| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2014 |
| Session | June |
| Marks | 7 |
| Topic | Tangents, normals and gradients |
| Type | Tangent meets curve/axis — further geometry |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question requiring standard techniques: finding where a curve crosses an axis (substitution), differentiation of a polynomial, finding a tangent equation, definite integration, and calculating an area using geometry. All steps are routine A-level procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.07a Derivative as gradient: of tangent to curve1.07m Tangents and normals: gradient and equations1.08d Evaluate definite integrals: between limits1.08e Area between curve and x-axis: using definite integrals |
**(i)** Show $7 \times 2 - 10 - 2^2 = 0$ OR solve $x^2 - 7x + 10 = 0$ to obtain $x = 2$ at least — B1 **[1]**
**(ii)** Obtain $\frac{\mathrm{d}y}{\mathrm{d}x} = 7 - 2x$ — B1
Obtain $y = 2$ and $\frac{\mathrm{d}y}{\mathrm{d}x} = 1$ at $x = 3$ — B1
Attempt equation of straight line — M1
Obtain $y = x - 1$ — A1
Substitute $x = 1$ and obtain $y = 0$ — A1 **[5]**
**(iii)** Obtain area of triangle $= 2$ — B1
Attempt integration — M1
Obtain $\left[\frac{7x^2}{2} - 10x - \frac{1}{3}x^3\right]$ — A1
Attempt to substitute limits of 2 and 3 — M1
Obtain $\frac{7}{6}$ — A1
Attempt subtraction from area of triangle — M1
Obtain $\frac{5}{6}$ with no decimals seen — A1 **[7]**6 The diagram shows the curve with equation $y = 7 x - 10 - x ^ { 2 }$ and the tangent to the curve at the point where $x = 3$.\\
\includegraphics[max width=\textwidth, alt={}, center]{29e924de-bedf-4719-bbfe-f5e0d3191d59-3_648_684_342_731}\\
(i) Show that the curve crosses the $x$-axis at $x = 2$.\\
(ii) Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and hence find the equation of the tangent to the curve at $x = 3$.
Show that the tangent crosses the $x$-axis at $x = 1$.\\
(iii) Evaluate $\int _ { 2 } ^ { 3 } \left( 7 x - 10 - x ^ { 2 } \right) \mathrm { d } x$ and hence find the exact area of the shaded region bounded by the curve, the tangent and the $x$-axis.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2014 Q6 [7]}}