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CBSE Class 12 Mathematics Solved Question Paper 2011

Long Answer Type

31.
Evaluate: $\int_{0}^{\frac{π}{2}} 2 \sin x \cos x \tan^{- 1} (\sin x) dx$

Consider the given integral

$I = \int_{0}^{\frac{π}{2}} 2 \sin x \cos x \tan^{- 1} (\sin x) dx Let t = \sin x \Rightarrow dt = \cos x dx When x = \frac{π}{2}, t = 1 When x = 0, t = 0 Now, \int 2 \sin x \cos x \tan^{- 1} (\sin x) dx = \int 2 t \tan^{- 1} t dt$

$= [\tan^{- 1} t] \int 2 t dt - \int [\frac{d}{dt} . (\tan^{- 1}) \int 2 t dt] dt = [\tan^{- 1} t] [2 . \frac{t^{2}}{2}] - \int (\frac{1}{1 + t^{2}} x 2 . \frac{t^{2}}{2}) dt = t^{2} \tan^{- 1} t - \int \frac{t^{2}}{1 + t^{2}} dt = t^{2} \tan^{- 1} t - \int [1 - \frac{1}{1 + t^{2}}] dt = t^{2} \tan^{- 1} t - t + \tan^{- 1} t$

$∴ I = \int_{0}^{\frac{π}{2}} 2 \sin x \cos x \tan^{- 1} (\sin x) dx = {[t^{2} \tan^{- 1} t - t + \tan^{- 1} t]}_{0}^{1} = [1^{2} \tan^{- 1} 1 - 1 + \tan^{- 1} 1] - [0^{2} \tan^{- 1} 0 - 0 + \tan^{- 1} 0] = [1 x \frac{π}{4} - 1 + \frac{π}{4}] - 0 = \frac{π}{4} - 1 + \frac{π}{4} = \frac{π}{2} - 1$

32.

Evaluate: $\int_{0}^{\frac{π}{2}} \frac{x \sin x \cos x}{\sin^{4} x + \cos^{4} x} dx$

33.

Find the equation of the plane which contains the line of intersection of the planes

$\vec{r} . (\hat{i} + 2 \hat{j} + 3 k) - 4 = 0, \vec{r} . (2 \hat{i} + \hat{j} - k) + 5 = 0$ and which is perpendicular to

the plane $\vec{r} . (5 \hat{i} + 3 \hat{j} - 6 k) + 8 = 0$ .

34.

A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftsman’s time in its making while a cricket bat takes 3 hours of machine time and 1 hour of craftsman’s time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftsman’s time. If the profit on a racket and on a bat is Rs20 and Rs 10 respectively, find the number of tennis rackets and crickets bats that the factory must manufacture to earn the maximum profit. Make it as an L.P.P and solve graphically.

35.

Suppose 5% of men and 0.25% of women have grey hair. A grey haired person is selected at random. What is the probability of this person being male? Assume that there are equal number of males and females.