Board Paper of Class 12-Commerce 2011 Maths Delhi(SET 3) - Solutions

Using properties of determinants, prove the following:

Find the values of a and b such that the following function f (x) is a continuous function:

Solve the following differential equation:

If two vectors are such that then find the value of

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

OR

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:

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.

A man is known to speak truth 3 out of 4 times. He throws a die and reports that it is a six.

Find the probability that it is actually a six.

Show that of all the rectangles of given area, the square has the smallest perimeter.

Using matrix method, solve the following system of equations:

OR

Using elementary transformations, find the inverse of the matrix

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.