[4 marks sum]

Question 14 Marks

Using the properties of proportion, solve for $x$ :
$\frac{\sqrt{6 x}+\sqrt{3 x+7}}{\sqrt{6 x}-\sqrt{3 x+7}}=11.$

Answer

Self

Question 24 Marks

If $\frac{\sqrt{x+15}+\sqrt{x-6}}{\sqrt{x+15}-\sqrt{x-6}}=\frac{7}{3}$, find the value of $x$

Answer

Self

Question 34 Marks

If $p=\frac{4 x y}{x+y}$, prove that $\frac{p+2 x}{p-2 x}+\frac{p+2 y}{p-2 y}=2$.

Answer

Self

Question 44 Marks

If $x=\frac{\sqrt{a+3 b}+\sqrt{a-3 b}}{\sqrt{a+3 b}-\sqrt{a-3 b}}$, prove that $3 b x^2-2 a x+3 b=0$

Answer

Self

Question 54 Marks

If $a x=b y=c z$, then prove that : $\frac{x^2}{y z}+\frac{y^2}{z x}+\frac{z^2}{x y}=\frac{b c}{a^2}+\frac{c a}{b^2}+\frac{a b}{c^2}$

Answer

Self

Question 64 Marks

If $a: b=c: d$, then show that $\left(a^2+c^2+a c\right):\left(a^2+c^2-a c\right)=\left(b^2+d^2+b d\right):\left(b^2+d^2-b d\right)$

Answer

Self

Question 74 Marks

If $a, b, c, d$ are in continued proportion, prove that
$\sqrt{a b}+\sqrt{b c}-\sqrt{c d}=\sqrt{(a+b-c)(b+c-d)} .$

Answer

Self

Question 84 Marks

If $a, b, c$ are in continued proportion, prove that $\frac{2 a^2+7 b^2-5 a b}{2 b^2+7 c^2-5 b c}=\frac{a}{c}$

Answer

Self

Question 94 Marks

If $b$ is the mean proportion between $a$ and $c$, prove that $\frac{a^2+b^2+c^2}{a^{-2}+b^{-2}+c^{-2}}=b^4$.

Answer

Self

Mathematics [4 marks sum]