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19 questions · timed · auto-graded

Question 21 Mark
Find the sub-triplicate ratio of 216: 343
Answer
Sub-triplicate ratio 216: 343 $=\sqrt[3]{216}: \sqrt[3]{343}=6: 7$
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Question 31 Mark
Find the sub-duplicate ratio of $9x^2a^4: 25y^6b^2$
Answer
The sub-duplicate ratio of $9 x^2 a^4: 25 y^6 b^2=$$\sqrt{9 x^2 a^4}: \sqrt{25 y^6 b^2}=3 x a^2: 5 y^3 b$
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Question 51 Mark
Find the duplicate ratio of $2 \sqrt{2}: 3 \sqrt{5}$
Answer
Duplicate ratio of $2 \sqrt{2}: 3 \sqrt{5}=(2 \sqrt{2})^2:(3 \sqrt{5})^2=8: 45$
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Question 61 Mark
If a : b = c : d, prove that: 5a + 7b : 5a – 7b = 5c + 7d : 5c – 7d.
Answer
Given $\frac{a}{b}=\frac{c}{d}$
$\Rightarrow \frac{5 a}{7 b}=\frac{5 c}{7 d}$ (Multiplying each side by 5/7)
$\Rightarrow \frac{5 a+7 b}{5 a-7 b}=\frac{5 c+7 d}{5 c-7 d}$ (By componendo and dividendo)

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Question 71 Mark
Using properties of proportion solve for x:
$\frac{3 x+\sqrt{9 x^2-5}}{3 x-\sqrt{9 x^2-5}}=5$
Answer
$\frac{3 x+\sqrt{9 x^2-5}}{3 x-\sqrt{9 x^2-5}}=5$
Applying componendo and dividendo
$\frac{3 x+\sqrt{9 x^2-5}+3 x-\sqrt{9 x^2-5}}{3 x \sqrt{9 x^2-5}-3 x+\sqrt{9 x^2-5}}=\frac{5+1}{5-1} $
$ \frac{6 x}{2 \sqrt{9 x^2-5}}=\frac{6}{4} $
$ \frac{3 x}{\sqrt{9 x^2-5}}=\frac{3}{2}$
$ \frac{x}{\sqrt{9 x^2-5}}=\frac{1}{2}$
Squaring both sides
$\frac{x^2}{9 x^2-5}=\frac{1}{4}$
$4 x^2-9 x^2=-5$
$-5 x^2=-5 $
$5 x^2=5 $
$ x^2=1 $
$x=\sqrt{1} $
$ x=1$
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Question 81 Mark
Find the reciprocal ratio of $\frac{x}{3}: \frac{y}{7}$
Answer
reciprocal ratio of $\frac{x}{3}: \frac{y}{7}=\frac{1}{\frac{x}{3}}: \frac{1}{\frac{y}{7}}=\frac{3}{x}: \frac{7}{y}=\frac{\frac{3}{x}}{\frac{7}{y}}=\frac{3 y}{7 x}=3 y: 7 x$
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Question 101 Mark
Find the sub-triplicate ratio of $x^3: 125 y^3$
Answer
Sub-triplicate ratio of $x^3: 125 y^3=125 y^3=\sqrt[3]{x^3}: \sqrt[3]{125 y^3}=x: 5 y$
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Question 111 Mark
Find the sub-triplicate ratio of 64: 27
Answer
Find the sub-triplicate ratio of $64: 27=\sqrt[3]{64}: \sqrt[3]{27}=4: 3$
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Question 121 Mark
Find the sub-duplicate ratio of $(x-y)^4:(x+y)^6$
Answer
Sub-duplicate ratio of $(x-y)^4:(x+y)^6$
$=\sqrt{(x-y)^4}: \sqrt{(x+y)^6}=(x-y)^2:(x+y)^3$
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Question 141 Mark
Find the triplicate ration of $\frac{m}{2}: \frac{n}{3}$
Answer
Triplicate ratio of $\frac{m}{2}: \frac{n}{3}$
$=\left(\frac{m}{2}\right)^3:\left(\frac{n}{3}\right)^3$
$=\frac{m^3}{8}: \frac{n^3}{27}$
$=\frac{\frac{m^3}{8}}{\frac{n^3}{27}}=27 m^3: 8 n^3$
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Question 161 Mark
Find the duplicate ratio of $3 \sqrt{3}: 2 \sqrt{5}$
Answer
Duplicate ratio of $3 \sqrt{3}: 2 \sqrt{5}=(3 \sqrt{3})^2:(2 \sqrt{5})^2=27: 20$
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Question 181 Mark
Find the compound ration of $2a: 3b$, $mn: x^2$​​​​​​​ and $x: n$
Answer
Required compound ratio $=2 a \times mn \times x : 3 b \times x^2 \times n ^`$
$=\frac{2 a \times m n \times x}{3 b \times x^2 \times n}$
$=\frac{2 a m}{3 b x}=2 a m: 3 b x$
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Question 191 Mark
Find the compound ratio of $2 : 3, 9: 14$ and $14: 27$
Answer
Required compound ration = 2 x 9 x 14 : 3 x 14 x 27
$=\frac{2 \times 9 \times 14}{3 \times 14 \times 27} $
$ =\frac{2}{9}=2: 9$
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[1 Mark Question Answer] - Mathematics STD 10 Questions - Vidyadip