Miscellaneous (Sets, Relations, Statistics & Mathematical Reasoning) |

If each of n observations is multiplied by a constant C, the standard deviation also gets multiplied by C.

If s. d. = 0, statements like (a) and (b) can not be given.

Given that x₁ < x₂ < x₃ < ....< x ₂₀₁

∴ Median of the given observation  th items

= 101th item = x₁₀₁

Now, deviations will be minimum if taken from the median

∴ Mean deviation will be min if k = x₁₀₁.

If any of the inequations hold, it must hold for any real numbers x₁, x₂, ..... x_n and any n∈N.

∴ let x₁ = 1, x₂ = 2, x₃ = 3; n = 3 then we can check none of the inequalities (a), (b) or (c) are satisfied.

S = {1, 2, 3, 4} Let P and Q be disjoint subsets of S
Now for any element a ∈ s, following cases are possible a ∈ P and a∉Q, a ∉P and a ∈ Q, a ∉P and a∉Q
⇒ For every element there are three option
∴ Total options = 34 = 81
Here P ≠Q except when P =Q = φ
∴ 80 ordered pairs (P, Q) are there for which P ≠ Q.
Hence total number of unordered pairs of disjoint subsets ==41

P = {θ : sinθ - cosθ = 2 cosθ}
sinθ = ( 2 + 1) cosθ , tanθ = 2 +1
Q = {θ : sinθ + cosθ = 2 sinθ} cosθ = ( 2 -1) sinθ or  tanθ = 2 +1
∴   P = Q