Learn Extracted exam questions A-Level Mathematics 9709 Mathematics November 2025 Question Paper 11
9709 Mathematics November 2025 Question Paper 11
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Find the set of values of the constant $k$ for which the quadratic equation $3kx^2 + (k+8)x + 3 = 0$ has two distinct real roots.
A geometric progression has first term $a$ and common ratio $\cos\theta$, where $0 < \theta < \tfrac12\pi$. It is given that the second term is 8 and the fifth term is $\tfrac18$.
Find the value of $\theta$. Give your answer correct to 3 significant figures.
Find the exact value of the sum to infinity.
In the expansion of $(px+3)^5 - \left(x^3 + \dfrac{p}{x}\right)^4$, the coefficient of $x^4$ is 216. Find the value of the positive constant $p$.
Express $1 - 6x - x^2$ in the form $a - (x+b)^2$, where $a$ and $b$ are constants.
The graph of $y = x^2$ is transformed to the graph of $y = 1 - 6x - x^2$ by a reflection followed by a translation of $\begin{pmatrix} m \\ n \end{pmatrix}$. Give details of the reflection and determine the values of $m$ and $n$.
Show that $\tan^4\theta - 1 \equiv \dfrac{1 - 2\cos^2\theta}{\cos^4\theta}$.
Hence solve the equation $\cos^2\theta(\tan^4\theta - 1) = 7$ for $0\degree < \theta < 180\degree$.
Functions f and g are defined by $f(x) = (x+3)^2 - 12$ for $x \geqslant 0$, and $g(x) = 2x - 5$ for $x \in \mathbb{R}$.
State the range of f.
Find an expression for $f^{-1}(x)$.
Solve the equation $\mathrm{gf}(x) = 69$.
The diagram shows a sector of a circle with centre $O$ and radius $r$ cm. The shaded region is bounded by the chord $AB$ and the arc $AB$. The size of angle $AOB$ is $\tfrac23\pi$ radians.
Show that the area of the shaded region is approximately $0.614r^2\text{ cm}^2$.
It is given that the radius of the circle is increasing at a rate of $0.4\text{ cm s}^{-1}$.
Find the rate of increase of the area of the shaded region at the instant when $r = 20$. Give your answer correct to 2 significant figures.
Find the rate of increase of the length of the arc $AB$. Give your answer correct to 2 significant figures.
The diagram shows the curve with equation $y = \tfrac12\sqrt{x}$ and the point $P$ with coordinates $\left(9, \tfrac32\right)$. The shaded region is bounded by the curve and the lines $x = 0$ and $y = \tfrac32$.
Find the area of the shaded region.
The shaded region is rotated through $360\degree$ about the $y$-axis. Find the exact volume of the solid produced.
An arithmetic progression has first term 2 and common difference $d$. The sum of the first $n$ terms is denoted by $S_n$.
It is given that $(S_2 - 1)$, $S_4$, $S_9$ are the first three terms of a second arithmetic progression. Find the value of $d$.
Hence find the difference between the values of the 15th terms of the two arithmetic progressions.
A circle has equation $x^2 + y^2 + 4y - 21 = 0$ and a straight line has equation $2x + y - 8 = 0$. The line intersects the circle at two points.
Find the coordinates of these two points of intersection.
The circle has centre $C$ and the two points of intersection are denoted by $A$ and $B$. Find the area of the triangle $ABC$.
A curve passes through the point $P(4, 3)$ and is such that $\dfrac{dy}{dx} = \dfrac{8}{x^2} - \dfrac{10}{(2x-3)^2}$.
Find the equation of the normal to the curve at $P$. Give your answer in the form $y = mx + c$.
Find the rate of change of the gradient of the curve when $x = 4$.
Given that the curve also passes through the point $(-1, q)$, find the value of $q$.