Learn Extracted exam questions A-Level Mathematics 9709 Mathematics November 2025 Question Paper 21
9709 Mathematics November 2025 Question Paper 21
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Find $\displaystyle\int 6\sin^2 x\,dx$.
Solve the equation $e^{2x}(e^{2x} - 8) = 48$.
Solve the equation $|2x - 3| = |5x + 2|$.
Hence solve the equation $|2\sec\theta - 3| = |5\sec\theta + 2|$ for $\pi < \theta < 2\pi$. Give your answer correct to 3 significant figures.
Solve the equation $\cot\theta\tan(\theta + 45\degree) = 7$ for $0\degree < \theta < 90\degree$.
The diagram shows the curve with equation $y = 8e^{-\frac{1}{2}x} - 1$. The curve meets the axes at the points $A$ and $B$. The shaded region is bounded by the curve and the line segment $AB$.
Show that the $x$-coordinate of $B$ is $6\ln 2$.
Find the area of the shaded region. Give your answer in the form $p\ln 2 - q$, where $p$ and $q$ are positive integers.
A curve has parametric equations $x = \tan\theta$, $y = \sin\theta - 2\sin^3\theta$, for $0 < \theta < \tfrac{1}{2}\pi$.
Show that $\dfrac{dy}{dx} = 6\cos^5\theta - 5\cos^3\theta$.
Find the equation of the normal to the curve at the point where it crosses the $x$-axis. Give your answer in the form $y = mx + c$, where $m$ and $c$ are exact constants.
The polynomial $p(x)$ is defined by $p(x) = 2x^4 + kx^3 + kx^2 + 17x + 18$, where $k$ is a constant. It is given that $(x + 2)$ is a factor of $p(x)$.
Find the value of $k$.
It is given that the equation $p(x) = 0$ has exactly two real roots, denoted by $\alpha$ and $\beta$, where $\alpha$ is an integer and $\beta$ is not an integer. State the value of $\alpha$ and show that $\beta$ satisfies the equation $x = \sqrt[3]{-2x - 4.5}$.
Show by calculation that $-1.4 < \beta < -1.0$.
Use an iterative formula, based on the equation in part (b), to find the value of $\beta$ correct to 3 significant figures. Give the result of each iteration to 5 significant figures.
The equation of a curve is $y = 4e^{1-2x}\sqrt{3x - 1}$. Find the exact coordinates of the stationary point of the curve.