1 Marks Question

Question 511 Mark

Without actual division, show that the following rational numbers is a non-terminating repeating decimal:$\frac{73}{\big(2^2\times3^3\times5\big)}$

Answer

$\frac{73}{\big(2^2\times3^3\times5\big)}$We know 2, 3 or 5 is not a factor of 73, so it is in its simplest form.
Moreover,$ (2^2 \times 3^3 \times 5) \neq (2^m \times 5^n)$
Hence, the given rational is non-terminating repeating decimal.

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Question 521 Mark

Classify the following number as rational or irrational:$\big(2-3\sqrt5\big)$

Answer

Let $2-3\sqrt5$ be rational.
Hence 2 and $2-3\sqrt5$ are rational.
$\therefore2-\big(2-3\sqrt5\big)=2-2+3\sqrt5$
$=3\sqrt5=$ rational $[\because$ Difference of two rational is rational$]$
$\therefore\frac{1}{3}\times3\sqrt5=\sqrt5=$ rational $[\because$ Product of two rational is rational$]$
This contradicts the fact that $\sqrt5$ is irrational.
The contradiction arises by assuming $2-3\sqrt5$ is rational.
Hence, $2-3\sqrt5$ is irrational.

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Question 531 Mark

State whether the given statement is true of false:
The sum of a rational and and irrational is irrational.

Answer

True.

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Question 541 Mark

Very-Short-Answer Questions:
Is it possible to have two numbers whose HCF is 25 and LCM is 520?

Answer

No, it is not possible since the LCM has to be a multiple of the HCF.
520 is not a multiple of 25

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Question 551 Mark

Without actual division, show that the following rational numbers is a terminating decimal. Express in decimal form:$\frac{23}{\big(2^3\times5^2\big)}$

Answer

$\frac{23}{2^3\times5^2}=\frac{23\times5}{2^3\times5^3}$$=\frac{115}{1000}=0.115$
We know either 2 or 5 is not a factor of 23, so it is in its simplest form.
Moreover, it is in the form of $(2^m\times 5^n).$
Hence, the given rational is terminating.

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Question 561 Mark

Without actual division, show that the following rational numbers is a terminating decimal. Express in decimal form:$\frac{15}{1600}$

Answer

$\frac{15}{1600}=\frac{15}{2^6\times5^2}=\frac{15\times5^4}{2^6\times5^6}$$=\frac{9375}{1000000}=0.009375$
We know either 2 or 5 is not a factor of 15, so it is in its simplest form.
Moreover, it is in the form of $(2^m\times 5^n).$
Hence, the given rational is terminating.

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Question 571 Mark

Classify the following number as rational or irrational:$\pi$

Answer

$\pi$ is an irrational number because it is a non-repeating and non-terminating decimal.

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Question 581 Mark

Classify the following number as rational or irrational:$\sqrt{21}$

Answer

$\sqrt{21}=\sqrt{3}\times\sqrt{7}$ is an irrational number because $\sqrt{3}$ and $\sqrt{7}$ are irrational and prime numbers.

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Question 591 Mark

Very-Short-Answer Questions:
Give an example of two irrationals whose product is rational.

Answer

Consider the irrational numbers, $\frac{1}{\sqrt2}$ and $\sqrt2$
$\frac{1}{\sqrt2}\times\sqrt{2}=1$
which is rational number.

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Question 601 Mark

The following numbers are irrational?
$\sqrt[3]{6}$

Answer

$\sqrt[3]{6}=\sqrt[3]{2}\times\sqrt[3]{3}$ is irrational.

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Question 611 Mark

Find the simplest form of:$\frac{473}{645}$

Answer

Prime factorisation of 473 and 645 is:
473 = 11 × 43
645 = 3 × 5 × 43
Therefore, $\frac{473}{645}=\frac{11\times\text{43}}{3\times5\times43}=\frac{11}{15}$
Thus, simplest form of $\frac{473}{645}$ is $\frac{11}{15}.$

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Question 621 Mark

Without actual division, show that the following rational numbers is a non-terminating repeating decimal:$\frac{9}{35}$

Answer

$\frac{9}{35}=\frac{9}{5\times7}$We know either 5 or 7 is not a factor of 9, so it is in its simplest form.
Moreover, (5 × 7) ≠ $(2^m\times 5^n).$
Hence, the given rational is non-terminating repeating decimal.

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Question 631 Mark

The following numbers are irrational?
$\frac{22}{7}$

Answer

$\frac{22}{7}$ is rational because it is in the form of $\frac{\text{p}}{\text{q}},\ \text{q}\neq0$

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Question 641 Mark

Classify the following number as rational or irrational:3.1416

Answer

3.1416 is a rational number because it is a terminating decimal.

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Question 651 Mark

Very-Short-Answer Questions:
Show that there is no value of n for which $(2^m\times 5^n).$ends in 5.

Answer

$2 \times 5 = 10$
$\Rightarrow (2 \times 5) = 10^n$
$\Rightarrow (2^n \times 5^n) = 10^n$
Since $(2^N \times 5^n).$ when combined will give 0 as the last digit, for value of n, it can never end in 5

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Question 661 Mark

Prove that $\frac{1}{\sqrt3}$ is irrational.Hint: $\frac{1}{\sqrt3} =\frac{1}{\sqrt3}\times\frac{\sqrt3}{\sqrt3}=\frac{1}{3}.\sqrt3$

Answer

Let $\frac{1}{\sqrt3}$ be rational.
$\therefore\frac{1}{\sqrt3}=\frac{\text{a}}{\text{b}},$ where a, b are positive integers having no common factor other than 1
$\therefore\sqrt3=\frac{\text{b}}{\text{a}}\dots(1)$
Since a, b are non-zero integers, $\frac{\text{b}}{\text{a}}$ is rational.
Thus, equation (1) shows that $\sqrt3$ is rational.
This contradicts the fact that $\sqrt3$ is rational.
The contradiction arises by assuming $\sqrt3$ is rational.
Hence, $\frac{1}{\sqrt3}$ is irrational.

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Question 671 Mark

Give an example of two irrationals whose sum is rational.
HINT: Take $\big(2+\sqrt3\big)$ and $\big(2-\sqrt3\big).$

Answer

Let $\big(2+\sqrt3\big),\big(2-\sqrt3\big)$ be two irrationals.
$\therefore\big(2+\sqrt3\big)+\big(2-\sqrt3\big)=4=$ rational number

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Question 681 Mark

Classify the following number as rational or irrational:1.535335333...

Answer

1.535335333... is an irrational number because it is a non-terminating and non-repeating decimal.

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Question 691 Mark

Classify the following number as rational or irrational:$\sqrt{6}$

Answer

Let $\sqrt6=\sqrt2\times\sqrt3$ be rational.
Hence, $\sqrt2,\ \sqrt{3}$ are both rational.
This contradicts the fact that $\sqrt2,\ \sqrt{3}$ are irrational.
The contradiction arises by assuming $\sqrt6$ is rational.
Hence, $\sqrt6$ is irrational.

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Question 701 Mark

Very-Short-Answer Questions:
If a and b are relatively prime then what is their HCF?

Answer

If a and b are relatively prime, it means they have no common factor other than 1.So, the HCF(a, b) = 1

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Question 711 Mark

State whether the given statement is true of false:
The sum of two irrationals is an irrational.

Answer

False.
Counter example:
$2+\sqrt3$ and $2+\sqrt3$ are two irrational numbers. But their sum is 4, which is a rational number.

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Question 721 Mark

State whether the given statement is true of false:
The product of two rationals is always rational.

Answer

True.

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Question 731 Mark

Without actual division, show that the following rational numbers is a non-terminating repeating decimal:$\frac{64}{455}$

Answer

$\frac{64}{455}=\frac{64}{5\times7\times13}$We know 5, 7 or 13 is not a factor of 64, so it is in its simplest form.
Moreover, (5 × 7 × 13) ≠ $(2^m\times 5^n).$
Hence, the given rational is non-terminating repeating decimal.

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Question 741 Mark

Classify the following number as rational or irrational:$3\sqrt{7}$

Answer

Let $3\sqrt7$ be rational.
$\therefore\frac{1}3{}\times3\sqrt{7}=\sqrt{7}=$ rational $[\because$ Product of two rational is rational$]$
This contradicts the fact that $\sqrt7$ is irrational.
The contradiction arises by assuming $3\sqrt7$ is rational.
Hence, $3\sqrt7$ is irrational.

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MATHS 1 Marks Question