AQA Paper 1 2024 June — Question 15 4 marks

Exam BoardAQA
ModulePaper 1 (Paper 1)
Year2024
SessionJune
Marks4
TopicReciprocal Trig & Identities

15
  1. Show that the expression $$\sin 2 \theta \operatorname { cosec } \theta + \cos 2 \theta \sec \theta$$ can be written as $$4 \cos \theta - \sec \theta$$ where \(\sin \theta \neq 0\) and \(\cos \theta \neq 0\)
    [0pt] [4 marks]
    15
  2. A student is attempting to solve the equation $$\sin 2 \theta \operatorname { cosec } \theta + \cos 2 \theta \sec \theta = 3 \text { for } 0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }$$ They use the result from part (a), and write the following incorrect solution: $$\sin 2 \theta \operatorname { cosec } \theta + \cos 2 \theta \sec \theta = 3$$ Step \(1 \quad 4 \cos \theta - \sec \theta = 3\)
    Step \(24 \cos \theta - \frac { 1 } { \cos \theta } - 3 = 0\)
    Step \(34 \cos ^ { 2 } \theta - 3 \cos \theta - 1 = 0\) Step \(4 \cos \theta = 1\) or \(\cos \theta = - 0.25\) Step \(5 \theta = 0 ^ { \circ } , 104.5 ^ { \circ } , 255.5 ^ { \circ } , 360 ^ { \circ }\) 15
    1. Explain why the student should reject one of their values for \(\cos \theta\) in Step 4. 15
  3. (ii) State the correct solutions to the equation $$\sin 2 \theta \operatorname { cosec } \theta + \cos 2 \theta \sec \theta = 3 \text { for } 0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }$$ Figure 2 below shows a 1.5 metre length of pipe. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{0320e0a6-adc0-440a-b1da-d1a49fe06179-26_335_693_502_740}
    \end{figure} The symmetrical cross-section of the pipe is shown below, in Figure 3, where \(x\) and \(y\) are measured in centimetres. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{0320e0a6-adc0-440a-b1da-d1a49fe06179-26_652_734_1247_717}
    \end{figure} Use the trapezium rule, with the values shown in the table below, to find the best estimate for the volume of the pipe.
    \(\boldsymbol { x }\)00.40.81.21.62
    \(\boldsymbol { y }\)- 3- 2.943- 2.752- 2.353- 1.5720