GATE Overflow Test Series | Mixed Subjects | Test 1 | Question: 19

Question

Given the following two sets

• $A=\{1,a^{2}-2a+3\}$
• $B=\{a+2,a^{2}+2,a^{2}-3\}$

If $A\cap B=\{6\}, a$ is ________

Answer 1

From $A \cap B = \{6\},$ we know that both sets $A$ and $B$ must have $6$ as an element. Therefore, for set $A$ we have,

$a^{2}-2a+3 = 6$

$\implies a^{2}-2a-3=0$

$\implies (a-3)(a+1)=0$

$\implies a=-1,3$

If the value of $a$ is $-1,$ set $B$ is

$B=\{a+2,a^{2}+2,a^{2}-3\}$

 $=\{1,3,-2\}$
 
Then $A\cap B=\{1\}\neq \{6\}.$

If the value of $a$ is $3,$ set $B$ is
$B=\{a+2,a^{2}+2,a^{2}-3\}$

$=\{5,11,6\}$

Then $A\cap B = \{6\}.$

Thus, the value of $a$ that satisfies $A \cap B = \{6\}$ is $a=3$.

So, the correct answer is $3$.

Comments

its a cool problem