Properties and Operations on Real Numbers

Q. The sum of two irrational numbers is always an irrational number.

1. True
2. False

Q.

The fifth root of $243$ is given by

Q. Find the value of ${125}^{-\frac{2}{3}}({125}^{\frac{4}{3}}-{125}^{\frac{2}{3}})$.

1. $25$
2. $625$
3. $24$
4. $600$

Q. Question 37
Without actually calculating the cubes, find the value of
(i) ${\left(\frac{1}{2}\right)}^3+{\left(\frac{1}{3}\right)}^3-{\left(\frac{5}{6}\right)}^3$

(ii) $(0.2{)}^3-(0.3{)}^3+(0.1{)}^3$

Q.

Which of the following is equivalent to $\frac{1}{\sqrt{6}}+\frac{1}{\sqrt{5}}$ ?

1. $\frac{5\sqrt{6}-6\sqrt{5}}{30}$

2. $\frac{\sqrt{6}\sqrt{5}}{30}$

3. $\frac{5\sqrt{6}+6\sqrt{5}}{30}$

4. $\frac{11}{30}$

Q. Question 1
Let x and y be rational and irrational numbers respectively. Is x + y necessarily an irrational number? Give an example in support of your answer.

Q. If $\frac{4+3\sqrt{5}}{4-3\sqrt{5}}=a+b\sqrt{5}$, then $(a,b)=$.........

1. $(\frac{-61}{29},-\frac{24}{29})$
2. $(\frac{61}{29},-\frac{-24}{29})$
3. $(\frac{61}{29},-\frac{24}{29})$
4. $(\frac{-61}{29},-\frac{-24}{29})$

Q. If $x=\frac{1}{\sqrt{3}+1}$ then the value of $x+\frac{1}{x}$ is

Q. If ${\{{\left({\left(x\right)}^5\right)}^2\times {\left(2\right)}^{-1}\}}^{\frac{1}{3}}=x^3,$ then find the value of $x$.

1. $\frac{2}{8}$
2. $\frac{1}{2}$
3. 2
4. 4

Q. If ${\{{\left({\left(x\right)}^5\right)}^2\times {\left(2\right)}^{-1}\}}^{\frac{1}{3}}=x^3,$ then find the value of $x$.

Q. Simplify-$\left(\right(6+\sqrt{6}\left)\right(6-\sqrt{6})$

1. 30
2. 40
3. 36
4. 63

Q. State whether True or False:$\sqrt[3]{3-125}$ is a surd

1. True
2. False

Q. If $x=\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$ and $y=\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}$, then find the value of $(x-y{)}^2$.

1. 60

Q. If a is an irrational number then $\frac{a}{\sqrt{7}}$_____.

1. is a rational number
2. is an irrational number
   
   
3. can be either rational or irrational
   
   

Q. If a is an irrational number then $\frac{a}{\sqrt{7}}$_____.

1. is a rational number
2. is an irrational number
   
   
3. can be either rational or irrational
   
   

Q. Which of the following is a true statement?

1. The product of two irrational numbers is an irrational number
2. The sum of two irrational numbers is an irrational number
3. Every real number is always rational
4. Every real number is either rational or irrational

Q. The rationalising factor of $\sqrt[3]{349}$ is

1. $\sqrt{49}$
2. $\sqrt[3]{37}$
3. none
4. 0

Q. Find the value of $a$ and $b$ in $\frac{2+\sqrt{3}}{2-\sqrt{3}}=a+b\sqrt{3}$

Q. Choose the correct answer from the alternatives given :
Express $\frac{(4+\sqrt{15}{)}^{3/2}+(4-\sqrt{15}{)}^{3/2}}{(6+\sqrt{35}{)}^{3/2}+(6-\sqrt{35}{)}^{3/2}}$ in rational form.

1. $\frac{7}{12}$
2. $\frac{7}{26}$
3. $\frac{7}{13}$
4. None of these

Q. Find the sum of the squares of the following: $\frac{\sqrt{3}}{\sqrt{2}+1},\frac{\sqrt{3}}{\sqrt{2}-1},\frac{\sqrt{2}}{\sqrt{3}}$

1. $\frac{56}{4}$
2. $18\frac{2}{3}$
3. $2\frac{18}{3}$
4. $3\frac{18}{2}$

Q. If ${\{{\left({\left(x\right)}^5\right)}^2\times {\left(2\right)}^{-1}\}}^{\frac{1}{3}}=x^3,$ then find the value of $x$.

1. $\frac{2}{8}$
2. $\frac{1}{2}$
3. 2
4. 4

Q. Express as a pure surd: $3\sqrt[4]{45}$

1. $\sqrt[2]{2405}$
2. $\sqrt[4]{4405}$
3. $\sqrt[4]{445}$
4. $\sqrt[4]{475}$

Q. Write the simplest form of rationalising factor for the given surd.
$4\sqrt{11}$

Q.

Are rational and irrational numbers real numbers?

Q.

Rational and irrational numbers taken together are known as __________.

Q. If $x=\frac{1}{\sqrt{3}+1}$ then the value of $x+\frac{1}{x}$ is

Q. If $x=3+2\sqrt{2}$, then the value of $(\sqrt{x}-\frac{1}{\sqrt{x}})$ = _____.

1. 2

Q. What is the value obtained when we expand $({1+\sqrt{3})}^2$?

1. 4
2. $4+2\sqrt{3}$
3. $2\sqrt{3}$
4. $3+2\sqrt{3}$

Q. Express as a pure surd: $5\sqrt{6}$

1. $\sqrt{150}$
2. $\sqrt{30}$
3. $\sqrt{180}$
4. $10$

Q.

The expression $\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}$ when simplified reduces to $a-b\sqrt{15}$.

Then a + b is .

1. 10
2. 5
3. 20
4. 15