Learn Extracted exam questions A-Level Mathematics 9709 Mathematics June 2025 Question Paper 21
9709 Mathematics June 2025 Question Paper 21
Source PDF on the left, extracted YAML on the right. Compare numbering, marks, options and text.
Given that $y = 6x\cos(x^2+1)$, find an expression for $\dfrac{dy}{dx}$.
Use logarithms to solve the inequality $4^x < 0.05$. Give your answer in the form $x < a$, where the value of $a$ is correct to 3 significant figures.
Solve the inequality $|3x+8| < 9$.
Hence state the integers that satisfy both of the inequalities in parts (a) and (b).
Sketch, on a single diagram, the graphs of $y = 3e^{-2x}$ and $y = \sec x$ for values of $x$ such that $0 \leqslant x < \tfrac12\pi$.
Show that the $x$-coordinate of the point of intersection of the two graphs satisfies the equation $x = \tfrac12\ln(3\cos x)$.
Use an iterative formula, based on the equation in part (b), to find the $x$-coordinate of the point of intersection correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
The diagram shows the curve with equation $y = 6e^{2x} - e^{3x}$. The shaded region is bounded by the axes and the curve.
Find the exact $x$-coordinate of the maximum point.
Find the area of the shaded region. Give your answer in the form $\dfrac{p}{q}$, where $p$ and $q$ are integers.
The polynomial $p(x)$ is defined by $p(x) = ax^3 + bx^2 - ax - 24$, where $a$ and $b$ are constants. It is given that $(2x-3)$ is a factor of $p(x)$ and that the remainder is $-15$ when $p(x)$ is divided by $(x+1)$.
Find the values of $a$ and $b$.
Hence factorise $p(x)$ completely.
Hence solve the equation $p(3\operatorname{cosec}\theta) = 0$ for $90\degree < \theta < 270\degree$.
The parametric equations of a curve are $x = \dfrac{2t+1}{3t+4}$, $y = 2\ln(3t+4)$, where $t > -\tfrac43$.
Show that $\dfrac{dy}{dx}$ can be expressed in the form $c(3t+4)$ and state the value of the constant $c$.
It is given that the gradient of the curve at the point $(a, \ln 100)$ is $m$. Find the values of $a$ and $m$.
State whether the curve represents a decreasing function or an increasing function or neither. Give a reason for your answer.
Prove that $\sin^2 2x + 4\cos^2 x\cos 2x \equiv 4\cos^4 x$.
Find the set of possible values of the constant $k$ for which the equation $\sin^2 2x + 4\cos^2 x\cos 2x + 5 = k$ has no real solutions.
Find the exact value of $\displaystyle\int_{-\frac13\pi}^{\frac13\pi} \sqrt{\sin^2 t + 4\cos^2\!\left(\tfrac12 t\right)\cos t}\;dt$.