CAT 2023 Quant was dominated by Arithmetic followed by Algebra. In Arithmetic, the questions were dominated by topics like Speed-time-distance, Mixture and Alligations. This year, there was a surprise. The questions from Geometry were relatively on the lower side as compared to the previous years. There were 8 TITA Qs this year. Overall this section was at a medium level of difficulty.
Question 1 : For a real number x, if 12,log3(2x−9)log34, and log5(2x+172)log54 are in an arithmetic progression, then the common difference is
12,log3(2x−9)log43,log5(2x+172)log45are in AP.
∵AB=A
12,(2x−9),(2x+172)are in A.P
2,(2x−9),(2x+172)are in A.P
If loglogA,loglogB,loglogC all in A.P, then A,B,C will be in G.P
∴2,(2x−9),(2x+172)are in G.P
(2x−9)2=2×(2x+172)
22x+81−9×2x+1=2x+1+17
22x−10×2x+1+64=0
(2x)2−20(2x)+64=0
Let y=2x
y2−20y+64=0
y2−16y−4y+64=0
y=16 or y=4
2x=16 or 2x=4
x=4 or x=2
if x=2;(22−9)=(−5)
But, since the argument of the log can't be negative, x=4
∴ The common ratio of the G.P is 2x−92=24−92=16−92=72
∴ The common difference of the AP=(72)
The question is " For a real number x, if 12,log3(2x−9)log34, and log5(2x+172)log54 are in an arithmetic progression, then the common difference is "
Choice C is the correct answer.
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