If \(A=\{x:\,x^2-5x+6=0\}\) and \(B=\{x:\,x^2-3x+2=0\}\), what is \(A\cap B\)?
Answer and explanation
Correct answer: \(\{2\}\)
Factor the first quadratic: \(x^2-5x+6=(x-2)(x-3)\), so its roots are 2 and 3 and \(A=\{2,3\}\). Factor the second quadratic: \(x^2-3x+2=(x-1)(x-2)\), so its roots are 1 and 2 and \(B=\{1,2\}\). The intersection consists only of elements common to both sets. The only common element is 2; therefore, \(A\cap B=\{2\}\), which is option B. Option A combines elements from both sets rather than finding common elements, option C includes 3 although it is not in B, and option D is incorrect because the intersection is not empty.
Frequently asked questions
What is the correct answer to this question?
\(\{2\}\)
Why is this the correct answer?
Factor the first quadratic: \(x^2-5x+6=(x-2)(x-3)\), so its roots are 2 and 3 and \(A=\{2,3\}\). Factor the second quadratic: \(x^2-3x+2=(x-1)(x-2)\), so its roots are 1 and 2 and \(B=\{1,2\}\). The intersection consists only of elements common to both sets. The only common element is 2; therefore, \(A\cap B=\{2\}\), which is option B. Option A combines elements from both sets rather than finding common elements, option C includes 3 although it is not in B, and option D is incorrect because the intersection is not empty.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Operations on Sets (Union, Intersection, Difference).
