Moderate -0.8 This is a straightforward application of differentiation requiring students to find dy/dx, set it equal to 2, and solve a quadratic equation. The polynomial differentiates easily, and the resulting equation 15x^4 + 12x^2 - 1 = 2 simplifies to a quadratic in x^2. While it requires multiple steps, each is routine and the question follows a standard textbook pattern with no problem-solving insight needed.
2. In this question you must show all stages of your working.
\section*{Solutions relying on calculator technology are not acceptable.}
A curve has equation
$$y = 3 x ^ { 5 } + 4 x ^ { 3 } - x + 5$$
The points \(P\) and \(Q\) lie on the curve.
The gradient of the curve at both point \(P\) and point \(Q\) is 2
Find the \(x\) coordinates of \(P\) and \(Q\).
M1: attempts differentiation with \(x^n \to x^{n-1}\) for one correct power. Allow \(x^5 \to x^4\) or \(x^3 \to x^2\) or \(x \to 1\). A1: correct expression, may be unsimplified
Sets \(\frac{dy}{dx} = 2\) and collects terms to one side to obtain a 3TQ in \(x^2\). Depends on first M1
\(\Rightarrow 3(5x^2 - 1)(x^2 + 1) = 0\)
ddM1
Correct attempt to solve 3TQ in \(x^2\) by factorising, quadratic formula, or completing the square. Depends on both previous M marks
\(\Rightarrow x = \pm\frac{1}{\sqrt{5}}\)
A1
Both values required. Exact equivalents such as \(\pm\frac{\sqrt{5}}{5}\), \(\pm\sqrt{\frac{3}{15}}\) acceptable. \(x = \frac{1}{\sqrt{5}}\) alone is A0. Ignore attempts to find \(y\) coordinates
## Question 2:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $y = 3x^5 + 4x^3 - x + 5 \Rightarrow \frac{dy}{dx} = 15x^4 + 12x^2 - 1$ | M1 A1 | M1: attempts differentiation with $x^n \to x^{n-1}$ for one correct power. Allow $x^5 \to x^4$ or $x^3 \to x^2$ or $x \to 1$. A1: correct expression, may be unsimplified |
| $15x^4 + 12x^2 - 1 = 2 \Rightarrow 15x^4 + 12x^2 - 3 = 0$ | dM1 | Sets $\frac{dy}{dx} = 2$ and collects terms to one side to obtain a 3TQ in $x^2$. Depends on first M1 |
| $\Rightarrow 3(5x^2 - 1)(x^2 + 1) = 0$ | ddM1 | Correct attempt to solve 3TQ in $x^2$ by factorising, quadratic formula, or completing the square. Depends on both previous M marks |
| $\Rightarrow x = \pm\frac{1}{\sqrt{5}}$ | A1 | Both values required. Exact equivalents such as $\pm\frac{\sqrt{5}}{5}$, $\pm\sqrt{\frac{3}{15}}$ acceptable. $x = \frac{1}{\sqrt{5}}$ alone is A0. Ignore attempts to find $y$ coordinates |
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2. In this question you must show all stages of your working.
\section*{Solutions relying on calculator technology are not acceptable.}
A curve has equation
$$y = 3 x ^ { 5 } + 4 x ^ { 3 } - x + 5$$
The points $P$ and $Q$ lie on the curve.\\
The gradient of the curve at both point $P$ and point $Q$ is 2\\
Find the $x$ coordinates of $P$ and $Q$.\\
\hfill \mbox{\textit{Edexcel P1 2021 Q2 [5]}}