A Level Mathematics 9709 Paper 33, May/June 2026

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9709/33 · May/June 2026

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Solve the equation $\ln 5 + \ln (3 x - 2) = 2 \ln (x - 4)$ . Give your answer in a simplified exact form. [4]

[4]

Find the set of values of $x$ satisfying the inequality $∣ 3^{2 x - 1} - 4 ∣ > 2$ . Give your answers correct to 3 significant figures. [4]

[4]

On an Argand diagram, shade the region which represents the complex numbers, $z$ , satisfying the inequalities $∣ z - 2 + i ∣ ⩽ 3$ and $∣ z ∣ ⩾ ∣ z - 2 + 2 i ∣$ . [5]

[5]

Solve the equation $\cos (2 θ - 45^{\circ}) + \sin (2 θ - 45^{\circ}) + 1 = 0$ for $- 180^{\circ} ⩽ θ ⩽ 180^{\circ}$ . [6]

[6]

The parametric equations of a curve are $x = \tan 2 t, y = \frac{1}{\sin 2 t} .$

Show that $\frac{d y}{d x}$ can be written as $\frac{A \cos 2 t}{\tan^{2} 2 t}$ , where $A$ is a constant to be found. [5]

[5]

Find an exact equation of the tangent to the curve at the point where $t = \frac{1}{8} π$ . [4]

[4]

Use the substitution $u = \sqrt{x}$ to find the exact value of $\int_{\frac{1}{16} π^{2}}^{\frac{1}{4} π^{2}} \frac{\cos (3 \sqrt{x})}{\sqrt{x}} d x$ . [6]

[6]

The square roots of $3 - (20 \sqrt{7}) i$ can be expressed in the Cartesian form $x + i y$ , where $x$ and $y$ are real and exact. By first forming a quartic equation, find the square roots of $3 - (20 \sqrt{7}) i$ . Give your answer in exact Cartesian form. [5]

[5]

By sketching a suitable pair of graphs, show that the equation $x^{3} = 4 - 2 x$ has exactly one real root. [1]

[1]

Show that, if a sequence of real values given by the iterative formula $x_{n + 1} = \frac{3 x_{n}^{3} + 4}{4 x_{n}^{2} + 2}$ converges, then it converges to the root of the equation in 8(a). [2]

[2]

Show by calculation that this root lies between 1 and 2. [2]

[2]

Use the iterative formula given in 8(b) to calculate this root correct to 2 decimal places. Give the result of each iteration to 5 decimal places. [3]

[3]

Let $f (x) = \frac{36 - 4 x - 3 x^{2}}{(3 x + 1) (x^{2} + 4)}$ .

Express $f (x)$ in partial fractions. [5]

[5]

Hence, obtain the expansion of $f (x)$ in ascending powers of $x$ , up to and including the term in $x^{3}$ . [4]

[4]

State the set of values of $x$ for which the expansion is valid. [1]

[1]

The variables $x$ and $y$ satisfy the differential equation $4 e^{2 x} \frac{d y}{d x} = x (y + 3)$ for $y > - 3$ . It is given that $y = - 2$ when $x = 0$ . Solve the differential equation and obtain an expression for $y$ in terms of $x$ . [8]

[8]

The points $A$ and $B$ have position vectors $(\begin{array}{c} 4 \\ - 1 \\ 2 \end{array})$ and $(\begin{array}{c} - 1 \\ - 3 \\ 5 \end{array})$ respectively. The line $t$ passes through the points $A$ and $B$ .

Find a vector equation for $t$ . [2]

[2]

The line $s$ has equation $r = (\begin{array}{c} 2 \\ 1 \\ 5 \end{array}) + μ (\begin{array}{c} - 3 \\ - 4 \\ 5 \end{array})$ . Find the acute angle between $s$ and $t$ . [3]

[3]

Find the length of the perpendicular from $A$ to $s$ . [5]

[5]

Mark scheme

9709/33 · May/June 2026

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1 · 4 marks

1 mark
Use correct law of logarithm for a power or for a sum
1 mark
Remove logarithms
A1
Obtain a correct quadratic equation, e.g. x^2 - 23x + 26 = 0
A1
Obtain final answer x = (23 + 5√17) / 2 only

2 · 4 marks

B1
State or imply non-modular inequalities 3^(2x - 1) - 4 < -2 and 3^(2x - 1) - 4 > 2, or (3^(2x - 1) - 4)^2 > 2^2 or corresponding pair of linear equations or quadratic equation
M1
Use correct method for solving an equation of the form 3^(2x - 1) = a where a > 0
A1
Obtain critical values x = 0.815 and 1.32 or exact equivalents
A1
State final answer x < 0.815, x > 1.32

3 · 5 marks

B1
Show a circle centre (2, -1)
1 mark
Show a circle with radius 3
B1
Show the point representing (2, -2) or the midpoint (1, -1) or the straight line passing through the origin and (2, -2) or the midpoint (1, -1) Or use y = x - 2 (SOI by diagram)
1 mark
Show the perpendicular bisector of the line joining (2, -2) and the origin, passing through or near (0, -2) and (2, 0) Or show the line y = x - 2 passing through (0, -2) and (2, 0)
B1
Shade the correct region

4 · 6 marks

B1
Use correct trig formulae for cos(2θ - 45°) or for sin(2θ - 45°)
1 mark
Use correct trig formulae and substitute values for cos 45 and sin 45
A1
Obtain correct simplified equation, e.g. √2 sin 2θ = -1
1 mark
Solve their equation in sin 2θ correctly to obtain a value for θ
A1
Obtain answer, e.g. -22.5
A1
Obtain all four answers -67.5°, -22.5°, 112.5° and 157.5° and no others in the interval

5a · 5 marks

B1
Obtain dx/dt = 2 sec^2 2t
M1
Attempt chain rule on (sin 2t)^(-1) for dy/dt, or attempt to use given formula
A1
Obtain dy/dt = -2(sin 2t)^(-2) cos 2t
M1
Use dy/dx = dy/dt × dt/dx
A1
Obtain dy/dx = -cos 2t / tan^2 2t

5b · 4 marks

B1
Obtain x = 1 and y = √2 when t = 1/8 π
1 mark
Substitute t = 1/8 π into their dy/dx of the correct form
1 mark
Form equation of the tangent with their (x, y) and their dy/dx
A1
Obtain equation of tangent √2 y + x = 3 or exact three-term equivalent

6 · 6 marks

B1
du/dx = 1/2 x^(-1/2) or dx/du = 2u
1 mark
Reach an integral of the form ∫ A cos 3u du
A1
Obtain ∫ 2 cos 3u du
1 mark
Obtain 2/3 sin 3u
1 mark
Substitute correct limits correctly in an expression of the form B sin 3u or B sin 3√x
A1
Obtain answer -2/3 - 1/3 √2

7 · 5 marks

1 mark
Square x + iy and equate real and imaginary parts to 3 and -20√7 respectively
A1
Obtain equations x^2 - y^2 = 3 and 2xy = -20√7
1 mark
Eliminate one variable correctly and find an equation in the other
A1
Obtain x^4 - 3x^2 - 700 = 0 or y^4 + 3y^2 - 700 = 0, or three-term equivalents
A1
Obtain answers ±(2√7 - 5i)

8a · 1 mark

B1
Sketch y = x^3 and y = 4 - 2x and justify the given statement

8b · 2 marks

M1
Express x_(n+1) = (3x_n^3 + 4) / (4x_n^2 + 2) as x = (3x^3 + 4) / (4x^2 + 2)
A1
Rearrange x = (3x^3 + 4) / (4x^2 + 2) to x^3 = 4 - 2x with full and correct working

8c · 2 marks

M1
Calculate the values of a relevant expression or pair of expressions at x = 1 and x = 2
A1
Complete the argument correctly with correct calculated values

8d · 3 marks

M1
Use the iterative formula correctly at least twice
A1
Obtain final answer 1.18
A1
Show sufficient iterations to 5 dp to justify 1.18 to 2 dp or show there is a sign change in the interval (1.175, 1.185)

9a · 5 marks

B1
State or imply the form A / (3x + 1) + (Bx + C) / (x^2 + 4)
M1
Use a correct method for finding a constant
A1
Obtain one of A = 9, B = -4 and C = 0
A1
Obtain a second value
A1
Obtain a third value

9b · 4 marks

M1
Use the correct method to find the first two terms of the expansion of (3x + 1)^(-1) or (x^2 + 4)^(-1) or (1 + 1/4 x^2)^(-1)
1 mark
Obtain correct unsimplified expansions up to the term in x^3 of each partial fraction
1 mark
Obtain correct unsimplified expansions up to the term in x^3 of each partial fraction
A1
9 - 28x + 81x^2 - 971/4 x^3

9c · 1 mark

B1
|x| < 1/3 or -1/3 < x < 1/3

10 · 8 marks

B1
Separate variables correctly
B1
Obtain term 4ln(y + 3)
M1
Commence integration by parts and reach axe^(-2x) - ∫ be^(-2x) dx
A1
Obtain -1/2 xe^(-2x) + ∫ 1/2 e^(-2x) dx
A1
Complete integration by parts and obtain -1/2 xe^(-2x) - 1/4 e^(-2x) [+ c]
M1
Use x = 0, y = -2 to evaluate a constant or as limits in a solution containing terms of the form a ln(y + 3), bxe^(-2x) and ce^(-2x)
A1
Obtain correct answer in any form
A1
Obtain y = exp(-1/8 xe^(-2x) - 1/16 e^(-2x) + 1/16) - 3

11a · 2 marks

M1
Use a correct method to form an equation for line t
A1
Obtain r = (4, -1, 2) + λ(-5, -2, 3)

11b · 3 marks

M1
Carry out correct process for evaluating the scalar product of (-3, -4, 5) and their (-5, -2, 3)
M1
Using the correct process for the moduli, divide the scalar product by the product of the moduli and evaluate the inverse cosine of the result
A1
Obtain answer 29.3° or 0.512^c

11c · 5 marks

B1
Find AQ (or QA) for a general point Q on s, e.g. (-2 - 3μ, 2 - 4μ, 3 + 5μ)
1 mark
Calculate the scalar product of AQ and the direction vector for s, and equate to 0
A1
Solve and obtain correct solution, e.g. μ = -13/50
1 mark
Carry out method to calculate AQ
A1
Obtain answer 3.69

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