CBSE Sample Paper

www.learnlive.in info@learnlive.in 011-45635788 SAMPLE PAPER -CLASS -XII -MATHEMATICS Q1. Let R be the relation in the set {1, 2, 3, 4} given by: R = {(1,2), (2,2), (1,1), (4,4), (1,3), (3,3), (3,2),} then check whether R is an equivalence relation. Q2. Write the principal value of tan -1 (1) + cos – 1    21 + sin – 1    21 SECTION – Awww.learnlive.in info@learnlive.in 011-45635788 Q3. If A =         sin cos cos sin such that A + A = I. Find the value of  ? Q4. Let A be square matrix of order 3, the find the value of kA in terms of A ?www.learnlive.in info@learnlive.in 011-45635788 Q5. Find the point where the line y = x + 1 is a tangent to the curve y2 = 4x. Q6. Find the angle between the vectors i j k    2  3 and i j k   3  2 www.learnlive.in info@learnlive.in 011-45635788 Q7. The Cartesian equation of a line is 2 6 7 4 3 5     x  y z . Write its vector form. Q8. Evaluate   dx x xx x 2 2 2 2 sin cos sin coswww.learnlive.in info@learnlive.in 011-45635788 Q9. Find a vector in the direction of i j k   5   2 which has magnitude 8 units.www.learnlive.in info@learnlive.in 011-45635788 Q10. Let A be a non singular square matrix of order 3 such that A = 3, find the value of A.adjA ?www.learnlive.in info@learnlive.in 011-45635788 Q11. Let f : N R be the function defined as f (x) = 4x2 + 12x + 15. Show that f : N S, where S is the range of f, is invertible. Find the inverse of f ? SECTION – Bwww.learnlive.in info@learnlive.in 011-45635788 Q12. Prove that            53 cos 13 5 sin 16 63 tan 1 1 1www.learnlive.in info@learnlive.in 011-45635788 Q13. Find intervals in which the following function is strictly increasing or decreasing: f(x) = 20 -9x + 6x2 -x3.www.learnlive.in info@learnlive.in 011-45635788www.learnlive.in info@learnlive.in 011-45635788 Q14. If x = a(cost + t sint) and y = b(sint -t cost), find www.learnlive.in info@learnlive.in 011-45635788www.learnlive.in info@learnlive.in 011-45635788 Q15. Water is leaking from a conical funnel at the rate o f5 cm3 /sec. If the radius of the bae of funnel is 5 cm and height 10 cm, find the rate at which is the water level is dropping when it is 2.5 cm from the top.www.learnlive.in info@learnlive.in 011-45635788www.learnlive.in info@learnlive.in 011-45635788 Q16. Evaluate       dx xx ex 2 2 ( 1)1www.learnlive.in info@learnlive.in 011-45635788 Q17. Evaluate   20 sin4 cos4 sin cos  dx x x x x xwww.learnlive.in info@learnlive.in 011-45635788www.learnlive.in info@learnlive.in 011-45635788 Q18. Find the coordinates of the foot of the perpendicular drawn from the point A(1, 8, 4) to the line joining the points B(0, -1, 3) and C (2, -3, -1).www.learnlive.in info@learnlive.in 011-45635788www.learnlive.in info@learnlive.in 011-45635788 Q19. If sum of two unit vectors is a unit vector, then prove that magnitude of their difference is 3www.learnlive.in info@learnlive.in 011-45635788 Q20. A factory has three machines X, Y and Z producing 1000, 2000 and 3000 bolts per day respectively. The machine X produces 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 to be defective. what is the probability that this defective bolt has been produced by the machine X?www.learnlive.in info@learnlive.in 011-45635788 Q21. For what values of a and b, the function f defined as:             5 2 , 1 11, 1 3 , 1 ( ) ax b ifx ifxax b ifx f x is continuous at x =1www.learnlive.in info@learnlive.in 011-45635788www.learnlive.in info@learnlive.in 011-45635788 Q22. Express the vector a = i j k   5  2  5 as sum of two vector s such that one is parallel to the vector b = i k  3  and other is perpendicular to b.www.learnlive.in info@learnlive.in 011-45635788www.learnlive.in info@learnlive.in 011-45635788 Q23. Using elementary row transformations, find the inverse of the matrix:            3 2 40 2 3 1 1 2 SECTION – Cwww.learnlive.in info@learnlive.in 011-45635788www.learnlive.in info@learnlive.in 011-45635788 Q24. A rectangle is inscribed in a semi – circle of radius r with one of its sides on diameter of semicircle. Find the dimensions of the rectangle so that its area is maximum. Also find the area.www.learnlive.in info@learnlive.in 011-45635788www.learnlive.in info@learnlive.in 011-45635788 Q25. Evaluate    20 (x2 x 2)dx as limit of sums.www.learnlive.in info@learnlive.in 011-45635788www.learnlive.in info@learnlive.in 011-45635788 Q26. Draw a rough sketch of the region enclosed between the circles x2 + y2 = 1 and (x-1)2 + y2 = 1. Using integration, find the area of the enclosed region.www.learnlive.in info@learnlive.in 011-45635788www.learnlive.in info@learnlive.in 011-45635788 Q27. A random variable X has the following probability distribution values of X: X 0 1 2 3 4 5 6 7 P(X) 0 k 2k 2k 3k K2 2k2 7k2+k Find each of the following: (i) K (ii) P(X < 6) (iii) P(X  6) (iv) P(0 < X < 5)www.learnlive.in info@learnlive.in 011-45635788www.learnlive.in info@learnlive.in 011-45635788 Q28. A firm manufacturers two types of products A & B, and sells them at a profit of Rs. 5 per unit of type A and Rs. 3 per unit of type B. Each product is processed on two machines M1 and M2. One unit of type A requires one minute of processing time on M1 and two minutes of processing time on M2; whereas one unit of type B requires one minute of processing time on M1 and one minute of processing time on M2. Machines M1 and M2 are respectively available for atmost 5 hours and 6 hours in a day. Find our how many units of each type of product should the firm produce a day in order to maximize the profit. Solve the problem graphically.www.learnlive.in info@learnlive.in 011-45635788www.learnlive.in info@learnlive.in 011-45635788www.learnlive.in info@learnlive.in 011-45635788 Q29. Find the coordinates of the foot of perpendicular and perpendicular distance drawn from the point (1, 3, 4) from the plane 2x – y + z + 3 = 0. Find also the image of this point in the plane.www.learnlive.in info@learnlive.in 011-45635788www.learnlive.in info@learnlive.in 011-45635788