ISC (Arts) Class 12 - CISCE Important Questions for Mathematics

For the curve y² = 2x³ – 7, the slope of the normal at (2, 3) is ______.

The intevral in which the function f(x) = 5 + 36x – 3x² increases will be ______.

A running track of 440 m is to be laid out enclosing a football field. The football field is in the shape of a rectangle with a semi-circle at each end. If the area of the rectangular portion is to be maximum,then find the length of its sides. Also calculate the area of the football field.

In which one of the following intervals is the function f(x) = x³ – 12x increasing?

Given that `1/y + 1/x = 1/12` and y decreases at a rate of 1 cms^–1, find the rate of change of x when x = 5 cm and y = 1 cm.

Mrs. Roy designs a window in her son’s study room so that the room gets maximum sunlight. She designs the window in the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12 m, find the dimensions of the window that will admit maximum sunlight into the room.

Sumit has bought a closed cylindrical dustbin. The radius of the dustbin is ‘r' cm and height is 'h’ cm. It has a volume of 20π cm³.

1. Express ‘h’ in terms of ‘r’, using the given volume.
2. Prove that the total surface area of the dustbin is `2πr^2 + (40π)/r`
3. Sumit wants to paint the dustbin. The cost of painting the base and top of the dustbin is ₹ 2 per cm² and the cost of painting the curved side is ₹ 25 per cm². Find the total cost in terms of ‘r’, for painting the outer surface of the dustbin including the base and top.
4. Calculate the minimum cost for painting the dustbin.

Find the interval in which the function f(x) = x²e^–x is strictly increasing or decreasing.

Find the equation to the tangent at (0, 0) on the curve y = 4x² – 2x³

A kite is being pulled down by a string that goes through a ring on the ground 8 meters away from the person pulling it. If the string is pulled in at 1 meter per second, how fast is the kite coming down when it is 15 meters high?

Evaluate : `int_0^4(|x|+|x-2|+|x-4|)dx`

Find : `int((2x-5)e^(2x))/(2x-3)^3dx`

Evaluate :

`int_e^(e^2) dx/(xlogx)`

Find the following integrals:

`int (x^3 - x^2 + x - 1)/(x - 1) dx`

By using the properties of the definite integral, evaluate the integral:

`int_0^(pi/4) log (1+ tan x) dx`

By using the properties of the definite integral, evaluate the integral:

`int_0^(pi/2) (sin x - cos x)/(1+sinx cos x) dx`

Evaluate `int(x^3+5x^2 + 4x + 1)/x^2  dx`

Evaluate `int tan^(-1) sqrtx dx`

Evaluate `int (x-1)/(sqrt(x^2 - x)) dx`

Evaluate `int_0^(pi/2) cos^2x/(1+ sinx cosx) dx`