## CBSE Previous Year Solved  Papers Class 12 Maths Delhi 2016

Time allowed: 3 hours                                                                                          Maximum Marks : 100
General Instructions:

1. All questions are compulsory.
2. Please check that this question paper contains 26 questions.
3. Questions 1-6 in Section A are very short-answer type questions carrying 1 mark each.
4. Questions 7-19 in Section B are long-answer I type questions carrying 4 marks each.
5. Questions 20-26 in Section C are long-answer II type questions carrying 6 marks each.
6. Please write down the serial number of the question before attempting it.

### SET I

SECTION — A

Note: Except for the following questions. All the remaining questions have been asked in previous set.
Question.1. Find the maximum value of

Solution.

Question.2. If A is a square matrix such that A2 = I, then find the simplified value of (A -1)3 + (A +1)3 – 7A.
Solution.

Question.3.

Solution.

Question.4.

Solution.

Question.5.

Solution.

Question.6. Find the vector equation of a plane which is at a distance , of 5 units from the origin and its normal vector is

Solution.

SECTION – B

Question.7. Prove that:

Solution.

OR

Solution.

Question.8. The monthly incomes of Aryan and Babban are in the ratio 3:4 and their monthly expenditures are in the ratio 5:7. If each saves Rs 15,000 per month, find their monthly incomes using matrix method. This problem reflects which value?
Solution. Let the monthly incomes of Aryan and Babban be 3x and 4x respectively.
Suppose their monthly expenditures are 5y and 7y respectively.
Since each saives Rs 15,000 per month.
Monthly saving of Aryan: 3x – 5y = 15,000
Monthly saving of Babban: 4x – 7y = 15,000

Question.9.

Solution.

OR

Solution.

Question.10. Find the values of p and q for which

is continuous at x = π/2.
Solution.

Question.11. Show that the equation of normal at any point on the curve x = 3 cost t – cos3 t and y = 3 sin t – sin3 t is 4(ycos3 t- sin3t) = 3 sin 4t.
Solution.

Question.12.

Solution.

OR

Solution.

Question.13.

Solution.

Question.14.

Solution.

Question.15. Find the particular solution of the differential equation (1 – y2) (1 + log x) dx + 2xy dy = 0, given that y = 0 when x = 1.
Solution.

Question.16. Find the general solution of the following differential equation:

Solution.

Question.17.

Solution.

Question.18. Find the vector and cartesian equations of the line through the point (1,2, – 4) and perpendicular to the two lines.

Solution.

Question.19. Three persons A, B and C apply for a job of Manager in a Private Company. Chances of their selection (A, B and C) are in the ratio 1:2:4. The probabilities that A, B and C can introduce changes to improve profits of the , company are 0.8,0.5 and 0.3 respectively. If the change does not take place, find the probability that it is due to the appointment of C.
Solution.

OR
A and B throw a pair of dice alternately. A wins the game if he gets a total of 7 and B wins the game if he gets a total of 10. If A starts the game’ then find the probability that B wins.
Solution.

SECTION – C

Question.20. Let/: N -> N be a function defined as/(x) = 9x2 + 6x-5. Show that: N—>S, where S is the range off, is invertible. Find the inverse of f and hence find f-1 (43) and f-1(163).
Solution.

Question.21.

Solution.

OR

Solution.

Question.22 Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is 4r/3. Also find maximum volume in terms of volume of the sphere.
Solution.

OR
Find the intervals in which f(x) = sin 3x – cos 3x, 0 < x < π, is strictly increasing or strictly decreasing.
Solution.

Question.23. Using integration find the area of the region {(x, y): x2 +y2 ≤2ax, y2 ≥ ax, x,y≥ 0}
Solution.

Question.24. Find the coordinate of the point P where the line through A(3, -4,-5) and B (2, – 3,1) crosses the plane passing through three points L (2,2,1), M (3,0,1) and N (4, -1,0). Also, find the ratio in which P divides the line segment AB.
Solution.

Question.25. An urn contains 3 white and 6 red balls. Four balls are drawn one by one with replacement from the urn. Find the probability distribution of the number of red balls drawn. Also find mean and variance of the distribution.
Solution.

Question.26. A manufacturer produces two products A and 6. Both the products are processed on two different machines. The available capacity of first machine is 12 hours and that of second machine is 9 hours per day. Each unit of product A requires 3 hours on both machines and each unit of product B requires 2 hours on first machine and 1 hour on second machine. Each unit of product A is sold at Rs 7 profit and B at a profit of Rs 4. Find the production level per day for maximum profit graphically.
Solution. Let the numbers of units of products A and B to be produced be x and y, respectively.