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ISC Class XII Board Exam 2024 : Mathematics

5 pages, 45 questions, 3 questions with responses, 3 total responses,

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SESHADRIPURAM PUBLIC SCHOOL PRE-BOARD EXAMINATION 2023-24 Subject: Mathematics Grade:XII Marks: 80 Date: 13.12.2023 Time:3Hrs Note: 1) Reading time: 15mins 2)Students must submit the answer scripts within the given time. 3) Marks will be deducted for untidy work. 4) Question paper consist of three sections A, B and C. 5) Students are required to attempt all the question from section A and all the questions either from section B or Section C. Section A (65 Mark2s) 1. In sub-parts (i) to (x) choose the correct options and in sub parts xi to xv answer the questions as instructed. i) Let : be defined by ( ) = + 1 then pre images of 17 and -3 respectively, are (a) , {4, 4} (b) {3,-3}, (c) {4}, (d) {4, 4}, {2, 2} ii) The value of tan (sec 2) + cot (cosec 3) is (a) 5 (b) 13 (c) 11 (d) 15 iii) If P(A)= , P(B)=0, then is (a) 0 (b) (c) not defined (d)1 iv) Function ( ) = | + 2| 1 has (a) -1 as minimum value (b) 2 as maximum value (c) -2 as maximum value (d) 1 as minimum value v) The relation R in the set of natural numbers N defined as = {( , ): = + 5 < 4} is (a) reflexive (b) symmetric (c) transitive (d) none of these vi) An integrating factor of the differential equation (1 + ) + = is (a) (b) 1 + (c) (1 + ) 7 8 vii) P and Q are two matrices such that [ + ] = . If = 3 0 which one of the following is Q 8 9 8 2 8 2 (a) (b) (c) 0 2 11 2 11 2 (d) 5 1 1 , 3 2 (d) 8 11 2 2 viii) For what value of n is the given a homogeneous differential equation? = (a) 1 b) 3 c) 0 d) 2 ix) Statement 1: The function : given by ( ) = 5 is one to one and onto. Statement 2: The function : given by ( ) = 5 is one to one and onto. Which one of the following options is correct? (a) Both statements are true (b) Both the statements are false (c) Statement 1 is true and Statement 2 is false (d) Statement 1 is false and statement2 is true sec tan x) Assertion(A) : The value =1 tan sec Reason(R): = + (a) Both A and R are true and R is the correct explanation of A (b) Both A and R are true and R is not the correct explanation of A (c) A is true but R is false d) A is false but R is true xi) Show that the function : given by ( ) = cos , is neither one-one nor onto. 4 2 2 1 xii) Find a matrix X such that 3 2 + = 0 where = , B= . 1 3 3 2 xiii) Write the sum and order of the differential equation 1 + . =9 xiv) The probability that at least one of the two events A and B occurs is 0.6 . If A and B occur simultaneously with probability 0.3 then evaluate ( ) + ( ). xv) Find the principle value of cos 2. (i) If f(x)=sin 2 cos 2 , then find . [2] Or (ii) If = cos + sin and = sin cos , then show that + =0 3. If the function in = . [2] 4. Show that the function ( ) = tan is always increasing in . 5. (i) Evaluate log|1 + | Or (ii) Evaluate . [2] [2] 6. Find the particular solution of the differential equation tan + (2 ) sec = 0 given that = , when x=0 7. Prove that cos[tan {cot(sin )}] = 8. Find the point on the curve 9 = , where the normal to the curve make equal intercept on the axes. [2] defined as ( ) = , then find [4] [4] 9. (i) Evaluate . [4] Or (ii) Evaluate 10. (i) A bag contains 4 white and two black balls. Another contains 3 white and 5 black [4] balls. If one ball is drawn from each bag. (a) find the probability that both are white. (b) find the probability that both are Black. (c) find the probability that one is bite and one is black. Or (ii) Once festival, two friends namely Shanti and Diya got identical gift bags containing 7 dark chocolates, 8 Milk chocolate, and five white chocolate each. (a)What is the probability of the year drawing dark chocolate from Shanti's bag? (b)while Shanti was away Diya transferred one chocolate from Shanti's bag to hers without looking what is the probability of drawing a dark chocolate from Diya's bag after the transfer? (c) Which one of the following probability is higher? Why? The probability of drawing a dark chocolate from the Diya s bag after the transfer or the probability of drawing a dark chocolate from Shanti's bag before the transfer. 11. The sum of three number is 6. If we multiply third number by 3 and add second number to it, we get 11. By adding first and third numbers, we get double of the second number. (i) Form a set of simultaneous equation for the above information. (ii) Solve the set of equations formed in(i) by matrix method. (iii) find the product of three numbers. [6] 12. (i) Solve + ( 1) = 0. Given, y=3-e, when = , then find when = 1. Or (ii) Evaluate [6] 13.(i) Prove that the semi Vertical angle of the right circular cone of a given volume and least curved surface area is cot 2. Or (ii) The volume of a closed rectangular metal box with a square base is 496 . The cost of polishing the outer surface of the box is rupees 4 per cm . Find the dimension of the box for the minimum cost of polishing it. 14. Three persons A B and C apply for a job of manager in a private company. chances of their selections (A, B and C) are in the ratio 1:2:4. The probability that A, B and C can introduce changes to improve profits of the company are 0.8, 0.5 and 0.3 respectively. (i) What is the probability of selecting person A? (ii) What is the probability of selecting person C? (iii) What is the probability that person B introduced the changes in the profits? (iv) If the changes does not take places, what is the probability that it is due to the appointment of C? [6] SECTION B [15marks] 15. In sub-parts (i) and (ii) choose the correct options and in sub-part (iii) to (v) answer the questions as instructed. [5 1] (i) The projection of = 3 + 5 on = 2 + 3 + is a) b) c) d) 14 (ii) If the lines (a) = and = (b) are perpendicular, then the value of is = 1= (c) (d) None of these (iii) Find the value of for which the points A(-1,3,2), B(-4, 2,-2) and C(5,5, ) are collinear. (iv) Find the distance of plane 3 4 + 12 = 3 from the origin. (v) If = , = and =k are unit vectors, what is the angle between ( ) ? 16.( i) If + = + , then show that the point P Q and R are collinear. [2] Or (ii) Using vectors, find the area of triangle whose vertices are A(1,2,3), B(2,-1,4) and C(4,5,-1) 17) (i) Find the area of the region bounded by the curve = + 2 and the lines = , = 0 and = 3. [4] Or (ii) Using integration find the area of the region bounded by the ellipse + = 1. 18) Find the distance of the point 345 from the plane + + = 2 measured parallel to the line 2x=y=z. SECTION C[15 Marks] [4] 19. In sub-parts (i) and (ii) choose the correct options and in sub-part (iii) to (v) answer the questions as instructed. [5 1] (i) At a toy factory, the production cost per day for number of toys is given by the following function ( ) = 0.0001 0.01 20 + 3500 Which of these represent the approximate additional cost to produce 201st toy? (a) C(200) (b) AC(201) (c) MC(200) (d) MC(201) (ii) The regression Coefficient for the data = 55, = 88, = 385, = 586 and = 10 is (a) 1.32 (b) 1.48 (c) 1.24 (d) none of these (iii) If two lines of regression are 4x + 3 Y + 7 =0 and 3x + 4y + 8=0, Find the correlation Coefficient between and . (iv) The demand for a certain product is represented by the equation = 300 + 24 in rupees where is the number of units and is the price per unit. Find the marginal revenue when 8 units are sold. (v) The cost function ( ) of a firm is given by ( ) = 2 4 + 5 . Find the average cost. 20. (i) Find the equation of two lines of regression for the following observations. (3,6), (4,5), (5,4), (6,3), (7,2). find an estimate of for x = 25 [2] Or (ii) The data for mark on Physics and History obtained by 10 students are given below Marks in Physics Marks in History 15 10 12 25 8 17 8 11 7 13 7 17 7 20 6 13 5 9 3 15 (a) find the line of regression in which physics is taken as the independent (b) A candidate has score 10 marks in Physics and but was absent from the History test. Estimate probable score for the latest test. 21. A manufacturing company makes two types of teaching aids A and B of mathematics for class XII. Each type of A requires 9 labour hours of fabricating and one 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 our week are 180 and 30 respectively. the company makes a profit of 80 on each piece of type A and 120 on each type of type B. How many pieces of type A and type B should be manufactured per week to get a maximum profit? Make it as an LPP and solve it graphically. What is the maximum profit for week? [4] 22. The following table shows the sales on advertisement expenditure of firm Sales Advertisement expenditure(in crore) (in crore) [4] Mean 40 6 Standard deviation 10 1.5 Coefficient of correlation, r=0.9. Estimate the likely sales for a proposed advertisement expenditure of 10 crore. Or The marginal revenue function of a commodity is given by = 9 2 + 4 . Find the demand function.

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