Digital SAT Hard Math Practice Flashcards

The parabola y = x² − 4x + 7 and the line y = 2x + k intersect at exactly one point. What is k?

Answer: −2

Set equal: x²−6x+(7−k)=0; discriminant = 36−4(7−k) = 36−28+4k = 8+4k = 0 → k = −2.

If 5^(x+1) = 25^(x−2), what is x?

Answer: 5

25^(x−2) = 5^(2(x−2)) = 5^(2x−4); so x + 1 = 2x − 4 → 5 = x.

A chord of a circle is 16 cm long and is 6 cm from the center. What is the radius of the circle?

Answer: 10

The perpendicular from the center to a chord bisects it: half-chord = 8; r² = 6² + 8² = 36 + 64 = 100 → r = 10.

A scatterplot shows a strong negative linear relationship between hours of TV watched (x) and GPA (y). The regression line is y = −0.3x + 4.2. Predict the GPA for a student who watches 8 hours of TV per day.

Answer: 1.8

y = −0.3(8) + 4.2 = −2.4 + 4.2 = 1.8.

The function f(x) = x³ − 3x + 2 is defined on [−2, 2]. What is the absolute maximum value of f on this interval?

Answer: 4

f′(x) = 3x² − 3 = 0 → x = ±1. f(−1) = −1 + 3 + 2 = 4; f(1) = 1 − 3 + 2 = 0; f(−2) = −8 + 6 + 2 = 0; f(2) = 8 − 6 + 2 = 4. Maximum = 4.

A bag has 3 red, 4 blue, and 5 green marbles. Two marbles are drawn without replacement. What is the probability that both are blue?

Answer: 1/11

P = (4/12) × (3/11) = 12/132 = 1/11.

If the mean of x, 2x, 3x, and 4x is 20, what is the value of x?

Answer: 8

(x+2x+3x+4x)/4 = 10x/4 = 2.5x = 20; solving gives x = 8.