Chapter 2 of 5

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XAT Quantitative Ability & Data Interpretation★ Roughly 4-6 of QA&DI's 28 questions, based on recent-paper analysis (XLRI does not publish an official sub-split) — lighter than arithmetic and geometry

Algebra

XAT's algebra is lighter than CAT's — fewer questions, and rarely the multi-variable systems CAT favours. What it does ask (quadratics, inequalities, logarithms, functions) rewards recognising a standard identity fast over grinding through algebraic manipulation from scratch.

⏱ 11 min readMedium difficulty✓ Free · exam-pattern aligned

Timed practiceMCQs from this chapter with instant scoring

FlashcardsFormulas & traps on spaced repetition

XAT PYQsSolved previous-year questions

By the end of this chapter you'll be able to…

1Use sum and product of roots to find symmetric functions of a quadratic's roots without computing the roots themselves
2Apply the discriminant to determine root nature and solve 'equal roots' style questions directly
3Apply the sign-flip rule correctly when multiplying or dividing an inequality by a negative number
4Use a sign chart to solve quadratic inequalities reliably
5Apply logarithm and index rules to simplify expressions before substituting numbers
6Apply the Part-1 skip-penalty arithmetic to decide when to guess versus skip an algebra question

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Why this chapter matters in XAT

XAT tests algebra comparatively lightly versus CAT's algebra-heavy style, and what it does ask rarely needs full equation-solving — recognising which standard identity (sum/product of roots, discriminant, sign-flip rule, log/index rules) applies is faster and more reliable than grinding through manipulation from scratch, especially given the five-option format often makes direct option-checking the fastest path to an answer.

Before you start — revise these

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Arithmetic (XAT QA&DI)

Comfort with fractions and percentage manipulation from that chapter transfers directly to simplifying algebraic expressions quickly here.

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XAT's overall marking scheme

The five-option format and 8-free-skip rule (docs/exam-briefs/xat-2026-brief.md) make direct option-checking a genuinely competitive strategy in this chapter specifically.

Algebra — XAT Quantitative Ability & Data Interpretation

Algebra at XAT level is rarely about solving a hard equation — it's about recognising which of a small set of standard identities and techniques the question is actually testing, then executing it cleanly. The questions that look intimidating usually collapse the moment the right identity is spotted.

1. What XAT actually asks

Algebra is estimated at roughly 4-6 of QA&DI's 28 questions across recent papers (see docs/exam-briefs/xat-2026-brief.md) — lighter than arithmetic and geometry, consistent with XAT's documented lean away from CAT's algebra-heavier style. The topic covers linear and quadratic equations, inequalities, logarithms and indices, and functions — permutations, probability, and sequences/series are covered separately under Modern Math.

Most XAT algebra questions are single-variable or reduce to one quickly. The five-option format means a question rarely needs you to fully solve for an unknown when checking each option directly against the given condition is faster.

2. Quadratic equations — the identities worth knowing cold

For $a x^{2} + b x + c = 0$ , the roots satisfy:

$sum of roots = - \frac{b}{a}, product of roots = \frac{c}{a}$

These two relationships solve a large share of XAT quadratic questions without ever computing the roots themselves — if a question asks for the sum or product of roots, or a symmetric function of the roots (like $α^{2} + β^{2}$ ), these identities get there directly.

$α^{2} + β^{2} = (α + β)^{2} - 2 α β$

The discriminant $D = b^{2} - 4 a c$ determines the nature of the roots: $D > 0$ gives two distinct real roots, $D = 0$ gives equal real roots, $D < 0$ gives no real roots (complex roots). A question asking "for what value of $k$ does this equation have equal roots" is really just asking you to set $D = 0$ and solve for $k$ — recognise this pattern immediately rather than attempting to factor first.

3. Inequalities — the sign-flip rule

Solving an inequality follows the same steps as an equation, with one critical exception: multiplying or dividing both sides by a negative number flips the inequality sign.

$- 2 x > 6 ⟹ x < - 3 (not x > - 3)$

For quadratic inequalities, factor first, then use a sign chart across the roots rather than reasoning verbally:

$x^{2} - 5 x + 6 > 0 ⟹ (x - 2) (x - 3) > 0 ⟹ x < 2 or x > 3$

The product of two factors is positive when both are positive or both are negative — testing one point in each of the three regions created by the roots ( $x < 2$ , $2 < x < 3$ , $x > 3$ ) confirms this quickly and avoids sign errors from reasoning about the inequality symbolically.

4. Logarithms and indices

$lo g_{a} (x y) = lo g_{a} x + lo g_{a} y, lo g_{a} (\frac{x}{y}) = lo g_{a} x - lo g_{a} y, lo g_{a} (x^{n}) = n lo g_{a} x$

$lo g_{a} x = \frac{l o g _{b} x}{l o g _{b} a} (change of base)$

Indices follow the same product-becomes-addition logic: $a^{m} \times a^{n} = a^{m + n}$ , $(a^{m})^{n} = a^{mn}$ , $a^{0} = 1$ for any $a \neq = 0$ , and $a^{- n} = \frac{1}{a ^{n}}$ . XAT frequently tests whether a candidate can simplify an expression with mixed bases and exponents into a single term before evaluating — always simplify fully before substituting numbers.

5. Functions

A function $f (x)$ maps each input to exactly one output. XAT tests two recurring skills: evaluating a composite function $f (g (x))$ by substituting the inner function's output as the outer function's input, working from the inside out; and finding a function's domain — values of $x$ for which the function is defined, most commonly excluding division by zero or a negative number under an even root.

$f (x) = x - 3 ⟹ domain: x \geq 3$

Worked examples

Question 1 of 3

Q1. If $α$ and $β$ are the roots of $2 x^{2} - 7 x + 3 = 0$ , find the value of $α^{2} + β^{2}$ .

Pick an option to check your answer.

Show explanation

Solution. Sum of roots $α + β = - \frac{- 7}{2} = \frac{7}{2}$ ; product $α β = \frac{3}{2}$ .

$α^{2} + β^{2} = (α + β)^{2} - 2 α β = (\frac{7}{2})^{2} - 2 (\frac{3}{2}) = \frac{49}{4} - 3 = \frac{49}{4} - \frac{12}{4} = \frac{37}{4}$ .

(d) is the trap of stopping after squaring the sum, forgetting to subtract $2 α β$ entirely. Answer: (a).

Question 2 of 3

Q2. For what value of $k$ does the equation $x^{2} - k x + 9 = 0$ have equal roots?

Pick an option to check your answer.

Show explanation

Solution. Equal roots require the discriminant $D = 0$ : $k^{2} - 4 (1) (9) = 0 ⟹ k^{2} = 36 ⟹ k = \pm 6$ .

(b) misses the negative root — $k = - 6$ satisfies the equation just as validly as $k = 6$ , since $(- 6)^{2} = 36$ too. (a) and (d) result from confusing this with the coefficient itself rather than solving $k^{2} = 36$ correctly. Answer: (c).

Question 3 of 3

Q3. Solve the inequality: $\frac{x - 1}{x + 2} > 0$ .

Pick an option to check your answer.

Show explanation

Solution. A fraction is positive when numerator and denominator share the same sign. Case 1: both positive — $x > 1$ and $x > - 2$ , giving $x > 1$ . Case 2: both negative — $x < 1$ and $x < - 2$ , giving $x < - 2$ .

Combining both cases: $x < - 2$ or $x > 1$ . Testing a point in the excluded middle region, $x = 0$ : $\frac{- 1}{2} < 0$ , confirming that region is correctly excluded. Answer: (b).

7. Common traps

Stopping after squaring the sum of roots without subtracting $2 α β$ when finding $α^{2} + β^{2}$ — a very common, deliberately planted five-option trap.
Missing the negative root when solving $k^{2} = c$ for a positive constant $c$ — both $+ c$ and $- c$ are valid solutions unless a stated constraint (like $k > 0$ ) rules one out.
Forgetting to flip the inequality sign when multiplying or dividing both sides by a negative number.
Reasoning about a quadratic inequality's sign verbally instead of using a sign chart across the roots — this is where most sign errors creep in.
Substituting numbers before fully simplifying a logarithmic or exponential expression — always reduce to a single term or the simplest form first.
Evaluating a composite function inside-out incorrectly — compute the innermost function first, then substitute that result into the outer function, never the reverse order.

8. When to guess, and why

Within your first 8 skips across all of Part 1, an algebra question requiring extensive manipulation under time pressure is a reasonable one to skip, particularly since checking the five given options directly against the condition is often faster than solving from scratch — if that direct-check approach also stalls, skip rather than grind. Beyond your 8th skip, a blind 1-in-5 guess has an expected value of exactly 0, while a blank costs -0.10 — mark your best remaining guess.

Because many algebra questions have options that are easy to individually verify (substitute back into the original equation or inequality), a partial check often eliminates two or three options even under time pressure, before you need to guess blind.

Summary

Algebra is roughly 4-6 of QA&DI's 28 questions — lighter than arithmetic and geometry, consistent with XAT's documented arithmetic-leaning style versus CAT.
Sum and product of roots ( $- b / a$ and $c / a$ ) solve most symmetric-function-of-roots questions without ever computing the roots themselves.
The discriminant $D = b^{2} - 4 a c$ determines root nature; "equal roots" questions are just "solve $D = 0$ " in disguise.
Multiplying or dividing an inequality by a negative number flips the sign — the single most common inequality error.
Use a sign chart across the roots for quadratic inequalities rather than reasoning verbally.
Logarithm and index rules convert products to sums and powers to multiples — always simplify fully before substituting numbers.
Composite functions evaluate inside-out: compute the innermost function first, then feed that result into the outer function.
Past your 8th free skip in Part 1, directly checking the five given options against the stated condition is often faster than solving from scratch, and usually narrows the field before a final guess is needed.

Key formulas & results

Everything to memorise for the exam hall, in one card. Screenshot this for revision.

Sum and product of roots

For ax² + bx + c = 0: sum of roots = −b/a, product of roots = c/a

Solves most symmetric-function-of-roots questions (like α²+β²) without ever computing the individual roots.

Sum of squares of roots

α² + β² = (α+β)² − 2αβ

The most commonly tested application of the sum/product identities — stopping after squaring the sum is the chapter's most common trap.

Discriminant

D = b² − 4ac. D>0: two distinct real roots. D=0: equal real roots. D<0: no real roots

'Equal roots' questions are just 'solve D=0' in disguise — recognise this pattern immediately.

Inequality sign-flip rule

Multiplying or dividing both sides of an inequality by a NEGATIVE number flips the inequality sign

The single most common inequality error — always check the sign of what you're multiplying/dividing by.

Logarithm rules

log(xy) = log x + log y; log(x/y) = log x − log y; log(xⁿ) = n log x; change of base: log_a x = log_b x / log_b a

Always simplify fully to a single term before substituting numeric values.

Index rules

aᵐ × aⁿ = a^(m+n); (aᵐ)ⁿ = a^(mn); a⁰=1; a⁻ⁿ = 1/aⁿ

Mixed-base, mixed-exponent expressions should always be simplified to a single term before evaluating.

XAT test series

Ready to be tested on this?

Algebra shows up in every XAT paper — sit a timed one and find out where you actually stand.

4–6Qs in paper

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Traps XAT sets — and how to dodge them

These are the exact option-traps and misreads that cost marks under negative marking.

WATCH OUT

✗ Stopping after squaring the sum of roots when finding α²+β²

✓ α²+β² = (α+β)² − 2αβ — the subtraction of 2αβ is essential and is the most commonly planted five-option trap in this question type.

WATCH OUT

✗ Missing the negative root when solving k² = c for a positive constant

✓ Both +√c and −√c are valid solutions unless a stated constraint (like k>0) explicitly rules one out — always state both roots unless told otherwise.

WATCH OUT

✗ Forgetting to flip the inequality sign when multiplying/dividing by a negative number

✓ Check the sign of what you're multiplying or dividing by every single time — this is worth a deliberate, explicit check, not intuition.

WATCH OUT

✗ Reasoning about a quadratic inequality's sign verbally instead of using a sign chart

✓ Factor the quadratic, mark the roots on a number line, and test one point in each resulting region — this is faster and far less error-prone than verbal sign reasoning.

WATCH OUT

✗ Substituting numbers into a log or index expression before fully simplifying

✓ Always reduce to a single term or simplest form using the product/quotient/power rules FIRST, then substitute — simplifying after substitution is slower and more error-prone.

WATCH OUT

✗ Evaluating a composite function outside-in instead of inside-out

✓ For f(g(x)), always compute g(x) first, then substitute that result into f — never attempt to evaluate f first.

Exam-pattern practice

PYQ-style questions with full solutions. Work through them as a readiness check — mark yourself honestly and get your gap report at the end.

Readiness check

Are you exam-ready for Algebra?

8 problems from this chapter. Try each one, reveal the worked solution, mark yourself honestly — get your gap report at the end.

8 questions~6 min

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•Algebra is roughly 4-6 of QA&DI's 28 questions — lighter than arithmetic and geometry, consistent with XAT's arithmetic-leaning style versus CAT.
•Sum of roots = −b/a, product of roots = c/a — solves symmetric-function-of-roots questions without computing individual roots.
•α²+β² = (α+β)² − 2αβ — forgetting the subtraction is the chapter's most common trap.
•Discriminant D=b²−4ac: D>0 two real roots, D=0 equal roots, D<0 no real roots — 'equal roots' means solve D=0.
•Multiplying or dividing an inequality by a negative number flips the sign — check this explicitly every time.
•Use a sign chart (test one point per region) for quadratic inequalities rather than verbal sign reasoning.
•Log and index rules convert products to sums and powers to multiples — simplify fully to a single term before substituting numbers.
•Composite functions evaluate inside-out: compute the innermost function first, then feed the result into the outer function.
•When solving k²=c, always check for both the positive and negative root unless a constraint rules one out.
•Past your 8th free skip in Part 1, directly checking the five given options against the stated condition is often faster than solving from scratch.

XAT question blueprint

How this topic is asked, tier by tier — so you can prep to the pattern.

Typical weightage: Roughly 4-6 of QA&DI's 28 questions (each worth +1/-0.25), based on recent-paper analysis — not an officially published XLRI split

Question style	Marks each	Typical count	What it tests
Quadratic equations	1	~2	Sum/product of roots, discriminant, symmetric functions of roots
Inequalities	1	~1	Sign-flip rule, quadratic inequality sign charts
Logarithms and indices	1	~1	Simplification using product/quotient/power rules
Functions	1	~1-2	Composite function evaluation, domain identification

Prep strategy

First pass: memorise the sum/product of roots identities, discriminant rule, and log/index rules until recall is automatic, not re-derived.
Second pass: drill quadratic inequalities specifically using the sign-chart method until it replaces any instinct to reason about signs verbally.
Final pass: practice recognising, within the first few seconds of reading a question, which standard identity or technique applies — this recognition speed matters more than raw algebraic manipulation speed for XAT's format.

Exam-hall strategy

Battle-tested tips from mentors and toppers for this topic under the sectional clock.

Recognise which standard identity a question is testing before attempting to solve algebraically from scratch — this alone resolves most XAT algebra questions faster.
For symmetric-function-of-roots questions, always reach for sum/product of roots rather than solving for the individual roots first.
Make the inequality sign-flip check an explicit, deliberate step, not something left to intuition under time pressure.
Use a sign chart with test points for every quadratic inequality — this is faster and more reliable than verbal reasoning.
Consider direct option-checking (substituting each of the five options back into the condition) as a legitimate, often-faster alternative to full algebraic solving.
Past your 8th free skip in Part 1, a partial substitution check against the given options usually narrows the field before a final guess is needed.

Beyond the exam

Where this skill shows up in the job you're competing for — and in life.

Break-even and cost-volume-profit analysis

Setting up and solving linear and quadratic relationships between cost, price and quantity is a direct business application of the equation-solving skills in this chapter.

Compound growth modelling

Logarithms are the standard tool for solving 'how long until this quantity doubles' style questions in finance and business growth modelling, directly using the log rules taught here.

Risk and constraint modelling

Inequalities are the standard mathematical language for expressing business constraints (budget limits, capacity limits), and the sign-flip rule matters in the same way when solving constraint systems for feasible ranges.

Pricing and demand functions

Composite functions model how one business variable (like cost) depends on another that itself depends on a third (like production volume depending on demand), directly using the inside-out evaluation skill from this chapter.

Where else this topic is tested

Prepare once, score in every exam that asks it.

CAT (Quantitative Ability)CAT tests algebra more heavily and often with more variables than XAT, so strong CAT algebra preparation typically covers XAT's lighter requirements comfortably

IIFT / SNAP Quantitative AbilityHigh overlap in the same core sub-topics: quadratics, inequalities, logarithms, functions

Bank PO / SSC CGL Quantitative AptitudeModerate overlap — these exams test algebra at a somewhat more basic level, useful for foundational practice

GMAT Quantitative ReasoningHigh conceptual overlap in algebraic fundamentals, within a different (computer-adaptive) format

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Questions aspirants ask

Pulled from the Q&A community and mentor sessions.

This reflects XAT's documented overall QA&DI character — heavier on arithmetic and geometry/mensuration, comparatively lighter on the algebra-heavy question styles (like multi-variable systems) CAT favours. Allocate study time proportionally if optimising specifically for XAT.

Use the identities whenever the question asks for a symmetric function of the roots (sum, product, sum of squares, or similar) — this is faster and avoids potential factoring errors. Only solve for the actual roots when the question specifically needs their individual values, such as checking a range constraint on one particular root.

Make the sign-flip check for negative multiplication/division an explicit, separate step every time, rather than trying to track it mentally while solving — and for quadratic inequalities specifically, always use a sign chart with test points rather than reasoning about the sign verbally.

Given XAT's five-option format, check whether substituting each option back into the original equation or condition is faster than a full algebraic solve — this is often true for higher-degree or more complex equations, and is a legitimate, efficient strategy, not a shortcut to be embarrassed about.

Memorise them. These rules are used constantly across arithmetic, algebra and even some DI questions, and re-deriving them under time pressure costs marks elsewhere — treat them with the same priority as the arithmetic shortcuts in the previous chapter.

More Quantitative Ability & Data Interpretation chapters

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Algebra

By the end of this chapter you'll be able to…

Before you start — revise these

Algebra — XAT Quantitative Ability & Data Interpretation

1. What XAT actually asks

2. Quadratic equations — the identities worth knowing cold

3. Inequalities — the sign-flip rule

4. Logarithms and indices

5. Functions

Worked examples

7. Common traps

8. When to guess, and why

Summary

Key formulas & results

Traps XAT sets — and how to dodge them