If \(A=\{x\in\mathbb{R}:x^2-4x+3>0\}\) and \(B=\{x\in\mathbb{R}:x>1\}\), what is \(A\cap B\)?
Answer and explanation
Correct answer: \((3,\infty)\)
Factor the quadratic: \(x^2-4x+3=(x-1)(x-3)\). Because the quadratic opens upward, it is positive outside its roots, so A is \((- infty,1)\cup(3, infty)\). Set B contains numbers greater than 1, represented by \((1, infty)\). Their common part is therefore only the interval \((3, infty)\). The endpoints 1 and 3 are excluded because the inequality is strict and the roots make the expression zero. Hence option A is correct.
Frequently asked questions
What is the correct answer to this question?
\((3,\infty)\)
Why is this the correct answer?
Factor the quadratic: \(x^2-4x+3=(x-1)(x-3)\). Because the quadratic opens upward, it is positive outside its roots, so A is \((- infty,1)\cup(3, infty)\). Set B contains numbers greater than 1, represented by \((1, infty)\). Their common part is therefore only the interval \((3, infty)\). The endpoints 1 and 3 are excluded because the inequality is strict and the roots make the expression zero. Hence option A is correct.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Operations on Sets (Union, Intersection, Difference).
