Commerce (English Medium) Class 12 - CBSE Important Questions for Mathematics

Evaluate: `int_0^π (x sin x)/(1 + cos^2x) dx`.

 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.

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 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]`

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`

Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is `(4r)/3`. Also find maximum volume in terms of volume of the sphere

Find the intervals in which the function `f(x) = x^4/4 - x^3 - 5x^2 + 24x + 12`  is (a) strictly increasing, (b) strictly decreasing

An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when the depth of the tank is half of its width. If the cost is to be borne by nearby settled lower-income families, for whom water will be provided, what kind of value is hidden in this question?

Find: `int(x+3)sqrt(3-4x-x^2dx)`

Find `int((3sintheta-2)costheta)/(5-cos^2theta-4sin theta)d theta`.

Find: `I=intdx/(sinx+sin2x)`

Solve the differential equation:

(1 + y²) dx = (tan⁻¹ y − x) dy

Find the shortest distance between the lines `vecr = (4hati - hatj) + lambda(hati+2hatj-3hatk)` and `vecr = (hati - hatj + 2hatk) + mu(2hati + 4hatj - 5hatk)`

A die, whose faces are marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let A be the event "number obtained is even" and B be the event "number obtained is red". Find if A and B are independent events.

Suppose a girl throws a die. If she gets 1 or 2 she tosses a coin three times and notes the number of tails. If she gets 3,4,5 or 6, she tosses a coin once and notes whether a ‘head’ or ‘tail’ is obtained. If she obtained exactly one ‘tail’, what is the probability that she threw 3,4,5 or 6 with the die ?