0606.14.5: Use differentiation to find gradients, tangents and normals: past paper questions

Real questions on 0606.14.5: use differentiation to find gradients, tangents and normals

The newest 20 of 81 written questions. Open one to see it on the real paper.

1. 0606/12 · May/June 2026 · Q5[4]
   Find, in exact form, the equation of the tangent to the curve $y=\ln ((2x-4{)}^5)$ at the point where $x=3$.See it on the paper →
2. 0606/13 · May/June 2026 · Q6b[4]
   In this question time is in seconds and lengths are in centimetres. You are given that $\cos 30^{\circ }=\frac{\sqrt{3}}{2}$, $\sin 30^{\circ }=\frac{1}{2}$ and $\tan 30^{\circ }=\frac{1}{\sqrt{3}}$. A water tank is shaped as a cone with its axis vertical. The angle between the vertical axis and the sloping edge of the co...See it on the paper →
3. 0606/13 · May/June 2026 · Q12a[2]
   A curve has equation $y=e^{\frac{2a}{x}}$ where $a$ is a positive constant. (a) Find $\frac{dy}{dx}$ in terms of $a$.See it on the paper →
4. 0606/13 · May/June 2026 · Q12b[5]
   12 A curve has equation $y={\mathrm{e}}^{\frac{2a}{x}}$ where $a$ is a positive constant. (b) Find the value of $a.$See it on the paper →
5. 0606/21 · May/June 2026 · Q10[10]
   10 A curve has equation $y=x^2\ln (3x^2+5)$. The normal to the curve at the point where $x=-1$ meets the $x$-axis at the point $P$ and the $y$-axis at the point $Q$. Find the area of the triangle $OPQ$, where $O$ is the origin. Give your answer correct to 3 significant figures.See it on the paper →
6. 0606/22 · May/June 2026 · Q12[4]
   12 A sphere has volume $V{\text{ cm}}^3$ and radius $r$ cm. Given that $r$ varies with time $t$, find $V$ at the instant when $\frac{\mathrm{d}V}{\mathrm{d}t}=9\frac{\mathrm{d}r}{\mathrm{d}t}$See it on the paper →
7. 0606/12 · February/March 2026 · Q3b[1]
   In this question, time $t$ is in seconds and displacement $s$ is in metres. A particle starts from rest at the origin $O$ and moves in a straight line with constant positive acceleration. When $t=4$, $s=6$. (b) Hence find the acceleration.See it on the paper →
8. 0606/12 · February/March 2026 · Q8b[5]
   A curve has equation $y=x{\mathrm{e}}^{2x+1}$. (b) The line $L$ is the tangent to the curve at the point where $x=\frac{1}{2}$. Find the exact coordinates of the two points where $L$ meets the axes.See it on the paper →
9. 0606/12 · October/November 2025 · Q4b[2]
   A function f is such that $f(x)=2x^2+5x+3$, for $x⩾p$, where $p$ is a constant. It is given that $f^{-1}$ exists. (b) Hence write down the coordinates of the stationary point on the curve $y=2x^2+5x+3$.See it on the paper →
10. 0606/12 · October/November 2025 · Q4c(i)[1]
    (c) (i) Write down the least possible value of $p.$See it on the paper →
11. 0606/13 · October/November 2025 · Q5[8]
    5 The normal to the curve $y=x^3+\frac{3}{2}{x}^2-2x+1$ at the point where $x=0$ cuts the curve again at two other points. Find the $x$-coordinates of these two points.See it on the paper →
12. 0606/11 · May/June 2025 · Q6b(i)[1]
    6 A curve has equation $y={\left(\frac{x^2-1}{x^2+1}\right)}^4$. (b) (i) Show that the curve has stationary points where $x=-1$, $x=0$ and $x=1.$See it on the paper →
13. 0606/12 · May/June 2025 · Q8a[5]
    It is given that $y=\frac{\ln (3x^2+16)}{x+2}$. (a) Find $\frac{dy}{dx}$ when $x=0$. Give your answer in the form $\ln p$, where $p$ is a constant.See it on the paper →
14. 0606/21 · May/June 2025 · Q10a(i)[3]
    A particle $P$ moves in a straight line. $t$ seconds after passing a fixed point, $O$, the acceleration of $P$, $a{\text{ ms}}^{-2}$, is given by $$a=t^2-2\text{for }0\le t\le 4$$ $$a=19-5{\text{e}}^{8-2t}\text{for }4\le t\le 10.$$ When $t=3$, the velocity of $P$ is $\frac{1}{3} \text{ ms}^{-...See it on the paper →
15. 0606/22 · May/June 2025 · Q4b[5]
    (b) A curve has equation $y=4\sin 2x\cos 2x$. The normal to the curve at the point where $x=\frac{\pi }{6}$ meets the $x$-axis at the point $P$. Find the exact coordinates of $P$.See it on the paper →
16. 0606/12 · February/March 2025 · Q10[9]
    10 The point $P$ lies on the curve $y=(5x+2{)}^{\frac{2}{3}}$. The $x$-coordinate of $P$ is 5. The normal to the curve at $P$ intersects the line $x+y=11$ at the point $Q$. The point $R$ is the reflection of $Q$ in the tangent to the curve at $P$. Find the coordinates of $R$.See it on the paper →
17. 0606/22 · February/March 2025 · Q2a[4]
    (a) Find the $x$-coordinates of the stationary points on the curve $y=\frac{1}{2}(3-2x)(x+2{)}^2$.See it on the paper →
18. 0606/22 · February/March 2025 · Q6b[2]
    It is given that $y=\frac{\ln (2x^2+1)}{x+2}$. (b) Given that $x$ increases from $2$ to $2+h$, where $h$ is small, find the approximate change in $y$.See it on the paper →
19. 0606/22 · February/March 2025 · Q9d[1]
    In this question, all lengths are in metres and time is in seconds. A particle, $P$, moves in a straight line such that $t$ seconds after passing through a fixed point $O$ its displacement, $s$, is given by $s=5\ln (2t+1)-5t$. (d) Find the acceleration when $t=4.5$.See it on the paper →
20. 0606/22 · October/November 2024 · Q2b[3]

    3-mark “Find” question — its maths and diagrams are best seen on the paper.

    See it on the paper →

What examiners said

Comprehensive responses • Many candidates offered a correct solution for their linear velocity-time graph from part (a). Limited responses • A few candidates who had drawn curves in part (a) were unable to offer a valid solution in this part as their graphs did not justify a constant acceleration. • A few candidates attempted to use suvat equations and sometimes this was done incorrectly. Knowled...

— Examiner report, February/March 2026, 0606/12 Q3(b) (Cambridge)

Fully correct responses showed that the denominator needed only to equal zero, then correctly deduced the three values of x by solving the resulting equations. Where verification was attempted, candidates often struggled to communicate clearly what they were attempting to show.

— Examiner report, May/June 2025, 0606/11 Q6(b)(i) (Cambridge)

Most candidates attempted differentiation of a quotient, usually using the correct form. Some chose to use the differentiation of a product which was just as acceptable provided it was a correct ( ) product. Errors usually occurred with the incorrect differentiation of ln 3 x 2 + 16 . The form of the final answer was given, however some candidates did not simplify their correct work sufficiently a...

— Examiner report, May/June 2025, 0606/12 Q8(a) (Cambridge)