Surds

Q.

Without actually performing the long division state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion.

(x).$\frac{77}{210}$.

Q.

Without actually performing the long division, find if $\frac{987}{10500}$ will have terminating or nonterminating repeating decimal expansion. Give a reason for your answer.

Q.

How do you simplify$i^{(-999)}$?

Q.

If $z_1=1+2i$ and $z_2=3+5i$, then the real part of $\frac{\overline{z_2}{z}_1}{z_2}$ is

1. $-\frac{31}{17}$

2. $\frac{17}{22}$

3. $-\frac{17}{31}$

4. $\frac{22}{17}$

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

1. True
2. False

Q. If $2^a=3^b=6^c$, then $c=$ .

1. $\frac{ab}{a+b}$
2. $\frac{a}{a+b}$
3. $\frac{a}{1+b}$
4. $\frac{abc}{a+b}$

Q.

The value of : $(a^{\frac{1}{8}}+a^{\frac{-1}{8}})(a^{\frac{1}{8}}-a^{\frac{-1}{8}})(a^{\frac{1}{4}}+a^{\frac{-1}{4}})(a^{\frac{1}{2}}+a^{\frac{-1}{2}})$ is -

1. 

2. 

3. 

4. 

Q.

The value of${\int }_0^{\infty }\frac{dx}{\left[a^2+x^2\right]}$ is

1. $\frac{\mathrm{\pi }}{2}$

2. $\frac{\mathrm{\pi }}{2a}$

3. $\frac{\mathrm{\pi }}{a}$

4. $\frac{1}{2a}$

Q. Which one of the following is an irrational number?
(a)√16/25, (b)√5, (c)3/9, √196

Q.

Prove that $\sqrt{3}$ is an irrational number. Hence, show that $7+2\sqrt{3}$ is also an irrational number.

Q. Question 2
Let x be rational and y be irrational. Is xy necessarily irrational? Justify your answer by an example.

Q.

$\sqrt{30}$ is a surd.

1. True

2. False

Q. If $x=9-4\sqrt{5}$ then find the value of $\sqrt{x}+\frac{1}{\sqrt{x}}$.

Q.

The least rationalising factor of 5$\sqrt{x}$ is ______.

1. $x$

2. $5$

3. $5\sqrt{x}$

4. $\sqrt{x}$

Q.

The value of${\int }_0^1\left[\frac{x^3}{1+x^8}\right]dx$ is

1. $\frac{\mathrm{\pi }}{8}$

2. $\frac{\pi }{4}$

3. $\frac{\pi }{16}$

4. $\frac{\pi }{6}$

5. $\frac{\pi }{12}$

Q. State whether the statement is true or false.
All surds are irrational but not all rational numbers are surds.

1. True
2. False

Q. Every real number is either rational or irrational.

1. True
2. False

Q. Prove that 3 is irrational.

Q.

Which of the following is an irrational number?

1. √36

2. √64

3. √81

4. √99

Q. Arrange the following surds in ascending order of their values from top to bottom.

1. $\sqrt[3]{324}$
2. $\sqrt{3}\times \sqrt{2}$
3. $2\sqrt{27}$
4. $\sqrt{72}$

Q.

Question 1 (iii)
Classify $\frac{2\sqrt{7}}{7\sqrt{7}}$ as rational or irrational.

Q.

$\sqrt[3]{1000}$ is equal to

1. $10$

2. $100$

3. $1$

4. none of these

Q. $\frac{7+3\sqrt{5}}{3+\sqrt{5}}-\frac{7-3\sqrt{5}}{3-\sqrt{5}}$ find the value $a+b\sqrt{5}$

Q.

Question 9(a)
Use the sign of >, < or = in the box to make the statements true.
$(-8)+(-4)\square (-8)-(-4)$

Q.

If $\frac{1120}{\sqrt{a}}=80$, then what is the value of $a$?

1. 14

2. 169

3. 196

4. 13

Q. Every real number is an irrational number?

Q. Prove that 2 is irrational?

Q. State whether the given statement is true or false:

1. True
2. False

Q.

Find the cube of $-3$.

Q. $3\sqrt[2]{23}$ is a surd.

1. true
2. false