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UPSC NDA I and II 2020 Math Solved Paper

Question Numbers: 1-3

Directions: Consider the following Venn diagram, where X, Y and Z are three sets. Let the number of elements in Z be denoted by

n(Z)

. which is equal to 90.

Question : 2 of 120

Marks: +1, -0

Solution:

Concept:

The expression

n(X) + n(Y) + n(Z) - n(X \cap Y) - n(Y \cap Z) - n(X \cap Z) + n(X \cap Y \cap Z)

simplifies to the total number of elements in the union of sets

X

Y

, and

Z

, using the inclusion‑exclusion principle.

Explanation:

From the Venn diagram, the given values are:

n(Z) = 90

n(X) = a + 46

and

n(Y) = b + 51

n(X \cap Y) = 16 + 18 = 34

n(Y \cap Z) = 17 + 18 = 35

n(X \cap Z) = 12 + 18 = 30

n(X \cap Y \cap Z) = 18

Now substitute these into the expression:

= (a + 46) + (b + 51) + 90 - 34 - 35 - 30 + 18

First, sum the positive parts:

a + b + 46 + 51 + 90 = a + b + 187

Then sum the negative parts:

34 + 35 + 30 = 99

So,

187 - 99 + 18 = 106

Thus, the expression simplifies to

a + b + 106

Answer:

Option D:

a + b + 106

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