UPSC Prelims 2026 CSAT, question 30

Topic: Arithmetic

Q30

Suppose $x,y$ and $z$ are variables taking positive real numbers as their possible values. It is given that $y$ is directly proportional to $x^2$ and $x$ is inversely proportional to $z$ . For $z=\frac{7}{25}$ , the values of $x$ and $y$ are $5$ and $50$ , respectively. If $y=98$ , what is $z$ equal to?

(a) $\frac{1}{7}$
(b) $\frac{1}{5}$ Correct answer
(c) $\frac{5}{7}$
(d) $1$

Solution.

Answer: (b) $\frac{1}{5}$

Since $y$ is directly proportional to $x^2$ ( $y = k_1 x^2$ ) and $x$ is inversely proportional to $z$ ( $x z = k_2$ ), substituting $x = 5, y = 50$ at $z = \frac{7}{25}$ gives $k_1 = \frac{50}{5^2} = 2$ and $k_2 = 5 \times \frac{7}{25} = \frac{7}{5}$ .
When $y = 98$ , we have $2 x^2 = 98 \Rightarrow x^2 = 49 \Rightarrow x = 7$ (since $x > 0$ ).
Substituting $x = 7$ into $x z = \frac{7}{5}$ gives $7 z = \frac{7}{5}$ , so $z = \frac{1}{5}$ .