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

Two collinear vectors having the same magnitude are equal.

Find the direction cosines of the vector `hati + 2hatj + 3hatk`.

Find the direction cosines of the vector joining the points A (1, 2, -3) and B (-1, -2, 1) directed from A to B.

Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are  `hati + 2hatj - hatk` and `-hati + hatj + hatk`  respectively, externally in the ratio 2:1.

Show that the points A, B and C with position vectors `veca = 3hati - 4hatj - 4hatk`, `vecb = 2hati - hatj + hatk` and `vecc = hati - 3hatj - 5hatk`, respectively form the vertices of a right angled triangle.

Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of the x-axis.

Find the value of x for which `x(hati + hatj + hatk)` is a unit vector.

If θ is the angle between two vectors `veca` and `vecb`, then `veca . vecb >= 0` only when ______.

Let `veca` and `vecb` be two unit vectors, and θ is the angle between them. Then `veca + vecb` is a unit vector if ______.

Find the rate of change of the area of a circle with respect to its radius r when r = 3 cm.

The volume of a cube is increasing at the rate of 8 cm³/s. How fast is the surface area increasing when the length of an edge is 12 cm?

The radius of a circle is increasing uniformly at the rate of 3 cm/s. Find the rate at which the area of the circle is increasing when the radius is 10 cm.

An edge of a variable cube is increasing at the rate of 3 cm/s. How fast is the volume of the cube increasing when the edge is 10 cm long?

A stone is dropped into a quiet lake and waves move in circles at the speed of 5 cm/s. At the instant when the radius of the circular wave is 8 cm, how fast is the enclosed area increasing?

The radius of a circle is increasing at the rate of 0.7 cm/s. What is the rate of increase of its circumference?

The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute. When x = 8 cm and y = 6 cm, find the rates of change of (a) the perimeter, and (b) the area of the rectangle.

A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is 15 cm.

A balloon, which always remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the later is 10 cm.