SAMPLE QUESTION PAPER MATHEMATICS (041) CLASS XII

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1 SAMPLE QUESTION PAPER MATHEMATICS (01) CLASS XII Time allowed: hours Maimum Marks: 100 General Instructions: (i) All questions are compulsory. (ii) This question paper contains 9 questions. (iii) Question 1- in Section A are very short-answer type questions carrying 1 mark each. (iv) Questions 5-1 in Section B are short-answertype questions carrying marks each. (v) Questions 1- in Section C are long-answer-i type questions carrying marks each. (vi) Questions -9 in Section D are long-answer-ii type questions carrying 6 marks each. Section A Questions 1 to carry 1 mark each. 1,,,. Let R be the equivalence relation on A A 1. Let A= a, brc, d iff a d b c. Find the equivalence class 1,. defined by. If A a ij is a matri of order, such that A 15 and Cij represents the cofactor of a ij, then find a1c1 ac. Give an eample of vectors. Determine whether the binary operation on the set N of natural numbers defined by a b ab is associative or not. Section B Questions 5 to 1 carry marks each 5. If 1 1 sin cos, then find the value of. 6. Find the inverse of the matri 5. Hence, find the matri P satisfying the 1 matri equation P. 5 1

2 7. Prove that if then cos cos 1 8. Find the approimate change in the value of =.00, when changes from = to 9. Find e 1 sin d 1 cos 10. Verify that a by 1 is a solution of the differential equation ( yy y ) yy Find the Projection (vector) of iˆ ˆj kˆ on iˆ ˆj kˆ. 1. If A and B are two events such that P A 0., PB 0.8 and 0.6 then find P A B. Section C Questions 1 to carry marks each. P B A, Find a and b, if the function given by is differentiable at 1 a b, if 1 f( ) 1, if 1 Determine the values of a and b such that the following function is continuous at = 0: sin, if 0 sin( a1) f ( ),if 0 sinb e 1,if 0 b

3 1 15. If ylog( ), then prove that ( 1) y ( 1) y Find the equation(s) of the tangent(s) to the curve y ( 1)( ) at the points where the curve intersects the ais. Find the intervals in which the function is strictly increasing or strictly decreasing. f ( ) log(1 ) log( ) 17. A person wants to plant some trees in his community park. The local nursery has to perform this task. It charges the cost of planting trees by the following formula: C( ) 5 600, Where is the number of trees and C() is the cost of planting trees in rupees. The local authority has imposed a restriction that it can plant 10 to 0 trees in one community park for a fair distribution. For how many trees should the person place the order so that he has to spend the least amount? How much is the least amount? Use calculus to answer these questions. Which value is being ehibited by the person? 18. Find sec d 1 cos ec 19. Find the particular solution of the differential equation : y y ye d ( y e ) dy, y(0) 1 Show that ( ) ( ) is a homogenous differential equation. Also, find the general solution of the given differential equation. 0. If are three vectors such that a b c 0, then prove that a b b c c a, and hence showthat a b c Find the equation of the line which intersects the lines y z 1 1 y z and and passes through the point (1, 1, 1). 1

4 . Bag I contains 1 white, black and red balls; Bag II contains white, 1 black and 1 red balls; Bag III contains white, black and red balls. A bag is chosen at random and two balls are drawn from it with replacement. They happen to be one white and one red. What is the probability that they came from Bag III.. Four bad oranges are accidentally mied with 16 good ones. Find the probability distribution of the number of bad oranges when two oranges are drawn at random from this lot. Find the mean and variance of the distribution. Section D Questions to 9 carry 6 marks each.. If the function f : be defined by f ( ) and g : by g then find f g and show that f g is invertible. Also, find ( ) 5, f g 1 1, hence find g 9 f. A binary operation is defined on the set of real numbers by a, if b 0 ab. If at least one of a and b is 0, then prove that a b ba. a b, if b 0 Check whether is commutative. Find the identity element for, if it eists. 5. If 1 A 1, then find 7 1 A and hence solve the following system of equations: y 7z 1, y z, y z If A 1 0 1, find the inverse of A using elementary row transformations 0 1 and hence solve the following matri equation XA Using integration, find the area in the first quadrant bounded by the curve y, the circle y and the y-ais

5 7. Evaluate the following: d cos 8. Evaluate ( ) d as the limit of a sum. r iˆ ˆj kˆ ( iˆ ˆj 9 kˆ) from the line measured parallel to the plane: y + z = A company produces two different products. One of them needs 1/ of an hour of assembly work per unit, 1/8 of an hour in quality control work and Rs1. in raw materials. The other product requires 1/ of an hour of assembly work per unit, 1/ of an hour in quality control work and Rs 0.9 in raw materials. Given the current availability of staff in the company, each day there is at most a total of 90 hours available for assembly and 80 hours for quality control. The first product described has a market value (sale price) of Rs 9 per unit and the second product described has a market value (sale price) of Rs 8 per unit. In addition, the maimum amount of daily sales for the first product is estimated to be 00 units, without there being a maimum limit of daily sales for the second product. Formulate and solve graphically the LPP and find the maimum profit.