**CBSE previous Year Solved Papers Class 12 Maths 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**

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

**SECTION – A**

**Question.1.**

** Solution.**

**Question.2. Write a unit vector in the direction of**

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** Solution.**

**Question.3.**

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** Solution.**

**Question.4. If matrix A = [12 3], write AA’, where A’ is transpose of matrix X**

** Solution.**

**Question.5. **

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** Solution.**

**Question.6. Using principal value, evaluate the following**

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** Solution.**

**Question.7.**

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** Solution.**

**Question.8.**

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** Solution.**

**Question.9.If the binary operation * on the set of integers Z, is defined by a * b = a + 3b ^{2}, then find the value of 2*4.**

**Solution.**Given a*b = a + 3b

^{2}

.

^{.}. 2*4 = 2 + 3(4)

^{2}= 2 + 3 x 16 = 50.

**Question.10.If A is an invertible matrix of order 3 and |A| =5, then ****find |adj.A|.**

** Solution.**

**SECTION – B**

**Question.11.**

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** Solution.**

**Question.12.**

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** Solution.**

**Solution.**

**Question.13.Find the value of A so that the lines**

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** perpendicular to each other.**

** Solution:** Given equations are

**Question.14.Solve the following differential equation:**

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** Solution.**

**Question.15.Find the particular solution, satisfying the given condition, for the following differential equation:**

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

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** Solution.**

**Question.17.A die is thrown again and again until three sixes are obtained. Find the probability of obtaining the third six in the sixth throw of the die.**

** Solution.**

**Getting a six third time at the 6th throw means outcome at 6th place is a six. Out of first five throws, two outcomes are sixes, three outcome are non sixes.**

**Question.18.Differentiate the following function w.r.t.**

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** Solution.**

**Question.19.**

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** Solution.**

**Solution.**

**Question.20. Prove that the relation R in the set A = {1,2,3,4,5} given by R={(a, b): \a-b\ is even}, is an equivalence relation.**

** Solution:** A = {1,2,3,4,5}

R = {(a, b): \ a – b \ is even}

For equivalence relation, the relation must be (i) reflexive, (ii) symmetric and (iii) transitive.

Now, R = {(1,3), (2,4), (3,5), (1,5), (1,1), (2,2), (3,3), (4,4), (5,5), (3,1), (4,2), (5,3), (5,1)}

(i) Reflexive: Since 1R1,2R2,3R3,4R4,5R5

.^{.}.R is reflexive.

(ii)Symmetric: Since

**Question.21.**

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** Solution.**

**Question.22.**

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** Solution.**

**SECTION – C**

**Question.23.Find the volume of the largest cylinder that can be inscribed in a sphere of radius r.**

** Solution:** Let the cylinder inscribed in the given sphere of radius r, have height h and base radius a.

In Right angled ΔAOB,

**A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8 m ^{3}. If building of tank costs Rs 70 per sq. metre for the base and Rs 45 per sq. metre for sides, what is the cost of least expensive tank ?**

**Solution:**

**Question.24.A diet is to contain at least 80 units of vitamin A and 100 units of minerals. Two foods F _{1} and F2 are available. Food F_{1} costs Rs. 4 per unit and F_{2} costs ? 6 per unit. One unit of food F_{1} contains 3 units of vitamin A and 4 units of minerals. One unit of food F_{2} contains 6 units of vitamin A and 3 units of minerals. Formulate this as a linear programming problem and find graphically the minimum cost for diet that consists of mixture of these two foods and also meets the minimal nutritional requirements.**

**Solution:**

**Question.25.Three bags contain balls as shownin the table below:**

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** A bag is chosen at random and two balls drawn from it. They happen to be white and red. What is the probability that they came from the III bag.**

** Solution.**

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

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** Solution.**

**Question.27.**

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** Solution.**

**Question.28.Using the method of integration, find the area of the region bounded by the lines**

** 2x + y = 4,3x – 2y = 6 and x – 3y + 5 = 0.**

** Solution.**

**Question.29. Find the equation of the plane passing through the point (-1,3,2) and perpendicular to each of the planes x + 2y + 3z = 5 and 3x + 3y + z = 0.**

** Solution :** Let the equation of the plane through (-1,3,2) be

a(x + 1)+ b(y-3) + c(z-2) = 0 …(1)

**SET II**

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

**SECTION-A**

**Question.2.**

**Solution.**

**Question.7.**

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** Solution.**

**Question.11.Differentiate the following function w.r.t:**

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** Solution.**

**Question.18.Find the value of A. so that the lines are perpendicular to each other.**

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** Solution.**

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

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** Solution.**The given differential equation is

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

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** Solution.**

**SECTION-C**

**Question.23.Two groups are competing for the position on the Board of Directors of a corporation. The probabilities that the first and the second groups will win are 0.6 and 0.4 respectively. Further, if the first group wins, the probability of introducing a new product is 0.7 and the corresponding probability is 0.3 if the second group wins. Find the probability that the new product was introduced by the second group.**

** Solution.**

**Question.26.Prove that the curves y ^{2} = 4x and x^{2} = 4y divide area of the square bounded by x = 0, x = 4, y = 4 and y = 0 into three equal parts.**

**Solution.**

**SET III**

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

**SECTION – A**

**Question.4.**

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** Solution.**

**Question.9.**

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** Solution.**

**SECTION-B**

**Question.15.Using properties of determinants, prove the following:**

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** Solution.**

**Question.17.**

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** Solution.**

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

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** Solution.** Given differential equation is

**Question.20.Find the value of X so that the following lines are perpendicular to each other**

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**SECTION-C**

**Question.24.Find the area of the region enclosed between the two circles x ^{2} + y^{2} = 9 and (x – 3) ^{2}+ y^{2} = 9.**

**Solution.**Equations of given circles are

**Equation (1) is a circle with centre at origin 0(0,0) and radius 3. Equations (2) is a circle with centre Q(3,0) and radius 3.**

**Required area of bounded region OPQRO between two circles = area of bounded region OPRO + area of bounded region PQRP = 2 (Area in 1st quadrant).**

**Question.27.There are three coins. One is a two headed coin (having head on both faces), another is biased coin that comes up tails 25% of the times and the third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows heads, what is the probability that it was the two headed coin ?**

** Solution.**