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This is one of our many quizzes on Irrational numbers. Click on this to access more such competitive quizzes on Number System.

Irrational numbers Quiz 2

mathematics

 Write the value of $'t'$from the equation, ${{\left( \sqrt{\dfrac{3}{5}} \right)}^{2t+\frac{9}{2}}}=\dfrac{3}{5}.$

 Solve power of $\dfrac{3}{5}$ in L.H.S. and equate it to power of $\dfrac{3}{5}$ on the R.H.S.

  $\text{Consider, }{{\left( \sqrt{\dfrac{3}{5}} \right)}^{2t+\frac{9}{2}}}=\dfrac{3}{5} \\ \Rightarrow {{\left[ {{\left( \dfrac{3}{5} \right)}^{\frac{1}{2}}} \right]}^{2t+\frac{9}{2}}}=\dfrac{3}{5} \\ \Rightarrow {{\left( \dfrac{3}{5} \right)}^{\dfrac{4t+9}{4}}}={{\left( \dfrac{3}{5} \right)}^{1}}\Rightarrow \dfrac{4t+9}{4}=1 \\ 4t+9=4\Rightarrow 4t=-4\Rightarrow t=\dfrac{-5}{4}$

 Write which of the following numbers is an irrational number? 

 Solve each number by finding square roots to check which number is irrational.

  $\sqrt{\dfrac{9}{4}}+\sqrt{\dfrac{36}{49}}=\dfrac{3}{2}+\dfrac{6}{7}=\dfrac{33}{14} \text{ is not an irrational number.} \\ \sqrt{\dfrac{144}{169}}-\sqrt{\dfrac{289}{49}}=\dfrac{12}{13}-\dfrac{17}{7}=\dfrac{-137}{91}\text{ is not an irrational number.} \\  \sqrt{\dfrac{100}{169}}+\sqrt{\dfrac{361}{225}}=\dfrac{10}{20}+\dfrac{19}{15}=\dfrac{1}{2}+\dfrac{19}{15}=\dfrac{53}{30}\text{ is not an irrational number.} \\ \sqrt{\dfrac{119}{29}}+\sqrt{\dfrac{64}{144}}=\sqrt{\dfrac{119}{29}}+\dfrac{8}{12}=\sqrt{\dfrac{119}{29}}+\dfrac{2}{3}\text{ is an irrational number as square root of }\sqrt{\dfrac{119}{29}}\text{ is not a rational number.} $

 Write the value of ${{\left( \dfrac{7}{9} \right)}^{\dfrac{\sqrt{3}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}}}\div {{\left( \dfrac{7}{9} \right)}^{\sqrt{15}}}.$

 First simplify $\dfrac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}$ by rationalising the denominator and then put its value in the given number.

  $\text{Consider, }{{\left( \dfrac{7}{9} \right)}^{\dfrac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}}}\div {{\left( \dfrac{7}{9} \right)}^{\sqrt{15}}} \\ \dfrac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}=\dfrac{{{\left( \sqrt{5}+\sqrt{3} \right)}^{2}}}{{{\left( \sqrt{5} \right)}^{2}}-{{\left( \sqrt{3} \right)}^{2}}} \\ \Rightarrow \dfrac{8+2\sqrt{15}}{2}=4+\sqrt{15} \\ \text{Substitute the above value in given number.} \\ \Rightarrow {{\left( \dfrac{7}{9} \right)}^{4+\sqrt{15}}}\div {{\left( \dfrac{7}{9} \right)}^{\sqrt{15}}}={{\left( \dfrac{7}{9} \right)}^{4+\sqrt{15}-\sqrt{15}}}={{\left( \dfrac{7}{9} \right)}^{4}}$

 Let $P=\sqrt{123}+\sqrt{119},$ then write the value of $p+\dfrac{1}{p}.$

 Find $\dfrac{1}{P}$ by rationalising the denominator and then add this value to value of $P$.

  $\text{Consider, }P=\sqrt{123}+\sqrt{119} \\ \Rightarrow \dfrac{1}{P}=\dfrac{1}{\sqrt{123}+\sqrt{119}}\times \dfrac{\sqrt{123}-\sqrt{119}}{\sqrt{123}-\sqrt{119}}=\dfrac{\sqrt{123}-\sqrt{119}}{123-119} \\ \Rightarrow \dfrac{\sqrt{123}-\sqrt{119}}{4} \\ \therefore P+\dfrac{1}{P}=\left( \sqrt{123}+\sqrt{119} \right)+\dfrac{\sqrt{123}-\sqrt{119}}{4} \\ \Rightarrow \dfrac{4\sqrt{123}+4\sqrt{119}+\sqrt{123}-\sqrt{119}}{4} \\ \Rightarrow \dfrac{5\sqrt{123}+3\sqrt{119}}{4}$

 If $K={{\left( \sqrt{7}+\sqrt{11} \right)}^{2}}+{{\left( \sqrt{7}-\sqrt{11} \right)}^{2}}$ and $L={{\left( \sqrt{3}+\sqrt{19} \right)}^{2}}+{{\left( \sqrt{3}-\sqrt{19} \right)}^{2}}$, write the value of $\dfrac{K +L}{K-L}$.

 Use the identity ${{\left( a+b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$ and ${{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$ to solve $K$ and $L$and then solve further.

  $\text{Given, }K={{\left( \sqrt{7}+\sqrt{11} \right)}^{2}}+{{\left( \sqrt{7}-\sqrt{11} \right)}^{2}} \\ \Rightarrow K=\left( 18+2\sqrt{77} \right)+\left( 18-2\sqrt{77} \right)=36 \\ \text{And, }L={{\left( \sqrt{3}+\sqrt{19} \right)}^{2}}+{{\left( \sqrt{3}-\sqrt{19} \right)}^{2}} \\ \Rightarrow 22+2\sqrt{57}+22-2\sqrt{57}=44 \\ \text{Now, }\dfrac{K+L}{K-L}=\dfrac{36+44}{36-44}=\dfrac{-80}{8}=-10$  