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CAT 2003 Slot 2 — QA Question 24

LogarithmsEasy
Passage / Data

Answer the following question based on the information given below.

Consider three circular parks of equal size with centres at A1, A2 and A3 respectively. The parks touch each other at the edge as shown in the figure (not drawn to scale). There are three paths formed by the triangles A1A2A3, B1B2B3 and C1C2C3, as shown. Three sprinters A, B, and C begin running from points A1, B1 and C1 respectively. Each sprinter traverses her respective triangular path clockwise and returns to her starting point.

If 13log3M+3log3N=1+log0.0085, then

Answer & solution

  • A

    M9=9N

  • N9=9M

  • C

    M3=3N

  • D

    N9=3M

Solution

13log3M+3log3N+1+log0.0085...(i)

Consider the R.H.S of (i):

1+log0.008⁡5=1+log⁡5log⁡0.008=1+log⁡(102)log⁡(81000)=1+log⁡10−log⁡2log⁡8−log⁡1000=1+log⁡10−log⁡23log⁡2−3log⁡10

=1+log⁡10−log⁡2−3(log⁡10−log⁡2)=1−13=23 ...(ii)

Consider the L.H.S of (i):

13log3⁡M+3log3⁡N

=log3⁡M1/3+log3⁡N3=log3⁡(M1/3N3) ....(iii)

Equating equations (ii) and (iii), we get,

log3⁡(M1/3N3)=23∴M1/3N3=32/3

Cubing both sides, we get,

M × N9 = 9

∴N9=9M

Hence, option (b).

CAT 2003 Slot 2 QA Q24: If 1 3 log 3 M + 3 log 3 N = 1 + log 0 . 008 5 , then — Solution | TheCATExam