GCSE Maths Revision: Five Diagnostic Questions with Worked Answers

Question: solve 3(2x − 1) = 21. Expand to obtain 6x − 3 = 21. Add 3 to both sides to get 6x = 24, then divide by 6: x = 4. Substitute into the original equation: 3(8 − 1) = 21.

If you got x = 3, you may have ignored the subtraction inside the bracket. Write one transformation per line and preserve equality. Your next revision task is another bracketed equation with a negative term, not repeating only this answer.

Question: solve x² − 5x + 6 = 0. Find two numbers whose product is 6 and sum is −5: −2 and −3. Therefore (x − 2)(x − 3) = 0, giving x = 2 or x = 3.

Factorising alone is not the full solution when the question asks you to solve. Explain why at least one factor must be zero. Substituting each solution into the original expression is a quick error check.

Question: find the line through (1, 3) and (3, 7). Its gradient is (7 − 3)/(3 − 1) = 2. Use y = 2x + c and substitute (1, 3): 3 = 2 + c, so c = 1. The equation is y = 2x + 1.

Keep the subtraction order consistent in numerator and denominator. Check the second point: 2×3 + 1 = 7. If the line does not pass through both points, revisit the gradient or intercept.

Question: similar shapes have corresponding lengths in the ratio 2:3. The smaller area is 20 cm². The length scale factor is 3/2; the area scale factor is (3/2)² = 9/4. The larger area is 20×9/4 = 45 cm².

Multiplying by 3/2 gives 30, which incorrectly uses a length factor for area. Draw a pair of squares as a check: scaling each of two dimensions multiplies the area twice. Volume would require a third factor, but this question asks for area.

Question: a bag has three red and two blue counters. Two are drawn without replacement. What is the probability both are red? First red is 3/5. After it is removed, two red counters remain out of four, so second red is 2/4. Multiply: 3/5×2/4 = 3/10.

The second fraction is not 3/5 because the first counter is not returned. A labelled tree diagram makes this change visible. If the question instead said with replacement, the answer would be 9/25.

Classify each mistake: concept, procedure, arithmetic or reading. For a concept gap, return to the underlying lesson and explain it in your own words. For arithmetic, show intermediate steps. For reading, underline conditions such as without replacement.

After reviewing a topic, solve a changed example without the notes. Later, mix it with other topics and use your board's official past papers and mark schemes. Marking your steps matters as well as checking the final number. This diagnostic is a starting point, not a complete GCSE revision plan.