Olive Defence

CDS Mathematics

Linear & Quadratic Equations

📐 Algebra · MC08 CDS Elementary Mathematics 🎯 High Priority

Linear and Quadratic Equations is the highest-scoring Algebra chapter in CDS, contributing 5–8 questions. The emphasis is on solving simultaneous equations cleanly, applying the quadratic formula confidently, and reading the discriminant to decide the nature of roots — without being asked to "compute" the full solution every time.

📌 CDS exam focus: (1) Simultaneous linear equations — substitution and elimination; (2) Quadratic factorisation — splitting middle term; (3) Quadratic formula for non-factorisable expressions; (4) Nature of roots — discriminant D = b²−4ac; (5) Sum and product of roots (Vieta's formulas); (6) Word problems leading to equations — age, number, geometry.

Topics at a Glance

① Linear Equations

One variable; transposing terms

② Simultaneous Equations

Substitution, elimination, cross-multiplication

③ Quadratic — Factorisation

Splitting middle term

④ Quadratic Formula

x = (−b ± √(b²−4ac)) / 2a

⑤ Nature of Roots

D>0 real distinct; D=0 equal; D<0 no real

⑥ Word Problems

Age, digits, geometry → equations

1. Linear Equations

1.1

Solving Linear Equations in One Variable

Foundation for all equation types

⚡ Method: Transpose, Simplify, Solve

Rule: What you do to one side, do to both sides. Transpose: move terms across = sign (sign changes) Example: 3x + 7 = 2x − 5 3x − 2x = −5 − 7 → x = −12 Example: (x+3)/4 − (x−1)/3 = 1 Multiply through by LCM(4,3)=12: 3(x+3) − 4(x−1) = 12 3x+9 − 4x+4 = 12 → −x = −1 → x = 1

2. Simultaneous Linear Equations

2.1

Three Methods — Choose the Fastest for Each Problem

Elimination is usually fastest in CDS MCQs

Substitution

Express x from eq1; sub into eq2
Best when one variable has coefficient 1
Then back-sub to find other variable

Elimination

Multiply equations to make coefficients equal
Add or subtract to eliminate one variable
Fastest for CDS — works for any pair

Cross-Multiplication

a₁x+b₁y=c₁ and a₂x+b₂y=c₂
x/(b₁c₂−b₂c₁) = y/(c₁a₂−c₂a₁) = 1/(a₁b₂−a₂b₁)
Useful for finding ratio x:y quickly

Worked Example — Elimination

Solve: 2x+3y=12 and 3x−y=1.
Multiply eq2 by 3: 9x−3y=3. Add to eq1: 11x=15 → x=15/11. Sub: 2(15/11)+3y=12 → 3y=12−30/11=102/11 → y=34/11.
Check: 3(15/11)−34/11=45/11−34/11=11/11=1 ✓

3. Quadratic Equations

3.1

Factorisation, Formula, Discriminant & Vieta's Formulas

Most tested Algebra topic in CDS — know all four aspects

⚡ Complete Quadratic Reference

Standard form: ax² + bx + c = 0 (a ≠ 0) QUADRATIC FORMULA: x = [−b ± √(b²−4ac)] / 2a DISCRIMINANT D = b²−4ac: D > 0 → Two distinct real roots D = 0 → Two equal (repeated) real roots: x = −b/2a D < 0 → No real roots (complex roots) VIETA'S FORMULAS (sum and product of roots α and β): α + β = −b/a (sum of roots) α × β = c/a (product of roots) FORM EQUATION FROM ROOTS: x² − (sum)x + (product) = 0 i.e. x² − (α+β)x + αβ = 0

Vieta's formulas let you find sum/product of roots WITHOUT solving the equation — very fast in MCQs.

Worked Examples

Find the nature of roots of 2x²−5x+4=0.
D=b²−4ac=25−32=−7<0. No real roots.

Roots of 3x²−7x+2=0 are α,β. Find α²+β².
α+β=7/3, αβ=2/3. α²+β²=(α+β)²−2αβ=49/9−4/3=49/9−12/9=37/9.

Form equation with roots 3 and −5.
Sum=−2, Product=−15. Equation: x²−(−2)x+(−15)=0 → x²+2x−15=0.

4. Inequations

4.1

Linear Inequations — Rules and Number Line

CDS tests basic inequation solving and solution sets

Rules for Inequalities

Add/subtract same value: inequality preserved
Multiply/divide by positive: inequality preserved
Multiply/divide by negative: inequality REVERSES
Example: −2x > 6 → x < −3 (divide by −2, reverse!)
Solution set: graph on number line; open circle for </>, closed for ≤/≥

Simultaneous Inequations

Solve each separately; find common region
x>2 and x<5 → 2<x<5
x≥1 and x≤3 and x is integer → x∈{1,2,3}
CDS: find number of integer solutions in a range
Always check boundary values when ≤ or ≥ used

📋 TOPIC-WISE PYQ

Equations — CDS Questions

Q1. If 2x+y=8 and x+2y=7, find x+y.

(a) 3 (b) 4 (c) 5 (d) 6

Answer: (c) 5
Add equations: 3x+3y=15 → x+y=5. (No need to find x and y separately!)

Q2. If the sum of roots of ax²+bx+c=0 is equal to the product of roots, find the relation.

(a) a=c (b) b=c (c) a=b (d) b=−a

Answer: (d) b=−a
Sum=−b/a, Product=c/a. Sum=Product → −b/a=c/a → −b=c → but answer should relate b and a... if c=a then −b/a=1 → b=−a. With c=a: b=−a.

Q3. The roots of x²−3x+2=0 are:

(a) 1, 2 (b) −1, −2 (c) 1, −2 (d) −1, 2

Answer: (a) 1, 2
(x−1)(x−2)=0 → x=1 or 2. Check: sum=3=3/1 ✓ product=2=2/1 ✓

Q4. For what value of k does kx²+2x+1=0 have equal roots?

(a) 1 (b) 2 (c) −1 (d) 0

Answer: (a) 1
Equal roots: D=0. b²−4ac=4−4k=0 → k=1.

Q5. The age of a father is three times his son's age. In 12 years it will be twice. Find present ages.

(a) Son=12, Father=36 (b) Son=10, Father=30 (c) Son=8, Father=24 (d) Son=6, Father=18

Answer: (a) Son=12, Father=36
Let son=x, father=3x. In 12 years: 3x+12=2(x+12) → 3x+12=2x+24 → x=12. Father=36.

Q6. The discriminant of 3x²−4x+3=0 indicates:

(a) Two equal roots (b) Two distinct real roots (c) No real roots (d) One root is zero

Answer: (c) No real roots
D=16−36=−20<0 → No real roots.

🔥 TRICKY QUESTIONS

Equations — Classic CDS Traps

🧩 T1. If α and β are roots of x²−px+q=0, form the equation whose roots are α/β and β/α.

Solution: qx²−(p²−2q)x+q=0.
New roots: α/β+β/α=(α²+β²)/αβ=((α+β)²−2αβ)/αβ=(p²−2q)/q.
Product: (α/β)(β/α)=1. New equation: x²−(p²−2q)/q·x+1=0 → qx²−(p²−2q)x+q=0.

🧩 T2. Solve: |x−3|+|x+2|=7.

Solution: x=4 or x=−3.
Case 1: x≥3: (x−3)+(x+2)=7 → 2x−1=7 → x=4 ✓ (satisfies x≥3)
Case 2: −2≤x<3: −(x−3)+(x+2)=7 → 5=7 → No solution.
Case 3: x<−2: −(x−3)−(x+2)=7 → 1−2x=7 → x=−3 ✓ (satisfies x<−2)

🧩 T3. The ratio of two numbers is 3:5. If 9 is added to each, ratio becomes 6:8. Find the numbers.

Solution: 9 and 15.
Let numbers = 3k and 5k. (3k+9)/(5k+9)=6/8=3/4.
4(3k+9)=3(5k+9) → 12k+36=15k+27 → 3k=9 → k=3.
Numbers: 3(3)=9 and 5(3)=15.

📐 Formula Sheet — MC08

Quadratic Formula

x=(−b±√(b²−4ac))/2a
D=b²−4ac
D>0: distinct real; D=0: equal; D<0: none

Vieta's Formulas

Sum α+β=−b/a
Product αβ=c/a
α²+β²=(α+β)²−2αβ
New eq: x²−(sum)x+product=0

Simultaneous

Elimination: match coefficients
Substitution: express and substitute
Add/subtract cleverly for sum/diff

Inequations

Multiply/divide by negative → reverse sign
Add/subtract: sign preserved
Open circle: strict < or >

⚡ Quick Revision Booster — MC08

Quadratic

Formula: (−b±√D)/2a
D=b²−4ac
Equal roots: D=0
No real: D<0

Vieta's

Sum=−b/a
Product=c/a
α²+β²=(sum)²−2(product)

Simultaneous Tips

Add equations first — often gives sum directly
Subtract for difference
Elimination: fastest method

Inequation Rule

÷ or × by negative → flip sign
Integer solutions: count carefully

Factorisation Check

ax²+bx+c: p×q=ac, p+q=b
Always verify by expanding

🚨 Traps

Adding simultaneous equations is faster than solving
For equal roots set D=0, not D>0
Inequation: negative divisor reverses

✏️ PRACTICE EXERCISE

Test Yourself — MC08

E1. Solve: 3x−4y=1 and x+y=3.

(a) x=13/7, y=8/7 (b) x=2, y=1 (c) x=1, y=2 (d) x=3, y=0

💡 From eq2: x=3−y. Substitute into eq1.

E2. If α,β are roots of 2x²−7x+3=0, find α²+β².

(a) 37/4 (b) 49/4 (c) 37/2 (d) 13/4

💡 α+β=7/2, αβ=3/2. α²+β²=(α+β)²−2αβ.

E3. For what value of k are 2x²+kx+3=0 equal roots?

(a) ±2√6 (b) ±6 (c) ±√6 (d) ±12

💡 Equal roots: D=0. k²−4(2)(3)=0.

E4. A number exceeds its square root by 20. Find the number.

(a) 16 (b) 25 (c) 36 (d) 49

💡 Let n−√n=20. Let √n=x → x²−x−20=0.

E5. If 5x+6 < 3x−2 and x is an integer, the greatest value of x is:

(a) −4 (b) −5 (c) −6 (d) −3

💡 5x+6<3x−2 → 2x<−8 → x<−4. Greatest integer: x=−5.

📝 Mock Tests 🎯 Subject Quizzes ✈️ Telegram