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

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SET-1 Series ONS U . Code No. U . 65/1/E 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 11 - 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 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 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/E Maximum Marks : 100 1 P.T.O. (i) (ii) U - (iii) U 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/E 2 U - SECTION - A 1 6 Question numbers 1 to 6 carry 1 mark each. a2b b2c 1. b2c c2a c2a a2b c2a a2b b2c a2b b2c c2a Write the value of b2c c2a a2b . c2a a2b b2c 2. 22 3 1 A5 2 5 24 2 3 B5 4 5 2 1 BA5(bij) , b211b32 2 3 22 3 1 If A5 and B5 4 5 and BA5(bij), find b211b32. 2 5 24 2 1 3. 233 U 1 2 Write the number of all possible matrices of order 233 with each entry 1 or 2. 4. (a, b, g) XZ - Write the coordinates of the point which is the reflection of the point (a, b, g) in the XZ-plane. 65/1/E 3 P.T.O. 5. S , S a2b , U U 1:3 a 13 b U U Find the position vector of the point which divides the join of points with position vectors a 13 b and a 2 b internally in the ratio 1 : 3. 6. ? a ?54, ? b ?53 a . b 56 3 , ? a3b ? If ? a ?54 , ? b ?53 and a . b 56 3, then find the value of ? a 3 b ? . U - SECTION - B 7 19 4 Question numbers 7 to 19 carry 4 marks each. 7. x 22x 1 21 x , x > 0 . 5 tan 2 21x 2 tan21 h 3 17 p 2sin21 2tan21 5 . 5 31 4 22x 1 21 x , x > 0 . 5 tan 1 x 2 2 2 Solve for x : tan21 OR 3 17 p Prove that 2sin21 2tan21 5 . 5 31 4 65/1/E 4 8. U U U U 8 , ` 10 U 16 U , ` 10 U m U U U ? On her birthday Seema decided to donate some money to children of an orphanage home. If there were 8 children less, every one would have got ` 10 more. However, if there were 16 children more, every one would have got ` 10 less. Using matrix method, find the number of children and the amount distributed by Seema. What values are reflected by Seema s decision ? 9. x5ecos2t U y5esin2t , S h dy y log x 52 . dx x log y [0, p] f(x)52 sinx1sin2x If x5ecos2t and y5esin2t, prove that dy y log x 52 . dx x log y OR Verify Mean Value theorem for the function f(x)52 sinx1sin2x on [0, p]. 10. f 1x e 21 , x 0 f ( x ) 5 e1 x 11 21 , x50 x50 U Show that the function f given by : 1x e 21 f ( x )5 e1 x 11 21 , if x 0 , if x50 is discontinuous at x50. 65/1/E 5 P.T.O. 11. y5 5x2 3 25 S U U U 4x22y1550 U Find the equation of the tangent line to the curve y5 5x2 3 25, which is parallel to the line 4x22y1550. 5 12. {?x21?1?x22?1?x23?} dx 1 : p x sinx 113cos2x dx 0 5 Evaluate : {?x21?1?x22?1?x23?} dx 1 OR Evaluate : p x sinx 113cos2x dx 0 13. dx 2 2x 112 (x 11)(x 1 4) Find : 14. 2x 11 (x2 11)(x2 1 4) dx (3x1 5) 5 1 4 x 2 2x 2 d x Find : (3x 1 5) 51 4 x 2 2x2 dx 65/1/E 6 15. U x dy 1 y 2 x 1 xy cotx 5 0; x 0. dx Solve the differential equation : x 16. dy 1 y 2 x1 xy cotx 5 0; x 0. dx U (x213xy1y2) dx2x2 dy50, x51 y50 Solve the differential equation : (x213xy1y2) dx2x2 dy50 given that y50, when x51. 17. ^ ^ ^ a1b U a2b b53 i1j22 k a1b a2b ^ ^ ^ a52 i2j13 k U U ^ ^ ^ Find the angle between the vectors a1b and a2b if a52 i2j13 k and ^ ^ ^ b53 i1j22 k , and hence find a vector perpendicular to both a1b and a2b . 65/1/E 7 P.T.O. 18. U y2 1 x2 1 z1 1 5 5 3 21 0 y x2 4 z 11 5 5 2 0 3 U U U Show that the lines y2 1 y x2 1 z1 1 x2 4 z 11 5 5 5 5 and intersect. 3 21 0 2 0 3 Find their point of intersection. 19. 7 U 4 U 4 l U U U U 2 , U U U C X U X 0 1 2 3 4 P(X) C 2C 2C 3C C 2 5 2C 6 2 2 7C 1C U U A committee of 4 students is selected at random from a group consisting of 7 boys and 4 girls. Find the probability that there are exactly 2 boys in the committee, given that at least one girl must be there in the committee. OR A random variable X has the following probability distribution : X 0 1 2 3 4 P(X) C 2C 2C 3C C 2 5 2C 6 2 Find the value of C and also calculate mean of the distribution. 65/1/E 8 2 7C 1C U - SECTION - C 20 26 6 Question numbers 20 to 26 carry 6 marks each. 20. R, A3A U A5{1, 2, 3, .........., 10} , U U (a, b) R (c, d) a1d5b1c ; a, b, c, d e A [(3, 4)] Show that the relation R defind by (a, b) R (c, d) a1d5b1c on the A3A, where A5{1, 2, 3, .........., 10} is an equivalence relation. Hence write the equivalence class [(3, 4)]; a, b, c, d e A. 21. U Z x a1x a2x a2x : a2x a1x a2x 5 0 a2x a2x a1x U U U 3 23 4 A 5 2 23 4 0 21 1 3x23y14z521, 2x23y14z520, 2y1z55 a1x a2x a2x Solve for x : a2x a1x a2x 5 0 , using properties of determinants. a2x a2x a1x OR 3 23 4 Using elementary row operations find the inverse of matrix A 5 2 23 4 0 21 1 and hence solve the following system of equations 3x23y14z521, 2x23y14z520, 2y1z55. 65/1/E 9 P.T.O. 22. h U a , - U 4 ph3tan2 a 27 f ( x) 5 4 sinx 2 x; 0 x 2p 21 cosx U U U U O 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 a is one-third that of the cone and the greatest volume of the cylinder is 4 ph3tan2 a . 27 OR Find the intervals in which the function f ( x ) 5 4 sinx 2 x; 0 x 2p is 21 cosx strictly increasing or strictly decreasing. 23. x- x21y259 (21, 2 2 ) U S U m U Using integration, find the area of the triangle formed by negative x-axis and tangent and normal to the circle x21y259 at (21, 2 2 ) . 24. P(4, 3, 2) x12y13z52 U U P Find the coordinates of the foot of perpendicular and perpendicular distance from the point P(4, 3, 2) to the plane x12y13z52. Also find the image of P in the plane. 65/1/E 10 C U U U 9 U A U U 25. A, B A, B and C throw a pair of dice in that order alternately till one of them gets a total of 9 and wins the game. Find their respective probabilities of winning, if A starts first. 26. U S U U A B U A U S U U ` 360 B U S U U ` 120 300 S U U ` 72,000 U B U S U U , A U S U U 200 A U S U U U ` 100 U B U S U U U ` 50 S U M m U A company manufactures two types of cardigans : type A and type B. It costs ` 360 to make a type A cardigan and ` 120 to make a type B cardigan. The company can make at most 300 cardigans and spend at most ` 72,000 a day. The number of cardigans of type B cannot exceed the number of cardigans of type A by more than 200. The company makes a profit of ` 100 for each cardigan of type A and ` 50 for every cardigan of type B. Formulate this problem as a linear programming problem to maximise the profit to the company. Solve it graphically and find maximum profit. 65/1/E 11 P.T.O.

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