## CBSE Previous Year Solved  Papers Class 12 Maths Delhi 2015

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

Note: Except for the following questions, All the remaining question have been asked in previous set.

SECTION – A

Question.1. Solution. Question.2. Solution. Question.3. If a line makes angles 90°, 60° and θ with x, y and z axis respectively, where θ is acute, then find θ.
Solution.  Question.4. Solution. Question.5. Find the differential equation representing the family A of curves υ = A/r + B, where A and B are arbitrary r constants.
Solution. Question.6. Find the integrating factor of the differential equation Solution.  SECTION – B

Question.7. Solution.  OR Solution.  Question.8. Solution.  Question.9. Solution.   OR Solution.  Question.10. Solution. Question.11. A bag ‘A’ contains 4 black and 6 red balls and bag ‘B’ contains 7 black and 3 red balls. A die is thrown. If 1 or 2 appears on it, then bag A is chosen, otherwise bag B. If two balls are drawn at random (without replacement) from the selected bag, find the probability of one of them being red and another black.
Solution. OR
An unbiased coin is tossed 4 times. Find the mean and variance of the number of heads obtained.
Solution. Let X denote the number of heads in the four tosses of the coin then, X is a random variable that can have values 0,1,2,3,4.  Question.12. Solution.  Question.13. Solution.  Question.14. If sin [cot-1 (x +1)] = cos(tan-1 x), then find x.
Solution. OR Solution.  Question.15. Solution.  Question.16. Solution.  Question.17.’ The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ?
Solution. Question.18. Solution.   Question.19. Three Schools A, B and C organized a mela for collecting funds for helping the rehabilitation of flood victims. They sold hand made fans, mats and plates from recycled material at a cost of Rs 25, Rs 100 and Rs 50 each. The number of articles sold are given below: Find the fund collected by each school separately by selling the above articles. Also find the total funds collected for the purpose. Write one value generated by the above situation.
Solution. The number of articles sold by each school can be represented by the 3 x 3 matrix SECTION-C

Question number 20 to 26 carry 6 marks each.
Question.20. Let N denote the set of all natural numbers and R be the relation on N x N defined by (a, b) R(c, d) if ad (b + c) = bc (a + d). Show that R is an equivalence relation.
Solution. We know that relation R will be an equivalence relation, if we prove it as a reflexive, symmetric and transitive relation.   Question.21. Using integration find the area of the triangle formed by positive x-axis and tangent and normal to the circle x2 + y2= 4 at (1, √3).
Solution.    OR Solution.    Question.22. Solve the differential equation: (tan-1 y-x)dy = (1 + y2) dx.
Solution. Same as solution Q. 23 (OR) Set 1 (Outside Delhi) up to eq.
x = tan-1 y -1 + ce -tan-1 y
OR Solution.   Question.23. Solution.   Question.24. Solution.   Question.25, Find the local maxima and local minima, of the function f(x) = sin x – cos x, 0 < x < 2π. Also find the local maximum and local minimum values.
Solution.   Question.26. Find graphically, the maximum value of z = 2x + 5y, subject to constraints given below:
2x + 4y ≤ 8
3x + y ≤ 6
x + y ≤ 4
x≥ 0, y ≤ 0.6
Solution. We first convert the inequalities into equations to obtain lines  