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

If the rate of change of volume of a sphere is equal to the rate of change of its radius then the surface area of a sphere is ____________.

A man 1.6 m tall walks at the rate of 0.3 m/sec away from a street light that is 4 m above the ground. At what rate is the tip of his shadow moving? At what rate is his shadow lengthening?

Find the direction ratio and direction cosines of a line parallel to the line whose equations are 6x − 12 = 3y + 9 = 2z − 2

The Cartesian equation of a line AB is: `(2x - 1)/2 = (y + 2)/2 = (z - 3)/3`. Find the direction cosines of a line parallel to line AB.

If the sum of numbers obtained on throwing a pair of dice is 9, then the probability that number obtained on one of the dice is 4, is ______.

Assertion (A): If a line makes angles α, β, γ with positive direction of the coordinate axes, then sin² α + sin² β + sin² γ = 2.

Reason (R): The sum of squares of the direction cosines of a line is 1.

A particle moves along the curve 3y = ax³ + 1 such that at a point with x-coordinate 1, y-coordinate is changing twice as fast at x-coordinate. Find the value of a.

A line l passes through point (– 1, 3, – 2) and is perpendicular to both the lines `x/1 = y/2 = z/3` and `(x + 2)/-3 = (y - 1)/2 = (z + 1)/5`. Find the vector equation of the line l. Hence, obtain its distance from the origin.

Equation of line passing through origin and making 30°, 60° and 90° with x, y, z axes respectively, is ______.

If A and B are two events such that `P(A/B) = 2 xx P(B/A)` and P(A) + P(B) = `2/3`, then P(B) is equal to ______.

If the equation of a line is x = ay + b, z = cy + d, then find the direction ratios of the line and a point on the line.

If the circumference of circle is increasing at the constant rate, prove that rate of change of area of circle is directly proportional to its radius.

If for two events A and B, P(A – B) = `1/5` and P(A) = `3/5`, then `P(B/A)` is equal to ______.

If points A, B and C have position vectors `2hati, hatj` and `2hatk` respectively, then show that ΔABC is an isosceles triangle.

If equal sides of an isosceles triangle with fixed base 10 cm are increasing at the rate of 4 cm/sec, how fast is the area of triangle increasing at an instant when all sides become equal?

Equation of a line passing through point (1, 2, 3) and equally inclined to the coordinate axis, is ______.

If a line makes angles of 90°, 135° and 45° with the x, y and z axes respectively, then its direction cosines are ______.

If for any two events A and B, P(A) = `4/5` and P(A ∩ B) = `7/10`, then `P(B/A)` is equal to ______.

Read the following passage:

Based on the above information, answer the following questions:

1. Find `P(L/C)` (1)
2. Find `P(overlineL/A)` (1)
3. (a) Find `P(A/L)` (2)
   OR
   (b) Find the probability that a randomly selected child is left-handed given that exactly one of the parents is left-handed. (2)

The median of an equilateral triangle is increasing at the ratio of `2sqrt(3)` cm/s. Find the rate at which its side is increasing.