Moderate -0.3 This is a standard textbook exercise requiring the identity sin²x = 1 - cos²x to convert to a quadratic, then routine solving. The technique is well-practiced in C2, though it requires multiple steps (substitution, factorization, and finding angles in a given range), making it slightly easier than average overall.
1 Show that the equation \(\sin ^ { 2 } x = 3 \cos x - 2\) can be expressed as a quadratic equation in \(\cos x\) and hence solve the equation for values of \(x\) between 0 and \(2 \pi\).
\(\cos 2x + 3\cos x - 3 = 0\) [or \(-\cos 2x - 3\cos x + 3 = 0\)]
dependent on award of previous method mark, must be correct for their quadratic
condone one sign error or constant term of \(-1\) (in LH version) or \(+1\) (in RH version)
M1
\(\cos x = \frac{-3 + \sqrt{21}}{2}\) or their equivalent
condone recovery from \(x = 0.791287847\ldots\) but M0 if no recovery
A1
\(\cos x =\) their \(0.79\) to \(0.7913\) soi
NB \(x = 0.65788395\ldots\)
A1
\([x =] 0.6578\) to \(0.66\) isw cao
ignore other values (eg \(-3.79\ldots\))
A0 for eg \(0.66\pi\) if \(0.66\) not seen separately
if A1A1 extra values in range incur a penalty of 1; ignore extra values outside range
A1
\([x =] 5.625\) to \(5.63\) isw cao
NB \(x = 5.625301357\ldots\)
no SC mark available if extra values in range
if A0A0 allow SC1 for \(37.69\) to \(37.7°\) and \(322\) to \(322.31°\) or for \((0.209 \text{ to } 0.21)\pi\) and \((1.79 \text{ to } 1.791)\pi\)
# Question 1
M1* | $1 - \cos 2x = 3\cos x - 2$ oe
M1* dep | $\cos 2x + 3\cos x - 3 = 0$ [or $-\cos 2x - 3\cos x + 3 = 0$]
| dependent on award of previous method mark, must be correct for their quadratic
| condone one sign error or constant term of $-1$ (in LH version) or $+1$ (in RH version)
M1 | $\cos x = \frac{-3 + \sqrt{21}}{2}$ or their equivalent
| condone recovery from $x = 0.791287847\ldots$ but M0 if no recovery
A1 | $\cos x =$ their $0.79$ to $0.7913$ soi
| NB $x = 0.65788395\ldots$
A1 | $[x =] 0.6578$ to $0.66$ isw cao
| ignore other values (eg $-3.79\ldots$)
| A0 for eg $0.66\pi$ if $0.66$ not seen separately
| if A1A1 extra values in range incur a penalty of 1; ignore extra values outside range
A1 | $[x =] 5.625$ to $5.63$ isw cao
| NB $x = 5.625301357\ldots$
| no SC mark available if extra values in range
| if A0A0 allow SC1 for $37.69$ to $37.7°$ and $322$ to $322.31°$ or for $(0.209 \text{ to } 0.21)\pi$ and $(1.79 \text{ to } 1.791)\pi$
1 Show that the equation $\sin ^ { 2 } x = 3 \cos x - 2$ can be expressed as a quadratic equation in $\cos x$ and hence solve the equation for values of $x$ between 0 and $2 \pi$.
\hfill \mbox{\textit{OCR MEI C2 Q1 [5]}}