**CBSE Previous Year Solved Papers Class 12 Maths Outside Delhi 2016**

**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 numbers 1 to 6 carry 1 mark each.**

** Question.1.**

**Solution.**

**Question.2. Use elementary column operation C _{2 }—>C_{2}+2C_{1} in the following matrix equation:**

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** Solution.**

**Question.3. Write the number of all possible matrices of order 2 x 2 with each entry 1,2 or 3.**

** Solution.** Total number of all possible matrices of order 2 x 2 with each entry 1,2 or 3 are 3^{4 }i.e., 81.

**Question.4.**

**Solution.**

**Question.5. Write the number of vectors of unit length perpendicular to both the vectors**

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** Solution.**

**Question.6. Find the vector equation of the plane with intercepts 3,-4 and 2 on x, y and z-axis respectively.**

** Solution.**

**SECTION – B**

**Question numbers 7 to 19 carry 4 marks each.**

** Question.7. Find the coordinates of the point where the line through the points A(3,4,1) and B(5,1,6) crosses the XZ plane. Also find the angle which this line makes with the XZ plane.**

** Solution.**

**Question.8.**

** **

** Solution.**

**Question.9. In a game, a man wins Rs 5 for getting a number greater than 4 and loses Rs 1 otherwise, When a fair die is thrown. The man decided to throw a die thrice but to quit as and when he gets a number greater than 4. Find the expected value of the amount he wins/loses.**

** Solution.** Let n denote the number of throws required to get a number greater than 4 and X denote the amount won/lost.

The man may get a number greater than 4 in the very first throw of the die or in second throw or in the third throw. Thus, we have the following probability distribution for X. ‘

**OR**

** A bag contains 4 balls. Two balls are drawn at random (without replacement) and are found to be white. What is the probability that all balls in the bag are white ?**

** Solution.** We know that the number of white balls can’t be less than 2.

**Question.10. Differentiate x ^{sin x} + (sin x)^{cos x} with respect to x.**

**Solution.**

**OR**

**Solution.**

**Question.11.**

**Solution.**

**Question.12. The equation of tangent at (2,3) on the curve y ^{2} = ax^{3} + b is y = 4x-5. Find the value of a and b.**

**Solution.**

**Question.13.**

**Solution.**

**Question.14.**

** **

** Solution.**

**OR**

**Question.15.**

** **

** Solution.**

**Question.16. Solve the differential equation:**

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** Solution.**

**Question.17. Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes.**

** Solution.**

This is the required differential equation representing the given family of circles.

**Question.18. Solve the equation for x: sin ^{-1} x + sin^{-1} (1 – x) = cos^{-1} x**

**Solution.**

**OR**

**Solution.**

**Question.19. A trust invested some money in two type of bonds. The first bond pays 10% interest and second bond pays 12% interest. The trust received Rs 2,800 as interest. However, if trust had interchanged money in bonds, they would have got Rs 100 less as interest. Using matrix method, find the amount invested by the trust. Interest received on this amount will be given to Help age India as donation. Which value is reflected in this question ?**

** Solution.**

**SECTION – C**

**Question numbers 20 to 26 carry 6 marks each.**

** Question.20. There are two types of fertilisers ‘A’ and ‘W. ‘A’ consists of 12% nitrogen and 5% phsophoric acid whereas ‘B’ consists of 4% nitrogen and 5% phsophoric acid. After testing the soil conditions, farmer finds that he needs at least 12 kg of nitrogen and 12 kg of phsophoric acid for his crops. If ‘A’ costs Rs 10 per kg and ‘B’ cost Rs 8 per kg, then graphically determine how much of each type of fertiliser should be used so that nutrient requirements are met at a minimum cost.**

** Solution.**

**Question.21. Five bad oranges are accidently mixed with 20 good ones. If four oranges are drawn one by one successively with replacement, then find the probability distribution of number of bad oranges drawn. Hence find the mean and variance of the distribution.**

** Solution.** Let X denote the number of bad oranges in a draw of 4 oranges from a group of 20 good oranges and 5 bad oranges. Since there are 5 bad oranges in the group, therefore X can take values, 0,1,2,3,4.

Now, P(X = 0) = Probability of getting no bad orange. P(X = 0) = Probability of getting 4 good oranges

**Question.22. Find the position vector of the loot of perpendicular and the perpendicular distance from the point P with position**

** **

** Also find image of P in the plane.**

** Solution.**

**Question.23. Show that the binary operation *on A = R-{-l) defined as a*b = a + b + ab for all a, b e A is commutative and associative on A. Also find the identity element of * in A and prove that every element of A is invertible.**

** Solution.**

**Question.24. Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is 6√3r.**

** Solution.**

**OR**

** If the sum of lengths of hypotenuse and a side of a right angled triangle is given, show that area of triangle is maximum, when the angle between them is π/3.**

** Solution.**

**Question.25. Prove that the curves y ^{2} = 4x and x^{2} = 4y divide the area of square bounded by x = 0, x = 4, y = 4 and y = 0 into three equal parts.**

**Solution.**

**Question.26. Using properties of determinants, show that ∆ABC is isosceles if:**

** **

** Solution.**

**OR**

** A shopkeeper has 3 varieties of pens ‘A’, ‘B’ and ‘C’. Meenu purchased 1 pen of each variety for a total of Rs 21. Jeevan purchased 4 pens of ‘A’ variety, 3 pens of ‘B’ variety and 2 pens of ‘C’ variety for Rs 60. While Shikha purchased 6 pens of’A’ variety, 2 pens of ‘W variety and 3 pens of ‘C’ variety for Rs 70. Using matrix method, find cost of each variety of pen.**

** Solution.**