IB Math Grade 9 — Core Problems

Q 01

Solving Linear Equations with Fractions

EASY

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Memory Point

CLEAR → COLLECT → SOLVE
First multiply both sides by LCD to clear fractions. Then collect x-terms.

▸ Worked Example

Solve: x/3 + 2 = x/2 − 1

Step 1 — Multiply everything by LCD = 6:
6(x/3) + 6(2) = 6(x/2) − 6(1)
2x + 12 = 3x − 6

Step 2 — Collect x-terms on one side:
12 + 6 = 3x − 2x
18 = x

Check — Substitute x = 18 back ✓

⚠️

Common Mistake: Forgetting to multiply the constant terms by LCD too! EVERY term gets multiplied.

▸ Your Turn

Solve: x/4 − 3 = x/6 + 1

💡 Hint: LCD of 4 and 6 is 12

Q 02

Expanding & Factoring Quadratics

TRICKY

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Memory Point

ac METHOD
For ax² + bx + c: find two numbers that multiply to ac and add to b

▸ Worked Example

Factor: 2x² + 7x + 3

a=2, b=7, c=3 → ac = 6
Find two numbers: multiply to 6, add to 7 → 6 and 1

Split middle term:
2x² + 6x + x + 3
= 2x(x + 3) + 1(x + 3)
= (2x + 1)(x + 3)

⚠️

Trap: When a ≠ 1, students forget to use "ac" and just use "c" — this gives wrong factor pairs!

▸ Your Turn

Factor: 3x² + 10x + 8

💡 ac = 24. Find two numbers that multiply to 24 and add to 10.

Q 03

Quadratic Formula — When to Use It

TRICKY

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Memory Point

"MINUS b, PLUS/MINUS root, OVER 2a"
Discriminant: b²−4ac → (+) two roots, (0) one root, (−) no real roots

x = (-b \pm \sqrt(b²-4ac)) / 2a

▸ Worked Example

Solve: x² − 5x + 3 = 0 (give exact answer)

a=1, b=−5, c=3
Discriminant = (−5)² − 4(1)(3) = 25 − 12 = 13

x = (5 ± √13) / 2
Two solutions: x = (5+√13)/2 or x = (5−√13)/2

⚠️

Trap: Forgetting the negative sign in "−b". If b = −5, then −b = +5!

▸ Your Turn

Solve using the quadratic formula: 2x² + 3x − 1 = 0

💡 First check: is the discriminant positive? (it is!)

Q 04

Simultaneous Equations (Substitution vs Elimination)

EASY

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Memory Point

SUBSTITUTE when one equation has "y =" already
ELIMINATE when coefficients can be matched quickly

▸ Worked Example (Elimination)

Solve: 3x + 2y = 12 and x − 2y = 4

Add the equations (2y cancels):
4x = 16 → x = 4

Substitute x=4 into 2nd equation:
4 − 2y = 4 → y = 0

Solution: x = 4, y = 0

⚠️

Trap: Students forget to check their answer in BOTH equations — always verify!

▸ Your Turn

Solve: 2x + y = 7 and x − y = 2

💡 Try adding these directly — what cancels?

Q 05

Domain & Range — Finding Restrictions

GOTCHA

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Memory Point

DENOMINATOR ≠ 0 and SQUARE ROOT ≥ 0
These are the only two restrictions you'll ever need!

▸ Worked Example

Find the domain of: f(x) = √(2x − 4) / (x − 7)

Restriction 1 (square root): 2x − 4 ≥ 0 → x ≥ 2

Restriction 2 (denominator): x − 7 ≠ 0 → x ≠ 7

Domain: x ≥ 2, x ≠ 7
In interval notation: [2, 7) ∪ (7, ∞)

⚠️

Trap: Students write x > 2 instead of x ≥ 2 for square roots. Zero IS allowed under the root!

▸ Your Turn

Find the domain of: g(x) = √(x + 5) / (x − 3)

💡 Two restrictions to check!

Q 06

Graph Transformations (Shifts & Reflections)

GOTCHA

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Memory Point

f(x) + k → UP k
f(x − h) → RIGHT h (OPPOSITE direction!)
−f(x) → FLIP over x-axis
f(−x) → FLIP over y-axis

▸ Worked Example

Start with f(x) = x². Describe g(x) = −(x−3)² + 5

• (x−3) → shift RIGHT 3

• − in front → REFLECT over x-axis

• +5 → shift UP 5

Vertex is at (3, 5), opens DOWNWARD

⚠️

Classic Trap: f(x−3) moves RIGHT, NOT left! The sign inside parentheses is OPPOSITE to direction.

▸ Your Turn

Describe: h(x) = (x+2)² − 4 compared to f(x) = x²

💡 What direction does (x+2) shift? What does −4 do?

Q 07

Inverse Functions — Finding f⁻¹(x)

TRICKY

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Memory Point

SWAP x and y → SOLVE for y
Inverse = mirror over y = x line. Input/output roles SWAP.

▸ Worked Example

Find the inverse of: f(x) = 3x − 5

Step 1 Write as y: y = 3x − 5

Step 2 Swap x and y: x = 3y − 5

Step 3 Solve for y: y = (x + 5)/3

Answer: f⁻¹(x) = (x + 5)/3

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Trap: f⁻¹(x) does NOT mean 1/f(x)! The −1 means "inverse", not a power.

▸ Your Turn

Find f⁻¹(x) for: f(x) = 2x + 7

💡 After finding the inverse, check: f(f⁻¹(x)) = x?

Q 08

Composite Functions f(g(x))

TRICKY

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Memory Point

INSIDE OUT — work from right to left
f(g(x)) means: first apply g, THEN apply f to the result

▸ Worked Example

f(x) = x² + 1, g(x) = 2x − 3. Find f(g(x)).

Step 1 Write g(x): 2x − 3

Step 2 Substitute into f: replace every x with (2x−3)
f(2x−3) = (2x−3)² + 1

Expand: = 4x² − 12x + 9 + 1 = 4x² − 12x + 10

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Trap: f(g(x)) ≠ g(f(x)) in general! The order matters — don't mix them up.

▸ Your Turn

If f(x) = x + 4 and g(x) = x², find g(f(x)).

💡 Apply f first (add 4), then square the result.

Q 09

The Sine Rule

EASY

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Memory Point

"SIDE over SINE of its OPPOSITE angle"
Use when you know: 2 angles + 1 side OR 2 sides + non-included angle

a/sin(A) = b/sin(B) = c/sin(C)

▸ Worked Example

Triangle with A = 40°, B = 65°, a = 8. Find b.

C = 180° − 40° − 65° = 75°

b/sin(65°) = 8/sin(40°)
b = 8 × sin(65°)/sin(40°) ≈ 11.27

▸ Your Turn

In a triangle: A = 30°, C = 80°, c = 15. Find side a.

Q 10

The Cosine Rule

TRICKY

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Memory Point

SAS → find side: a² = b² + c² − 2bc·cos(A)
SSS → find angle: cos(A) = (b²+c²−a²)/2bc

▸ Worked Example

b = 5, c = 7, A = 60°. Find side a.

a² = 5² + 7² − 2(5)(7)cos(60°)
a² = 25 + 49 − 70 × 0.5 = 74 − 35 = 39
a = √39 ≈ 6.24

⚠️

Trap: Students mix up Sine Rule and Cosine Rule. Use Cosine when you have SAS (two sides + included angle) or SSS.

▸ Your Turn

a = 6, b = 9, c = 10. Find angle A.

💡 Use the rearranged formula: cos(A) = (b²+c²−a²)/2bc

Q 11

Exact Trig Values (30°, 45°, 60°)

GOTCHA

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Memory Point

"1 2 3 / √1 √2 √3 / 2"
sin(30°)=½, sin(45°)=√2/2, sin(60°)=√3/2
cos goes in REVERSE ORDER

▸ Cheat Table

θ	sin θ	cos θ	tan θ
30°	1/2	√3/2	1/√3
45°	√2/2	√2/2	1
60°	√3/2	1/2	√3

⚠️

Trap: Students swap sin and cos! Remember: sin goes UP (30→45→60) and cos goes DOWN.

▸ Your Turn

Without a calculator: Find the exact value of sin(60°)·cos(30°) + cos(60°)·sin(30°)

💡 Use the table above. Recognize the expansion pattern?

Q 12

Circle Theorems — Angles & Arcs

GOTCHA

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Memory Point

CENTRE angle = 2 × CIRCUMFERENCE angle (same arc)
SEMICIRCLE angle = 90° always
CYCLIC QUAD: opposite angles add to 180°

▸ Worked Example

Angle at centre = 110°. Find angle at circumference on same arc.

Angle at circumference = 110° / 2 = 55°

▸ Your Turn

In a cyclic quadrilateral, one angle is 72°. Find the opposite angle.

💡 Opposite angles in a cyclic quad are supplementary (add to 180°)

Q 13

Mean, Median, Mode — Which Measure?

EASY

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Memory Point

MEAN = average (sensitive to outliers)
MEDIAN = middle value (best for skewed data)
MODE = most frequent (best for categories)

▸ Worked Example

Data: 3, 7, 7, 9, 100. Find mean, median, mode.

Mean = (3+7+7+9+100)/5 = 126/5 = 25.2 (distorted by 100)

Median = middle value = 7 (more representative)

Mode = 7 (appears most)

→ With the outlier 100, median is best measure here

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Trap: Forgetting to SORT the data before finding median! Always order from smallest to largest first.

▸ Your Turn

Data: 5, 2, 8, 2, 15, 4, 2. Find the mean, median, and mode.

💡 Sort first! Which measure would you use for house prices in a neighbourhood?

Q 14

Probability — AND / OR Rules

GOTCHA

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Memory Point

AND (∩) → MULTIPLY (for independent events)
OR (∪) → ADD, then SUBTRACT overlap
P(A∪B) = P(A) + P(B) − P(A∩B)

▸ Worked Example

P(Rain) = 0.4, P(Cold) = 0.3, P(Rain AND Cold) = 0.15.
Find P(Rain OR Cold).

P(Rain ∪ Cold) = 0.4 + 0.3 − 0.15 = 0.55

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Trap: Forgetting to subtract the overlap for OR! Without subtracting, you count the intersection TWICE.

▸ Your Turn

P(A) = 0.5, P(B) = 0.4, P(A∩B) = 0.2. Find P(A∪B) and P(A∩B').

💡 For A∩B': A occurring but NOT B = P(A) − P(A∩B)

Q 15

Standard Deviation — Spread of Data

TRICKY

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Memory Point

SUBTRACT MEAN → SQUARE → AVERAGE → ROOT
Large SD = data spread out. Small SD = data clustered near mean.

▸ Worked Example

Data: 2, 4, 4, 4, 5, 5, 7, 9. Find the standard deviation.

Mean = (2+4+4+4+5+5+7+9)/8 = 40/8 = 5

Deviations from mean (then squared):
(−3)²=9, (−1)²=1, (−1)²=1, (−1)²=1, 0²=0, 0²=0, 2²=4, 4²=16

Variance = (9+1+1+1+0+0+4+16)/8 = 32/8 = 4

SD = √4 = 2

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Trap: Dividing by n−1 (sample) vs n (population). In IB Grade 9, usually use n for population SD.

▸ Your Turn

Find the standard deviation of: 1, 3, 5, 7, 9

💡 Mean = 5. Square the differences, average them, then root.

Q 16

Arithmetic Sequences — Finding the nth Term

EASY

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Memory Point

uₙ = u₁ + (n−1)d
u₁ = first term, d = common difference (constant gap)

▸ Worked Example

Sequence: 5, 9, 13, 17, ... Find the 20th term.

u₁ = 5, d = 4

u₂₀ = 5 + (20−1)(4) = 5 + 76 = 81

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Trap: Using n instead of (n−1). The formula says (n−1)d, not nd!

▸ Your Turn

Sequence: 3, 8, 13, 18, ... Which term equals 103?

💡 Set uₙ = 103 and solve for n.

Q 17

Geometric Sequences — nth Term & Ratio

TRICKY

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Memory Point

uₙ = u₁ × rⁿ⁻¹
r = common ratio (MULTIPLY each time). Find r: r = u₂/u₁

▸ Worked Example

Sequence: 2, 6, 18, 54, ... Find the 7th term.

u₁ = 2, r = 6/2 = 3

u₇ = 2 × 3⁶ = 2 × 729 = 1458

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Trap: Using rⁿ instead of rⁿ⁻¹! Geometric and arithmetic formulas both use (n−1), not n.

▸ Your Turn

Sequence: 80, 40, 20, 10, ... Find the 8th term and find which term is less than 1.

💡 r = 1/2. For the second part, set uₙ < 1 and solve.

Q 18

Sum of Arithmetic Series

TRICKY

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Memory Point

Sₙ = n/2 × (2u₁ + (n−1)d)
Or: Sₙ = n/2 × (first + last) — use this when you know both ends!

▸ Worked Example

Find the sum of the first 15 terms of: 4, 7, 10, 13, ...

u₁ = 4, d = 3, n = 15

S₁₅ = 15/2 × (2(4) + (14)(3)) = 7.5 × (8 + 42) = 7.5 × 50 = 375

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Trap: Using (n−1)d but also writing n/2 as n/2 correctly. Don't confuse the n here — it's the number of terms, not the index.

▸ Your Turn

Find the sum of all even numbers from 2 to 100.

💡 First term = 2, last term = 100, d = 2. How many terms are there?

Q 19

Exponential Growth & Decay

GOTCHA

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Memory Point

A = A₀ × (1 + r)ᵗ for growth (r is positive %)
A = A₀ × (1 − r)ᵗ for decay (r is positive %)
Always convert % to decimal first!

▸ Worked Example

A car costs $20,000 and depreciates at 15% per year. Find its value after 5 years.

A = 20000 × (1 − 0.15)⁵

A = 20000 × (0.85)⁵

A = 20000 × 0.4437... = $8,874

⚠️

Trap: Using 15% = 15 instead of 0.15 in the formula. ALWAYS convert % to decimal!

▸ Your Turn

A bacteria population of 500 grows at 20% per hour. How many bacteria after 4 hours?

💡 Use growth formula. After getting the answer, is it growth or decay?

Q 20

Exponent Laws — The Full Set

GOTCHA

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Memory Point

SAME BASE → ADD when multiplying, SUBTRACT when dividing
NEGATIVE exponent → FLIP (reciprocal)
FRACTIONAL exponent → ROOT: a^(1/n) = ⁿ√a

▸ All Laws Cheat Sheet

aᵐ × aⁿ = aᵐ⁺ⁿ

aᵐ ÷ aⁿ = aᵐ⁻ⁿ

(aᵐ)ⁿ = aᵐⁿ

a⁰ = 1

a⁻ⁿ = 1/aⁿ

a^(m/n) = ⁿ√(aᵐ)

Example: Simplify 8^(2/3)
= (8^(1/3))² = (∛8)² = 2² = 4

⚠️

Trap: (a+b)² ≠ a² + b². The exponent law only works for MULTIPLICATION inside, not addition!

▸ Your Turn

Simplify without a calculator:
a) 27^(2/3) b) 16^(−3/4) c) (2x³y²)³

💡 For b), negative means flip first OR flip at the end!

CORE
PROBLEMS

① Algebra

Linear & Quadratic Equations

② Functions

Functions & Graphs

③ Geometry & Trig

Geometry & Trigonometry

④ Statistics

Statistics & Probability

⑤ Sequences

Sequences & Series

COREPROBLEMS

① Algebra

Linear & Quadratic Equations

② Functions

Functions & Graphs

③ Geometry & Trig

Geometry & Trigonometry

④ Statistics

Statistics & Probability

⑤ Sequences

Sequences & Series

CORE
PROBLEMS