CBSE SAMPLE PAPER-04 FIRST PRE-BOARD EXAMINATON, 2008-09 CLASS - XII MATHEMATICS [Time : 3 hrs.] [M. M.: 100] General Instructions:(i) All questions are compulsory. (ii) This question paper consists of 29 questions divided into three sections A, B and C. Section A consists 10 questions of 1 mark each, section B is of 12 questions of 4 marks each and section C is 7 questions of 6 marks each. There is no overall choice. However an internal choice has been provided I n4 questions (iii) of 4 marks and 3 questions of 6 marks each. SECTION – A 1. IF A is a square matrix of order 3 such that |adj A | = 64. Find |A|. 4 d y  dy  2. Write the order and degree of the following differential equation   + 3 y 2 = 0 dx  dx  2 3. Find the point on the curve y = x2 – 2x + 3 where the tangent is parallel to x-axis. 1 3   y 4. Find the value of x and y if   +  0 x  1 0  5 6  = . 2  1 8  2π  5. Find the principal value of cos-1  cos 3  6. Cartesian equation of a line are 2π  −1   + sin  sin 3    .  2x −1 4 − y z = = . Write the vector equation of the line. 4 7 2 7. Find | a × b | if a = ɵi + 2 ɵj + 3kɵ , bɵ = 5ɵi + 3 ɵj − 2 kɵ . 8. If θ be angle between two vectors aɵ and bɵ with magnitude 3 aɵ .bɵ = 3. and 2 respectively and Find sin θ Material downloaded from http://myCBSEguide.com and http://onlineteachers.co.in Portal for CBSE Notes, Test Papers, Sample Papers, Tips and Tricks 9. If f(x) = 4x + 3 be a real valued function. Find a real function g(x), such that gof = lR. 1 0  0 1  10. If A =  and B =   . Show that matrix multiplication is not commutative.  0 −1 1 0  SECTION – B π  x −1  −1  x + 1  11. Solve for x, tan −1   + tan   = ,| x |< 1.  x−2  x−2 4 OR Prove that sin −1 5 3 63 + cos −1 = tan −1 . 13 5 16 12. Using properties of determinants prove the following: a + b + 2c c c a b + c + 2a a =2(a+b+c) 3. b b c + a + 2b 13. Show that the relation R defined by (a, b) R(c, d) a + b = b + c on the set N × N is an equivalence relation. 14. Solve the differential equation (x + 2y2) dy – ydx = 0, y = 1 when x =0, cos2 xdy + ydx = tan xdx, y = 1 when x = 0. 15. Form the differential equation of the family of circles touching the y-axis at origin. 16. Evaluate ∫ ex 5 − 4e x − e 2 x dx. OR  1  Evaluate ∫ log(log x) + dx. (log x) 2   17. For what value of k is the following function is continuous at x = 2? 2 x + 1 , if x < 2  f(x) = k , if x = 2 3 x − 1 , if x > 2  d2y π 1  18. If x = a  cos t + log tan  and y = a sin t. Find at t = . 2 dx 4 2  Material downloaded from http://myCBSEguide.com and http://onlineteachers.co.in Portal for CBSE Notes, Test Papers, Sample Papers, Tips and Tricks 19. Using differentials find the approximate value of f(3.05) if f(x) = 3x2 = 15x = 15. OR Find the intervals in which the following function f(x) is (a) increasing (b) decreasing f(x) = 2x3 – 9x2 + 12x + 15. 20. Twelve cards numbered 1 to 12 are placed in a box mixed up thoroughly and then a card is drawn at random form the box. If it is known that the number on the drawn card is more than 4. Find the probability that it is an odd number. ( ) ( 21. Find a vector of magnitude 7 units perpendicular to each of the vectors aɵ + bɵ and aɵ − bɵ ) where a = ɵi + ɵj + kɵ and b = 2ɵi + 3 ɵj + kɵ . 22. Find Cartesian equation of the line parallel to the line x −1 3 − y z +1 = = and passing through 5 2 4 the point (3, 0, -4). Also find the distance between these two lines. SECTION – C 23. The cost of 4kg onion, 3 kg wheat and 2kg rice is Rs.60. The cost of 2kg onion 4kg wheat is Rs.42. The cost of 2kg wheat and 3kg rice is Rs.40. Find cost of each item per kg by matrix method. OR Evaluate ∫ (x 3 2 1 ) + x + 2 dx as the limit of sums. 24. A window has the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12m. Find the dimension of the rectangle that will produce the largest area of the window. OR Show that the semi-vertical angle of the right circular cone of given total surface are and 1 maximum volume is sin −1   . 3 25. Find the area of that part of the circle x2 + y2 = 16 which is exterior to the parabola y2 = 6x. Material downloaded from http://myCBSEguide.com and http://onlineteachers.co.in Portal for CBSE Notes, Test Papers, Sample Papers, Tips and Tricks 26. Find the distance of the point (-2, 3, -4) form the line x + 2 2 y + 3 3z + 4 = = measured parallel 3 4 5 to the plane 4x + 12y – 3z + 1 = 0. 27. A diet for a sick person must contain at least 4000 units of vitamins, 50 units of mineral and 1400 units of calories. Two foods A and B are available at a cost of Rs.5 and Rs.4 per unit respectively. One unit of food A contains 200 units of vitamins, 1 units of minerals and 40 units of calories, while one units of the food B contains 100 units of vitamins, 2 units of minerals and 40 units of calories. Find what combination of the foods A and B should be used to have least cost, but it must satisfy the requirement of the sick person. From the question of LPP and solve it graphically. 28. In a bulb factory machines A, B and C manufacture 60%, 30% and 10% bulbs respectively. 1%, 2% and 3% of the bulbs produced respectively by A, B and C are found to be effective. A bulb is picked up at random from the total production and found to be defective. Find the probability that this bulb was produced by the machine B. 29. Evaluate ∫ π 0 log (1 + cos x ) dx OR Evaluate ∫ π 0 xdx . a sin x + b 2 cos 2 x 2 2 Material downloaded from http://myCBSEguide.com and http://onlineteachers.co.in Portal for CBSE Notes, Test Papers, Sample Papers, Tips and Tricks