Learn Extracted exam questions A-Level Mathematics 9709 Mathematics November 2025 Question Paper 12
9709 Mathematics November 2025 Question Paper 12
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Express $9x^2 - 36x + 8$ in the form $p(x+q)^2 + r$, where $p$, $q$ and $r$ are constants.
Hence find the set of values of the constant $k$ for which the equation $9x^2 - 36x + 8 = k$ has no real roots.
Find the exact roots of the equation $9x^2 - 36x + 8 = -15$.
Find the term independent of $x$ in the expansion of $\left(2x^2 - \dfrac{3}{x}\right)^6$.
The graph of $y = f(x)$ is transformed to the graph of $y = f(3x) + 2$. Describe fully the two transformations which have been combined to give the resulting graph.
A different graph has equation $y = g(x)$. This graph is stretched by scale factor 3 in the $y$-direction and then reflected in the $y$-axis. Write down the equation of the transformed graph in terms of the function g.
The equation of a curve is such that $\dfrac{dy}{dx} = kx^3 + \dfrac{2}{x^2}$, where $k$ is a constant. The curve passes through the point $S(2, 20)$ and the gradient of the curve at $S$ is $\dfrac{65}{2}$.
Find the value of $k$.
The coordinates of a point $T$ on the curve are $(1, t)$. Find the value of $t$.
The equation of a curve is $y = 4x^{\frac12} - x$. The curve has a maximum point when $x = a$ and crosses the $x$-axis at the point with coordinates $(b, 0)$, where $b > 0$. The shaded region is bounded by the curve, the line $x = a$ and the $x$-axis (see diagram).
Find the value of $a$.
Find the exact area of the shaded region.
Sketch the graph of $y = 3\sin x + 2$ for $0 \leqslant x \leqslant 2\pi$.
Determine the number of solutions in the interval $0 \leqslant x \leqslant 2\pi$ of each of the following equations.
$3\sin x + 2 = x$
$3\sin x + 2 = 5 - x$
Solve the equation $3\sin x + 2 = 5\cos^2 x - 1$ for $0 \leqslant x \leqslant 2\pi$.
The coordinates of the points $P$ and $Q$ are $(1, 1)$ and $(7, 11)$ respectively. The line segment $PQ$ forms a diameter of a circle.
Find the equation of the circle.
Find the equation of the tangent to the circle at the point $Q$.
The other point on the circle with $x$-coordinate 7 is $R$. Find the coordinates of the point of intersection of the tangent at $Q$ with the tangent at $R$.
The first three terms of a geometric progression are $a$, $b$ and $c$ respectively, where $a$, $b$ and $c$ are positive constants. The first three terms of an arithmetic progression are $a$, $b$ and $-3c$ respectively.
Show that $a^2 - 10ac + 9c^2 = 0$.
It is now given that $a = 9$ and $c$ takes the smaller of its two possible values.
Find the sum to infinity of the geometric progression.
Find the sum of the first 20 terms of the arithmetic progression.
The function f is defined by $f(x) = \dfrac{4}{(3x-6)^2} + \dfrac{1}{(3x-6)^3}$ for $x > 2$.
Find an expression for $f'(x)$ and hence determine whether f is an increasing function, a decreasing function or neither.
State whether $f^{-1}$ exists. Give a reason for your answer.
The function g is defined by $g(x) = 4x - 3$ for $x > a$. Find the range of g in terms of the constant $a$.
Find the set of values of $a$ for which the composite function $\mathrm{fg}$ exists.
The diagram shows a circle with centre $A$ and radius $r$ passing through points $B$, $C$ and $D$. A larger circle of radius $s$ has centre $C$ and passes through $B$ and $D$. The length $BD$ is also $s$.
Show that $s = \sqrt3\, r$.
Find an expression for the area of the shaded region. Give your answer in the form $(a + b\pi)r^2$, where $a$ and $b$ are constants to be found.