**Answer:** Sketch of $y = |2x + 1|$ | **Marks:** M1 | **Guidance:** Condone no intercept labels, but must be a 'V' shape with vertex on −ve x axis

**Answer:** $y = −x$ and two intersections indicated | **Marks:** A1 | **Guidance:** (none)

**Answer:** $x = -1$ (not from ww, condone $(-1, 1)$) | **Marks:** B1 | **Guidance:** squaring: $(2x+1)^2 = x^2 \Rightarrow 3x^2 + 4x + 1 = 0 \Rightarrow (3x+1)(x+1) = 0$, $x = -1, -1/3$

**Answer:** $x = -1/3$ (not from ww, condone $(-1/3, 1/3)$) | **Marks:** B1 | **Guidance:** (none)

**Total Marks:** [4]

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## Question 5(i)

**Answer:** $dV/dh = 4 \frac{1}{2}(h^3 + 1)^{-1/2} \cdot 3h^2$ | **Marks:** M1 | **Guidance:** their deriv of $4u^{1/2}$ × their deriv of $h^3+1$

**Answer:** (correct) | **Marks:** A1 | **Guidance:** (none)

**Answer:** Substituting $h = 2$ into their derivative | **Marks:** M1 | **Guidance:** (none)

**Answer:** when $h = 2$, $dV/dh = 8$ | **Marks:** A1cao | **Guidance:** (none)

**Total Marks:** [4]

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## Question 5(ii)

**Answer:** $dV/dt = 0.4$ | **Marks:** B1 | **Guidance:** soi

**Answer:** $dV/dt = dV/dh \times dh/dt$ | **Marks:** M1 | **Guidance:** o.e. Any correct chain rule in $V$, $h$, $t$ (or $r$)

**Answer:** $0.4 = 8 \times dh/dt \Rightarrow dh/dt = 0.05$ (m per min) | **Marks:** A1cao | **Guidance:** 0.05 or 1/20

**Total Marks:** [3]

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## Question 6(i)

**Answer:** $2\cos 2y \, dy/dx = 1$ | **Marks:** M1 | **Guidance:** $k\cos 2y \, dy/dx = 1$ or $dy/dy = k\cos 2y$, $k\cos 2y \, dy = dx$

**Answer:** $\Rightarrow dy/dx = 1/(2\cos 2y)$ | **Marks:** A1 | **Guidance:** (none)

**Answer:** when $x = 1\frac{1}{2}$, $y = \pi/12$, $dy/dx = 1/(2\cos(\pi/6)) = 1/\sqrt{3}$ | **Marks:** M1* | **Guidance:** substituting $y = \pi/12$ *dep 1st M1 or $\sqrt{3}/3$

**Total Marks:** [4]

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## Question 6(ii)

**Answer:** $2y = \arcsin(x - 1)$ | **Marks:** M1 | **Guidance:** (none)

**Answer:** $\Rightarrow y = \frac{1}{2}\arcsin(x - 1)$ | **Marks:** A1 | **Guidance:** or $\frac{1}{2}\sin^{-1}(x-1)$

**Answer:** translation of 1 unit in positive $x$-direction | **Marks:** B1 | **Guidance:** allow 'shift', but not 'move', vector only is B0

**Answer:** [one-way] stretch s.f. $\frac{1}{2}$ in $y$-direction | **Marks:** B1 | **Guidance:** not 'shrink', 'squash' etc; transformations can be in either order

**Total Marks:** [4]

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