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If X and Y are two sets, then X ∩ (X∪Y)^{c} equals. (1979)
The expression is equal to (1980)
Select the correct alternative in each of the following. Indicate your choice by the appropriate letter only.Let S be the standard deviation of n observations. Each of the n observations is multiplied by a constant c. Then the standard deviation of the resulting number is (1980)
If each of n observations is multiplied by a constant C, the standard deviation also gets multiplied by C.
The standard deviation of 17 numbers is zero. Then (1980)
If s. d. = 0, statements like (a) and (b) can not be given.
Consider any set of 201 observations x_{1}, x_{2}, ....x_{200}, x_{201}.It is given that x_{1}< x_{2}<....< x_{200} <x_{201}. Then the mean deviation of this set of observations about a point k is minimum when k equals (1981  2 Marks)
Given that x_{1} < x_{2} < x_{3} < ....< x _{201}
∴ Median of the given observation th items
= 101th item = x_{101}
Now, deviations will be minimum if taken from the median
∴ Mean deviation will be min if k = x_{101}.
If x1, x2,..............., xn are any real numbers and n is any postive integer, then (1982  2 Marks)
If any of the inequations hold, it must hold for any real numbers x_{1}, x_{2}, ..... x_{n} and any n∈N.
∴ let x_{1} = 1, x_{2} = 2, x_{3} = 3; n = 3 then we can check none of the inequalities (a), (b) or (c) are satisfied.
Let S={1, 2, 3, 4} . The total number of unordered pairs of disjoint subsets of S is equal to (2010)
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
Let P = {θ : sinθ – cosθ = cosθ} and Q = {θ : sinθ + cosθ = sinθ} be two sets. Then (2011)
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
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