| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2016 |
| Session | June |
| Marks | 7 |
| Topic | Chain Rule |
| Type | Determine if function is increasing/decreasing |
| Difficulty | Moderate -0.3 Part (i) is straightforward algebraic verification requiring common denominators. Part (ii) is a standard quotient rule differentiation. Part (iii) requires showing the derivative is negative, which follows directly from part (ii) with minimal analysis. This is a routine multi-part calculus question with no novel insights required, making it slightly easier than average. |
| Spec | 1.02y Partial fractions: decompose rational functions1.07o Increasing/decreasing: functions using sign of dy/dx1.07q Product and quotient rules: differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\frac{3(x+1)+(x+2)}{(x+2)(x+1)} = \frac{4x+5}{x^2+3x+2}\) | M1, A1 [2] | Attempt to add fractions using common denominator. Simplify to obtain given answer. |
| Answer | Marks | Guidance |
|---|---|---|
| \(A(x+1) + B(x+2) = 4x + 5\) so \(A = 3\) and \(B = 1\). | M1, A1 | Use partial fractions on RHS. Obtain given answer. |
| (ii) \(-\frac{3}{(x+2)^2} - \frac{1}{(x+1)^2}\) | M1, A1, A1 [3] | Differentiate both terms on the LHS, or any other valid method. Obtain one correct term. Obtain fully correct \(f'(x)\). |
| Answer | Marks | Guidance |
|---|---|---|
| (iii) Denominators always \(+\)ve as \((x+k)^2 > 0\). Numerators always \(-\)ve, and \(\frac{\text{−ve}}{\text{+ve}}\) is \(-\)ve | M1, A1 [2] | State, or imply, that "decreasing" implies \(f'(x) < 0\), and make some attempt to use this. Conclude convincingly that \(f'(x) < 0\) for all \(x\) (CWO, A0 if incorrect \(f'(x)\)). |
**(i)** $\frac{3(x+1)+(x+2)}{(x+2)(x+1)} = \frac{4x+5}{x^2+3x+2}$ | M1, A1 [2] | Attempt to add fractions using common denominator. Simplify to obtain given answer.
**OR**
$A(x+1) + B(x+2) = 4x + 5$ so $A = 3$ and $B = 1$. | M1, A1 | Use partial fractions on RHS. Obtain given answer.
**(ii)** $-\frac{3}{(x+2)^2} - \frac{1}{(x+1)^2}$ | M1, A1, A1 [3] | Differentiate both terms on the LHS, or any other valid method. Obtain one correct term. Obtain fully correct $f'(x)$.
Quotient rule: M1 – attempt quotient rule; A1 – correct unsimplified expression; A1 – correct simplified expression.
**(iii)** Denominators always $+$ve as $(x+k)^2 > 0$. Numerators always $-$ve, and $\frac{\text{−ve}}{\text{+ve}}$ is $-$ve | M1, A1 [2] | State, or imply, that "decreasing" implies $f'(x) < 0$, and make some attempt to use this. Conclude convincingly that $f'(x) < 0$ for all $x$ (CWO, A0 if incorrect $f'(x)$).\begin{enumerate}[label=(\roman*)]
\item Show that $\frac{3}{x+2} + \frac{1}{x+1} \equiv \frac{4x+5}{x^2+3x+2}$. [2]
\item Differentiate $\frac{4x+5}{x^2+3x+2}$ with respect to $x$. [3]
\item Hence show that the function given by
$$f(x) = \frac{4x+5}{x^2+3x+2}, \quad x \neq -1, x \neq -2,$$
is a decreasing function. [2]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2016 Q5 [7]}}