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CBSE Class 12 Board Exam 2016 : Mathematics (Series S)

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SET-1 Series ONS U . Code No. U . 65/1/S U U U- S - c U U Roll No. Candidates must write the Code on the title page of the answer-book. U - c U 12 - U U U U U- S - c U U U - 26 U M U , - 15 U - U q 10.15 10.15 10.30 U - U U - S U U Please check that this question paper contains 12 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 26 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 U 3 U 100 Time allowed : 3 hours 65/1/S Maximum Marks : 100 1 P.T.O. (i) (ii) U - (iii) U 1 - 6 - U U 1 U (iv) U 26 7 - 19 - U I U U 4 20 - 26 - U II U U 6 U (v) U U (vi) U U U General Instructions : (i) All questions are compulsory. (ii) Please check that this question paper contains 26 questions. (iii) Questions 1 - 6 in Section A are very short-answer type questions carrying 1 mark each. (iv) Questions 7 - 19 in Section B are long-answer I type questions carrying 4 marks each. (v) Questions 20 - 26 in Section C are long-answer II type questions carrying 6 marks each. (vi) Please write down the serial number of the question before attempting it. 65/1/S 2 U - SECTION - A 1 6 1 Question numbers 1 to 6 carry 1 mark each. 1. U a b 4 1 a5 , b5 3 2 U 1 a3b 5 3 U , a.b 1 1 4 If vectors a and b are such that a 5 , b 5 and a 3 b 5 3 , then find 2 3 a.b . 2. a b ? a b , a2 2 b If a and b are unit vectors, then what is the angle between a and b for a 2 2 b to be a unit vector ? 3. r . (2 i 2 3 j 1 6 k ) 2 4 5 0 r . (6 i 2 9 j 118 k )130 5 0 U Find the distance between the planes r . (2 i 23 j 1 6 k )2 4 5 0 and r . (6 i 2 9 j 118 k )130 5 0 . 65/1/S 3 P.T.O. 4. A ?A ? 5 5 , ?AA T ? If A is a square matrix such that ?A?5 5 , write the value of ?AA T ? . 5. 1 2 A 5 3 21 1 24 3 22 B5 , ?AB? 2 1 24 , find ?AB?. and B 5 3 21 3 22 1 If A 5 6. 0 3 2 25 A5 0 4a KA 5 28 5b , k a 0 3 0 4a and KA 5 find the values of k and a. 2 25 28 5b If A 5 U - SECTION - B 7 19 4 Question numbers 7 to 19 carry 4 marks each. 7. ( sin 2x )x1 sin21 3x x 1 1 x2 2 1 2 x2 tan21 1 1 x 2 1 1 2 x2 65/1/S cos21x2 4 Differentiate ( sin 2x )x1 sin21 3x with respect to x. OR 1 1 x2 2 1 2 x2 Differentiate tan21 1 1 x2 1 1 2 x2 8. with respect to cos21x2. p k sin 2 ( x 11), x [ 0 x50 f ( x )5 tan x 2 sin x , x > 0 x3 U , k p k sin 2 ( x 11), x [ 0 Find k, if f ( x )5 is continuous at x50. tan x 2sin x , x > 0 x3 9. ay25x3 x am2 , U U Find equation of normal to the curve ay25x3 at the point whose x coordinate is am2. 10. 12sin x sin x (11sin x) dx Find : 65/1/S 12sin x sin x (11sin x) dx 5 P.T.O. 11. [log (log x )1 Find : [log (log x )1 p 12. 1 ( log x )2 1 ( log x )2 ] dx ] dx sin2 x dx sin x 1 cos x 0 2 1 cot21 (1 2 x 1 x2 ) dx 0 p sin2 x sin x 1 cos x dx 0 Evaluate : 2 OR 1 ( ) Evaluate : cot21 1 2 x 1 x2 dx 0 13. U ( x 1 1) dy 2 y 5 e3x ( x 1 1)3 dx Solve the differential equation : ( x 1 1) dy 2 y 5 e3x ( x 1 1)3 dx 65/1/S 6 14. U x x 2y e y dx 1 y 22x e y dy 50 x x y y Solve the differential equation : 2y e dx 1 y 22x e dy 50 15. U U, l U U Z 50 . 50 . , U U 10 . U 20 . U 5300 2 U U U ? Ishan wants to donate a rectangular plot of land for a school in his village. When he was asked to give dimensions of the plot, he told that if its length is decreased by 50 m and breadth is increased by 50 m, then its area will remain same, but if length is decreased by 10 m and breadth is decreased by 20 m, then its area will decrease by 5300 m2. Using matrices, find the dimensions of the plot. Also give reason why he wants to donate the plot for a school. 16. h 3 17 p 2sin21 2 tan21 5 5 31 4 U x 3 cos (tan21x )5sin cot21 4 3 17 p Prove that 2sin21 2 tan21 5 5 31 4 OR 21 3 21 Solve the equation for x : cos (tan x )5sin cot 4 65/1/S 7 P.T.O. 17. A U B A 3 U 4 B 4 U 3 U ( S ) U B There are two bags A and B. Bag A contains 3 white and 4 red balls whereas bag B contains 4 white and 3 red balls. Three balls are drawn at random (without replacement) from one of the bags and are found to be two white and one red. Find the probability that these were drawn from bag B. 18. U p, q, r, s a , b c b 5 s i 13 j 1 4 k 5 6 a5b1 c a 5 p i 1 q j 1 r k, c 53 i 1 j 2 2 k Given that vectors a , b , c form a triangle such that a 5 b 1 c . Find p, q, r, s such that area of triangle is 5 6 where a 5 p i 1 q j 1 r k, b 5s i 13 j 14 k and c 53 i 1 j 22 k . 19. A (3, 2, 1), B (4, 2, 22) C (6, 5, 21) U U l A (3, 2, 1), B (4, 2, 22), C (6, 5, 21) D (l, 5, 5) U n 5 i 1 j 13k U U 65/1/S 4 11 r 5(2i 22 j 23 k )1l(3 i 14 j 13 k ) 8 U Find the equation of plane passing through the points A (3, 2, 1), B (4, 2, 22) and C (6, 5, 21) and hence find the value of l for which A (3, 2, 1), B (4, 2, 22), C (6, 5, 21) and D (l, 5, 5) are coplanar. OR Find the co-ordinates of the point where the line r 5(2i 22 j 23 k )1l(3 i 14 j 13 k ) meets the plane which is perpendicular to 4 from origin. 11 the vector n 5 i 1 j 13k and at a distance of U - SECTION - C 20 26 6 Question numbers 20 to 26 carry 6 marks each. 20. f (x)54x2112x115 m U U f : N S ( S, f U U ) f f21(31) f21(87) f : N N Let f : N N be a function defined as f (x)54x2112x115. Show that f : N S is invertible (where S is range of f). Find the inverse of f and hence find f21(31) and f21(87). 21. U Z U h : (b 1 c)2 a2 bc (c 1 a)2 b2 ca 5(a 2 b) (b 2 c) (c 2 a) (a 1 b 1 c) (a2 1 b2 1 c2 ) (a 1 b)2 c2 ab U U 2 21 3 A 5 25 3 1 23 2 3 65/1/S 9 P.T.O. Using properties of determinants, prove that : (b 1 c)2 a2 bc (c 1 a)2 b2 ca 5(a 2 b) (b 2 c) (c 2 a) (a 1 b 1 c) (a2 1 b2 1 c2 ) (a 1 b)2 c2 ab OR Using elementary row operations, find the inverse of the following matrix : 2 21 3 A 5 25 3 1 23 2 3 22. U U f(x)5x428x3122x2224x121 U U U U O f (x)5sec x1log cos2 x, 0 < x < 2p Determine the intervals in which the function f (x)5x428x3122x2224x121 is strictly increasing or strictly decreasing. OR Find the maximum and minimum values of f (x)5sec x1log cos 2 x, 0 < x < 2p. 23. U {(x, y) : y2[6ax x21y2[16a2} Using integration find the area of the region {(x, y) : y2[6ax and x21y2[16a2} 65/1/S 10 24. U U x 21 y 11 z 5 5 21 2 3 x y 22 z 11 5 5 4 22 6 c U U U , U x22 y 21 z 22 5 5 3 1 5 c U U ? Find the equation of the plane containing two parallel lines x 21 y 11 z x y 22 z 11 5 5 5 and 5 . Also, find if the plane thus obtained 21 4 22 6 2 3 contains the line 25. x22 y 21 z 22 5 5 or not. 3 1 5 U 80 U A U 100 l F1 U F2 ` 5 U ` 6 F1 U A 4 U 3 , F2 U A 3 U 6 U M M U l Z U U U A diet is to contain at least 80 units of Vitamin A and 100 units of minerals. Two foods F1 and F2 are available costing ` 5 per unit and ` 6 per unit respectively. One unit of food F1 contains 4 units of vitamin A and 3 units of minerals whereas one unit of food F2 contains 3 units of vitamin A and 6 units of minerals. Formulate this as a linear programming problem. Find the minimum cost of diet that consists of mixture of these two foods and also meets minimum nutritional requirement. 65/1/S 11 P.T.O. 26. U Z U ( S ) Z X U U U , X U U U Three numbers are selected at random (without replacement) from first six positive integers. If X denotes the smallest of the three numbers obtained, find the probability distribution of X. Also find the mean and variance of the distribution. 65/1/S 12

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