HL MATH 2

Purpose: Entrance test for entry into Higher Mathematics in September 2024.
Candidate Information Needed:
- Name (in capitals)
- School
Courses Studied Options: Candidates to indicate any studied or currently study from the following:
- Additional Mathematics OCR
- Additional Mathematics CIE
- Further Mathematics AQA
- Further Pure Mathematics Edexcel
- MYP Extended
Exam Structure:
- Reading Time: 5 minutes
- Exam Time: 40 minutes
- Equipment: Candidates should bring a pen, pencil, and eraser.
- Calculator Policy: Calculators are NOT allowed.
- Question Format: Four questions, each exploring different mathematical topics.
- Scoring: Each question is worth 15 marks.
- Correct answers with no or poor workings will receive zero marks.
- The final mark consists of the two highest-scored questions.
- Candidates are advised to focus on completing two full questions instead of attempting all four questions.

Task: Express the quadratic expression $3xx^2 + 6xx + 5$ in the form $a(x + b)^2 + c$, where $a, b, c$ are integers.
Marking: Worth 2 marks.

Sub-task i: Factorise the following expression: $15xx^2 + 30xx + 25$.
Sub-task ii: Factorise the following expression: $18xx^3 + 36xx^2 + 30xx$.
Sub-task iii: Factorise the following expression: $9xx^4 + 18xx^3 + 15xx^2$.
Marking: Each factorisation sub-task is worth a total of 2 marks (6 marks total).

Task: Explain why the equation $9xx^4 + 36xx^3 + 66xx^2 + 60xx + 25 = 0$ has no solutions.
Marking: Worth 4 marks.

Sub-task i: Using the previous answers, justify that the function $f(x) = 49xx^4 + 36xx^3 + 66xx^2 + 60xx + 25$ has a maximum value and find the maximum of $f(x)$. (4 marks)
Sub-task ii: Sketch the graph of $f(x)$ indicating where it meets the axes. (3 marks)

Sub-task i: Prove that the difference between two consecutive squares is always an odd number. (1 mark)
Sub-task ii: Show that the expression $\frac{2n+1}{n^2(n+1)^2}$ can be formatted as $\frac{1}{n^2} - \frac{1}{(n+1)^2}$. (3 marks)
Sub-task iii: Using previous results, evaluate the sum $3 \frac{12 \times 22 + 5 \times 22 \times 32 + 7 \times 32 \times 42 + … + 199 \times 992 \times 1002}{4}$ (4 marks).

Sub-task i: Proof that the difference between two consecutive squares is always one less than or one more than a multiple of 4. (2 marks)
Sub-task ii: Find the value of the series $124^2 - 123^2 + 122^2 - 121^2 + … + 4^2 - 3^2 + 2^2 - 1^2$. (5 marks)
- Hint Provided: The formula $S_n = n^2 [2a + (n-1)d]$ is useful, where $a$ is the first term in the series, $d$ is the common difference, and $n$ is the term number in the series.

Task: Explain why $QQRR = PPSS$ where $P$ and $Q$ are chords and $∠PPPP = θ$. Justification necessary. (3 marks)

Task: Show that the area ratio of triangles $ riangle PPPP$ and $ riangle SSSS$ can be represented as $|PPSS|^2$. (4 marks)

Task: Sketch graphs for $y = sin(θ)$ and $y = cos(θ)$ for the range $-180° ≤ θ ≤ 360°$, carefully labeling the graphs. (1 mark)

Task: Given that $PPPP$ is the diameter of the circle, show that the ratio of triangular areas formed by two chords can be expressed in terms of a single trigonometric function only. (7 marks)

Sub-task i: Find the inverse function $f^{-1}(x)$ given $f(x) = x - 1$. Fill in the inverse function. (1 mark)
Sub-task ii: State the domain and range of $f(x)$. Fill in the domain and range. (2 marks)

Task: Find the following composite functions:
- Sub-task i: Calculate $f^2(x)$. (1 mark)
- Sub-task ii: Calculate $f^3(x)$. (1 mark)

Task: Given the equation $F(x) + F\left(x - 1\right) = 1 + x$, find the function $F(x)$ as a single fraction in terms of $x$. (7 marks)