GATE CSE 2018 | Question: GA-7

Question

 If $pqr \ne 0$ and $p^{-x}=\dfrac{1}{q},q^{-y}=\dfrac{1}{r},r^{-z}=\dfrac{1}{p},$ what is the value of the product $xyz$ ?

• A. $-1$
• B. $\dfrac{1}{pqr}$
• C. $1$
• D. $pqr$

Comments

got 1

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same here ans should be 1

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you have type wrong question

p−x=1/q

q−y=1/r

r−z=1/p

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pqr !=0, p^-x = 1/q, q^-y = 1/r, r^-z = 1/p

Taking log both sides,

-x(log p) = -log q,

-y(log q) = -log r,

-z(log r) = -log p

=> x = log q/ log p

=> y = log r/ log q

=> z = log p/ log r

multiplying x,y and z,

(x)(y)(z) = 1

Answer 1

By simple maths:
multiply all,
p^(-x)q(-y)r^(-z) = 1/pqr
compare exponent, x==1, y==1, z==1.

xyz = 1*1*1 = 1

NOTE: (if p=q=r=1 then pqr!=0 it can take any value of x,y,z. :-P)

Comments

thnks

Answer 2

Take log on both sides and try to find out x . Similarly find y and then z.

-xlogp=log1(/q)

thus , x=(log(1/q) / log p)
Similarly find y and z.

and take the product.

Use of formula : log (m/n )=log m - log n

Answer 3

Answer is (C)

This method helps when log doesn't come to our mind in the exam : )

Here we have options as (B)1/pqr and (D)pqr

so, we try to bring some relation with p,q and r.

Given: p^-x  = 1/q =>  1/p^x  =   1/q

                            =>  (r^-z)^x =   1/q  (Given: r^-z = 1/p => p = 1/ r^-z => p^x= (1)^x/ (r^-z)^x )

                            =>   (1/q^-y )^-xz = 1/q ( Given: q^-y = 1/r => r= 1/q ^-y )

                           =>    1/(q)^xyz =  1/q  (we know 1 power anything is "1")

                           => xyz =1 (C)

Comments

p^x = q     q^y = r       r^z=p

log_pq =x      log_qr =y     log_rp =z

xyz =  log_pq * log_qr * log_rp

       = log_pr * log_rp

      = log_pp

      =   1

Answer 4

 

p^x = q     q^y = r       r^z=p 

log_pq =x      log_qr =y     log_rp =z 

xyz =  log_pq * log_qr * log_rp 

       = log_pr * log_rp 

      = log_pp 

      =   1