**CBSE previous Year Solved Papers Class 12 Maths Outside Delhi 2014**

**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. If R = {(x, y): x + 2y = 8} is a relation on N, write the range of R.**

** Solution.**

**Question.2.**

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** Solution.**

**Question.3. If A is a square matrix such that A ^{2} = A, then write the value of 7A – (I + A)^{3}, where I is an identity matrix.**

**Solution.**

**Question.4.**

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** Solution.**

**Question.5.**

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** Solution.**

**Question.6.**

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** Solution.**

**Question.7.**

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** Solution.**

**Question.8.**

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** Solution.**

**Question.9.**

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** Solution.**

**Question.10.**

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** Solution.**

**SECTION – B**

**Question.11.**

**Solution.**

**Question.12.**

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** Solution.**

**OR**

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** Solution.**

**Question.13. Using properties of determinants, prove that:**

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** Solution.**

**Question.14.**

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** Solution.**

**Question.15.**

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** Solution.**

**Question.16. Find the value (s) of x for which y = [x(x – 2)] ^{2} is an increasing function.**

**Solution.**

**OR**

**Solution.**

**Question.17.**

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** Solution.**

**OR**

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** Solution.**

**Question.18. Find the particular solution of the differential equation**

** dy/dx= 1 + x + y + xy, given that y = 0 when x = 1.**

** Solution.**

**Question.19.**

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** Solution.**

**Question.20.**

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** Solution.**

**OR**

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** Solution.**

**Question.21.**

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** Solution.**

**Question.22. An experiment succeeds thrice as often as it fails. Find the probability that in the next five trials, there will be at least 3 successes.**

** Solution.**

**SECTION – C**

**Question.23. Two schools A and B want to award their selected students on the values of sincerity, truthfulness and helpfulness. The school A wants to award Rs x each, Rs y each and Rs z each for the three respective values to 3,2 and 1 students respectively with a total award money of Rs 1, 600. School B wants to spend Rs 2,3000 to award its 4,1 and 3 students on the respective values (by giving the same award money to the three values as before). If the total amount of award for one prize on each value is Rs 900, using matrices, find the award money f or each value. Apart from these three values, suggest one more value which should be considered for award.**

** Solution.**

Apart from the three values, sincerity, truthfulness and helpfulness, another value for award should be discipline.

**Question.24. Show that the altitude of the right circular cone of maximum volume that can be described in a sphere of radius r is 4r/3. Also show that the maximum volume 3 of the cone is 8/27 of the volume of the sphere.**

** Solution.**

**Question.25.**

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** Solution.**

**Question.26. Using integration, find the area of the region bounded by the triangle whose vertices are (- 1, 2), (1,5) and (3,4).**

** Solution.**

**Question.27. Find the equation of the plane through the line of intersection of the planes x + y + z=1 and 2x+3y+4z=5 which is perpendicular to the plane x – y + z = 0. Also find the distance of the plane obtained above, from the origin.**

** Solution .** Equation of any plane through the line of intersection of the planes.

**OR**

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** Solution.**

**Question.28. A manufacturing company makes two types of teaching aids A and B of Mathematics for class XII. Each type of A requires 9 labour hours of fabricating and 1 labour hour for finishing. Each type of B requires 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available per week are 180 and 30 respectively. The company makes a profit of Rs 80 on each piece of type A and Rs 120 on each piece of type B. How many pieces of type A and type B should be manufactured per week to get a maximum profit ? Make it as an LPP and solve graphically. What is the maximum profit per week ?**

** Solution.**

**Question.29. There are three coins. One is a two-headed coin (having head on both faces), another is a biased coin that comes up heads 75% of the times and third is also a biased coin that comes up tails 40% of the times. One of the three coins is chosen at random and tossed, and it shows heads. What is the probability that it was the two-headed coin ?**

** Solution.** Let A be the two headed coin, B be the biased coin showing up heads 75% of the times and C be the biased coin showing up tails 40% (i.e., showing up heads 60%) of the times.

**OR**

** Two numbers are selected at random (without replacement) from the first six positive integers. Let X denote the larger of the two numbers obtained. Find the probability distribution of the random variable X, and hence find the mean of the distribution.**

** Solution.**

**SET II**

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

**SECTION – A**

**Question.9.**

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** Solution.**

**Question.10.**

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** Solution.**

**SECTION – B**

**Question.19. Using properties of determinants, prove that**

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** Solution.**

**Question.20.**

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** Solution.**

**Question.21. Find the particular solution of the differential equation x(1 + y ^{2}) dx – y(1 + x^{2}) dy = 0, given that y = 1 when x = 0.**

**Solution.**

**Question.22.**

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** Solution.**

**SECTION – C**

**Question.28.**

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** Solution.**

**Question.29. Prove that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is 2R/√3. Also find the maximum volume.**

** Solution.**

**SET III**

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

**SECTION – A**

**Question.9.**

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** Solution.**

**Question.10.**

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** Solution.**

**SECTION-B**

**Question.19. Using properties of determinants, prove that:**

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** Solution.**

**Question.20.**

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** Solution.**

**Question.21. Find the particular solution of the differential equation log (dx/dy) =3x + 4y given that y = 0 when x = 0.**

** Solution.**

**Question.22.**

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** Solution.**

**SECTION-C**

**Question.28. If the sum of the lengths of the hypotenuse and a side of a right triangle is given, show that the area of the triangle is maximum, when the angle of between them is 60°.**

** Solution.**

**Question.29.**

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** Solution.**