12-A Second Floor, Yusuf Sarai New Delhi – 110016 Ph.: 46150335 Mob.: 9810004225 Sample Paper – 8 (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 5 3 1. For what value of , the matrix has no inverse? 1 2. Let A be a non singular matrix of order 3 3 such that |A| = 5. What is |Adj A|? 8 2 7 5 3. Write the matrix x if 3x – 6 0 0 0 4. Let * be the binary operation defined on R, then of a*b = (a b) 2 , which the value of (1*2)*3 3 1 5. Write values of : sin sin 1 2 3 1 dx 6. Write values of : 5 x 2 x log x /4 7. Write values of : x 4 sin x dx /4 8. What is the value of iˆ.( ˆjxkˆ) ˆj.(iˆxkˆ) kˆ.( ˆjxiˆ) ? 9. What is the angle which 3iˆ 6 ˆj 2kˆ makes with z-axis. 15 10. For what value of , a 2iˆ 6 ˆj 5kˆ is parallel to b 3iˆ ˆj kˆ ? 2 Section – B x y z 11. Using properties of determinants, prove that x 2 y 2 z 2 ( x y )( y z )( z x)( xy yz zx ) yz zx xy x3 | x 3 | a of 12. For what values of a and b fx a b of x3 | x 3 | 2b of Sample Paper (Maths) – 8 (XII) 1 x3 x 3 is continuous at x = 3 x3 www.eeducationalservices.com OR 2 d y dx 2 13. Considered f : R+ [–5, ) given by f(x) = 9x2 + 6x – 5. Show that f is invertible function also find the inverse of the function f. 14. Using differentials, find the approximate value of 25.3 OR Find the intervals in which the function defined as f (x) = sin x – cos x, 0 < x < 2 is strictly. If x = a cos3, y = sin3, then find x x 15. Solve the differential equation : y.e y .dx ( xe y y )dy 16. Solve the differential equation : cos2x dy = (tan x – y) dx 17. There are 5% defective items in a large bulk of items. What is the probability that a sample of 10 items will include not more than one defective item. OR Let x denote larger of the two numbers, when two numbers are chosen from first six positive integers. Find the probability distribution of x 18. Find the equation of plane passing through the points (2, 3, – 4) and (1, – 1, 3) and parallel to x-axis. OR Find the equation of plane passing through (1 – 2, 1) and perpendicular to planes r .(2iˆ 2 ˆj 2kˆ) 0 and r .(iˆ ˆj 2kˆ) 4 19. Find a vector whose magnitude is 4 and which makes angles with x=axis, with y-axis and an 4 3 obtuse with z-axis. 20. Solve for x : sin – 1 (1 – x) – 2 sin – 1x = dy dx sin x cos x 2 21. If xy + yx = a, find 22. Evaluate 9 16 sin 2 x dx Section – C 23. Solve the system of linear equations : 4x – 5y – 11z = 12, x – 3y + z = 1, 2x + 3y – 7z = 2 OR 3 1 1 1 2 2 –1 If A = 15 6 5 and B = 1 3 0 find (AB) – 1 5 2 2 0 2 1 24. Using the method of integration, find the area of the region bounded by parabola y2 = x and line x + y = 2. x tan x 25. Evaluate dx sec x tan x 0 OR 3 Evaluate ( x 2 2 x 5)dx, as limit of sum. 1 26. A helicopter is flying along the curve y = x2 + 2. A soldier is placed at the point (3, 2). Find the shortest distance between the soldier and the helicopter. 27. A girl throw a die, if she gets a 5 or 6, she tosses a coin three times and notes the number of heads. If she gets a 1, 2, 3 or 4 she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one head, what is eh probability that she threw 1, 2, 3 or 4 with the die? Sample Paper (Maths) – 8 (XII) 2 www.eeducationalservices.com 28. Find the position vector of foot of perpendicular drawn from point (2iˆ 3 ˆj 5kˆ) to plane r .(6iˆ 3 ˆj 5kˆ) 74 0. 29. A company manufactures two types of toys A and B. Type A required 5 minutes each for cutting and 10 minutes each for assembling. Type B required 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours available for cutting and 4 hours available for assembling in a day. The profit is Rs.50 each on type A and Rs.60 each on type B. How many toys of each type should the company manufacture in a day to maximum the profit? Sample Paper (Maths) – 8 (XII) 3 www.eeducationalservices.com

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