Standard +0.8 This question requires knowledge that the maximum of a sin x + b cos x is √(a² + b²), then solving simultaneous equations involving trigonometric substitution. It combines multiple A-level techniques (trigonometric identities, simultaneous equations with surds) and requires insight beyond routine application, making it moderately challenging but accessible to strong students.
A curve has equation
$$y = a \sin x + b \cos x$$
where \(a\) and \(b\) are constants.
The maximum value of \(y\) is 4 and the curve passes through the point \(\left(\frac{\pi}{3}, 2\sqrt{3}\right)\) as shown in the diagram.
\includegraphics{figure_6}
Find the exact values of \(a\) and \(b\).
[6 marks]
Question 6:
6 | ( x±a)
Compares with Rcos or
( x±a)
Rsin
by forming an identity e.g.
( x+a)≡asinx+bcosx
Rsin
OE
or
Differentiates correctly and
equates to zero CAO PI by
acosx=bsinx
PI by
R = 4 or a2 +b2 =16 | 3.1a | M1 | ( x+a)=asinx+bcosx
Rsin
R =4
π
4sin +a =2 3
3
π
a=
3
π
a =4cos =2
3
π
b=4sin =2 3
3
Deduces R = 4
or
a2 +b2 =16 | 2.2a | A1
Forms a correct equation for a
PI by
correct a
or
Forms the equation shown
below
a 3 b
2 3 = + OE
2 2
Must substitute correct exact
values for the trig functions | 1.1b | B1
Solves their equation to obtain
any correct value of a
Correct values are shown below
π
( x±a)
a= or 0 for Rsin
3
π
( x±a)
a= ± for Rcos
6
or
Eliminates a variable correctly
from their two equations – must
obtain a correct simplified
equation | 1.1a | M1
Deduces a =2 | 2.2a | R1
Deduces b=2 3 | 2.2a | R1
Total | 6
Q | Marking instructions | AO | Mark | Typical solution
A curve has equation
$$y = a \sin x + b \cos x$$
where $a$ and $b$ are constants.
The maximum value of $y$ is 4 and the curve passes through the point $\left(\frac{\pi}{3}, 2\sqrt{3}\right)$ as shown in the diagram.
\includegraphics{figure_6}
Find the exact values of $a$ and $b$.
[6 marks]
\hfill \mbox{\textit{AQA Paper 2 2019 Q6 [6]}}