Board Paper of Class 12-Commerce 2010 Maths (SET 2) - Solutions

Prove the following:

OR

Solve for x:

A family has 2 children. Find the probability that both are boys, if it is known that

(i) at least one of the children is a boy,

(ii) the elder child is a boy.

Show that the relation S in the set A = {x ∈ Z: 0 ≤ x ≤ 12} given by S = {(a, b): a, b ∈ Z, |a − b| is divisible by 4} is an equivalence relation. Find the set of all elements related to 1.

If , then find the value of A² − 3A + 2I.

Find the points on the line at a distance of 5 units from the point P (1, 3, 3).

OR

Find the distance of the point P (6, 5, 9) from the place determined by the points A (3, −1, 2), B (5, 2, 4) and C (−1, −1, 6).

If and , find a vector of magnitude 6 units which is parallel to the vector

OR

Let and Find a vector which is perpendicular to both and and = 18.

Solve the following differential equation:

|x| ≠ 1

OR

Solve the following differential equation:

Evaluate:

If , −1 ≤ x ≤1, then show that

Show that the following differential equation is homogeneous, and then solve it:

If find

Evaluate the following:

Find the equations of the tangent and the normal to the curve at

Find the equation of the plane passing through the point P (1, 1, 1) and containing the line. Also, show that the plane contains the line

Using properties of determinants, prove the following

OR

Find the inverse of the following matrix using elementary operations

A bag contains 4 balls. Two balls are drawn at random, and are found to be white. What is the probability that all balls are white?

Find the area of the circle 4x² + 4y² = 9 which is interior to the parabola x² = 4y.

OR

Using integration, find the area of the triangle ABC, coordinates of whose vertices are A (4, 1), B (6, 6) and C (8, 4)

One kind of cake required 300g of flour and 15g of fat, another kind of cake requires 150g of flour and 30g of fat. Find the maximum number of cakes which can be made from 7.5 kg of flour and 600g of fat, assuming that there is no shortage of the other ingredients used in making the cakes. Make it as an L.P.P and solve it graphically.

If the length of three sides of a trapezium other than the base is 10 cm each, find the area of the trapezium, when it is maximum.