DIGITAL SAT MATH PROBLEMS AND SOLUTIONS
(Part - 12)

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Problem 1 :

x² + (2p – 1)x + p² = 0

In the quadratic equation above, p is a constant. If the roots of the equation are real, then which of the following is true?

A) p ≥ 1

B) p ≤ 4

C) p ≥ 1/4

D) p ≤ 1/4

Solution :

Problem 2 :

For the function f, f(0) = 86, and for each increase in x by 1, the value of f (x) decreases by 80%. What is the value of f (2)?

Solution :

Problem 3 :

In the xy-plane, a line with equation 2y = 4.5 intersects a parabola at exactly one point. If the parabola has equation y = -4x² + bx, where b is a positive constant, what is the value of b ?

Solution :

Problem 4 :

The scatterplot shows the relationship between two variables, x and y. A line of best fit for the data is also shown.

At x = 32, which of the following is closest to the y-value predicted by the line of best fit?

A) 0.4

B) 1.5

C) 2.4

D) 3.3

Solution :

Problem 5 :

In a group, 40% of the items are red. Of all the red items in the group, 30% also have stripes. What percentage of the items in the group are red and have stripes?

A) 10%

B) 12%

C) 70%

D) 75%

Solution :

Problem 6 :

The density of a certain type of wood is 353 kilograms per cubic meter. A sample of this type of wood is in the shape of a cube and has a mass of 345 kilograms. To the nearest hundredth of a meter, what is the length of one edge of this sample?

A) 0.98

B) 0.99

C) 1.01

D) 1.02

Solution :

Problem 7 :

The expression 90y⁵ - 54y⁴ is equivalent to ry⁴(15y - 9), where r is a constant. What is the value of r?

Solution :

Problem 8 :

If x/y = 4 and 24x/ny = 4, what is the value of n?

Solution :

Problem 9 :

A bowl contains 20 ounces of water. When the bowl is uncovered, the amount of water in the bowl decreases by 1 ounce every 4 days. If 9 ounces of water remain in this bowl, for how many days has it been uncovered?

A) 3

B) 7

C) 36

D) 44

Solution :

Problem 10 :

If 9(4 - 3x) + 2 = 8(4 - 3x) + 18, what is the value of 4 - 3x?

A) -16

B) -4

C) 4

D) 16

Solution :

Problem 11 :

In the xy-plane above, line ℓ is defined by the equation 3x + 4y = 12 and OP is perpendicular to line ℓ. What is the length of OP?

A) 1.6

B) 2.4

C) 3.2

D) 3.6

Solution :

Problem 12 :

If f(x) = 2^x and f(x – 1) = f(x) – 8, what is the value of x?

A) 2

B) 4

C) 6

D) 8

Solution :

Problem 13 :

If f[(x - 2)/2] = 5x - 10, which of the following represents f(x)?

A) f(x) = 10x - 5

B) f(x) = 10x + 5

C) f(x) = 10x

D) f(x) = 10x + 10

Solution :

Problem 14 :

In the figure above, PA and PB are tangent to circle O with radius AO = 6. If the measure of ∠APB is 60°, which of the following is closest to the area of the shaded region?

A) 20

B) 25

C) 28

D) 30

Solution :

Problem 15 :

f(x) = x² + mx + 4

In the function above, f(x) > 0 for all values of x. Which of the following could be the value of m?

A) 8

B) 6

C) 4

D) 2

Solution :

Problem 16 :

A small shop can manufacture 4 windows every 5 hours. At that rate, how long will it take to manufacture 7 windows?

A) 8 hr

B) 8 hr 15 min

C) 8 hr 30 min

D) 8 hr 45 min

Solution :

Problem 17 :

If X is 20% of Y and Y is 30% of Z, then what percent of Z is X?

A) 5%

B) 6%

C) 8%

D) 12%

Solution :

Problem 18 :

The graph of the function g in the xy-plane is shown above. If f is another function defined in the same xy-plane and f(1) = 1, then g could be which of the following?

A) f - 1

B) f - 2

C) f + 1

D) f + 2

Solution :

Problem 19 :

This year, Roger beat Rafael in 25% of their tennis matches. If Rafael won 18 matches, how many matches did Roger win?

A) 2

B) 3

C) 4

D) 6

Solution :

Problem 20 :

One calorie is equivalent to 4.184 joules of energy. One calorie also represents the amount of energy required to raise the temperature of 1 gram of water by 1° Celsius. Based on this information, raising the temperature of 1 gram of water by 120° Fahrenheit would require how many joules of energy, to the nearest hundredth? (1° Celsius = 33.8° Fahrenheit)

A) 12.76

B) 13.88

C) 14.32

D) 14.85

Solution :