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

If x^m . yⁿ = (x + y)^m+n, prove that `("d"^2"y")/("dx"^2)` = 0

If y = `sqrt(sinx + y)`, then `"dy"/"dx"` is equal to ______.

For the curve `sqrt(x) + sqrt(y)` = 1, `"dy"/"dx"` at `(1/4, 1/4)` is ______.

Prove that the function f(x) = tanx – 4x is strictly decreasing on `((-pi)/3, pi/3)`

Determine for which values of x, the function y = `x^4 – (4x^3)/3` is increasing and for which values, it is decreasing.

Show that f(x) = 2x + cot^–1x + `log(sqrt(1 + x^2) - x)` is increasing in R

Show that for a ≥ 1, f(x) = `sqrt(3)` sinx – cosx – 2ax + b ∈ is decreasing in R

Show that f(x) = tan^–1(sinx + cosx) is an increasing function in `(0, pi/4)`

The interval on which the function f(x) = 2x³ + 9x² + 12x – 1 is decreasing is ______.

Let the f : R → R be defined by f (x) = 2x + cosx, then f : ______.

y = x(x – 3)² decreases for the values of x given by : ______.

The function f(x) = 4 sin³x – 6 sin²x + 12 sinx + 100 is strictly ______.

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

The function f(x) = tanx – x ______.

The values of a for which the function f(x) = sinx – ax + b increases on R are ______.

The function f(x) = `(2x^2 - 1)/x^4`, x > 0, decreases in the interval ______.

Evaluate the following:

`int x/(sqrt(x) + 1) "d"x`  (Hint: Put  `sqrt(x)` = z)

Evaluate the following:

`int sqrt(("a" + x)/("a" - x)) "d"x`

Evaluate the following:

`int x^(1/2)/(1 + x^(3/4)) "d"x`   (Hint: Put `sqrt(x)` = z⁴)

The value of cos²15° - cos²30° + cos²45° - cos²60° + cos²75° is ______.