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RI · 2026 · Prelims · Paper 1

Question 3

Part (a) · Find a related rate from a volume function

Part (b) · Link related rates using implicit differentiation

Original hemisphere diagram for RI 2026 Paper 1 Question 3

Water is poured at a rate of $0.5\ \mathrm{m}^3$ per minute into an empty container in the form of an open hemisphere with inside radius $3\ \mathrm{m}$ (see diagram). At time $t$ minutes after the start, the radius of the water surface is $r\ \mathrm{m}$ and the volume of the water in the hemispherical container is $\dfrac{1}{3}\pi h^2(9-h)$ , where $h\ \mathrm{m}$ is the depth of the water. When the depth of the water is $1\ \mathrm{m}$ ,

(a)

find the rate of change of the depth of water,

[3]

(b)

find the rate of change of the radius of the water surface.

[3]

HCI · 2026 · Prelims · Paper 2

Question 8

Part (a) · Draw a scatter plot and interpret correlation

Part (b) · Justify the direction of a regression line

Part (c) · Identify an outlier in a scatter plot

Part (d) · Fit a regression line and calculate correlation

Part (e) · Extrapolate using regression and assess reliability

A biologist is investigating the relationship between the light intensity level, $x$ (in arbitrary units), and the growth rate, $y$ (in mm/month), of a coral species. Samples of this coral species are placed in a controlled environment and exposed to 10 different light intensity levels for a fixed period. The growth rates of the samples are measured. The results are shown in the table below.

Light intensity level and coral growth rate
$x$	1	2	3	4	5	6	7	8	9	10
$y$	4.0	6.2	9.4	13.2	7.8	15.0	18.0	21.1	23.0	28.0

Light intensity level and coral growth rate, paired values
$x$	$y$
1	4.0
2	6.2
3	9.4
4	13.2
5	7.8
6	15.0
7	18.0
8	21.1
9	23.0
10	28.0

The product moment correlation coefficient between $x$ and $y$ is 0.960, correct to 3 significant figures.

(a)

Draw the scatter diagram for these values, labelling the axes clearly. What do the scatter diagram and product moment correlation coefficient suggest about the relationship between the light intensity level and growth rate? Justify your answer.

[3]

(b)

The biologist wishes to find the light intensity level required to achieve a target coral growth rate of 25 mm/month. Suggest, in the context of the question, why the regression line of $y$ on $x$ should be used for this purpose.

[2]

(c)

The biologist checks the results and one of the values of $y$ appears to be an outlier. Indicate the corresponding point on your diagram in part (a) by labelling it $P$ .

[1]

Omit $P$ for the rest of the question.

(d)

Use your calculator to find the equation of the least squares regression line suggested in part (b). State the value of the product moment correlation coefficient.

[3]

(e)

Use the equation of the least squares regression line to estimate the growth rate of the coral species when the light intensity level is 12. Explain whether you would expect this estimate to be reliable.

[2]

ASRJC · 2026 · Paper 1

Question 1

Question 1 · Recover a function by reversing graph transformations

A graph with equation $y=\mathrm{f}(x)$ undergoes, in succession, the following transformations:

I:

A translation of 3 units in the negative $y$ -direction;

II:

A translation of 1 unit in the negative $x$ -direction;

III:

A scale parallel to the $x$ -axis by a scale factor of $\dfrac{1}{2}$ ;

IV:

A reflection in the $y$ -axis.

The equation of the resulting curve is $y=\ln(2-4x)-2$ .

Determine the equation of function $\mathrm{f}$ , showing your workings clearly.

[4]

RI · 2026 · Prelims · Paper 1

Question 1

Question 1 · Determine parametric coefficients from stationary points

A curve has parametric equations

$x=t^2+1,\qquad y=at^3+bt^2+ct,$

where $t>-2$ and $a$ , $b$ and $c$ are constants.

The curve has stationary points at $t=-1$ and $t=2$ . Given that the curve passes through the point $(5,-2)$ , find the values of $a$ , $b$ and $c$ .

[4]

HCI · 2026 · Prelims · Paper 2

Question 1

Part (a)(i) · Determine a parameter from an oblique asymptote

Part (a)(ii) · Sketch a rational curve and its key features

Part (a)(iii) · Determine when a circle and curve do not intersect

Part (b) · Track coordinates through successive graph transformations

(a)

The curve $C$ has equation $y=\dfrac{kx^2-kx+68}{x-2}$ , where $k$ is a constant.

(i)

If the equation of the oblique asymptote of $C$ is $y=2x+2$ , state the value of $k$ .

[1]

Use the value of $k$ found in part (a)(i) for the rest of the question.

(ii)

Sketch $C$ , showing clearly the coordinates of any axial intercepts and stationary points, and the equations of any asymptotes.

[3]

(iii)

Find the range of values of $r$ , where $r>0$ , such that the equation $(x-8)^2+\left(\dfrac{kx^2-kx+68}{x-2}-10\right)^2=r^2$ has no real solutions for $x$ .

[2]

(b)

The graph of $y=f(x)$ is transformed onto the graph of $y=g(x)$ in a sequence of three transformations. The graph of $y=f(x)$ is first transformed onto the graph of $y=f(x)+1$ , followed by a reflection in the $x$ -axis and is finally transformed onto the graph of $y=g(x)$ by transformation $T$ . A point $P$ with coordinates $(a,b)$ lies on the graph of $y=f(x)$ . $P$ is mapped onto a point $Q$ with coordinates $(a-1,m)$ where $Q$ lies on the graph of $y=g(x)$ . Find $m$ in terms of $b$ and describe $T$ .

[3]

ASRJC · 2026 · Paper 1

Question 5

Part (a) · Evaluate a definite integral with a modulus

Part (b)(i) · Verify a logarithmic–trigonometric derivative

Part (b)(ii) · Integrate by parts and rearrange a recurring integral

(a)

Find the exact value of $\displaystyle\int_{-1}^{2}|\mathrm{e}^{-x}-x-1|\,\mathrm{d}x$ . Show your workings clearly.

[3]

(b)(i)

Show that $\dfrac{\mathrm{d}}{\mathrm{d}x}\left[\ln(\sec x+\tan x)\right]=\sec x$ for $0<x<\dfrac{\pi}{2}$ .

[1]

(ii)

Hence, using integration, show that

$\displaystyle\int\sec^3x\,\mathrm{d}x=\dfrac{1}{2}\bigl(\sec x\tan x+\allowbreak\ln(\sec x+\tan x)\bigr)\allowbreak+C$ for $0<x<\dfrac{\pi}{2}$ .

[3]

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