EJU Mathematics Course 1 Practice Flashcards

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

Answer: 13

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

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

Answer: 11

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

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

Answer: 3

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

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

Answer: 2√7

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.

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

Answer: 29

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.

A bag contains 3 red and 5 blue balls. Two balls are drawn at random without replacement. What is the probability that both are red?

Answer: 3/28

P(both red) = C(3,2)/C(8,2) = 3/28.

The point P(3, -1) is reflected across the line y = x. What are the coordinates of the image P'?

Answer: (-1, 3)

Reflection across y = x swaps the x and y coordinates. So P(3, -1) maps to P'(-1, 3).