Learn Extracted exam questions A-Level Mathematics 9709 Mathematics June 2025 Question Paper 11
9709 Mathematics June 2025 Question Paper 11
Source PDF on the left, extracted YAML on the right. Compare numbering, marks, options and text.
Solve the equation $6\sin\theta = 1 + \dfrac{2}{\sin\theta}$ for $-180\degree < \theta < 180\degree$.
The equation of a curve is such that $\dfrac{dy}{dx} = 4(2x-5)^3 - 9x^{\frac12}$. The curve passes through the point $A\left(4, -\tfrac{11}{2}\right)$.
Find the gradient of the normal to the curve at the point $A$.
Find the equation of the curve.
The third term of a geometric progression is 18 and the sum of the first three terms is 26. It is given that the common ratio is negative.
Find the tenth term of the progression. Give your answer correct to 3 significant figures.
Find the exact value of the sum to infinity of the progression.
The diagram shows the curve with equation $y = 5x^{\frac32} - 20x$ and the line with equation $y = x - 16$. The $x$-coordinates of the points of intersection of the curve and line are 1 and 16. Find the area of the shaded region between the curve and the line.
Find the first three terms, in ascending powers of $x$, in the expansion of each of the following expressions.
$(2 - px)^5$
$\left(1 - \tfrac12 x\right)^4$
Given that the coefficient of $x^2$ in the expansion of $(2 - px)^5\left(1 - \tfrac12 x\right)^4$ is 93, find the possible values of the constant $p$.
The equation of a curve is $2x^2 - kxy + 2 = 0$ and the equation of a line is $y = px + 3$, where $k$ and $p$ are constants.
Given that $k = 2$ and $p = 11$, find the coordinates of the points of intersection of the curve and the line.
Given instead that $p = 4$, find the set of values of $k$ for which the curve and the line do not intersect.
The equation of a curve is $y = 4x^2 + \dfrac{9}{x^2} - 8$.
A point $P$ is moving along the curve in such a way that its $y$-coordinate is decreasing at 5 units per second. Find the rate at which the $x$-coordinate of point $P$ is changing when $x = 2$.
Find the coordinates of the stationary points of the curve and determine their nature.
The circle with equation $x^2 + y^2 - 6x + 10y - 27 = 0$ intersects the line $x = -2$ at the points $P$ and $Q$. Find the area of the triangle formed by the tangents to the circle at $P$ and $Q$, and the line $x = -2$.
The diagram shows a sector $ABC$ of a circle with centre $A$ and radius $r$ cm. The angle $BAC$ is $\alpha$ radians, where $0 < \alpha < \tfrac12\pi$.
It is given that the area of the triangle $ABC$ is $4\text{ cm}^2$ and the area of the sector $ABC$ is $8\alpha\text{ cm}^2$. Find the exact area of the shaded segment.
It is given instead that the length of the chord $BC$ is $\tfrac{1}{\sqrt2}r$ cm but the area of the triangle $ABC$ is still $4\text{ cm}^2$. Find the area of the shaded segment. Give your answer correct to 3 significant figures.
The functions f and g are defined by $f(x) = \sqrt{x}$ for $x \geqslant 0$, and $g(x) = 3\sqrt{x+2} - 5$ for $x \geqslant -2$.
Describe fully a sequence of transformations which transforms the graph of $y = f(x)$ to the graph of $y = g(x)$. You should make clear the order in which the transformations are applied.
The diagram shows the graph of $y = g(x)$. On the diagram sketch the graph of $y = g^{-1}(x)$ together with any relevant mirror line.
Find an expression for $g^{-1}(x)$.
State the range of $g^{-1}$.
The function h is defined by $h(x) = x - 2$ for $x \geqslant 0$. Find the value of $g^{-1}h(4)$.
Explain why the composite function $hg^{-1}$ cannot be formed.