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MATHEMATICS [4 marks sum]

Indices · ICSE STD 9 (ICSE - English Medium). 15 questions.

Questions

[4 marks sum]

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15 questions · timed · auto-graded

Question 14 Marks
If $1960=2^a \times 5^b \times 7^c$, find the values of $a, b, c$.
Hence, calculate the value of $\left(2^{-a} \times 5^{-c} \times 7^b\right)$.
Answer
$(a=3, b=1, c=2), \frac{7}{200}$
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Question 34 Marks
If $\frac{9^n \times 3^2 \times 3^n-(27)^n}{3^{3 m} \times 2^3}=3^{-3}$, prove that $(m-n)=1$.
Answer
self
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Question 44 Marks
If $a, b, c$ are positive real numbers, show that:
\[\sqrt{a^{-1} b} \cdot \sqrt{b^{-1} c} \cdot \sqrt{c^{-1} a}=1\]
Answer
self
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Question 54 Marks
If $a b c=1$, prove that : $\frac{1}{1+a+b^{-1}}+\frac{1}{1+b+c^{-1}}+\frac{1}{1+c+a^{-1}}=1$.
Answer
self
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Question 64 Marks
If $1960=2^a \times 5^b \times 7^c$, find the values of $a, b, c$.
Hence, calculate the value of $\left(2^{-a} \times 5^{-c} \times 7^b\right)$.
Answer
$(a=3, b=1, c=2), \frac{7}{200}$
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Question 84 Marks
If $\frac{9^n \times 3^2 \times 3^n-(27)^n}{3^{3 m} \times 2^3}=3^{-3}$, prove that $(m-n)=1$.
Answer
self
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Question 94 Marks
If a, b, c are positive real numbers, show that:
$\sqrt{a^{-1} b} \cdot \sqrt{b^{-1} c} \cdot \sqrt{c^{-1} a}=1$
Answer
self
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Question 104 Marks
If $a b c=1$, prove that : $\frac{1}{1+a+b^{-1}}+\frac{1}{1+b+c^{-1}}+\frac{1}{1+c+a^{-1}}=1$.
Answer
self
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Question 114 Marks
Prove that : $\frac{1}{1+x^{b-a}+x^{c-a}}+\frac{1}{1+x^{a-b}+x^{c-b}}+\frac{1}{1+x^{b-c}+x^{a-c}}=1$.
Answer
self
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Question 124 Marks
Prove that: $\frac{a^{-1}}{\left(a^{-1}+b^{-1}\right)}+\frac{a^{-1}}{\left(a^{-1}-b^{-1}\right)}=\frac{2 b^2}{\left(b^2-a^2\right)}$
Answer
self
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Question 134 Marks
Prove that:
$\left(\frac{x^a}{x^b}\right)^{(a+b-c)} \cdot\left(\frac{x^b}{x^c}\right)^{(b+c-a)} \cdot\left(\frac{x^c}{x^a}\right)^{c+a-b}=1$.
Answer
self
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Question 144 Marks
Prove that:
$\left(\frac{x^a}{x^b}\right)^{\frac{1}{a b}} \cdot\left(\frac{x^b}{x^c}\right)^{\frac{1}{b c}} \cdot\left(\frac{x^c}{x^a}\right)^{\frac{1}{a c}}=1$
Answer
self
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Question 154 Marks
Prove that:
$\left(\frac{x^a}{x^b}\right)^{(a+b)} \cdot\left(\frac{x^b}{x^c}\right)^{(b+c)} \cdot\left(\frac{x^c}{x^a}\right)^{(c+a)}=1$
Answer
self
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[4 marks sum] - MATHEMATICS STD 9 Questions - Vidyadip