For what value of x is the following matrix a skew symmetric matrix ? 0 [6 6 17/3 0 ] 17/3 0 (xv) If A = {2, 4, 6,8} and B = {1, 2, 3}, then find the number of relations from A to B. Q2. If y = 2sinx + 3cosx, prove that y + y = 0 [2] OR Differentiate y = (sin3x)2x 3 with respect to 5x3 + 7 Q3. Let R be a relation defined on the set of natural numbers N as R = {(x, y) : x N, y N, 2x + y = 24} Determine whether R is an equivalence relation or not. [2] OR Show that the relation R in the set {3, 6, 9, 12}defined by R = { ( 3, 6), (3, 12), (4, 4), (6, 12) } is transitive but not symmetric. Q4. Evaluate lim 0 [2] Q5. Solve the differential equation : y(1 + x) dx + x (1 + y) dy = 0 Q6. Integrate (sec3x tanx) with respect to x. Q7. Show that tan 1 ( 1 3 2) = 1 [2] [2] 1+2 cos 1 ( 2+ ) 2 Q8. Solve the differential equation : y + 2y = 4x, given that y = 1 when x = 0 [4] [4] OR Solve the differential equation : (y2 x2)dx 2xy dy = 0 Q9. Mathews can hit a target three times in five shots, Keshav two times in five shots and Chetan three times in four shots. They fire a volley. What is the probability that two shots hit the target? [4] Q10. Evaluate ( )2 [4] OR Evaluate 5 3 ( 2)(3 +1) Q11. Using matrix method, solve the following system of equations : [6] OR Using properties of determinants, prove that 2 | 2 2 2 2 | = - (sinx siny) (siny sinz) (sinz sinx) 2 Q12. A conical tent of given capacity has to be constructed. Find the ratio of the height to the radius of the base for the minimum amount of canvas required for the tent. [6] /2 Q13. Evaluate 0 4 + 4 [6] OR Evaluate 0 log(1 + ) Q14. A fair die is rolled. If 1 turns up, a ball is picked up at random from bag A. If 2 or 3 turns up, a ball is picked up from bag B. If 4, 5, or 6 turn up, a ball is picked up from bag C. Bag A contains 3 red & 2 white balls; bag B contains 3 red & 4 white balls; bag C contains 4 red & 5 white balls. The die is rolled, a bag is picked and a ball is drawn. (i) What are the chances of drawing a red ball? (ii) If the ball drawn is red, what are the chances that bag B was picked up? [6] SECTION B (15 Marks) Q15. [5 X 1 = 5] (i) The vector in the direction of vector 2 + 2 and magnitude 9 is (A) 3(2 + 2 ) (B) 9 (2 + 2 ) (C) (2 + 2 )/3 (ii) The straight line 3 2 = +6 3 (D) (2 + 2 )/9 , z = 4 is (A) Parallel to x axis (B) parallel to y axis (C) parallel to z axis (D) perpendicular to z axis (iii) If | | = 5, | | = 7 and . = 12, find | X | (iv) Find the length of the perpendicular drawn from the point P(0, 3, 2) on the plane x 3y +7z = 5. (v) Find the angle between the vectors 3 + + 4 and 5 2 Q16. In any triangle ABC, prove by vector method that a2 = b2 + c2 2bc cosA. [2] OR Prove by vector method that if the diagonals of a rectangle are perpendicular, then the rectangle is a square. Q17. Find the image of the point (2, - 1, 5) in the line 11 10 = +2 4 = +8 11 . [4] Q18. Draw a rough sketch & find the area enclosed between the curves x2 + y = 9, the x axis and the lines x + 1 = 0 and x 2 = 0. OR Find the area enclosed between the curves x2 = 8y and y2 = 8x [4] SECTION C (15 Marks) Q19. [5 X 1=5] (i) (ii) (iii) The total cost function for a production of x units of a commodity is given by 3x 2 28x + 108, then the number of units produced for which MC = AC is (A) 16 (B) 24 (C) 36 (D) 6 2 2 If for 6 observations of pairs (x, y), = 252, = 180, x = 318, y = 184 and xy = 199, then the value of bxy is (A) 0.323 (B) 0.458 (C) 0.323 (D) -0.458 3 2 The cost function of a firm is given by C(x) = 5x + 3x 100. Find its marginal cost function. (iv) (v) Calculate the coefficient of correlation for bivariate data, with byx = - 1.5 and bxy = 0.5 For manufacturing a certain item, the fixed cost is Rs9400 and the variable cost of producing each unit is Rs 25. Find the average cost of producing 47 units. Q20. If C(x) = 5x + 350 and R(x) = 50x x2 are respectively the total cost and total revenue functions for a company that produces and sells x units of a particular product. Find the values of x that would produce profit. [2] OR The fixed cost of a new product is Rs30,000 and the variable cost per unit is Rs1200. If the demand function is p(x) = 4900 100x, find the break even value(s). Q21. The marks obtained by 10 candidates in Physics and Chemistry are given below. Estimate the probable score for Chemistry if the marks obtained in Physics are 28. [4] Marks in Physics Marks in Chemistry 15 19 19 22 13 12 20 17 18 23 21 25 17 19 14 21 11 14 12 8 Q22. A diet is to contain atleast 90 units of proteins and 100 units of vitamins. Two foods A and B are available. Food A costs Rs30 per unit and food B costs Rs150 per unit. One unit of food A contains 30 units of proteins and 50 units of vitamins. One unit of food B contains 90 units of proteins and 50 units of vitamins. 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. [4] OR A manufacturing company makes two products A and B. Each piece of product A requires 30 hours of labour of skill set 1 and 15 hour of labour of skill set 2. Each piece of product B requires 50 hours of labour of skill set 1 and 6 hours of labour of skill set 2. The maximum labour hours available for skill set 1 & 2 are 150 and 30 hours respectively. The company makes a profit of Rs5000 on each piece of product A and Rs3000 on each piece of product B. How many pieces of each product should be manufactured per week to realise a maximum profit. What is the maximum profit per week?
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