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ISC Class XII Sample / Model Paper 2026 : Mathematics

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T5017028 MATHEMATICS Maximum Marks: 80 Time Allotted: Three Hours Reading Time: Additional Fifteen minutes Instructions to Candidates 1. You areallowed an additional fifteen minutes for only rcading the paper. 2. You must NOT start writing during reading time. 3. The Question Paper has 11 printed pages and one blank page. 4. TheQuestion Paper is divided into three sections and has 22 questions in all. 5. Section A is compulsory and has fourteen questions. 6. You are required to attempt all questions either from Section B or Section C. 7. Section B and Section C have four questions each. 8. Internal choices have becen provided in two questions of 2 marks, two questions of 4 marks and two questions of 6 marks in Section A. 9. Internal choices have been provided in one question of 2 marks and one question of 4 marks each in Section B and Section C. 10. While attempting Multiple Choice Questions in Section A, B and C, you are required to write only ONE option as the answer. 11. Allworkings, including rough work, should be done on the same page as, and adjacent to, the rest of the answer. 12. Mathematical tables and graph papers are provided. 13. The intended marks for questions or parts of questions are given in the brackets [). Instructionto Supervising Examiner 1. Kindly read aloud the instructions given aboveto all thecandidates present in the examination hall. 1225-860 Copyright reserved. 1 Turn over SECTION A - 65 MARKS Question 1 (xv), answer the correct options and in subparts (xii) to the choose (xi) to (i) subparts In questions as instructed. () (11) If A= 0 |1| then Al6 is: (a) Unit matr1x (b) Null matrix (c) Diagonal matrix (d) Skew matrix equation? Which of the following is a homogenous differential 4) dx = 0 (a) (4x+ 6y + 5) dy - (3y' + 2x + (b) (xy) dx - (x + y') dy = 0 (c) (x+ 2y') dx + 2ry dy = 0 y' dx + (r'- xy - y') dy = 0 Consider the graph of the function f(x) shownbelow: (d) (111) Y 4 2 X Statement 1: The function flx) is increasing in(;2). Statement 2: The function f(x) is strictly increasing in ;1). Which of the following is correct with respect to the above statements? (b) Statement l is true and Statement 2 is false. Statement 2is true and Statement l is false. (c) Both the statements are true. (d) Both the statements are false. (a) 1225-860 |1| (iv) rlx- 1 x2 + 1 dx is equal to: 2 (a) 3 1 (b) 3 -2 (c) 3 (d) (v) Assertion: Consider the two events A and B such that n(A) = n(B) and [1] P)-P). Reason: The events A and B are mutually exclusive. (a) Both Assertion and Reason are true and Reason is the correct explanation for Assertion. (b) Both Assertion and Reason are true but Reason is not the correct explanation for Assertion. (vi) (c) Assertion is true and Reason is false. (d) Assertion is false and Reason is true. The existence of unique solution of the system of equations x +y=1 and [1] 5x + ky= 2 depends on: (a) A only (b) k (c) (d) (vii) both k and k only A cylindrical popcorn tub of radius 10 cm is being filled with popcorns at the rate [1] of 314 cm per minute. The level of the popcorns in the tub is increasing at the rate of: (a) (b) (c) (d) lcm/minute 0-1 cm/minute 1:lcm/minute 0-5 cm/minute x+2 (viii) If f(x)=-x-2 x<0 0sx<1 x1 then the number of point(s) of discontinuity of f(x), is / are: (a) 1225-860 (b) 3 (c) (d) 2 0 3 Turn over (ix) Assertion: If SetA has m elements, Set B has n elements and n <m, then the number of one-one function(s) from A B is zero. Reason: A function f:A Bis defined only if allelements in Set A have an image in Set B. (a) Both Assertion and Reason are true and Reason is the correct explanation for Assertion. (b) Both Assertion and Reason are true but Reason is not the correct explanation for Assertion. (x) (c) Assertion is true and Reason is false. (d) Assertion is false and Reason is true. Let X be a discrete random variable. The probability distribution of X is given [1| below: X 30 10 -10 P(X) 1 3 1 5 10 2 Then E(X)will be: (xi) (xii) (a) 1 (b) 4 (c) 2 (d) 30 Statement 1: If 'A' is an invertible matrix, then (A')-1= (A-1)2 Statement 2: If 'A' is an invertible matrix, then JA-|= |A|-1 (a) Statement 1is true and Statement 2 is false. (b) Statement 2 is true and Statement 1 is false. (c) Both the statements are true. (d) Both the statements are false. [1] Write the smallest equivalence relation from the set A to A, where A={1,2, 3}. 0 1 (xiil) For what value of x, is A 3 a skew symmetric matrix? 0 -3 (xiv) Three critics review a book. Odds in favour of the book are 5:2, 4:3 and 3:4 respectively for the three critics. Find the probability that all critics are in favour of the book. (xv) 1225-860 Evaluate: 5 V2x+7 [1] dx 4 |1| Question 2 Find the point on the curve y = 2x- 6x 4 at which the tangent is parallel to the x-axis. (2) Question 3 Find the value of tan x - cot x, if (tan-'x)? (cot-lx) = [2] Question 4 (i) Ifx' = exy, prove that dy dx logx (1+logx)2 OR (ii) Iff*) = log(1 + x) + ,? show that f(x) attains its minimum value at x = 0. 1+x [2] Question 5 Three shopkeepers Gaurav, Rizwan and Jacob use carry bags made of polythene, handmade paper and newspaper. The number of polythene bags, handmade bags and newspaper bags used by Gaurav, Rizwan and Jacob are (20, 30, 40), (30, 40, 20) and (40, 20, 30) respectively. One polythene bag costs 1, one handmade bag is for 5 and one newspaper bag costs 2. Gaurav, Rizwan and Jacob spendEA, I BandE Crespectively on these carry bags. Using the concepts of matrices and determinants, answer the following questions: () (ii) Represent the above information in Matrix form. Findthe values of A, B and C. (2] Question 6 (i) Differentiate sin1 with respect to x. OR (i) Show that tan-1x+ tan1 y = Cis the general solution of the differential equation (1 + x) dy + (1 + y') dx = 0 [4] Question 7 1 1 = 0, using properties of determinant. If x+y+z = 0 then show that X z3 1225-860 5 Turn over Question 8 () 14] COSX Evaluate: 3cosx- 5 dx OR (ii) Evaluate: f(log x) dx Question 9 (i) ( logy) then show that (1 +x*)+ (2x- a) =0 dx2 If x = tan (4] dx OR (ii) The graph off(x) =-x + 27x - 2 is given below: Y B (C, d) (0, 0) X A (a, b) (a) Find the slope of the above graph. [1] (b) Find the co-ordinates of turning points, A and B. (c) [21 Evaluate f"(-2), f(0) and f'(3) and arrange them in ascending order. Question 10 Pia, Sia and Dia displayed their paintings in an art exhibition. The three artists displayed 15, 5and 10 of their paintings respectively. A person bought three paintings from the exhibition. (i) Find the probability that he bought one painting from each of them. [2] (ii) Find the probability that he bought all the three paintings from the same person. [21 Question 11 (6] (i) 37 Prove: xdx 1+ sinx =(V2 - 1)n 4 OR 1225-860 Evaluate: (i) J-1)2a? +1) r [6] Question 12 () Solve the differential equation: (x+ 5y ) dy dx =y when x = 2 and y = 1 OR (ii) Find the particular solution of the differential equation: (x'- 2y')dx + 2xydy = 0, when x = 1and y = 1 Question 13 Observe the two graphs, Graph 1 and Graph 2 given belowand answer the questions that follow. X (0,0) (0,0) X Graph 2 Graph 1 [1] () Which one of the graphs represents y = sin' x ? (ii) Write the domain and range of y =sin' x. (111) Prove that sin 1 + sin [2] 2 (iv) Find the value of tan-|2sin(2 cos-1) [2] Question 14 [6] An international conference takes place in a metropolitan city. International leaders, scientists and industrialists participate in it. The organisers of the conference appoint three agencies namely X, Y and Z for the security of the participants. The track record of the success of X, Y and Z in providing security services is 99%, 98-5% and 98% respectively. The organisers assign the responsibility of ensuring the security of 1000 people to agency X, 2000 people to agency Yand 3000 people to agency Z. At the end of the conference, one participant goes missing from the conference room. What is the probability that the missing participant was placed under the responsibility of the security agency X? 1225-860 7 Turn over SECTION B - 15 MARKS Question I5 In subparts (i) and (ii) choose the correct options and in subparts (iii) and (iv), answer the questions as instructed. () Assertion: ( + b)+ (5- a) = 2(a + b2) [1| Reason: Dot product of any two vectors is commutative. (a) Both Assertion and Reason are true and Reason is the correct explanation for Assertion. (b) BothAssertion and Reason are true but Reason is not the correct explanation for Assertion. (c) Assertion is true and Reason is false. (d) Assertion is false and Reason is true. (ii) The angle between the two planes x + y + 2z = 9 and 2x - y t z = 15 is: (a) (b) |1| 2 3 (c) Tt (d) 37 4 (iii) Show that points P(-2, 3, 5), Q(1, 2, 3) and R(7, 0, -1)are collinear. (iv) Two honeybees are flying parallel to each other in the garden to collect the nectar. The path traced by the bees is given in the form ofastraight line. The equation of the path traced by one honeybee is Y= (+ 2j + 3k) + (2i + 3 + 4k). (a) Write the above-mentioned equation in cartesian form. (b) Find the equation of the path traced by the other honeybee passing through the point (2, 4, 5). 1225-860 [1| E [1] Question l6 and (i) Find the equation of the plane passing through the points (2, 2, -1), (3, 4, 2) [21 (7, 0, 6). OR (1) Find the equation of the plane passing through the points (2, 3, 1), (4, -5, 3) and parallel to x-axis. Question 17 OB = i + 2j - 2k and Consider the position vectors of A, B and C as O = 2i- 2f + k, 0C= 2i -jt4k () Calculate AB and BC. (1) Find the projection of AB on BC. (111) Find the area of the triangle ABC whose sides are AB and BC. |1] [1| [21 Question 18 () The equation y= 4-x represents a parabola. (a) Make a rough sketch of the graph of the given function. (b) Determine the area enclosed between the curve, the x-axis, the lines x =0 and [1] (2) X=2. (c) [1] Hence, find the area bounded by the parabola and the x-axis. OR (ii) A farmer has a field bounded by three lines x +2y = 2, y-x =1, 2x +y = 7. Using integration, find the area of the region bounded by these lines. (4] SECTION C-15 MARKS Question 19 In subparts (i)and (ii) choose the correct options and in subpart (ii), answer the questions as instructed. (i) The total revenue received from the sale of 'x' units of a product is 1] R(x) =36x +3x+5. Then, the actual revenue for selling the 10 item will be: (a) 27 (b) 90 (c) 93 (d) 33 1225-860 Turn over (11) Read the following statements and choose the correct option. the regression coefficients are of the same sign. (I) The correlation coefficient and regression The correlation coefficient is the arithmetic mean between the (II) coefficients. equal to 1. (III) The product of two regression coefficients is always greater than unity. (V) Both the regression coefficients cannot be numerically (a) Only (IV) is correct. (b) Only () and (II) are correct. Only(I) and (IV) are correct. (c) (d) Only (III) and (IV) are correct. (iii) Consider the following data: 2 1 6 3 6 4 1 y - |1| (a) Calculate ~ and (b) Complete the table. |1| (c) Calculate bxy [2| Question 20 (i) Find the regression line of best fit from the following data. Lx =24, 2 y = 44, }xy = 306, Ex = 164, }y = 576, n= 4 OR (ii) Two ines of regression are given as 4x + 3y + 7 = 0and 3x + 4y + 8= 0. Identify the line of regression of x on y. Question 21 i) Autensil manufacturer produces 'x' dinner sets per week and sells each set at where x 600-p The cost of production of 'x'sets is x2 + 78x + 2000 p, (a) Write the revenue function. (b) (c) Write the profit function. [1] Calculate the number of dinner sets to be produced and sold per week to ensure maximum profit. [2] OR 1225-860 10 (ii) The Average Cost of producing 'x' units of commodity is given by: AC= 30 + 200 5000 50 (a) Find the Cost function. (b) (c) Find the Marginal Cost function. |1] Find the Marginal Average Cost function. [1] [1] MC-AC (d) Verify that (AC) = dx Question 22 Twodifferent types of books have to be stacked in the shelf of a library. The first type of book weighs 1 kg and has a thickness of 6 cm. The second type of book weighs 15 kg and has a thickness of 4 cm. The shelf is 96 cm long and can support a maximum weight of 21 kg. How should both the types of books be placed in the shelf to include the maximum number of books? Formulate aLinear Programming Problem and solve it graphically. 1225-860 11 14)

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