COMC COMC Number Theory and Algebra 2 — Questions and Answers
Question 1: If x + y = 5 and xy = 6, what is the value of x² + y²?
- 13 (Correct answer)
- 11
- 19
- 25
Correct answer: 13
Using x² + y² = (x+y)² − 2xy = 25 − 12 = 13.
Question 2: For what value of k does kx² − 6x + 3 = 0 have exactly one real solution?
- 3 (Correct answer)
- 2
- 4
- 1
Correct answer: 3
Equal roots require discriminant = 0: (−6)² − 4(k)(3) = 36 − 12k = 0, giving k = 3.
Question 3: If the roots of x² − px + q = 0 are 3 and 5, what is p² − 4q?
- 4 (Correct answer)
- 1
- 9
- 16
Correct answer: 4
By Vieta's formulas p = 8 and q = 15, so p² − 4q = 64 − 60 = 4.
Question 4: What is the maximum value of the expression 6x − x² − 5?
- 4 (Correct answer)
- 3
- 6
- 1
Correct answer: 4
Completing the square gives −(x−3)² + 4, so the maximum value is 4 at x = 3.
Question 5: If a/b = 3/4 and a + b = 28, what is the value of a − b?
- −4 (Correct answer)
- 4
- −7
- 7
Correct answer: −4
Setting a = 3k and b = 4k gives 7k = 28, so k = 4, a = 12, b = 16, and a − b = −4.
Question 6: What is the value of (1 + √3)(1 − √3)?
- −2 (Correct answer)
- 2
- −4
- 4
Correct answer: −2
By the difference of squares identity: (1+√3)(1−√3) = 1² − (√3)² = 1 − 3 = −2.
COMC COMC Number Theory and Algebra 2
If x + y = 5 and xy = 6, what is the value of x² + y²?