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Core Practice · Grade 9

IB Math
Key Problems

20 essential questions from every major unit.
Enter only a number or simple word — no special symbols needed.

1
Algebra — Linear Equations
Solve for \(x\):   \(3x - 4 = 17\)
ISOLATE x · move numbers to right · divide last
Quick Example
\(2x - 1 = 9\) → \(2x = 10\) → \(x = 5\)
Answer
💡 Explanation
\(3x - 4 = 17\) → add 4 to both sides → \(3x = 21\) → divide by 3 → \(x = 7\)
2
Algebra — Expanding Brackets
Expand and simplify:   \((x+2)(x+3)\)
FOIL: First · Outer · Inner · Last
Quick Example
\((x+1)(x+4)\) → \(x^2 + 4x + x + 4\) → \(x^2 + 5x + 4\)
Answer
💡 Explanation
First: \(x \cdot x = x^2\) · Outer: \(3x\) · Inner: \(2x\) · Last: 6 → \(x^2 + 5x + 6\). Type: x2+5x+6
3
Algebra — Factorising Quadratics
Factorise:   \(x^2 + 7x + 10\)
Find two numbers: SUM = 7, PRODUCT = 10
Quick Example
\(x^2 + 5x + 6\) → need sum 5, product 6 → 2 and 3 → \((x+2)(x+3)\)
Answer
💡 Explanation
Sum = 7, Product = 10 → numbers are 2 and 5 → \((x+2)(x+5)\). Type: (x+2)(x+5)
4
Functions — Substitution
If \(f(x) = 2x^2 - 3\), find \(f(-2) + f(1)\).
PLUG IN: replace x with number, mind the negative signs
Quick Example
\(f(x)=3x+1\), \(f(2) = 7\). Negative: \(f(-2)=3(-2)+1=-5\)
Answer
💡 Explanation
\(f(-2)=2(4)-3=5\),  \(f(1)=2(1)-3=-1\) → Wait, recheck: \(f(-2)=2(-2)^2-3=8-3=5\), \(f(1)=2-3=-1\) → \(5+(-1)=\mathbf{4}\)... Actually: \(f(-2)=8-3=5\), \(f(1)=2-3=-1\), sum = 4. Hmm, let me restate: answer is 4.
5
Coordinate Geometry — Gradient
Find the gradient of the line through \((1, 2)\) and \((3, 8)\).
GRADIENT = rise/run = (y2 - y1) / (x2 - x1)
Quick Example
Points \((0,0)\) and \((2,6)\): gradient \(= \frac{6-0}{2-0} = 3\)
Answer
💡 Explanation
\(\frac{8-2}{3-1} = \frac{6}{2} = 3\). The gradient is 3.
6
Coordinate Geometry — y-intercept
The line passes through \((2, 5)\) with gradient 3. What is the \(y\)-intercept?
y = mx + c → plug in point to find c
Quick Example
Gradient 2, point \((1,3)\): \(3=2(1)+c\) → \(c=1\)
Answer
💡 Explanation
\(5 = 3(2) + c\) → \(5 = 6 + c\) → \(c = -1\). Answer: -1
7
Geometry — Pythagoras Theorem
A right triangle has legs 5 and 12. Find the hypotenuse.
a² + b² = c² · hypotenuse is ALWAYS opposite the right angle
Quick Example
Legs 3 and 4: \(3^2+4^2=9+16=25\) → \(c=5\)
Answer
💡 Explanation
\(5^2 + 12^2 = 25 + 144 = 169\) → \(\sqrt{169} = 13\). Answer: 13
8
Trigonometry — Exact Values
What is the exact value of \(\sin 30°\)?
SOH-CAH-TOA · sin30=0.5, sin45=0.707, sin60=0.866
Quick Example
\(\cos 60° = 0.5\)  ·  \(\tan 45° = 1\)  ·  \(\sin 90° = 1\)
Answer
💡 Explanation
\(\sin 30° = \frac{1}{2} = 0.5\). This is a must-memorize value.
9
Statistics — Mean
Find the mean of:   \(3, 5, 6, 7, 9\)
MEAN = sum / count · add all, then divide
Quick Example
Data: 2, 4, 6 → Sum = 12, Count = 3 → Mean = 4
Answer
💡 Explanation
\(3+5+6+7+9 = 30\), divided by 5 values → Mean = 6
10
Statistics — Median
Find the median of:  \(9, 3, 7, 1, 6\)   (unsorted)
SORT FIRST · median = middle value · if even, average middle two
Quick Example
Data: 5, 2, 8 → sorted: 2, 5, 8 → middle = 5
Answer
💡 Explanation
Sorted: 1, 3, 6, 7, 9 → middle (3rd of 5) = 6
11
Probability — Basic
A fair coin is flipped. What is the probability of getting heads?
P = favourable / total · always between 0 and 1
Quick Example
Die: P(rolling 2) = 1/6 ≈ 0.167  ·  P(even) = 3/6 = 0.5
Answer
💡 Explanation
1 favourable outcome (heads), 2 total → P = 1/2 = 0.5
12
Number — Laws of Indices
Simplify:   \(2^5 \div 2^2\)
DIVIDE = SUBTRACT powers · a^m / a^n = a^(m-n)
Quick Example
\(3^4 \div 3^2 = 3^{4-2} = 3^2 = 9\)
Answer
💡 Explanation
\(2^5 \div 2^2 = 2^{5-2} = 2^3 = 8\). Answer: 8
13
Number — Standard Form
\(5 \times 10^3 \times 2 \times 10^2 = a \times 10^n\). Find \(n\).
MULTIPLY: add the powers · check coefficient stays 1-9
Quick Example
\(3 \times 10^2 \times 2 \times 10^3 = 6 \times 10^5\) → \(n = 5\)
Answer
💡 Explanation
\(5 \times 2 = 10\), \(10^3 \times 10^2 = 10^5\) → product \(= 10 \times 10^5 = 10^6\) ... re-expressed: \(1 \times 10^6\). But asking for \(n\) when written as \(1 \times 10^6\): \(n=6\). Alternatively in simplified form: \(n=5\) refers to sum of exponents. Correct \(n\) value here is 6.
14
Sequences — nth Term
The sequence is:   \(5, 8, 11, 14, \ldots\)   Write the \(n\)th term.
nth term = dn + (first term - d) · d is the common difference
Quick Example
2, 5, 8, 11 → \(d=3\), starts at 2 → nth term \(= 3n - 1\)
Answer
💡 Explanation
\(d=3\). First term is 5, so \(a = 5 - 3 = 2\). nth term = \(3n + 2\). Check: \(n=1\): \(5 ✓\). Type: 3n+2
15
Geometry — Circle Area
Find the area of a circle with radius 7. Use \(\pi \approx \frac{22}{7}\).
Area = π r² · remember RADIUS not diameter
Quick Example
Radius 3: \(A = \pi \times 9 \approx 28.3\)
Answer
💡 Explanation
\(A = \frac{22}{7} \times 7^2 = \frac{22}{7} \times 49 = 22 \times 7 = 154\). Answer: 154
16
Algebra — Inequalities
Solve:   \(2x - 1 > 5\)
Same as equation BUT: flip sign when dividing by NEGATIVE
Quick Example
\(3x + 2 > 11\) → \(3x > 9\) → \(x > 3\)
Answer
💡 Explanation
\(2x - 1 > 5\) → \(2x > 6\) → \(x > 3\). Type: x>3
17
Algebra — Simultaneous Equations
Solve the system. What is \(x\)?
\(x + y = 5\)   and   \(x - y = -1\)
ELIMINATION: add or subtract equations to remove one variable
Quick Example
\(x+y=7\), \(x-y=3\) → add → \(2x=10\) → \(x=5\)
x = ?
💡 Explanation
Add both equations: \((x+y)+(x-y)=5+(-1)\) → \(2x=4\) → \(x=2\). Answer: 2
18
Number — Ratio and Proportion
Divide 45 in the ratio \(2 : 1\). What is the larger part?
PARTS: find total parts first, then multiply · 2:1 = 2 parts, 1 part
Quick Example
Split 30 in ratio 1:2 → total 3 parts → each part = 10 → sizes: 10 and 20
Answer
💡 Explanation
Total parts = 3. Each part = \(45 \div 3 = 15\). Larger share (2 parts) = \(2 \times 15 = 30\). Answer: 30. Wait — re-reading: "what is the larger part?" → 30. Corrected expected answer is 30.
19
Algebra — Quadratic Solutions
One solution of \(x^2 - 5x + 6 = 0\) is \(x = 3\). What is the other solution?
Two roots multiply to give c, add to give -b · or just factorise
Quick Example
\(x^2-3x+2=0\) → \((x-1)(x-2)=0\) → roots: 1 and 2
Other x
💡 Explanation
\(x^2-5x+6=(x-2)(x-3)=0\) → roots are 2 and 3. Other root: 2
20
Geometry — Transformations
Reflect the point \((3, 4)\) in the \(y\)-axis. Write the new coordinates.
y-axis reflection: flip the x sign · (x,y) → (-x, y)
Quick Example
\((5, 2)\) reflected in y-axis → \((-5, 2)\)  ·  x-axis reflection flips y: \((5,-2)\)
Answer
💡 Explanation
Reflecting in y-axis: \((3,4)\) → \((-3, 4)\). x changes sign, y stays same. Type: (-3,4)
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