Science (English Medium) कक्षा ११ - CBSE Question Bank Solutions for Mathematics

If p₁ and p₂ are the lengths of the perpendiculars from the origin upon the lines x sec θ + y cosec θ = a and x cos θ − y sin θ = a cos 2 θ respectively, then

If p be the length of the perpendicular from the origin on the line x/a + y/b = 1, then

If `(x/3+1, y-2/3)` = `(5/3,1/3),`find the values of x and y.

When A = Φ, then number of elements in P(A) is ______.

Power set of the set A = {1, 2} is ______.

Given A = {1, 2, 3, 4, 5}, S = {(x, y) : x ∈ A, y ∈ A}. Find the ordered pairs which satisfy the conditions given below:

x + y = 5

Given A = {1, 2, 3, 4, 5}, S = {(x, y) : x ∈ A, y ∈ A}. Find the ordered pairs which satisfy the conditions given below:

x + y < 5

Given A = {1, 2, 3, 4, 5}, S = {(x, y) : x ∈ A, y ∈ A}. Find the ordered pairs which satisfy the conditions given below:

x + y > 8

Express the following functions as set of ordered pairs and determine their range.
f : X → R, f(x) = x³ + 1, where X = {–1, 0, 3, 9, 7}

State True or False for the following statement.

The ordered pair (5, 2) belongs to the relation R = {(x, y) : y = x – 5, x, y ∈ Z}.

Prove that `(tanA + secA  - 1)/(tanA - secA + 1) = (1 + sinA)/cosA`

If `(2sinalpha)/(1 + cosalpha + sinalpha)` = y, then prove that `(1 - cosalpha + sinalpha)/(1 + sinalpha)` is also equal to y.
`["Hint": "Express" (1 - cosalpha + sinalpha)/(1 + sinalpha) = (1 - cosalpha + sinalpha)/(1 + sinalpha) * (1 + cosalpha + sinalpha)/(1 + cosalpha + sinalpha)]`

If m sinθ = n sin(θ + 2α), then prove that tan(θ + α)cotα = `(m + n)/(m - n)`

[Hint: Express `(sin(theta + 2alpha))/sintheta = m/n` and apply componendo and dividendo]

If cos(α + β) = `4/5` and sin(α – β) = `5/13`, where α lie between 0 and `pi/4`, find the value of tan2α.
[Hint: Express tan2α as tan(α + β + α – β)]

If tanx = `b/a`, then find the value of `sqrt((a + b)/(a - b)) + sqrt((a - b)/(a + b))`

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?