Operations on Rational and Irrational numbers

Q.

If 14.287628762876........ can be represented in $\frac{p}{q}$ form, then 'p - q ' is ________.

1. 132863

2. 142862

3. 142876

4. 142863

Q.

Which of the following is a correct statement?

1. Sum of two irrational numbers is always irrational.

2. Sum of a rational and irrational numbers is always an irrational number.

3. Square of an irrational number is always a rational number.

4. Sum of two rational numbers can never be an integer.

Q.

Factorize the following expression: $121b^2-88bc+16c^2$

Q.

Expand ${\left(\frac{1}{2}a-\frac{1}{4}b+2\right)}^2$.

Q.

$\sqrt[n]{n2}\times \sqrt[n]{n3}=\sqrt[n]{n6}$

1. True

2. False

Q. Identify whether the given number is a rational or irrational number:

Q.

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

1. $\frac{8-2\sqrt{3}}{2\sqrt{3}-1}$

2. $\frac{\sqrt{3}}{2}$

3. $\frac{1}{5}$

4. $\frac{\sqrt{2}}{3}$

Q.

Find the value of $:$ $124.52$ $-$ $113.290$

Q.

Expand$(x-2y-3z{)}^2$.

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

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

Q. $\sqrt{10}\times \sqrt{15}=$ ______

1. $6\sqrt{5}$
2. $5\sqrt{6}$
3. $25$
4. $10\sqrt{5}$

Q. Express $0.8+0.\overline{7}+0.4\overline{3}$ in the form $\frac{p}{q},$ where p and q are integers and $q\ne 0$.

1. $\frac{182}{90}$
2. $\frac{181}{90}$
3. $\frac{90}{182}$
4. $\frac{180}{90}$

Q.

What is the value of (a + b) (a – b)?

1. $a^2+b^2-2ab$

2. $a^2+b^2$

3. $a^2+b^2+2ab$

4. $a^2-b^2$

Q. The value of $\frac{4+\sqrt{5}}{4-\sqrt{5}}+\frac{4-\sqrt{5}}{4+\sqrt{5}}$

1. $\frac{11}{23}$
2. $\frac{42}{11}$
3. $\frac{42}{10}$
4. $0$

Q. If $1^3+2^3+...+9^3=2025$, then $(0.11{)}^3+(0.22{)}^3+...(0.99{)}^3$ will be _________.

1. 0.2695
2. 2.695
3. 3.695
4. 0.3695

Q.

The number ${(1+\sqrt{3})}^2$ is a/an ____________.

1. natural number

2. rational number

3. irrational number

4. integer

Q. If $\sqrt{2}=1.414,\sqrt{3}=1.732,\sqrt{5}=2.236$ and $\sqrt{6}=2.449,$ find the value of $\frac{2+\sqrt{3}}{2-\sqrt{3}}+\frac{2-\sqrt{3}}{2+\sqrt{3}}+\frac{\sqrt{3}-1}{\sqrt{3}+1}$ .

1. 18.428
2. 14.268
3. 16.629
4. 14.662

Q. If $x=2+\sqrt{3},xy=1,$ then find $\frac{x}{2-x}+\frac{y}{2-y}$.

1. $-3$
2. $2\sqrt{3}$
3. $-2$
4. $\frac{4}{\sqrt{3}}$

Q.

If we take the sum of an irrational number and a rational number, which of the following options hold true regarding the resulting sum?

1. It may be irrational.

2. It is always irrational.

3. It is always rational.

4. It may be natural.