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CBSE Class 12 Board Exam 2015 : Mathematics

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SET 1 . Series : SSO/C Code No. . 65/1 - - Roll No. Candidates must write the Code on the title page of the answer-book. - 8 - - - - 26 , - 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 26 questions. Please write down the Serial Number of the question before attempting it. 15 minutes 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. : 3 ] MATHEMATICS Time allowed : 3 hours ] [ : 100 [ Maximum Marks : 100 : (i) (ii) - 26 (iii) - 1 6 - 1 (iv) - 7 19 - I 4 (v) - 20 26 - II 6 (vi) 65/1 1 [P.T.O. General Instructions : (i) All questions are compulsory. (ii) Please check that this Question Paper contains 26 Questions. (iii) Questions 1 to 6 in Section-A are Very Short Answer Type Questions carrying one mark each. (iv) Questions 7 to 19 in Section-B are Long Answer I Type Questions carrying 4 marks each. (v) Questions 20 to 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. SECTION A 1 6 1 Question numbers 1 to 6 carry 1 mark each. 1. 3 a + 2 b a = ^i + ^j 2k^ ^ ^ ^ b = 2i 4 j + 5k ^ 1 ^ ^ Write the direction ratio s of the vector 3 a + 2 b where a = i + j 2k and ^ ^ ^ b = 2i 4j + 5k. 2. a = 2^i + 3^j + 2k^ ^ ^ ^ b = 2i + 2j + k ^ ^ ^ 1 ^ ^ ^ Find the projection of the vector a = 2i + 3j + 2k on the vector b = 2i + 2j + k. 3. (1, 2, 3) ^ ^ ^ r ( i + 2j 5k) + 9 = 0 Write the vector equation of the line passing through (1, 2, 3) and perpendicular to the ^ ^ 1 ^ plane r ( i + 2j 5k) + 9 = 0. 4. /2 < x < x 3 2 sin x 1 2 sin x 1 3 2 sin x In the interval /2 < x < , find the value of x for which the matrix 1 2 sin x is singular. 65/1 2 5. dy dx = x3 e 2y Find the solution of the differential equation 6. dy 3 2y =x e . dx dy + y = e 2 x dx Write the integrating factor of the differential equation dy x + y = e 2 x . dx 1 x 1 SECTION B 7 19 4 Question numbers 7 to 19 carry 4 marks each. 7. ` 35,000 - 8% , 10% . . . ( ) ` 35,000 ` 3,200 ? ? 4 A trust fund has ` 35,000 is to be invested in two different types of bonds. The first bond pays 8% interest per annum which will be given to orphanage and second bond pays 10% interest per annum which will be given to an N.G.O. (Cancer Aid Society). Using matrix multiplication, determine how to divide ` 35,000 among two types of bonds if the trust fund obtains an annual total interest of ` 3,200. What are the values reflected in this question ? 8. 2 4 6 5 A = 7 3 1 2 4 - 4 2 4 6 5 as the sum of a symmetric and skew Express the matrix A = 7 3 1 2 4 symmetric matrix. /OR 1 2 2 3 A = 1 4 , B = 1 3 , (AB) 1 = B 1 A 1 1 2 2 3 , verify that (AB) 1 = B 1 A 1. ,B= If A = 1 3 1 4 65/1 3 [P.T.O. 9. x : a+x a x a x a x a+x a x a x a x a+x 4 =0 a+x Using properties of determinants, solve for x : a x a x a x a+x a x a x a x a+x =0 /4 10. . log (1 + tan x) dx . : 4 0 /4 . log (1 + tan x) dx. Evaluate . 0 11. : . . (x2 x dx + 1) (x 1) 4 1 2 : . sin 1 x dx . (1 x2)3/2 0 . x Find dx. 2 . (x + 1) (x 1) OR 1 2 . sin 1 x Find dx. . (1 x2)3/2 0 12. 52 4 (i) 4 ? (ii) 2 ? 65/1 4 4 Four cards are drawn successively with replacement from a well shuffled deck of 52 cards. What is the probability that (i) all the four cards are spades ? (ii) only 2 cards are spades ? OR A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability distribution of number of successes. Hence find the mean of the distribution. 13. b + c ,d] = [a , b, d] : [ a , + [a , c , d] 4 Prove that [ a , b + c , d ] = [ a , b , d ] + [ a , c , d ] 14. r = 2^i 5^j + ^ ^ ^ ^ ^ ^ ^ ^ ^ k + (3 i + 2j + 6k) r = 7 i 6k + ( i + 2j + 2k) 4 Find the shortest distance between the following lines : ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ r = 2i 5j + k + (3 i + 2j + 6k) and r = 7i 6 k + ( i + 2j + 2k) 15. 2tan 1 2 + tan 1 7 = sin 1 31 25 2 1 1 4 x : tan 1 1 + x = 2 1 x 1 tan 1 x, x > 0 1 1 31 Prove that 2 tan 1 + tan 1 = sin 1 2 7 25 2 OR 1 x 1 Solve for x : tan 1 = tan 1 x, x > 0 1 + x 2 16. f(x) = (x2 4x + + 2), x 0 x=0 6 , x > 0 : x=0 4 (x2 + 2), if x 0 is continuous at + 6 , if x > 0 x = 0 ? Hence check the differentiability of f(x) at x = 0. For what value of the function defined by f(x) = 4x 65/1 5 [P.T.O. 17. dy If x = aet (sin t + cos t) and y = aet (sin t cos t), prove that 18. y = Aemx + Benx , If y = Aemx + Benx, show that 19. x+y x = aet (sin t + cos t) y = aet (sin t cos t) , dx = x y . : . . . Find . 4 dy x + y . = dx x y d2y dy (m + n) + mny = 0 2 dx dx 4 dy d2y (m + n) + mny = 0. 2 dx dx x+3 dx 5 4x 2x2 4 x+3 dx. 5 4x 2x2 SECTION C 20 26 6 Question numbers 20 to 26 carry 6 marks each. 20. A = {1, 2, 3, 4, 5} R = {(a, b) : |a b|, 2 } 6 Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b) : |a b| is divisible by 2} is an equivalence relation. Write all the equivalence classes of R. 21. {(x, y) : y2 4x, 4x2 + 4y2 9} 4y = 3x2 2y = 3x + 12 Find the area of the region {(x, y) : y2 4x, 4x2 + 4y2 9}, using integration. OR Using integration, find the area enclosed by the parabola 4y = 3x2 and the line 2y = 3x + 12. 65/1 6 6 22 y= 4 y x sin2 x y dx + x dy = 0 , x=1 , 6 dy dx 3y cot x = sin 2x , , x = 2 , y = 2 Solve the differential equation y x sin2 x y dx + x dy = 0 given y = 4 when x = 1. OR Solve the differential equation 23. dy 3y cot x = sin 2x given y = 2 when x = . dx 2 ^ ^ ^ r (2 i + 5j + 3k) = 9 ^ ^ ^ z- r (2 i + 2j 3k) = 7 x- : 6 Find the vector and cartesian equations of the plane passing through the line of intersection of the planes ^ ^ ^ ^ ^ ^ r (2 i + 2j 3k) = 7, r (2 i + 5j + 3k) = 9 such that the intercepts made by the plane on x-axis and z-axis are equal. 24 2 3 5 5 1 3 , ? 6 In answering a question on a multiple choice test, a student either knows the answer or 3 2 guesses. Let be the probability that he knows the answer and be the probability that 5 5 he guesses. Assuming that a student who guesses at the answer will be correct with 1 probability , what is the probability that the student knows the answer given that he 3 answered it correctly ? 65/1 7 [P.T.O. 25 A 2 B 3 3 A 2 B ` 24 ` 18 10 , ? 6 A manufacturer produces nuts and bolts. It takes 2 hours work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and 2 hours on machine B to produce a package of bolts. He earns a profit of ` 24 per package on nuts and ` 18 per package on bolts. How many packages of each should be produced each day so as to maximize his profit, if he operates his machines for at the most 10 hours a day. Make an L.P.P. from above and solve it graphically ? 26 x , 3, x 2x , x x The sum of surface areas of a sphere and a cuboid with sides , x and 2x, is constant. 3 Show that the sum of their volumes is minimum if x is equal to three times the radius of sphere. __________ 65/1 8 6

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