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ISC Class XII Prelims 2025 : Mathematics (St. Stephen's School, Dum Dum, Kolkata (Calcutta))

7 pages, 54 questions, 0 questions with responses, 0 total responses,

Lelouch Lamperouge
M. C. Kejriwal Vidyapeeth (MCKV), Liluah, Howrah

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t. ephen's Sthool, Dumoum Pre Board Examination: 2024 -25 Class: l2 Subject : Mathematics Full Marks :80 Time:3 Hrs (Candidates are allowed additional 15 minutes for only reading the paper.) Attempt allthe questions from Section A andall the questions either from Section B or from Section C. SECTION A -(65 MARKS) Question 1 (15x1) In subparts i) tox) choose the correct options and in subparts xi) to xv) answer the questions as instructed. i) Let R be a relation on the set N of natural numbers defined by nRm ifn divides m. Then Ris a) reflexive and symmetric b) transitive and symmetric c)equivalence d) reflexive, transitive but not symmetric ii) If f: R R is defined by fx) =x, then a)f is one-one onto - b) f is many one onto d) f is neither one-one nor onto. c)f is one-one but not onto ii)The value of cot( cosec'2+ cos a)1 iv)if a)x=y b) 0 2 ) is c) -1 d) V3 is a symmetric matrix, then b) x=0 c) y=0 v) If Ais asquare matrix such that A'=|, Then A is d) x*y a) A+| vi) The b) A c)0 d) 2A derivative of log x with respect to-1 is b) - d) -x vii)The integrating factor of the differential equation x -y= 2x is a)e-* b)e* d)' c)x vi)Adie is thrown and a card is selected from a pack of 52 cards. Ihe probability of getting an even number on the die and a spadeplaying card is c); d); ix) The points of discontinuity of the function f(x) = X+2 2x2-x-1 a) 1, - b) -1 are )-; d) ;1 x) f(x) =1+x2 is increasing in a)(- 1, oo) b) (-1,1) c) (-oo -1) d) none of these xi) If the function f:R R, defined by f(x)= 2x -5 is invertible, find f. 1 k 2 xii) If = A 1 2 5 \2 1 is not invertible, then find the value of k. 1. xi) Evaluate e5logx dx. Xiv) If Aand Bare events of a random experiment such that P(AUB)=4/5, P(B)= 2/5, P(A'UB') =7/10, find the value of P(A). xv) Twoballs are drawn at random with replacement from a bag containing 10 green and8 blue balls.Find the probabilitythat one of thenm isgreen and another is blue. [2] Question 2 sin x a)lf y= tan1 1+cos X find dy dx OR b)Jthe function x- 62x' +ax +9 attains its maximum value at x=1, on the Anterval [0,2), then find the value of a. (2] Question 3 1-cos2ax value of 'a'. x#0 ; is continuous at x=0, find the x2 (8 X=0 [2] Question 4 (1+y')(1+tlog x) Solve the differential equation dy dx [2] Question 5 a) Evaluate: log -1)dx. OR 2y? b) Evaluate : y2+4 dy Question 6 f f(x)=Icos x -sin x1, find f). Qyesfion 7 If tan1 yz + Xr tan:12 + tan:1 Zr yr 7prove that x +y'+z' =r?. [4] Qyestion 8 a)lf y=e,prove that d'y - 2a dy dx2 da +(a' +b )y=0. OR b)Evaluate: fcos(log x) dx. [4] Question 9 +y=0. Solve: (2x - 10y) dy dx [4) Quest on 10 a)A bag contains 8 white and 7 black balls. A ball is drawn at random from the bag andput into another bag whichcontains 5 white and 4 black balls. Nowa ball is drawn randomly from the second bag. What is the probability that it is white ? OR b)A problem in Mathematics is given to three students A, B, and C. Their chances of solving the sum are that i) 1 1 2' 3 and 1 respectively. Find the probability exactly two students will solve the problem. ii) at least two students will solve the problem. Question 11 If A= 3 11 [6] 1 1 -1 linear equations : -1 find A. Hence find the solution of the system of x+y-z =1; 3x+y -22 =3; and x-y-z=-1. Question 12 a) Evaluate : [6] 2x+3 x3+x2-2x dx. OR b)Find the particular solution of the differential equation (x'+y') dx - 2xy dy =0; given that y= 0 when x=1. Question 13 (6] aFind the points on the curve y 2x- 15x+ 36x -21 at which the tangents to the curve are parallel to the Xaxis. Also find the equations of the tangents to the curve at these points. OR b) Find the volume of the largest cylinder that can be inscribed in a sphere of radius 7 cm. [6] Question 14 Acoin is tossed four times. Let Xdenotes the number of heads. Find the probability distribution of X. Also find its Mean and Variance. [15] SECTION-B [5x 1] Question 15 answer Insubparts i) to i) choose the correct options and insubparts iii)to v) the questions as instructed. i)f =x(i +j+ k)is aunit vector, then 1 1) x=y3 2)x= 3) I l =1 4)1 l=t1. a) only 1) is true b) 1) and 3) are true c) 2)and 3) are true d) 1) and 4)are true ii) The value of y for which the vectors 3i - 6 + k and 2i -4 +yk are parallel is 3 ii) Find the equation of the plane passing through the points (2,0,0), (0,3,0) and (0,0,4). iv) If a line makes an angle of 45 with each ofY and Zaxis, then find the angle it makes with X axis. v) Find a vector in the direction ofthe vector @ =2i-3j, whose magnitude is 7 units. [2] Question 16 a)lf a= 3i + +2k and b= 2 - 2f+ 4k, then finda unit vector perpendicular to both the vectors and b. OR b)Find p, if the scalar projection of d= pi +j +4kon b- 2i +6j +3k is 4units. Question 17 [4] a)Findthe co-ordinates of the point where the line through the points A(3,4,1) and B(S,1,6) crosses the XY- plane. OR b)Find the image of the point (2,-1,5) in the line +2 10 == -4 z+8 Question 18 [4) Using integration, find the area of the region bounded by the curves *4=1 and -+=1. 3 2 4 SECTION -C (15] Que_tion 19 [5x 1] In subparts i) to i) choose the correct options and in subparts ii) to v) answer the questions as instructed. i)lf the demand function is x= 24-2p then revenue function in terms of pis 3 A{(12p-') b) 2(12p-p') )p-3p') d) none of these i)lf C(x)= Rs. 26000 + 30x and each product is sold at Rs.43, then breakeven point is c) 3000 b)1000 2000 d) none of these. i)The demand function for acertainproduct is given by x= 160- 4p, then find theprice and quantity at which marginal revenue is zero. 4 iv)ifr=Fby = o, =4, then find o. v)lf =5, =12, and by, = 0.95,then find regressionlineof yon x. Qyefon 20 a)Let the cost of afirn isgiven by the equation C(x)- 300x 10x, whereCis cost and xis output. Calculate output at which the nagialcost is minimuni Also calculate the minimum marginal cost. |2) OR b)The marginal cost of producing xunits of an electric appliance is given by MC= xVx + 1 and thecost of producing3 units of appliance is Rs. 6000. Find the cost function. [4] Question 21 regression a)Findthe regression co-efficient br and by and the two lines of from the given data: 5 6 X 5 6 Also compute the correlation co-efficient. OR b)The following observations are given: (1,4), (2,3), (3,2), (4,12), (5,10), (16,14), (7,16), (8,6) (9,18). value ofx when Estimate the value of y when the value of x is 10 and also the the value of y is 5. [4] Question 22 Use agraph paper tosolve the following problem: Amanufacturer manufactures two types of toys, A and B. Three machines are needed for manufacturing the toys. The time in minutes required for manufacturing each cup on the machines is given below: Types of Toys Machine | A 12 B 6 Time in minutes Machine I| 18 Machine ll 6 Each machine is available for a maximum 6 hours per day. If the profit on each toy of type Ais Rs. 90and that on each toy of type Bis Rs. 60, find the number of toys of each type that should be manufactured in a day to get a maximum profit.

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