Board Paper of Class 12-Science 2013 Maths All India(SET 2) - Solutions

Differentiate the following with respect to x :

Evaluate :

OR

Evaluate :

Using properties of determinants, prove the following:

Show that:

OR

Solve the following equation :

Consider f : R₊→ [4, ∞) given by f(x) = x² + 4. Show that f is invertible with the inverse f⁻¹ of f given by, where R₊ is the set of all non-negative real numbers.

Find the value of k, for which

is continuous at x = 0.

OR

If x = a cos³θ and y = a sin³θ, then find the value of at .

Show that the lines

are intersecting. Hence find their point of intersection.

OR

Find the vector equation of the plane through the points (2, 1, −1) and (−1, 3, 4) and perpendicular to the plane x − 2y + 4z = 10.

The probabilities of two students A and B coming to the school in time are respectively. Assuming that the events, ‘A coming in time’ and ‘B coming in time’ are independent, find the probability of only one of them coming to the school in time.

Write at least one advantage of coming to school in time.

If x^y = e^{x − y}, prove that

Evaluate:

Evaluate:

If , then find the value of λ, so that are perpendicular vectors.

Find the equation of the plane passing through the line of intersection of the planes and , whose perpendicular distance from origin is unity.

OR

Find the vector equation of the line passing through the point (1, 2, 3) and parallel to the planes.

In a hockey match, both teams A and B scored same number of goals up to the end of the game, so to decide the winner, the referee asked both the captains to throw a die alternately and decided that the team, whose captain gets a six first, will be declared the winner. If the captain of team A was asked to start, find their respective probabilities of winning the match and state whether the decision of the referee was fair or not.

A manufacturer considers that men and women workers are equally efficient and so he pays them at the same rate. He has 30 and 17 units of workers (male and female) and capital respectively, which he uses to produce two types of goods A and B. To produce one unit of A, 2 workers and 3 units of capital are required while 3 workers and 1 unit of capital is required to produce one unit of B. If A and B are priced at Rs 100 and Rs 120 per unit respectively, how should he use his resources to maximise the total revenue? Form the above as an LPP and solve graphically.

Do you agree with this view of the manufacturer that men and women workers are equally efficient and so should be paid at the same rate?

Find the area of the greatest rectangle that can be inscribed in an ellipse.

OR

Find the equations of tangents to the curve 3x² − y² = 8, which pass through the point.

The management committee of a residential colony decided to award some of its members (say x) for honesty, some (say y) for helping others and some others (say z) for supervising the workers to keep the colony neat and clean. The sum of all the awardees is 12. Three times the sum of awardees for cooperation and supervision added to two times the number of awardees for honesty is 33. If the sum of the number of awardees for honesty and supervision is twice the number of awardees for helping others, using matrix method, find the number of awardees of each category. Apart from these values, namely, honesty, cooperation and supervision, suggest one more value which the management of the colony must include for awards.

Find the area of the region {(x, y): y²≤ 6ax and x² + y²≤ 16a²} using method of integration.

Show that the differential equation is homogeneous. Find the particular solution of this differential equation, given that when x = 1.