Range of squared harmonic expression

A question is this type if and only if it asks for the maximum, minimum, or range of values of an expression involving the square of a harmonic form, such as (a·sin + b·cos)² or expressions like c + d·(harmonic)².

5 questions · Standard +0.1

1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc
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CAIE P3 2021 November Q2
5 marks Moderate -0.3
2
  1. Express \(5 \sin x - 3 \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Give the exact value of \(R\) and give \(\alpha\) correct to 2 decimal places.
  2. Hence state the greatest and least possible values of \(( 5 \sin x - 3 \cos x ) ^ { 2 }\).
Edexcel C3 Q16
8 marks Standard +0.3
  1. Express \(1.5 \sin 2x + 2 \cos 2x\) in the form \(R \sin (2x + \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{1}{2}\pi\), giving your values of \(R\) and \(\alpha\) to 3 decimal places where appropriate. [4]
  2. Express \(3 \sin x \cos x + 4 \cos^2 x\) in the form \(a \cos 2x + b \sin 2x + c\), where \(a\), \(b\) and \(c\) are constants to be found. [2]
  3. Hence, using your answer to part (a), deduce the maximum value of \(3 \sin x \cos x + 4 \cos^2 x\). [2]
Edexcel C4 Q1
8 marks Moderate -0.3
  1. Express \(1.5 \sin 2x + 2 \cos 2x\) in the form \(R \sin (2x + \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{1}{2}\pi\), giving your values of \(R\) and \(\alpha\) to 3 decimal places where appropriate. [4]
  2. Express \(3 \sin x \cos x + 4 \cos^2 x\) in the form \(a \cos 2x + b \sin 2x + c\), where \(a\), \(b\) and \(c\) are constants to be found. [2]
  3. Hence, using your answer to part (a), deduce the maximum value of \(3 \sin x \cos x + 4 \cos^2 x\). [2]
AQA Paper 2 Specimen Q5
9 marks Standard +0.3
  1. Determine a sequence of transformations which maps the graph of \(y = \cos \theta\) onto the graph of \(y = 3\cos \theta + 3\sin \theta\) Fully justify your answer. [6 marks]
  2. Hence or otherwise find the least value and greatest value of $$4 + (3\cos \theta + 3\sin \theta)^2$$ Fully justify your answer. [3 marks]
SPS SPS FM Pure 2021 June Q7
9 marks Standard +0.3
  1. Determine a sequence of transformations which maps the graph of \(y = \cos \theta\) onto the graph of \(y = 3\cos \theta + 3\sin \theta\) Fully justify your answer. [6 marks]
  2. Hence or otherwise find the least value and greatest value of $$4 + (3\cos \theta + 3\sin \theta)^2$$ Fully justify your answer. [3 marks]