Commerce (English Medium) कक्षा १२ - CBSE Important Questions for Mathematics

Find the distance of the point (−1, −5, −10) from the point of intersection of the line `vecr=2hati-hatj+2hatk+lambda(3hati+4hatj+2hatk) ` and the plane `vec r (hati-hatj+hatk)=5`

If a * b denotes the larger of 'a' and 'b' and if a∘b = (a * b) + 3, then write the value of (5)∘(10), where * and ∘ are binary operations.

Find the value of `tan^(-1) sqrt3 - cot^(-1) (-sqrt3)`

 If A is a square matrix such that A² = I, then find the simplified value of (A – I)³ + (A + I)³ – 7A.

if the matrix A =`[(0,a,-3),(2,0,-1),(b,1,0)]` is skew symmetric, Find the value of 'a' and 'b'

If A = `[(2,-3,5),(3,2,-4),(1,1,-2)]` find A⁻¹. Using A⁻¹ solve the system of equations

2x – 3y + 5z = 11
3x + 2y – 4z = – 5
x + y – 2z = – 3

Find the area of the region in the first quadrant enclosed by the x-axis, the line y = x and the circle x² + y² = 32.

Find the differential equation representing the family of curves `y = ae^(bx + 5)`. where a and b are arbitrary constants.

Find the vector equation of the line passing through the point A(1, 2, –1) and parallel to the line 5x – 25 = 14 – 7y = 35z.

Find the magnitude of each of two vectors `veca` and `vecb` having the same magnitude such that the angle between them is 60° and their scalar product is `9/2`

If θ is the angle between two vectors `hati - 2hatj + 3hatk and 3hati - 2hatj + hatk` find `sin theta`

A black and a red dice are rolled. 

Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4.

Let X be a random variable which assumes values  x₁ , x₂, x₃ , x₄ such that  2P (X = x₁) = 3P (X = x₂) = P (X = x₃) = 5P (X = x₄). Find the probability distribution of X.

Let A = {x ∈ Z : 0 ≤ x ≤ 12}. Show that R = {(a, b) : a, b ∈ A, |a – b| is divisible by 4}is an equivalence relation. Find the set of all elements related to 1. Also write the equivalence class [2]

Show that the function f: ℝ → ℝ defined by f(x) = `x/(x^2 + 1), ∀x in R`is neither one-one nor onto. Also, if g: ℝ → ℝ is defined as g(x) = 2x - 1. Find fog(x)

Prove that `3sin^(-1)x = sin^(-1) (3x - 4x^3)`, `x in [-1/2, 1/2]`

Using elementary row transformations, find the inverse of the matrix A = `[(1,2,3),(2,5,7),(-2,-4,-5)]`

Using properties of determinants, prove that `|(1,1,1+3x),(1+3y, 1,1),(1,1+3z,1)| = 9(3xyz + xy +  yz+ zx)`

Using elementary row transformations, find the inverse of the matrix A = `[(1,2,3),(2,5,7),(-2,-4,-5)]`

If x = a cos θ + b sin θ, y = a sin θ − b cos θ, show that `y^2 (d^2y)/(dx^2)-xdy/dx+y=0`