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Maths 1 Marks Question

Real Numbers · Gujarat Board (GSEB) STD 10 (GSEB - English Medium). 74 questions.

Questions · Page 2 of 2

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, $\left(2^2 \times 3^3 \times 5\right) \neq\left(2^m \times 5^n\right)$
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 $(2^m× 5^n).$
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 $(2^m× 5^n).$
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:
$473 = 11 \times 43$
$645 = 3 \times 5 \times 43$
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, $(5 × 7) ≠ (2^m× 5^n)$
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 $\left(2^n \times 5^n\right)$ ends in $5$.
Answer
$ 2 \times 5=10 $
$ \Rightarrow(2 \times 5)=10^n $
$ \Rightarrow\left(2^n \times 5^n\right)=10^n$
$\text { Since }\left(2^n \times 5^n\right) \text { when combined will give } 0 \text { as the last digit, for value of } n \text {, it can never end in } 5$
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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, $(5 \times 7 \times 13) ≠ \left(2^m \times 5^n\right)$
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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