Commerce (English Medium) इयत्ता १२ - CBSE Question Bank Solutions

Integration of \[\frac{1}{1 + \left( \log_e x \right)^2}\] with respect to log_e x is

\[\int\frac{8x + 13}{\sqrt{4x + 7}} \text{ dx }\]

\[\int\frac{1 + x + x^2}{x^2 \left( 1 + x \right)} \text{ dx}\]

Find \[\frac{dy}{dx}\] at \[t = \frac{2\pi}{3}\] when x = 10 (t – sin t) and y = 12 (1 – cos t).

If xy = e^(x – y), then show that `dy/dx = (y(x-1))/(x(y+1)) .`

If logy = tan^–1 x, then show that `(1+x^2) (d^2y)/(dx^2) + (2x - 1) dy/dx = 0 .`

Differentiate \[\tan^{- 1} \left( \frac{\sqrt{1 + x^2} - 1}{x} \right) w . r . t . \sin^{- 1} \frac{2x}{1 + x^2},\]tan-11+x2-1x w.r.t. sin-12x1+x2, if x ∈ (–1, 1) .

If x = sin t and y = sin pt, prove that \[\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} + p^2 y = 0\] .

Find : \[\int\left( 2x + 5 \right)\sqrt{10 - 4x - 3 x^2}dx\] .

\[\text { If } y = \left( x + \sqrt{1 + x^2} \right)^n , \text { then show that }\]

\[\left( 1 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = n^2 y .\]

Find the minimum value of (ax + by), where xy = c².

Let f : N → ℝ be a function defined as f(x) = 4x² + 12x + 15. Show that f : N → S, where S is the range of f, is invertible. Also find the inverse of f.

If y = x^x, prove that \[\frac{d^2 y}{d x^2} - \frac{1}{y} \left( \frac{dy}{dx} \right)^2 - \frac{y}{x} = 0 .\]

Find the shortest distance between the lines

Find the shortest distance between the lines 

Find the shortest distance between the lines

Differentiate sin(log sin x) ?