| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Session | Specimen |
| Marks | 4 |
| Topic | Sine and Cosine Rules |
| Type | Algebraic side lengths |
| Difficulty | Standard +0.8 This question requires applying the cosine rule to obtain an algebraic expression, then manipulating an inequality involving that expression to derive a quadratic inequality and solve it. The multi-step algebraic manipulation (substituting the cosine rule expression into an inequality, rearranging to standard form, and solving the quadratic inequality) combined with the need to interpret the solution in context makes this moderately challenging, though the individual techniques are standard A-level content. |
| Spec | 1.05b Sine and cosine rules: including ambiguous case |
**(i)** $1 = 4 + k^2 - 2(2)(k)\cos C$ correct use of cosine rule M1
Obtain $\cos C = \frac{k^2+3}{4k}$ A1 **[2]**
**(ii)** From their $\frac{k^2+3}{4k} < \frac{7}{8}$ M1
Obtain $2k^2 - 7k + 6 < 0$ AG A1
Attempt to find critical values by factorizing *OR* correct use of the quadratic formula M1
$1.5 < k < 2$ A1 **[4]**4
The diagram shows triangle $A B C$, in which $A B = 1$ unit , $A C = k$ units and $B C = 2$ units .\\
(i) Express $\cos C$ in terms of $k$.\\
(ii) Given that $\cos C < \frac { 7 } { 8 }$, show that $2 k ^ { 2 } - 7 k + 6 < 0$ and find the set of possible values of $k$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 Q4 [4]}}