Q1.
The decimal expansion of \( \frac{33}{8} \) is
Q2.
Express 0.\(\overline6 \) in the form of \( \frac{p}{q} \), where p and q are integers and q \(\not=0\).
\( \frac{2}{3} \)
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Q3.
Who were the mathematicians that established the correspondence between real numbers and points on the number line?
Q4.
The value of 1.999…. in the form \( \frac{p}{q} \), where p and q are integers and q \(\not=0\). is:
Q5.
Simplify the expression: (3 - \( \sqrt{3} \))(3 + \( \sqrt{3} \))
6
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Q6.
Find the value of x, if \( 3^x \) = 81.
4
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Q7.
What is the value of : 15\( \sqrt{15} \) ÷ 3\( \sqrt{3} \) ?
5 \( \sqrt{5} \)
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Q8.
If x = \( \sqrt{7} \) - \( \sqrt{6} \), then find the value of \( \frac{1}{x} \).
\( \sqrt{7} \) + \( \sqrt{6} \)
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Q9.
Find the least rationalizing factor of \( \sqrt{63} \) .
\( \sqrt{7} \)
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Q10.
The number pi (π) has a never-ending decimal expansion. what is the value of π to 10 decimal places?
3.1415926535
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Q11.
The product of (8 + 3\(\sqrt{2} \) ) and (8 ⎼ 3\(\sqrt{2} \) ) is
Q12.
An irrational number between 0ꞏ3101 and 0ꞏ333… is
Q13.
The irrational number between 2 and 2.5 is
Q14.
Archimedes showed that 3.140845 < π < 3.142857. Find the difference between the upper and lower bounds of his approximation.
0.002012
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Q15.
Express 3\(\frac{1}{8} \) in decimal form and state what kind of decimal expansion it has.
3.125 Terminating
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Q16.
Given that \( \frac{1}{7} \)= \( 0.\overline{142857} \). Find the decimal expansion of \( \frac{2}{7} \) without actually performing the long division.
\( 0.\overline{285714} \)
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Q17.
Find the value of (\(1^3 + 2^3 + 3^3)^{3/2} \)
216
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Q18.
The irrational number \(\sqrt{2} \) , discovered by Hippasus, has a never-ending decimal expansion. What is its value up to 10 decimal places?
1.4142135623
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Q19.
If x = 2 + \(\sqrt{3} \) , then find the value of x + \( \frac{1}{x} \)
4
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Q20.
Find the value of \( 125^ \frac{1}{3} \)
5
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Q21.
If \( 10000^x \) = 10, then find the value of x.
\( \frac{1}{4} \)
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Q22.
Find the value of \( 4^ \frac{3}{2} \)
Q23.
If x = 9 + 4\(\text{}\sqrt{5} \), then what is the value of x + \( \frac{1}{x} \)
18
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Q24.
What is the rationalizing factor of \(\text{}\sqrt{5} \) +\(\text{} \sqrt{3} \) ?
Q25.
Simplify the expression:
(\(\text{} \sqrt{3} \) + \(\text{} \sqrt{7} )\ ^2 \)
10 + 2\(\text{} \sqrt{21} \)
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Q26.
1.\(\overline{54} \) when expressed in the form of \( \frac{p}{q} \), where p and q are integers and q \(\not=0\) is
Q27.
0 .\(\overline{72} \) when expressed in the form of \( \frac{p}{q} \), where p and q are integers and q \(\not=0\) is
Q28.
Observe the given figure of a square. Which theorem can be used to find the length of the diagonal?
Q29.
The value of 0ꞏ55555… in the form \( \frac{p}{q} \), where p and q are integers and q \(\not=0\) is
Q30.
The square roots of positive integers are
Q31.
π is
Q32.
After rationalising the denominator of \( \frac{1}{(\text{} \sqrt {5 } \ ) - (\text{} \sqrt{3 } \ )} \), the denominator obtained is
Q33.
What is the value of (\( 2^ \frac{2}{3} \) x \( 2^ \frac{1}{3} \))?
2
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Q34.
Which fraction is equivalent to \( \frac{1}{2} \)?
Q35.
Which of the following is equal to \( \frac{1}{\sqrt{7}\ - 2}\) after rationalising its denominator?
Q36.
If \(x\sqrt{8} + 5\sqrt{8} = 12\sqrt{8}\), then which of the following is the value of x?
Q37.
Which of the following is an irrational number?
Q38.
The decimal expansion of the number \(\sqrt{2} \) is
Q39.
Find the repeating block in the decimal expansion of \( \frac{1}{27} \).
0370
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Q40.
Simplify the expression: (5 + \(\sqrt{5} \) )(5 ⎼ \(\sqrt{5} \) )
20
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Q41.
Without performing long division, what is the maximum number of digits in the repeating block of the decimal expansion of \(\frac{1}{17} \)?
16
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Q42.
To how many decimal places did Aryabhata correctly compute the value of π, and what value did he obtain?
Four decimal places; π=3.1416
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Q43.
Observe the numbers 169, 250, 343, and 256. Identify the numbers whose square roots are rational.
169 and 256
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Q44.
Which mathematician showed that \(\sqrt{3} \) is irrational. Mention the approximate year as well.
Theodorus of Cyrene, around 425 BC.
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Q45.
Simplify \(\frac{6}{8} \) so that numerator and denominator become co-prime.
\(\frac{3}{4} \)
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Q46.
What do the fractions \(\frac{1}{2} \) = \(\frac{2}{4} \) = \(\frac{10}{20} \) represents?
Equivalent fractions
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Q47.
Which of these numbers can be written as \(\frac{p}{q} \) with q = 1?
Q48.
The product of a non-zero rational number and an irrational number is
Q49.
What is the value of \(\sqrt{32} \) ÷ \(\sqrt{2} \)?
4
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Q50.
Find the value of \(( 343)^ \frac{-1}{3} \)?
\(\frac{1}{7} \)
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Q51.
Observe the numbers 9, 27, 125, and 36. Identify the numbers whose square roots are irrational.
27 and 125
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Q52.
Which of the following is irrational?
Q53.
What is the set of numbers {…, –3, –2, –1, 0, 1, 2, 3, …} called?
Q54.
A rational number is expressed in the form of \(\frac{p}{q} \), where p and q are integers. Which of the following cannot be the value of q?
Q55.
Which fraction is not in the simplest form?
Q56.
Which of these fractions is in the simplest form?
Q57.
Which number is both an integer and a rational number?
Q58.
How many rational numbers are there between − 5 and − 4?
infinite
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Q59.
Which of the following is a whole number but not a natural number?
Q60.
The decimal representation of the rational number is
Q61.
If the decimal representation of a number is non-terminating and non-repeating, what type of number is it?
An irrational number
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Q62.
Which of the following is a terminating decimal?
Q63.
The decimal expansion of \(\frac{329}{400} \) is
Q64.
What is the decimal representation of \(\frac{10}{3}\)?
3.333...
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Q65.
Which of the following is an irrational number?
Q66.
Rationalise the denominator of \( \frac{2}{(\text{} \sqrt {5 } \ ) - (\text{} \sqrt{3 } \ )} \)ꞏ
\(\sqrt{5} \) +\(\sqrt{3} \)
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Q67.
Which of the following variables represent a rational number?
Q68.
.Which of the following is equal to 0.66666...?
Q69.
If x = 3 + \(\sqrt{8} \) , then find the value of x + \( \frac{1}{x} \)
6
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Q70.
A rational number between \(\frac{1}{9} \) and \(\frac{4}{9} \) is
Q71.
Which of the following is an irrational number between 2.11243... and 2.12243?
Q72.
Every rational number is
Q73.
The rational number \( 0.\overline{3} \) can also be written
Q74.
The product of (\(\sqrt{5} \) +\(\sqrt{7} \) ) and (\(\sqrt{5} \) - \(\sqrt{7} \) ) is
Q75.
Which of the following statements is true?
Q76.
Find the decimal form of 3\(\frac{3}{8} \)
3.375
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Q77.
Which of the following is a rational number between 4 and 5?
Q78.
Out of the following, the irrational number is
Q79.
Find the rationalizing factor of 7 - 2\(\sqrt{3} \).
7 + 2\(\sqrt{3} \)
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Q80.
Which of the following can be written in the form \( \frac{p}{q} \)?
Q81.
Which of the following numbers is irrational?
Q82.
Which of the following statements is true?
Q83.
A number whose decimal expansion neither terminates nor repeats is called:
Q84.
Peter draws the spiral of irrational numbers on a paper. What is the length of OE in the spiral?
\(\sqrt{5} \)
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Q85.
Vasu represents \(\sqrt{4.5} \) on the number line PW. The length of TS = 1 unit. His representation is shown below. Which letter represent the number 0 of the number line?
Q86.
Which of the following is an equivalent fraction to \( \frac{3}{5} \)?
Q87.
Which number has a non-terminating, repeating decimal expansion?
Q88.
The product of two irrational numbers is:
Q89.
12 \(\sqrt{12} \) divided by 3 \(\sqrt{3} \) is equal to
Q90.
In the given figure, OABC is a square with each side of length 1 unit, and ∠OAB = 90°. Using the Pythagorean theorem, find the length of the diagonal OB.
\(\sqrt{2} \)
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Q91.
Who was the first mathematician to prove that π is an irrational number, and in which year was this proof established?
Johann Heinrich Lambert in 1761
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Q92.
Which of the following is an irrational number between 2 and 3?
Q93.
In 1761, Lambert proved that π is irrational. His work was further refined by another mathematician in 1794. What was the name of that mathematician?
Adrien-Marie Legendre
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Q94.
Starting with a unit square whose diagonal OB=√2, a perpendicular line segment BD of length 1 unit is drawn from B. What irrational number is represented by the hypotenuse OD?
\(\sqrt{3} \)
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Q95.
If \(\sqrt{2} \) and \(\sqrt{3} \) are constructed successively on the number line using perpendicular unit segments, which number will be obtained in the next step?
2
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Q96.
Without constructing \(\sqrt{2} \), \(\sqrt{3} \), and \(\sqrt{4} \) successively, determine the lengths of the two perpendicular sides of a right triangle that can be used to construct √5 directly on the number line.
1 unit and 2 units
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Q97.
Observe the following rational numbers: \( \frac{1}{2} \) = 0.5, \( \frac{11}{25} \) = 0ꞏ44, \( \frac{9}{20} \) = 0ꞏ45. All of these have terminating decimals. What special property must the denominators satisfy?
The denominators must have only power of 2 or powers of 5 or both as prime factors.
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Q98.
If \( \frac { 2 + \sqrt{3}}{2 - \sqrt{3}} \) = a + b\(\sqrt{3} \), find the values of ‘a’ and ‘b’.
a = 7, b = 4
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Q99.
Which of the following is a rational number between \(\sqrt{2} \) and \(\sqrt{3} \) ?
Q100.
Express \( 0.\overline{001} \) in the form of \(\frac{p}{q} \), where p and q are integers and q≠0.
\( \frac{1}{999} \)
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Q101.
If a=2 - \(\sqrt{3} \) ,then find the value of a + \( \frac{1}{a} \)
4
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Q102.
Which of the following is an irrational number?
Q103.
Assertion (A): The sum of two irrational numbers is always irrational.
Reason (R): Irrational numbers cannot be expressed in the form \( \frac{p}{q} \), where p and q are integers and q≠0.
Q104.
Which of the following pairs has a rational product but an irrational sum?
Q105.
Express 0.6 + \( 0.\overline{7} \) in the form of \(\frac{p}{q} \), where p and q are integers and q≠0.
\( \frac{62}{45} \)
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Q106.
What is the simplest form of the expression \(\frac{\sqrt {2} + 1}{\sqrt{2} - 1 } \)?
3 + 2\(\sqrt{2} \)
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Q107.
If \( x^{-2} \) = \(\frac{1}{9} \), then what is the value of x?
3
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Q108.
Which of the following is equal to x?
Q109.
Simplify: \(\sqrt[4]{81} \) - 8 \(\sqrt[3]{216} \) + 15 \(\sqrt[5]{32} \) + \(\sqrt{225} \)
0
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Q110.
What is the value of \( 16^{-\frac{1}{4}} \) x \(\sqrt[4]{16} \)?
1
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Q111.
Simplify the following expression:
\( 3(243)^{-\frac{1}{5}} \) + \( 6^2 \)\( (216)^{-\frac{2}{3}} \) + \( 4^3 \)\( (256)^{-\frac{3}{4}} \)
3
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Q112.
Find the value of \( 27^{\frac{1}{3}} \) - \( 27^{\frac{2}{3}} \)
-6
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Q113.
Simplify the expression: \( 27^{-\frac{1}{3}} \) x \( 27^{-\frac{1}{3}} \)
\(\frac{1}{9} \)
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Q114.
Simplify the expression: \(\frac{\left(x^{a-b}\right)\times\left(x^{b-c}\right)}{x^{c-a}} \)
\( x^{2(a-c)} \)
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Q115.
Find the value of the expression \( \left\{7\left(64^{\frac{1}{3}}+27^{\frac{1}{3}}\right)\right\}^{\frac{1}{2}}\)
7
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Q116.
If the \(\frac{p}{q} \) form of 0.72 is \(\frac{m}{n} \), then find the value of (m + n).
43
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Q117.
Which of the following is a rational number between - 2 and - 5?
Q118.
\( \text{If } a = 2^{\frac{1}{3}} \text{ and } b = 2^{\frac{2}{3}}, \text{ find the value of } a^3 b^3. \)
8
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Q119.
\( \text{If } a = 2+\sqrt{3} \text{ and } b = 2-\sqrt{3}, \text{ find } a^2+b^2. \)
14
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Q120.
Find the value of x, if
3 + \( 2^x \) = \( \left(64\right)^{\frac{1}{2}} + \left(27\right)^{\frac{1}{3}} \)
3
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Q121.
In the figure, OA=1 unit and AB=1 unit, with ∠OAB=90∘. Find the length of OB.
\(\sqrt{2} \)
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Q122.
Which of the following are the three rational numbers between 3 and 4?
Q123.
What is the simplest form of the decimal 0.272727...?
\(\frac{3}{11} \)
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Q124.
A rational number between \( \frac{1}{4} \) and \( \frac{1}{2} \) is
Q125.
Assertion (A): The decimal expansion of \( \frac{1}{8} \) is terminating.
Reason (R): The denominator 8= \( 2^3 \) has only 2 as its prime factor.
Q126.
Using the successive square-root construction method, \(\sqrt{7} \) can be located on the number line after locating:
Q127.
Find the product of the following:
\( \left(\frac{ \sqrt{3} - 1}{2}\right)\times\left(\frac{\sqrt{3} + 1}{3}\right) \)
\( \frac{1}{3} \)
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Q128.
Simplify: \( \left(\sqrt{2}+\frac{1}{\sqrt{2}}\right)^2 \)
\( \frac{9}{2} \)
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Q129.
If \( \frac{\sqrt{3}-1}{\sqrt{3}+1}=a + b\sqrt{3} \), then find the values of a and b.
a = 2, b = -1
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Q130.
Assertion (A): Every integer is a rational number.
Reason (R): Any integer 'n' can be written as \(\frac{n}{1} \) where 1 ≠ 0