20 essential problems covering the most frequently misunderstood topics. Built for self-study.
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Unit 1 · Algebra & Equations
Q 01
Expand and simplify: \((x + 3)(x - 5)\)
⚡FOIL · First Outer Inner Last
Mini Example
\((a+b)(c+d) = ac + ad + bc + bd\)
e.g. \((x+1)(x+2) = x^2 + 2x + x + 2 = x^2 + 3x + 2\)
📘 Explanation
Use FOIL: \((x+3)(x-5) = x \cdot x + x \cdot (-5) + 3 \cdot x + 3 \cdot (-5)\)
\(= x^2 - 5x + 3x - 15 = x^2 - 2x - 15\) Trap: Many students forget the sign: \(+3 \times -5 = -15\), not \(+15\).
Q 02
Solve for \(x\): \(\dfrac{2x - 1}{3} = x + 2\)
⚡CROSS · multiply both sides by denominator first
📘 Explanation
Multiply both sides by 3: \(2x - 1 = 3(x + 2) = 3x + 6\)
\(-1 - 6 = 3x - 2x \Rightarrow -7 = x\) Trap: Forgetting to distribute the 3 on the right side is the #1 error here.
Q 03
Factor completely: \(x^2 - 9\)
⚡DOTS · Difference Of Two Squares: \(a^2-b^2=(a+b)(a-b)\)
⚡SUM-PRODUCT · find two numbers: sum = –b, product = c
Key Idea
For \(x^2 + bx + c = 0\), find \(m\) and \(n\) such that \(m + n = -5\) and \(m \times n = 6\).
📘 Explanation
Find two numbers that multiply to \(6\) and add to \(-5\): those are \(-2\) and \(-3\).
So \(x^2 - 5x + 6 = (x-2)(x-3) = 0\), giving \(x = 2\) or \(x = 3\). Trap: Both numbers must be negative here. \(-2 + -3 = -5\) ✓ and \(-2 \times -3 = 6\) ✓.
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Unit 2 · Linear & Quadratic Functions
Q 05
The gradient of the line passing through \((1, 4)\) and \((3, 10)\) is:
📘 Explanation
\(m = \dfrac{10 - 4}{3 - 1} = \dfrac{6}{2} = 3\) Trap: Students flip the fraction: \(\frac{x_2-x_1}{y_2-y_1}\). Always put \(\Delta y\) on TOP (rise over run).
Q 06
The equation of a line with gradient \(-2\) passing through \((0, 5)\) is:
⚡y = mx + b · slope-intercept form
📘 Explanation
\(y = mx + b\): \(m = -2\), \(b = 5\) (y-intercept is the point where \(x=0\)).
So \(y = -2x + 5\). Trap: Students swap \(m\) and \(b\), writing \(y = 5x - 2\) or forget the negative sign.
Q 07
The vertex of the parabola \(y = (x - 2)^2 + 3\) is:
⚡VERTEX FORM · \(y = a(x-h)^2 + k\) → vertex \((h, k)\) — watch the SIGN of h!
📘 Explanation
In \(y = (x - h)^2 + k\), the vertex is \((h, k)\).
Here \(h = 2\) (from \(x - 2\)) and \(k = 3\). Vertex = \((2, 3)\). Trap: \(x - 2\) means \(h = +2\), NOT \(-2\)! The minus sign inside is part of the formula.
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Unit 3 · Number & Indices
Q 08
Simplify: \(\dfrac{x^5 \cdot x^{-2}}{x^3}\)
⚡INDICES · ADD multiply, SUBTRACT divide, same base only
📘 Explanation
Numerator: \(x^5 \cdot x^{-2} = x^{5+(-2)} = x^3\)
Then: \(\dfrac{x^3}{x^3} = x^{3-3} = x^0 = 1\) Trap: Many forget that \(x^0 = 1\) for any nonzero \(x\). Also, adding a negative exponent means subtracting!
Q 09
Evaluate: \(27^{2/3}\)
⚡FRACTIONAL EXP · \(a^{m/n} = (\sqrt[n]{a})^m\) → ROOT first, then POWER
📘 Explanation
\(27^{2/3} = (\sqrt[3]{27})^2 = 3^2 = 9\)
Step 1: Cube root of 27 → \(\sqrt[3]{27} = 3\).
Step 2: Square it → \(3^2 = 9\). Trap: Students multiply first: \(27^2 = 729\), then try to cube-root → error! Always ROOT first.
Q 10
Write \(0.000045\) in standard form (scientific notation).
⚡SCI-NOT · \(a \times 10^n\) where \(1 \le a < 10\), negative n = small number
📘 Explanation
Move decimal 5 places right: \(0.000045 \to 4.5\), so exponent is \(-5\).
Answer: \(4.5 \times 10^{-5}\). Trap: Small numbers → NEGATIVE exponent. Large numbers → POSITIVE exponent. C and D don't have \(1 \le a < 10\).
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Unit 4 · Geometry & Trigonometry
Q 11
In a right triangle, opposite = 5, hypotenuse = 13. What is \(\sin \theta\)?
📘 Explanation
\(\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}} = \dfrac{5}{13}\)
Note: adjacent = \(\sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12\) (Pythagorean triple 5–12–13). Trap: \(\frac{12}{13}\) is \(\cos\theta\) and \(\frac{5}{12}\) is \(\tan\theta\) — make sure you identify which ratio!
Q 12
The area of a triangle with base 8 cm and height 5 cm is:
⚡HALF-BH · Area = \(\frac{1}{2} \times base \times height\)
📘 Explanation
\(A = \dfrac{1}{2} \times 8 \times 5 = \dfrac{40}{2} = 20 \text{ cm}^2\) Trap: Students forget the \(\frac{1}{2}\) and get 40 (rectangle area). Height must be PERPENDICULAR to base.
Q 13
Two lines are parallel. One has equation \(y = 3x - 1\). Which is parallel to it?
⚡PARALLEL = SAME SLOPE · different y-intercept only
📘 Explanation
Parallel lines have the SAME gradient. The slope of \(y = 3x - 1\) is \(3\).
A: \(y = 3x + 7\) has slope 3 ✓. B has slope \(-\frac{1}{3}\) (perpendicular!). C has slope 2. D: rearranged is \(y = -3x + 4\). Trap: Option B is the PERPENDICULAR gradient (negative reciprocal): \(-\frac{1}{3}\).
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Unit 5 · Statistics & Probability
Q 14
Data set: 3, 7, 7, 9, 12, 15, 15, 15, 20. What is the mode?
⚡MODE = MOST · Mean=average, Median=middle, Mode=most frequent
📘 Explanation
Count occurrences: 7 appears twice, 15 appears THREE times. Mode = 15.
Median (middle of 9 values, index 5) = 12. Mean ≠ mode. Trap: Students often pick 7 (they stop counting early). Always check all values!
Q 15
A bag has 4 red and 6 blue marbles. You pick one at random. What is \(P(\text{red})\)?
⚡PROBABILITY · favorable ÷ total outcomes (always between 0 and 1)
📘 Explanation
Total marbles = \(4 + 6 = 10\). Red = 4.
\(P(\text{red}) = \dfrac{4}{10} = \dfrac{2}{5}\) Trap: Option B uses only red+blue in denominator incorrectly. D is \(P(\text{blue})\)!
Q 16
The mean of 5 numbers is 12. One number is removed and the new mean of the 4 remaining is 10. What was the removed number?
⚡REVERSE MEAN · total = mean × count → find missing value
📘 Explanation
Original total: \(5 \times 12 = 60\). New total: \(4 \times 10 = 40\).
Removed number = \(60 - 40 = 20\). Trap: Students subtract the means \((12 - 10 = 2)\) instead of the totals. Always work with SUMS.
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Unit 6 · Simultaneous Equations & Inequalities
Q 17
Solve simultaneously: \(x + y = 7\) and \(x - y = 3\)
⚡ELIMINATION · add or subtract equations to cancel one variable
📘 Explanation
Add: \(2x = 10 \Rightarrow x = 5\). Substitute: \(5 + y = 7 \Rightarrow y = 2\).
Check with 2nd eq: \(5 - 2 = 3\) ✓ Trap: Students confuse which variable they solved for. Always LABEL \(x\) and \(y\) clearly.
Q 18
Solve: \(-3x + 1 > 10\)
⚡FLIP RULE · dividing/multiplying by a NEGATIVE → flip inequality sign!
📘 Explanation
\(-3x + 1 > 10 \Rightarrow -3x > 9 \Rightarrow x < -3\) (flip when dividing by \(-3\)!) Trap: The most common error in inequalities. When you divide or multiply by a NEGATIVE, the \(>\) becomes \(<\).
Q 19
The sum of two consecutive integers is 47. What are they?
⚡CONSECUTIVE · n and n+1 → set up equation, solve
📘 Explanation
Let the integers be \(n\) and \(n+1\). Then \(n + (n+1) = 47 \Rightarrow 2n + 1 = 47 \Rightarrow n = 23\).
Integers: 23 and 24. Check: \(23 + 24 = 47\) ✓ Trap: D has two equal numbers — not consecutive! Consecutive means differing by exactly 1.
Q 20
Which value of \(x\) satisfies BOTH inequalities: \(x > 1\) AND \(x \leq 4\)?
⚡AND = INTERSECTION · both must be true simultaneously
📘 Explanation
We need \(x > 1\) AND \(x \leq 4\), so \(1 < x \leq 4\).
A: \(x = 1\) fails because \(1 \not> 1\) (strict inequality). B: \(5 > 4\), fails. D: \(0 \not> 1\), fails.
C: \(x = 4\) satisfies \(4 > 1\) ✓ and \(4 \leq 4\) ✓. Answer: C Trap: Strict \(>\) means 1 is NOT included. \(\leq\) means 4 IS included.