## Question of the Week – Week 25

Solve $3^xe^{4x-1}=5$, giving your answer in the form $\frac{a+\ln(b)}{c+\ln(d)}$.

## Question of the Week – Week 24

Differentiate $y=2x^3$ from first principles.

## Question of the Week – Week 23

Solve $2\tan(2x)-3\sin(2x)=0$ on the interval $-180^\circ\leq x\leq 180^\circ$.

See Solution.

## Question of the Week – Week 21

The line $l_1$ has equation $4y+3=2x$.  Points $A\left(\frac{19}{2},4\right)$ and B (to the left of A) lie on $l_1$ such that $AB=\sqrt{80}$. The line $l_2$ passes through $C(2,4)$, is perpendicular to $l_1$ and intersects $l_1$ at the point $D$. The point $E$ lies on $l_2$ such that the length of $CDE=3$ times the length of $CD$. Find the area of the quadrilateral $ACBE$.

## Question of the Week – Week 20

Use the first 3 terms of the binomial expansion of $\left(1+\frac{x}{4}\right)^8$ (in ascending powers of x) to find an approximation for $1.025^8$.

## Question of the Week – Week 19

Sketch the graph of $y=(x+1)(x-k)^2$ where $k\textgreater 0$, labelling any points of intersection with the axes.

## Question of the Week – Week 18

Prove by contradiction that $\sqrt{2}$ is irrational.

## Question of the Week – Week 17

Given that $8^x\times5^{2x}=2^{2x+2}\times5^{x-1}$, find the value of $10^x$.

## Question of the Week – Week 16

Given that (x-3) is a factor of $f(x)=2x^3-5x^2-9x+18$, find all the exact solutions of $g(y)=0$ where $g(y)=2\left(3^{3y}\right)-5\left(3^{2y}\right)-9\left(3^y\right)+18$.

## Question of the Week – Week 15

Show that $\int_{k}^{2k}\frac{2}{(2x-k)^2}\,dx$ is inversely proportional to k.