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ISC Class XII Prelims 2026 : Mathematics (Bombay Scottish School, Mahim, Mumbai)

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Vamika bhati
St. Joseph's School, Greater Noida

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PRELIMINARY ASSESSMENT MATHEMATICS Grade : 12 Max. Marks : 80 Date : 15.01.24 No. of Questions : 22 Duration : Three hours No. of Pages: 10 ________________________________________________________________________ [Answers to this paper must be written on the answer booklet provided] The first 15 minutes is allotted for reading the question paper. The intended marks for questions or parts of questions are given in the brackets [ ]. ________________________________________________________________________ This Question Paper consists of three sections A, B and C. Candidates are required to attempt all questions from Section A and all questions EITHER from Section B OR Section C. Section A: Internal choice has been provided in two questions of two marks each, two questions of four marks each and two questions of six marks each. Section B: Internal choice has been provided in one question of two marks and one question of four marks. Section C: Internal choice has been provided in one question of two marks and one question of four marks. 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 Question 1. [1x15] In subparts (i ) to (x) choose the correct options and in subparts (xi) to (xv), answer the questions as instructed i. If there are two values of a which makes the determinant 1 2 5 !2 1! = 86, then the sum of these numbers is 0 4 2 a. b. c. d. 4 5 -4 9 ii. If (7 4 ) = (7 4 ) + , then value of a is a. -4 b. - c. 3 d. 7 iii. If tan?@ ( ) = 2 , then is equal to (a) /3 (b) /4 (c) /6 (d) None of these iv. The order and degree of the differential equation DF @/K DE F + HDG I DG E a) b) c) d) 2 2 2 2 and and and and + @/L = 0 respectively, are not defined 1 3 4 v. If P (A) = 0.8, P (B) = 0.5 and P (B|A) = 0.4, what is the value of P (A B)? (a) 0.32 (b) 0.25 (c) 0.1 (d) 0.5 vi. If the points (3, -2), (x, 2), (8, 8) are collinear, then find the value of x. (a) 2 (b) 3 (c) 4 (d) 5 vii. The set of points where the function f given by ( ) = |2 1| , is differentiable is a) R b) R - {1/2} c) (0, ) d) None viii. a. b. c. d. The above function is continuous for all real values The above function is not a continuous function The above function is continuous for all real values except 0. None ix. Statement 1; : given by ( ) = U , is neither injective nor surjective. Statement 2; : given by ( ) = L , is neither injective nor surjective a. b. c. d. Both the statements are true. Both the statements are false. Statement 1 is true and Statement 2 is false. Statement 1 is false and Statement 2 is true. 0 x. Assertion [A] : matrix W 3 2 Reason [R] : A matrix A is 3 2 3 5Y is a skew symmetric matrix. 5 0 a skew symmetric if / = a. Both A and R are individually true and R is the correct explanation of A. b. Both A and R are individually true and R is not the correct explanation of A. c. 'A' is true but 'R' is false d. Both A and R are false. xi. Identify the nature of the function : @ U. xii. If = [ \, 0 < < /2 , + / = , find the value of x. xiii. If A={ 1,1,3}, then write the smallest equivalence relation on A. xiv. Given that the events A and B are such that P(A) = 1/2, P (A B) = 3/5, and P(B) = p. Find p if they are mutually exclusive. xv. A die is rolled. If the outcome is an odd number, what is the probability that it is prime? Question 2. [2] a. Differentiate w.r.t x : b U + U + 1 OR b. Find the equation of the normal to the curve = U + 2 G + 2 at (0,4). Question 3 e/K f [2] 4 3 . Question 4 Find the values of x , for which ( ) = 2 L 15 U + 36 + 1 is an increasing function. [2] Question 5 [2] a. Evaluate: G?U Ggh OR b. Evaluate : @gG G.jkl(GgmnoG) Question 6 [2] Write the function in the simplest form @gG E ?@ tan?@ p G Question 7 q [4] Solve the following equation for x. sin?@ 4 + sin?@ 3 = /2 Question 8 Evaluate: [4] @ G (LG?@)E Question 9 [4] a. If x = a sec3 and y = a tan3 , find DE F DG E at = /3. OR b. For what choice of a and b, is the function U , f(x) = t + , > differentiable at = ? Question 10 [4] a. Suppose a girl throws a dice. If she gets 1 or 2, she tosses a coin three times and notes the number of tails. If she gets 3, 4, 5 or 6, she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one tail , what is the probability that she threw 3, 4, 5 or 6 with the die? OR b. Three machines E1, E2 and E3 in a certain factory producing electric bulbs, produce 50%, 25% and 25% respectively, of the total daily output of electric bulbs. It is known that 4% of the bulbs produced by each of machines E1 and E2 are defective and that 5% of those produced by machine E3 are defective. If one defective bulb is picked up at random from a day s production, calculate the probability that it is from machine E3. Question 11 1 If = W2 0 1 3 1 [6] 0 2 4Y and = W 4 2 2 2 2 1 4 4Y 5 Find AB and hence solve the system of equations = 3, 2 + 3 + 4 = 17, + 2 = 7 Question 12 [6] a. Solve the differential equation: z b { = . OR b. UjklUG?|njG }?|njE G?KjklG Question 13 [6] a. If the sum of the lengths of the hypotenuse and a side of a right angled triangle is given as k , show that the area of the triangle is maximum e when the angle between them is L . OR b. A closed rectangular box is made of aluminium sheet of negligible thickness, and the length of the box is twice its breadth. If the capacity of the box is 243 cm3 ,compute the dimensions of the box of least surface area. Question 14 [6] a. Two numbers are selected at random from the first six positive integers. Let X denote the larger of the two numbers obtained. Find the probability distribution of X. Find the mean of this distribution. SECTION B Question 15 [1x5] = = , then = i. If OACB is a parallelogram with a. + b. c. @ d. @ U U ) ( ( ) ii. If are inclined at an angle of 120 and | | = 1, = 2, then U z + 3 { z3 { = a. b. c. d. 225 250 275 300 iii. Can the numbers @ , @ , @ U U U be the direction cosines of a straight line? Give reasons. iv. find the length of the perpendicular from the origin to the plane . z3 4 12 { + 39 = 0. v. find the angle between the lines 2 = 3 = and 6 = = 4 Question 16 [2] a. If = 3 + + 2 = 2 2 + 4 , then find the sine of the angle between . OR b. Find the shortest distance between the lines whose vector equations are = z4 + 2 { + z + 2 3 { and = z2 + { + z3 + 2 4 {. Question 17 [4] a. A variable plane which remains at a constant distance 3p from the origin cuts the coordinate axes at A,B,C. Show that the locus of the centroid of the triangle ABC is ?U + ?U + ?U = ?U . OR b. Show that the points ( + 3 ) and 3( + + ) are equidistant from the plane z5 + 2 7 { + 9 = 0. Question 18 [4] Find the area bounded by the curve = and the line = 2 + 3 in the first quadrant and the x-axis. SECTION C Question 19 i. ii. [1x5] A company manufactures a product with cost function given by ( ) = 400 + 3 and the revenue received on the sale of x units is given by U + 3 . Then breakeven point is a. 25 b. 40 c. 20 d. 50 @ U If G = 5, = U and FG = then the value of F = a. b. Uf Uf c. Uf d. Uf iii. iv. v. Given = 2 + 4 and = + 6 are the lines of regression of x on y and y on x respectively. Find the value of k, if = 0.5. Find the mean of x and y, if the lines of regression are given by 4 + 3 + 7 = 0 and 3 + 4 + 8 = 0. The demand for a commodity is represented by = 300 5 , at what price the marginal revenue is 0? Question 20 [2] a. Find the relationship between the slopes of marginal revenue curve and the average revenue curve for the demand function = . OR a. Let the cost function of a firm be given by = 300 10 U + G L . Calculate output at which the marginal cost is minimum. Question 21 [4] The marks obtained by 10 candidates in English and Mathematics respectively are given below English 20 13 18 21 11 12 17 14 19 15 Mathematics 17 12 23 25 14 8 19 21 22 19 Estimate the probable score of Mathematics, if the marks obtained in English are 24. Question 22 [4] a. Solve the following linear programming problem graphically: Maximise = 7 + 10 Subject to constraints: 4 + 6 240, 6 + 3 240, 10, 0, 0. OR b. A cooperative society of farmers has 50 hectares of land to grow two crops X and Y. The profits from crops X and Y per hectare are estimated as 10,500 and 9000 respectively. To control weeds, a liquid herbicide has to be used for crops X and Y at rates of 20 Litres and 10 Litres per hectare respectively. Further, no more 800 Litres of herbicide should be used in order to protect the crop. How much land should be allocated to each crop so as to maximise the total profit of the society? Formulate the above LPP mathematically and then solve it graphically.

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