N5 Essential Skills National 5 Mathematics Practice Exam 2026 - Paper 1 - (Non-calculator)

Sep 30, 2026

Practice exam paper for National 5 Mathematics (N5), Paper 1, designed for a 1-hour session with total marks of 40. The questions require all working to be shown, prohibit the use of a calculator, and instruct students to state units where appropriate and use blue or black ink. Topics include algebraic expansion and simplification, functions and solving for inputs, indices and radicals, statistics with median and interquartile range, trigonometry ordering and justification, geometry and sine rule relationships, coordinate geometry, scatter graphs with line of best fit, and analytic geometry with parabolas and circle geometry.

LanguageEnglish

CategoryExam

FormatPDF

Pages8

Reading Time20 min

AccessFree to download

Practice Exam Instructions

Allowed materials and marking guidance

Question Paper

Algebra and functions

Surds and indices

Statistics

Trigonometry

Geometry and bearings

Coordinate geometry and graphs

N5 Essential Skills National 5 Mathematics Practice Exam 2026 | Paper 1 | (Non-calculator) | | --- | --- | | Time | 1 hour | Total marks – 40 Attempt ALL questions. You may NOT use a calculator. To earn full marks, you must show your working in your answers. State the units for your answer where appropriate. Use blue or black ink. Marks available are harsh but I wanted as much course coverage as possible. FORMULAE LIST The roots of Sine rule Cosine rule Area of a triangle Volume of a sphere Volume of a cone Volume of a pyramid Standard deviation: or 􀜽􀝔2 + 􀜾􀝔 + 􀜿 = 0 are 􀝔 = −􀜾~~ ~~±~~ ~~√(􀜾2􀜽2~~ ~~−~~ ~~4􀜽􀜿) 􀜽 sin 􀜣 = 􀜾 􀜿 ~~ ~~ = ~~ ~~ sin 􀜤 sin 􀜥 􀜽 2 = 􀜾 2 + 􀜿 2 − 2􀜾􀜿 cos 􀜣 or cos 􀜣 = 􀜾~~ ~~2~~ ~~+􀜿2􀜾2􀜿~~ ~~−~~ ~~􀜽~~ ~~2 􀜣 = 12 􀜽􀜾 sin 􀜥 􀜸 = 43 􀟨􀝎3 􀜸 = 13 􀟨􀝎2 ℎ 􀜸 = 13􀜣ℎ 􀝏 = √∑~~ ~~(􀝔􀝊~~ ~~−~~ ~~􀝔1)2 􀝏 = √∑~~ ~~􀝔~~ ~~2􀝊−−(∑1􀝔~~􀝊~~)2 , where 􀝊 is the sample size. Attempt ALL questions Total marks – 40 1. Evaluate 2 4 ÷ 1 2 9 3. Give your answer in its simplest form. 2 2. Expand and simplify (􀝔 − 3)(2􀝔2 − 3􀝔 + 1) . 3 3. A function is defined as 􀝂(􀝔) = 􀝔 2 − 􀝔 + 3. Given that 􀝂(􀝌) = 5 , find the value(s) of 􀝌 . 3 4. Express √18 + 4√2 − √50 3 5. The ages of 10 customers in a coffee shop are listed below. 13 16 16 17 21 29 36 45 49 51 Calculate the median and interquartile range of the ages. 3 6. Write the following in order, starting with the smallest. sin 60° sin 180° sin 330° Justify your answer. 2 V 7. In triangle STV • ST = 8 centimetres • SV = 6 centimetres • angle VST = 30° Given that 􀝏􀝅􀝊30° = 12 , show that sin 􀜵􀜸􀜶 = 23. S T 3 1 8. (a) Expand and simplify 􀝔 (􀝔2 − 􀝔 −1) 2 (b) Hence, evaluate when 􀝔 = 9. 2 9. Given 􀢇 = ( 1) and 􀢈 = ( 2) Calculate |2􀢇 − 􀢈| . 3 10. In a cycling race, riders start at separate times. The scatter graph shows the relationship between the time spent cycling, T minutes, and the distance remaining to the finish, D kilometres. A line of best fit has been drawn. Point A represents a cyclist who has been cycling for 2 minutes and is 30 km from the finish. Point B represents a cyclist who has been cycling for 8 minutes and is 12 km from the finish. 􀜶 (a) Find the equation of the line of best fit in terms of D and T. 3 (b) Use your equation to estimate how far a cyclist is from the finish after 5 minutes. 1 11. A parabola has equation 􀝕 = 􀝔 2 + 4􀝔 + 3. (a) Write the equation in the form 􀝕 = (􀝔 + 􀜽)2 + 􀜾 . 2 (b) Sketch the curve 􀝕 = 􀝔 2 + 4􀝔 + 3 , showing the coordinates of the turning point and the point of intersection with the 􀝕 −axis. 2 12. The diagram shows the circle with centre O. • QR is a tangent at point M. • MN is a diameter. • Angle MPQ is 113° . Calculate the size of shaded angle MQP. 3 􀜱 􀜯 􀜳 􀜰 􀜴 13. The diagram shows the straight lines 4􀝔 − 6􀝕 = 17 and 3􀝔 + 􀝕 = 10. 􀝕 4􀝔 − 6􀝕 = 17 􀝔 􀜲 3􀝔 + 􀝕 = 10 Establish the coordinates of the point of intersection, P. 3 [ END OF QUESTION PAPER ]

FAQs

Is a calculator allowed for this exam paper?

No calculator is allowed. Answers must be completed without a calculator.

What do students need to include to earn full marks?

Students must show their working, state units where appropriate, and attempt all questions.

What kinds of topics are covered in Paper 1?

The paper covers algebraic manipulation and functions, surds and radicals, statistics (median and interquartile range), trigonometry, and geometry/graph questions including a scatter graph and a parabola.

Sep 30, 2026

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