Learn Extracted exam questions A-Level Mathematics 9709 Mathematics November 2025 Question Paper 22
9709 Mathematics November 2025 Question Paper 22
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Solve the equation $\ln(3x + 5) - \ln(x - 2) = 4$. Give your answer in an exact form.
Solve the equation $2\tan^2\theta + 3\sec\theta = 18$ for $-180\degree < \theta < 180\degree$.
Solve the inequality $|3x - 4| \leqslant |2x + 5|$.
Hence find the largest integer $N$ satisfying the inequality $|3 \times 7^{0.01N} - 4| \leqslant |2 \times 7^{0.01N} + 5|$.
The polynomial $p(x)$ is defined by $p(x) = x^4 - 10x^3 + 20x^2 - 30x + 40$.
Find the quotient when $p(x)$ is divided by $(x^2 + 3)$ and show that the remainder is $-11$.
Hence find the real roots of the equation $p(x) + 11 = 0$. Give your answers in exact form.
The diagram shows the curve with equation $y = 4\cos 2x + 8\sin x$ for $0 \leqslant x \leqslant \pi$. The maximum points on the curve are denoted by $A$ and $B$, and the shaded region is bounded by the line segment $AB$ and the curve.
Find the coordinates of $A$ and $B$.
Find the exact area of the shaded region.
It is given that $\displaystyle\int_{-2a}^{a}\left(\tfrac{1}{2}e^{2x} + \tfrac{1}{4}e^{-x}\right)dx = 5$, where $a$ is a positive constant.
Show that $a = \tfrac{1}{2}\ln\!\left(10 + \tfrac{1}{2}e^{-a} + \tfrac{1}{2}e^{-4a}\right)$.
Hence show by calculation that the value of $a$ lies between 1.0 and 1.2.
Use the iterative formula $a_{n+1} = \tfrac{1}{2}\ln\!\left(10 + \tfrac{1}{2}e^{-a_n} + \tfrac{1}{2}e^{-4a_n}\right)$ to find the value of $a$ correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
A curve has equation $5x^2 y + 4e^{2y} - 7x + 10 = 0$.
Find an expression in terms of $x$ and $y$ for $\dfrac{dy}{dx}$ and hence find the gradient of the curve at the point for which $y = 0$.
Show that there is no point on the curve at which the tangent is parallel to the $y$-axis.