The number of integral solutions of the equation ( x 2 - 10 ) ( x 2 - 3 x - 10 ) = 1 is

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4 — Easy An expression $A^B=1$ in exactly three ways: (i) base $A=1$; (ii) exponent $B=0$ (with $A
e 0$); (iii) base $A=-1$ with $B$ even. Check each case for integer $x$. 1 Case base $=1$: $x^2-10=1$ $$x^2=11\ \Rightarrow\ x=\pm\sqrt{11}\quad	ext{(not integer, rejected)}$$ 2 Case exponent $=0$: $x^2-3x-10=0$ $$(x-5)(x+2)=0\ \Rightarrow\ x=5\ 	ext{or}\ x=-2$$ Both give base $x^2-10
e 0$, so valid. 2 solutions. 3 Case base $=-1$: $x^2-10=-1\Rightarrow x^2=9\Rightarrow x=\pm 3$. Need the exponent

CAT 2022 Slot 2 — QA Question 15

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