A Level Mathematics 9709 Paper 12, Oct/Nov 2025

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9709/12 · Oct/Nov 2025

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Express $9 x^{2} - 36 x + 8$ in the form $p (x + q)^{2} + r$ , where $p$ , $q$ and $r$ are constants.

[2]

Hence find the set of values of the constant $k$ for which the equation $9 x^{2} - 36 x + 8 = k$ has no real roots.

[1]

Find the exact roots of the equation $9 x^{2} - 36 x + 8 = - 15$ .

[2]

The graph of $y = f (x)$ is transformed to the graph of $y = f (3 x) + 2$ . Describe fully the two transformations which have been combined to give the resulting graph.

[3]

A different graph has equation $y = g (x)$ . This graph is stretched by scale factor 3 in the $y$ -direction and then reflected in the $y$ -axis. Write down the equation of the transformed graph in terms of the function $g$ .

[2]

The equation of a curve is such that $\frac{d y}{d x} = k x^{3} + \frac{2}{x^{2}}$ , where $k$ is a constant. The curve passes through the point $S (2, 20)$ and the gradient of the curve at $S$ is $\frac{65}{2}$ .

Find the value of $k$ .

[1]

The coordinates of a point $T$ on the curve are $(1, t)$ . Find the value of $t$ .

[5]

The equation of a curve is $y = 4 x^{\frac{1}{2}} - x$ . The curve has a maximum point when $x = a$ and crosses the $x$ -axis at the point with coordinates $(b, 0)$ , where $b > 0$ . The shaded region is bounded by the curve, the line $x = a$ and the $x$ -axis (see diagram).

Find the value of $a$ .

[3]

Find the exact area of the shaded region.

[5]

Sketch the graph of $y = 3 \sin x + 2$ for $0 ⩽ x ⩽ 2 π$ .

[2]

b(i)

Determine the number of solutions in the interval $0 ⩽ x ⩽ 2 π$ of each of the following equations. $3 \sin x + 2 = x$

[1]

b(ii)

$3 \sin x + 2 = 5 - x$

[1]

Solve the equation $3 \sin x + 2 = 5 \cos^{2} x - 1$ for $0 ⩽ x ⩽ 2 π$ .

[5]

The coordinates of the points $P$ and $Q$ are $(1, 1)$ and $(7, 11)$ respectively. The line segment $P Q$ forms a diameter of a circle.

Find the equation of the circle.

[4]

Find the equation of the tangent to the circle at the point $Q$ .

[3]

The other point on the circle with $x$ -coordinate 7 is $R$ . Find the coordinates of the point of intersection of the tangent at $Q$ with the tangent at $R$ .

[4]

The first three terms of a geometric progression are $a$ , $b$ and $c$ respectively, where $a$ , $b$ and $c$ are positive constants. The first three terms of an arithmetic progression are $a$ , $b$ and $- 3 c$ respectively.

Show that $a^{2} - 10 a c + 9 c^{2} = 0$ .

[3]

b(i)

It is now given that $a = 9$ and $c$ takes the smaller of its two possible values. Find the sum to infinity of the geometric progression.

[5]

b(ii)

Find the sum of the first 20 terms of the arithmetic progression.

[3]

The function $f$ is defined by $f (x) = \frac{4}{(3 x - 6)^{2}} + \frac{1}{(3 x - 6)^{3}}$ for $x > 2$ .

Find an expression for $f^{'} (x)$ and hence determine whether $f$ is an increasing function, a decreasing function or neither.

[4]

State whether $f^{- 1}$ exists. Give a reason for your answer.

[1]

The function $g$ is defined by $g (x) = 4 x - 3$ for $x > a$ . Find the range of $g$ in terms of the constant $a$ .

[1]

Find the set of values of $a$ for which the composite function $f g$ exists.

[2]

The diagram shows a circle with centre $A$ and radius $r$ passing through points $B$ , $C$ and $D$ . A larger circle of radius $s$ has centre $C$ and passes through $B$ and $D$ . The length $B D$ is also $s$ .

Show that $s = \sqrt{3} r$ .

[2]

Find an expression for the area of the shaded region. Give your answer in the form $(a + b π) r^{2}$ , where $a$ and $b$ are constants to be found.

[7]

Mark scheme

9709/12 · Oct/Nov 2025

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1a · 2 marks

B2
{9}{(x - 2)^2}{-28}

1b · 1 mark

1 mark
k < -28

1c · 2 marks

M1
9(x - their 2)^2 - their 28 = -15 => (x - 2)^2 = 13/9 or x = (36 +- sqrt(36^2 - 4 * 9 * 23)) / (2 * 9)
A1
x = 2 +- sqrt(13)/3 or (6 +- sqrt(13))/3 or 2 +- sqrt(13/9) or (36 +- sqrt(468))/18 oe

3a · 3 marks

B1
Translation (0, 2), or Translation 2 'parallel to the y-axis' or 'in the y-direction' or 'vertically'.
B2
{Stretch} {factor 1/3} {parallel to x-axis or in x-direction or horizontally}

3b · 2 marks

B2
[y = ] 3g(-x)

4a · 1 mark

B1
[k = ] 4

4b · 5 marks

2 marks
{k/4 x^4} + {(2x^(-2+1))/(-2+1)} [+ c]
M1
20 = (their k)/4 * 2^4 - 2/2 + c
A1
c = 5
A1
[t = ] 4

5a · 3 marks

B1
[dy/dx = ] 2x^(-1/2) - 1
M1
2x^(-1/2) - 1 = 0 => x^(1/2) = 2
A1
[a = ] 4

5b · 5 marks

B1
[x = or b = ] 16
B2
[integral(4x^(1/2) - x)dx = ] {4/(3/2) x^(3/2)} + {- 1/2 x^2}
M1
[Area = ] (8/3 * 16^(3/2) - 1/2 * 16^2) - (8/3 * 4^(3/2) - 1/2 * 4^2)
1 mark
{128/3 - 40/3} = 88/3 or 29 1/3

6a · 2 marks

B2
A sine wave curve passing through (0, 2), maximum at (1/2 pi, 5), minimum at (3/2 pi, -1), ending at (2 pi, 2).

6b(i) · 1 mark

B1
1

6b(ii) · 1 mark

B1
3

6c · 5 marks

M1
3sin x + 2 = 5(1 - sin^2 x) - 1
1 mark
5sin^2 x + 3sin x - 2 [= 0]
A1
sin x = 0.4 and -1
A2
x = 0.41, 2.73, 3/2 pi AWRT or 0.13pi, 0.87pi, 1.5pi or 59/450 pi, 391/450 pi, 3/2 pi

7a · 4 marks

B1
[Centre is] (4, 6)
B1
sqrt(3^2 + 5^2) or 3^2 + 5^2 or sqrt(6^2 + 10^2) or 6^2 + 10^2 oe
M1
(x - their 4)^2 + (y - their 6)^2 = (their radius)^2
A1
(x - 4)^2 + (y - 6)^2 = 34 or x^2 + y^2 - 8x - 12y + 18 = 0

7b · 3 marks

B1
[Gradient of PQ =] (11 - 1)/(7 - 1) or 5/3 oe
M1
[Gradient of tangent =] -3/5
A1
(y - 11)/(x - 7) = -3/5 oe e.g. y = -3/5 x + 76/5

7c · 4 marks

B1
y-coordinate of R is 1
B1
[Equation of tangent at R is] (y - 1)/(x - 7) = 3/5 oe eg y = 3/5 x - 16/5
M1
3/5(x - 7) + 1 = -3/5(x - 7) + 11 => x = ... or y = ...
A1
x = 46/3 oe and y = 6

8a · 3 marks

B1
-3c - b = b - a or 2b = a - 3c or b/a = c/b or b^2 = ac oe
M1
4ac = (a - 3c)^2 or 2sqrt(ac) = a - 3c oe
A1
=> a^2 - 6ac + 9c^2 = 4ac => a^2 - 10ac + 9c^2 = 0

8b(i) · 5 marks

M1
81 - 90c + 9c^2 = 0 => [9](c - 9)(c - 1) = 0 or [(90 +- sqrt(5184))/18 = (90 +- 72)/18]
B1
c = 1
M1
r^2 = (their 1)/9 => r = [1/3], or b^2 = their 1 * 9 => b = 3 => r = [1/3]
1 mark
S_infinity = 9 / (1 - 1/3)
A1
S_infinity = 27/2 oe

8b(ii) · 3 marks

B1
d = -6
M1
S_20 = 20/2 (2 * 9 + (20 - 1) * (-6))
A1
S_20 = -960

9a · 4 marks

B3
(4 * 3 * (-2))/(3x - 6)^3 + (1 * 3 * (-3))/(3x - 6)^4 or 4 * 3 * (-2)(3x - 6)^(-3) + 1 * 3 * (-3)(3x - 6)^(-4)
B1
Decreasing.

9b · 1 mark

1 mark
f^(-1) exists because f is a decreasing function Or f^(-1) exists because it is one-to-one, or passes the horizontal line test.

9c · 1 mark

B1
g(x) > 4a - 3

9d · 2 marks

M1
Either 4a - 3 > 2 allow with x or a or >= Or 3 * (4a - 3) - 6 > 0 allow with x or a or >=
A1
a > 5/4 or a >= 5/4 oe

10a · 2 marks

M1
s^2 = r^2 + r^2 - 2r^2 cos(2pi/3) or r/sin(pi/6) = s/sin(2pi/3) or s = 2r * cos(pi/6) or (r/2)^2 + (s/2)^2 = r^2 or (3r/2)^2 + (s/2)^2 = s^2 oe
A1
=> E.g. s = 2r * sqrt(3)/2 => s = sqrt(3)r

10b · 7 marks

B1
[Angle BAD, BAC or CAD =] 2pi/3 or 120 deg
B1
[Sector BAD =] 1/2 r^2 * 2pi/3 oe
B1
[Sector BCD =] 1/2 s^2 * pi/3 oe
1 mark
= 3/6 pi r^2 or 1/2 pi r^2
B1
Either [Area triangle ABD, ABC or ACD =] 1/2 r^2 * sin(2pi/3) or r/2 * s/2 oe = sqrt(3)/4 r^2 Or [Area triangle CBD =] 1/2 s^2 * sin(pi/3) or 3r/2 * s/2 oe = (3sqrt(3))/4 r^2
1 mark
= sqrt(3)/4 r^2 or = (3sqrt(3))/4 r^2
B1
(sqrt(3)/2 - pi/6)r^2 or e.g. ((12sqrt(3) - 4pi)/24)r^2 oe in the required exact form

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