Board Paper of Class 12-Commerce 2012 Maths All India(SET 3) - Solutions

Using properties of determinants, show that

Evaluate:

Or

Evaluate:

A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2 cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall?

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

OR

Differentiate with respect to x.

Show that f: N → N, given by is both one-one and onto.

OR

Consider the binary operations *: R × R → R and o: R × R → R defined as and a * b = |a − b| and a o b = a for all a, b ∈ R. Show that ‘*’ is commutative but not associative, ‘o’ is associative but not commutative.

Two cards are drawn simultaneously (without replacement) from a well-shuffled pack of 52 cards. Find the mean and variance of the number of red cards.

Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes.

OR

Find the particular solution of the differential equation

Find the coordinates of the point where the line through the points (3, −4, −5) and (2, −3, 1) crosses the plane 3x + 2y + z + 14 = 0

Find the particular solution of the following differential equation :

Prove the following:

Evaluate:

OR

Evaluate:

Prove that the radius of the right circular cylinder of greatest curved surface area which can be inscribed in a given cone is half of that of the cone.

OR

An open box with a square base is to be made out of a given quantity of cardboard of area c² square units. Show that the maximum volume of the box is cubic units.

Find the area of the region. .

A dietician wishes to mix two types of foods in such a way that the vitamin contents of the mixture contains atleast 8 units of vitamin A and 10 units of vitamin C. Food I contains 2 units/kg of vitamin A and 1 unit/kg of vitamin C while Food II contains 1 unit/kg of vitamin A and 2 units/kg of vitamin C. It costs Rs 5 per kg to purchase Food I and Rs 7 per kg to purchase Food II. Determine the minimum cost of such a mixture. Formulate the above as a LPP and solve it graphically.

Suppose a girl throws a die. If she gets a 5 or 6, she tosses a coin times and notes the number of heads. If she gets 1, 2, 3, or 4 she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one head, what is the probability that she threw 1, 2, 3, or 4 with the die?

Find the coordinates of the foot of perpendicular and the length of the perpendicular drawn from the point P (5, 4, 2) to the line . Also find the image of P in this line.

Using matrices, solve the following system of equations: