Board Paper of Class 12-Commerce 2011 Maths Delhi(SET 1) - Solutions

Consider the binary operation * on the set {1, 2, 3, 4, 5} defined by a * b = min. {a, b}. Write the operation table of the operation *.

Prove the following:

OR

Find the value of

Using properties of determinants, prove that

Find the value of ‘a’ for which the function f defined as

is continuous at x = 0

Differentiate.

OR

If x = a (θ − sinθ), y = a (1 + cosθ), find .

Sand is pouring from a pipe at the rate of 12 cm³/s. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4 cm?

OR

Find the points on the curve x² + y² − 2x − 3 = 0 at which the tangents are parallel to x-axis.

Solve the following differential equation:

e^x tan y dx + (1 − e^x) sec²y dy = 0

Solve the following differential equation:

Find a unit vector perpendicular to each of the vector and where

Find the angle between the following pair of lines:

and check whether the lines are parallel or perpendicular.

Probabilities of solving a specific problem independently by A and B are and respectively. If both try to solve the problem independently, find the probability that (i) the problem is solved (ii) exactly one of them solves the problem.

Using matrix method, solve the following system of equations:

OR

Using elementary transformations, find the inverse of the matrix

Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.

Using integration find the area of the triangular region whose sides have equations y = 2x + 1, y = 3x + 1 and x = 4.

OR

Find the equation of the plane which contains the line of intersection of the planes and which is perpendicular to the plane

A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftsman’s time in its making while a cricket bat takes 3 hours of machine time and 1 hour of craftsman’s time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftsman’s time.

If the profit on a racket and on a bat is Rs 20 and Rs 10 respectively, find the number of tennis rackets and cricket bats that the factory must manufacture to earn the maximum profit. Make it as an L.P.P. and solve graphically.

Suppose 5% of men and 0.25% of women have grey hair. A grey haired person is selected at random. What is the probability of this person being male? Assume that there are equal number of males and females.