GATE ECE 2016 Set 1 | GA | Question: 9

Question

If $q^{-a}  = \frac{1}{r}$ and $r^{-b}=\frac{1}{s}$ and $s^{-c} = \frac{1}{q}$ ,the value of $abc$ is______.

• A. $(rqs)^{-1}$
• B. $0$
• C. $1$
• D. $r+q+s$

Comments

solve using log

Answer 1

Multiply all terms,we will get more faster than above method.

$q^{-a}\times r^{-b}\times s^{-c}=\frac{1}{r}\times\frac{1}{s}\times\frac{1}{q}$

$q^{-a}\times r^{-b}\times s^{-c}=q^{-1}\times r^{-1}\times s^{-1}$

Compare the terms

$a=1,b=1,c=1$

So,$abc=1$

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Suppose they ask $a+b+c=?$

We can simply put the values of $a,b$ and $c$ and get the answer

$a+b+c=1+1+1=3$

Correct Answer: $C$

Comments

Option C

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ans is 1

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@Rahul Jain25 Nice approach.Really astute move.

Answer 2

Given that,

$\large q^{-a} = \dfrac{1}{r},\;r^{-b}= \dfrac{1}{s},\; s^{-c} =\dfrac{1}{q}$

Now, $\large q^{-a} = \dfrac{1}{r}$

$\large\Rightarrow \dfrac{1}{q^{a}} =\dfrac{1}{r}$

$\large\Rightarrow q^{a} = r$

Similarly, $\large r^{b} = s$ and $\large s^{c} = q$

In $\large q^{a} = r$, we replace $q$ by $\large s^{c}$ and $r$ by $\large s^{\dfrac{1}{b}}$

We get, $\large s^{ca} = s^{\dfrac{1}{b}}$

Since, the bases are equal, we can work with the exponents.

So, we get, $ac$ = $\dfrac{1}{b}$ $\Rightarrow abc = 1$

Comments

@Samujjal Das Really clever approach.

Answer 3

$q^{-a} = \dfrac{1}{r} , \quad r^{-b} = \dfrac{1}{s} , \quad s^{-c} = \dfrac{1}{q}$

$q^{-a} = \dfrac{1}{r} \Rightarrow log(q^{-a}) = log\Big(\dfrac{1}{r}\Big) \Rightarrow - \, a \cdot log(q) = - \, log(r) \Rightarrow \boxed{a = \dfrac{log(r)}{log(q)}}$

$r^{-b} = \dfrac{1}{s} \Rightarrow log(r^{-b}) = log\Big(\dfrac{1}{s}\Big) \Rightarrow - \, b \cdot log(r) = - \, log(s) \Rightarrow \boxed{b = \dfrac{log(s)}{log(r)}}$

$s^{-c} = \dfrac{1}{q} \Rightarrow log(s^{-c}) = log\Big(\dfrac{1}{q}\Big) \Rightarrow - \, c \cdot log(s) = - \, log(q) \Rightarrow \boxed{c = \dfrac{log(q)}{log(s)}}$

$abc = \dfrac{log(r)}{log(q)} \times \dfrac{log(s)}{log(r)} \times \dfrac{log(q)}{log(s)}$

$\boxed{abc = 1}$

Answer 4

Correct Answer: C

Comments

Great !!!!!!