DIGITAL SAT MATH PROBLEMS AND SOLUTIONS
(Part - 49)

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Problem 1 :

f(x) = ax² + bx + c

The quadratic function f is defined as shown. In the xy-plane, the two x-intercepts of the graph of f(x) are (k, 0) and (11, 0), where k is a constant. If f(4) = f(12), What is the value of k?

A) 1

B) 3

C) 5

D) 7

Solution :

Problem 2 :

The expression 8x(2x² – 5) + 2(20x – 1) is equivalent to 2(ax³– 1), where a is a constant. What is the value of a?

Solution :

Problem 3 :

h(x) = 2(x – 4)² – 32

The quadratic function h is defined as shown. In the xy-plane, the graph of y = h(x) intersects the x-axis at the points (0, 0) and (t, 0) where t is a constant. What is the value of t?

A) 1

B) 2

C) 4

D) 8

Solution :

Problem 4 :

The function f is defined by f(x) = 3^x. The function g is a increasing linear function. In the xy-plane, the graphs of y = f(x) and y = g(x) intersect at two points (a, j) and (b, k), where j < k. When g(x) > f(x), which of the following must be true?

A) x > k

B) x < j

C) x < j or x > b

D) a < x < b

Solution :

Problem 5 :

If √3x² - 6x - 24 = 0, what is the value of x² - 2√3x?

A) √3

B) 3

C) 8

D) 8√3

Solution :

Problem 6 :

A right square pyramid has a total surface area of 19200 square inches and the combined surface area of the four lateral faces of the pyramid is 9984 square inches. What is the height of the pyramid, inches?

Solution :

Problem 7 :

In the sube above, the length of diagonal PQ is 12. What is the surface area of the cube?

A) 27

B) 64

C) 150

D) 288

Solution :

Problem 8 :

On a car trip, Adam drove 50 miles more than half the number of miles Benjamin drove. Benjamin drove his car for 5 hours at the rate of 40 miles per hour. If Adam also had taken the same time, what is the speed of Adam in miles per hour?

Solution :

Problem 9 :

A cellular phone company offers two plans A and B as given below.

Plan A :

A fixed fee $25, first 30 minutes free calls and $0.20 is charged for each additional minute.

Plan B :

A fixed fee $40, first 2 hours free calls and $0.45 is charged for each additional minute.

What is the number of minutes, when the total cost is the same for both plans?

Solution :

Problem 10 :

A tiger is 500 feet from its prey. It starts to sprint towards its prey at 88 feet per second. At the same time, the prey starts to sprint in the same direction at p feet per second. The tiger catches its prey in 20 seconds. The graphs shown above represent this relationship. Based on the graphs, what is the value of p?

A) 50

B) 63

C) 70

D) 72

Solution :

Problem 11 :

The diagonal of a rectangle is 2 units more than its length. If the area of the rectangle is 48 square units, what is the perimeter of the rectangle ?

A) 28 units

B) 32 units

C) 36 units

D) 40 units

Solution :

Problem 12 :

A square is inscribed in a circle. If the radius of the circle is 5, what is the area of the region of the circle which is not covered by the square?

A) 25π – 2

B) 5(π – 5)

C) 5(5π – 1)

D) 25(π – 2)

Solution :

Problem 13 :

If the difference between a number and its positive square root is 12, what is the number?

Solution :

Problem 14 :

P(t) = b + at

Jessie purchased a micro oven for $750. After 10 years, the value of the oven will be $0. The value P of the oven during year t is modeled by the equation above, where a and b are constants. In how many years, will the value of the micro oven be decreased by $180?

A) 2.4

B) 4

C) 4.5

D) 5

Solution :

Problem 15 :

A tire repair center uses the formula above to calculate the pressure of tire, where d is the diameter of the tire. Which of the following expresses d in terms of the other variables?

Solution :

Problem 16 :

In a 3-4-5 right triangle, if the length of the hypotenuse is 17, what is the length of the perpendicular line segment drawn from one of the vertices to hypotenuse?

Solution :

Problem 17 :

x³ + px² – 12x + 4 = 0

For the above equation, if x = –2 is one of the solutions and p is a constant, what is the sum of all the solutions?

A) 5

B) 6

C) 7

D) 8

Solution :

Problem 18 :

A gym is offering a promotion where the first month is free, the second month is half off the regular price and the remaning months are regularly priced. If the regular price for a month is $24.50, which function f gives the total cost, in dollars, of x months of gym use during this promotion, where x ≥ 2?

A) f(x) = 24.50(x – 1) + 12.25

B) f(x) = 24.50(x – 2) + 12.25

C) f(x) = 24.50(x – 1) + 12.25(x – 2)

D) f(x) = 24.50(x – 2) + 12.25(x – 1)

Solution :