Question

How many distinct positive integer-valued solutions exist to the equation $(x^2-7x+11)^{(x^2-13x+42)}=1$?

• A. 6
• B. 8
• C. 2
• D. 4

Answer 1

Answer: (A)

6

Answer 2

$(x^2-7x+11)^{(x^2-13x+42)}=1$
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: $x^2-13x+42 = 0 ⇒ x = 6,7$

case 2: $x^2-7x+11 = 1 ⇒ x^2-7x+10 = 0 ⇒ x = 2 \;or\; 5$

case 3: x^2-7x+11 = -1 ⇒ x^2-7x+12 = 0$$\ ⇒ x=4 \;or \ 3
Thus, the number of solutions are $6$.

So, the correct option is (A): $6$