Question
If $a$ and $b$ are different positive primes such that $(\text{a}+\text{b})^{-1}(\text{a}^{-1}+\text{b}^{-1})=\text{a}^\text{x}\text{b}^\text{y},$ find $\text{x}+\text{y}+2.$
✓
Answer
$(\text{a}+\text{b})^{-1}(\text{a}^{-1}+\text{b}^{-1})=\text{a}^\text{x}\text{b}^\text{y}$
$\Rightarrow\frac{1}{\text{a}+\text{b}}\times\Big(\frac{1}{\text{a}}+\frac{1}{\text{b}}\Big)=\text{a}^\text{x}\text{b}^\text{y}$
$\Rightarrow\frac{1}{\text{a}+\text{b}}\times\frac{\text{a}+\text{b}}{\text{ab}}=\text{a}^\text{x}\text{b}^\text{x}$
$\Rightarrow\frac{1}{\text{ab}}=\text{a}^\text{x}\text{b}^\text{y}$
$\Rightarrow(\text{ab})^{-1}=\text{a}^\text{x}\text{b}^\text{y}$ $\Rightarrow\text{a}^{-1}\times\text{b}^{-1}=\text{a}^\text{x}\times\text{b}^\text{y}$
$\Rightarrow\text{x}=-1$ and $\text{y}=-1$
$\therefore\text{x}+\text{y}+2=-1-1+2=0$
$\Rightarrow\frac{1}{\text{a}+\text{b}}\times\Big(\frac{1}{\text{a}}+\frac{1}{\text{b}}\Big)=\text{a}^\text{x}\text{b}^\text{y}$
$\Rightarrow\frac{1}{\text{a}+\text{b}}\times\frac{\text{a}+\text{b}}{\text{ab}}=\text{a}^\text{x}\text{b}^\text{x}$
$\Rightarrow\frac{1}{\text{ab}}=\text{a}^\text{x}\text{b}^\text{y}$
$\Rightarrow(\text{ab})^{-1}=\text{a}^\text{x}\text{b}^\text{y}$ $\Rightarrow\text{a}^{-1}\times\text{b}^{-1}=\text{a}^\text{x}\times\text{b}^\text{y}$
$\Rightarrow\text{x}=-1$ and $\text{y}=-1$
$\therefore\text{x}+\text{y}+2=-1-1+2=0$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Simplify the following products:
$(\text{x}^3 -3\text{x}^2-\text{x})(\text{x}^2-3\text{x} + 1)$
→$(\text{x}^3 -3\text{x}^2-\text{x})(\text{x}^2-3\text{x} + 1)$
Assuming that $x, y, z$ are positive real numbers, simplify the following: $\Big(\frac{\sqrt{2}}{\sqrt{3}}\Big)^5\Big(\frac{6}{7}\Big)^2$
→ Evaluate:
$\frac{(25)^{\frac{5}{2}}\times(729)^{\frac{1}{3}}}{(125)^{\frac{2}{3}}\times(27)^{\frac{2}{3}}\times8^{\frac{4}{3}}}$
→$\frac{(25)^{\frac{5}{2}}\times(729)^{\frac{1}{3}}}{(125)^{\frac{2}{3}}\times(27)^{\frac{2}{3}}\times8^{\frac{4}{3}}}$
In fig lines $XY$ and $MN$ intersect at $O$ If $\angle POY = 90$^\circ and $a:b = 2:3$ find $\angle c.$
→Three angles of a quadrilateral are respectively equal to $110^\circ , 50^\circ $ and $40^\circ $. Find its fourth angle.
→Factorize:
$(x+2)\left(x^2+25\right)-10 x^2-20 x$
→$(x+2)\left(x^2+25\right)-10 x^2-20 x$
Prove that: $\Big(\frac{64}{125}\Big)^{-\frac{2}{3}}+\frac{1}{\Big(\frac{256}{625}\Big)^\frac{1}{4}}+\Big(\frac{\sqrt{25}}{\sqrt[3]{64}}\Big)^0=\frac{61}{16}$
→Find the volume and surface area of a sphere whose radius is: $\big(\text{Take}\ \pi=\frac{22}{7}\big). 5m$
→Metal spheres, each of radius $2\ cm$, are packed into a rectangular box of internal dimensions $16cm \times 8cm \times 8cm.$ When $16$ spheres are packed the box is filled with preservative liquid. Find the volume of this liquid. Give your answer to the nearest integer. $[\text{Use }\pi=3.14]$
→What are:
$i.$ Primary data,
$ii. $Secondary data?
Which of the two$-$the primary or the secondary data$-$is more reliable and why?
→$i.$ Primary data,
$ii. $Secondary data?
Which of the two$-$the primary or the secondary data$-$is more reliable and why?