Question of the Week – Week 13

Solve the simultaneous equations:

y=x-4,\hspace{20pt}2x^2-xy=8

giving your answers exactly.

Question of the Week – Week 12

Given that

(5-c)^2\textgreater 12,

find the set of all possible values of c.

Question of the Week – Week 11

Find all the values for x such that

\frac{\log_2(32)+\log_2(16)}{\log_2(x)}=\log_2{(x)}

Question of the Week – Week 10

Solve the following equation:

y-10y^{0.5}+23=0


Question of the Week – Week 9

Simplify

  1. (2\sqrt{5})^2
  2. \frac{\sqrt{2}}{2\sqrt{5}-3\sqrt{2}}


Question of the Week – Week 8

4x-5-x^2=q-(x+p)^2

where p and q are integers. Find the values of p and q.

Question of the Week – Week 7

Solve

\sqrt{2}\left(\cos(x)+\sin(x)\right)=\frac{1}{\cos(x)-\sin(x)}

for 0\leq x\leq 2\pi.

Question of the Week – Week 6

The equation kx^2+4kx+3=0, where k is a constant, has no real roots.

Prove that \hspace{30pt}0\textless k \hspace{3pt} \textless \hspace{3pt}\frac{3}{4}.

Question of the Week – Week 5

Simplify \log_x(81)\times\log_3(x).

Question of the Week – Week 4

The first three terms of a geometric sequence are

7k-5, 5k-7, 2k+10,

where k is a constant. Show that

11k^2-130k+99=0.

Given that k is not an integer, show that \hspace{10pt}k=\frac{9}{11}.

Question of the Week – Week 3

Prove that for all positive x and y:

\sqrt{xy}\leq \frac{x+y}{\sqrt{2}}.

Question of the Week – Week 2

Solve 2^{2x+1}-17\left(2^x\right)=-8.

Question of the Week – Week 1

Do you know the correct way to solve the following inequality?

\frac{2}{x-5}\geq 6,\hspace{15pt}x\ne 5