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ISC Class XII Prelims 2025 : Mathematics (St. James' School, Kolkata)

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EXAMINATION 2024-25 FINAL REHEARSAL MATHEMATICS Time : 3 hrs. F.M.: 80 questions attempt ALL are required to Candidates and C. three sectionsA, B Section B OR Section C Paper consists of from Class : 12 Date: 7.1, 25 The Question guestions EITHER and ALL from Section A (15 x 1=15] Marks ) SECTION A( 65 instructed. the questionsas answer (xv), (xi) to optionsandinsubparts correct the (x)cho0se Insubparts(0) to aRb iffa>b, then Ris by defined none of these nos.) equivalence relation (d) ofall real (set R on relation an (c) transitive ()Let Ris a symmetric (b) reflexive & Question 1: (a) reflexive & answer: Choose thecorrect f:R R. (i ) Let f(x) =5x+3, (b) f is only onto onto la) fis one- one& onto (c) fis one-one but not function nor onto. (d)fis neither one -one |42 1 6| (ii) The value of the determinant |28 7 4 is : l14 3 2l (d)none of these (c) 12 (b) 1 (a) 0 then |3A| is: (iv) If Ais a matrix of order 3 x3, () 314| (b)9 |4| (a) 27 |A| 2x*- 6x- 4 at (v) The point on the curve y= (d) none of these is : which tangent is parallelto X- axis 1) then A' is equal to : (vi) fA = ) (vi) The order and the degree of differential (b) 2 and1 (a) 1and 2 y 0are: equation ( + 3 dx? (c) 3 and 1 (vii) Which of the following is correct ? (a) Determinant is a square matrix (d) 1 and 3 (b) Determinant is a number associated to a matrix matrix (c) Determinant isa number associated to a square (d)none of these (ix) Let Abe a non-singular matrix of order 2. Assertion: adj (adj A) =A Reason : ladj A| =|A| the correct explanation for Assertion (a) Both Assertion and Reason are true and Reason is correct explanation for Assertion (b) Both Assertion and Reason are true but Reason is not the (d) Assertion is false and Reason is true. (c) Assertion is true and Reason is false () The value of cos-'(cos (a) 71 6 (b) 13n1 6 13r is (c) 6 (d) none of these COS X (xi) Evaluate : J0[2. cos x+ sinx dx 1- cosX (xii) If y= log find 1+ cosx dx 1 1 1 (xii) If AandBare events such that P(A) = ;,P(B) = and P(ANB) =-, then find P(A'B) 3 4 (xiv) Find the slope of the tangent to the curve x=t +3t - 8andy =2t'- 2t - 5at the point (/2,-1). (xv) if A=( and A is symmetric, then write the relation between x and y. Question 2: Show that: sin + cos-1 = tan-'(-) [2] Question 3: y= tan-1 , prove that dy 2 dx 1+x2 [21 Question 4: Evalyete : ftan? dx Evaluate : OR 2 1-x dx [2] Que_tion 5: Solve the differential equation : log =ax +by (2] Question 6: Prove that the function f(x) = x-6x+12 x+5 is increasing on R Question 7: If the function f(x) = x 3| + x 4,then show that f is not differentiable at x=3 and x=4 21 [4] Question 8: Ifx= tan( logy) prove that :(1 +x)y2 +(2x- a) y, =0 [4] Question 9: Using properties of derminants, prove that: ja'+1 ab ac ab b2 +1 bc ac bc c + 1| (4] =1+a' + b2 + c? OR Eusluote : x2 + 1 dx Question 10: From a lot of 1O items containing 3 defect ves, a sample of 4 items is drawn at random. Let the random variable X denote the number of defective items in the sample. If the sample is drawn randomly, find (a)the probability distribution of X (b) P(X s1) (c) P (0<X<2) (d) E(X) [4] Question 11 : [6] Solve the following system of linear equations by matrix method: X-2 y= 10, 2x-y-z= 8,-2 y+z=7 Question 12: Snow that the rectangle of maximum perimeter which can be inscribed in a circle of radius 10cm is a square of side 10v2 cm. [6] OR Mow that the surface area of a closed cuboid with it is a cube sauare base and given volume is minimum when Question 13: [6] In a test, an examinee either guesses or copijes or knows the answer to multiple choice question with four choices. The probability that he makes a guess is 3 and the probability that he copies the answer is~: The probability that his answer is correct, given that he copied it, is 1 G.Find the probabilitythat he knewthe answer to the question, given that he correctly answered it. Question 14: [6] Solve the differential equation : dx -2xy +y=0, given x=e and y=e SECTION B(15 Marks) Question 15: [5] In subparts (i) to (i) choose the correct options and in subparts (ii) to (v), answer the questions as instructed. (i) The value of for which the vectors 3 i-6+k and 2i- 4f+2k are parallel is (b); (a) 3 5 (ii) If the direction cosines of a line are k, k, k, then (a) k>o (c) k=1 (b) 0< k<1 (iii) Find the value of k for which lines (d) k= or x +3 Z-5 -3 5 and x+1 y-2 Z-5 -6 2 10 are parallel. (iv) Find the equation of a line parallel to x- axis and passing through origin. (v) Find the area of the parallelogram whose adjacent sides are i+k and 2i+ j+ k [2] Question 16: If l < =8,Jb| =3and | xb| = 12, then find the value of .b OR Find the unit vector perpendicular to thetwo vectors a= i+2j- k and b =2i+3 +k [4] Question 17: Show that the lines X-4 y+3 1 -4 Z+1 7 z+10 X-1 and 2 intersect. Also find the -3 coordinates of their point of intersection. OR 1-z Find the equation of the plane containing the line -2 3 1 and the point (-1, 0,2) Th Qe ues Question 18: from 1,25 If the area bounded by the parabola y'=x and the straight line x=4 is divided into two equal parts by the line x= a, then find the value of a. OR Find the area bounded by the circle x+ y=16 exterior to the parabola y = 6x SECTION C(15 Marks ) Question 19: In subparts (i) to (i) choose the correct options and in subparts (ii) to (v), answer the questions as instructed. ()) If cost function C(x) f afirm is given by C(x)= 3 x-6x + 5,then marginal cost at x=2 is (a) 6 (b) 8 (d) none of these (c) 14 (ii) Objective function of LPP is : (a) arelation between the variables (b)a linear constraint (d) none of these (c) a function to be optimized (ii) Find the slope of the marginal cost function if C(x) =ax+bx+c. (iv) Find the marginal average cost (MAC) if C(x) = 10 x-7x +3x (v) Find the marginal revenue at x =5 when R(x) = 36 x+3x+5. Question 20: [2] The revenue function is given by R(x) = 100 x (a) the demand function X. Find (b) marginal revenue function OR The average cost function is given by AC= x+5 +.Find the total cost and the marginal costfunctions of x. Question 21; The random variables have regression lines 3x + 2y 26 =0and 6x +y-31 = 0. Calculate: (a) mean values of x and y (b) co-efficient of correlations OR The demand function is x = 24 -2 p 3 where x is the number of units demanded and p is the price per unit. Find: (a) the revenue function Rin terms of p (b) the price and the number of units demanded for which the revenue is maximum (4] Question 22: Acompany manufactures two types of toys Aand B. Atoyof type Arequires 5minutes for cutting and 10minutes for assembling. Atoy of type Brequires &minutes for cutting and 8minutes for assembling. There are 3hours available for cutting and 4 hours available for assembling the toys in a day. The profit is Rs. 50 each on a toy Aand Rs. 60 each on a toy of type B. How many toys of each type should the company manufacture in a day to maximize the profit ? Solvetheproblem graphically (Use graph sheet). X

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