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

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

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

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

If `y = sin^-1 x + cos^-1 x , "find"  dy/dx`

If e^y ( x +1)  = 1, then show that  `(d^2 y)/(dx^2) = ((dy)/(dx))^2 .`

Find `(dy)/(dx) , if y = sin ^(-1) [2^(x +1 )/(1+4^x)]`

If `(sin "x")^"y" = "x" + "y", "find" (d"y")/(d"x")`

Find:
`int_(-pi/4)^0 (1+tan"x")/(1-tan"x") "dx"`

If y = (log x)^x + x^log x, find `"dy"/"dx".`

If e^y = y^x, then show that `"dy"/"dx" = (logy)^2/(log y - 1)`.

Show that a matrix which is both symmetric and skew symmetric is a zero matrix.

Express the matrix A as the sum of a symmetric and a skew-symmetric matrix, where A = `[(2, 4, -6),(7, 3, 5),(1, -2, 4)]`

Let A = `[(2, 3),(-1, 2)]`. Then show that A² – 4A + 7I = O. Using this result calculate A⁵ also.