Commerce (English Medium) इयत्ता १२ - CBSE Important Questions

Using properties of determinants, prove that 

`|(a^2 + 2a,2a + 1,1),(2a+1,a+2, 1),(3, 3, 1)| = (a - 1)^3`

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

Using properties of determinants, prove that `|(x,x+y,x+2y),(x+2y, x,x+y),(x+y, x+2y, x)| = 9y^2(x + y)`

Using properties of determinants, prove that

`|[b+c , a ,a  ] ,[ b , a+c, b ] ,[c , c, a+b ]|` = 4abc 

If f(α) = `[(cosα, -sinα, 0),(sinα, cosα, 0),(0, 0, 1)]`, prove that f(α) . f(– β) = f(α – β).

If for a square matrix A, A² – A + I = 0, then A^–1 equals ______.

Read the following passage:

Based on the above information, answer the following questions:

1. Convert the given above situation into a matrix equation of the form AX = B. (1)
2. Find | A |. (1)
3. Find A^–1. (2)
   OR
   Determine P = A² – 5A. (2)

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`

If `y=tan^(−1) ((sqrt(1+x^2)+sqrt(1−x^2))/(sqrt(1+x^2)−sqrt(1−x^2)))` , x²≤1, then find dy/dx.

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 y = sin (sin x), prove that `(d^2y)/(dx^2) + tan x dy/dx + y cos^2 x = 0`

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

If `tan ((x + y)/(x - y))` = k, then `dy/dx` is equal to ______.

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

Find the equations of the tangent and normal to the curve x = a sin³θ and y = a cos³θ at θ=π/4.

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 ?