Mathematics Class IX

Question 1 of 20

1. Question
1 point(s)
Which of the following is not a rational number?
- $ \sqrt{3} $
- $ \sqrt{25} $
- $ \sqrt{4} $
- $ \sqrt{36} $
Correct

Incorrect

Hint

$\sqrt{3}$ is not a rational number because 3 is not a perfect square, whereas 4, 25 and 36 are perfect square numbers and their square root is a rational number.
Question 2 of 20

2. Question
1 point(s)
What type of number is $ (\sqrt{3}+\sqrt{5})^{2} $?
- Rational number
- Irrational number
- Decimal number
- Fraction
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} (\sqrt{3}+\sqrt{5})^{2} &= 3+2\sqrt{15}+5 \\ &= 8+2\sqrt{15} \end{split}\end{equation*}$

The result obtained after solving the given expression is $8+\sqrt{15}$ in which 8 is a rational number and $8\sqrt{15}$ is an irrational number. So the given number is an irrational number.
Question 3 of 20

3. Question
1 point(s)
What is the rationalizing factor of $ \sqrt[4]{pq^{2}r^{3}} $?
- $ \sqrt[4]{p^{2}q^{3}r} $
- $ \sqrt[4]{p^{2}qr^{3}} $
- $ \sqrt[4]{p^{3}q^{2}r} $
- $ \sqrt[4]{pq^{2}r^{3}} $
Correct

Incorrect

Hint

Rationalizing factor of $\sqrt[n]{x^{y}} = \sqrt[n]{x^{n-y}}$ . Therefore, the rationalizing factor of $\sqrt[4]{pq^{2}r^{3}} = \sqrt[4]{p^{3}q^{2}r}$ .
Question 4 of 20

4. Question
1 point(s)
What is the product of $ \sqrt[3]{25} $ and $ \sqrt[3]{35} $?
- $ 7\sqrt[3]{5} $
- $ 14\sqrt[3]{5} $
- $ 15\sqrt[3]{7} $
- $ 5\sqrt[3]{7} $
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} \sqrt[3]{25} \times \sqrt[3]{35} &= \sqrt[3]{25 \times 35} \\ &= \sqrt[3]{125\times7} \\ &= 5\sqrt[3]{7} \end{split}\end{equation*}$
Question 5 of 20

5. Question
1 point(s)
What is the value of $ p-q^{p-q} $ if $ p=3 $ and $ q=-3 $?
- $ 726 $
- $ 729 $
- $ -729 $
- $ -726 $
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} p-q^{p-q} &= 3-(-3)^{3-(-3)} \\ &= 3-(-3)^{6} \\ &= 3-729 \\ &= -726 \end{split}\end{equation*}$
Question 6 of 20

6. Question
1 point(s)
What is the value of $ x $ if $ 8^{x+2} = 4160-8^{x} $?
- $ -1 $
- 2
- 1
- $ -2 $
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} 8^{x+2} &= 4160-8^{x} \\ 8^{x} \times 64&= 4160-8^{x} \\ 8^{x} (64+1)&= 4160 \\ 8^{x} &= 64 \\ 8^{x} &= 8^{2} \\ x &= 2 \end{split}\end{equation*}$
Question 7 of 20

7. Question
1 point(s)
What is the value of $ x+\frac{1}{x} $ if $x=2-\sqrt{3}$?
- $ 2\sqrt{3} $
- $ -2\sqrt{3} $
- 4
- $ -4 $
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} \frac{1}{x} &= \frac{1}{2-\sqrt{3}} \times \frac{2+\sqrt{3}}{2+\sqrt{3}} \\ \frac{1}{x} &= 2+\sqrt{3} \\ x+\frac{1}{x} &= 2-\sqrt{3}+2+\sqrt{3} \\ &= 4 \end{split}\end{equation*}$
Question 8 of 20

8. Question
1 point(s)
What is the value of $ \frac{1}{x^{2}}+\frac{1}{y^{2}}$ if $x=8+3\sqrt{7}$ and $xy=1 $?
- 254
- 48
- $ -48\sqrt{7} $
- $ 48\sqrt{7} $
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} xy &= 1 \\ y &= \frac{1}{x} \\ y &= \frac{1}{8+3\sqrt{7}} \times \frac{8-3\sqrt{7}}{8-3\sqrt{7}} \\ y &= 8-3\sqrt{7} \\ \frac{1}{x^{2}}+\frac{1}{y^{2}} &= y^{2}+x^{2} \\ &= (8-3\sqrt{7})^2+(8+3\sqrt{7})^2 \\ &= 254 \end{split}\end{equation*}$
Question 9 of 20

9. Question
1 point(s)
What is the value of $ x^{3}-4x^{2}-x+10 $ if $x=\frac{1}{\sqrt{5}-2}$?
- $ -10 $
- $ -12 $
- 12
- 10
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} x &= \frac{1}{\sqrt{5}-2} \times \frac{\sqrt{5}+2}{\sqrt{5}+2} \\ x &= \sqrt{5}+2 \\ x-2 &= \sqrt{5} \\ (x-2)^{2} &= (\sqrt{5})^2 \\ x^{2}-4x+4 &= 5 \\ x^{2}-4x-1 &= 0 \\ x(x^{2}-4x-1) + 10 &= x(0)+10 \\ x^{3}-4x^{2}-x+10 &= 10 \end{split}\end{equation*}$
Question 10 of 20

10. Question
1 point(s)
What is the value of $ p^{p}+q^{q} $ if $p=2$ and $q=5$?
- 29
- 27
- $ -27 $
- $ -29 $
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} p^{p}+q^{q} &= 2^{2}+5^{5} \\ &= 4+25 \\ &= 29\end{split}\end{equation*}$
Question 11 of 20

11. Question
1 point(s)
Which of the following is the simplest form of $ \frac{5\sqrt{2}+2\sqrt{5}}{5\sqrt{2}-2\sqrt{5}}$?
- $ \frac{7+2\sqrt{10}}{3} $
- $ \frac{2+7\sqrt{10}}{3} $
- $ \frac{10+2\sqrt{5}}{3} $
- $ \frac{5+2\sqrt{10}}{3} $
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} \frac{5\sqrt{2}+2\sqrt{5}}{5\sqrt{2}-2\sqrt{5}} &= \frac{5\sqrt{2}+2\sqrt{5}}{5\sqrt{2}-2\sqrt{5}} \times \frac{5\sqrt{2}+2\sqrt{5}}{5\sqrt{2}+2\sqrt{5}}\\ &= \frac{50+20\sqrt{10}+20}{50-20} \\ &= \frac{7+2\sqrt{10}}{3} \end{split}\end{equation*}$
Question 12 of 20

12. Question
1 point(s)
If $ p$ and $ q $ are integers and it is given that $ \sqrt{pq} = 6$, then which of the following can’t be the value of $ p+q $?
- 13
- 11
- 15
- 20
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} \sqrt{pq} &= 6 \\ pq &= 36 \end{split}\end{equation*}$

Possible values of $(p,q) = (1,36), (2,18), (3,12), (4,9), (6,6), (9,4), (12,3), (36,1)$ Therefore, out of the given options 11 cannot be equal to $p+q$ .
Question 13 of 20

13. Question
1 point(s)
If $ x= \sqrt[3]{\sqrt{6}-\sqrt{5}}$ then what is the value of $ x^{3}+\frac{1}{x^{3}} $?
- $ \sqrt{5} $
- $ \sqrt{6} $
- $ 2\sqrt{6} $
- $ 2\sqrt{5} $
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} x &= \sqrt[3]{\sqrt{6}-\sqrt{5}} \\ x^{3} &= \sqrt{6}-\sqrt{5} \\ \frac{1}{x^{3}} &= \frac{1}{\sqrt{6}-\sqrt{5}} \times \frac{\sqrt{6}+\sqrt{5}}{\sqrt{6}+\sqrt{5}} \\ \frac{1}{x^{3}} &= \sqrt{6}+\sqrt{5} \\ x^{3}+\frac{1}{x^{3}} &= \sqrt{6}-\sqrt{5} + \sqrt{6}+\sqrt{5} \\ &= 2\sqrt{6} \end{split}\end{equation*}$
Question 14 of 20

14. Question
1 point(s)
Which of the following symbol is used to represent the set of all positive integers?
- R
- Z
- Q
- N
Correct

Incorrect

Hint

All the positive integers are basically Natural numbers. Therefore, the symbol N is used to represent them.
Question 15 of 20

15. Question
1 point(s)
If $ a+b\sqrt{5} = \frac{\sqrt{5}+1}{\sqrt{5}-2} $ then what is the value of $ ab $?
- 21
- 15
- 35
- 105
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} a+b\sqrt{5} &= \frac{\sqrt{5}+1}{\sqrt{5}-2} \times \frac{\sqrt{5}+2}{\sqrt{5}+2} \\ a+b\sqrt{5} &= 5+2\sqrt{5}+\sqrt{5}+2 \\ a+b\sqrt{5} &= 7+3\sqrt5} \\ ab &= 7\times 3 \\ ab &= 21 \\ \end{split}\end{equation*}$
Question 16 of 20

16. Question
1 point(s)
If $ \frac{a}{b}+\frac{b}{a} = 1 $ then what is the value of $ a^{3}+b^{3} $?
- 0
- 1
- $ -1 $
- $ -2 $
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} \frac{a}{b}+\frac{b}{a} &= 1 \\ a^{2}+b^{2}-ab &= 0 \\ (a+b)(a^{2}-ab+b^{2}) &= (a+b)(0) \\ a^{3}+b^{3} &= 0 \end{split}\end{equation*}$
Question 17 of 20

17. Question
1 point(s)
What is the positive square root of $ 8+2\sqrt{7} $?
- $ 1-\sqrt{7} $
- $ 1+\sqrt{7} $
- $ \sqrt{7} $
- $ \sqrt{7}-1 $
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} \sqrt{8+2\sqrt{7}} &= a+b\sqrt{7} \\ 8+2\sqrt{7} &= a^{2}+7b^{2}+2ab\sqrt{7} \end{split}\end{equation*}$

Comparing the equation, we get $a^{2}+7b^{2}=8$ and $2ab\sqrt{7}=2\sqrt{7}$ . Therefore, $a=b=1$ . The square root of $8+2\sqrt{7} = 1+\sqrt{7}$ .
Question 18 of 20

18. Question
1 point(s)
What is the value of $ x $ in the equation $ \frac{5^{5x} \times 25^{2} \times 15625}{5^{3x}} = 5^{8} $?
- 2
- 3
- 0
- 1
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} \frac{5^{5x} \times 25^{2} \times 15625}{5^{3x}} &= 5^{8} \\ \frac{5^{5x} \times 5^{4} \times 5^{6}}{5^{3x}} &= 5^{8} \\ 5^{2x} &= 1 \\ 5^{2x} &= 5^{0} \\ x &= 0 \end{split}\end{equation*}$
Question 19 of 20

19. Question
1 point(s)
What is the value of $ x $ if $ 5^{3x-4} \cdot 7^{x+1} = 8575 $?
- $ -1 $
- $ -2 $
- 1
- 2
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} 5^{3x-4} \cdot 7^{x+1} &= 8575 \\ 5^{3x-4} \cdot 7^{x+1} &= 5^{2} \cdot 7^{3} \end{split}\end{equation*}$

Comparing both the sides, we get

(2) $\begin{equation*}\begin{split} 3x-4 &= 2 \\ x &= 2 \end{split}\end{equation*}$

and

(3) $\begin{equation*}\begin{split} x+2 &= 3 \\ x &= 2 \end{split}\end{equation*}$

Therefore, $x=2$
Question 20 of 20

20. Question
1 point(s)
What is the value of $ \frac{1}{\sqrt{10}-3} – \frac{1}{3-\sqrt{8}} + \frac{1}{\sqrt{8}-\sqrt{7}} $?
- $ \sqrt{10}+6+2\sqrt{7} $
- $ \sqrt{10}-\sqrt{7} $
- $ \sqrt{10}-6-2\sqrt{7} $
- $ \sqrt{10}+\sqrt{7} $
Correct

Incorrect

Hint

(1) $\begin{equation*}\begin{split} \frac{1}{\sqrt{10}-3} - \frac{1}{3-\sqrt{8}} + \frac{1}{\sqrt{8}-\sqrt{7}} &= \frac{1}{\sqrt{10}-3}\times\frac{\sqrt{10}+3}{\sqrt{10}+3} - \frac{1}{3-\sqrt{8}}\times\frac{3+\sqrt{8}}{3+\sqrt{8}} + \frac{1}{\sqrt{8}-\sqrt{7}}\times\frac{\sqrt{8}+\sqrt{7}}{\sqrt{8}+\sqrt{7}} \\ &= \sqrt{10}+3-3-\sqrt{8}+\sqrt{8}+\sqrt{7} \\ &= \sqrt{10}+\sqrt{7} \end{split}\end{equation*}$