DIGITAL SAT MATH PROBLEMS AND SOLUTIONS
(Part - 47)

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Problem 1 :

(ax + b)(5x² - bx + 4) = 20x³ - 9x² - 2x + 12

The equation is above is true for all x, where a and b are constants. What is the value of ab?

A) 18

B) 20

C) 24

D) 40

Solution :

Problem 2 :

Function f is defined by f(x) = -a^x + b, where a and b are constants. In the xy-plane, the graph of y = f(x) - 12 has a y-intercept at (0, -75/7). The product of a and b is 320/7. What is the value of a?

Solution :

Problem 3 :

The table shows the distribution of students in a certain college by age group. If a person in this college is chosen at random, which of the following is closest to the probability of selecting a student who is 35 years or older given that the person is at least 23 years old?

A) 0.08

B) 0.29

C) 0.33

D) 0.47

Solution :

Problem 4 :

f(x) = -2x² + bx + c

In the given quadratic function, a and c are constants. The graph of y = f(x) is a parabola that has a vertex at the point (h, k), where h and k are constants. If k < 0 and f(-5) = f(7), which of the following must be true?

I. c < -1

II. b ≥ 3

A) I only

B) II only

C) I and II

D) Neither I or nor II

Solution :

Problem 5 :

A certain park has an area of 11,863,808 square yards. What is the area in square miles of this park? (1 mile = 1,760 yards)

A) 1.96

B) 3.83

C) 3,444.39

D) 6,740.8

Solution :

Problem 6 :

The xy-plane above shows the graphs of two quadratic functions and a rectangle. Points A, B, C and D lie on the graphs of y = x² and y = -x² repsectively. If the area of rectangle ABCD is 108, what is the length of BC?

Solution :

Problem 7 :

5s – 2t – 1 = –a

–8s + bt – 2 = 2

In the system of equations above, a and b are constants. If the system has infinitely many solutions, what is the value of a?

Solution :

Problem 8 :

In the figure above, circle O is tangent to the sides of triangle ABC. If AB = 10 and BC = 6, what is the circumference of the circle to the nearest tenth?

Solution :

Problem 9 :

The complete graph of the function f is shown above. Which of the following are equal to 2?

I. f(-2)

II. |f(-1)|

III. f(4.7)

A) I only

B) I and III only

C) II and III only

D) I, II and III

Solution :

Problem 10 :

x² – 4x + 5 = (x – 1)(x – 2) + ax + b

In the equation above, a and b are constants. If the equation is true for all values of x, what are the values of a and b?

A) a = –3, b = –1

B) a = 3, b = –1

C) a = –1, b = 3

D) a = 3, b = 1

Solution :

Problem 11 :

The function y = x(8 – x) is graphed in the xy-plane. The length of AB of isosceles triangle ABC is 4 and AB is parallel to the x-axis. What is the area of the triangle?

Solution :

Problem 12 :

In the figure above, DE || FG || AC and AF = FD = DB. if the area of AFGC is 20, what is the area of the shaded region?

Solution :

Problem 13 :

A piece of iron rod costs $60. If the rod was 2 meters shorter and each meter costs $1 more, the cost would remain unchanged. What is the length of the rod?

Solution :

Problem 14 :

In the figure above, if line ℓ has a slope of –½, what is the area of the shaded region, in square units?

A) 28

B) 32

C) 36

D) 40

Solution :

Problem 15 :

For all integers n, if n is defined by the equation

Which of the following is equivalent to *10?

A) *3

B) *9

C) *20

D) *90

Solution :

Problem 16 :

x² + y² – 2x – 2y = 7

y = k

In the system of equations above, k is a constant. For which of the following values of k, does the system of equations have exactly two real solutions?

A) k = 6

B) k = 5

C) k = 4

D) k = 3

Solution :

Problem 17 :

In the right triangle ABC above, AC = 10 and the value of sin A = 0.4. What is the length of DC?

A) 1.6

B) 2.4

C) 2.5

D) 2.8

Solution :

Problem 18 :

In the xy-plane above, the area of triangle PRS is 10. What is the value of k?

A) 4

B) 5

C) 6

D) 7

Solution :

Problem 19 :

In the figure above, OP and OS are radii and the length of AC of rectangle ABCO is 10. If the measure of angle is ACO is 60°, which of the following is the closest to the area of the shaded region?

A) 30

B) 32

C) 35

D) 40

Solution :