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CBSE Class 12 Board Exam 2019 : Mathematics (Series 1)

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SET 1 Series : BVM/1 Code No. 65/1/1 - - Roll No. Candidates must write the Code on the title page of the answer-book. - - - - - 29 , - 15 - 10.15 10.15 10.30 - - Please check that this question paper contains 11 printed pages. Code number given on the right hand side of the question paper should be written on the title page of the answer-book by the candidate. Please check that this question paper contains 29 questions. Please write down the Serial Number of the question before attempting it. 15 minute time has been allotted to read this question paper. The question paper will be distributed at 10.15 a.m. From 10.15 a.m. to 10.30 a.m., the students will read the question paper only and will not write any answer on the answer-book during this period. 11 MATHEMATICS 3 100 Time allowed : 3 hours 65/1/1 Maximum Marks : 100 1 [P.T.O. (i) (ii) (iii) (iv) (v) - 29 , , 4 8 11 6 , - 1 , 3 , 3 3 , , General Instructions : (i) All questions are compulsory. (ii) This question paper contains 29 questions divided into four sections A, B, C and D. Section A comprises of 4 questions of one mark each, Section B comprises of 8 questions of two marks each, Section C comprises of 11 questions of four marks each and Section D comprises of 6 questions of six marks each. (iii) All questions in Section A are to be answered in one word, one sentence or as per the exact requirement of the question. (iv) There is no overall choice. However, internal choice has been provided in 1 question of Section A, 3 questions of Section B, 3 questions of Section C and 3 questions of Section D. You have to attempt only one of the alternatives in all such questions. (v) Use of calculators is not permitted. You may ask logarithmic tables, if required. SECTION A 1 4 1 Question numbers 1 to 4 carry 1 mark each. 1. A B 3 |A| = 2 AB = 2I , |B| If A and B are square matrices of the same order 3, such that |A| = 2 and AB = 2I, write the value of |B|. 65/1/1 2 2. d f(x) = x + 1 , dx (fof) (x) d If f(x) = x + 1, find dx (fof) (x). 3. d2y dy 2 4 x2 dx2 = 1 + dx Find the order and the degree of the differential equation 4. x2 2 4 d2y dy = 1 + dx . dx2 x- , y- z- : 90 , 135 , 45 - (3, 4, 5) 2^i + 2^j 3k^ If a line makes angles 90 , 135 , 45 with the x, y and z axes respectively, find its direction cosines. OR Find the vector equation of the line which passes through the point (3, 4, 5) and is ^ parallel to the vector 2^i + 2^j 3k. SECTION B 5 12 2 Question numbers 5 to 12 carry 2 marks each. 5. * R a * b = ab + 1 (i) - (ii) - , ? Examine whether the operation * defined on R by a * b = ab + 1 is (i) a binary or not. (ii) if a binary operation, is it associative or not ? 65/1/1 3 [P.T.O. 6. A 2A 3B + 5C = O, 2 2 0 B = 3 1 4 2 0 2 C= 7 1 6 2 2 0 and Find a matrix A such that 2A 3B + 5C = O, where B = 3 1 4 2 0 2 . C= 7 1 6 7. : . sec2x . tan2x + 4 dx. . sec2x Find : . dx. tan2x + 4 8. : 1 sin 2x dx, < x < . 4 2 . : . 1 .sin (2x) dx. . Find : . 1 sin 2x dx, 4 < x < 2 OR . Find : .sin 1(2x) dx. 9. y = e2x (a + bx), a, b , Form the differential equation representing the family of curves y = e2x (a + bx), where a and b are arbitrary constants. 65/1/1 4 , 3 10. ^ a = 2^i + 3^j + k, b = ^i 2^j + k^ c = 3^i + ^j + 2k^ , [ a b c ] If the sum of two unit vectors is a unit vector, prove that the magnitude of their difference is 3. OR ^ b = ^i 2^j + k^ and ^ find [ If a = 2^i + 3^j + k, c = 3^i + ^j + 2k, a b c ]. 11. 1, 2, 3 4, 5, 6 , A B A B A die marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let A be the event number is even and B be the event number is marked red . Find whether the events A and B are independent or not. 12. : , (i) 5 (ii) 5 , - ? X P(X) , k : k , x = 0 2k, x = 1 P(X = x) = 3k, x = 2 0, k A die is thrown 6 times. If getting an odd number is a success , what is the probability of (i) 5 successes ? (ii) atmost 5 successes ? OR The random variable X has a probability distribution P(X) of the following form, where k is some number. k2k,, ifif xx == 01 P(X = x) = 3k, if x = 2 0, otherwise Determine the value of k . 65/1/1 5 [P.T.O. SECTION C 13 23 4 Question numbers 13 to 23 carry 4 marks each. 13. R = {(a, b) : a b} R , f : N N, f(x) = x2 + x + 1, , f : N S, S f , Show that the relation R on but not symmetric. defined as R = {(a, b) : a b}, is reflexive, and transitive OR Prove that the function f : N N, defined by f(x) = x2 + x + 1 is one-one but not onto. Find inverse of f : N S, where S is range of f. 14. : tan 1 4x + tan 1 6x = 4 Solve : tan 1 4x + tan 1 6x = 4. 15. 16. a2 + 2a 2a + 1 1 , 2a + 1 a + 2 1 = (a 1)3. 3 3 1 2 + 2a 2a + 1 1 a Using properties of determinants, prove that 2a + 1 a + 2 1 = (a 1)3. 3 3 1 y dy x+y log (x2 + y2) = 2 tan 1 x , dx = x y . dy xy yx = ab , dx dy x + y y If log (x2 + y2) = 2 tan 1 x , show that dx = x y . OR dy If xy yx = ab, find dx . 65/1/1 6 17. d2y If y = (sin 1x)2, prove that (1 x2) 18. dy y = (sin 1x)2 , (1 x2) dx2 x dx 2 = 0. y = 3x 2 d2y dy x dx 2 = 0. dx2 - 4x 2y + 5 = 0 Find the equation of tangent to the curve y = 3x 2 which is parallel to the line 4x 2y + 5 = 0. Also, write the equation of normal to the curve at the point of contact. 19. . 3x + 5 : dx. .x2 + 3x 18 . 3x + 5 Find : . 2 dx. x + 3x 18 20. a a . . f(x) dx = .f(a x) dx, . 0 0 . x sin x dx .1 + cos2x 0 a a . . . x sin x Prove that .f(x) dx = .f(a x) dx, hence evaluate . dx. 1 + cos2x 0 0 0 65/1/1 7 [P.T.O. : x dy y dx = 21. x2 + y2 dx , y = 0 x = 1. dy : (1 + x2) dx + 2xy 4x2 = 0 , y(0) = 0. Solve the differential equation : x dy y dx = x2 + y2 dx, given that y = 0 when x = 1. OR dy Solve the differential equation : (1 + x2) dx + 2xy 4x2 = 0, subject to the initial condition y(0) = 0. ^ ^i + ^j + k, 22. 2^i + 5^j, 3^i + 2^j 3k^ ^i 6^j k^ : A, B, C D AB CD AB CD ^ 2^i + 5^j, 3^i + 2^j 3k^ and ^i 6^j k^ respectively are the position If ^i + ^j + k, vectors of points A, B, C and D, then find the angle between the straight lines AB and CD. Find whether AB and CD are collinear or not. 23. : 1 x 7y 14 z 3 7 7x y 5 6 z = = = 1 = 5 , 3 2 3 1 x 7y 14 z 3 7 7x y 5 6 z Find the value of , so that the lines 3 = = 2 and = 1 = 5 3 are at right angles. Also, find whether the lines are intersecting or not. 65/1/1 8 SECTION D 24 29 6 Question numbers 24 to 29 carry 6 marks each. 24. 1 1 1 A = 1 0 2 3 1 1 , A 1 : : x+y+z=6 x + 2z = 7 3x + y + z = 12 : 1 2 2 A = 1 3 0 0 2 1 1 1 1 If A = 1 0 2 , find A 1. Hence, solve the system of equations 3 1 1 x + y + z = 6, x + 2z = 7, 3x + y + z = 12. OR Find the inverse of the following matrix using elementary operations. 1 2 2 A = 1 3 0 0 2 1 25. 2 m 8 m3 ` 70/m2 ` 45/m2 ? A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8 m3. If building of tank costs ` 70 per square metre for the base and ` 45 per square metre for the sides, what is the cost of least expensive tank ? 65/1/1 9 [P.T.O. 26. , ABC , A(2, 5), B (4, 7) C(6, 2) ABC , x- x2 + y2 = 8x y2 = 4x Using integration, find the area of triangle ABC, whose vertices are A(2, 5), B(4, 7) and C(6, 2). OR Find the area of the region lying above x-axis and included between the circle x2 + y2 = 8x and inside of the parabola y2 = 4x. 27. (2, 2, 1), (3, 4, 2) (7, 0, 6) : (4, 3, 1) ( 1, 3, 4) r = (^i + ^j) + (^i + 2^j k^ ) (2, 1, 4) Find the vector and Cartesian equations of the plane passing through the points (2, 2 1), (3, 4, 2) and (7, 0, 6). Also find the vector equation of a plane passing through (4, 3, 1) and parallel to the plane obtained above. OR Find the vector equation of the plane that contains the lines r = (^i + ^j) + (^i + 2^j k^ ) and the point ( 1, 3, 4). Also, find the length of the perpendicular drawn from the point (2, 1, 4) to the plane, thus obtained. 65/1/1 10 28. A, B C A 1% , B 5% C 7% A 50% , B 30% C 20% , A A manufacturer has three machine operators A, B and C. The first operator A produces 1% of defective items, whereas the other two operators B and C produces 5% and 7% defective items respectively. A is on the job for 50% of the time, B on the job 30% of the time and C on the job for 20% of the time. All the items are put into one stockpile and then one item is chosen at random from this and is found to be defective. What is the probability that it was produced by A ? 29. 5 10 A B 2 A 2 B 1 - 3 8 A ` 15 B ` 10 A B , ? A manufacturer has employed 5 skilled men and 10 semi-skilled men and makes two models A and B of an article. The making of one item of model A requires 2 hours work by a skilled man and 2 hours work by a semi-skilled man. One item of model B requires 1 hour by a skilled man and 3 hours by a semi-skilled man. No man is expected to work more than 8 hours per day. The manufacturer s profit on an item of model A is ` 15 and on an item of model B is ` 10. How many of items of each model should be made per day in order to maximize daily profit ? Formulate the above LPP and solve it graphically and find the maximum profit. __________ 65/1/1 11 [P.T.O. 65/1/1 12

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