MCQ 11 Mark
Which of the following is rational?
- A
$\sqrt{3}$
- B
$\pi$
- C
$\frac{4}{0}$
- ✓
$\frac{0}{4}$
AnswerCorrect option: D. $\frac{0}{4}$
$a. \sqrt{3}=1.732 \ ...=$ Non$-$terminating and non$-$repeating number, hence irrational
$b. \pi=3.14 \ ...$ also can not be terminated to $\frac{\text{p}}{\text{q}}$ form, and is non$-$terminating and non$-$repeating in nature.
Hence, irrational.
$c. \frac{4}{0}$ is not a rational number because this is in the form $\frac{\text{p}}{\text{q}}$ where $p$ and $q$ are integers but $q = 0$
$d. \frac{0}{4}$ follows the defination of rational number.
Hence, correct option is $(d)$.
View full question & answer→MCQ 21 Mark
An irrational number between $2$ and $2.5$ is:
- A
$\sqrt{11}$
- ✓
$\sqrt{5}$
- C
$\sqrt{22.5}$
- D
$\sqrt{12.5}$
AnswerCorrect option: B. $\sqrt{5}$
$\sqrt{4}=2$ and $\sqrt{6.25}=2.5$
Option $(a), (c)$ and $(d)$: $\sqrt{11},\sqrt{22.5}$ and $\sqrt{12.5},$
all are greater than $\sqrt{6.25}$
$⇒$ Out of interval $(2, 2.5)$
Option $(b):$ $\sqrt{4}<\sqrt{5}<\sqrt{6.25}$
$\Rightarrow$ lies in the interval $(2, 2.5)$
Hence, option $(b)$ is correct.
View full question & answer→MCQ 31 Mark
The number $0.\overline{32}$ when expressed in the form $\frac{\text{p}}{\text{q}}$ $\big($p, q are integers and $\text{q}\neq0\big),$ is:
- A
$\frac{8}{25}$
- ✓
$\frac{29}{90}$
- C
$\frac{32}{99}$
- D
$\frac{32}{199}$
AnswerCorrect option: B. $\frac{29}{90}$
Let $\text{x}=0.\overline{32}=0.32222..(1)$
Now, $10\text{x}=3.2222=3.\overline{2}...(2),$
Subtracting equation $(2)$ and $(3)$, we get
$90\text{x}=29$
$\Rightarrow\text{x}=\frac{29}{90}$
Hence, option $(b)$ is correct.
View full question & answer→MCQ 41 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.
If two irrational numbers i.e. $\sqrt{2},\sqrt{5},2+\sqrt{3},2-\sqrt{3}$ etc. are added it is not necessary that sum comes out to be an irrational number always, or a rational nnumber always, or a rational number always...
Since $\sqrt{2}+\sqrt{5}=$ an irrational number
$2+\sqrt{\not\text{3}}+2-\sqrt{\not\text{3}}=4=$ a rational number
So we see that $\sqrt{2}$ and $\sqrt{5}$ are irrational numbers, and their sum is also irrational.
But $2+\sqrt{3}$ and $2-\sqrt{3}$ are also irrational numbers, and their sum is rational number $'4'$.
So sum of two irrational numbers can be either an irrational number or a rational number depending which numbers are being added.
So options $(a)$ and $(b)$ are totally wrong, because they are not 'always' true.
Option $(c)$ is correcrt because sum can be either irrational or rational and option $(c)$ is verifying this statement.
Option $(d)$ - again it is not always true, if we add two irrational numbers like $2+\sqrt{3}$ and $2-\sqrt{3}.$
Sum is an integer $= 4$, but if we add $\sqrt{3}$ and $\sqrt{3},$ sum is $2\sqrt{3}$ which is not an integer but again an irrational number.
So option $(d)$ is also incorrect.
Hence, correct option is $(c)$.
View full question & answer→MCQ 51 Mark
The number $1.\overline{27}$ in the form $\frac{\text{p}}{\text{q}},$ where $p$ and $q$ are integers and $\text{q}\neq0,$ is:
- A
$\frac{14}{9}$
- ✓
$\frac{14}{11}$
- C
$\frac{14}{13}$
- D
$\frac{14}{15}$
AnswerCorrect option: B. $\frac{14}{11}$
Let $\text{x}=1.\overline{27}=1.272727...(1)$
Now, $100\text{x}=127.272727=127.\overline{27}...(2)$
Subtracting equation $(1)$ from $(2)$, we get
$99\text{x}=126$
$\Rightarrow\text{x}=\frac{126}{99}=\frac{14}{11}$
Hence, option $(b)$ is correct.
View full question & answer→MCQ 61 Mark
Every point on a number line represents:
AnswerOn number line, we have $-\infty$ to $\infty$ numbers, consisting $-\infty...-4,-3,-2,-1,0,1,2,3,4...\infty,$ $1.12,1.14$ and $1.41406532, 3.146201286295...$ etc. That means on number line, there are natural nmbers $(1, 2, 3, 4 ...)$, integers, rational numbers $\frac{1}{2},\frac{1}{3},1.3333,$ irrational numbers $1.4148625385...$
But if we see every number as a complete family, it becomes
Real numbers (any number which can be represent on Real axes)
So, every point on the number line reoresent a unique real number which contains every type.
Hence, option $(a)$ is correct.
View full question & answer→MCQ 71 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. $\frac{3}{11}$
$a. \frac{39}{24}=1.625=$ Terminating Decimal
$b. \frac{3}{16}=0.1875=$ Terminating Decimal
$c. \frac{3}{11}=0.27272727 \ ...=$ Non$-$Terminating Decimal
$d. \frac{137}{25}=5.48=$ Terminating Decimal
Hence, option $(c)$ is correct.
View full question & answer→MCQ 81 Mark
The number $1.\overline{3}$ in the form $\frac{\text{p}}{\text{q}},$ where $p$ and $q$ are integers and $\text{q}\neq0,$ is:
- A
$\frac{33}{100}$
- B
$\frac{3}{10}$
- ✓
$\frac{1}{3}$
- D
$\frac{3}{100}$
AnswerCorrect option: C. $\frac{1}{3}$
Let $\text{x}=1.\overline{3}=0.3333..(1)$
Now, $10\text{x}=3.3333=3.\overline{3}...(2)$
Subtracting equation $(1)$ from $(2)$, we get
$9\text{x}=3$
$\Rightarrow\text{x}=\frac{3}{9}=\frac{1}{3}$
$\Rightarrow0.\overline{3}=\frac{1}{3}$
Hence, option $(c)$ is correct.
View full question & answer→MCQ 91 Mark
The number of consecutive zeros in $2^3 \times 3^4 \times 5^4 \times 7$, is:
Answer$5 \times 2 = 10 \Rightarrow $ one $5$ and one $2$ make one zero, so$ 5 \times 2 \times 5 \times 2 = 100$
Numbers of pairs of $5$ and $2$ will be equal to the number of consecutive zeros in the given number.
In the given number, there are three $2's$ and four $5's$.
So number of pairs of $5$ and $2$ are only three.
So there will be three consecutive zeros in the given number
Hence, option $(a)$ is correct.
View full question & answer→MCQ 101 Mark
Which of the following is irrational?
- A
$0.14$
- B
$0.14\overline{16}$
- C
$0.\overline{1416}$
- ✓
$0.1014001400014...$
AnswerCorrect option: D. $0.1014001400014...$
$a. 0.14=\frac{14}{100},$ which is a Rational number
$b. 0.14\overline{16}$ is non$-$terminating but repeating, hence a rational number
$c. 0.\overline{1416}$ is non$-$terminating but repeating, hence a rational number
$d. 0.1014001400014...$ is non$-$terminating as well as non$-$repeating number, which is irrational in nature.
Hence, correct option is $(d)$.
View full question & answer→MCQ 111 Mark
Which of the following statements is true?
- A
Product of two irrational numbers is always irrational.
- ✓
Product of a 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 rational and an irrational number is always irrational.
$a.$ Is incorrect, Product of two irrational numbers is not always irrational, it can be also rational sometimes.
when an irrational number is multiplied to itself, or multiplied by another irrational, that product becomes a perfect square.
Example:
$\sqrt{2}\times\sqrt{2}=2 ($Rational$)$
$\sqrt{2}\times\sqrt{8}=\sqrt{16}=\pm4 ($Rational$)$
$b.$ Is correct, because when a rational number is multiplied to an irrational number, it can not make an irrational number terminating or Non$-$terminating Repeating.
Product again becomes a Non$-$terminating Non$-$Repeating number.
as: $2\times\sqrt{3}=2\sqrt{3}$
$\frac{2}{3}\times\sqrt{3}=\frac{2}{\sqrt{3}}$
So, product of a rational number and an irrational number is always an irrational, because irrational number is just changed in magnitude not in properties.
$c.$ Is incorrect, Sum of two irrational numbers can be an irrational number.
i.e. if we add $\sqrt{2}$ and $\sqrt{3},$ we will get $\sqrt{2}+\sqrt{3}$ which is also an irrational.
$d.$ Is incorrect, Sum of an integer and a rational number can be a integer.
Because all integers are rational numbers and also we can say some rational numbers are integers.
So their sum with integer would be a integer
i.e. $2 + 3 = 5$
Hence, correct option is $(b)$.
View full question & answer→MCQ 121 Mark
The value of $0.\overline{23}+0.\overline{22}$ is:
- ✓
$0.\overline{45}$
- B
$0.\overline{43}$
- C
$0.\overline{45}$
- D
$0.45$
AnswerCorrect option: A. $0.\overline{45}$
Let $\text{x}=0.\overline{23}=0.232323...(1)$
Now, $\text{y}=0.\overline{22}=0.22222...(2)$
Adding equation $(1)$ and $(2)$, we get
$\text{x}+\text{y}=0.454545=0.\overline{45}$
$\Rightarrow0.\overline{23}+0.\overline{22}=0.\overline{45}$
Hence, option $(a)$ is correct.
View full question & answer→MCQ 131 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.
Decimal Expansion of a Rational number is not only terminating,
It can be either terminating like $\frac{1}{2}=0.5$ or non-terminating Repeating like $\frac{1}{3}=0.3333333......$ So option $(a)$ is not true alone.
Now we know that Non-Terminating numbers are of two types:
One is Non-Terminating Repeating and other is Non-Terminating Non-Repeating.
The Decimal expansion of a Rational number matches one of it's kind i.e Non-Terminating Repeating of Non-Terminating numbers.
So Rational number does not consist both the kinds of Non-Terminating numbers.
Hence, they are not Non-Terminating numbers.
An irrational number is always Non-Terminating in nature, but again not of both of it's kinds.
The decimal Expansion of an irrational number is Non-Terminating Non-Repeating in Nature.
So from all above points and theory we can conclude an Irrational number is Non-Terminating but Non-Repeating in nature
i.e. $\sqrt{2}=1.4142135623730...$
So, option $(d)$ is correct.
View full question & answer→MCQ 141 Mark
The number $0.318564318564318564 ........$ is:
Answer$0.318564318564318564 \ ...=0.\overline{318564}$ is a Non-terminating repeating Number.
Hence, it is a rational number.
So, correct option is $(c)$.
View full question & answer→MCQ 151 Mark
$23.\overline{43}$ when expressed in the form $\frac{\text{p}}{\text{q}}$ $\big($p, q are integers and $\text{q}\neq0\big),$ is:
- ✓
$\frac{2320}{99}$
- B
$\frac{2343}{100}$
- C
$\frac{2343}{999}$
- D
$\frac{2320}{199}$
AnswerCorrect option: A. $\frac{2320}{99}$
Let $\text{x}=23.\overline{43}=23.434343...(1)$
Now, $100\text{x}=2343.43333...(2)$
Subtracting equation $(1)$ from $(2)$, we get
$99\text{x}=2320$
$\Rightarrow\text{x}=\frac{2320}{99}$
Hence, option $(a)$ is correct.
View full question & answer→MCQ 161 Mark
$0.\overline{001}$ when expressed in the form $\frac{\text{p}}{\text{q}}$ $\big($$p, q$ are integers and $\text{q}\neq0\big),$ is:
- A
$\frac{1}{1000}$
- B
$\frac{1}{100}$
- C
$\frac{1}{1999}$
- ✓
$\frac{1}{999}$
AnswerCorrect option: D. $\frac{1}{999}$
Let $\text{x}=0.\overline{001}=0.001001001...(1)$
Now, $1000\text{x}=001.001001001...(2)$
Subtracting equation $(1)$ from $(2)$, we get
$999\text{x}=1$
$\Rightarrow\text{x}=\frac{1}{999}$
Hence, option $(d)$ is correct.
View full question & answer→MCQ 171 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.
$a.$ Is incorrect, because sum of two irrational numbers is not an irrational number always.
It can also be a rational number
i.e. if we add $2+\sqrt{3}$ and $2-\sqrt{3},$ sum comes out to be $2+\sqrt{\not\text{3}}+2-\sqrt{\not\text{3}}=4,$ which is a rational number.
$b.$ Is correct, if a rational number is added to an irrational number means to a Non$-$ terminating$-$repeating number, the sum will also be non$-$terminating and Non$-$repeating number,
i.e an irrational number.
Example: a rational number $'2\ '$ and an irrational no $'\sqrt{3}'$ is added, sum $=2+\sqrt{3}$ which is again a non$-$terminating and non-repeating number,
hence an irrational number always.
$c.$ Is incorrect, Square of an irrational number is not necessarily a rational number.
Again it can be either a rational or irrational.
i.e $(\sqrt{2})^2=2 ($Rational$)$
$(2+\sqrt{3})^2=4+3+2\times2\sqrt{3}=7+4\sqrt{3} ($irrational$)$
$d.$ Is incorrect, Sum of two rational numbers can be an integer and a rational number both.
i.e $\frac{1}{2}+\frac{1}{4}=\frac{3}{4} ($Rational number$)$
Hence, correct option is $(b)$.
View full question & answer→MCQ 181 Mark
Which of the following is irrational?
- A
$\sqrt{\frac{4}{9}}$
- B
$\frac{4}{5}$
- ✓
$\sqrt{7}$
- D
$\sqrt{81}$
AnswerCorrect option: C. $\sqrt{7}$
$a.$ Is incorrect, because $\sqrt{\frac{4}{9}}=\frac{\sqrt{4}}{\sqrt{9}}=\pm\frac{2}{3} ($Rational$)$
$b.$ Is also incorrect, as $\frac{4}{5}$ is in the form of $\frac{\text{P}}{\text{Q}}(\text{Q}\neq0), ($Rational$)$
$c.$ Is incorrect, because $\sqrt{7}$ is a non$-$terminating and Non$-$Repeating number.
$d.$ Is incorrect, because $\sqrt{81}=\pm9 ($Rational$)$
Hence, correct option is $(c)$.
View full question & answer→MCQ 191 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
$3$
- D
$30$
AnswerCorrect option: B. $\frac{3}{10}$
$\frac{1}{3}=0.33333...$ (a Non - termnating number)
Now, if we remove $3$ from denominator it will terminate.
So, if we multiply by $\frac{3}{10}$
i.e. $\frac{1}{3}\times\frac{1}{3}=\frac{1}{10}=0.1$ ( terminates after one place of decimal)
By multiplying by $\frac{1}{10},$ $3$ does not replaces.
By multiplying by $3$, we get $1$, which is not terminating after one place of decimal
And, by multiplying by $30$, we get $10$, again not terminating after one palce of decimal.
Hence, option $(b)$ is correct.
View full question & answer→MCQ 201 Mark
Which of the following is irrational?
- A
$0.15$
- B
$0.01516$
- C
$0.\overline{1516}$
- ✓
$0.5015001500015$.
AnswerCorrect option: D. $0.5015001500015$.
$a. 0.15=\frac{15}{100}=$ Rational number
$b. 0.1516=\frac{1516}{100000}=$ Rational number
$c. 0.\overline{1516}$ is a Non$-$terminating Repeating number $=$ Rational number.
$d. 0.5015001500015$. is a Non$-$terminating, Non$-$Repeating decimal number, So is a irrational number.
Hence, option $(d)$ is correct.
View full question & answer→MCQ 211 Mark
If n is a natural number, then $\sqrt{\text{n}}$ is:
- A
- B
always an irrational number.
- C
always an irrational number.
- ✓
sometimes a natural number and sometimes an irrational number.
AnswerCorrect option: D. sometimes a natural number and sometimes an irrational number.
$a.$ Is incorrect, because $\sqrt{\text{n}}$ can not be always a natural numbe
i.e. if $\text{n}=2, \ \sqrt{\text{n}}=\sqrt{2} ($not a natural no.$)$
$b.$ Is incorrect, similiarly, if $n = 2, 5, ….$ Or any odd no. or not perfect square, $\sqrt{\text{n}}=\sqrt{2},\sqrt{5},\sqrt{7}$ are Non$-$terminating and non$-$repeating, So irrational in nature, So, not always a rational number.
$c.$ Is also incorrect, $\sqrt{\text{n}}$ can aslo be rational or say a natural number.
If $n = 4, 9, 16...$ or any perfect square number then $\sqrt{\text{n}}=2,3,4...$ natural numbers.
$d.$ Is fully correct because if $n$ is any odd number or non$-$perfect square number then $\sqrt{\text{n}}$ would be irrational, but if $n$ is a perfect square number $\sqrt{\text{n}}$ then will be a natural number.
If $n = 2, 3, 5, 8 ...\sqrt{\text{n}}=\sqrt{2},\sqrt{3},\sqrt{8}... ($irrational$)$
If $n = 4, 9, 16 ... = 2, 3, 4 ... ($Natural number$)$
So, correct option is $(d)$.
View full question & answer→MCQ 221 Mark
A rational number between $\sqrt{2}$ and $\sqrt{3}$ is
Answer(c)
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 231 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 241 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 251 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 261 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 271 Mark
An irrational number between $\sqrt{2}$ and $\sqrt{3}$ is
AnswerCorrect option: D. $\sqrt{\sqrt{2} \times \sqrt{3}}$
(d)
$\sqrt{a b}$ is a real number between two real numbers a and b. Therefore, $\sqrt{\sqrt{2} \times \sqrt{3}}$ is an irrational number between $\sqrt{2}$ and $\sqrt{3}$.
View full question & answer→MCQ 281 Mark
An irrational number between 5 and 6 is
- A
$\frac{1}{2}(5+6)$
- B
$\sqrt{5+6}$
- ✓
$\sqrt{5 \times 6}$
- D
AnswerCorrect option: C. $\sqrt{5 \times 6}$
(c)
We observe that $\frac{1}{2}(5+6)$ is a rational number between 5 and 6. So, option (a) is in correct.
$\sqrt{5+6}=\sqrt{11}=3.3166247 \ldots$ is an irrational number not lying between 5 and 6.
$\sqrt{5 \times 6}$ is an irrational number lying between $\sqrt{5}$ and $\sqrt{6}$.Hence, option (c) is correct.
View full question & answer→MCQ 291 Mark
An irrational number between $\frac{1}{7}$ and $\frac{2}{7}$ is
- A
$\frac{1}{2}\left(\frac{1}{7}+\frac{2}{7}\right)$
- B
$\frac{1}{7} \times \frac{2}{7}$
- ✓
$\sqrt{\frac{1}{7} \times \frac{2}{7}}$
- D
AnswerCorrect option: C. $\sqrt{\frac{1}{7} \times \frac{2}{7}}$
(c)
We know that $\frac{a+b}{2}$ and $\sqrt{a b}$ are real numbers between any two real numbers. So $\frac{1}{2}\left(\frac{1}{7}+\frac{2}{7}\right)$ and $\sqrt{\frac{1}{7} \times \frac{2}{7}}$ are real numbers between 1/7 and 2/7.
We observe that $\frac{1}{2}\left(\frac{1}{7}+\frac{2}{7}\right)$ and $\frac{1}{7} \times \frac{2}{7}$ are rational numbers.
So, options (a) and (b) are incorrect. Clearly $\sqrt{\frac{1}{7} \times \frac{2}{7}}=\frac{\sqrt{2}}{7}$ is an irrational number between 1/7 and 2/7 So, option (c) is is correct.
View full question & answer→MCQ 301 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 311 Mark
If the number x = 1.242424 is expressed in the simplest form $\frac{p}{q}$, then p + q equals
Answer(d)
We have,
$x=1.242424 \ldots$
$\Rightarrow \quad x=1 . \overline{24}=1+0.2 \overline{4}=1+\frac{24}{99}=1+\frac{8}{33}=\frac{41}{33}$
$\therefore \quad p=41$ and $q=33 \Rightarrow p+q=41+33=74$
View full question & answer→MCQ 321 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 331 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 341 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 351 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 361 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 371 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 381 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 391 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 401 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 411 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 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
An irrational number between 2 and 2.5 is
- A
$\sqrt{11}$
- ✓
$\sqrt{5}$
- C
$\sqrt{22.5}$
- D
$\sqrt{12.5}$
AnswerCorrect option: B. $\sqrt{5}$
View full question & answer→MCQ 441 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 451 Mark
$0 . \overline{001}$ when expressed in the form $\frac{p}{q}(p, q$ are integers, $q \neq 0)$, is
- A
$\frac{1}{1000}$
- B
$\frac{1}{100}$
- C
$\frac{1}{1999}$
- ✓
$\frac{1}{999}$
AnswerCorrect option: D. $\frac{1}{999}$
View full question & answer→MCQ 461 Mark
$23 . \overline{43}$ when expressed in the form $\frac{p}{q}(p, q$ are integers $q \neq 0)$, is
- ✓
$\frac{2320}{99}$
- B
$\frac{2343}{100}$
- C
$\frac{2343}{999}$
- D
$\frac{2320}{199}$
AnswerCorrect option: A. $\frac{2320}{99}$
View full question & answer→MCQ 471 Mark
$0.3 \overline{2}$ when expressed in the form $\frac{p}{q}(p, q$ are integers $q \neq 0)$, is
- A
$\frac{8}{25}$
- ✓
$\frac{29}{90}$
- C
$\frac{32}{99}$
- D
$\frac{32}{199}$
AnswerCorrect option: B. $\frac{29}{90}$
View full question & answer→MCQ 481 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 491 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 501 Mark
Every point on a number line represents
MCQ 511 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 521 Mark
If n is a natural number, then $\sqrt{n}$ is
- A
- B
- C
always an irrational number
- ✓
sometimes a natural number and sometimes an irrational number
AnswerCorrect option: D. sometimes a natural number and sometimes an irrational number
View full question & answer→MCQ 531 Mark
The number 0.318564318564318564 ………….. Is:
MCQ 541 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 551 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 561 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 571 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 581 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 591 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 601 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 611 Mark
The product of a non-zero rational number with an irrational number is
MCQ 621 Mark
A rational number between - 3 and 3 is
MCQ 631 Mark
Which of the following numbers is irrational?
AnswerCorrect option: C. $\sqrt{8}$
View full question & answer→MCQ 641 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 651 Mark
How many digits are there in the repeating block of digits in the decimal expansion of $\frac{17}{7}$ ?
View full question & answer→MCQ 661 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 671 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 681 Mark
A number is an irrational if and only if its decimal representation is
- A
- B
non-terminating and repeating
- ✓
non-terminating and non-repeating
- D
AnswerCorrect option: C. non-terminating and non-repeating
View full question & answer→MCQ 691 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 701 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 711 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→MCQ 721 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→