MCQ 11 Mark
The decimal expansion of the number $\sqrt{2}$ is
- A
- B
- C
non-terminating recurring
- ✓
non-terminating non-recurring
AnswerCorrect option: D. non-terminating non-recurring
(d)
$\sqrt{2}$ is an irrational number. So, its decimal expansion is non-terminating non-recurring. Hence, option (d) is correct.
View full question & answer→MCQ 21 Mark
Answer(c)
We observe $\frac{2}{3}$ is a rational number but it is neither a natural number nor an integer nor a whole number. But, it is a real number. In fact every rational number is a real number. Hence, option (c) is correct.
View full question & answer→MCQ 31 Mark
Between two rational numbers
- A
there is no rational number
- B
there is exactly one rational number
- ✓
there are infinitely many rational numbers
- D
there are only rational numbers and no irrational numbers.
AnswerCorrect option: C. there are infinitely many rational numbers
(c)
Between two distinct rational numbers there are infinitely many rational as well as irrational numbers. In fact, this holds for any two distinct real numbers. Hence, option (c) is correct and remaining are false.
View full question & answer→MCQ 41 Mark
A rational number between $\sqrt{2}$ and $\sqrt{3}$ is
Answer(c) 1.5
We know that $\sqrt{2}=1.41421356 \ldots$ and $\sqrt{3}=1.732050807 \ldots$
We find that $\frac{\sqrt{2}+\sqrt{3}}{2}$ and $\frac{\sqrt{2} \cdot \sqrt{3}}{2}$ are irrational numbers. So, options (a) and (b) are incorrect.Clearly, 1.5 is a rational number between $\sqrt{2}$ and $\sqrt{3}$.
View full question & answer→MCQ 51 Mark
Which one of the following statements is true?
- A
The sum of two irrational numbers is always an irrational number
- B
The sum of two irrational numbers is always a rational number
- ✓
The sum of two irrational numbers may be a rational number or an irrational number
- D
The sum of two irrational numbers is always an integer
AnswerCorrect option: C. The sum of two irrational numbers may be a rational number or an irrational number
View full question & answer→MCQ 61 Mark
Which one of the following is a correct statement?
- A
Decimal expansion of a rational number is terminating
- B
Decimal expansion of a rational number is non-terminating
- C
Decimal expansion of an irrational number is terminating
- ✓
Decimal expansion of an irrational number is non-terminating and non-repeating
AnswerCorrect option: D. Decimal expansion of an irrational number is non-terminating and non-repeating
View full question & answer→MCQ 71 Mark
Which of the following statements is true?
(a) The square of an irrational number is always rational
(b)$\frac{\sqrt{12}}{\sqrt{3}}$ is not a rational number as $\sqrt{12}$ and $\sqrt{3}$ are not integers
(c) $\frac{\sqrt{15}}{\sqrt{3}}$ is written in the form $\frac{p}{q}, q \neq 0$ and so it is a rational number.
(d) If x is rational and y is an irrational number then xy is not necessarily irrational number.
- A
The square of an irrational number is always rational
- B
$\frac{\sqrt{12}}{\sqrt{3}}$ is not a rational number as $\sqrt{12}$ and $\sqrt{3}$ are not integers
- C
$\frac{\sqrt{15}}{\sqrt{3}}$ is written in the form $\frac{p}{q}, q \neq 0$ and so it is a rational number.
- ✓
If x is rational and y is an irrational number then xy is not necessarily irrational number.
AnswerCorrect option: D. If x is rational and y is an irrational number then xy is not necessarily irrational number.
(d)
Consider irrational number $x=\sqrt{\sqrt{3}}$.We find that $x^2=\sqrt{3}$ is also an irrational number. So, statement in option (a) many not be true.
$\frac{\sqrt{12}}{\sqrt{3}}=\sqrt{\frac{12}{3}}=\sqrt{4}=2$ is a rational number. So, statement in option (b) is not true.
$\frac{\sqrt{15}}{\sqrt{3}}=\sqrt{\frac{15}{3}}=\frac{\sqrt{5}}{1}$ is an irrational number. So, statement in option (c) is not true.
We find that 0 is rational and $\sqrt{3}$ is an irrational number. But, $0 \times \sqrt{3}=0$ is rational. Hence, statement in option (d) is true.
View full question & answer→MCQ 81 Mark
Which of the following statements is true?
(a) $\frac{\sqrt{2}}{3}$ is a rational number
(b) There are infinitely many integers between any two integers
(c) Number of rational numbers between 15 and 18 is finite.
(d) There are numbers which cannot be written in the form $\frac{p}{q}, q \neq 0, p, q$ both are integers
- A
$\frac{\sqrt{2}}{3}$ is a rational number
- B
There are infinitely many integers between any two integers
- C
Number of rational numbers between 15 and 18 is finite.
- ✓
There are numbers which cannot be written in the form $\frac{p}{q}, q \neq 0, p, q$ both are integers.
AnswerCorrect option: D. There are numbers which cannot be written in the form $\frac{p}{q}, q \neq 0, p, q$ both are integers.
(d)
$\frac{\sqrt{2}}{3}=\frac{1}{3} \times \sqrt{2}$ is an irrational number. So, statement in option (a) is not true. There is no integer between integers 7 and 8. So, statement is option (b) is not true. There are infinitely many rational numbers between any two distinct integers. So, statement in option (c) is not true.$\sqrt{\frac{3}{2}}$ can be written as $\frac{\sqrt{3}}{\sqrt{2}}$ where $\sqrt{3}$ and $\sqrt{2}$ are not integers. So, statement in option (d) is true.
View full question & answer→MCQ 91 Mark
Which of the following statements is true?
- A
Product of two irrational numbers is always irrational
- ✓
Product of a non-zero rational and an irrational number is always irrational
- C
Sum of two irrational numbers can never be irrational
- D
Sum of an integer and a rational number can never be an integer
AnswerCorrect option: B. Product of a non-zero rational and an irrational number is always irrational
View full question & answer→MCQ 101 Mark
Which of the following statements is true?
- A
$\pi$ and $\frac{22}{7}$ are both rationals
- B
$\pi$ and $\frac{22}{7}$ are both irrationals
- C
$\pi$ is rational and $\frac{22}{7}$ is irrational
- ✓
$\pi$ is irrational and $\frac{22}{7}$ is rational
AnswerCorrect option: D. $\pi$ is irrational and $\frac{22}{7}$ is rational
View full question & answer→MCQ 111 Mark
Which of the following statements is / are correct?
(i) Every integer is a rational number
(ii) Every rational number is an integer
(iii) A real number is either rational or irrational number
(iv) Every whole number is a natural number.
MCQ 121 Mark
Which of the following rational numbers is equivalent to a decimal that terminates?
- A
$\frac{1}{3}$
- B
$\frac{2}{3}$
- ✓
$\frac{3}{8}$
- D
$\frac{5}{6}$
AnswerCorrect option: C. $\frac{3}{8}$
(c)
If the denominator of a rational number is not expressible in the form $2^m \times 5^n$ where m, n are non-negative integers, then its decimal representation is non-terminating and recurring.Therefore, $\frac{1}{3}, \frac{2}{3}$ and $\frac{5}{6}=\frac{5}{2 \times 3}$ have non-terminating recurring decimal representation. Only $\frac{3}{8}=\frac{3}{2^3 \times 5^0}$ has terminating decimal representation.
View full question & answer→MCQ 131 Mark
Which of the following numbers is irrational?
AnswerCorrect option: C. $\sqrt{8}$
View full question & answer→MCQ 141 Mark
Which of the following numbers can be represented as non-terminating, repeating decimals?
- A
$\frac{39}{24}$
- B
$\frac{3}{16}$
- ✓
$\frac{3}{11}$
- D
$\frac{137}{25}$
AnswerCorrect option: C. $\frac{3}{11}$
View full question & answer→MCQ 151 Mark
Which of the following is true about $1=0 \overline{3}$ ?
- ✓
x is a rational number, because x can be expressed in the form $\frac{p}{q}$, by solving the equation 10x = 3 - x
- B
x is a rational number because x can be expressed in the form $\frac{p}{q}$ by solving the equation 10x = 3 - x.
- C
x is an irrational number because x can be expressed in the form $\frac{p}{q}$ by solving the equation 10x = 3 - x.
- D
x is an irrational number because y can be expressed in the form $\frac{p}{q}$ by solving the equation 10x = 3 - x.
AnswerCorrect option: A. x is a rational number, because x can be expressed in the form $\frac{p}{q}$, by solving the equation 10x = 3 - x
View full question & answer→MCQ 161 Mark
Which of the following is rational?
- A
$\sqrt{3}$
- B
$\pi$
- C
$\frac{4}{0}$
- ✓
$\frac{0}{4}$
AnswerCorrect option: D. $\frac{0}{4}$
View full question & answer→MCQ 171 Mark
Which of the following is irrational?
- A
- B
- C
$0 . \overline{1516}$
- ✓
View full question & answer→MCQ 181 Mark
Which of the following is irrational?
- A
- B
$0.14 \overline{16}$
- C
$0 . \overline{1416}$
- ✓
View full question & answer→MCQ 191 Mark
Which of the following is irrational?
- A
$\sqrt{\frac{4}{9}}$
- B
$\frac{4}{5}$
- ✓
$\sqrt{7}$
- D
$\sqrt{81}$
AnswerCorrect option: C. $\sqrt{7}$
View full question & answer→MCQ 201 Mark
Which of the following is equivalent to $0.5 \overline{782} ?$
- A
$\frac{5770}{9990}$
- B
$\frac{5772}{9990}$
- ✓
$\frac{5777}{9990}$
- D
$\frac{5782}{9990}$
AnswerCorrect option: C. $\frac{5777}{9990}$
(c)
Let $x=0.5 \overline{782}$
Then, $\quad 10 x=5 . \overline{782} \Rightarrow 10 x=5+\frac{782}{999} \Rightarrow 10 x=\frac{5777}{999} \Rightarrow x=\frac{5777}{9990}$
View full question & answer→MCQ 211 Mark
Which of the following is a rational number?
- A
$\sqrt{3}+1$
- B
$\pi$
- C
$2 \sqrt{3}$
- ✓
$0$
View full question & answer→MCQ 221 Mark
Which of the following is a rational number?
MCQ 231 Mark
Which of the following is an irrational number?
- A
$\frac{\sqrt{12}}{\sqrt{3}}$
- B
$\frac{\sqrt{18}}{\sqrt{2}}$
- ✓
$\frac{\sqrt{42}}{\sqrt{7}}$
- D
$\frac{\sqrt{45}}{\sqrt{5}}$
AnswerCorrect option: C. $\frac{\sqrt{42}}{\sqrt{7}}$
(c)
Using the result $\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}$, we obtain
$\frac{\sqrt{12}}{\sqrt{3}}=\sqrt{\frac{12}{3}}=\sqrt{4}=2, \frac{\sqrt{18}}{\sqrt{2}}=\sqrt{\frac{18}{2}}=\sqrt{9}=3, \frac{\sqrt{42}}{\sqrt{7}}=\sqrt{\frac{42}{7}}=\sqrt{6}$ and $\frac{\sqrt{45}}{\sqrt{5}}=\sqrt{\frac{45}{5}}=\sqrt{9}=3$.
Clearly, $\frac{\sqrt{42}}{\sqrt{7}}=\sqrt{6}$ is an irrational number.
View full question & answer→MCQ 241 Mark
Which of the following is an irrational number?
- A
- B
$0.13 \overline{15}$
- C
$0 . \overline{1315}$
- ✓
View full question & answer→MCQ 251 Mark
Which of the following is an irrational number?
MCQ 261 Mark
Which of the following is a correct statement?
- A
Sum of two irrational numbers is always irrational
- ✓
Sum of a rational and irrational number is always an irrational number
- C
Square of an irrational number is always a rational number
- D
Sum of two rational numbers can never be an integer
AnswerCorrect option: B. Sum of a rational and irrational number is always an irrational number
View full question & answer→MCQ 271 Mark
When $0 . \overline{001}$ is expressed in the form p/q, where p and q are integers not having any common factor except 1 then q is equal
Answer(c)
Using rule given on page 1, we obtain $0 . \overline{001}=\frac{1}{999}$.
Hence, q = 999
View full question & answer→MCQ 281 Mark
The value of $2 . \overline{45}+0 . \overline{36}$ in the simple form is
- A
$\frac{67}{33}$
- B
$\frac{24}{11}$
- ✓
$\frac{31}{11}$
- D
$\frac{167}{110}$
AnswerCorrect option: C. $\frac{31}{11}$
(c)
$2 . \overline{45}+0 . \overline{36}=2+0 . \overline{45}+0 . \overline{36}=2+\frac{45}{99}+\frac{36}{99}=2+\frac{5}{11}+\frac{4}{11}=\frac{31}{11}$
View full question & answer→MCQ 291 Mark
The value of $2 . \overline{36}+0 . \overline{23}$ when expressed in the simplest form is
- ✓
$\frac{257}{99}$
- B
$\frac{238}{99}$
- C
$\frac{247}{99}$
- D
$\frac{275}{99}$
AnswerCorrect option: A. $\frac{257}{99}$
(a)
$2 \cdot \overline{36}+0 . \overline{23}=2+0 . \overline{36}+0 . \overline{23}=2+\frac{36}{99}+\frac{23}{99}=2+\frac{59}{99}=\frac{257}{99}$
View full question & answer→MCQ 301 Mark
The value of $0 . \overline{4}$ in the form $\frac{p}{q}$, where p and q are integers and $q \neq 0$, is
- ✓
$\frac{4}{9}$
- B
$\frac{2}{5}$
- C
$\frac{1}{5}$
- D
$\frac{4}{5}$
AnswerCorrect option: A. $\frac{4}{9}$
View full question & answer→MCQ 311 Mark
The value of $0 . \overline{2} \times 0 . \overline{5}$ is
- A
- B
$\frac{10}{9}$
- ✓
$\frac{10}{81}$
- D
$\frac{10}{99}$
AnswerCorrect option: C. $\frac{10}{81}$
View full question & answer→MCQ 321 Mark
The value of $0 . \overline{23}+0 . \overline{22}$ is
- ✓
$0 . \overline{45}$
- B
$0 . \overline{43}$
- C
$0 . \overline{54}$
- D
AnswerCorrect option: A. $0 . \overline{45}$
View full question & answer→MCQ 331 Mark
The value of 0.9999... when expressed as a fraction in the simplest form is
- A
$\frac{1}{9}$
- B
$\frac{8}{9}$
- ✓
- D
$\frac{10}{9}$
Answer(c)
$0.9999 \ldots=0 . \overline{9}=\frac{9}{9}=1$.
Let x = 0.9999 Then,
$\begin{aligned} & 10 x=9.9999 \ldots \\ \Rightarrow \quad & 10 x-x=(9.9999 \ldots)-(0.9999 \ldots) \Rightarrow 9 x=9 \Rightarrow x=1\end{aligned}$
View full question & answer→MCQ 341 Mark
The sum of $0 . \overline{2}$ and $0 . \overline{5}$ is
- A
$\frac{7}{10}$
- ✓
$\frac{7}{9}$
- C
$\frac{7}{99}$
- D
$\frac{3}{10}$
AnswerCorrect option: B. $\frac{7}{9}$
View full question & answer→MCQ 351 Mark
The smallest rational number by which $\frac{1}{3}$ should be multiplied so that its decimal expansion terminates after one place of decimal, is
- A
$\frac{1}{10}$
- ✓
$\frac{3}{10}$
- C
- D
AnswerCorrect option: B. $\frac{3}{10}$
View full question & answer→MCQ 361 Mark
The simplest form of $1 . \overline{6}$ is
- A
$\frac{8}{5}$
- ✓
$\frac{5}{3}$
- C
$\frac{833}{500}$
- D
$\frac{7}{6}$
AnswerCorrect option: B. $\frac{5}{3}$
View full question & answer→MCQ 371 Mark
The simplest form of $0.12 \overline{3}$ is
- A
$\frac{41}{330}$
- ✓
$\frac{37}{300}$
- C
$\frac{41}{333}$
- D
AnswerCorrect option: B. $\frac{37}{300}$
(b)
Let $x=0.12 \overline{3}$.Then, $100 x=12 . \overline{3} \Rightarrow 100 x=12+0 . \overline{3} \Rightarrow 100 x=12+\frac{3}{9} \Rightarrow 100 x=12+\frac{1}{3} \Rightarrow 100 x=\frac{37}{3} \Rightarrow x=\frac{37}{300}$
View full question & answer→MCQ 381 Mark
The representation of $1 . \overline{3}$ in the form p/q is
- ✓
$\frac{4}{3}$
- B
$\frac{5}{3}$
- C
$\frac{5}{4}$
- D
AnswerCorrect option: A. $\frac{4}{3}$
(a)
\[1 . \overline{3}=1+0 . \overline{3}=1+\frac{3}{9}=1+\frac{1}{3}=\frac{4}{3}\]
[Using rule given on page 1]
View full question & answer→MCQ 391 Mark
There is a number x such that $x^2$ is irrational but $x^4$ is rational. Then x can be
- A
$\sqrt{5}$
- B
$\sqrt{2}$
- C
$\sqrt[3]{2}$
- ✓
$\sqrt[4]{5}$
AnswerCorrect option: D. $\sqrt[4]{5}$
View full question & answer→MCQ 401 Mark
The product of any two irrational numbers is
- A
always an irrational number
- B
- C
- ✓
sometimes rational, sometimes irrational
AnswerCorrect option: D. sometimes rational, sometimes irrational
(d)
We find that the product of irrational numbers $\sqrt{3}$ and $\frac{2}{5} \sqrt{3}$ is $\frac{6}{5}$,which is a rational number. So, product of two irrational numbers need not be always an irrational number. The product of $\sqrt{3}$ and $2+\sqrt{3}$ is $2 \sqrt{3}+3$, which is an irrational number. So, the product of two irrational numbers need not always be a rational number. The product of irrational numbers $\frac{1}{3}+\sqrt{2}$ and $\frac{1}{3}-\sqrt{2}$ is $-\frac{5}{3}$ which is not an integer. So, option (c) is incorrect. In fact, the product is sometimes rational and sometimes irrational.
View full question & answer→MCQ 411 Mark
The product of a non-zero rational number with an irrational number is
MCQ 421 Mark
The number of consecutive zeros in $2^3 \times 3^4 \times 5^4 \times 7$, is
View full question & answer→MCQ 431 Mark
The number $1 . \overline{27}$ in the form $\frac{p}{q}$, where p and q are integers and $q \neq 0$, is
- A
$\frac{14}{9}$
- ✓
$\frac{14}{11}$
- C
$\frac{14}{13}$
- D
$\frac{14}{15}$
AnswerCorrect option: B. $\frac{14}{11}$
View full question & answer→MCQ 441 Mark
The number $0 . \overline{3}$ in the form $\frac{p}{q}$, where pand q are integers and $q \neq 0$, is
- A
$\frac{33}{100}$
- B
$\frac{3}{10}$
- ✓
$\frac{1}{3}$
- D
$\frac{3}{100}$
AnswerCorrect option: C. $\frac{1}{3}$
View full question & answer→MCQ 451 Mark
The number 0.318564318564318564 ………….. Is:
MCQ 461 Mark
The decimal representation of a rational number cannot be
- A
- B
- C
non-terminating repeating
- ✓
non-terminating non-repeating
AnswerCorrect option: D. non-terminating non-repeating
(d)
The decimal representation of rational number $\frac{2}{5}$ is 0.4, which is terminating. So, option (a) is not true. The decimal representation of $\frac{2}{3}$ is 0.66666... which is non-terminating repeating. So, option (b) and (c) are not true.
A rational number can not have non-terminating non-repeating decimal representation.
Hence, option (d) is correct.
View full question & answer→MCQ 471 Mark
The decimal expansion that a rational number cannot have is
- A
- B
$0.25 \overline{28}$
- C
$0 . \overline{2528}$
- ✓
View full question & answer→MCQ 481 Mark
The decimal expansion of a rational number is
- A
terminating or non-terminating non-repeating
- ✓
terminating or non-terminating repeating
- C
terminating and repeating
- D
AnswerCorrect option: B. terminating or non-terminating repeating
View full question & answer→MCQ 491 Mark
On a number line, $\frac{3}{\sqrt{18}}$ is halfway located between 0 and $\sqrt{a}$.What is the value of a?
Answer(a)
A number located halfway between 0 and $\sqrt{a}$ on the number line is $\frac{0+\sqrt{a}}{2}=\frac{1}{2} \sqrt{a}$.
$\therefore \quad \frac{1}{2} \sqrt{a}=\frac{3}{\sqrt{18}} \Rightarrow \frac{1}{2} \sqrt{a}=\frac{3}{3 \sqrt{2}} \Rightarrow \frac{1}{2} \sqrt{a}=\frac{1}{\sqrt{2}} \Rightarrow \sqrt{a}=\frac{2}{\sqrt{2}}=\sqrt{2} \Rightarrow a=2$
View full question & answer→MCQ 501 Mark
If the some of the rational numbers between 7 and 11 are written in the form $\frac{m}{6}$, then integer values of m lie between
View full question & answer→