12-A Second Floor, Yusuf Sarai New Delhi – 110016 Ph.: 46150335 Mob.: 9810004225 Sample Paper – 6 (Mathematics) Time : 3 hours Maximum Marks : 100 General Instructions : (a) All questions are compulsory. (b) The question paper consists of 29 questions divided into three sections A, B and C. Section A contains 10 questions of 1 mark each, section B is 12 questions of 4 marks each and section C is of 7 questions of 6 marks each. (c) There is no overall choice. However internal choice has been provided in section B and C only. (d) Use of calculator is not permitted. However you may ask for mathematical tables. Section – A 1. Form the differential equation satisfied by the family of curves given by x2 + y2 = 2 ax.      2. If  is the angle between any two vectors a and b and a.b  a  b then find the angle .   3. If a is a unit vector and (x – a) . (x + a) = 8, then find x 4. Find the vector and Cartesian equation of line passing through the point A (2, –1, 1) and parallel to the line joining B (–1, 4, 1) and C (1, 2, 2). 5. Prove that if a plane has the intercepts a, b, c and is at a distance of p from the origin then 1  1  1  1 . 2 2 2 2 a b c p 6 6. Evaluate :  sec x dx 5 1 7. Given P (A  B) = , P (A  B) = , P (B`) = ½. Find P(A) and P(B) also find whether events A and B 6 3 are independent or not. 8. The volume of a cube is increasing at a constant rate. Prove that the increase in surface area varies inversely as the length of the edge of the cube. d2y 9. Solution of the differential equation  0 is given by: dx 2 (a) y = 0 (b) y = ax (c) y = ax + b (d) none of these 10. Write the degree and order of xyy” + x (y`)2 – y y` = 0. Section – B 2 x y2 11. Find the area bounded by the ellipse 2  2  1 and the ordinate x = 0 and x = ae where b2 = a2 (1 – e2) a b and e < 1. x2  y2 12. Show that the family of curves for which the slope of the tangent at any point (x, y) on its , is 2 xy given by x2 – y2 = Cx.      13. Let a  iˆ  4 ˆj  2kˆ, b  3iˆ  2 ˆj  7 kˆ Find a vector d which is perpendicular to both the vector a and b  and c .d  15. OR The two adjacent sides of a parallelogram are 2iˆ  4 ˆj  5kˆ and iˆ  2 ˆj  3kˆ. Find the unit vector parallel to its diagonal. Also find its area. Sample Paper (Maths) – 6 (XII) 1 www.eeducationalservices.com 14. Find the shortest distance between the two lines whose vector equation is given by  r  iˆ  ˆj   2iˆ  ˆj  kˆ  r  2iˆ  ˆj  kˆ   3iˆ  5 ˆj  2kˆ     2 15. Evaluate the integral  ( x 2  1)dx as limit of a sum. 0  /2 16. By using the properties of definite integrals evaluate the integral  (2 log sin x  log sin 2 x)dx 0 1 17. Evaluate :  ( x. sin 1 x )dx 0 18. Evaluate : 5x  ( x  1)( x 2  9) dx OR x2 dx  6x  5 19. A factory has three machines X, Y, Z producing 1000, 2000 and 3000 bolts per day respectively. The machine X produced 1% defective bolts, Y produces 1.5% and Z produces 2% defective bolts. At the end of a day a bolt is drawn at random and is found defective. What is the probability that the defective bolt is produced by machine X. 20. A coin is tossed until a head appears or until it has been tossed three times, Given that ‘head’ does not occur on the first toss, what is the probability that the coin is tossed thrice? x2 y2 21. Find the points on the curve   1 at which the tangents are : 4 25 (a) parallel to the x-axis (b) parallel to the y-axis OR Find the approximate value of f(2.01) where f(x) = 4x2 + 5x + 2. 22. Find the intervals in which the function f(x) = 2 log (x – 2) – x2 + 4x + 1 is increasing. Section – C 2 23. Find the area of the region {(x, y) : y  4x, 4x2 + 4y2  9}. dy 24. Solve the Differential equation tan y  tan x  cos y. cos 2 x dx   25. Find the cosine and the sine of the angle between the vectors a  2iˆ  ˆj  3kˆ and b  4iˆ  2 ˆj  2kˆ. 26. If a young man rides his motorcycle at 25 km/hr he had to spend Rs.2 per km on petrol. If he rides at a faster speed of 40 km/hr the cost increases at Rs.5 per km. He has Rs.100 to spend on petrol and whishes to find what is the maximum distance he can travel with in one hour. Express this as a Linear Programming Problem and solve it graphically. 27. A pair of dice is rolled twice. Let X denote the number of times, ‘a total of 9 is obtained’. Find the mean and variance of the random variable X. OR A bag contains 7 Red, 4 White and 5 Black balls. If four balls are drawn one by one with replacement what is the probability that : (a) None is white (b) All are white (c) Only two are white (d) At leas two are white 28. Find the length and the foot of the perpendicular from the point (1, 1, 2) to the plane r.2iˆ  2 ˆj  4 kˆ   5  0. OR  Find the equation of the plane passing through the line of intersection of the planes r iˆ  2 ˆj  3kˆ  4  0   and r 2iˆ  ˆj  kˆ  5  0 and is perpendicular to the plane r . 5iˆ  3 ˆj  6kˆ  8  0.  2x 2       29. Show that the height of a cylinder which is open at the top having a given surface area and greatest volume is equal to the radius of the base. Sample Paper (Maths) – 6 (XII) 2 www.eeducationalservices.com