Science (English Medium) Class 12 - CBSE Important Questions

Why was the 'Mahanavami Dibba' a centre of main Vijayanagara rituals? Explain.

Explain the role of village panchayats in the Mughal rural society.

“The power of the Jotedars was more effective than that of the zamindars.” Justify the statement with suitable arguments.

"Art and Literature as much as the writing of history have helped in keeping alive the memory of 1857." Explain the statement in reference to Rani Lakshmibai.

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

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`

Differentiate `tan^(-1) ((1+cosx)/(sin x))` with respect to x

if `(x^2 + y^2)^2 = xy` find `(dy)/(dx)`

If y = x^x, prove that `(d^2y)/(dx^2)−1/y(dy/dx)^2−y/x=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 equations of the tangent and the normal, to the curve 16x² + 9y² = 145 at the point (x₁, y₁), where x₁ = 2 and y₁ > 0.

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)`

Evaluate : `int_1^3 (x^2 + 3x + e^x) dx` as the limit of the sum.