# Class 9 Maths Worksheet for Number System

Here we are trying to give the Class 9 Maths Worksheet for Number System. It tests the basic concepts and at the same time makes the student comfortable with the questions

Question 1
Find the value of each of the Following
a. $16^{1/4}$
b. $625^{-3/4}$
c. $2 \sqrt {3} + 8 \sqrt {18}$
d. $15 \sqrt {6} + \sqrt {216}$
e. $\frac {10^{5/2}}{10^{3/2}}$
f. $5^{3/2} \times 7^{3/2}$
g. $\frac {\sqrt {162}}{ \sqrt {2}}$
h. $\frac {(1+ \sqrt {2})}{(1- \sqrt {2})}$

Solutions

a. $16^{1/4} = (2^4)^{1/4} = 2$
b. $625^{-3/4} = (5^4)^{-3/4}= 5^{-3} = \frac {1}{125}$
c. $2 \sqrt {3} + 8 \sqrt {18}= 2 \sqrt {3} + 24 \sqrt {2} = \sqrt {2}( \sqrt {6} + 24)$
d. $15 \sqrt {6} + \sqrt {216} =15 \sqrt {6} + \sqrt {6^2 \times 6}= 15 \sqrt {6} + 6 \sqrt {6}=21 \sqrt {6}$
e. $\frac {10^{5/2}}{10^{3/2}} = 10^{\frac {5}{2} - \frac {3}{2}} = 10^1 = 10$
f. $5^{3/2} \times 7^{3/2} = (125)^{1/2} \times (343)^{1/2} = \sqrt {42875}$
g. $\frac {\sqrt {162}}{ \sqrt {2}} = \frac {\sqrt {81 \times 2}}{ \sqrt {2}} =\frac { 9 \times \sqrt {2}}{\sqrt {2}}= 9$
h. $\frac {(1+ \sqrt {2})}{(1- \sqrt {2})} = \frac {(1+ \sqrt {2})}{(1- \sqrt {2})} \times \frac {(1+ \sqrt {2})}{(1+ \sqrt {2})} = -( 1 + \sqrt {2})^2 = -(3+ 2\sqrt {2})$

Question 2
Simplify the below expression
a. $\frac {1+ \sqrt {2}}{1 - \sqrt {3}} + \frac {1- \sqrt {2}}{1 + \sqrt {3}}$

b.$\frac {5+ \sqrt {2}}{5 - \sqrt {2}} + \frac {5- \sqrt {2}}{5 + \sqrt {2}}$

c. $\frac {1}{1 - \sqrt {2} + \sqrt {3}}$

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

Solutions
a.$\frac {1+ \sqrt {2}}{1 - \sqrt {3}} + \frac {1- \sqrt {2}}{1 + \sqrt {3}}$
=$\frac {(1+ \sqrt {2})(1 + \sqrt {3}) + (1- \sqrt {2})(1 - \sqrt {3})}{ 1 -3}$
$= -\frac {1}{2} \times [1 + \sqrt {2} + \sqrt {3} + \sqrt {6} + 1 -\sqrt {2} + \sqrt {3} - \sqrt {6}]$
$=-\frac {1}{2} \times [2 + 2 \sqrt {3}]= -(1 + \sqrt {3})$

b.$\frac {5+ \sqrt {2}}{5 - \sqrt {2}} + \frac {5- \sqrt {2}}{5 + \sqrt {2}}$
$= \frac {(5+ \sqrt {2})(5 + \sqrt {2}) + (5- \sqrt {2})(5- \sqrt {2})}{5-2}$
$= \frac {1}{3} \times [25 + 2 + 10 \sqrt {2} + 25 + 2 -10 \sqrt {2}]=18$

c. $\frac {1}{1 - \sqrt {2} + \sqrt {3}}$
$= \frac {1}{1 - \sqrt {2} + \sqrt {3}} \times \frac {1 - \sqrt {2} - \sqrt {3}}{1 - \sqrt {2} - \sqrt {3}}$
$= \frac {1 - \sqrt {2} - \sqrt {3}}{ (1-\sqrt {2})^2 - 3}$
$= \frac {1 - \sqrt {2} - \sqrt {3}}{1 + 2 -2 \sqrt {2} -3}$
$=-\frac {1 - \sqrt {2} - \sqrt {3}}{2 \sqrt {2}}$
$=-\frac {1 - \sqrt {2} - \sqrt {3}}{2 \sqrt {2}} \times \frac {\sqrt {2}}{\sqrt {2}}$
$= -\frac {1}{4} \times [\sqrt {2} -2 - \sqrt {6}]$

d. $\frac {1}{\sqrt {3} - \sqrt {2}}$
$= \frac {1}{\sqrt {3} - \sqrt {2}} \times \frac {\sqrt {3} + \sqrt {2}}{\sqrt {3} + \sqrt {2}}$
$= \sqrt {3} + \sqrt {2}$
Question 3
Find the value of a and b such that

$\frac {5 + \sqrt {3}}{7 + 2 \sqrt {3}} = a- b \sqrt {3}$

Solution
$\frac {5 + \sqrt {3}}{7 + 2 \sqrt {3}}$
$= \frac {5 + \sqrt {3}}{7 + 2 \sqrt {3}} \times \frac {7 - 2 \sqrt {3}}{7 - 2 \sqrt {3}}$
$= \frac {1}{37} \times [35 -10 \sqrt {3} + 7 \sqrt {3} -6]$
$= \frac {1}{37} \times [29 -3 \sqrt {3}]$
$= \frac {29}{37} - \frac {3}{37} \sqrt {3}$

Now
$\frac {29}{37} - \frac {3}{37} \sqrt {3}= a- b \sqrt {3}$
so $a=\frac {29}{37}$ and $b=\frac {3}{37}$

Question 4
If $p = 1 + \sqrt {3}$
Find the value of
i. $p^2 + \frac {1}{p^2}$
ii. $p^4 + \frac {1}{p^4}$
Solution
$p = 1 + \sqrt {3}$
$p^2 = (1 + \sqrt {3})^2 = 1 + 3 + 2 \sqrt {3}= 2(2 + \sqrt {3})$
$p^4 = 4(2 + \sqrt {3})^2 = 4(4 + 3 + 4 \sqrt {3})= 4(7 + 4\sqrt {3})$
i. $p^2 + \frac {1}{p^2}$
$=2(2 + \sqrt {3}) + \frac {1}{2(2 + \sqrt {3})}$
$= 2(2 + \sqrt {3}) + \frac {1}{2(2 + \sqrt {3})} \times \frac {2 - \sqrt {3}}{2 - \sqrt {3}}$
$= 2(2 + \sqrt {3}) + \frac {2 - \sqrt {3}}{2}$
$= \frac {10 + 3 \sqrt {3}}{2}$

ii. $p^4 + \frac {1}{p^4}$
$=(p^2 + \frac {1}{p^2})^2 -2$
$=(\frac {10 + 3 \sqrt {3}}{2})^2 -2$
$= \frac {100 + 27 + 60 \sqrt {3} -8}{4}$
$= \frac {119 + 60 \sqrt {3}}{4}$
Question 5
State whether the following statements are true or false
(i) $\frac {2}{\sqrt {5}}$ is a rational number.
(ii) There are infinitely many integers between any two integers.
(iii) Number of rational numbers between 11 and 13 is finite.
(iv) There are numbers which cannot be written in the form  p/q , q ≠ 0 , p, q both   are integers.
(v) The square of an irrational number is always rational.
(vi)$\frac {\sqrt {18}}{\sqrt {2}}$ is not a rational number as $\sqrt {18}$ and $\sqrt {2}$ are not integers.
Solution
5.
i. False
ii. false
iii. false
iv) True
v. False
vi. False

• Notes
• NCERT Solutions
• Assignments
• Revision sheet

Reference Books for class 9 Math

Given below are the links of some of the reference books for class 9 Math.

1. Mathematics for Class 9 by R D Sharma One of the best book for studying class 9 level mathematics. It has lot of problems to be solved.
2. Secondary School Mathematics for Class 9 by R S Aggarwal This is also as good as R.D. Sharma. Either this or the book by R.D. Sharma will do. I find book R.S. Aggarwal little bit more challenging than the one by R.D. Sharma.
3. Pearson IIT Foundation Series - Maths - Class 9 Buy this book if you want to challenge yourself further and want to prepare for JEE foundation.
4. Pearson IIT Foundation Physics, Chemistry & Maths combo for Class 9 Only buy if you are prepared to study extra topics and want to take your studies a step further. You might need help to understand topics in these books.

You can use above books for extra knowledge and practicing different questions.

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