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CBSE Class 12 Sample / Model Paper 2023 : Mathematics : Sample Paper maths

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KENDRIYA VIDYALAYA GACHIBOWLI, GPRA CAMPUS, HYD 32 SAMPLE PAPER TEST 01 FOR BOARD EXAM (2022-23) SUBJECT: MATHEMATICS (041) MAX. MARKS : 80 CLASS : XII DURATION: 3 HRS General Instructions: 1. This Question paper contains - five sections A, B, C, D and E. Each section is compulsory. However, there are internal choices in some questions. 2. Section A has 18 MCQ s and 02 Assertion-Reason based questions of 1 mark each. 3. Section B has 5 Very Short Answer (VSA)-type questions of 2 marks each. 4. Section C has 6 Short Answer (SA)-type questions of 3 marks each. 5. Section D has 4 Long Answer (LA)-type questions of 5 marks each. 6. Section E has 3 source based/case based/passage based/integrated units of assessment (4 marks each) with sub parts. SECTION A Questions 1 to 20 carry 1 mark each. 1. For any matrix A = [aij], if cij denotes its cofactors then find the value of a11c12 + a12c22 + a13c32. (a) 1 (b) -1 (c) 0 (d) none of these 1 2 3 1 7 11 2. If , then write the value of k. 3 4 2 5 k 23 (a) 17 (b) -17 (c) 13 (d) -13 3. The magnitude of each of the two vectors a and b , having the same magnitude such that the angle between them is 60 and their scalar product is 9/2, is (a) 2 (b) 3 (c) 4 (d) 5 1 4. If f ( x) x 2 sin , where x 0, then the value of the function f at x = 0, so that the function is x continuous at x = 0, is (a) 0 (b) -1 (c) 1 (d) None of these 2 5. The value of e 0 (a) dx is 1 sin x (b) 0 (c) 3 (d) /2 6. If m and n are the order and degree, respectively of the differential equation 3 2 2 d y dy y x 3 2 xy sin x , then write the value of m + n. dx dx (a) 1 (b) 2 (c) 3 (d) 4 7. Corner points of the feasible region for an LPP are (0, 2), (3, 0), (6, 0), (6, 8) and (0, 5). Let F =4x + 6y be the objective function. The minimum value of F occurs at (a) Only (0, 2) (b) Only (3, 0) (c) the mid-point of the line segment joining the points (0, 2) and (3, 0) (d) any point on the line segment joining the points (0, 2) and (3, 0) Prepared by: M. S. KumarSwamy, TGT(Maths) Page - 1- 8. The projection of the vector 2 i 3 j 2k (a) 10/ 6 (b) 10/ 3 (c) 5/ 6 sin 2 x cos2 x 9. Evaluate: dx sin 2 x cos 2 x (a) tanx cotx + C (c) tanx + cotx + C i 2 j k is (d) 5/ 3 (b) tanx + cotx + C (d) tanx cotx +C 2 1 10 10. If x y , find the value of x. 3 1 5 (a) 1 (b) 2 (c) 3 (d) 4 11. Feasible region (shaded) for a LPP is shown in the given figure. The maximum value of the Z = 0.4x + y is (a) 45 (b) 40 2x 5 3 = 0, find x. 5x 2 9 (a) 13 (b) 3 (c) 50 (d) 41 (c) -13 (d) 3 12. If 13. If A is a non-singular matrix of order 3 and |A| = 4, find |adj A|. 1 4 14. If A and B are two independent events with P(A) = 3/5 and P(B) = 4/9, then find P ( A B) . (a) 1/9 (b) 2/9 (c) 1/3 (d) 4/9 (a) 4 15. If (b) 16 (c) 64 (d) dy y sin 2 x , y(0) = 1, then solution is dx (a) y = esin 2 x (b) y = sin2x (c) y = cos2x d2y 16. If y = 5 cos x 3 sin x, then is equal to: dx 2 (a) y (b) y (c) 25y Prepared by: M. S. KumarSwamy, TGT(Maths) (d) y = ecos 2 x (d) 9y Page - 2- 17. If the equation of a line AB is AB. (a) 1, 2, 4 (b) 1, 2, 4 x 3 y 2 z 5 , find the direction ratios of a line parallel to 1 2 4 (c) 1, 2, 4 (d) 1, 2, 4 18. If a line makes angles , , with the positive direction of co-ordinates axes, then find the value of sin2 + sin2 + sin2 . (a) 1 (b) 2 (c) 3 (d) 4 ASSERTION-REASON BASED QUESTIONS In the following questions, a statement of assertion (A) is followed by a statement of Reason (R). Choose the correct answer out of the following choices. (a) Both A and R are true and R is the correct explanation of A. (b) Both A and R are true but R is not the correct explanation of A. (c) A is true but R is false. (d) A is false but R is true. x 5 y 4 z 6 19. Assertion: If the cartesian equation of a line is , then its vector form is 3 7 2 r 5i 4 j 6k (3i 7 j 2k ) Reason: The cartesian equation of the line which passes through the point ( 2, 4, 5) and x 3 y 4 z 8 x 3 y 4 z 8 parallel to the line given by is . 3 5 6 2 4 5 1 1 1 20. Assertion (A): The domain of the function 2 is , , 2 2 1 Reason (R): ( 2) = 4 SECTION B Questions 21 to 25 carry 2 marks each. 21. Prove that the Greatest Integer Function f : R R, given by f(x) = [x] is neither one-one nor onto. Where [x] denotes the greatest integer less than or equal to x. OR 1 If sin sin 1 cos 1 x =1, then find the value of x. 5 1 cos kx x sin x , if x 0 22. Find the value (s) of k so that the following function f ( x ) is continuous at 1 , if x 0 2 x = 0. 23. If | a b |2 ( a.b) 2 144 and | a | 4 , then find the value of | b | . 24. Find the angle between the vectors a i j k and b i j k . OR x 3 y 1 z 5 Find the coordinates of the point where the line cuts the XY plane. 3 1 5 Prepared by: M. S. KumarSwamy, TGT(Maths) Page - 3- 1 x 1 x dy 1 25. If y = sin 1 , then show that 2 dx 2 1 x 2 SECTION C Questions 13 to 22 carry 3 marks each. 26. In a group of 50 scouts in a camp, 30 are well trained in first aid techniques while the remaining are well trained in hospitality but not in first aid. Two scouts are selected at random from the group. Find the probability distribution of number of selected scouts who are well trained in first aid. OR An urn contains 5 white and 8 white black balls. Two successive drawing of three balls at a time are made such that the balls are not replaced before the second draw. Find the probability that the first draw gives 3 white balls and second draw gives 3 black balls. x sin x dx 27. Evaluate: 1 cos2 x 0 OR 3 Evaluate: | x 2 2 x | dx. 1 28. Evaluate: 9x 2 dx . 6 x 10 29. Find the general solution of the following differential equation; x dy (y + 2x2)dx = 0 OR dy Solve: 1 x y xy dx 30. Solve the following Linear Programming Problem graphically: Maximise Z = x + 2y subject to the constraints: x + 2y 100; 2x y < 0; 2x + y 200; x, y 0 31. Evaluate: e x (1 sin x ) dx (1 cos x) SECTION D Questions 32 to 35 carry 5 marks each. 32. Find the area of the region bounded by the parabola y2 = 8x and the line x = 2. 33. Find the shortest distance between the lines r 3 i 2 j 4k ( i 2 j 2k ) and r 5 i 2 j (3 i 2 j 6k ) . If the lines intersect find their point of intersection. OR Find the vector equation of the line passing through (1, 2, 4) and perpendicular to the two lines: x 8 y 19 z 10 x 15 y 29 z 5 and 3 16 7 3 8 5 34. Let N be the set of all natural numbers and let R be a relation on N N defined by (a,b)R(c,d) such that ad=bc for all (a,b),(c,d) N N. Show that R is an equivalence relation on N N. OR Show that the relation S in the set R of real numbers, defined as S = {(a, b) : a, b R and a b3} is neither reflexive, nor symmetric, nor transitive. Prepared by: M. S. KumarSwamy, TGT(Maths) Page - 4- 1 1 0 2 2 4 35. Given A = 2 3 4 and B = 4 2 4 , verify that BA = 6I, use the result to solve the 0 1 2 2 1 5 system x y = 3, 2x + 3y + 4z = 17, y + 2z = 7. SECTION E(Case Study Based Questions) Questions 35 to 37 carry 4 marks each. 36. Case-Study 1: Read the following passage and answer the questions given below. The temperature of a person during an intestinal illness is given by ( ) = 0.1 + + 98.6,0 12, m being a constant, where f(x) is the temperature in F at x days. (i) Is the function differentiable in the interval (0, 12)? Justify your answer. (ii) If 6 is the critical point of the function, then find the value of the constant (iii) Find the intervals in which the function is strictly increasing/strictly decreasing. OR (iii) Find the points of local maximum/local minimum, if any, in the interval (0, 12) as well as the points of absolute maximum/absolute minimum in the interval [0, 12]. Also, find the corresponding local maximum/local minimum and the absolute maximum/absolute minimum values of the function. 37. Case-Study 2: Read the following passage and answer the questions given below. An architect designs a building for a multinational company. The floor consists of a rectangular region with semicircular ends having a perimeter of 200 m as shown here: (i) If x and y represents the length and breadth of the rectangular region, then find the relation between the variable. (ii) Find the area of the rectangular region A expressed as a function of x. (iii) Find the maximum value of area A. Prepared by: M. S. KumarSwamy, TGT(Maths) Page - 5- OR The CEO of the multi-national company is interested in maximizing the area of the whole floor including the semi-circular ends. Find the value of x for this to happen. 38. Case-Study 3: Read the following passage and answer the questions given below. In an office three employees Vinay, Sonia and Iqbal process incoming copies of a certain form. Vinay process 50% of the forms. Sonia processes 20% and Iqbal the remaining 30% of the forms. Vinay has an error rate of 0.06, Sonia has an error rate of 0.04 and Iqbal has an error rate of 0.03. (i) Find the conditional probability that an error is committed in processing given that Sonia processed the form. (ii) Find the probability that Sonia processed the form and committed an error. (iii) The manager of the company wants to do a quality check. During inspection he selects a form at random from the days output of processed forms. If the form selected at random has an error, find the probability that the form is not processed by Vinay. OR If the form selected at random has an error, find the probability that the form is processed by Sonia Prepared by: M. S. KumarSwamy, TGT(Maths) Page - 6-

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