| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Session | Specimen |
| Marks | 4 |
| Topic | Tangents, normals and gradients |
| Type | Normal meets curve/axis — further geometry |
| Difficulty | Standard +0.8 This question requires finding tangent/normal equations and their intersection point. While the calculus is straightforward (differentiating sin x), part (iii) demands solving simultaneous equations with exact trigonometric values, requiring careful algebraic manipulation. The multi-step nature and need for exact answers elevate this above routine differentiation exercises, but it remains a standard A-level technique without requiring novel insight. |
| Spec | 1.05f Trigonometric function graphs: symmetries and periodicities1.05o Trigonometric equations: solve in given intervals1.07m Tangents and normals: gradient and equations |
(i) Graph of correct sinusoidal shape [B1]
Correct $x$-intercepts and range indicated [B1] **2 marks**
(ii) Gradient of tangent at $P\left(\frac{1}{4}\pi, 1\right)$ is $\sqrt{2}\cos(\frac{1}{4}\pi) = 1$ [M1]
Equation of tangent at $P$ is $y - 1 = x - \frac{1}{4}\pi$ [A1] **2 marks**
(iii) Equation of normal at $P$ is $y - 1 = -(x - \frac{1}{4}\pi)$ [M1A1]
$R$ has coordinates $\left(\frac{1}{2}\pi, 1 - \frac{1}{4}\pi\right)$ [A1A1] **4 marks**4 (i) Sketch the graph of $y = \sqrt { 2 } \sin x$ for $0 \leqslant x \leqslant 2 \pi$.
The points $P$ and $Q$ on the graph have $x$-coordinates $\frac { 1 } { 4 } \pi$ and $\frac { 3 } { 4 } \pi$, respectively.\\
(ii) Determine the equation of the tangent to the curve at $P$.
The normals to the curve at $P$ and $Q$ intersect at the point $R$.\\
(iii) Determine the exact coordinates of $R$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 Q4 [4]}}