Learn Extracted exam questions A-Level Mathematics 9709 Mathematics June 2025 Question Paper 22
9709 Mathematics June 2025 Question Paper 22
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Show that $\displaystyle\int_2^{11} \dfrac{8}{4x+1}\,dx = \ln a$, where $a$ is an integer to be found.
Sketch on the same diagram the graphs of $y = |2x-9|$ and $y = 4x-5$.
Solve the inequality $|2x-9| < 4x-5$.
Find the coordinates of the stationary points of the curve with equation $y = \dfrac{8x}{2x+3} - 6x + 5$.
The diagram shows parts of the curves with equations $y = 4e^{-2x}$ and $y = 1 + 0.5\sin 3x$. Point $P$ is a point of intersection of the curves, and the shaded region is bounded by the two curves and the $y$-axis.
Show that the $x$-coordinate of $P$ satisfies the equation $x = -0.5\ln(0.25 + 0.125\sin 3x)$.
Use an iterative formula, based on the equation in part (a), to find the $x$-coordinate of $P$ correct to 4 significant figures. Use an initial value of 0.5 and give the result of each iteration to 6 significant figures.
Hence find the area of the shaded region. Give your answer correct to 2 significant figures.
The polynomial $p(x)$ is defined by $p(x) = ax^4 + bx^3 + 13x^2 - 35x + 15$, where $a$ and $b$ are constants. It is given that $(2x-1)$ and $(x-3)$ are factors of $p(x)$.
Find the values of $a$ and $b$.
Hence factorise $p(x)$.
Find the least positive value of $\theta$ in radians such that $p(\cot 2\theta) = 0$.
A curve has equation $(x^2-3)\ln y + 6x = 14$.
Show that there is no point on the curve at which the $y$-coordinate is $e^{-1}$.
Find the equation of the tangent to the curve at the point $(2, e^2)$. Give your answer in the form $y = mx + c$, where $m$ and $c$ are exact constants.
Express $4\cos\theta\sin(\theta+30\degree)$ in the form $R\cos(2\theta-\alpha) + k$, where $R > 0$, $0\degree < \alpha < 90\degree$ and $k$ is a constant.
Hence solve the equation $12\cos 2\phi\sin(2\phi+30\degree) = 5$ for $0\degree < \phi < 90\degree$.