**CBSE previous Year Solved Papers Class 12 Maths Outside Delhi 2009**

**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. Find the value of x, if**

** **

** Solution.**

**Question.2. Let * be a binary operation on N given by ****a*b = HCF (a, b),a,b ∈ N. ****Write the value of 22 * 4.**

** Solution:**

**Question.3.**

**Solution:**

**Question.4.**

**Solution:**

**Question.5.**

**Solution:**

**Question.6. **

**Solution:**

**Question.7. Find the value of x from the following:**

**Question.8. Find the value of p if**

** **

** Solution:**Given

**Question.9. Write the direction cosines of a line equally inclined to**

** the three coordinate axes.**

**Question.10.**

** **

** Solution:**Given

**SECTION – B**

**Question.11.The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cnVminute. When x = 8 cm and y = 6 cm, find the rate of change of (a) the perimeter, (b) the area of the rectangle.**

** Solution:**Given

**OR**

** Find the intervals in which the function f given by f (x) = sin x + cos x, 0 ≤ x ≤ 2π is strictly increasing or strictly decreasing.**

**Question.12.**

** Solution:**

**Question.13.**

**Solution: CaseI:**

**Question.14.**

**Solution:**

**Question.15.**

**Solution:**Given

**Question.16.On a multiple choice examination with three possible answer (out of which only one is correct) for each of the five questions, what is the probability that a candidate would get four or more correct answers just by guessing?**

** Solution:** Given

**Question.17. Using properties of determinants, prove the following:**

** **

** Solution:**

**Question.18.Solve the following differential equation:**

**Solution:** Given differential equation is

**Question.19. Solve the following differential equation:**

**Question.20. Find the shortest distance between the following two lines:**

** **

** Solution:**

**Question.21.Prove the following:**

**Question.22.**

**Solution:**

**SECTION – C**

**Question.23. Find the equation of the plane determined by the points A(3, – 1, 2), B(5, 2, 4) and C(- 1, – 1, 6). Also find the distance of the point P(6,5,9) from the plane.**

** **

** Solution:**

**Question.24. Find the area of the region included between the parabola y ^{2} = x and the line x + y = 2.**

**Solution:**Given equation probla and line are

**Question.25**

**Solution:**

**Question.26.Using matrices, solve the following system of equations:**

** **

** Solution:**

**Question.27. Coloured balls are distributed in three bags as shown in the following table:**

** **

** A bag is selected at random and then two balls are randomly drawn from the selected bag. They happen to be black and red. What is the probability that they came from bag I**

** Solution:**Let ** **E_{2} , ** **E_{2} , ** **E_{3}be the following events.

**Question.28. A dealer wishes to purchase a number of fans and sewing machines. He has only Rs 5,760 to invest and has a space for at most 20 items. A fan costs him Rs 360 and a sewing machine Rs 240. His expectation is that he can sell a fan at a profit of Rs 22 and a sewing machine at a profit of Rs 18. Assuming that he can sell all the items that he can buy, how should he invest his money in order to maximize the profit ? Formulate this as a linear programming problem and solve it graphically.**

** Solution:** Let the dealer buy x fans and y sewing machines. The LPP is to maximize profit

The feasible region is OBPCO which is shaded in the figure.

P is the point of intersection of the lines x + y = 20 and 3x + 2y = 48.

Solving these equations we get point P (8,12).

The vertices of the feasible region are 0(0, 0), B(0,20), P(8,12), C(16,0)

The value of objective function Z = 22x + 18y at these vertices are as follows:

The maximum profit is Rs 392 when 8 fans and 12 sewing machines are purchased.

**Question.29. If the sum of the lengths of the hypotenuse and a side of a right-angled is given, show that the area of the triangle is maximum when the angle between them is π/360**

** Solution:** Let, Base PQ = a

Hypotenuse, PR = b

Let 0 be the angle between them.

**SET II**

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

**SECTION – A**

**Question.2**

** **

** Solution:**

**Question.5.**

** **

** Solution:**

**Question.11.**

** **

** Solution:**

**Question.18. Find the shortest distance between the following two lines:**

**Question.19. Form the differential equation of the family of circles touching the y-axis at origin.**

** Solution :** The centre of the circle touching the y-axis at the origin lies on x-axis.

Let C (a, 0) be the centre of the circle and its radius is a Now, the equation of the family of circle with centre (a, 0) and radius a is

**Question.21. Using properties of determinants, prove the following:**

** **

** Solution :**

**SECTION – C**

**Question.25. Find the area of the region included between the parabola 4y = 3×2 and the line 3x – 2y +12 = 0.**

** Solution:** Given parabola

**SET III**

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

**SECTION – A**

**Question.7**

** **

** Solution:**

**Question.10. Find the value of x from the following:**

**SECTION – B**

**Question.13. Find the shortest distance between the following two lines:**

**Question.14. Form the differential equation representing the family of curves given by (x – a) ^{2}+ 2y^{2} = a^{2}, where a is an arbitrary constant.**

**Solution:**Given family of curves is

**Question.16. Using properties of determinants, prove the following:**

** **

** Solution:**Taking L.H.S.

**Question.18.**

** **

** Solution:**

**SECTION – C**

**Question.23. Find the area of the region bounded by the curves y ^{2} = 4ax and x^{2} = 4ay.**

**Solution:**The equations of parabola are

**Question.26. A man is known to speak the truth 3 out of 5 times. He throws a die and reports that it is a number greater than 4. Find the probability that it is actually a number greater than 4.**

** Solution:** Let E_{1} be the event that a number greater than 4 and E_{2} be the event that a number is not greater than 4.