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

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")`

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.

If A and B are symmetric matrices of the same order, then (AB′ –BA′) is a ______.

If A and B are two skew-symmetric matrices of same order, then AB is symmetric matrix if ______.

If A = `[(0, 1),(1, 1)]` and B = `[(0, -1),(1, 0)]`, show that (A + B)(A – B) ≠ A² – B² 

Show that A′A and AA′ are both symmetric matrices for any matrix A.

If A = `[(cosalpha, sinalpha),(-sinalpha, cosalpha)]`, and A^–1 = A′, find value of α

If the matrix `[(0, "a", 3),(2, "b", -1),("c", 1, 0)]`, is a skew symmetric matrix, find the values of a, b and c.

If A, B are square matrices of same order and B is a skew-symmetric matrix, show that A′BA is skew-symmetric.

Express the matrix `[(2, 3, 1),(1, -1, 2),(4, 1, 2)]` as the sum of a symmetric and a skew-symmetric matrix.