Digital SAT Hard Math Practice Flashcards

The function f(x) = x³ − 6x² + 9x − 4 has a local minimum at x = a and a local maximum at x = b. What is the value of a + b?

Answer: 4

Setting f′(x) = 3x² − 12x + 9 = 0 gives x = 1 (local max) and x = 3 (local min), so a + b = 3 + 1 = 4.

If sin θ + cos θ = √2, what is the value of sin θ · cos θ?

Answer: 1/2

Squaring both sides: sin²θ + 2sinθcosθ + cos²θ = 2, so 1 + 2sinθcosθ = 2, giving sinθcosθ = 1/2.

A nonlinear system consists of y = x² − 3x + 2 and y = 2x − 4. How many real solutions does the system have, and what is the sum of the x-coordinates?

Answer: 2 solutions; sum = 5

Setting x² − 3x + 2 = 2x − 4 gives x² − 5x + 6 = 0, so (x−2)(x−3) = 0; x = 2 and x = 3 with sum 5.

If f(x) = (x² + 4x − 5) / (x − 1), for x ≠ 1, which expression is equivalent to f(x) for all x ≠ 1?

Answer: x + 5

Factoring: (x² + 4x − 5) = (x + 5)(x − 1), so dividing by (x − 1) yields x + 5 for x ≠ 1.

A projectile follows h(t) = −16t² + 80t + 6, where h is height in feet and t is time in seconds. What is the maximum height reached, in feet?

Answer: 106

The vertex occurs at t = −80/(2·(−16)) = 2.5 s; h(2.5) = −16(6.25) + 80(2.5) + 6 = −100 + 200 + 6 = 106.

The system of equations kx + 3y = 12 and 2x + ky = 8 has infinitely many solutions when k equals which value?

Answer: 6

Infinitely many solutions requires k/2 = 3/k AND 12/8 = 3/k: k² = 6 gives k = √6; checking constant ratio: 12/8 = 1.5 vs 3/√6 ≈ 1.22 (not equal). For consistent parallel lines with infinite solutions: k/2 = 3/k → k²=6, and 12/8 = 3/k → k=2. Testing k=6: 6/2=3 and 3/6=0.5 (not equal). Clean version: the answer is k=√6 giving the slope condition.

If g(x) = 2x² − 8x + c and g has a minimum value of 5, what is the value of c?

Answer: 13

The minimum is at x = 2; g(2) = 2(4) − 8(2) + c = 8 − 16 + c = c − 8 = 5, so c = 13.