Digital SAT Hard Math Flashcards

If f(x) = x² - 6x + 9 and g(x) = x - 3, for what value of x does f(x)/g(x) = 7?

Answer: x = 10

f(x)/g(x) = (x-3)²/(x-3) = x-3 for x ≠ 3. Setting x - 3 = 7 gives x = 10.

A population of bacteria triples every 4 hours. If there are 200 bacteria initially, which expression gives the number after t hours?

Answer: 200 · 3^(t/4)

Every 4 hours the count multiplies by 3, so after t hours there are t/4 intervals of tripling, giving 200 · 3^(t/4).

The system of equations 3x + ky = 12 and 6x + 4y = 24 has infinitely many solutions. What is the value of k?

Answer: 2

For infinitely many solutions, the equations must be proportional. Multiplying the first by 2 gives 6x + 2ky = 24. For this to match 6x + 4y = 24, we need 2k = 4, so k = 2.

A quadratic equation has roots at x = 2 + √5 and x = 2 - √5. Which of the following could be the equation?

Answer: x² - 4x - 1 = 0

Sum of roots = 4 (coefficient of x is -4), product of roots = (2)² - (√5)² = 4 - 5 = -1 (constant term is -1).

If 2^(x+1) = 5^(x-1), what is the value of x? (Round to the nearest hundredth.)

Answer: x ≈ 4.31

Taking log of both sides: (x+1)ln2 = (x-1)ln5. Solving: x(ln2-ln5) = -ln5-ln2, so x = (ln5+ln2)/(ln5-ln2) ≈ 4.31.

A circle has center (3, -2) and passes through the point (7, 1). What is the area of the circle?

Answer: 25π

Radius = √((7-3)² + (1-(-2))²) = √(16+9) = √25 = 5. Area = π·5² = 25π.

The function h(t) = -16t² + 64t + 6 models the height (in feet) of a projectile t seconds after launch. What is the maximum height reached?

Answer: 70 feet

Maximum occurs at t = -64/(2·(-16)) = 2. h(2) = -16(4) + 64(2) + 6 = -64 + 128 + 6 = 70.

If |3x - 7| = 2x + 1, which of the following gives all solutions for x?

Answer: x = 8 or x = 6/5

Case 1: 3x-7 = 2x+1 → x = 8 (check: |17| = 17 ✓). Case 2: 3x-7 = -(2x+1) → 5x = 6 → x = 6/5 (check: |18/5-7| = 17/5, 2(6/5)+1 = 17/5 ✓).