Commerce (English Medium) Class 12 - CBSE Question Bank Solutions for Mathematics

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?

A particle moves along the curve 6y = x³ +2. Find the points on the curve at which the y-coordinate is changing 8 times as fast as the x-coordinate.

The radius of an air bubble is increasing at the rate  `1/2`  cm/s. At what rate is the volume of the bubble increasing when the radius is 1 cm?

A balloon, which always remains spherical, has a variable diameter  `3/2 (2x +   1)` Find the rate of change of its volume with respect to x.

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?

The total cost C(x) in rupees associated with the production of x units of an item is given by C(x) = 0.007x³ – 0.003x² + 15x + 4000. Find the marginal cost when 17 units are produced

The total revenue in rupees received from the sale of x units of a product is given by R(x) = 13x² + 26x + 15. Find the marginal revenue when x = 7.

The rate of change of the area of a circle with respect to its radius r at r = 6 cm is ______.

The total revenue in rupees received from the sale of x units of a product is given by R(x) = 3x² + 36x + 5. The marginal revenue, when x = 15 is ______.

The two equal sides of an isosceles triangle with fixed base b are decreasing at the rate of 3 cm per second. How fast is the area decreasing when the two equal sides are equal to the base?

The sum of the perimeter of a circle and square is k, where k is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle.

Find the area of the region bounded by the curve y² = x and the lines x = 1, x = 4 and the x-axis.

Find the area of the region bounded by y² = 9x, x = 2, x = 4 and the x-axis in the first quadrant.

Find the area of the region bounded by x² = 4y, y = 2, y = 4 and the y-axis in the first quadrant.

Find the area of the region bounded by the ellipse  `x^2/16 + y^2/9 = 1.`

Find the area of the region bounded by the ellipse `x^2/4 + y^2/9 = 1.`

Find the area of the region in the first quadrant enclosed by x-axis, line x = `sqrt3` y and the circle x² + y² = 4.

Find the area of the smaller part of the circle x² + y² = a² cut off by the line  `x = a/sqrt2`

The area between x = y² and x = 4 is divided into two equal parts by the line x = a, find the value of a.

Find the area of the region bounded by the parabola y = x2 and y = |x| .