Edexcel C3 — Question 5 10 marks

Exam BoardEdexcel
ModuleC3 (Core Mathematics 3)
Marks10
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicReciprocal Trig & Identities
TypeSketch reciprocal function graphs
DifficultyStandard +0.3 This question requires sketching a transformed secant function and finding x-intercepts. While it involves reciprocal trig functions (a C3 topic), the transformations are straightforward (horizontal shift and vertical translation), and finding zeros requires only basic algebraic manipulation of sec(x) = -2. The main challenge is correctly identifying asymptotes from cos(x - π/6) = 0, but this is a standard technique for C3 students.
Spec1.02n Sketch curves: simple equations including polynomials1.02w Graph transformations: simple transformations of f(x)1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs
5. (a) Sketch the graph of \(y = 2 + \sec \left( x - \frac { \pi } { 6 } \right)\) for \(x\) in the interval \(0 \leq x \leq 2 \pi\). Show on your sketch the coordinates of any turning points and the equations of any asymptotes.
(b) Find, in terms of \(\pi\), the \(x\)-coordinates of the points where the graph crosses the \(x\)-axis.
(a)
AnswerMarks
[Graph showing curve with turning points at \((\frac{\pi}{6}, 3)\) and \((\frac{7\pi}{6}, 1)\), vertical asymptotes at \(x = \frac{2\pi}{3}\) and \(x = \frac{5\pi}{3}\)]M2 A3
(b)
AnswerMarks Guidance
\(2 + \sec(x - \frac{\pi}{6}) = 0, \quad \sec(x - \frac{\pi}{6}) = -2, \quad \cos(x - \frac{\pi}{6}) = -\frac{1}{2}\)M1
\(x - \frac{\pi}{6} = \pi - \frac{\pi}{3}, \pi + \frac{\pi}{3} = \frac{2\pi}{3}, \frac{4\pi}{3}\)B1 M1
\(x = \frac{5\pi}{6}, \frac{3\pi}{2}\)A2 (10) 
**(a)**
[Graph showing curve with turning points at $(\frac{\pi}{6}, 3)$ and $(\frac{7\pi}{6}, 1)$, vertical asymptotes at $x = \frac{2\pi}{3}$ and $x = \frac{5\pi}{3}$] | M2 A3 |

**(b)**
$2 + \sec(x - \frac{\pi}{6}) = 0, \quad \sec(x - \frac{\pi}{6}) = -2, \quad \cos(x - \frac{\pi}{6}) = -\frac{1}{2}$ | M1 |
$x - \frac{\pi}{6} = \pi - \frac{\pi}{3}, \pi + \frac{\pi}{3} = \frac{2\pi}{3}, \frac{4\pi}{3}$ | B1 M1 |
$x = \frac{5\pi}{6}, \frac{3\pi}{2}$ | A2 | (10)
5. (a) Sketch the graph of $y = 2 + \sec \left( x - \frac { \pi } { 6 } \right)$ for $x$ in the interval $0 \leq x \leq 2 \pi$.

Show on your sketch the coordinates of any turning points and the equations of any asymptotes.\\
(b) Find, in terms of $\pi$, the $x$-coordinates of the points where the graph crosses the $x$-axis.\\

\hfill \mbox{\textit{Edexcel C3  Q5 [10]}}
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