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

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SET-1 Series ONS U . Code No. U . 65/1/C 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/C 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/C 2 U - SECTION - A 1 6 1 Question numbers 1 to 6 carry 1 mark each. 1. x e N U If x e N and 2. x 13 2 2 58 2 3x 2x , x x 13 2 2 5 8 , then find the value of x. 2 3 x 2x U S C2 C212C1 U U 2 1 3 1 1 0 5 2 0 2 0 21 1 Use elementary column operation C2 C212C1 in the following matrix equation : 2 1 3 1 1 0 5 2 0 2 0 21 1 3. U 232 , 1, 2 3 , Write the number of all possible matrices of order 232 with each entry 1, 2 or 3. 65/1/C 3 P.T.O. 4. S , S , U U 2 : 1 U 3 a 22 b 2 a 13 b Write the position vector of the point which divides the join of points with position vectors 3 a 22 b and 2 a 13 b in the ratio 2 : 1. 5. a 52 i 1 j 12 k b 5j 1k U Write the number of vectors of unit length perpendicular to both the vectors a 52 i 1 j 12 k and b 5 j 1 k . 6. U , x, y U z- U 3, 24 U 2 U U Find the vector equation of the plane with intercepts 3, 24 and 2 on x, y and z-axis respectively. U - SECTION - B 7 19 4 Question numbers 7 to 19 carry 4 marks each. 7. U A(3, 4, 1) U B(5, 1, 6) U U XZ U U U XZ Find the coordinates of the point where the line through the points A(3, 4, 1) and B(5, 1, 6) crosses the XZ plane. Also find the angle which this line makes with the XZ plane. 65/1/C 4 8. U 2 i 2 4 j 25 k U 2 i 1 2 j 13 k Z U Z U U The two adjacent sides of a parallelogram are 2 i 24 j 25 k and 2 i 12 j 13 k . Find the two unit vectors parallel to its diagonals. Using the diagonal vectors, find the area of the parallelogram. 9. ` 5 4 ` 1 U 3 U U U U U c m U / U U 4 U S U ? In a game, a man wins ` 5 for getting a number greater than 4 and loses ` 1 otherwise, when a fair die is thrown. The man decided to throw a die thrice but to quit as and when he gets a number greater than 4. Find the expected value of the amount he wins/loses. OR A bag contains 4 balls. Two balls are drawn at random (without replacement) and are found to be white. What is the probability that all balls in the bag are white ? 65/1/C 5 P.T.O. 10. xsinx1(sinx)cosx x y52 cos(logx)13 sin(logx) , h d2 y dy 2 x 1x 1 y 50 2 dx dx Differentiate xsinx1(sinx)cosx with respect to x. OR d2 y dy If y52 cos(logx)13 sin(logx), prove that x2 2 1 x 1 y 50 . dx dx 11. x5a sin 2t(11cos 2t) y5b cos 2t(12cos 2t) , t 5 p U 4 If x5a sin 2t(11cos 2t) and y5b cos 2t(12cos 2t), find 12. y25ax31b b (2, 3) U S U U dy dx dy p at t 5 . dx 4 y54x25 , a The equation of tangent at (2, 3) on the curve y25ax31b is y54x25. Find the values of a and b. 13. 2 4 x 2 dx x 1 x 22 x2 dx Find : 4 x 1 x2 2 2 65/1/C 6 14. p 2 2 x sinsin x 1 cosx dx 0 3 2 : x cos px dx 0 Evaluate : p 2 sin2 x sinx 1 cosx dx 0 OR Evaluate : 3 2 x cos px dx 0 15. (3x 1 1) 4 2 3x 2 2x 2 d x Find : (3x 1 1) 4 2 3x 2 2x2 dx 16. U y1x dy dy 5x2y dx dx Solve the differential equation : y1x 65/1/C dy dy 5x2y dx dx 7 P.T.O. 17. m Z U S U Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes. 18. x sin21x1sin21(12x)5cos21x y x cos21 1 cos21 5a a b , h xy y2 x2 2 2 cos a1 5sin2 a 2 2 ab a b Solve the equation for x : sin21x1sin21(12x)5cos21x OR y xy y2 x x2 cosa1 2 5sin2 a If cos21 1 cos21 5a , prove that 2 22 a b ab a b 19. US U U U ( ) U 10% U U U 12% US U ` 2,800 U US U - U U , ` 100 US U U - ? A trust invested some money in two type of bonds. The first bond pays 10% interest and second bond pays 12% interest. The trust received ` 2,800 as interest. However, if trust had interchanged money in bonds, they would have got ` 100 less as interest. Using matrix method, find the amount invested by the trust. Interest received on this amount will be given to Helpage India as donation. Which value is reflected in this question ? 65/1/C 8 U - SECTION - C 20 26 6 Question numbers 20 to 26 carry 6 marks each. 20. U A U B A 12% U U 5% S U U B 4% U U 5% S U U ^ S U 12 . . U U 12 . . S U U A ` 10 . . U B ` 8 . . m U U U U U There are two types of fertilisers A and B . A consists of 12% nitrogen and 5% phosphoric acid whereas B consists of 4% nitrogen and 5% phosphoric acid. After testing the soil conditions, farmer finds that he needs at least 12 kg of nitrogen and 12 kg of phosphoric acid for his crops. If A costs ` 10 per kg and B cost ` 8 per kg, then graphically determine how much of each type of fertiliser should be used so that nutrient requirements are met at a minimum cost. U U 5 U U U U U U S , U U U U U 21. 20 Five bad oranges are accidently mixed with 20 good ones. If four oranges are drawn one by one successively with replacement, then find the probability distribution of number of bad oranges drawn. Hence find the mean and variance of the distribution. 65/1/C 9 P.T.O. 22. P, S 2 i 13 j 1 4 k ( ) r . 2 i 1 j 1 3 k 2 26 5 0 S U P U Find the position vector of the foot of perpendicular and the perpendicular distance from the point P with position vector 2 i 13 j 14 k to the plane ( ) r . 2 i 1 j 1 3 k 2 26 5 0 . Also find image of P in the plane. 23. m U A5R2{21} U a, b e A a*b5a1b1ab m U U A * h A Show that the binary operation * on A5R2{21} defined as a*b5a1b1ab for all a, b e A is commutative and associative on A. Also find the identity element of * in A and prove that every element of A is invertible. 24. h m , r , U 6 3r , p 3 Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is 6 3 r . OR If the sum of lengths of hypotenuse and a side of a right angled triangle is given, show that area of triangle is maximum, when the angle between them is 65/1/C p . 3 10 25. h y254x U x254y, U U U U x50, x54, y54 U y50 m U U h Prove that the curves y254x and x254y divide the area of square bounded by x50, x54, y54 and y50 into three equal parts. 26. U Z U h 1 11 cosA 1 11 cosB 1 11 cosC DABC 50 m cos2 A 1 cosA cos2B 1 cosB cos2C 1 cosC U U A , B U C U - ` 21 U A U 4 , B U 3 U C U 2 ` 60 U A U 6 , B U 2 U C U 3 ` 70 U U Using properties of determinants, show that DABC is isosceles if : 1 11 cosA 1 11 cosB 1 11 cosC 50 cos2 A 1 cosA cos2B 1 cosB cos2C 1 cosC OR A shopkeeper has 3 varieties of pens A , B and C . Meenu purchased 1 pen of each variety for a total of ` 21. Jeevan purchased 4 pens of A variety, 3 pens of B variety and 2 pens of C variety for ` 60. While Shikha purchased 6 pens of A variety, 2 pens of B variety and 3 pens of C variety for ` 70. Using matrix method, find cost of each variety of pen. 65/1/C 11 P.T.O.

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