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CBSE Class 12 Mathematics Solved Question Paper 2008

Long Answer Type

31.
Evaluate: $\int_{- a}^{a} \sqrt{\frac{a - x}{a + x}} dx$

$I = \int_{- a}^{a} \sqrt{\frac{a - x}{a + x}} dx = \int_{- a}^{a} \frac{a - x}{\sqrt{a^{2} - x^{2}}} dx = \int_{- a}^{a} \frac{a}{\sqrt{a^{2} - x^{2}}} dx - \int_{- a}^{a} \frac{x}{\sqrt{a^{2} - x^{2}}} dx = I_{1} + I_{2} Where I_{1} = \int_{- a}^{a} \frac{a}{\sqrt{a^{2} - x^{2}}} dx, which is the integral of an even function And I_{2} = \int_{- a}^{a} \frac{x}{\sqrt{a^{2} - x^{2}}} dx, which is the integral of an even function, and so I_{2} = 0$

Now,

$I = I_{1} = \int_{- a}^{a} \frac{a}{\sqrt{a^{2} - x^{2}}} dx = 2 \int_{0}^{a} \frac{a}{\sqrt{a^{2} - x^{2}}} dx = 2 a \int_{0}^{a} \frac{1}{\sqrt{a^{2} - x^{2}}} dx = 2 a . {[\sin^{- 1} (\frac{x}{a})]}_{0}^{a} = 2 a . [\sin^{- 1} 1 - \sin^{- 1} 0] = 2 a (\frac{π}{2}) = πa$

32.

Find the equation of the plane passing through the point (−1, − 1, 2) and perpendicular to each of the following planes: 2x + 3y – 3z = 2 and 5x – 4y + z = 6

33.

Find the equation of the plane passing through the points (3, 4, 1) and (0, 1, 0) and parallel to the line $\frac{x + 3}{2} = \frac{y - 3}{7} = \frac{z - 2}{5}$

34.

A factory owner purchases two types of machines, A and B for his factory. The requirements and the limitations for the machines are as follows:

Machine	Area occupied	Labour force	daily output ( in units )
A	1000 m²	12 men men	60
B	1200 m²	8 men	40

He has maximum area of 9000 m2 available, and 72 skilled labourers who can operate both the machines. How many machines of each type should he buy to maximise the daily output?

35.

An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of an accident involving a scooter, a car and a truck are 0.01, 0.03 and 0.15 respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver?