Science (English Medium) Class 11 - CBSE Question Bank Solutions

Prove that cosθ `cos  theta/2 - cos 3theta cos  (9theta)/2` = sin 7θ sin 8θ.

[Hint: Express L.H.S. = `1/2[2costheta cos  theta/2 - 2 cos 3theta cos  (9theta)/2]`

If cosα + cosβ = 0 = sinα + sinβ, then prove that cos2α + cos2β = -2cos(α + β).
[Hint: (cosα + cosβ)² - (sinα + sinβ)² = 0]

Find the value of the expression `3[sin^4 ((3pi)/2 - alpha) + sin^4 (3pi + alpha)] - 2[sin^6 (pi/2 + alpha) + sin^6 (5pi - alpha)]`

cos2θ cos2Φ + sin²(θ – Φ) – sin²(θ + Φ) is equal to ______.

If |z₁| = |z₂|, is it necessary that z₁ = z₂?

If f(z) = `(7 - z)/(1 - z^2)`, where z = 1 + 2i, then |f(z)| is ______.

If the seventh terms from the beginning and the end in the expansion of `(root(3)(2) + 1/(root(3)(3)))^n` are equal, then n equals ______.

The equations of the lines passing through the point (1, 0) and at a distance `sqrt(3)/2` from the origin, are ______.

Evaluate `lim_(x -> 2) 1/(x - 2) - (2(2x - 3))/(x^3 - 3x^2 + 2x)`

Evaluate `lim_(x -> 0) (sqrt(2 + x) - sqrt(2))/x`

Find the positive integer n so that `lim_(x -> 3) (x^n - 3^n)/(x - 3)` = 108.

Evaluate `lim_(x -> pi/2) (secx - tanx)`

Evaluate `lim_(x -> 0)  (sin(2 + x) - sin(2 - x))/x`

Evaluate `lim_(x -> pi/6) (2sin^2x + sin x - 1)/(2sin^2 x - 3sin x + 1)`

Evaluate `lim_(x -> 0) (tanx - sinx)/(sin^3x)`

Evaluate `lim_(x -> a) (sqrt(a + 2x) - sqrt(3x))/(sqrt(3a + x) - 2sqrt(x))`

Evaluate `lim_(x -> 0) (cos ax - cos bx)/(cos cx - 1)`

Find the derivative of f(x) = `sqrt(sinx)`, by first principle.

`lim_(x -> 0) sinx/(x(1 + cos x))` is equal to ______.

`lim_(x -> pi/2) (1 - sin x)/cosx` is equal to ______.