Board Paper of Class 12-Commerce 2008 Maths Delhi(SET 1) - Solutions

(i) Is the binary operation* defined on set N, given by for all, commutative?

(ii) Is the above binary operation* associative?

Prove the following:
$=\frac{\mathrm{\pi }}{4}$

Let .Express A as the sum of two matrices such that one is symmetric and the other is skew symmetric.

OR

If, verify that A²−4A − 5I = 0

For what value of kis the following function continuous at x= 2?

Differentiate the following with respect to x:

Find the equation of tangent to the curve x= sin 3t, y = cos 2t, at t =

Solve the following differential equation:

(x²− y²) dx + 2 xy dy = 0

Giventhat y = 1 when x = 1

OR

Solve the following differential equation:

, if y = 1 when x = 1

Solve the following differential equation:

If and , find a vector such that and

OR

If and and, show that the angle between and is 60°

Find the shortest distance between the following lines:

and

OR

Find the point on the line at a distance from the point (1, 2, 3).

A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability distribution of the number of successes.

Using properties of determinants, prove the following

Show that the rectangle of maximum area that can be inscribed in a circle is a square.

OR

Show that the height of the cylinder of maximum volume that can be inscribed in a cone of height his .

Using integration find the area of the region bounded by the parabola y²= 4x and the circle 4x²+ 4y²= 9.

Evaluate:

Find the equation of the plane passing through the point (−1, − 1, 2) and perpendicular to each of the following planes:

OR

Find the equation of the plane passing through the points (3, 4, 1) and (0, 1, 0) and parallel to the line

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:

He has maximum area of 9000 m²available, and 72 skilled labourers who can operate both the machines. How many machines of each type should he buy to maximise the daily output?

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.