Find an expression for $V_T$ in the form $A \cos (Bt+C)$, where $A, B$ and $C$ are real constants.

Hence write down the maximum voltage in the circuit.

METHOD 1

recognizing that the real part is distributive           (M1)

$V_T=\text{Re}\left(2{\text{e}}^{3t\text{i}}+5{\text{e}}^{3t\text{i}+4\text{i}}\right)$

$=\text{Re}\left({\text{e}}^{3t\text{i}}\left(2+5{\text{e}}^{4\text{i}}\right)\right)$           (A1)

(from the GDC)  $2+5{\text{e}}^{4\text{i}}=3.99088\dots {\text{e}}^{-1.89418\dots \text{i}}$           (A1)

Note: Accept arguments differing by $2\mathrm{\pi }$ e.g. $4.38900\dots )$.

therefore $V_T=3.99 \cos \left(3t-1.89\right) \left(3.99088\dots \cos \left(3t-1.89418\dots \right)\right)$          A1

Note: Award the last A1 for the correct values of $A, B$ and $C$ seen either in the required form or not. If method used is unclear and answer is partially incorrect, assume Method 2 and award appropriate marks eg. (M1)A1A0A0 if only $A$ value is correct.

 

METHOD 2

converting given expressions to cos form          (M1)

$V_T=2 \cos 3t+5 \cos \left(3t+4\right)$

(from graph) $A=3.99 \left(3.99088\dots \right)$         A1

$V_T=3.99 \cos \left(Bt+C\right)$

either by considering transformations or inserting points

$B=3$         A1

$C=-1.89 \left(-1.89418\dots \right)$         A1

Note: Accept arguments differing by $2\mathrm{\pi }$ e.g. $4.38900\dots$.

(so, $V_T=3.99 \cos \left(3t-1.89\right) \left(3.99088\dots \cos \left(3t-1.89418\dots \right)\right)$ )

Note: It is possible to have $A=3.99$, $B=-3$ with $C=1.89$  OR  $A=-3.99$, $B=3$ with $C=1.25$  OR  $A=-3.99$, $B=-3$ with $C=-1.25$ due to properties of the cosine curve.

 

[4 marks]

maximum voltage is $3.99 \left(3.99088\dots \right)$ (units)         A1

 

[1 mark]

The crucial step in this question was to realize that $\text{Re}\left(2{\text{e}}^{\text{3}t\text{i}}\text{)+Re}\right(5{\text{e}}^{\text{(3}t+4)\text{i}}\text{)=Re}\left(2{\text{e}}^{\text{3}t\text{i}}\text{+}5{\text{e}}^{\text{(3}t+4)\text{i}}\right)$. Candidates who failed to do this step were usually unable to obtain the required result.

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