**CBSE Previous Year Solved Papers Class 12 Maths Outside 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**

**SECTION – A**

**Question.1. Write the value of ∆ =**

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** Solution.**

**Question.2. Write the sum of the order and degree of the following differential equation:**

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** Solution.**

**Question.3. Write the integrating factor of the following differential equation:**

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** Solution.**

**Question.4.**

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** Solution.**

**Question.5. Write a unit vector perpendicular to both the vectors**

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** Solution.**

**Question.6. The equations of a line are 5x – 3 = 15y + 7 = 3- 10z. Write the direction cosines of the line.**

** Solution.**

**SECTION – B**

**Question.7. To promote the making of toilets for women, an organization tried to generate awareness through (i) house calls (ii) letters, and (iii) announcements. The cost for each mode per attempt is given below: (i)Rs 50 (ii) Rs 20 (iii)Rs 40**

** The number of attempts made in three villages X, Y and Z are given below:**

** (i) (ii) (iii)**

** 400 300 100**

** 300 250 75**

** 500 400 150**

** Find the total cost incurred by the organization for the three villages separately, using matrices.**

** Write one value generated by the organization in the society.**

** Solution.** The number of attempts made in three villages X, Y and Z can be represented by the 3 x 3 matrix.

Hence the total cost incurred by the organization for the three villages separately are Rs 30,000, Rs 23,000 and Rs 39,000.

The organization in the society generated the value of cleanliness for the women welfare.

**Question.8.**

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** Solution. Given**

**OR**

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** Solution.**

**Question.9. Using properties of determinants, prove the following:**

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** Solution.**

**Question.10. Find the adjoint of the matrix.**

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** Solution.**

**Question.11. Show that the function f(x) = | x-1 | + | x +1|, for all x ϵ R, is not differentiable at the points x=-1 and x = 1.**

** Solution.**

**Question.12.**

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** Solution.**

**Question.13.**

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** Solution.**

**Question.14.**

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** Solution.**

**OR**

**Solution.**

**Question.15.**

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** Solution.**

**Question.16.**

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** Solution.**

**Question.17.**

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** Solution.**

**Question.18. Find the equation of a line passing through the point (1,2, -4) and perpendicular to two lines**

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** Solution.** Let the direction ratios of required line be a, b, c, t since, the line is perpendicular to

**OR**

** Find the equation of the plane passing through the points (-1, 2, 0), (2, 2, -1) and parallel to the line**

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** Solution.**

**Question.19. Three cards are drawn successively with replacement from a well shuffled pack of 52 cards. Find the probability distribution of the number of spades. Hence find the mean of the distribution.**

** Solution.** Let X denote the number of spades when three cards are drawn, then, X is a random variable that can take values 0,1,2,3:

Let E be the event when spade card is drawn,

**OR**

** For 6 trials of an experiment, let X be a binomial variate which satisfies the relation 9P(X = 4) = P(X =2). Find the probability of success.**

** Solution.** Let p denote the probability of getting success and q be the probability of failure.

**SECTION – C**

**Question.20. Consider f: R+ -> [- 9, ∞ ] given by f(x) = 5x ^{2 } + 6x – 9. Prove that f is invertible with**

**Solution.**

**OR**

**Solution.**

**Question.21. Find the value of p for which the curves x ^{2}= 9p(9 – y) and x^{2} = p(y +1) cut each other at right angles.**

**Question.22. Using integration, prove that the curves y ^{2} = 4x and x^{2} = 4y divide the area of the square bounded by x = 0, x = 4, y = 0 and y = 4 into three equal parts.**

**Solution.**

**Question.23.**

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** Solution.**

**OR**

** Find the particular solution of the differential equation (tan ^{-1}y -x) dy = ( 1 + y^{2}) dx, given that x = 1, when y = 0.**

**Solution.**

**Question.24. Find the distance of the point P(3,4, 4) from the point, where the line joining the points A(3, – 4, – 5) and B(2, -3,1) intersects the plane 2x + y + z = 7.**

** Solution.** Equation of the line joining the points A(3, – 4, – 5) and B(2, – 3,1) is

**Question.25. A company manufactures three kinds of calculators A, B and C in its two factories I and II, The company has got an order for manufacturing at least 6400 calculators of Kind A, 4000 of kind B and 4800 of kind C. The daily output of factory I is of 50 calculators of kind A, 50 calculators of kind B, and 30 calculators of kind C. The daily output of factory II is of 40 calculators of kind A, 20 of kind B and 40 of kind C. The cost per day to run factory I is Rs 12,000 and of factory II is Rs 15,000. How many days do the two factories have to be in operation to produce the order with the minimum cost ? Formulate this problem as an LPP and solve it graphically.**

** Solution.**

**Question.26. In a factory which manufactures bolts, machines A, B and C manufacture respectively 30% 50% and 20% of the bolts. Of their outputs 3,4 and 1 percent respectively are defective bolts. A bolt is drawn at random from the product and is found to be defective. Find the probability that this is not manufactured by machine B.**

** Solution.**