EJU EJU Mathematics Course 1 Practice 1

Question 1

A quadratic equation x² - 5x + 6 = 0 has roots α and β. What is the value of α² + β²?

Accepted Answer

13

Answer

By Vieta's formulas, α + β = 5 and αβ = 6. So α² + β² = (α + β)² - 2αβ = 25 - 12 = 13.

Question 2

If log₂(x) = 3 and log₂(y) = 5, what is log₂(x²y)?

Accepted Answer

11

Answer

log₂(x²y) = 2·log₂(x) + log₂(y) = 2(3) + 5 = 6 + 5 = 11.

Question 3

A function f(x) = x² - 4x + 7. What is the minimum value of f(x)?

Accepted Answer

3

Answer

Completing the square: f(x) = (x-2)² + 3. The minimum value is 3, occurring at x = 2.

Question 4

In triangle ABC, if angle A = 60°, b = 4, and c = 6, what is the value of side a using the law of cosines?

Accepted Answer

2√7

Answer

a² = b² + c² - 2bc·cos(A) = 16 + 36 - 2(4)(6)cos(60°) = 52 - 48(0.5) = 52 - 24 = 28. So a = √28 = 2√7.

Question 5

The sum of the first n terms of an arithmetic sequence is Sₙ = 3n² + 2n. What is the 5th term of the sequence?

Accepted Answer

29

Answer

The nth term is aₙ = Sₙ - Sₙ₋₁ = (3n² + 2n) - (3(n-1)² + 2(n-1)) = 6n - 1. For n=5: a₅ = 6(5) - 1 = 29.

EJU - Examination for Japanese University Practice Test