By solving the defining equations step by step and carefully using intersection and union, we determine that the set (P ∩ S) ∪ Q contains exactly three elements.

Question statement and notation

Let ℝ be the set of all real numbers.

The sets are defined as:

• P = { x ∈ ℝ : (x − 1)(x² + 1) = 0 }
• Q = { x ∈ ℝ : x² − 9x + 2 = 0 }
• S = { x ∈ ℝ : x = 5y for some y ∈ ℝ }

The task is to find the number of elements in the set (P ∩ S) ∪ Q.

Step 1: Find set P

To find P, solve the equation (x − 1)(x² + 1) = 0.

• From x − 1 = 0 ⇒ x = 1.
• From x² + 1 = 0 ⇒ x² = −1, which has no real solution.

Therefore, the only real solution is x = 1, so P = {1}.

Step 2: Describe set S

The set S is given by S = { x ∈ ℝ : x = 5y for some y ∈ ℝ }, which means S is the set of all real multiples of 5.

To compute P ∩ S, check whether the element 1 (the only element of P) belongs to S. If 1 ∈ S, there must exist a real y such that 1 = 5y, giving y = 1/5, which is a real number. Hence 1 ∈ S.

Therefore, P ∩ S = {1}.

Step 3: Find set Q

To find Q, solve the quadratic equation x² − 9x + 2 = 0 using the quadratic formula.

The roots are:

x = [9 ± √(9² − 4·1·2)] / 2 = [9 ± √(81 − 8)] / 2 = [9 ± √73] / 2.

Thus Q contains two distinct real numbers:

• α = (9 + √73) / 2
• β = (9 − √73) / 2

Hence Q = { (9 + √73) / 2, (9 − √73) / 2 }.

Step 4: Form (P ∩ S) ∪ Q

We already have:

• P ∩ S = {1}
• Q = { (9 + √73) / 2, (9 − √73) / 2 }

The union is:

(P ∩ S) ∪ Q = {1} ∪ { (9 + √73) / 2, (9 − √73) / 2 } = { 1, (9 + √73) / 2, (9 − √73) / 2 }.

All three elements are distinct real numbers, so the cardinality of (P ∩ S) ∪ Q is 3, meaning the set contains exactly three elements.

Explanation of all options

• (A) exactly two elements – Incorrect:
  The union set has three distinct elements, not two.
• (B) exactly three elements – Correct:
  (P ∩ S) ∪ Q = {1, (9 + √73) / 2, (9 − √73) / 2}, which has three elements.
• (C) exactly four elements – Incorrect:
  Neither P nor Q nor their combinations yield four distinct real solutions; only three unique elements appear in the final union.
• (D) infinitely many elements – Incorrect:
  Although S is infinite, P ∩ S = {1} is finite, and the union with Q (which is finite) remains finite.