Arts (English Medium) कक्षा १२ - 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 ____________.

Polio drops are delivered to 50 K children in a district. The rate at which polio drops are given is directly proportional to the number of children who have not been administered the drops. By the end of 2^nd week half the children have been given the polio drops. How many will have been given the drops by the end of 3^rd week can be estimated using the solution to the differential equation `"dy"/"dx" = "k"(50 - "y")` where x denotes the number of weeks and y the number of children who have been given the drops.

State the order of the above given differential equation.

Write the sum of the order and the degree of the following differential equation:

`d/(dx) (dy/dx)` = 5

Find: `int (x + 1)/((x^2 + 1)x) dx`

The value of ‘k’ for which the function f(x) = `{{:((1 - cos4x)/(8x^2)",",  if x ≠ 0),(k",",  if x = 0):}` is continuous at x = 0 is ______.

If m and n, respectively, are the order and the degree of the differential equation `d/(dx) [((dy)/(dx))]^4` = 0, then m + n = ______.

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

Define the relation R in the set N × N as follows:

For (a, b), (c, d) ∈ N × N, (a, b) R (c, d) if ad = bc. Prove that R is an equivalence relation in N × N.

Given a non-empty set X, define the relation R in P(X) as follows:

For A, B ∈ P(X), (4, B) ∈ R iff A ⊂ B. Prove that R is reflexive, transitive and not symmetric.

Find the general solution of the following differential equation:

`(dy)/(dx) = e^(x-y) + x^2e^-y`

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.

Degree of the differential equation `sinx + cos(dy/dx)` = y² is ______.

Anti-derivative of `(tanx - 1)/(tanx + 1)` with respect to x is ______.

The function f(x) = x |x| 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 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.