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ISC Class XII Prelims 2026 : Mathematics (The Doon School, Dehradun)

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PRACTICS PAPER SECTION A -65 MARKS Question 1 [15 X 1 = 15] In subparts (i) to (x) choose the correct options and in subparts (xi) to (xv), answer the queSti0ns as instructed. (i) The value of tan 2 (seC 1 2) (a)13 (ii) If y x2 (c)l (d) 5 2 2 (d) - x2 (b)-2.. x2 The function f(x) = 8x3 - 60x2 + 144x + 27 is a strictly decreasing function in the interval (a) [-5,6) (b)[-co,2) (c) ( 3, co) (d) (2,3) 2 3 3 2 The degree of the differential equation d y + 3 (dy) = x 2 (d Y) is 3 2 dx dx dx (a) 1 (b) 2 If x = acos0,y = a sin 0, then (a) -1 (vi) (cosec- 1 3) is d y = log e (2 ), then -2 is e dx X (iv) 2 (6)11 (a):. + cot (b)1 a (c)3 d y dx at 0 (d) 4 = will be 2 (c) 0 (d) 2 If J; f(x)dx = J0 f(x)dx + J0a f(k - x)dx, then the value of k is (a) a a (b) 2a (c) independent of a (d)3a Ves (vi) ) Assertion(A): If tan-'x + tan- y = then x+y+xy = 1 Reason(R): tanx + tan y= tan \1-2y) Which of the following options is correct? (a) Both Aand Rare true, and Ris the correct explanation of A. (b) Both Aand Rare true, and Ris not the correct explanation of A. (c) Ais true, but Ris false. (d) Ais false, but Ris true. (vii) Statement 1: Iff e*( d =ef() +c, then the function f(x) is Statement 2: e*f(x) -f'()] = e*f(x) +c Which of the following options is correct? (a) Both statements are true (b) Both statements are false (c) Statement 1is true, but Statement 2 is false (d) Statement 1is false,but Statement 2 is true (X) The slope of the tangent to the curve y= 2cos'3x at x =is 6 (a) 1 ( ) (c)0 (d) 2 For what value of n is the given a homogeneous differential equation? dy x-y* dx (a) 1 (xi) (b) -1 x"y + xyn (b) 2 (c) 3 (d) 4 What do you think about the continuity of the functionf(r) = 2x- at x =0. (xii) Calculate the integrating factor of the differential equation: (x+ 1)-y = esr(x +1) dx (xii) Differentiate -cos (sin x) with respect to x Find the minimum value of the function f(x) =xlogx (xv) Evaluate: lx- 5|dx Question 2 - Find the points on the curve y = 1+x2 for which the tangent is parallel to the x-axis. [2] Auestion 3 |1-cos2x dx Evaluate: f taml 1+cos2x Sxsec'x dx OR (2] Question 4 How fast the volume of 3 Ifan edge of a variable cube is increasing at the rate ofcm/sec. (2) the cube is increasing when the edge is 5cm long? uestion 5 a) Find the derivative of log(1 +x)with respect to tan x. (2] OR b) If x + y = 4,then prove that ydx =x Question 6" 2] Determine the values of 'x for which the function f(x) = -2x' - 8x is increasing. Question7 a)v[howthat 2tan-1+sec-( +2tan-"()= [4]5olve OR for x: (tan'r)' + (cot-' )? = 518 Question 8 x sinxcosx Evaluate using the properties of definite integral, JR Jo costx+sintx dx [4] Question 9 a) Ify = x*, prove that d y dx? -()--0 OR [4]If y = cos (logx) + bsin (log x), prove thatx+x +y=0 dx2 dx Question 10 Find the value of "a for which the function f(x) is a continuous function at x =0 asin(+1), xs0 2 fx) =} tan x-sin x ,X > 0 [4] Question 11 a) Evaluate: - secx -dx 1+cosec x OR 6x+5 VEvaluate: f V6tx-2x2 dx [6] Question 12 a) Solve the following differential equation: (1+ y')dx - (tany -x)dy = 0 OR [6] 8 Solve the following differential equation: (1-)+(1+ ei) = 0, when x=0,y=1 ind the l e from the Question 13 In aresidential society comprising of 100 houses, there were 60 children between the ages of Que. 10-15 years. They were inspired by their teachers to start composting to ensure that biodear. -adable waste is recycled. For this purpose, instead child doing it for only his/her house, children convinced the Residents' welfare association to do it as a society initiative. For this they identified asquare area in the local park. Local authorities charged amount of 50 per square metre for space so that there is no misuse of the space and the Resident welfare association takes it seriously. The Association hired a labourer to dig out 250 m and he charged 400 x(depth) . a) If the side of the square plot is xmetres, and its depth is hmetres, then calculate the cost Cfor the pit in terms of h. b) Find the value of h (in m) for which= 0. dh d'c c) Express dh2 in terms of h. d) Calculate the value of x (in m) for minimum cost. [6] Question 149 If cos-1 X ;tcos=0, prove that 9x - 12 xy cose + 4y = 36 sin 0 3 [6] SECTION B- 15 MARKS Question 15 : [5 x 1 = 5] Insub parts i)to i) choose the correct options and in sub parts ii) tov), answer the questions as instructed. i) The angle between the planes 2x -y +z =6 and x +y+ 2z =7 is: a) i) i) iv) b) c) 6 The projection of the vector i+j+ kalong vector j is : a) 1 c) 2 b)3 d) -2 If and bare two vectors such that | x b|= .b, then what is the angle between and b. Find the value of ksothat the lines== 3 2k 2 and 3k 1 - are perpendicular toeach 7 other. v) If Obe the origin and the coordinates of Pbe (1,2, -3), then find the equation of the plane passing through P and perpendicular to OP. Question 16: Find the area of the parallelogram whose adjacent sides are determined by the vectors = -j+ 3k and b=2i- 7j + k OR [2] Using vector algebra, find the area of AABC, where BC = j+ 3k and C = -i- 21 Question 17: Find the equation of the plane passing through the point (1, 7, -7) and containing the line having equation X+1 1 OR [4] Find the image of the point from the point (2, 1, -5) to (2, 1, -5) in the the line. Question 18: line 11 10 y* =-11. Also, find the length of the perpendicular -4 Find the area of the region bounded by y=x +2, y =x, x =0and x = 3. SECTION C- 15 MARKS [4] Question 19 : [5x1=5] In subparts () and (ii) choose the correct options and in subparts (iii) to (v), answer the questions as instructed. i) Assertion(A): If the cost function of a firm is qiven byC= 2x +x 4, then the average cost is given by 2x +1-* Reason(R): Average cost (AC) = Which of the following options is correct? a) Both A and R are true, and R is the correct explanation of A. b) Both Aand Rare 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. The feasible region for an LPP is shown below: i) (4,10) (0,8) (6,8) (6,5) (0,0) (5,0) -X Let Z=3x+4y be the objective function. Maximum of Z occurs at a) (0,0) iii) b) (4,10) c) (6,8) d) (6,5) If the regression line of x on y is 9x + 3y - 46 = 0 and of y on x is 3x+ 12y - 7=0, then calculate the correlation coefficient. iv) The revenue function is given by R(x) = 100x -x-x. Find the Marginal revenue function. Acompany finds that the daily cost of producing x item ofa product is given by C(x) = 210x + 7000. If each item is sold for 280, find the minimum number of units that must be produced and sold daily. \guestion 20 a) The demand function of a monopolist is given by x = 300 - 4p. Calculate the quantity at which the Marginal revenue will be O. OR b) i) ii) [2] Afirm has the cost function,C =-7x + 111x + 50 and demand function, x = 100 -D. 3 Write the total revenue function in terms ofx Formulate the total profit P in terms of x. Question 21 ice cream temperature, and the number of average the that includes a) Shown below is a data set of six days. in a small town over the course per day by an ice cream truck Cohes sold Cones sold Temperature ("F) X 180 65 250 75 160 60 210 70 300 80 220 90 Find the line of regression of y on x. [4] OR b) From the given data: < Variable X Mean 6 8 Standard 4 6 deviation And correlation coefficient: calculate: i) ii) ii) Regression coefficients byy and b,xy Regression line of x on y Most likely value of x when y = 14 Question 22 The Welham Girls' School Mathematics department wants to organize two after-schoolworkshops: Trigonometry (Workshop-A) and Calculus (Workshop-B). The school has limited room-hours this month: Math Lab: 90 hours AVC: 80 hours Auditorium: 50 hours Each Trigonometry session uses 2 Math lab hr, 1 AVC hr, 1Auditorium hr. Each Calculus session uses 1Math lab hr, 2 AVChr, 1Auditorium hr. The school earns reward from CISCE at 48 points per session of Trigonometry and 40 points per session of Calculus. Formulate this situation as an LPP and solve it graphically to find how many sessions of each workshop should the department schedule to maximize reward from CISCE? [4] **************k****k**

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