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ISC Class XII Prelims 2020 : Select (Don Bosco School, Park Circus, Kolkata)

8 pages, 23 questions, 1 questions with responses, 1 total responses,

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SET 1 Series : GBM/1/C Code No. 65/1/1 Roll No. Candidates must write the Code on the title page of the answer-book. - 8 - - - - 29 , - 15 - 10.15 10.15 10.30 - - Please check that this question paper contains 8 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. MATHEMATICS Time allowed : 3 hours (i) (ii) (iii) (iv) (v) 65/1/1 100 Maximum Marks : 100 29 , , 4 8 11 6 , 3 3 , 1 [P.T.O. General Instructions : (i) All questions are compulsory. (ii) This question paper consists of 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 3 questions of four marks each and 3 questions of six marks each. You have to attempt only one of the alternatives in all such questions. (v) Use of calculators is not permitted. You may ask for logarithmic tables, if required. SECTION A 1 4 1 Question numbers 1 to 4 carry 1 mark each. 1. A B : 3 2 2 4 (AB) Let A and B are matrices of order 3 2 and 2 4 respectively. Write the order of matrix (AB). 2. y = sin x (0, 0) - Write the equation of tangent drawn to the curve y = sin x at the point (0, 0). 3. 1 k : x(1 + log x) dx 1 Find : x(1 + log x) dx 4. a b b a Write the angle between the vectors a b and b a. 65/1/1 2 SECTION B 5 12 2 Question numbers 5 to 12 carry 2 marks each. 5. R2 R2 + R1 2 3 1 0 8 3 1 4 2 1 9 4 In the following matrix equation use elementary operation R2 R2 + R1 and write the equation thus obtained. 2 3 1 0 8 3 1 4 2 1 9 4 6. k k , 2 x 3x 10 f ( x) ,x 2 x 2 k , x 2 x = 2 Find the value of k for which the function 2 x 3x 10 f ( x) ,x 2 x 2 k , x 2 is continuous at x = 2. 7. r, 3 / h, 2 / r = 9 h = 6 , k The radius r of a right circular cone is decreasing at the rate of 3 cm/minute and the height h is increasing at the rate of 2 cm/minute. When r = 9 cm and h = 6 cm, find the rate of change of its volume. 8. k : Find : 9. x 2 2 x dx x 2 2 x dx y2 = 4ax k Find the differential equation of the family of curves y2 = 4 ax. 65/1/1 3 [P.T.O. dy 2y e 3 x k : dx Find the general solution of the differential equation dy 2y e 3 x dx 10. 11. 10^i + 3^j, 12^j 5^j ^i + 11^j , , k If the points with position vectors 10^i + 3^j, 12^i 5^j and ^i + 11^j are collinear, find the value of . 12. 1200 200 80 ` 400 ` 200 ` 3000 A firm has to transport atleast 1200 packages daily using large vans which carry 200 packages each and small vans which can take 80 packages each. The cost for engaging each large van is ` 400 and each small van is ` 200. Not more than ` 3,000 is to be spent daily on the job and the number of large vans cannot exceed the number of small vans. Formulate this problem as a LPP given that the objective is to minimize cost. SECTION C 13 23 4 Question numbers 13 to 23 carry 4 marks each. 13. tan 1 Prove that : tan 1 : a 14. 1 x 1 x 1 cos 1 x, 1 x 1 x 4 2 1 x 1 x 1 cos 1 x, 1 x 1 x 4 2 1 x 1 2 b y c z a x b c z 0 a x b y c 1 x 1 2 a b c x y z x, y, z 0 A k 1 2 A 2 1 65/1/1 4 k b y c z a If a x b c z 0 , then using properties of determinants, find the value of a x b y c a b c , where x, y, z 0. x y z OR Using elementary operations, find the inverse of the following matrix A 1 2 . A 2 1 15. x = a (cos + sin ) y = a (sin cos ) , d2y dx 2 If x = a (cos + sin ) and y = a (sin cos ), then find 16. y = cos (x + y), 2 x 0 k d2y . dx 2 - k x + 2y = 0 Find the equation of tangent to the curve y = cos (x + y), 2 x 0 , that is parallel to the line x + 2y = 0. 17. k : 2 x 5 dx 3 x 13 x 10 /4 k : 0 Find : 3x 2 1 dx cos x 4 sin 2 x 2 x 5 dx 13 x 10 OR /4 Evaluate : 0 18. k : Find : 65/1/1 1 dx cos x 4 sin 2 x 2 x 2dx ( x 1)( x 2 1) x 2dx ( x 1)( x 2 1) 5 [P.T.O. 19. k : y dy y x cos y cos x x dx x Find the general solution of the following differential equation : y dy y x cos y cos x x dx x 20. A, B, C D : ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ 4i + 3j + 3k, 5i + xj + 7k, 5i + 3j 7i + 6j + k , , x k ^ ^ ^ ^ ^ ^ ^ ^ If four points A, B, C and D with position vectors 4 i + 3 j + 3k, 5i + xj + 7k, 5i + 3 j ^ ^ ^ and 7 i + 6 j + k respectively are coplanar, then find the value of x. 21. p k 1 x 7 y 14 z 3 3 2p 1 7 7 x 5 y 11 z 3p 1 7 x + y + z = 1 2x + 3y + 4z = 5 k y- : z : Find the value of p so that the lines 1 x 7 y 14 z 3 7 7 x 5 y 11 z and 3 2p 1 3p 1 7 are at right angles OR Find the equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 and twice of its y-intercept is equal to three times its z-intercept. 22. : : z = 6x + 3y 4 x y 80 x 5y 115 : 3x 2y 150 x 0, y 0 Solve the following Linear Programming problem graphically : Minimize : z = 6x + 3y 4 x y 80 x 5y 115 Subject to the constraints : 3x 2y 150 x 0, y 0 65/1/1 6 23. 60 A : B : C : A , 30 B C A 0.002 B 0.02 C 0.20 , k C C ? There are three categories of students in a class of 60 students : A : Very hard working students B : Regular but not so hard working C : Careless and irregular 10 students are in category A, 30 in category B and rest in category C. It is found that probability of students of category A, unable to get good marks in the final year examination is, 0.002, of category B it is 0.02 and of category C, this probability is 0.20. A student selected at random was found to be the one who could not get good marks in the examination. Find the probability that this student is of category C. What values need to be developed in students of category C ? SECTION D 24 29 6 Question numbers 24 to 29 carry 6 marks each. 24. f ( x) 4x 3x 4 f : 4 3 f : Range of f (f ) : f 4 3 4 3 f 1 k A = A (a, b) (c, d) = (a + c, b + d) A , , k 4 3 4x . Show that, in 3x 4 4 4 f : 3 Range of f, f is one-one and onto. Hence find f 1. Range f 3 . OR Let A = and be the binary operation on A defined by (a, b) (c, d) = (a + c, b + d). Show that is commutative and associative. Find the identity element for on A, if any. Let f : 65/1/1 be a function defined as f ( x) 7 [P.T.O. 25. 26. 2 1 1 8 10 A k , : 1 0 A 1 2 5 3 4 9 22 15 2 1 1 8 10 Find matrix A, if 1 0 A 1 2 5 3 4 9 22 15 k f(x) = sin x + cos x, 0 x 2 f h , : k Find the intervals in which the function f given by f(x) = sin x + cos x, 0 x 2 is strictly increasing or strictly decreasing. OR Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height h and semi-vertical angle , is one-third that of the cone. Hence find the greatest volume of the cylinder. 27. {(x, y) : y2 4x, 4x2 + 4y2 9} k Using integration find the area of the region {(x, y) : y2 4x, 4x2 + 4y2 9}. 28. x 8 y 19 z 10 3 16 7 k x 38 y 29 z 5 3 8 5 Find the equation of plane containing the lines x 38 y 29 z 5 . 3 8 5 29. x 8 y 19 z 10 3 16 7 and 8 k : (i) 5 (ii) 6 (iii) 6 52 - k A fair coin is tossed 8 times, find the probability of (i) exactly 5 heads (ii) at least six heads (iii) at most six heads OR Three cards are drawn successively with replacement from a well shuffled pack of 52 cards. Find the mean and variance of the number of red cards. __________ 65/1/1 8

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