Science (English Medium) Class 12 - CBSE Important Questions

If `[[x-y,z],[2x-y,w]]=[[-1,4],[0,5]]` find the value of x+y.

If A is a 3 × 3 invertible matrix, then what will be the value of k if det(A^–1) = (det A)^k

Show that all the diagonal elements of a skew symmetric matrix are zero.

Let A = `((2,-1),(3,4))`, B = `((5,2),(7,4))`, C= `((2,5),(3,8))` find a matrix D such that CD − AB = O

Given `A = [(2,-3),(-4,7)]` compute `A^(-1)` and show that `2A^(-1) = 9I - A`

If x = a sin 2t (1 + cos2t) and y = b cos 2t (1 – cos 2t), find the values of  `dy/dx `at t = `pi/4`

Differentiate the function with respect to x.

`(sin x)^x + sin^(-1) sqrtx`

if `x^m y^n = (x + y)^(m + n)`, prove that `(d^2y)/(dx^2)= 0`

If x = a (2θ – sin 2θ) and y = a (1 – cos 2θ), find `dy/dx` when `theta = pi/3`

If x = A cos 4t + B sin 4t, then `(d^2x)/(dt^2)` is equal to ______.

Find the intervals in which the function f(x) = 3x⁴ − 4x³ − 12x² + 5 is

(a) strictly increasing

(b) strictly decreasing

If the sum of lengths of hypotenuse and a side of a right angled triangle is given, show that area of triangle is maximum, when the angle between them is π/3.

Find the intervals in which f(x) = sin 3x – cos 3x, 0 < x < π, is strictly increasing or strictly decreasing.

The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ?

Find the local maxima and local minima, of the function f(x) = sin x − cos x, 0 < x < 2π.

Find the value(s) of x for which y = [x(x − 2)]² is an increasing function.

Show that the function f(x) = 4x³ - 18x² + 27x - 7 is always increasing on R.

The total cost C(x) associated with the production of x units of an item is given by C(x) = 0.005x³ – 0.02x² + 30x + 5000. Find the marginal cost when 3 units are produced, whereby marginal cost we mean the instantaneous rate of change of total cost at any level of output.

Find the dimensions of the rectangle of perimeter 36 cm which will sweep out a volume as large as possible, when revolved about one of its sides. Also, find the maximum volume.

A particle moves along the curve 3y = ax³ + 1 such that at a point with x-coordinate 1, y-coordinate is changing twice as fast at x-coordinate. Find the value of a.