CBSE Question Paper 2013 class 12 Mathematics

• All question are compulsory.
• The question paper consists of 29 questions divided into three sections A, B and C. Section A comprises of 10 questions of one mark each, Section B comprises of 12 questions of four marks each and Section C comprises of 7 questions of six marks each.
• All questions in Section A are to be answered in one words, one sentence or as per the exact requirement of the questions.
• There is no overall choice. However, internal choice has been provided in 4 questions of four marks each and 2 questions of six marks each. You have to attempt only one of the alternatives in all such questions.
• Use of calculators is not permitted. You may ask for logarithmic tables, if required.

Section A

1. Write the differential equation representing the family of curves y = mx, where m is an arbitrary constant.

2. If A_ij is the cofactor of the element a_ij of the determinant  then write the value of a₃₂.A₃₂.

3. Write the value of .

4. For what value of x, is the matrix  a skew-symmetric matrix ?

5. Write the principal value of

6. The money to be spent for the welfare of the employees of a firm is proportional to the rate of change of its total revenue (marginal revenue). If the total revenue (in rupees) received from the sale of x units of a product is given by R(x) = 3x² + 36x + 5, find the marginal revenue, when x = 5, and write which value does the question indicate.

7. Find the length of the perpendicular drawn from the origin to the plane 2x – 3y + 6z + 21 = 0.

8. Find , if for a unit vector

9. If matrix  and A² = pA, then write the value of p.

10. A and B are two points with position vectors  and  respectively. Write the position vector of a point P which divides the line segment AB internally in the ratio 1:2.

Section B

11. Differentiate the following with respect to x:

12. Evaluate:

            OR

Evaluate:

13. Using properties of determinants, prove the following:

14. Show that:

OR

Solve the following equation:

15. Consider  given by f(x) = x² + 4. Show that f is invertible with the inverse f^-1 of f given by where R₊ is the set all non-negative real numbers.

16. Find the value of k, for which

f(x)=

is continuous at x = 0.

OR

If  and  then find the value of  at

17. Show that the lines

are intersecting. Hence find their point of intersection.

OR

Find the vector equation of the plane through the point (2, 1, -1) and (-1, 3, 4) and perpendicular to the plane x – 2y + 4z = 10.

18. The probabilities of two students A and B coming to the school in time are  and  respectively. Assuming that the events, ‘A coming in time’ and ‘B coming in time’ are independent, find the probability of only one of them coming to the school in time.

Write at least one advantage of coming to school in time.

19. If  prove that

20. Evaluate:

21. Evaluate:

22. If  and  then find the value of so that  and  are perpendicular vectors.

Sector C

23. Find the equation of the plane passing through the line of intersection of the planes and  whose perpendicular distance from origin is unity.

OR

Find the vector equation of the line passing through the point (1, 2, 3) and parallel to the planes  and

24. In a hockey match, both teams A and B scored same number of goals up to the end of the game, so to the game, so to decide the winner, the referee asked both the captain to throw a die alternately and decided that the team, whose captain gets a six first, will be declared the winner. If the captain of team A was asked to start, find their respective probabilities of winning the match and state whether the decision of the referee was fair or not.

25. A manufacturer considers that men and women workers are equally efficient and so he pays them at the same rate. He has 30 and 17 units of workers (male and female) and capital respectively, which he uses to produce two types of goods A and B. To produce one unit of A, 2 workers and 3 units of capital are required while 3 workers and 1 unit of capital is required to produce one unit of B. If A and B are priced at Rs 100 and Rs 120 per unit respectively, how should he use his resources to maximize the total revenue? Form the above as an LPP and solve graphically.

Do you agree with this view of the manufacturer that men and women workers are equally efficient and so should be paid at the same rate?

26. Find the area of the greatest rectangle that can be inscribed in an ellipse

OR

Find the equations of tangents to the curve 3x² – y² = 8, which pass through the point

27. The management committee of a residential colony decided to award some of its members (say x) for honesty, some (say y) for helping other and some others (say z) for supervising the works to keep the colony neat and clean. The sum of all the awardees is 12. Three times the sum of awardees for cooperation and supervision added to two times the number of awardees for honesty is 33. If the sum of the number of awardees for honesty and supervision is twice the number of awardees for helping others, using matrix method, find the number of each category. Apart from these values, namely, honesty cooperation and supervision, suggest one more value which the management of the colony must include for awards.

28. Find the area of the region  and  using method of integration.

29. Show that the differential equation is homogeneous. Find the particular solution of this differential equation, given that  when x = 1.