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

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SET-1 Series ONS U . Code No. U . 65/1/N 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/N Maximum Marks : 100 1 P.T.O. (i) (ii) U - (iii) - 1 - 6 - U U 1 U (iv) U- U 7 - 19 - U I (v) U- U 20 - 26 - U II (vi) U U U 26 U U 4 U U 6 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/N 2 U - SECTION - A 1 6 1 Question numbers 1 to 6 carry 1 mark each. 1. sina cosa A5 2 sin a cosa , A 1 A T 5 2 I2 0<a< p 2 c U U a , AT, A U sina cosa p If A 5 , find a satisfying 0 < a < 2 sin a cos a 2 when A 1 A T 5 2 I2 ; where AT is transpose of A. 2. A 333 ?3A?5k?A? , k If A is a 333 matrix and ?3A?5k?A?, then write the value of k. 3. k U U x1y1z52 2x1y2z53 3x12y1kz54 m ? For what values of k, the system of linear equations x1y1z52 2x1y2z53 3x12y1kz54 has a unique solution ? 65/1/N 3 P.T.O. 4. ( ) r . 2 i 1 j 2 k 255 0 m U U U U ( ) Write the sum of intercepts cut off by the plane r . 2 i 1 j 2 k 2 55 0 on the three axes. 5. l U m , ( i 13 j 19k )3(3 i 2l j 1mk )5 0 Find l and m if ( i 13 j 19k )3(3 i 2l j 1mk )5 0 . 6. a 54 i 2 j 1 k a1b U , , b 52 i 22 j 1 k U If a 54 i 2 j 1 k and b 52 i 22 j 1 k , then find a unit vector parallel to the vector a 1 b . U - SECTION - B 7 19 4 Question numbers 7 to 19 carry 4 marks each. 7. x tan21(x21)1tan21x1tan21(x11)5tan213x. h 65/1/N 6 x 2 8 x3 1 4x 21 2 tan21 5 tan 2x ; ?2 x? < 3 . 2 2 1212x 12 4 x tan21 4 Solve for x : tan21(x21)1tan21x1tan21(x11)5tan213x. OR 6 x 2 8 x3 1 4x 21 2 tan21 5 tan 2x ; ?2 x? < 3 . 2 2 1212x 12 4 x Prove that tan21 8. U S U 10 U 3 c U U U ` 145 , 3 U 10 c U U U ` 180 U c U U U - U U l U S U 5 c U U U ` 2 c U U l ? - ? A typist charges ` 145 for typing 10 English and 3 Hindi pages, while charges for typing 3 English and 10 Hindi pages are ` 180. Using matrices, find the charges of typing one English and one Hindi page separately. However typist charged only ` 2 per page from a poor student Shyam for 5 Hindi pages. How much less was charged from this poor boy ? Which values are reflected in this problem ? 9. sin(a 1 1)x 1 2 sinx , x <0 x f (x) 5 , x50 2 1 1 bx2 1 , x >0 x x50 U a b sin(a 1 1)x 1 2 sinx , x <0 x 2 , x50 If f ( x ) 5 1 1 bx2 1 , x >0 x is continuous at x50, then find the values of a and b. 65/1/N 5 P.T.O. 10. x cos(a1y)5cosy h , h sina dy cos2 (a 1 y ) 5 dx sina d2 y dy y 1 sin2(a 1 ) 50 . dx dx2 6x 2 4 1 2 4 x2 y 5 sin21 5 , dy dx dy cos2 (a 1 y ) If x cos(a1y)5cosy then prove that 5 . dx sina Hence show that sina d2 y dy y 1 sin2(a 1 ) 50. dx dx2 OR dy 21 Find if y 5 sin dx 11. 6x 2 4 1 2 4 x2 5 y5x312x24 S U U , U x114y1350 U Find the equation of tangents to the curve y5x 3 12x24, which are perpendicular to line x114y1350. 65/1/N 6 12. (2x 25) e2x (2x 23)3 dx 2 2x 1 x 1 1 dx ( x 1 1)(x 1 2) Find : (2x 25) e2x (2x 23)3 dx OR Find : 13. x2 1 x 1 1 ( x2 1 1)(x 1 2) dx 2 x 22 1 1 5 Evaluate : 14. x2 2 x2 x 22 1 1 5 ( x 1 3) dx . dx . 3 2 4 x 2 x 2 dx . Find : ( x 1 3) 3 2 4 x 2 x2 dx . 65/1/N 7 P.T.O. 15. U c U , y51 x50 , dy x 1 y cosx 52 dx 11sinx Find the particular solution of differential equation : x 1 y cosx dy 52 dx 11sinx given that y51 when x50. 16. U 2y ex/y dx1(y22x ex/y)dy50 c U , x50 , y51 Find the particular solution of the differential equation 2y ex/y dx1(y22x ex/y)dy50 given that x50 when y51. 17. U A(4, 5, 1), B(0, 21, 21), C(3, 9, 4) U D(24, 4, 4) Show that the four points A(4, 5, 1), B(0, 21, 21), C(3, 9, 4) and D(24, 4, 4) are coplanar. 18. A(21, 8, 4) B(0, 21, 3) C(2, 23, 21) U U U U BC A Find the coordinates of the foot of perpendicular drawn from the point A(21, 8, 4) to the line joining the points B(0, 21, 3) and C(2, 23, 21). Hence find the image of the point A in the line BC. 65/1/N 8 19. X 4 2 Y 3 3 U ( S ) Z Y B U - U U U 10 U A U U A A bag X contains 4 white balls and 2 black balls, while another bag Y contains 3 white balls and 3 black balls. Two balls are drawn (without replacement) at random from one of the bags and were found to be one white and one black. Find the probability that the balls were drawn from bag Y. OR A and B throw a pair of dice alternately, till one of them gets a total of 10 and wins the game. Find their respective probabilities of winning, if A starts first. U - SECTION - C 20 26 6 Question numbers 20 to 26 carry 6 marks each. 20. U Z U ( S ) Z X U X U U U Three numbers are selected at random (without replacement) from first six positive integers. Let X denote the largest of the three numbers obtained. Find the probability distribution of X. Also, find the mean and variance of the distribution. 65/1/N 9 P.T.O. 21. A5R3R U *, A (a, b) * (c, d)5(a1c, b1d) m U U m U * A * A (a, b) e A Let A5R3R and * be a binary operation on A defined by (a, b) * (c, d)5(a1c, b1d) Show that * is commutative and associative. Find the identity element for * on A. Also find the inverse of every element (a, b) e A. 22. p 0, 2 h y5 4 sinu 2u, u 2 1 cosu U U cos21 1 3 Prove that y 5 4 sinu 2 u is an increasing function of u on 2 1 cosu p 0, 2 . OR Show that semi-vertical angle of a cone of maximum volume and given slant height is cos21 1 . 3 23. U , (2, (4, 3) U (1, 2) 22), Using the method of integration, find the area of the triangular region whose vertices are (2, 22), (4, 3) and (1, 2). 65/1/N 10 24. U , ( ( ) r . i 2 2 j 13 k 24 50 ) r . 22 i 1 j 1 k 155 0 U U U m U x- U U y- U U U Find the equation of the plane which contains the line of intersection of the planes ( ( ) r . i 2 2 j 1 3 k 2 4 5 0 and ) r . 22 i 1 j 1 k 15 5 0 and whose intercept on x-axis is equal to that of on y-axis. 25. ` 50,000 U U U U ( ) A U B U , U 10% U 9% U U A ` 20,000 U U B ` 10,000 U U A U, U B U U S m U A retired person wants to invest an amount of ` 50,000. His broker recommends investing in two type of bonds A and B yielding 10% and 9% return respectively on the invested amount. He decides to invest at least ` 20,000 in bond A and at least ` 10,000 in bond B . He also wants to invest at least as much in bond A as in bond B . Solve this linear programming problem graphically to maximise his returns. 65/1/N 11 P.T.O. 26. U Z U h ( x 1 y )2 zx zy zx ( z 1 y )2 xy zy xy ( z 1 x )2 5 2xyz ( x 1 y 1 z)3 1 0 2 A 5 0 2 1 2 0 3 A326A217A1kI35O , k Using properties of determinants, prove that ( x 1 y )2 zx zy zx ( z 1 y )2 xy zy xy ( z 1 x )2 5 2xyz ( x 1 y 1 z)3 OR 1 0 2 If A 5 0 2 1 and A326A217A1kI35O find k. 2 0 3 65/1/N 12

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