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

Prove that  y = `(4sin theta)/(2 + cos theta) - theta` is an increasing function of θ in `[0, pi/2]`

Prove that the logarithmic function is strictly increasing on (0, ∞).

Prove that the function f given by f(x) = x² − x + 1 is neither strictly increasing nor strictly decreasing on (−1, 1).

Which of the following functions are strictly decreasing on `(0, pi/2)`?

1. cos x
2. cos 2x
3. cos 3x
4. tan x

On which of the following intervals is the function f given byf(x) = x¹⁰⁰ + sin x –1 strictly decreasing?

Find the least value of a such that the function f given by f (x) = x² + ax + 1 is strictly increasing on [1, 2].

Let I be any interval disjoint from (−1, 1). Prove that the function f given by `f(x) = x + 1/x` is strictly increasing on I.

Prove that the function f given by f(x) = log sin x is strictly increasing on `(0, pi/2)` and strictly decreasing on `(pi/2, pi)`

Prove that the function f given by f(x) = log cos x is strictly decreasing on `(0, pi/2)` and strictly increasing on `((3pi)/2, 2pi).`

Prove that the function given by f (x) = x³ – 3x² + 3x – 100 is increasing in R.

The interval in which y = x² e^–x is increasing is ______.

Find the intervals in which the function f given by `f(x) = (4sin x - 2x - x cos x)/(2 + cos x)` is (i) increasing (ii) decreasing.

Find the intervals in which the function f given by `f(x) = x^3 + 1/x^3 x != 0`, is (i) increasing (ii) decreasing.

Let f be a function defined on [a, b] such that f '(x) > 0, for all x ∈ (a, b). Then prove that f is an increasing function on (a, b).

Find the area of the region bounded by the curve y² = x and the lines x = 1, x = 4 and the x-axis.

Find the area of the region bounded by y² = 9x, x = 2, x = 4 and the x-axis in the first quadrant.

Find the area of the region bounded by x² = 4y, y = 2, y = 4 and the y-axis in the first quadrant.

Find the area of the region bounded by the ellipse  `x^2/16 + y^2/9 = 1.`

Find the area of the region bounded by the ellipse `x^2/4 + y^2/9 = 1.`

Find the area of the region in the first quadrant enclosed by x-axis, line x = `sqrt3` y and the circle x² + y² = 4.