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

Solve the differential equation `[e^(-2sqrtx)/sqrtx - y/sqrtx] dx/dy = 1 (x != 0).`

Find a particular solution of the differential equation `dy/dx + y cot x = 4xcosec x(x != 0)`, given that y = 0 when `x = pi/2.`

Show that the function given by f(x) = sin x is

1. strictly increasing in `(0, pi/2)`
2. strictly decreasing in `(pi/2, pi)`
3. neither increasing nor decreasing in (0, π)

Find the intervals in which the function f given by f(x) = 2x² − 3x is

1. strictly increasing
2. strictly decreasing

Find the intervals in which the function f given by f(x) = 2x³ − 3x² − 36x + 7 is

1. Strictly increasing
2. Strictly decreasing

Find the intervals in which the following functions are strictly increasing or decreasing:

x² + 2x − 5

Find the intervals in which the following functions are strictly increasing or decreasing:

10 − 6x − 2x²

Find the intervals in which the following functions are strictly increasing or decreasing:

−2x³ − 9x² − 12x + 1

Find the intervals in which the following functions are strictly increasing or decreasing:

 (x + 1)³ (x − 3)³

Find the intervals in which the following functions are strictly increasing or decreasing:

6 − 9x − x²

Show that y = `log(1+x) - (2x)/(2+x), x> -  1`, is an increasing function of x throughout its domain.

Find the values of x for  `y = [x(x - 2)]^2` is an increasing function.

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].