2025 Paper 1

The diagram shows the graph of $\displaystyle y=f(x)$, with stationary points at (0, 3) and (4, 0).

On the diagram in your answer booklet, sketch the graph of $\displaystyle y=f(-x)+3$.

The diagram shows a right-angled triangle with angle q.

(a) Determine the value of:
(i) $\displaystyle \sin 2q$
(ii) $\displaystyle \cos 2q$.
A second right-angled triangle has angle $\displaystyle r$ as shown.

(b) Find the value of $\displaystyle \sin(2q-r)$

(a) Show that $\displaystyle (x+3)$ is a factor of $\displaystyle 5x^{3}+16x^{2}-x-12$
(b) Hence, or otherwise, solve
$\displaystyle 5x^{3}+16x^{2}-x-12=0.$

Find the coordinates of the points of intersection of the line with equation $\displaystyle y=x+1$ and the circle with equation
$\displaystyle x^{2}+y^{2}-2x+6y-15=0$

The vectors u and v are such that:
$\displaystyle u=\begin{pmatrix}1\\ 1\\ 0\end{pmatrix}$
$\displaystyle v=\begin{pmatrix}1\\ 3\\ k\end{pmatrix}$
the angle between u and v is $\displaystyle 45^{\circ}$.
Find the value of k, where $\displaystyle k \gt 0$.

The equation $\displaystyle 9x^{2}+3kx+k=0$ has two real and distinct roots.
Determine the range of values for k.
Justify your answer.

Given that:
$\displaystyle \frac{dy}{dx}=6~\cos~x+8~\sin~2x$
and $\displaystyle y=4$ when $\displaystyle x=\frac{\pi}{6}$
express y in terms of x.

A function, f, is defined on the set of real numbers.
The derivative of f is $\displaystyle f^{\prime}(x)=(x+5)(2-x)$
(a) Find the x-coordinates of the stationary points on the curve with equation $\displaystyle y=f(x)$ and determine their nature.
It is known that:
•  f is a cubic function
•  $\displaystyle f(0) \lt 0$
•  the equation $\displaystyle f(x)=0$ has exactly one solution. The solution lies between -10 and 10.
(b) Draw a sketch of a possible graph of $\displaystyle y=f(x)$ on the diagram in your answer booklet.