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

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.

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.

Using Integration, find the area of triangle whose vertices are (– 1, 1), (0, 5) and (3, 2).

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 ______.

Let A = {3, 5}. Then number of reflexive relations on A is ______.

The sum of the order and the degree of the differential equation `d/dx[(dy/dx)^3]` is ______.

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

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.

Read the following passage:

Based on the above information, answer the following questions:

1. How many relations are possible from B to G? (1)
2. Among all the possible relations from B to G, how many functions can be formed from B to G? (1)
3. Let R : B `rightarrow` B be defined by R = {(x, y) : x and y are students of the same sex}. Check if R is an equivalence relation. (2)
   OR
   A function f : B `rightarrow` G be defined by f = {(b₁, g₁), (b₂, g₂), (b₃, g₁)}. Check if f is bijective. Justify your answer. (2)