Enforcement Officer / Accounts Officer (EO/AO) 2020 recruitment test, advertisement 421. Transcribed from the Commission's own booklet (series A, NMJX-U-GTA); answer as per the official UPSC final answer key. This question is in Part B. Answer last reviewed 24 August 2026.

If ‘a’ varies as ‘b’, then which of the following statements is/are correct ? 1. n^th root of a^2b varies as (2n)^th root of a^4b^2. 2. a/b^2 varies inversely as b. Select the correct answer using the code given below :

(a)1 only
(b)2 only
(c)Both 1 and 2
(d)Neither 1 nor 2

Answer

Why

Correct — C, (c) Both 1 and 2. 'a varies as b' is the textbook phrase for direct variation, and it means a = kb for some non-zero constant k that stays fixed while a and b change. Write that down first; both statements are settled from it. Statement 1 is true, and it is true whether or not a varies as b — it is an algebraic identity. The two quantities are the nth root of a-squared-b and the (2n)th root of a-to-the-fourth-b-squared, the ordinals printed as true superscripts in the booklet and written here with carets. The whole of it turns on noticing that a^4b^2 is the square of a^2b. Taking a root of order 2n of a square leaves you exactly where a root of order n of the unsquared quantity does: (a^4b^2) raised to 1/(2n) is ((a^2b)^2) raised to 1/(2n), which is (a^2b) raised to 2/(2n), which is (a^2b) raised to 1/n. The two expressions are the same number, so each varies as the other with constant of proportionality 1 — and nothing in 'varies as' requires that constant to be anything but 1. Statement 2 is also true, and this one does use the variation. The expression is printed as a stacked fraction, a over b squared, set as a two-storey fraction with a horizontal rule and written inline with a solidus in the transcription. Substitute a = kb: a/b^2 = kb/b^2 = k/b. Since k is fixed, a/b^2 is a constant multiple of 1/b, which is precisely what 'varies inversely as b' asserts. Both statements stand, so the code is Both 1 and 2.

Why the others are wrong

(a)1 only — This accepts the identity in statement 1 and then mishandles statement 2, and the slip is nearly always the same one: reading a/b^2 straight off the page as varying inversely as b squared, because b squared is what sits in the denominator. It does sit there — but a is not a constant, and a carries a factor of b inside it. Put a = kb in and one of the two powers of b in the denominator is consumed: kb/b^2 = k/b. The general rule is worth taking away from this item. Before deciding how an expression varies with a variable, replace every other varying quantity in it by its own expression in that variable. Reading exponents off the printed form is only safe when everything else in the expression is genuinely constant, and here it is not.
(b)2 only — This rejects statement 1, and the rejection comes from letting the appearance of the two expressions stand in for their value. They look different and they are equal. a^4b^2 is nothing more than a^2b squared, and a root of order 2n undoes that square exactly as far as a root of order n leaves the unsquared quantity — both come to (a^2b) raised to the power 1/n for every positive a and b. There is a second route to rejecting statement 1, and it is a misunderstanding of the vocabulary rather than of the algebra: a belief that 'varies as' demands a constant other than one, or that two quantities which are simply equal cannot be said to vary as each other. Equality is proportionality with the constant equal to 1, and the statement is satisfied.
(d)Neither 1 nor 2 — Both statements survive, so choosing this requires two independent mistakes — but the item is built to invite exactly that, and it is worth seeing how. The two statements are true for entirely different reasons. Statement 1 is an identity in the exponents and does not use the given variation at all; statement 2 uses the variation and nothing else. A candidate hunting for one principle that settles both finds none, and concludes that the whole item is a trick with no true statement in it. That is a reading of the question's design, not of the mathematics. Take the statements one at a time, with a different tool for each: simplify the exponents for the first, substitute a = kb for the second.

Concept

Variation is proportionality given a vocabulary. 'a varies as b', written a is proportional to b, means a = kb with k a fixed non-zero constant, so the ratio a/b never changes. 'a varies inversely as b' means a = k/b, so the product ab never changes. 'a varies jointly as b and c' means a = kbc. The single most useful consequence, and the one statement 1 leans on, is that proportionality survives being raised to any fixed power: if a varies as b then a^m varies as b^m for any m, whole or fractional, and an mth root is just the power 1/m. Two exponent facts do the rest of the work here — (x^p)^q = x^(pq), and the mth root of x is x raised to 1/m. Together they collapse the (2n)th root of a square into the nth root of the thing that was squared. The trap in statement 2 is a different one and it is conceptual, not procedural: an expression's exponents tell you how it varies only once every symbol in it other than the variable of interest has been made genuinely constant.

This is one of only five items on this paper that print a numbered statement list, and one of four that ask which of the numbered statements is or are correct — the paper is overwhelmingly single-fact recall, so the format itself is a signal to slow down. The code options are the standard four, and they are exhaustive: exactly one of them must be right, and each statement has to be judged on its own before the code is chosen. The item rewards the discipline of writing a = kb on the page instead of reasoning about the printed shapes, and it punishes the opposite habit — inspecting exponents where they sit and inferring a proportionality from them.

Key facts

'a varies as b' means a = kb for a fixed non-zero constant k, so the ratio a/b is constant.
'a varies inversely as b' means a = k/b, so the product ab is constant.
a^4b^2 is the square of a^2b, so its (2n)th root equals the nth root of a^2b — the two quantities in statement 1 are equal.
Statement 1 holds for all positive a and b whether or not a varies as b; it is an identity, not a consequence of the variation.
With a = kb, the expression a/b^2 becomes k/b, which is inverse variation in b — not in b squared.
If a varies as b then a^m varies as b^m for any fixed m, including fractional powers such as roots.
Equality is the special case of proportionality with the constant equal to 1, so equal quantities do vary as each other.
The booklet sets a/b^2 as a stacked two-storey fraction and the ordinals 'th' as true superscripts; the transcription writes them with a solidus and carets.

Two statements, both true — for entirely different reasons

The premise, written down first→‘a varies as b’ is the textbook phrase for direct variation: a = kb for some fixed non-zero constant k that stays put while a and b change, so the ratio a/b never changes. Both statements are settled from that one line.

Statement 1 — TRUE, and an identity→a^4b^2 is nothing more than the SQUARE of a^2b. A root of order 2n undoes that square exactly as far as a root of order n leaves the unsquared quantity: (a^4b^2)^(1/2n) = ((a^2b)^2)^(1/2n) = (a^2b)^(2/2n) = (a^2b)^(1/n). The two expressions are the same number, so each varies as the other with a constant of proportionality of 1 — and nothing in ‘varies as’ requires that constant to be anything but 1. This statement holds for all positive a and b whether or not a varies as b.

Statement 2 — TRUE, and this one does use the variation→Substitute a = kb: a/b^2 = kb/b^2 = k/b. Since k is fixed, a/b^2 is a constant multiple of 1/b, which is precisely what ‘varies inversely as b’ asserts.

The trap in statement 2→Reading a/b^2 straight off the page as varying inversely as b SQUARED, because b squared is what sits in the denominator. It does sit there — but a is not a constant, and a carries a factor of b inside it, so one of the two powers of b in the denominator is consumed.

Why ‘Neither 1 nor 2’ is on the page→Because the two statements are true for entirely different reasons: statement 1 is an identity in the exponents and does not use the given variation at all, while statement 2 uses the variation and nothing else. A candidate hunting for one principle that settles both finds none, and may conclude that neither holds.

The vocabulary, and the one consequence that matters→‘a varies as b’ means a = kb. ‘a varies inversely as b’ means a = k/b, so the product ab is constant. ‘a varies jointly as b and c’ means a = kbc. The single most useful consequence, and the one statement 1 leans on, is that proportionality survives being raised to any fixed power: if a varies as b then a^m varies as b^m for any m, whole or fractional — and an mth root is just the power 1/m.

Study next

Direct, inverse and joint variation, and how to find the constant from one given pair of valuesLaws of indices, including fractional and negative exponents and the root-as-power identityRatio and proportion, including componendo and dividendoStatement-and-code items in general — judging each statement independently before reading the optionsProblems where one quantity varies partly as another and partly as a constant, which is not proportionality

Common traps

Reading a/b^2 as inverse variation in b squared without first substituting a = kb.
Assuming two expressions that are written differently must have different values.
Believing 'varies as' rules out the case where the constant of proportionality is 1.
Choosing a code option before both statements have been tested separately.
Losing the superscripts. Read 'n^th' as an ordinal and 'a^2' as a power; on the page these are set differently and only the typography distinguishes them.

Where an EPFO EO/AO paper puts a mathematics item into the statement-and-code format, it usually pairs one statement that is a pure algebraic identity with one that depends on the condition given in the stem. The four code options are always the same, so nothing can be read off the option set, and the only route is to evaluate each numbered statement on its own terms.

Related PYQs

EPFO_EOAO_2020_Q25Open & attempt →

Let n ( > 1) be a composite natural number whose square root is not an integer. Consider the following statements : 1. n has a factor which is greater than 1 but less than the square root of n. 2. n has a factor which is greater than the square root of n but less than n. Which of the statements given above is/are correct ?

(a) 1 only
(b) 2 only
(c) Both 1 and 2
(d) Neither 1 nor 2

Answer(c) Both 1 and 2

The paper's other mathematical statement-and-code item, and its twin in construction: two numbered statements about a general algebraic situation, the same four code options, and both statements true. It is a useful reminder that in this format the two statements are judged separately and can perfectly well both stand.

Practice

practice — not a real PYQ
If ‘p’ varies as ‘q’, then which of the following statements is/are correct ? 1. p^2 varies as q^2. 2. p − q varies inversely as q. Select the correct answer using the code given below :
- (a)1 only
- (b)2 only
- (c)Both 1 and 2
- (d)Neither 1 nor 2
Answer(a) 1 only
practice — not a real PYQ
If ‘y’ varies inversely as ‘x’ and y = 8 when x = 3, then what is the value of y when x = 12 ?
- (a)2
- (b)4
- (c)6
- (d)32
Answer(a) 2

Why