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.

Question of the Week – Week 22

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.