IPMAT Indore 2022 - QA (MCQ) Q32 Explanation

Explanation:

(b)

Given 0 < θ < 𝜋/4 , or 0 < θ < 45°

a= sinθ^sinθ log2 cosθ

b= cosθ^sinθ log2 sinθ

c= sinθ^cosθ log2 cosθ

d= sinθ^sinθ log2 sinθ

In order to find median of a, b, c, d, we need to arrange them in increasing or decreasing order. Then the mean value of middle two numbers will be the median of all four values.

Let us take θ = 30°

Then sin30° = 1/2 = 0.5

cos 30° = √3/2 = 1.732/2 = 0.866 = 0.87

log2 sin30° = log2 1/2 = log2 1− log2 2 = 0 − 1 = −1

log2 cos30° = log2 √3/2 = log2 √3 − log2 2 = 1/2 log2 3 − 1 = 1/2 × 1.5 − 1

= 0.75 − 1

= - 0.25

(We Assume log2 2 < log2 <log2 4]

(Or, 1 < log2 3 < 2)

a = 0.5^0.5 log2 cos30° = 0.5^0.5 (−0.25) 

b = 0.87^0.5 log2 sin30° = 0.87^0.5 (−1)

c = 0.5^0.87 log2 cos30° = 0.5^0.87 × (−0.25)

d = 0.5^0.5 log2 sin30° = 0.5^0.5 × (−1)

All the numbers are negative.

So, the number with greatest magnitude (value) will be the least and the number with least magnitude (value) will be highest.

e.g. - 100 < - 2

Now comparing 'a' and 'd’ clearly [a >d]

Now comparing 'a' and 'c',

We know, 0.5^0.87 < 0.5^0.5

∴ 0.5^0.87 × (-0.25)  > 0.5^0.5 × (−0.25)

∴ c > a

Now comparing 'd' and 'b', we get

We know 0.87^0.5 > 0.5^0.5

0.87^0.5 × (−1) < 0.5^0.5 × (-1)

b < d

or, d > b

Combining all inequalities, we get c > a > d > b

Thus, median = a+d/2.