Question 1

By rationalising the denominator, write the expression  $\frac{2+\sqrt3}{5-\sqrt3}$ in the form $a+b\sqrt3$

Accepted Answer

$\frac{13}{22}+\frac{7}{22}\sqrt3$

Answer

$\frac{1+3\sqrt3}{34-10\sqrt3}$

Answer

$\frac{1}{34}+\frac{3}{34}\sqrt3$

Answer

$\frac{5}{2}$

Question 2

Solve the equation $\frac{81^x}{3}=9^{x+1}$

Accepted Answer

$\frac{3}{2}$

Answer

$\frac{5}{2}$

Answer

1

Answer

3

Question 3

Find $x$ if $\frac{8^{2x+1}}{4^x}=32$

Accepted Answer

$\frac{1}{2}$

Answer

$\frac{3}{2}$

Answer

$\frac{7}{4}$

Answer

$\frac{8}{5}$

Question 4

Simplify as far as possible $\frac{6}{3+\sqrt7}-\frac{4}{3-\sqrt7}$

Accepted Answer

$3-5\sqrt7$

Answer

$\frac{3-5\sqrt7}{8}$

Answer

$\frac{6-3\sqrt7}{2}$

Answer

$\frac{6-5\sqrt7}{2}$

Question 5

Solve $\frac{27^x}{3^{x-1}}=3\sqrt3$

Accepted Answer

$\frac{1}{4}$

Answer

$\frac{5}{8}$

Answer

$-\frac{1}{4}$

Answer

$\frac{5}{4}$

Question 6

Solve the simultaneous equations 1: $4x-3y=11$ and 2: $27^x$x$9^{y+3}=3\sqrt3$

Accepted Answer

$x=\frac{1}{2}, y=-3$

Answer

$x=\frac{57}{2}, y=-51$

Answer

$x=\frac{88}{17}, y=\frac{55}{17}$

Answer

$x=\frac{76}{17}, y=\frac{39}{17}$

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