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CBSE Class 12 Mathematics Solved Question Paper 2008

Long Answer Type

31.

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

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?

Let x and y respectively be the number of machines A and B, which the factory owner should buy.

Now according to the given information, the linear programming problem is:

maximise Z = 60x + 40y

Subject to the constraints

1000x + 1200 y $\leq$ 9000

$\Rightarrow 5 x + 6 y \leq 45 . . . . . . . . . . . . (1) 12 x + 8 y \leq 72 \Rightarrow 3 x + 2 y \leq 18 . . . . . . . . . . . . . (2) x \geq 0, y \geq 0 . . . . . . . . . . . . . (3)$

The inequalities (1), (2), and (3) can be graphed as:

The shaded portion OABC is the feasible region.

The value of Z at the corner points are given in the following table.

corner point	Z = 60x + 40y
0(0, 0)	0
A $(0, \frac{15}{2})$	300
B $(\frac{9}{4}, \frac{45}{8})$	360 $\to$	maximum
C (6, 0)	360 $\to$	Maximum

The maximum value of Z is 360 units, which is attained at B $(\frac{9}{4}, \frac{45}{8})$ , and c( 6, 0 ).

Now the number of machines cannot be in fraction.

Thus, to maximise the daily output, 6 machines of type A and no machine of type B need to be bought.

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?