Algebra

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Algebra — Key Facts for CLAT Quantitative Techniques • Linear Equation: $ax + b = 0 \Rightarrow x = -\dfrac{b}{a}$; when $a \neq 0$ • Quadratic Equation: $ax^2 + bx + c = 0$. Solutions: $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ • Discriminant: $D = b^2 - 4ac$

• $D > 0$: Two distinct real roots
• $D = 0$: Two equal real roots
• $D < 0$: No real roots (complex conjugates) • Sum and Product of Roots (for $ax^2 + bx + c = 0$):
• Sum: $\alpha + \beta = -\dfrac{b}{a}$
• Product: $\alpha \beta = \dfrac{c}{a}$ • Simultaneous Equations (2 variables): Solve by substitution or elimination. • Identity: $(a+b)^2 = a^2 + 2ab + b^2$; $(a-b)^2 = a^2 - 2ab + b^2$; $a^2 - b^2 = (a+b)(a-b)$

⚡ Exam Tip: For CLAT Quantitative Techniques, the algebra section focuses on word problems. The key is translating English to equations: “more than” → +, “times” → ×, “less than” → −, “is” → =, “what fraction/part” → variable.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Algebra — CLAT Study Guide

Core Concept: CLAT Quantitative Techniques algebra covers linear and quadratic equations, identities, and word problems. Unlike JEE or CAT, CLAT tests algebra primarily through application in word problems rather than pure algebraic manipulation.

Essential Identities:

• $(a+b)^2 = a^2 + 2ab + b^2$
• $(a-b)^2 = a^2 - 2ab + b^2$
• $a^2 - b^2 = (a+b)(a-b)$
• $(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$
• $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$
• $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$

Linear Equations — Word Problems:

Worked Example 1: A vendor sells mangoes at ₹40 per kg and apples at ₹25 per kg. He mixes 8 kg of mangoes with some apples. If the mixture is sold at ₹35 per kg with no profit/loss, how many kg of apples were added?

• Let $x$ = kg of apples
• Cost of mangoes = $8 \times 40 = ₹320$
• Cost of apples = $25x$
• Total cost = $320 + 25x$
• Mixture weight = $8 + x$ kg
• Selling price per kg = ₹35 → Total SP = $35(8+x)$
• No profit/loss: $35(8+x) = 320 + 25x$
• $280 + 35x = 320 + 25x$ → $10x = 40$ → $x = 4$ kg

Quadratic Equations — Discriminant Analysis:

Worked Example 2: For which value of $k$ does the equation $x^2 - 4x + k = 0$ have equal roots?

• Equal roots: $D = 0$
• $D = (-4)^2 - 4(1)(k) = 16 - 4k = 0$ → $k = 4$

Worked Example 3: The sum of two numbers is 24 and their product is 143. Find the numbers.

• Let numbers be $x$ and $24-x$
• $x(24-x) = 143$ → $24x - x^2 = 143$ → $x^2 - 24x + 143 = 0$
• $(x-13)(x-11) = 0$ → $x = 13$ or $11$; other number = 11 or 13
• Numbers are 13 and 11.

Common Student Mistakes: Forgetting to check that $a \neq 0$ in linear equations, misapplying the quadratic formula sign ($b$ vs $-b$), and in word problems, not defining the variable clearly before translating to equations.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Algebra — Comprehensive CLAT Notes

Theoretical Foundation: CLAT Quantitative Techniques algebra is designed for law aspirants who need practical quantitative skills for legal contexts (e.g., calculating damages, interest rates, profit-sharing). The emphasis is on word problems that simulate real situations, not on solving complex polynomial equations.

Simultaneous Equations — Two Variables:

Method 1 — Elimination: Multiply equations to match coefficients, then add/subtract.

Worked Example: $3x + 4y = 25$ and $5x - 2y = 4$. Find $x$ and $y$.

• Multiply second equation by 2: $10x - 4y = 8$
• Add to first: $3x + 4y + 10x - 4y = 25 + 8$ → $13x = 33$ → $x = 33/13 ≈ 2.54$
• Substitute: $3(33/13) + 4y = 25$ → $99/13 + 4y = 325/13$ → $4y = 226/13$ → $y = 226/52 = 113/26 ≈ 4.35$

Method 2 — Substitution: Express one variable in terms of the other from equation 1, substitute into equation 2.

Inequalities — Basic Principles:

• When multiplying/dividing by a negative number, the inequality sign reverses.
• $ax + b < 0$ → $x < -b/a$ if $a > 0$; $x > -b/a$ if $a < 0$
• Graphical representation on number line

Indices/Exponents:

• $a^m \times a^n = a^{m+n}$
• $a^m \div a^n = a^{m-n}$
• $(a^m)^n = a^{mn}$
• $a^0 = 1$ (where $a \neq 0$)
• $a^{-m} = 1/a^m$

Logarithms (less common but appears occasionally):

• $\log_a (xy) = \log_a x + \log_a y$
• $\log_a (x/y) = \log_a x - \log_a y$
• $\log_a (x^m) = m \log_a x$
• Change of base: $\log_a x = \dfrac{\log_b x}{\log_b a}$

CLAT PYQ Pattern (2019-2024):

• 2024: 8-10 questions from Quantitative Techniques total; algebra word problems = 3-4 questions
• Most common algebra question types: mixture/alligation, ages, speed-distance-time, percentage-based profit/loss
• Quadratic equations appear once every 2 years
• Simultaneous equations with 2 unknowns appear yearly (elimination method)
• Difficulty level: Moderate — calculators are not allowed but numbers are manageable (often designed to cancel out nicely)

Strategy: Focus on word problems (translating English to equations) and the four basic operations on equations. Master the discriminant concept — it appears frequently in CLAT Quantitative Techniques.

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