2024 Paper 2

Triangle ABC has vertices $\displaystyle A(-3,8)$, $\displaystyle B(-1,-6)$ and $\displaystyle C(11,0)$.

(a)  Find the equation of the median through B.
(b)  Find the equation of L, the line perpendicular to BC passing through C.
(c)  Determine the coordinates of the point of intersection of the median through B and the line L.

A curve has equation
$\displaystyle y=\frac{8}{x^{3}}$
where $\displaystyle x \gt 0.$
Find the equation of the tangent to this curve at the point where $\displaystyle x=2$.

The coordinates of points D, E and F are given by $\displaystyle D(2,-3,4)$, $\displaystyle E(1,1,-2)$ and $\displaystyle F(3,2,1).$
(a)  Express $\displaystyle \overrightarrow{ED}$ and $\displaystyle \overrightarrow{EF}$ in component form.
(b)  (i) Calculate $\displaystyle \overrightarrow{ED}.\overrightarrow{EF}.$
      (ii) Hence, or otherwise, calculate the size of angle DEF.

The diagram shows the graph of a quartic function $\displaystyle y=f(x)$.
A maximum turning point occurs at (-1, 3).
The graph of $\displaystyle y=f(x)$ also has a point of inflection at $\displaystyle x=2$.

(a)  Determine the coordinates of the maximum turning point on the graph of $\displaystyle y=f(x-4)+2$.
(b)  On the diagram in your answer booklet, sketch the graph of $\displaystyle y=f^{\prime}(x)$.

Evaluate
$\displaystyle \int_{0}^{\frac{\pi}{7}}\sin 5x dx$

Two variables, x and y, are connected by the equation $\displaystyle y=ax^{b}$.
The graph of $\displaystyle \log_{5}y$ against $\displaystyle \log_{5}x$ is a straight line as shown.

Find the values of a and b.

The diagram shows the curve with equation $\displaystyle y=x^{3}-6x^{2}+11x$ intersecting the curve with equation $\displaystyle y=6+4x-2x^{2}$ at $\displaystyle x=2$

Calculate the shaded area.

Functions f and g are defined on $\displaystyle \mathbb{R}$, the set of real numbers, by:
$\displaystyle f(x)=2x^{2}-18$
$\displaystyle g(x)=x+1$
(a)  Find an expression for $\displaystyle f(g(x)).$
(b)  Find the values of x for which
$\displaystyle \frac{1}{f(g(x))}$
is undefined.

(a)  Determine the coordinates of the stationary points on the curve with equation
$\displaystyle y=\frac{1}{3}x^{3}-x^{2}-3x+1$
(b)  Hence, determine the greatest and least values of y in the interval $\displaystyle -1\le x\le6$.

The circle $\displaystyle C_{1}$ has equation $\displaystyle x^{2}+y^{2}+18x-2y-8=0$.
(a)  Find the centre and radius of $\displaystyle C_{1}.$
A second circle, $\displaystyle C_{2},$ touches $\displaystyle C_{1}$ internally.
The centre of $\displaystyle C_{2}$ is (-6, 0).

(b)  Determine the equation of $\displaystyle C_{2}$.

The number of electric vehicles worldwide can be modelled by
$\displaystyle N=6.8e^{kt}$
where:
•  N is the estimated number of vehicles in millions
•  t is the number of years since the end of 2020
•  k is a constant.
(a)  Use the model to estimate the number of electric vehicles worldwide at the end of 2020.
At the end of 2030, it is estimated there will be 125 million electric vehicles worldwide.
(b)  Determine the value of k.

The diagram shows the graph of $\displaystyle y=f(x)$, where $\displaystyle f(x)$ is a quartic function.

Express $\displaystyle f(x)$ in the form
$\displaystyle f(x)=k(x+a)^{2}(x+b)(x+c)$