| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2022 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Solving quadratics and applications |
| Type | Substitution to solve disguised quadratic |
| Difficulty | Moderate -0.3 Part (a) is a routine substitution problem (let u = √y) leading to a straightforward quadratic equation. Part (b) applies the same result to a trigonometric context requiring basic inverse tan calculations. While it involves multiple steps and two parts, the techniques are standard for P1 level with no novel insight required, making it slightly easier than average. |
| Spec | 1.02b Surds: manipulation and rationalising denominators1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| \(6y+2-7y^{1/2}\ [=0]\) | \*M1 | OE Rearrange to a 3-term quadratic. |
| \(\left(2y^{\frac{1}{2}}-1\right)\left(3y^{\frac{1}{2}}-2\right)[=0]\) or e.g. \((2u-1)(3u-2)[=0]\) | DM1 | Or use of formula or completing the square. |
| \([y^{1/2}=]\frac{1}{2},\frac{2}{3}\) | A1 | Answers only SC B1 if DM1 not scored. |
| \([y=]\frac{1}{4},\frac{4}{9}\) | A1 | Answers only SC B1 if DM1 not scored. |
| Answer | Marks | Guidance |
|---|---|---|
| Use of \(\tan x =\) *their* \(y\) values | M1 | Must have at least 2 values of \(y\) from part (a). |
| \(x=14[.0], 24[.0]\), \(x=194[.0], 204[.0]\) | A1, A1 FT | FT for \(180+\) angle (twice). AWRT |
## Question 5(a):
$6y+2-7y^{1/2}\ [=0]$ | **\*M1** | OE Rearrange to a 3-term quadratic.
$\left(2y^{\frac{1}{2}}-1\right)\left(3y^{\frac{1}{2}}-2\right)[=0]$ or e.g. $(2u-1)(3u-2)[=0]$ | **DM1** | Or use of formula or completing the square.
$[y^{1/2}=]\frac{1}{2},\frac{2}{3}$ | **A1** | Answers only **SC B1** if DM1 not scored.
$[y=]\frac{1}{4},\frac{4}{9}$ | **A1** | Answers only **SC B1** if DM1 not scored.
**4 total**
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## Question 5(b):
Use of $\tan x =$ *their* $y$ values | **M1** | Must have at least 2 values of $y$ from part **(a)**.
$x=14[.0], 24[.0]$, $x=194[.0], 204[.0]$ | **A1, A1 FT** | FT for $180+$ angle (twice). AWRT
**3 total**
---5
\begin{enumerate}[label=(\alph*)]
\item Solve the equation $6 \sqrt { y } + \frac { 2 } { \sqrt { y } } - 7 = 0$.
\item Hence solve the equation $6 \sqrt { \tan x } + \frac { 2 } { \sqrt { \tan x } } - 7 = 0$ for $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2022 Q5 [7]}}