Pearson Edexcel International GCSE Mathematics A Paper 1HR Higher Tier May 2024 Study Notes

General Examination Information

• Total Marks: 100.

• Time Duration: 2 hours.

• Paper Reference: 4MA1/1HR (Higher Tier).

• Date: Thursday 16 May 2024.

• Required Tools: Ruler graduated in centimeters and millimeters, protractor, pair of compasses, pen, HB pencil, eraser, calculator, and optional tracing paper.

• Core Instructions:     * Show all stages of working; correct answers without sufficient working may be awarded no marks.     * Calculators are permitted.     * Nothing should be written on the formulae page; such markings will not gain credit.

Mathematical Formulae - Higher Tier

Arithmetic Series

• Sum to $n$ terms: $S_n = \frac{n}{2} [2a + (n - 1)d]$

Algebra

• Area of a Trapezium: $\frac{1}{2}(a+b)h$

• The Quadratic Equation: For $ax^2 + bx + c = 0$ where $a \neq 0$, the solutions are: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Trigonometry and Geometry

• Sine Rule: $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$

• Cosine Rule: $a^2 = b^2 + c^2 - 2bc\cos(A)$

• Area of a Triangle: $\frac{1}{2}ab\sin(C)$

Mensuration

• Volume of a Cone: $\frac{1}{3}\pi r^2 h$

• Curved Surface Area of a Cone: $\pi rl$

• Volume of a Prism: $\text{area of cross section} \times \text{length}$

• Volume of a Cylinder: $\pi r^2 h$

• Curved Surface Area of a Cylinder: $2\pi rh$

• Volume of a Sphere: $\frac{4}{3}\pi r^3$

• Surface Area of a Sphere: $4\pi r^2$

Number and Arithmetic Properties

• The Mean Calculation (Problem 1): Five cards have numbers $16, 15, 3, 2, 9$. To find the sixth card so the mean of all six is $11$, calculate the total sum: $11 \times 6 = 66$. The sum of the known cards is $16 + 15 + 3 + 2 + 9 = 45$. The missing card is $66 - 45 = 21$.

• Mixed Fractions (Problem 4): Requirement to show that $2\frac{1}{3} \times 5\frac{1}{4} = 12\frac{1}{4}$.

• Compound Interest (Problem 5): Slavomir invests $5200\text{ euros}$ for $4\text{ years}$ at $2.5\%$ interest per year. The formula used is $5200 \times (1.025)^4$. The final amount must be rounded to the nearest euro.

• Highest Common Factors and Prime Factors (Problem 11):     * Given $A = 2^5 \times 5^7 \times 7^5$ and $B = 2^3 \times 5^3 \times 7^4$.     * Evaluate the HCF of $5A$ and $2B$.     * Calculate the value of $AB^2$ as a product of prime factors.

Algebra and Functions

• Simplification and Expansion (Problem 7):     * $g^9 \div g^2 = g^7$     * Expand $5k(4k^2 + 3) = 20k^3 + 15k$

• Quadratic Equations:     * Factorize $x^2 - 2x - 63$. Finding factors of $-63$ that sum to $-2$: $(x - 9)(x + 7)$.     * Solve $x^2 - 2x - 63 = 0$ to find $x = 9$ or $x = -7$.

• Inequalities: Solve 7 - 2y < 3(y - 1).

• Linear Equations (Problem 9): Determine the equation of Line L from a grid in the form $y = mx + c$.

• Simultaneous Equations (Problem 12):     1. $4x + 3y = 9.6$     2. $6x + 5y = 16.8$

• Algebraic Manipulation (Problem 14):     * Expand and simplify $(x-4)(x+3)(x+1)$.     * Simplify algebraically: $(a^3 b^9)^{\frac{1}{3}}$.

• Index Laws and Surds (Problem 17):     * Solve $k^4 \times (k^{12})^5 = k^n$ to find $n$.     * Express $\frac{7}{2 - \sqrt{3}}$ in the form $c + d\sqrt{3}$ by rationalizing the denominator.

• Completing the Square (Problem 20):     * Express $2x^2 - 11x + 9$ in the form $a(x - b)^2 + c$.     * Find the minimum point $P$ of the curve $y = (2x^2 - 11x + 9)^2$.

Geometry and Trigonometry

• Cylinder Mensuration (Problem 6):     * Volume $V = 1208\,cm^3$, Radius $r = 8\,cm$. Calculate height $h$.     * Given density $1.25\,g/cm^3$, calculate the mass of the cylinder in kilograms ($kg$).

• Trapezium Perimeter (Problem 8):     * Trapezium $ABCD$ with right angles at $A$ and $D$. $AD = 15\,cm$, $DC = 14\,cm$. Area is $360\,cm^2$. Work out the perimeter by first finding the length of $AB$ and then the slanted side $BC$.

• Circle Theorems (Problem 13): A, B, C, and D are points on a circle with centre O. Given angle $BCD = 128^{\circ}$, determine angle $OBD$. Reason: Opposite angles in a cyclic quadrilateral sum to $180^{\circ}$; angle at the center is twice the angle at the circumference.

• Isosceles Triangle Area (Problem 15): Triangle $EFG$ where $EF = GF$ and $\angle EFG = 130^{\circ}$ . Given Area = $74\,cm^2$, find length $EF$ using $\text{Area} = \frac{1}{2} ab \sin(C)$.

• Similarity (Problem 18): Vases A and B have heights of $30\,cm$ and $12\,cm$ respectively. Area scale factor is the square of the linear scale factor. Given the difference in surface area is $178.5\,cm^2$, calculate the surface area of Vase A.

• 3D Geometry (Problem 22): Cuboid with dimensions $8 \times 12 \times 20\,cm$. Calculate angle $BMP$ where $M$ is the midpoint of base $ADEH$ and $P$ is the midpoint of edge $CF$.

Statistics and Probability

• Biased Spinner (Problem 2):     * Probabilities for numbers 1 to 5: $2x, 0.27, 0.04, x, 0.12$.     * Total probability sum: $2x + 0.27 + 0.04 + x + 0.12 = 1$.     * Calculate $x$ and then use it to estimate landing on an odd number (1, 3, or 5) over $400$ spins.

• Ratio and Profit (Problem 3):     * Total loaves: $200$. White to Brown ratio = $3:2$.     * Profit on White: $40\%$ of $\pounds 1.50$. Profit on Brown: $60\%$ of $\pounds 1.75$.     * Calculate total profit for all loaves.

• Interquartile Range (Problem 10): Data set of goals: $0, 1, 2, 2, 3, 4, 4, 6, 7, 9, 11$. Identify the median, lower quartile ($Q_1$), and upper quartile ($Q_3$) to find $IQR = Q_3 - Q_1$.

• Histogram Construction (Problem 16):     * Trees of heights $h$ metered. Frequency density is calculated as $\text{Frequency} \div \text{Class Width}$.     * Intervals: $0-2$, $2-5$, $5-10$, $10-20$, $20-35$.

• Probability and Algebra (Problem 21):     * Bag contains $25$ counters: $6$ blue, $x$ orange (x > 9), and remainder are pink ($25 - 6 - x = 19 - x$).     * Probability of picking one orange and one pink counter is $\frac{22}{75}$.     * Solve algebraically for $x$ and determine the probability of picking two pink counters.

Advanced Algebra and Calculus

• Stationary Points (Problem 19):     * Curve $C$: $y = \frac{x^3}{3} - \frac{x^2}{2} - 12x + 5$.     * Find the stationary points by finding $\frac{dy}{dx}$ and setting it to zero ($x^2 - x - 12 = 0$).     * Solve for $x$ coordinates of points A and B. Identify A where x_A > x_B.

• Arithmetic Sequences (Problem 23):     * Sequence terms: $(4x - 1)$, $(x + 14)$, $(7x + 9)$. (Note: Transcript suggests terms related to $x$).     * Calculate the value of $x$ based on common difference $d$, then find the sum of the first $40$ terms using $S_n$.