Module 1 · JEE Mathematics
Algebra — The Detective's Toolkit
No formulas. No memorization. Just stories, animations, and "AHA!" moments.
🔍 Detective
📦 Sets & Funcs
🌀 Complex
⛲ Quadratics
🔐 P&C
✨ Binomial
📊 Matrices
🏗️ Induction
📏 Inequalities
🔢 Sequences
🎯 Practice
🔍 Part 1: The Missing Number Mystery
📦 Part 2: Sets — The Toy Boxes
👥 Part 3: Relations & Functions — Friendship Maps & Vending Machines
🌀 Part 4: Complex Numbers — The 2D Numbers
⛲ Part 5: Quadratics — The Water Fountain
🔐 Part 6: P&C — The Password and The Pizza Toppings
✨ Part 7: Binomial Theorem — The Shortcut
📊 Part 8: Matrices — The Spreadsheet of Numbers
🏗️ Mathematical Induction — The Domino Effect
📏 Linear Inequalities — The Number Line Zone
🔢 Sequences & Series — Patterns That Grow
🎯 Algebra Problem-Solving Toolkit
⚠️ Common Algebra Mistakes (Don't Fall For These!)
Similarly: (a-b)² = a² - 2ab + b².
√(a² + b²) is the hypotenuse (Pythagoras). It's always less than a + b unless one is 0.
🔥 Advanced Worked Examples (JEE Level)
🎯 Practice Zone: Test Your Understanding (No Formulas!)
Algebra Is Just Detective Work
A detective finds a cryptic note: "2x + 3 = 11". The x is a hidden number — a criminal's code. The detective must find it using the clues.
The detective thinks: "If I try x = 1, then 2×1 + 3 = 5. Nope, not 11. Try x = 2: 2×2 + 3 = 7. Closer! Try x = 3: 9. Try x = 4: 8 + 3 = 11! Found it!"
🔎 The Core Idea
Algebra = Finding Hidden Numbers Using Clues
Every equation is a mystery. The variable (x) is the unknown. The equal sign says "both sides are the same truth."
Balance rule: Whatever you do to one side, do to the other. Like a scale — if you add 3kg to the left, you must add 3kg to the right to stay balanced.
Balance rule: Whatever you do to one side, do to the other. Like a scale — if you add 3kg to the left, you must add 3kg to the right to stay balanced.
💡 Algebra is NOT about moving terms and changing signs. It's about KEEPING THE SCALE BALANCED.
⚖️ The Balance Scale — Keep It Level!
The scale shows "2x + 3" on the left and "11" on the right. To find x, remove 3 from both sides (2x = 8), then split both sides in half (x = 4). The scale stays balanced at every step!
🌟 First Big Idea
An equation is a balanced scale. Whatever you do to one side, do to the other. That's ALL of algebra in one sentence.
2x + 3 = 11 → 2x = 8 → x = 4. The detective cracked the case!
Sets Are Just Labeled Boxes Full of Things
Imagine you have two labeled boxes: Box A contains numbers {1, 2, 3}. Box B contains {3, 4, 5}. That's all a set is — a collection of things with a label.
🧸 Sets = Toy Boxes
Union (∪): Pour both boxes into one big box. A ∪ B = {1, 2, 3, 4, 5}. The 3 only appears once — sets don't have duplicates!
🔄 Intersection — What's in BOTH Boxes?
Intersection (∩): The glowing center shows what's in BOTH boxes. Only 3 appears in both A and B. So A ∩ B = {3}.
🔄 De Morgan's Law — The Complement Dance
The complement is everything outside the box. De Morgan's laws tell us how complements of unions and intersections relate — like saying "not (A or B)" is the same as "not A and not B".
Relations Are Connections, Functions Are Vending Machines
A relation is just a connection between things. Like a friendship map: Alice knows Bob, Bob knows Charlie. Arrows go from one person to another.
A function is a special kind of relation — a vending machine. Each button (input) gives EXACTLY ONE snack (output). Not zero, not two — exactly one.
👫 Relations = Friendship Arrows
A relation is just arrows connecting things. Alice knows Diana and Eve. Bob knows Diana. Charlie knows Frank. Any arrow pattern is allowed!
🥤 Functions = Vending Machines
Domain = buttons you can press (inputs). Codomain = all snacks in the machine (possible outputs). Range = snacks that actually drop for some button. Each button → exactly one snack. That's a function!
🧩 The Vending Machine Test
Is It a Function? The "One Button → One Snack" Rule
Function: Button 1 gives chips, Button 2 gives cookies. Every button gives exactly one snack. ✅
Not a function: Button 1 gives chips AND cookies. One input → two outputs. ❌
Still a function: Button 1 and Button 2 both give chips. Different inputs → same output is fine!
One-to-one: Every button gives a different snack.
Onto: Every snack in the machine comes out for SOME button.
Not a function: Button 1 gives chips AND cookies. One input → two outputs. ❌
Still a function: Button 1 and Button 2 both give chips. Different inputs → same output is fine!
One-to-one: Every button gives a different snack.
Onto: Every snack in the machine comes out for SOME button.
🧠 Functions are the strict parent of relations: "One input, exactly one output. No exceptions!"
🔄 One-to-One vs Onto — Visual Matching
One-to-one: Every input maps to a unique output (no two share the same). Onto: Every possible output is used. Neither: Some outputs unused AND some outputs shared.
What If Numbers Lived on a Plane?
Regular numbers live on a line. 1, 2, 3, 0, -1, -2... all on one straight line. But what about √(-1)?
No regular number squared gives -1. So mathematicians invented a NEW number: i = √(-1). Now numbers live on a PLANE!
🧙 The Invention of i
"What If We Pretend √(-1) Exists?"
Imagine you're playing Mario and you fall off the left edge of the screen. Where do you go? You don't disappear — you wrap around to a new dimension!
That's i. It's not on the regular number line. It's a NEW direction — the "up-down" dimension. Every complex number z = a + bi is like a coordinate (a, b) on a 2D map.
That's i. It's not on the regular number line. It's a NEW direction — the "up-down" dimension. Every complex number z = a + bi is like a coordinate (a, b) on a 2D map.
💡 Real numbers are 1D (left-right). Complex numbers are 2D (left-right + up-down). That's it!
🗺️ The Complex Plane — Numbers in 2D
Watch the point dance around the complex plane! Real part (a) = horizontal position (left-right). Imaginary part (b) = vertical position (up-down). The modulus |z| = distance from origin = √(a² + b²) — just the Pythagorean theorem!
🔄 Multiplying by i = Rotating by 90°
Watch the purple point at z = 2 + i. Multiply by i once → it rotates 90° (green). Multiply by i again → another 90° (amber). Four multiplications by i = full 360° rotation back to start!
🌟 Key Insight
i is just a "90° rotation operator." Multiplying by i doesn't stretch — it SPINS.
i² = -1 means "two 90° rotations = one 180° flip = negative"
Why Do Parabolas Make a U?
Imagine a water fountain shooting water straight up. The water rises, slows down, stops, then falls back down. The path it traces? A beautiful U-shape called a parabola.
⛲ The Water Fountain Parabola
A fountain's water traces a parabola. If a > 0, the U opens upward (like a smile). If a < 0, the U opens downward (like a frown — water falling).
🎮 Interactive Parabola — See How a, b, c Change the Shape
Slide the values of a, b, c in y = ax² + bx + c and watch the parabola change!
🔍 The Discriminant Tells You If the Parabola Touches the Ground
D > 0: Parabola crosses the x-axis twice (two real roots). D = 0: Just touches at one point (one repeated root). D < 0: Never touches the x-axis (no real roots).
🌟 Quadratic Intuition
Roots = where the parabola crosses the x-axis. Discriminant = tells you how many crossings.
The quadratic formula x = [-b ± √(b²-4ac)]/(2a) is just: "start at the vertex, walk left/right to find the crossings."
Does Order Matter?
That's the ONE question that separates permutations from combinations.
🔐 The Password Problem (Permutation)
"I need a 3-letter password from {A, B, C, D}"
ABC and CBA are DIFFERENT passwords. If someone tries "ABC" but your password is "CBA", they don't get in! Order matters! → This is a permutation.
Number of 3-letter passwords from 4 letters: 4 × 3 × 2 = 24.
Number of 3-letter passwords from 4 letters: 4 × 3 × 2 = 24.
🧠 Permutations: "Order matters" = each arrangement is unique
🌳 Tree Diagram — All 3-Letter Passwords Branching Out
Each branch is a different password. ABC is one path (highlighted). CBA is a completely different path. Order matters → each ordering is unique!
🍕 The Pizza Toppings Problem (Combination)
"I need 2 toppings from {Pepperoni, Mushrooms, Onions, Olives}"
Pepperoni + Mushrooms is the SAME pizza as Mushrooms + Pepperoni. Order doesn't matter! → This is a combination.
Combinations are permutations divided by the number of ways to rearrange: C(4,2) = 24 ÷ 2 = 6.
Combinations are permutations divided by the number of ways to rearrange: C(4,2) = 24 ÷ 2 = 6.
🧠 Combinations: "Order doesn't matter" = {A,B} is the same as {B,A}
🍕 Pizza Toppings — Same Combinations Collapsing
PM and MP are different permutations but the SAME combination — same pizza! Combinations collapse permutations that are just rearrangements of the same items.
Without the Shortcut, (x+y)⁵ Takes Forever
Imagine multiplying (x + y)(x + y)(x + y)(x + y)(x + y) by hand. That's 32 terms to compute and combine! But there's a pattern — Pascal's Triangle gives you the coefficients instantly.
🔺 Pascal's Triangle — The Coefficient Shortcut
Pascal's Triangle is a cheat code. Each number is the sum of the two numbers above it. Row 5 gives the coefficients of (x+y)⁵. No tedious multiplication needed!
A Matrix Is Just a Grid
Think of a matrix like an Excel spreadsheet — a grid of numbers organized in rows and columns. You can add them, subtract them, and even transform shapes with them!
📊 Adding Matrices = Adding Matching Slots
Adding matrices = add numbers in the SAME position. Top-left with top-left, top-right with top-right. Like adding two spreadsheets cell by cell!
🔢 The Determinant — The "Magic Number" of a 2×2 Matrix
The determinant of a 2×2 matrix = (a × d) − (b × c). Green shows the product you ADD, red shows the product you SUBTRACT. If the determinant is 0, the matrix squishes space into a line — you can't undo that!
If You Can Prove the First Case and the Step, You've Proved Everything
🏗️ The Domino Principle
"If you push the first domino, it knocks over the second, which knocks over the third... if ALL dominoes are spaced properly, they ALL fall."
Imagine a line of dominoes stretching to infinity. You want to prove they ALL fall. You need two things:
1. Base case: Push the FIRST domino. It falls. (P(1) is true)
2. Induction step: If any domino falls, it knocks over the NEXT one. (P(k) → P(k+1))
That's it! Once you have both, every domino MUST fall. The chain reaction takes care of the rest.
1. Base case: Push the FIRST domino. It falls. (P(1) is true)
2. Induction step: If any domino falls, it knocks over the NEXT one. (P(k) → P(k+1))
That's it! Once you have both, every domino MUST fall. The chain reaction takes care of the rest.
🧠 Induction = Base case + Induction step = Proof for ALL natural numbers
🏗️ 5 Dominoes Falling — The Induction Chain Reaction
Like dominoes, induction works in a chain. Base case: Prove P(1) falls. Induction step: Prove P(k) → P(k+1). Then ALL dominoes from 1 to infinity fall!
Worked Example: Sum of First n Natural Numbers
Claim: 1 + 2 + 3 + ... + n = n(n+1)/2 for all n ≥ 1.
1
Base case: Prove P(1) is true
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P(1): 1 = 1(1+1)/2 = 2/2 = 1. ✅
The left side is just 1. The right side is 1×2÷2 = 1. They match! The first domino falls.
The left side is just 1. The right side is 1×2÷2 = 1. They match! The first domino falls.
2
Induction hypothesis: Assume P(k) is true
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Assume: 1 + 2 + 3 + ... + k = k(k+1)/2
This is our "domino k just fell" assumption. We'll use this to prove domino k+1 falls.
This is our "domino k just fell" assumption. We'll use this to prove domino k+1 falls.
3
Induction step: Show P(k) → P(k+1)
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We want: 1 + 2 + ... + k + (k+1) = (k+1)(k+2)/2
Starting from the hypothesis:
[1 + 2 + ... + k] + (k+1) = k(k+1)/2 + (k+1)
= (k+1)(k/2 + 1) = (k+1)(k+2)/2 ✅
The chain holds! If domino k falls, domino k+1 falls. By induction, the formula is true for ALL n. 🎯
Starting from the hypothesis:
[1 + 2 + ... + k] + (k+1) = k(k+1)/2 + (k+1)
= (k+1)(k/2 + 1) = (k+1)(k+2)/2 ✅
The chain holds! If domino k falls, domino k+1 falls. By induction, the formula is true for ALL n. 🎯
Instead of "Exactly This", We Have "Anything in This Range"
An equation says x = 5 — exactly one answer. An inequality says x > 5 — an ENTIRE ZONE of answers. Think of it like a "number line neighborhood" instead of a single address.
📏 The Core Rule
"Everything works the same as equations... EXCEPT when you multiply or divide by a negative!"
Solving inequalities is almost identical to solving equations. You add, subtract, multiply, and divide both sides. BUT there's one crucial difference:
If you multiply or divide by a NEGATIVE number, the inequality sign FLIPS.
Think about it: -2 < 5 is true. Multiply both sides by -1: 2 < -5 is FALSE. You must flip: 2 > -5. ✅
If you multiply or divide by a NEGATIVE number, the inequality sign FLIPS.
Think about it: -2 < 5 is true. Multiply both sides by -1: 2 < -5 is FALSE. You must flip: 2 > -5. ✅
🧠 Multiply/divide by negative → FLIP the sign. Always. No exceptions.
📏 The Number Line Zone — Open vs Closed Circles
Closed circle (●): The value IS included (≤ or ≥). Open circle (○): The value is NOT included (< or >). The shaded region pulses to show the solution range.
🔄 The Sign Flip in Action
Watch closely: -2x < 6 → divide by -2 → sign FLIPS! Result: x > -3. This is the single most forgotten rule in algebra. Never skip the flip!
Worked Example with Graphical Solution
Question: Solve -3 ≤ 2x - 1 < 5 and show on the number line.
1
Add 1 to all three parts
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-3 + 1 ≤ 2x - 1 + 1 < 5 + 1
-2 ≤ 2x < 6
-2 ≤ 2x < 6
2
Divide all three parts by 2 (positive → NO flip)
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-2/2 ≤ 2x/2 < 6/2
-1 ≤ x < 3
Since we divided by +2, the inequality signs stay the same!
-1 ≤ x < 3
Since we divided by +2, the inequality signs stay the same!
3
Graph the solution
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x is in [-1, 3): closed circle at -1 (include), open circle at 3 (exclude).
x ∈ [-1, 3)
Arithmetic vs Geometric: Equal Steps vs Multiplied Steps
Sequences are patterns with a rule. Two types dominate JEE: Arithmetic Progressions (add the same number each time) and Geometric Progressions (multiply by the same number each time).
🔢 AP vs GP
"AP is like climbing stairs (equal steps). GP is like compound interest (multiplying)."
Arithmetic Progression (AP): 2, 5, 8, 11, 14... Each term = previous + 3. The common difference d = 3.
Geometric Progression (GP): 2, 4, 8, 16, 32... Each term = previous × 2. The common ratio r = 2.
AP grows linearly (straight line if graphed). GP grows exponentially (curves up or down fast).
Geometric Progression (GP): 2, 4, 8, 16, 32... Each term = previous × 2. The common ratio r = 2.
AP grows linearly (straight line if graphed). GP grows exponentially (curves up or down fast).
🧠 AP adds (linear). GP multiplies (exponential). The formula sheets follow from these simple rules.
📏 AP — Equal Steps on the Number Line
An AP takes equal steps. Here a = 2, d = 3: 2, 5, 8, 11, 14... Each term is the previous one plus the common difference d. Graph a straight line.
📈 GP — Exponential Growth (r > 1) and Decay (0 < r < 1)
A GP multiplies by r each step. Here a=1, r=2: 1, 2, 4, 8, 16... (exponential growth). If 0<r<1, bars shrink (exponential decay). If |r|<1, the infinite sum converges to a/(1-r).
Special Series — Summation Formulas
Three sums appear constantly in JEE. Memorize them, but understand the visual proof:
🧊 Sum of Squares — Visual Proof (Stacked Squares)
Special series: The sum of the first n cubes equals the square of the sum of first n numbers! Σk = n(n+1)/2, Σk² = n(n+1)(2n+1)/6, Σk³ = [n(n+1)/2]².
4 Strategies That Crack Almost Any Problem
🧠 Strategy 1
"When Stuck, Try Small Values"
Substitute n = 1, 2, 3 to spot the pattern. Most sequences and series problems give themselves away after 3 terms.
Example: Find 1×2 + 2×3 + 3×4 + ... + n(n+1). Try n=1: 2. n=2: 2+6=8. n=3: 2+6+12=20. Pattern: 2, 8, 20... = n(n+1)(n+2)/3? Check: n=1 → 2✅, n=2 → 8✅. You've guessed the formula!
Example: Find 1×2 + 2×3 + 3×4 + ... + n(n+1). Try n=1: 2. n=2: 2+6=8. n=3: 2+6+12=20. Pattern: 2, 8, 20... = n(n+1)(n+2)/3? Check: n=1 → 2✅, n=2 → 8✅. You've guessed the formula!
💡 Small values reveal big patterns. Always start with n=1,2,3.
🧠 Strategy 2
"Symmetry Is Your Friend"
Symmetric expressions like (a+b)² + (a-b)² or (x-1)(x-2)(x-3)(x-4) can be simplified by pairing terms.
Example: (x+1)(x+2)(x+3)(x+4). Instead of multiplying all four, pair them: [(x+1)(x+4)] × [(x+2)(x+3)] = (x²+5x+4)(x²+5x+6). Now substitute y = x²+5x and it becomes (y+4)(y+6) = y²+10y+24. Done!
Example: (x+1)(x+2)(x+3)(x+4). Instead of multiplying all four, pair them: [(x+1)(x+4)] × [(x+2)(x+3)] = (x²+5x+4)(x²+5x+6). Now substitute y = x²+5x and it becomes (y+4)(y+6) = y²+10y+24. Done!
💡 Symmetric pairing + substitution = massive simplification.
🧠 Strategy 3
"The Substitution Trick"
Replace complicated expressions with simple letters. The problem usually becomes a basic equation or quadratic.
Example: Solve (x²+3x)² + 2(x²+3x) - 8 = 0. Let t = x²+3x. Then t² + 2t - 8 = 0 → (t+4)(t-2) = 0 → t = -4 or t = 2 → x²+3x+4=0 or x²+3x-2=0. Now solve two easy quadratics instead of one terrifying quartic!
Example: Solve (x²+3x)² + 2(x²+3x) - 8 = 0. Let t = x²+3x. Then t² + 2t - 8 = 0 → (t+4)(t-2) = 0 → t = -4 or t = 2 → x²+3x+4=0 or x²+3x-2=0. Now solve two easy quadratics instead of one terrifying quartic!
💡 When something repeats, give it a name. t-substitution is magic.
🧠 Strategy 4
"Work Backwards from the Answer Choices"
In JEE and other MCQ tests, the answer is literally in front of you. Plug each option back and check which works — sometimes it's faster than solving.
Example: For which k does x² + 2x + k = 0 have no real roots? Options: (A) k=0 (B) k=1 (C) k=2 (D) k=3. Check D = 4-4k. For no real roots, D < 0 → k > 1. So check k=2 and k=3 — the boundary reveals the answer!
Example: For which k does x² + 2x + k = 0 have no real roots? Options: (A) k=0 (B) k=1 (C) k=2 (D) k=3. Check D = 4-4k. For no real roots, D < 0 → k > 1. So check k=2 and k=3 — the boundary reveals the answer!
💡 In multiple choice, "backwards" is often faster than "forwards."
These Traps Trip Up Even Top Students
⚠️ Mistake 1
Forgetting to Flip the Inequality Sign
This is the #1 algebra mistake in JEE. When you multiply or divide an inequality by a negative number, the sign FLIPS.
Wrong: -2x < 6 → x < -3 ❌
Right: -2x < 6 → x > -3 ✅
Test: x = -2 gives -2(-2)=4 < 6 ✅ but x = -4 gives -2(-4)=8 < 6 ❌. So x > -3, not < -3!
Rule: Multiply/divide by negative → FLIP THE SIGN.
Wrong: -2x < 6 → x < -3 ❌
Right: -2x < 6 → x > -3 ✅
Test: x = -2 gives -2(-2)=4 < 6 ✅ but x = -4 gives -2(-4)=8 < 6 ❌. So x > -3, not < -3!
Rule: Multiply/divide by negative → FLIP THE SIGN.
⚠️ Always check: "Am I multiplying/dividing by a negative? Then FLIP!"
⚠️ Mistake 2
(a+b)² ≠ a² + b² — Don't Forget 2ab
(a+b)² = a² + 2ab + b²
Many students forget the middle term. Visual proof: a square of side (a+b) has area (a+b)². It's made of a², b², and TWO rectangles of area ab. So total = a² + 2ab + b².Similarly: (a-b)² = a² - 2ab + b².
⚠️ (a+b)² = a² + b² is FALSE. The 2ab term is essential!
⚠️ Mistake 3
√(a² + b²) ≠ a + b — The Pythagorean Trap
√(a² + b²) = distance, NOT a + b
If a = 3 and b = 4, √(3² + 4²) = √25 = 5, but a + b = 7. Completely different!√(a² + b²) is the hypotenuse (Pythagoras). It's always less than a + b unless one is 0.
⚠️ Square roots do NOT distribute over addition: √(a²+b²) ≠ a+b
⚠️ Mistake 4
Cancelling Terms Across a Sum, Not a Factor
Wrong: (x² + 3x) / x = x² + 3 ❌ (cancelled one x from the first term only)
Right: (x² + 3x) / x = x(x + 3)/x = x + 3 ✅ (factor x first, then cancel)
Rule: You can ONLY cancel a common FACTOR, not terms added.
✅ (ab+ac)/a = b+c | ❌ (a+b)/a = b
Right: (x² + 3x) / x = x(x + 3)/x = x + 3 ✅ (factor x first, then cancel)
Rule: You can ONLY cancel a common FACTOR, not terms added.
✅ (ab+ac)/a = b+c | ❌ (a+b)/a = b
⚠️ "Factor before you cancel" — never cancel across a plus sign!
Multi-Step Problems That Combine Multiple Concepts
Example 1: Complex Number + Inequality
If |z - 3 + 4i| ≤ 5, find the minimum and maximum possible values of |z|.
1
Understand the geometry
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|z - (3 - 4i)| ≤ 5 means z lies inside or on a circle with center C = (3, -4) and radius 5.
Distance of center from origin = |3 - 4i| = √(3² + (-4)²) = √25 = 5.
Distance of center from origin = |3 - 4i| = √(3² + (-4)²) = √25 = 5.
2
Visualize on the complex plane
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The center C is exactly 5 units from the origin. The circle has radius 5.
So the origin lies ON the boundary of the circle!
Minimum |z|: The closest point of the circle to origin = the origin itself → min |z| = 0.
Maximum |z|: The farthest point is diametrically opposite the origin. Distance from origin to center = 5, plus radius = 5 → max |z| = 10.
So the origin lies ON the boundary of the circle!
Minimum |z|: The closest point of the circle to origin = the origin itself → min |z| = 0.
Maximum |z|: The farthest point is diametrically opposite the origin. Distance from origin to center = 5, plus radius = 5 → max |z| = 10.
3
Final answer
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0 ≤ |z| ≤ 10
The origin lies exactly on the circle, so the minimum distance is 0. The farthest point is 10 units away (diameter through origin). This combines complex modulus, circle geometry, and inequalities.
Example 2: Sum of Squares + Induction
Prove that 1² + 2² + 3² + ... + n² = n(n+1)(2n+1)/6 for all n ≥ 1 by induction.
1
Base case n = 1
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LHS = 1² = 1
RHS = 1(2)(3)/6 = 6/6 = 1 ✅
Base case holds. Domino 1 falls.
RHS = 1(2)(3)/6 = 6/6 = 1 ✅
Base case holds. Domino 1 falls.
2
Induction hypothesis: assume true for n = k
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Assume: 1² + 2² + ... + k² = k(k+1)(2k+1)/6
3
Prove for n = k+1
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[1² + 2² + ... + k²] + (k+1)² = k(k+1)(2k+1)/6 + (k+1)²
= (k+1)[k(2k+1)/6 + (k+1)]
= (k+1)[(2k² + k + 6k + 6)/6]
= (k+1)(2k² + 7k + 6)/6
= (k+1)(k+2)(2k+3)/6
= (k+1)[(k+1)+1][2(k+1)+1]/6 ✅
By induction, the formula holds for all natural numbers n. 🎯
= (k+1)[k(2k+1)/6 + (k+1)]
= (k+1)[(2k² + k + 6k + 6)/6]
= (k+1)(2k² + 7k + 6)/6
= (k+1)(k+2)(2k+3)/6
= (k+1)[(k+1)+1][2(k+1)+1]/6 ✅
By induction, the formula holds for all natural numbers n. 🎯
Example 3: Quadratic + Determinant + Inequality
For what values of k does the equation x² + kx + 9 = 0 have real roots AND both roots are positive?
1
Condition 1: Real roots → D ≥ 0
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D = b² - 4ac = k² - 4(1)(9) = k² - 36
For real roots: k² - 36 ≥ 0 → k² ≥ 36 → |k| ≥ 6
So k ≤ -6 or k ≥ 6.
For real roots: k² - 36 ≥ 0 → k² ≥ 36 → |k| ≥ 6
So k ≤ -6 or k ≥ 6.
2
Condition 2: Both roots positive
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For a quadratic ax² + bx + c = 0 with real roots:
• Sum of roots = -b/a = -k > 0 → k < 0
• Product of roots = c/a = 9 > 0 (automatically true)
So we need k < 0.
• Sum of roots = -b/a = -k > 0 → k < 0
• Product of roots = c/a = 9 > 0 (automatically true)
So we need k < 0.
3
Combine both conditions
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Condition 1: k ≤ -6 or k ≥ 6
Condition 2: k < 0
Intersection: k ≤ -6
Condition 2: k < 0
Intersection: k ≤ -6
k ∈ (-∞, -6]
This problem combines quadratics (discriminant, sum/product of roots), inequalities (two conditions intersected), and careful logical reasoning — a classic JEE multi-step question.
Practice Zone: Think, Don't Compute
These questions test if you truly understand the ideas. No calculation needed — just reasoning!