MATH HL 1

SIXTH FORM ENTRANCE TEST - MATHEMATICS Higher Level

Test Information

• Test Type: Six Form Entrance Test for entry into Lower Sixth (Year 12) in September 2025.

• Name and School: Fields provided for candidates to input their name (in capitals) and school.

Additional Courses Studied

• Candidates are required to circle if they have studied or are currently studying any of the following courses:

  • Additional Mathematics OCR

  • Additional Mathematics CIE

  • Further Mathematics AQA

  • Further Pure Mathematics Edexcel

  • MYP Extended

Exam Details

• Reading Time: 5 minutes

• Exam Time: 40 minutes

• Equipment Needed: Pen, pencil, eraser

• Calculator Policy: Calculators are NOT allowed.

• Structure: The exam consists of four questions, each covering different topics.

• Marks Distribution: 15 marks awarded per question.

• Scoring Policy: Correct answers with no or poor workings will receive zero marks. Final mark based on the two highest-scored questions.

• Advice: Candidates are not expected to finish all questions, thus, advised to complete only two full questions.

QUESTION 1: FACTORISATION AND EQUATIONS

a. Factorisation of Expressions

1. Factorise the expression:
   $6x^2 - 11x + 3$

   • Marks: [2 marks]

2. Factorise the expression:
   $10x^2 - 21xy + 9y^2$

   • Marks: [3 marks]

b. Solving an Equation

• Solve the following equation by squaring both sides: $ext{√}x + 22 = 2 + x$

  • Marks: [4 marks]

c. Solving Simultaneous Equations

• Solve the simultaneous equations:

  1. $4x + 3y = 49$

  2. $ext{√}5x + ext{√}5y = ext{√}17x + 7y$

  • Hint: Consider methods used in parts a. and b.

  • Marks: [6 marks]

QUESTION 2: SIMILARITY

a. Triangle Similarity Explanation

• Explain why triangles ABC and DEC are similar, given that AB and DE are parallel.

  • Marks: [1 mark]

b. Calculating Length in Similar Triangles

• Find the length of $x$ in similar triangles where one side measures 6 cm, another is $x$ cm, and a third side measures 5 cm.

  • Marks: [2 marks]

c. Expression for Height

• Write an expression for height $h$ in terms of $x$ and $y$ for similar triangles.

  • Given triangle dimensions: 6 cm, $x$ cm, and $y$ cm.

  • Marks: [2 marks]

d. Finding Height in Right-Angled Triangles

• Using similar triangles, find the height $h$ in a diagram where the triangles are right-angled with a base of 60 cm and height 20 cm.

  • Marks: [5 marks]

e. Value Calculation in a Right Triangle

• In the right triangle below, find the value of $x + y + z$. ![Diagram not provided in transcript]

  • Marks: [5 marks]

QUESTION 3: SURDS AND GEOMETRY

a. Expansion and Simplification

• Expand and simplify: $(3 + 5 ext{√}2)^2$

  • Marks: [2 marks]

b. Finding Square Root of an Expression

• By letting: $43 + 24 ext{√}3 = (a + b ext{√}3)^2$

  • Find $ext{√}(43 + 24 ext{√}3)$ where $a$ and $b ∈ ℤ^+$.

  • Marks: [5 marks]

c. Side Length of Triangle in Circle

• The circles below have a radius of 1 cm contained within equilateral triangles.

  • Find the exact length of one side of the triangle in pattern 1.

  • Note: You may use the fact that $an(30°) = rac{1}{ ext{√}3}$.

  • Marks: [2 marks]

d. Area of Triangle

• Find the area of the triangle in pattern 3.

  • Marks: [6 marks]

QUESTION 4: SEQUENCES

Note on Arithmetic Sequence

• Formula for the sum of an arithmetic sequence:
  $S<em>n = rac{n}{2} [2u</em>1 + (n - 1)d]$

Sequence Terms

1. Given that:

   • $T_1 = 1$

   • $T_2 = 1 + 2 = 3$

   • $T_3 = 1 + 2 + 3 = 6$

   • Write down the next two terms $T<em>4$ and $T</em>5$.

   • Marks: [2 marks]

b. General Expression for Sequence

• By considering the sum to $n$ terms of an arithmetic sequence, find a general expression for $T_n$.

  • Marks: [2 marks]

c. Finding Value of S17

• Find the value of $S_{17}$ given that:

  • $S_1 = 1$

  • $S_2 = 2 + 3$

  • $S_3 = 4 + 5 + 6$

  • Marks: [5 marks]

d. General Expression for Sn

• Find a general expression for $S<em>n$ given that it is of the form: $a</em>n^3 + b<em>n^2 + c</em>n + d$

  • Hint:

  • For $n = 1$, we get: $a + b + c + d = 1$.

  • For $n = 2$, we get: $8a + 4b + 2c + d = 5$.

  • Marks: [6 marks]