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General
Easy

Question

If l, m, n are the p to the power of text th end text end exponent comma q to the power of text th end text end exponent comma r to the power of text th end text end exponent terms of a G.P which are +ve, then open vertical bar table attributes columnalign left end attributes row cell l o g invisible function application l end cell p 1 row cell l o g invisible function application m end cell q 1 row cell l o g invisible function application n end cell r 1 end table close vertical bar equals

  1. 3    
  2. 2    
  3. 1    
  4. 0    

hintHint:

The formula for the nth term of the geometric progression is:
an = arn-1
where, a is the first term
  • r is the common ratio
  • n is the number of the term which we want to find.

The correct answer is: 0


    Given : l, m, n are the p to the power of text th end text end exponent comma q to the power of text th end text end exponent comma r to the power of text th end text end exponent terms of a G.P 
     We know that in GP 
    a subscript n space equals space a r to the power of n minus 1 end exponent

    where,
    • a is the first term
    • r is the common ratio
    • n is the number of the term which we want to find.
    Using this :
    p to the power of t h end exponent space t e r m space equals space l space equals space space equals space a r to the power of P minus 1 end exponent
q to the power of t h end exponent space t e r m space equals space m space equals space space equals space a r to the power of Q minus 1 end exponent
r to the power of t h end exponent space t e r m space equals space n space equals space space equals space a r to the power of R minus 1 end exponent


    Substituting these values in :
    rightwards double arrow open vertical bar table attributes columnalign left left left end attributes row cell log invisible function application l end cell p 1 row cell log invisible function application m end cell q 1 row cell log invisible function application n end cell r 1 end table close vertical bar equals open vertical bar table attributes columnalign left left left end attributes row cell log left parenthesis a r to the power of P minus 1 end exponent right parenthesis end cell p 1 row cell log left parenthesis a r to the power of Q minus 1 end exponent right parenthesis end cell q 1 row cell log left parenthesis a r to the power of R minus 1 end exponent right parenthesis end cell r 1 end table close vertical bar

    We know that
    rightwards double arrow log open parentheses a b close parentheses space equals space log a space plus log b space a n d
rightwards double arrow log a to the power of m space equals space m log a
    Using these formulas and further simplifying
    rightwards double arrow open vertical bar table attributes columnalign left left left end attributes row cell log a space plus left parenthesis P minus 1 right parenthesis log space r right parenthesis end cell p 1 row cell log a space plus left parenthesis Q minus 1 right parenthesis log space r right parenthesis end cell q 1 row cell log a space plus space left parenthesis R minus 1 right parenthesis log space r right parenthesis end cell r 1 end table close vertical bar

R subscript 2 space rightwards arrow R subscript 2 space minus space R subscript 1 space a n d space R subscript 3 rightwards arrow space R subscript 3 space minus space R subscript 1
    rightwards double arrow open vertical bar table attributes columnalign left left left end attributes row cell log a space plus left parenthesis P minus 1 right parenthesis log space r right parenthesis end cell p 1 row cell left parenthesis Q minus P right parenthesis log space r end cell cell q space minus p end cell 0 row cell left parenthesis R minus P right parenthesis log space r end cell cell r space minus space p end cell 0 end table close vertical bar

T a k i n g space o u t space c o m m o n space f r o m space R subscript 2 space a n d space R subscript 3
rightwards double arrow left parenthesis Q minus P right parenthesis left parenthesis R minus P right parenthesis open vertical bar table attributes columnalign left left left end attributes row cell log a space plus left parenthesis P minus 1 right parenthesis log space r right parenthesis end cell p 1 row cell log space r end cell 1 0 row cell log space r end cell 1 0 end table close vertical bar

I f space 2 space r o w s space a r e space e q u a l space t h e n space t h e i r space d e t e r m i n a n t space i s space 0

rightwards double arrow left parenthesis Q minus P right parenthesis left parenthesis R minus P right parenthesis open vertical bar table attributes columnalign left left left end attributes row cell log a space plus left parenthesis P minus 1 right parenthesis log space r right parenthesis end cell p 1 row cell log space r end cell 1 0 row cell log space r end cell 1 0 end table close vertical bar space equals space 0

    Thus,  open vertical bar table attributes columnalign left end attributes row cell l o g invisible function application l end cell p 1 row cell l o g invisible function application m end cell q 1 row cell l o g invisible function application n end cell r 1 end table close vertical bar equals 0

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