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ISC Class XII Prelims 2025 : Mathematics (Don Bosco School (DBS), Siliguri)

4 pages, 55 questions, 2 questions with responses, 2 total responses,

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DON B0SCO SCHOOL, SILIGURI PRE BOARD EXAMINATION MATHEMATICS CLASS 12 DATE: 13.01.2025 TIMES: 3 HRS FM: 80 Answer to the paper must be written on th paper proviaded separately. You will NOT be allowed to write during the first 15 minutes. This time is to be spent in reading the question paper. The time given at the head of this paper is the tine allowed for writing the answvers. The Question Paper consists of three sections A, Band C. Candidates are required to attempt all questions from Section Aand allquestions EITHER fiom Section BOR Section C. All working, incuding rough work, should be done on the same sheet as, and adjacent to the rest of the answer. The intended marks for questions or pats of questions are given in the brackets [ J SECTION A- 65 MARKS Question 1 (1x15=15) Choose the correct option for the following questions. i) IfA is a skew-symmetric matrix, then Ais a c. Null matrix d. Cannot be determined a. Skew symmetric matrix b. Symmetric matrix i) sin- sin(-2) is equal to 1 a. C. -2 2 d. None of the above ii) What is the value of f(x + Vz) dx b log(1+ Vz) +C a. 2 log(1 + Vx) + C iv) The general solution of differential equation v) If P(A) = 5 vi) 5 2 c. 3 3 4 16x 6 9x 8 12x 11 2xy what is the value of P(B/A) 11 d. None of the above C. b. 10x a. 0 7 5 2 Evaluate y -* dx d. Both A and B are false. P(B) =; and P(AUB) = b. a. dy d. None of the above b. x? + y = CGx explanation of A. a. x'+ y' = Gxy c. x -y = Cx 6 C. 2log (r+ 1) +C d. None of the dbu. G vii) If arelation R on the set {1,2,3} be defined by R= {(1,2)), then R is d. None of the above C. symmetric b reflexive a atransitive viii) If y= cos |1+cos x ,find dy 2 dy 3 d 2 d. -cosec(logx) +c ix) Assertion (A): ifA is a square matrix, then A. adi(A) = adj(A). A = det(A). I a dx 2 b. dx =3 C. Reason (R): The adjoint of matrix A, denoted as adi(A), is defined asthe transpose.of the cofactor of matrix A. a. Both A and R are true and R is the coTectexplanation of A b. BothA and R are true but R is not the correct explanation of A C. A is true. R is false d. A is false, R is true x) Find the value of k for which fkx +5, when xs2 ,is continuous x-1, when x> 2 b. k =-20 a. k= -12 at x = 2 c. k=-2 Page 1 d. None of the above. This Question paper consists of 4 printed pages. xi) Two numbers are selected at random from the integers 1 through 9. If the sum is even, find the probability that both the numbers are odd. 2 x) If A= ( 1.,find (A- 21)(A -3), xii) If y= log( )then find xiv) Solve : where Iis the unit matrix. dy dx dy = sinx-x dx xv) Two dice are thrown. Find the probability of getting an odd number on the first die and a multiple of 3 on the other. Question 2 i) Find dy if xy +y' = tanx+y dx [2] OR i) Prove that f(x) =x -sin(x) is an increasing function. [2] Question 3 Evaluate : [2] a Question 4 2 Find the equation of tangent to the curve xi + y+ = 2 at (1, !) Question5 a. Evaluate : sin x (2] dx [21 Evaluate : Sdx 1+ ex [2] 1+sin x OR o. Question 6 Solve for x: tan1 2x + tan-13x = [2] 4 Question7 Show that : 2tan-b + b Question 8 dx Evaluate: J:3 sin x+ 4 cos x tan) (a cos + b) COs (4] 14] dx eMestion 9 a. Astoneis dropped into a quiet lake and waves move in circles at a speed of 3.5 cm per second. At the instant when the radius of the circular wave is 7.5 cm, how fast is the enclosed area increasing? [4] OR log y = sin-x,show that (1-x) dy'. - xdx,- y = 0. [4] destion 10 a. In a class of 60 students, 30opted for singing, 32 opted for dancing and 24 opted for both. If one of these students is selected at random, find the probability that: i.the student opted for singing or dancing. ii. the student has opted neither singing or dancing. ii. The selected student has opted singing but not for dancing. [4] OR Three individuals A, B, and Chave applied for ajob at aprivate firm. Their chances of selection are in the ratio 1:2:4. Theprobabilities that A, B, andC can bring about changes to boost the company's profits are 0.8. 0.5, and 0.3., Kespectively. If the desiredjchang doesnt bccur, find the probability [4] that it is due to the appointment of C Page 2 This Question paper consists of4 printed pages. 11 Qudtion The cost of 4kg onion, 3 kg whcat and 2 kg rice 1s 60. The cost of 1 kg onion, 2 kg wheat and 3 kg rice is 45. The cost of 6kg onion, 2 kg wheat and 3 kg rice is 70. Find the cost of each item per kg by matrix method. [6] Question 12 (6] Solve : x log xy +y=2 log x OR b. Show that J2 log (tan x+ cot x) dx = n log 2 and hence deduce the value of dx. 1+x [6] Question 13 a. Prove that the radiusof the right circular cylinder of greatest curved surface area which can be inscribed in a given cone is half of that of the Cone. OR b. Two sides of a triangle have lengths 'a' and 'b' and the angle between P them is 0. What value of 0 will maximize the area of the triangle? Find b the maximum area of the triangle also. R Question 14 An unbiased die is thrown twice. Asuccess is getting asix. Find the probability distribution of the number of successes. [6] SECTION B 15 MARKS Question 15 In subparts (i) and (ii) choose options and in subparts (ii) to(v), answer the questions as instructed. [5] i. The area betweenthe curve y =2sin7x, 0<xsT and the x-axis is d. none of the above b. 5sq.units C. 6sq.units a. 4 sq. units i. The vector equation of a plane which is at a distance ofl44 units from the origin is a. 7 C. 9 b. 8 d. none of the above iii. Findthe area of the parallelogram whose adjacent sides are given by the vectors =3i- /+4k and b =-j+k. iv. Findthe angle between the line *= =andthe plane 2x -- 13y - 5z = 4 2 v. Find the Cartesian equation of the line passing through the points (-1,0,2) and (3,4,6). Question 16 a. Show that: (xb)' = | . [2) .b| OR b. Show that .(b+ )x( +26 +3 = l 6 el Question 17 a. Find the shortest distance between the lines X-1 y+3 -16 7 and Z-3 3 [4] = 8= -6 OR b. Find the Cartesian equation of the plane passing through the intersection of the planesr. (2i- 16j) - 12 = 0 and 3x-y-4z-0 and at a unit distance from the origin. Page 3 This Question paper consists of 4 printed pages. [4] Question 18 Find the area of the following region fo.)+%sis+} [4] SECTION C- 15 MARKS Question 19 (51 In subparts (i)and (ii) choose options and in subparts (iii) to (v), answer the questions as instructed. 41+ i. Find the maximum profit that a connpany can make, if the profit function is given by P(x) = 24x -18 x2 a. 42 b. 49 C. 69 d. None of the above C. 160 d. none of the above 1.6 and by = 0.5,find the most likely value of ywhen ii. By using the data ~ = 25, = 30; byy X-60? a. 55 b. 86 ii. If 5,x =y=20,x?= Ey = 90,xy = 76. Find the regression equation treating yas independent variable. C(x) = 30x + 250. Determine the iv. The price of a commodity is fixed at Rs 55 and its cost function is break -even point. v. The demand function fora monopolist is given by x = 100 4p. Find the price and quantity at which MR = 0. Question 20 1000- 15x a. The demand function for a certain commodity is given by p= What is the total revenue from the sale of five units? OR x,0<x< 25. [2] demanded and p is the price per b. The demand function is x-24-2p /3 where x is the number of units unit. Find: (i)The rev nue function R in terms of p. the revenue is maximum (ii) The price and the number of units demanded which Question 21 2x = 30 , The following were obtained with respect to two variables x and y: 199, (2] y= 42, )xy = x = 184, Ey = 318 ,n=6, find the following : . the regression coefficients. i. correlationcoefficients between x and y i . the regression equation of y on x, i. the likely value of ywhen x= 10. [41 OR distribution are x +2y -5 =0and 2x + 3y -8 = 0and bivariate a of lines regression two The b. [4] coefficient of correlation. the variance of x is 12. Find the variance of y and the Qaestion 22 Solve the following inear programming problem graphically: x+y Minimize Z = 3x + 4y subject to the constraints 3, 2x + y >4, x,y Page 4 This Question paper consists of 4 printed pages. 0 [4]

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