Equations

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Equations — Quick Facts for CAT

Linear Equations in One Variable: $ax + b = 0$. Solution: $x = -b/a$. Simple. No tricks — just isolate the variable.

Quadratic Equations: $ax^2 + bx + c = 0$. Solutions: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

• Discriminant $D = b^2 - 4ac$:
  • $D > 0$: two distinct real roots
  • $D = 0$: one repeated real root
  • $D < 0$: no real roots (two complex conjugates)
• Sum of roots $\alpha + \beta = -b/a$
• Product of roots $\alpha\beta = c/a$

Roots and Coefficients: If $\alpha$ and $\beta$ are roots of $ax^2 + bx + c = 0$:

• $\alpha + \beta = -b/a$
• $\alpha\beta = c/a$
• $x^2 - (\alpha+\beta)x + \alpha\beta = 0$ is the equation with these roots

⚡ Exam tip: For word problems leading to quadratic equations, always check if both solutions make sense in context. Discard any negative or impossible solutions.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Equations — CAT QA Study Guide

Simultaneous Linear Equations: Two variables: $a_1 x + b_1 y = c_1$ $a_2 x + b_2 y = c_2$

Solution by elimination or Cramer’s rule: $x = \frac{c_1 b_2 - c_2 b_1}{a_1 b_2 - a_2 b_1}$; $y = \frac{a_1 c_2 - a_2 c_1}{a_1 b_2 - a_2 b_1}$. If $a_1 b_2 = a_2 b_1$, the lines are parallel (no solution) or coincident (infinite solutions).

Three Variables: Use elimination:

1. Pair equations to eliminate one variable → two equations in two variables
2. Solve those → find two variables
3. Substitute back → find the third

Quadratic Word Problems: Example: “A rectangular garden is 3 m longer than it is wide. Its area is 180 m². Find dimensions.” Let width = $x$ m, length = $x+3$ m. $x(x+3) = 180$. $x^2 + 3x - 180 = 0$. $(x+15)(x-12) = 0$. $x = 12$ (positive). Width = 12 m, Length = 15 m.

Nature of Roots:

• Both positive: $\alpha + \beta > 0$ and $\alpha\beta > 0$
• Both negative: $\alpha + \beta < 0$ and $\alpha\beta > 0$
• Opposite signs: $\alpha\beta < 0$
• Sum = 0: $\alpha + \beta = 0$ → $b = 0$
• Product = 1: $\alpha\beta = 1$ → $c/a = 1$

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Equations — Comprehensive CAT QA Notes

Cubic Equations: $x^3 + px^2 + qx + r = 0$. If $\alpha, \beta, \gamma$ are roots:

• $\alpha + \beta + \gamma = -p$
• $\alpha\beta + \beta\gamma + \gamma\alpha = q$
• $\alpha\beta\gamma = -r$

To solve $x^3 - 6x^2 + 11x - 6 = 0$: Note that $x = 1$: $1 - 6 + 11 - 6 = 0$ ✓. So $(x-1)$ is a factor. Divide: $x^3 - 6x^2 + 11x - 6 = (x-1)(x^2 - 5x + 6) = (x-1)(x-2)(x-3)$. Roots: 1, 2, 3.

Rational Root Theorem: For $ax^n + … + c = 0$, any rational root $p/q$ (in lowest terms) has $p$ dividing $c$ and $q$ dividing $a$.

Quadratic Inequations: Solve $ax^2 + bx + c > 0$:

1. Find roots (if real)
2. The quadratic opens upward ($a > 0$) or downward ($a < 0$)
3. Determine the sign in each interval

Example: $x^2 - 5x + 6 < 0$. Factor: $(x-2)(x-3) < 0$. Roots: 2, 3. Opens upward. So it’s negative between the roots: $2 < x < 3$.

Absolute Value Equations: $|x - 3| = 5$. Case 1: $x - 3 = 5$ → $x = 8$. Case 2: $x - 3 = -5$ → $x = -2$.

$|x - 2| + |x - 5| = 8$. Three cases:

• $x < 2$: $-(x-2) - (x-5) = 8$ → $-2x + 7 = 8$ → $x = -0.5$ (valid since $x < 2$)
• $2 \leq x < 5$: $(x-2) - (x-5) = 8$ → $3 = 8$ → no solution
• $x \geq 5$: $(x-2) + (x-5) = 8$ → $2x - 7 = 8$ → $x = 7.5$ (valid since $x \geq 5$) Solutions: $x = -0.5, 7.5$.

Logarithmic Equations: Example: $\log_2(x+1) + \log_2(x-1) = 3$. $\log_2[(x+1)(x-1)] = 3$. $(x+1)(x-1) = 8$. $x^2 - 1 = 8$. $x^2 = 9$. $x = 3$ or $x = -3$. Check: $x = -3$: $\log_2(-2)$ is undefined. Discard. $x = 3$: $\log_2 4 + \log_2 2 = 2 + 1 = 3$ ✓. Solution: $x = 3$.

Exponential Equations: Example: $3^{2x+1} - 10 \cdot 3^x + 3 = 0$. Let $3^x = y > 0$. $3 \cdot y^2 - 10y + 3 = 0$. $3y^2 - 10y + 3 = 0$. $(3y - 1)(y - 3) = 0$. $y = 1/3$ or $y = 3$. $3^x = 1/3 = 3^{-1}$ → $x = -1$. $3^x = 3$ → $x = 1$. Solutions: $x = -1, 1$.

Inequalities — AM-GM: AM-GM inequality: $\frac{a+b}{2} \geq \sqrt{ab}$ for $a, b > 0$. Equality when $a = b$. Useful for proving certain expressions are always greater/less than something.

System of Non-Linear Equations: Example: $x + y = 8$ and $xy = 15$. From first: $y = 8 - x$. Substitute: $x(8-x) = 15$. $8x - x^2 = 15$. $x^2 - 8x + 15 = 0$. $(x-3)(x-5) = 0$. So $x = 3$ or $5$. Corresponding $y = 5$ or $3$. Solutions: $(3,5)$ and $(5,3)$.

Word Problems — CAT Level:

Example: “A boat goes 20 km upstream and 20 km downstream in 5 hours. The speed of the stream is 3 km/h. Find the speed of the boat in still water.” Let boat speed in still water = $v$ km/h. Upstream speed = $v - 3$. Downstream speed = $v + 3$. Time upstream: $20/(v-3)$. Time downstream: $20/(v+3)$. Total: $20/(v-3) + 20/(v+3) = 5$. $20[(v+3)+(v-3)]/(v^2-9) = 5$. $20(2v)/(v^2-9) = 5$. $40v = 5(v^2-9) = 5v^2 - 45$. $5v^2 - 40v - 45 = 0$. $v^2 - 8v - 9 = 0$. $(v-9)(v+1) = 0$. $v = 9$ km/h (reject $v = -1$).

JAMB/CAT Pattern: CAT QA equations questions test: quadratic equations (nature and sum/product of roots, forming equations from given roots), solving equations with absolute values, and word problems. A frequent CAT question type: “If $\alpha$ and $\beta$ are roots of $x^2 - 5x + 6 = 0$, find the equation whose roots are $(\alpha+2)$ and $(\beta+2)$.” Answer: Sum = $(\alpha+\beta) + 4 = 5 + 4 = 9$. Product = $\alpha\beta + 2(\alpha+\beta) + 4 = 6 + 10 + 4 = 20$. Equation: $x^2 - 9x + 20 = 0$.

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📊 CAT Exam Essentials

Detail	Value
Sections	VARC (24 Qs), DILR (20 Qs), QA (22 Qs)
Time	2 hours (40 min per section)
Total	66 questions, 198 marks
Marking	+3 correct, −1 wrong (MCQ); no penalty for TITA
Mode	Computer-based, multiple sessions
Percentile	Normalized — 99+ needed for top IIMs

🎯 High-Yield Topics for CAT

• Reading Comprehension — 16-20 marks in VARC
• Para Summary + Odd Sentence — 8-12 marks
• DI Sets (Tables + Caselets) — 10-15 marks in DILR
• Arithmetic (Percentages + Profit/Loss) — 8-12 marks in QA
• Geometry + Mensuration — 6-10 marks
• Logarithm + Sequences — 6-10 marks

📝 Previous Year Question Patterns

• Q: “The passage is primarily concerned with…” [2024 VARC — RC passage]
• Q: “If f(x) = x² - 5x + 6, the value of f(3) is…” [2024 QA — Arithmetic]
• Q: “How many ways can 5 people be arranged around a round table…” [2024 DILR — Circular]

💡 Pro Tips

• VARC is the top priority — strong RC skills can push you to 99+ percentile quickly
• DILR: attempt 2 full sets out of 4-5 sets — accuracy matters more than coverage
• QA: arithmetic (time-speed-work) + geometry carry ~40% of QA marks
• Take 3-4 full mocks before the exam to find your section-wise pacing

🔗 Official Resources

• IIM CAT Official
• [CAT Syllabus](https://iimcat.ac.in/exam pattern)

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