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Question : 80 of 86

Marks: +1, -0

If

x y^{p-1}=a, x y^{q-1}=b,

x y^{r-1}=c

a^{q-r} b^{r-p} c^{p-q}

is equal to

1
0
$p+q+r-1$
None of these

Solution:

Concept:

Express

a

,

b

,

c

in terms of

x

and

y

, then simplify the exponent sum.

Explanation:

1. Given:

a = x y^{p-1}

,

b = x y^{q-1}

,

c = x y^{r-1}

.

2. Substitute into

a^{q-r} b^{r-p} c^{p-q}

:

a^{q-r} = (x y^{p-1})^{q-r} = x^{q-r} y^{(p-1)(q-r)}

b^{r-p} = (x y^{q-1})^{r-p} = x^{r-p} y^{(q-1)(r-p)}

c^{p-q} = (x y^{r-1})^{p-q} = x^{p-q} y^{(r-1)(p-q)}

3. Multiply: Exponent of

x

is

(q-r)+(r-p)+(p-q)=0

, so

x^0=1

.

4. Exponent of

y

:

S = (p-1)(q-r)+(q-1)(r-p)+(r-1)(p-q)

.

Expand:

S = (pq-pr-q+r)+(qr-qp-r+p)+(rp-rq-p+q)=0

.

So

y^0=1

.

5. Product is

1 \times 1 = 1

.

Answer:

1

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