CAT 2020 Exam Shift 1 Solved Paper

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Question : 59 of 76

Marks: +1, -0

How many distinct positive integer-valued solutions exist to the equation

(x^{2}-7 x+11)^{(x^{2}-13 x+42)}=1 ?

6
8
2
4

Solution:

(x^{2}-7 x+11)^{(x^{2}-13 x+42)}=1

We know if

a^{b}=1

⇒ a=1

and

{b}

is any number
or

a=-1

and

b

is even

a>0

and

b

is 0
case

1: x^{2}-13 x+42=0 ⇒ x=6,7

case

2: x^{2}-7 x+11=1 ⇒ x^{2}-7 x+10=0 ⇒ x=2

or 5
case

3: x^{2}-7 x+11=-1 ⇒ x^{2}-7 x+12=0

⇒ {x}=4

or 3
Hence number of solutions are 6

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