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CBSE Class 12 Mathematics Solved Question Paper 2012

Long Answer Type

31.
Find the equation of the plane determined by the point A( 3, - 1, 2 ), B( 5, 2, 4 ) and C( -1, -1, 6 ) and hence find the distance between the plane and the point P( 6, 5, 9 ).

We know that, equation of a plane passing through 3 points,

$|\begin{matrix} x - x_{1} & y - y_{1} & z - z_{1} \\ x_{2} - x_{1} & y_{2} - y_{1} & z_{2} - z_{1} \\ x_{3} - x_{1} & y_{3} - y_{1} & z_{3} - z_{1} \end{matrix}| = 0 \Rightarrow |\begin{matrix} x - 3 & y + 1 & z - 2 \\ 2 & 3 & 2 \\ - 4 & 0 & 4 \end{matrix}| = 0 \Rightarrow (x - 3) (12 - 0) - (y + 1) (8 + 8) + (z - 2) (0 + 12) = 0 \Rightarrow 12 x - 36 - 16 y - 16 + 12 z - 24 = 0 \Rightarrow 12 x - 16 y + 12 z - 76 = 0 \Rightarrow 3 x - 4 y + 3 z - 19 = 0$

Also, perpendicular distance of P ( 6, 5, 9 ) to the plane 3 x - 4 y + 3 z - 19 = 0

$= \frac{|3 \times 6 - 4 \times 5 + 3 \times 9 - 19|}{\sqrt{9 + 16 + 9}} = \frac{6}{\sqrt{34}} units$

32.

Prove that $\int_{0}^{\frac{π}{4}} \sqrt{\tan x} + \sqrt{\cot x} dx = \sqrt{2} . \frac{π}{2}$

33.

Evaluate $\int_{1}^{3} 2 x^{2} + 5 x dx$ as a limit of sum.

34.

A girl throws a die. If she get a 5 OR 6, she tosses a coin three times and notes the number of heads. If she gets 1, 2, 3 OR 4, she tosses a coin two times and notes the number of heads obtained. If she obtained exactly two heads, what is the probability that she threw 1, 2, 3 OR 4 with the die?