EOY Practice Paper: End-of-Year Examination — question paper (print)

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Secondary 3 · Additional Mathematics (G3)

EOY Practice Paper

End-of-Year Examination

Duration: 90 minutes

Total: 60 marks

Name:

Class:

Date:

READ THESE INSTRUCTIONS FIRST

Write your name, class and register number in the spaces at the top of this page.
Answer all questions.
Show all necessary working. Omission of essential working will result in loss of marks.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.
The number of marks is given in brackets [ ] at the end of each question or part question.
The total number of marks for each paper is shown on its cover.

Original practice material written for the MOE Additional Mathematics syllabus. It is not an official MOE or school examination paper. The full Sec 3 year (Chapters 1–7).

For Examiner’s Use
Section A	/ 60
Total	/ 60

Section A: Structured questions[60 marks]

Answer all questions in the spaces provided. Show all necessary working — omission of essential working will result in loss of marks. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless told otherwise. You may use lg 2 = 0.301, lg 3 = 0.477 and lg 5 = 0.699.

1
The function f is defined by f(x) = 5 + 4x − 2x².
1. (a)
  Express f(x) in the form a(x + b)² + c.
  [3]
2. (b)
  State the maximum value of f(x) and the value of x at which it occurs.
  maximum =[2]
3. (c)
  Explain why f(x) can never be greater than your answer to (b).
  [2]
2
Do not use a calculator in this question.
1. (a)
  Simplify (√7 − 3)(2 + √7), giving your answer in the form p + q√7.
  [3]
2. (b)
  Rationalise the denominator of 12/(√7 + 2).
  [3]
3
The polynomial f(x) = x³ − 6x² + 9x − 4.
1. (a)
  Show that (x − 1) is a factor of f(x), and factorise f(x) completely.
  [4]
2. (b)
  Write down all the solutions of f(x) = 0.
  [2]
4
Express (4x² + 15x + 13)/((x + 1)(x + 2)²) in partial fractions.
1. Show the substituted values and the coefficient comparisons.
  [7]
5
Write down, in ascending powers of x, the first four terms of the expansion of (2 + x)⁶.
1. Show how each coefficient is formed from the expansion.
  [3]
6
Hence find the coefficient of x³ in the expansion of (2 + x)⁶(1 − x).
1. Identify every product of terms that contributes to the x³ term.
  [3]
7
Solve the equation 5ˣ = 40, giving your answer correct to 3 significant figures.
1. Take logarithms and show every step.
  [4]
8
Solve the equation ln(x + 6) = 2, giving your answer correct to 3 significant figures.
1. Convert to index form and give the answer to 3 significant figures.
  [3]
9
The equation of a curve is x² + y² − 6x + 2y + 6 = 0.
1. (a)
  Show that the curve is a circle, and find its centre and radius.
  [4]
2. (b)
  The point P(5, −1) lies on the circle. Find the equation of the tangent at P.
  [3]
3. (c)
  Determine whether the point (1, 1) lies inside or outside the circle.
  [2]
10
Solve the equation 2 cos x = −1 for 0° ≤ x ≤ 360°.
1. State the quadrant reasoning as well as the answers.
  [3]
11
Prove the identity (1 − sin²θ)/sin θ = cosec θ − sin θ.
1. Transform one side only, quoting each identity you use.
  [3]
12
For the curve y = 3 sin(2x)° + 1, write down the amplitude and the period.
1. Quote the period formula you use.
  [3]
13
Express 210° in radians, giving your answer in terms of π, and write down the exact value of tan(5π/6).
1. Keep exact values in exact form.
  [3]

— End of paper —