Science (English Medium) Class 12 - CBSE Important Questions

Find the inverse of the following matrix, using elementary transformations: 

`A= [[2 , 3 , 1 ],[2 , 4 , 1],[3 , 7 ,2]]`

If `"A" = [(1,1,1),(1,0,2),(3,1,1)]`, find A^-1. Hence, solve the system of equations x + y + z = 6, x + 2z = 7, 3x + y + z = 12.

If A = [a_ij] is a square matrix of order 2 such that a_ij = `{(1","  "when i" ≠ "j"),(0","  "when"  "i" = "j"):},` then A² is ______.

If `[(2, 0),(5, 4)]` = P + Q, where P is symmetric, and Q is a skew-symmetric matrix, then Q is equal to ______.

If `|(α, 3, 4),(1, 2, 1),(1, 4, 1)|` = 0, then the value of α is ______.

Using the matrix method, solve the following system of linear equations:

`2/x + 3/y + 10/z` = 4, `4/x - 6/y + 5/z` = 1, `6/x + 9/y - 20/z` = 2.

 If x=a sin 2t(1+cos 2t) and y=b cos 2t(1−cos 2t), find `dy/dx `

if x^x+x^y+y^x=a^b, then find `dy/dx`.

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

If sin y = xsin(a + y) prove that `(dy)/(dx) = sin^2(a + y)/sin a`

If `log (x^2 + y^2) = 2 tan^-1 (y/x)`, show that `(dy)/(dx) = (x + y)/(x - y)`

If x^y - y^x = a^b, find `(dy)/(dx)`.

If f(x) = x + 1, find `d/dx (fof) (x)`

If x = `e^(x/y)`, then prove that `dy/dx = (x - y)/(xlogx)`.

The function f(x) = x |x| is ______.

If y = `sqrt(ax + b)`, prove that `y((d^2y)/dx^2) + (dy/dx)^2` = 0.

If f(x) = `{{:(ax + b; 0 < x ≤ 1),(2x^2 - x; 1 < x < 2):}` is a differentiable function in (0, 2), then find the values of a and b.

The derivative of x^2x w.r.t. x is ______.

If f(x) = `{{:(x^2"," if x ≥ 1),(x"," if x < 1):}`, then show that f is not differentiable at x = 1.

If y = tan x + sec x then prove that `(d^2y)/(dx^2) = cosx/(1 - sinx)^2`.