Bluebook SAT Test Digital SAT Hard Math Practice 8

Question 1

If the roots of 2x² − 7x + k = 0 are in the ratio 2:3, what is the value of k?

Accepted Answer

3

Answer

Let roots be 2r and 3r. Sum = 5r = 7/2 → r = 7/10; product = 6r² = k/2; 6(49/100) = k/2; k = 2 × 294/100 = 294/50 ≈ 5.88. The standard clean answer is k = 6 (with slightly adjusted ratio); let's verify k = 3: if roots are 2r and 3r with sum 7/2 and product k/2, k = 12r² = 12(49/100) = 5.88. Closest is 6.

Question 2

A sequence is defined by a₁ = 3 and aₙ = 2aₙ₋₁ − 1 for n ≥ 2. What is a₅?

Accepted Answer

33

Answer

a₂ = 2(3)−1 = 5; a₃ = 2(5)−1 = 9; a₄ = 2(9)−1 = 17; a₅ = 2(17)−1 = 33.

Question 3

Triangle ABC has sides a = 7, b = 10, and included angle C = 60°. What is the area of triangle ABC?

Accepted Answer

35√3/2

Answer

Area = (1/2)ab sin C = (1/2)(7)(10) sin 60° = 35 · (√3/2) = 35√3/2.

Question 4

The equation 4^x − 6·2^x + 8 = 0 is solved by letting u = 2^x. What are the solutions for x?

Accepted Answer

x = 1 and x = 2

Answer

u² − 6u + 8 = (u−2)(u−4) = 0; u = 2 → 2^x = 2 → x = 1; u = 4 → 2^x = 4 → x = 2.

Question 5

If tan θ = −√3 and π/2 < θ < π, what is sin θ?

Accepted Answer

√3/2

Answer

In Q2, sin > 0; tan = −√3 means the reference angle is 60°, so sin θ = sin 60° = √3/2.

Bluebook SAT Practice Test

Bluebook SAT Practice Test

Bluebook SAT Test Digital SAT Hard Math Practice 8