Last reviewed: 7 November 2026
Quick answer: These eight original KCSE-style mathematics questions show the working, units and checks as well as the final answer. The 2026 label identifies this practice edition, not an examination year source. This is not a copy of a 2025 or 2026 paper, and all worked solutions on this page are free.
Original practice, not an examination paper. No verified 2025 paper or official answers are supplied here. The year in this page title identifies the revision edition only. Use the existing Mathematics past-paper practice page for the site’s separate paper-format resources; check each resource’s description before using it.
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How to use these worked questions
Attempt each task before reading its answer. Keep your working or first draft, compare it with the explanation, then record one correction to practise again. These are individual exercises, not a complete timed paper. No official marks, grade prediction or claim of exam coverage is attached.
Questions with worked answers
1. Simultaneous equations
Question: Solve 2x + y = 11 and x − y = 1.
Worked answer: Add the equations to eliminate y: 3x = 12, so x = 4. Substitute in x − y = 1: 4 − y = 1, so y = 3. Check in the other equation: 2(4) + 3 = 11. Thus the solution is x = 4 and y = 3. Show the elimination step so the method is clear, rather than giving only a pair of numbers.
Check your work: When subtracting equations, subtract every term, including negative terms.
2. Quadratic equation
Question: Solve x² − 7x + 12 = 0 by factorisation.
Worked answer: Find two numbers whose product is 12 and sum is −7: they are −3 and −4. Thus (x − 3)(x − 4) = 0. At least one factor must be zero, giving x = 3 or x = 4. Check: 9 − 21 + 12 = 0 and 16 − 28 + 12 = 0. Both values satisfy the original equation.
Check your work: A product equal to zero does not require both factors to equal zero at the same time.
3. Compound interest
Question: KSh 10,000 is invested at 10% per year, compounded annually, for two years. Find the final balance and the interest earned.
Worked answer: A = P(1 + r)ⁿ = 10,000(1.10)² = KSh 12,100. Interest earned = 12,100 − 10,000 = KSh 2,100. A year-by-year check gives KSh 11,000 after year one; year-two interest is KSh 1,100, not KSh 1,000. The formula assumes no deposits, withdrawals, fees or taxes and the stated annual compounding.
Check your work: The final amount and the interest are different quantities; label both.
4. Right-angled triangle
Question: A ladder 5 m long reaches a point 4 m above level ground on a vertical wall. Find the distance from its foot to the wall.
Worked answer: The ladder is the hypotenuse. By Pythagoras, d² + 4² = 5². Hence d² = 25 − 16 = 9, so d = 3 m. Use the positive root because a physical distance cannot be negative. Check the familiar 3–4–5 triangle: 3² + 4² = 25. The model assumes a right angle between the wall and ground.
Check your work: Do not add 5² and 4² when 5 m is already the hypotenuse.
5. Probability without replacement
Question: A bag contains three red balls and two blue balls. Two balls are drawn at random without replacement. Find the probability that both are red.
Worked answer: The probability of a red ball first is 3/5. After a red ball has been removed, two red balls remain among four balls, so the next probability is 2/4. Multiply along this sequence: (3/5)(2/4) = 6/20 = 3/10. There is no second ordering to add because both outcomes in the event are red.
Check your work: Without replacement changes the second denominator from 5 to 4.
6. Equation of a line
Question: Find the equation of the straight line through (2, 3) and (6, 11).
Worked answer: Gradient m = (11 − 3)/(6 − 2) = 8/4 = 2. Use y = mx + c and the point (2, 3): 3 = 2(2) + c, so c = −1. Therefore y = 2x − 1. Substitute the second point: 11 = 2(6) − 1, confirming the equation. Keep the same point order in the numerator and denominator.
Check your work: A gradient is a change in y divided by a change in x, not y divided by x at one point.
7. Volume of a cylinder
Question: A cylinder has radius 3 cm and height 10 cm. Find its volume in terms of π and approximately to one decimal place.
Worked answer: V = πr²h = π × 3² × 10 = 90π cm³. Using a calculator value of π gives approximately 282.7 cm³. The exact answer is 90π cm³; rounding should be performed at the end if a decimal is required. A radius is half a diameter, but the question already supplies the radius, so it must not be halved again.
Check your work: Volume uses cubic units; cm² would describe area.
8. Grouped-data estimate
Question: A frequency table has class intervals 0 ≤ x < 10, 10 ≤ x < 20 and 20 ≤ x < 30, with frequencies 2, 3 and 5. Estimate the mean.
Worked answer: Use class midpoints 5, 15 and 25. The weighted total is (2 × 5) + (3 × 15) + (5 × 25) = 10 + 45 + 125 = 180. Total frequency = 2 + 3 + 5 = 10. Estimated mean = 180/10 = 18. It is an estimate because the individual observations within each interval are unknown.
Check your work: Do not divide the midpoint total by the number of classes when class frequencies differ.
What to revise next
If an answer was wrong, identify whether the gap was subject knowledge, interpretation of the instruction or the method used. Redo that question without looking, then try a different question on the same topic. A correct final answer does not replace a clear explanation where the task asks for one.
Read the Mathematics revision guide and return to the paper-format practice resource. The eight examples on this page remain free. If you want additional paid practice, a KSh 20 Day Pass is available. It does not purchase a verified 2025 examination paper or official marking scheme.
Frequently asked questions
Are these actual 2025 or 2026 examination questions?
No. They are original practice questions. The 2026 label is the revision edition, not the source year of an examination paper.
Are the worked answers free?
Yes. All eight questions and explanations on this page are free without a login.
Is this a complete examination paper?
No. It is a focused collection of exercises. It does not reproduce the structure or full coverage of an examination.
Does a Day Pass buy an official marking scheme?
No. A Day Pass provides access to the site’s paid original practice and explanations, not a verified official paper or marking scheme.
Reference for concepts
OpenStax textbook reference. This is a reference for general concepts, not a source of examination questions or an endorsement of these exercises.
Revision edition prepared 7 November 2026. Examples and model responses are original CBC Guide practice.