Find intersection or crossing points

A question is this type if and only if it requires finding where a harmonic curve intersects with axes or equals a specific value, giving coordinates.

3 questions · Standard +0.3

Sort by: Default | Easiest first | Hardest first
Edexcel C3 2008 January Q7
13 marks Standard +0.3
  1. A curve \(C\) has equation
$$y = 3 \sin 2 x + 4 \cos 2 x , - \pi \leqslant x \leqslant \pi$$ The point \(A ( 0,4 )\) lies on \(C\).
  1. Find an equation of the normal to the curve \(C\) at \(A\).
  2. Express \(y\) in the form \(R \sin ( 2 x + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give the value of \(\alpha\) to 3 significant figures.
  3. Find the coordinates of the points of intersection of the curve \(C\) with the \(x\)-axis. Give your answers to 2 decimal places.
OCR C3 Q7
11 marks Standard +0.3
7. \includegraphics[max width=\textwidth, alt={}, center]{6cdbc2bc-8863-4003-a218-44552d75d137-2_556_777_246_468} The diagram shows the curve \(y = \mathrm { f } ( x )\) which has a maximum point at ( \(- 45,7\) ) and a minimum point at \(( 135 , - 1 )\).
  1. Showing the coordinates of any stationary points, sketch the curve with equation \(y = 1 + 2 \mathrm { f } ( x )\). Given that $$f ( x ) = A + 2 \sqrt { 2 } \cos x ^ { \circ } - 2 \sqrt { 2 } \sin x ^ { \circ } , \quad x \in \mathbb { R } , \quad - 180 \leq x \leq 180 ,$$ where \(A\) is a constant,
  2. show that \(\mathrm { f } ( x )\) can be expressed in the form $$\mathrm { f } ( x ) = A + R \cos ( x + \alpha ) ^ { \circ }$$ where \(R > 0\) and \(0 < \alpha < 90\),
  3. state the value of \(A\),
  4. find, to 1 decimal place, the \(x\)-coordinates of the points where the curve \(y = \mathrm { f } ( x )\) crosses the \(x\)-axis.
Edexcel C3 Q7
14 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7c3dd501-0545-4166-aaf9-5e1ac1f369c5-4_552_771_248_470} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve \(y = \mathrm { f } ( x )\) which has a maximum point at ( \(- 45,7\) ) and a minimum point at \(( 135 , - 1 )\).
  1. Showing the coordinates of any stationary points, sketch on separate diagrams the graphs of
    1. \(y = \mathrm { f } ( | x | )\),
    2. \(y = 1 + 2 \mathrm { f } ( x )\). Given that $$f ( x ) = A + 2 \sqrt { 2 } \cos x ^ { \circ } - 2 \sqrt { 2 } \sin x ^ { \circ } , \quad x \in \mathbb { R } , \quad - 180 \leq x \leq 180 ,$$ where \(A\) is a constant,
  2. show that \(\mathrm { f } ( x )\) can be expressed in the form $$\mathrm { f } ( x ) = A + R \cos ( x + \alpha ) ^ { \circ } ,$$ where \(R > 0\) and \(0 < \alpha < 90\),
  3. state the value of \(A\),
  4. find, to 1 decimal place, the \(x\)-coordinates of the points where the curve \(y = \mathrm { f } ( x )\) crosses the \(x\)-axis.