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STD :- XII SUB :- MATHS DATE :- 08 - 01- 2020 ST.GREGORIOS HIGH SCHOOL PRELIMINARY EXAMINATION (2019 20) TIME :- 3HRS MAX.MARKS:- 100 (Candidates are allowed additional 15 minutes for only reading the paper. They must not start writing during this time.) 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 the working including rough work, should be done on the same sheet as, and adjacent to, the rest of the answer. The intended marks for the questions are given in brackets[ ]. Mathematical tables and graph paper are provided. Slide rule may be used. SECTION A [10 2] Question 1. (i) (ii) (iii) (iv) (v) For the operation * defined below, determine whether * is commutative or + ab associative. On Z , define a * b = 2 . 3 2 k For what value of k, the matrix is not invertible? 5 1 Using determinants, find the equation of the line joining the points (1, 2) and (3, 6). x2 y2 1 c , then prove that dy y . If tan x2 y2 dx x Evaluate e x (1 tan x tan 2 x)dx . 3 d4y 1 dy . (vi) Find order and degree of the differential equation: 1 dx 4 dx x cos x sin x lim (vii) By applying L Hospital s Rule, find the limits of . 2 x 0 x sin x (viii) A die is thrown twice and the sum of the numbers appearing is observed to be 6. What is the conditional probability that the number 4 has appeared at least once?. (ix) A speaks truth in 75% cases and B speaks truth in 80% cases. Find the probability that they contradict each other while narrating a certain fact. (x) Solve : sin 1 cos(sin 1 x) . 3 QUESTION 2. Show that the function f : R R defined by f ( x) x , x R , is x2 1 neither one-one nor onto. [4] OR Show that the relation S in the set R of real numbers, defined as 3 S= {(a, b) : a, b R and a < b } is neither reflexive , nor symmetric, nor transitive. QUESTION 3. By using the properties of determinants, prove that 3a a b a c b a 3b c a c b b c 3(a b c)(ab bc ca ). [4] 3c QUESTION 4. 5 2 1 1 2 tan 1 . Prove that 2 tan 1 sec 1 5 8 4 7 Question 5. [4] A circular cone, with semi- vertical angle 450 , is fixed with its axis vertical and its 3 vertex downwards. Water is poured into the cone at the rate of 2 cm per minute. Find the rate at which the depth of the water is increasing when the depth is 4 cm. [4] OR 1 Show that the function f given by f (x) = tan (sin x + cos x), x > 0 is always an increasing function in 0, . 4 Question 6. t d2y If x a[cost log tan ] , y = a sin t , find at t . [4] 2 4 dx 2 Question 7. Evaluate (5 x 2) dx . 3x 2 2 x 1 [4] OR Evaluate dx 3 2 sin x cos x Question 8. Prove that the function f(x) = A x 1A , x R is continuous at x = 1 but not differentiable. [4] Question 9. Find the particular solution, satisfying the given condition, for the following x x y x y dx 0 , y =1, when x = 0. differential equation : e 1 1 e y dy Question 10. [4] A card from a pack of 52 cards is lost. From the remaining cards of the pack, two cards are drawn and are found to be both diamonds. Find the probability of the lost card being a diamond. [4] Question 11. Solve the following system of equations by matrix method. 3x 2y + 3z = 8 2x + y z = 1 4x 3y + 2z = 4 [6] Question 12. Prove that the volume of the largest cone that can be inscribed in a sphere of 8 radius R is of the volume of the sphere. [6] 27 OR A tank with a rectangular base and rectangular sides, open at the top is to be 3 constructed so that its depth is 2 m and volume is 8 m . If building of the tank costs Rs 70 per sq. metre for the base and I 45 per square metre for the sides. What is the cost of the least expensive tank? Question 13. Prove that 2 x sin x cos x 2 dx 16 . 0 cos4 x sin 4 x [6] Question 14. Let a pair of dice be thrown and the random variable X be the sum of the numbers that appear on the two dice. Find the mean and standard deviation of X. [6] OR Three persons A , B, C throw a pair of dice in succession in the same order till one of them gets a total of 7 and wins the game, find their respective probabilities of winning. Section B Question 15. (a) If a is a unit vector and (x a). (x+ a) = 99, then what is the value of | x |? (b) Find the projection of a i 3 j k along b 2i 3 j 6k [2] [2] (c) Find the angle between the two straight lines r 8 i 16 j 10k (3i 16 j 7k ) and r 15i 29 j 5k (3i 8 j 5k ) . [2] Question 16. For any three vectors a , b , c , show that a b , b c c a are coplanar . [4] OR If the vertices of triangle are A(2, 1, 1), B(1, 3, 5) and C(3, 4, 4), prove by vector method that it is a right angled triangle. Question 17. Find the shortest distance between the lines r ( 1) i ( 1) j (1 )k and r (1 ) i (2 1) j ( 2)k . [4] OR Find the equation of the plane through the intersection of the planes x + 3 y + 6 = 0 and 3 x y 4z = 0 and whose perpendicular distance from origin is unity. Question 18. Using the method of integration, find the area of the region bounded by the lines 2x + y = 4, 3x 2y = 6 and x 3y + 5 = 0. [6] Section C Question 19. (a) (b) The fixed cost of a new product is I 18000 and the variable cost is I 550. If the demand function is p(x) = 4000 150x, find the break even values. Find the regression coefficient of y on x for the following data : xy 306 , x 2 164 , y 2 574 , n = 4 . [2] x 24 , y 44 , [2] (c) 3 2 The total cost function is given by the equation, C = x + 2x 3.5 x . Find the Marginal average cost function. [2] Question 20. If the two regression lines of a bivariate distribution are 4x 5y + 33 = 0 and 20x 9y 107 = 0, calculate : (i) x and y , the arithmetic means of x and y respectively. (ii) estimate the value of x when y = 7 (iii) find variance of y when 3 . x [4] OR Calculate (i) Coefficients of regression and (ii) regression equations, for the following data: Price (x) 78 89 97 Demand (y) 125 137 156 Estimate the price when demand is 100 69 112 59 107 79 136 68 123 61 108 Question 21. The total cost and the demand functions of an item are given by x3 7 x 2 111x 50 and p = 100 x respectively. Write the total revenue C(x) = 3 function and the profit function. Find the profit maximizing level of output x and the maximum profit . [4] OR 2 The marginal cost function MC for a production is given by MC = and the 4x 9 fixed cost is I 2000. Find the total cost function and average cost of producing 10 units of the output. Question 22. A furniture firm manufactures chairs and tables, each requiring the use of three machines A, B and C. Production of one chair requires 2 hours on machine A, 1 hour on machine B and 1 hour on machine C. Each table requires 1 hour each on machine A and B and 3 hours on machine C. The profit obtained by selling one chair is I 30 while by selling one table the profit is I 60. The total time available per week on machine A is 70 hours, on machine B is 40 hours and on machine C is 90 hours. How many chairs and tables should be made per week so as to maximise profit? Formulate the problems as a L.P.P. and solve it graphically.? [6]
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