Standard +0.8 This question requires understanding of absolute value functions, solving trigonometric inequalities, and identifying where the expression inside the absolute value changes sign. Students must split into cases (where sin x + 1/2 ≥ 0 and < 0), solve sin x ≥ 0 and sin x ≤ -1 respectively, then combine solutions carefully using set notation. The multi-step reasoning and case analysis elevates this above routine trigonometry questions, though it remains accessible to well-prepared Further Maths students.
The function f is defined by
$$f(x) = \left|\sin x + \frac{1}{2}\right| \quad (0 \leq x \leq 2\pi)$$
Find the set of values of \(x\) for which
$$f(x) \geq \frac{1}{2}$$
Give your answer in set notation.
[5 marks]
Question 7:
7 | Sketches graph of y=sinx+1
2
PI by graph of y= sinx+1
2
or
considers one equation or
inequality without modulus sign
eg sinx+ 1 = 1 or
2 2
( sinx+ 1)2 = 1
2 4 | 3.1a | M1 | 3π
{ x:0≤ x≤π} { 2π}
2
Obtains the set of values
0≤ x≤π
Condone 0< x<π | 2.2a | A1
Obtains a graph of y= sinx+1
2
with the correct shape
or
Obtains the other equation or
inequality without modulus sign
eg sinx+ 1 =−1
2 2
or
Obtains two critical values from
a quadratic in sinx | 1.1a | M1
Obtains 3π/2 | 1.1b | A1
Obtains a completely correct
answer, and expresses it using
set notation.
3π
eg [ 0,π] ,2π
2
3π
{x:0≤x≤π} x:x= {x:x=2π}
2
Condone
3π
x:0≤x≤π, ,2π
2 | 2.5 | A1
Total | 5
Q | Marking instructions | AO | Marks | Typical solution
The function f is defined by
$$f(x) = \left|\sin x + \frac{1}{2}\right| \quad (0 \leq x \leq 2\pi)$$
Find the set of values of $x$ for which
$$f(x) \geq \frac{1}{2}$$
Give your answer in set notation.
[5 marks]
\hfill \mbox{\textit{AQA Further Paper 1 2023 Q7 [5]}}