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CBSE Class 12 Mathematics Solved Question Paper 2009

Long Answer Type

31.
Evaluate: $\int_{0}^{π} \frac{e^{cosx}}{e^{cosx} + e^{- cosx}}$

$Let I = \int_{0}^{π} \frac{e^{cosx}}{e^{cosx} + e^{- cosx}} dx Using \int_{0}^{a} f (x) = \int_{0}^{a} f (a - x) dx I = \int_{0}^{π} \frac{e^{\cos (π - x)}}{e^{\cos (π - x)} + e^{- \cos (π - x)}} . dx 2 I = \int_{0}^{π} \frac{e^{- \cos x} + e^{\cos x}}{e^{\cos x} + e^{- \cos x}} . dx I = \frac{1}{2} \int_{0}^{π} dx = \frac{1}{2} [π - 0] I = \frac{π}{2}$

32.

Evaluate: $\int_{0}^{\frac{π}{2}} (2 \log \sin x - \log \sin 2 x) dx$

33.

Prove that the curves y²= 4x and x²= 4y divide the area of the square bonded by x = 0, x = 4, y = 4, and y = 0 into three equal parts.

34.

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.

35.

A diet is to contain at least 80 units of Vitamin A and 100 units of minerals. Two foods F1 and F2 are available. Food F1 cost Rs. 4 per unit and F2 costs Rs. 6 per unit. One unit of food F1 contains 3 units of Vitamin A and 4 units of minerals. One unit of food F2 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 minerals nutritional requirements.