Pearson Edexcel GCSE (9-1) Mathematics Higher Tier Paper 1 (Non-Calculator) Study Notes

Examination Overview and General Instructions

This document covers the Pearson Edexcel Level 1/Level 2 GCSE (9–1) Mathematics Paper 1 (Non-Calculator) Higher Tier, specifically the Wednesday 6 November 2024 session. The paper reference is 1MA1/1H. The total time allotted for this examination is 1 hour 30 minutes, and the paper is worth a total of 80 marks.

Candidates are required to have a ruler graduated in centimetres and millimetres, a protractor, a pair of compasses, a pen, an HB or B pencil, an eraser, and the enclosed Formulae Sheet. Tracing paper may also be used. All answers must be written in black ink or ball-point pen, except for diagrams and graphs, which must be drawn with a dark pencil (HB or B). It is a mandatory requirement to show all stages of working. Calculators are strictly prohibited for this paper. Diagrams provided in the paper are not accurately drawn unless otherwise specified.

Basic Arithmetic, Fractions, and Estimates

Question 1 requires the calculation of a decimal division: $818.4 \div 1.2$. Candidates are expected to perform the division without the aid of a calculator, typically by multiplying both terms by 10 to clear the decimal places, resulting in $8184 \div 12$.

Question 3 addresses fraction operations in two parts. Part (a) asks for the subtraction of mixed numbers: $3\frac{1}{2} - 1\frac{1}{6}$. The result must be expressed as a mixed number. Part (b) is a proof-style question where the student must show that $5\frac{1}{4} \div 2\frac{1}{3} = 2\frac{1}{4}$. Full working is required to demonstrate the division of improper fractions.

Question 5 involves estimation in a real-world context involving speed and distance. A car travels $4.96\,miles$ at an average speed of $30.4\,miles\,per\,hour$. In part (a), candidates must work out an estimate for the time taken in minutes. In part (b), students must determine whether their estimate is an underestimate or an overestimate and provide a detailed reason justifying their choice based on the rounding performed in the calculation.

Probability and Statistics

Question 2 focuses on a biased dice with outcomes for 3, 4, 5, and 6 given. The probability of landing on 3 is $0.10$, on 4 is $0.30$, on 5 is $0.05$, and on 6 is $0.25$. Karim assumes the probabilities of landing on 1 and 2 are identical. In part (a), assuming Karim is correct and given 500 rolls, candidates must estimate how many times the dice will land on 2. In part (b), the scenario changes: Karim is wrong, and the probability of landing on 2 is actually greater than the probability of landing on 1. Candidates must explain how this specific information impacts the numerical estimate calculated in part (a).

Question 9 utilizes cumulative frequency data from 60 students in a test. The highest recorded mark was 74 and the lowest was 8. In part (a), candidates must use the cumulative frequency graph and the provided high/low marks to construct a box plot on a provided grid. In part (b), a pass mark of 40 is established. Sian claims that 30% of the 60 students passed the test. Candidates must verify if Sian is correct by showing their working, which involves reading the cumulative frequency at the mark of 40 to determine the number of students who scored above it.

Question 19 presents a chess tournament probability tree. In the semi-finals, player A plays B, and player C plays D. The winners move to the final. The following probabilities are provided: $P(\text{A wins against B}) = 0.6$, $P(\text{A wins against C}) = 0.5$, $P(\text{A wins against D}) = 0.3$, and $P(\text{C wins against D}) = 0.2$. The task is to calculate the total probability that player A wins the entire tournament by accounting for the different possible paths to victory.

Geometry, Trigonometry, and Vectors

Question 4 concerns a parallelogram ABCD. The internal angles are given in terms of a variable $e$, specifically angle A is marked as $e$ and another angle as $3e$. Candidates must find an expression in terms of $e$ for the size of angle CAD, providing a geometric reason for every stage of the calculation (e.g., opposite angles in a parallelogram are equal, or co-interior angles sum to $180^{\circ}$).

Question 6 describes a pentagon with five angles: $a, b, c, d, e$. The relationships between them are: angle $a =$ angle $c$, angle $b = 155^{\circ}$, angle $d = 3c$, and angle $e = 2c$. Using the sum of interior angles for a pentagon ($(5-2) \times 180^{\circ} = 540^{\circ}$), candidates must work out the exact size of angle $a$.

Question 14 involves non-right-angled trigonometry in triangle ABC. Given an angle of $30^{\circ}$ at B, $45^{\circ}$ at C, and the side length $AC = 3\sqrt{2}\,cm$, the student is asked to calculate the length of side AB. This typically requires the application of the Sine Rule: $\frac{a}{\sin(A)} = \frac{b}{\sin(B)}$.

Question 15 explores vector geometry in a parallelogram OXYZ where $\vec{OY} = \mathbf{a}$ and $\vec{OZ} = \mathbf{b}$. M is located on OY such that the ratio $OM:MY = 1:3$. N is located on OZ such that $ON:NZ = 1:2$. The final task is to work out the ratio of the vectors $XN:MN$.

Question 17 involves circle theorems for points A, B, and C on a circle with centre O. CD is a tangent to the circle. Given that angle $BCD = 40^{\circ}$ and angle $OAB = 3 \times \text{angle } OAC$, candidates must determine the size of angle ACD. All circle theorems used in the proof must be explicitly stated (e.g., Alternate Segment Theorem, angles at the centre, or radius-tangent perpendicularity).

Algebra, Functions, and Graphs

Question 10 tests indices. Part (a) asks for the evaluation of $25^{\frac{1}{2}} \times 2^{3}$. Part (b) asks for the numerical value of $\left( 32 \right)^{-\frac{3}{5}}$, requiring knowledge of both fractional and negative indices.

Question 11 covers factorisation. In part (a), Kate makes a statement regarding factorising $x^2 + 5x + 6$ into $(x + a)(x + b)$, claiming the sum of $a$ and $b$ must be 6 and the product must be 5. Candidates must explain the error in her logic (identifying that the requirements are actually swapped). Part (b) requires fully factorising $2m^2 - 2$, and part (c) requires fully factorising $ax + bx - ay - by$ via grouping.

Question 13 involves the algebraic comparison of the volumes of two cuboids, A and B. Cuboid A has dimensions $(x+3), (x-1), \text{ and } (x+2)$. Cuboid B has dimensions $2x, (x-3), \text{ and } (2x-1)$. Given that the volume of cuboid A is $142\,cm^3$ greater than the volume of cuboid B, the task is to set up and solve an equation to find the value of $x$.

Question 18 focuses on functions. Given $f(x) = \frac{x - 5}{3} + 4$, part (a) requires finding the inverse function $f^{-1}(x)$. Part (b) provides $g(x) = (x - 1)^2$ and $h(x) = 1 - 2x$, asking for the composite function value $gh(5)$.

Question 20 deals with coordinate geometry. A circle C has the equation $x^2 + y^2 = 4$. The task is to find the equation of the tangent to the circle at the point $(p, 1)$, with the condition p > 0. The final answer must be in the form $y = ax + b$ where $a$ and $b$ are integers.

Mathematical Applications in Context

Question 7 displays a linear graph representing the volume of water ($V\,litres$) in a tank over time ($t\,seconds$). Candidates are asked to identify what the gradient of this graph represents in a physical context (e.g., the rate of change of volume or flow rate).

Question 8 involves a solid triangular prism resting on a floor. The formula for pressure ($\text{pressure} = \frac{\text{force}}{\text{area}}$) is provided. The face in contact with the floor is a rectangle of width $2\,m$. The pressure is given as $80\,newtons/m^2$ and the force is given as $720\,newtons$. From this, the length of the prism must be calculated.

Question 12 explores similarity in three-dimensional shapes. Spheres A, B, and C are described. Sphere A has a volume of $64\,cm^3$ and Sphere B has a volume of $125\,cm^3$. The radius of Sphere C is $50\%$ of the radius of Sphere B. Students must work out the ratio of the surface area of Sphere A to the surface area of Sphere C, expressed in the integer form $a:b$.

Question 16 focuses on surds. Part (a) asks to rationalise the denominator of $\frac{15}{\sqrt{5}}$ and simplify. Part (b) requires writing an expression involving surds, given as $(7\sqrt{5} - 2) \times \frac{1}{2\sqrt{3}}$, into the form $a\sqrt{b} - \frac{\sqrt{c}}{3}$ (or similar integer-based forms).