**CBSE previous Year Solved Papers Class 12 Maths Outside Delhi 2010**

**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.**

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** Solution.**

**Question.2.Write the principal value of sec ^{-1} (-2).**

**Solution.**

**Question.3.What positive value of x makes the following pair of determinants equal ?**

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** Solution.**

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**Question.4.**

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** Solution.**

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**Question.5.Write the adjoint of the following matrix:**

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** Solution.**

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**Question.6.Write the value of the following integral:**

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** Solution.**

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**Question.7. A is a square matrix of order 3 and | A | =7. Write the value of | adj. A |.[1]**

** Solution.**

**Question.8.Write the distance of the following plane from the origin**

** :2x – y + 2z +1 = 0**

** Solution.**Given plane is2x – y + 2z+1 = 0 .

Its distance from origin is

**Question.9. Write a vector of magnitude 9 units in the direction of vector**

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** Solution.**

**Question.10.**

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** Solution.**

**Question.11. A family has 2 children. Find the probability that both are boys, if it is known that**

** (i) at least one of the children is a boy,**

** (ii)the elder child is a boy.**

** Solution:** Let eldest boy and girl be represented by capital B and G respectively and youngest boy and girl be denoted by small letter b and g respectively.

**Question.12. Show that relation Sin the set A={x ∈ Z:0 ≤ x ≤ 12} given**

** by S = {(a, b): a, b∈ Z, | a – b | is divisible by 4} is an equivalence relation. Find the set of all elements related to 1.**

** Solution:** Reflexivity:

**Question.13. Prove the following**

**Solution**.

**Question.14. Express the following matrix as the sum of a symmetric and skew symmetric matrix, and verify your result:**

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** Solution.**

**Question.15.**

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** Solution.**

**Solution.** Given that

**Question.16.**

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** Solution.**

**Find the distance of the point P(6,5,9) from the plane determined by the points A(3, – 1, 2), B(5,2,4) and C(-1, -1,6).**

** Solution.**

**Question.17. Solve the following differential equation:**

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** Solution.**

** Solution.**

**Question.18.Show that the differential equation is homogeneous and solve it.**

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** Solution.**

**Question.19.**

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** Solution.**

**Question.20. Evaluate the following:**

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** Solution.**

**Question.21.**

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** Solution.**

**Question.22.**

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** Solution.**

**Question.23. Using properties of determinants, prove the following:**

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** Solution.**

**Solution.**

**Question.24. A bag contains 4 balls. Two balls are drawn at random and are found to be white. What is the probability that all balls are white ?**

** Solution:** Consider the following events:

E_{1} = The bag contains two white and two other colour balls.

E_{2} = The bag contains three white and one other coloured ball. –

E_{3} = The bag contains all white balls.

A = Two ball drawn are white.

Clearly,

**Question.25.One kind of cake requires 300g of flour and 15g of fat, another kind of cake requires 150g of flour and 30g of fat. Find the maximum number of cakes which can be made from 7.5 kg of flour and 600g of fat, assuming that there is no shortage of the other ingredients used in making the cakes. Make it as an L.P.P. and solve it graphically.**

** Solution:** Let x be the number of cakes of first kind and y is of second kind. We make the following table from the given data:

**Question.26.Find the coordinates of the foot of the perpendicular**

** and the perpendicular distance of the point P(3, 2,1) from the plane 2x-y + z +1 = 0. Find also the image of the point in the plane.**

** Solution.**Equation of the plane is given as

**Question.27. Find the area of the circle 4x ^{2}+4y^{2}=9 Which is interior to the parabola x^{2}=4y**

**Solution.**Given curves are

and

Clearly, the first one represents a circle with centre at origin and radius at 1.5 umts. The second one represents a parabola opening upwards.

**OR**

**Using integration, find the area of the triangle ABC, coordinates of whose vertices are A(4,1), B(6,6) and C(8, 4).**

**Solution: First we find the equation of the sides of ΔABC**

**Question.28.If the length of three sides of a trapezium other than the base is 10 cm each, find the area of the trapezium, when it is maximum.**

** Solution:** Let ABCD be the given trapezium.

**Question.29. Find the intervals in which the following function f(x) = 20 – 9x + 6x ^{2} – x^{3} is**

**(a) strictly increasing,**

**(b) strictly decreasing.**

**Solution.**

**SET II**

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

**SECTION – A**

**Question.10.**

**Solution.**

**SECTION – B**

**Question.11.Prove the following:**

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** Solution.**

**Solution.** We have,

**Question.14.**

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** Solution.**

**Question.18.**

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** Solution.**

**Question.20. Show that the following differential equation is homogeneous, and then solve it:**

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** Solution.**

**SECTION – C**

**Question.23. Find the equations of the tangent and the normal to the**

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** Solution.**

**Question.24.Find the equation of the plane passing through the point P(1,1,1) and containing the line that the plane contains**

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** Solution.**

**SET III**

**Note: Except for the following questions, all the remaining questions have been asked in previous sets.**

**SECTION – A**

**Question.6.**

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** Solution.**

**Question.7.**

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** Solution.**

**SECTION – B**

**Question.11. Show that the relation S defined on the set N x N by (a, b) S (c, d) ⇒ a + d = b + c is an equivalence relation.**

** Solution.**

**Question.15. For the following matrices A and B, verify that (AB)’ = B’A’ where**

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** Solution.** Givsen that,

**Question.17.Solve the following differential equation:**

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** Solution.**

**Solution.**

**Question.20.**

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** Solution.**

**SECTION – C**

**Question.23. Using matrices, solve the following system of equations:**

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** Solution.** We have,

**Solution.**

**Question.25. Show that the volume of the greatest cylinder that can be inscribed in a cone of height ‘h’ and semi-vertical**

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** Solution.**Let us consider height of cone = h

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