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

**Time allowed: 3 hours Maximum Marks : 100 **

**General Instructions:**

**All questions are compulsory.****Please check that this question paper contains 26 questions.****Questions 1-6 in Section A are very short-answer type questions carrying 1 mark each.****Questions 7-19 in Section B are long-answer I type questions carrying 4 marks each.****Questions 20-26 in Section C are long-answer II type questions carrying 6 marks each.****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.**

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** Solution.**

**Question.2.**

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** 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.**

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** 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**

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** Solution.**

**SECTION – B**

**Question.7.**

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** Solution.**

**OR**

**Solution.**

**Question.8.**

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** Solution.**

**Question.9.**

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** Solution.**

**OR**

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** Solution.**

**Question.10.**

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** 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.**

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** Solution.**

**Question.13.**

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** Solution.**

**Question.14. If sin [cot ^{-1} (x +1)] = cos(tan^{-1} x), then find x.**

**Solution.**

**OR**

**Solution.**

**Question.15.**

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** Solution.**

**Question.16.**

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** 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.**

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** 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 x ^{2} + y^{2}= 4 at (1, √3).**

**Solution.**

**OR**

**Solution.**

**Question.22. Solve the differential equation: (tan ^{-1} y-x)dy = (1 + y^{2}) 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.**

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** Solution.**

**Question.24.**

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** 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

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