If U = {x : x ∈ Z, -8 ≤ x ≤ 8} and A = {x : x ∈ U, x² - 2x - 15 ≤ 0}, how many elements are in A′?
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Answer and explanation
Correct answer: 8
Factor the quadratic: x² − 2x − 15 = (x − 5)(x + 3). The inequality (x − 5)(x + 3) ≤ 0 holds for −3 ≤ x ≤ 5. Therefore, A = {−3, −2, −1, 0, 1, 2, 3, 4, 5}, which has 9 integers. The universal set U contains all integers from −8 through 8, so n(U) = 17. Hence n(A′) = n(U) − n(A) = 17 − 9 = 8.
Frequently asked questions
What is the correct answer to this question?
8
Why is this the correct answer?
Factor the quadratic: x² − 2x − 15 = (x − 5)(x + 3). The inequality (x − 5)(x + 3) ≤ 0 holds for −3 ≤ x ≤ 5. Therefore, A = {−3, −2, −1, 0, 1, 2, 3, 4, 5}, which has 9 integers. The universal set U contains all integers from −8 through 8, so n(U) = 17. Hence n(A′) = n(U) − n(A) = 17 − 9 = 8.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Complement of a Set and Its Properties.
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