DIGITAL SAT MATH PROBLEMS AND SOLUTIONS
(Part - 40)

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Problem 1 :

Given that f(x) = 3(x - a)(x - b)(x - c), where a, b and c are constants. When a < x < b, the value of f(x) is positive, the graph of y = f(x) in the xy-plane contains the point (r, s), where r and s are constants. If s = 8, which of the following could be true?

i) r < a

ii) b < r < c

iii) r > c

A) i only

B) iii only

C) i and iii

D) ii and iii

Solution :

Problem 2 :

In the expression above, b and c are integers. If the expression is equivalent to x + b and x ≠ b, which of the following could be the value of c?

A) 4

B) 6

C) 8

D) 10

Solution :

Problem 3 :

The graph of y = f(x) is shown. Which of the following best represents the function of f(x)?

A) x(x - 1)(x + 1)(x + 3)²

B) x(x + 1)(x - 1)(x + 3)

C) (x - 3)²(x - 1)(x + 1)

D) (x + 3)²(x + 1)(x - 1)

Solution :

Problem 4 :

Which of the following expressions is equivalent to

Solution :

Problem 5 :

3x + by = 36

ax - 7y = 24

If the given system of equations has no solution, what is the value of ab?

Solution :

Problem 6 :

The function f is defined by f(x) = |x² – 8| - |x – 20|, and the function g is defined by g(x) = |x³ – 12| - |x – 16|. If p = f(-6) and q = g(3), where p and q are constants, what is the value of 172^p^{– q}?

Solution :

Problem 7 :

For the exponential function g, the value of g(3) is k, where k is a constant. Which of the following definitions of the function g shows the value of k as the coefficient or the base?

A) g(x) = 18(1.01)^x^{- 1}

B) g(x) = 18(1.01)^x^{- 3}

C) g(x) = 18(1.01)^x^{+ 1}

D) g(x) = 18(1.01)^x^{+ 3}

Solution :

Problem 8 :

In the figure above, O and P are the centers of two semicircles of radius r. If the length of the perimeter is 8π + 16 what is the value of r?

A) 2

B) 4

C) 6

D) 8

Solution :

Problem 9 :

If x¹² = 5000 and x¹¹/y = 10, what is the value of xy?

A) 500

B) 100

C) 50

D) 10

Solution :

Problem 10 :

If f is a linear function for which f(10) – f(5) = 10, what is the value of f(20) – f(8)?

Solution :

Problem 11 :

A semicircle is shown in the xy-plane above. If the semicircle intersects the y-axis at point R, what is the value of b?

A) 3

B) 4

C) 5

D) 6

Solution :

Problem 12 :

x² + (k + 1)x + 16 = (x + h)²

In the equation above, k and h are positive constants. If the equation is true for all real numbers of x, what is the value of k?

Solution :

Problem 13 :

In the right triangle above, the value of sin C is 0.6 and the length of BC is 20. What is the length of AD?

A) 7.2

B) 8.0

C) 9.6

D) 10

Solution :

Problem 14 :

Squares ABCD and DEFG with integer-length sides of b and a respectively are shown in the figure above. If the area of the shaded region is 28, what is the area of the square ABCD?

Solution :

Problem 15 :

In the xy-plane above, points A and B lie on line ℓ. If AB = 5, what is the y-intercept of the line?

A) (0, 5/2)

B) (0, 8/3)

C) (0, 3)

D) (0, 10/3)

Solution :

Problem 16 :

A consumer analyst believes that a new car will lose 18 percent of its value every year. After n years, the value of a new car that costs $20,000 is modeled by V = 20,000 ⋅ Cⁿ, where V is the value of the car after n years. To the nearest dollar, what is the value of the car 5 years after it was purchased ?

Solution :

Problem 17 :

A) -1/4

B) 1/4

C) -1/2

D) 1/2

Solution :

Problem 18 :

In the figure above, the diameter of the semicircle is 10 and the length of CD of rectangle ABCD is 3. What is the length of BC?

A) 4

B) 6

C) 7

D) 8

Solution :

Problem 19 :

In the triangle above, the length of DB is 4 and the length of CD is 6. What is the area of triangle ACD?

A) 24

B) 27

C) 30

D) 39

Solution :

Problem 20 :

C(t) = 15 + 0.15(t – K)

The cost of using a smart phone is $15 for the first 200 minutes and $0.15 for additional minute. The cost C is modeled by the equation above, where t is the length of time in minutes and K is a constant. If a customer paid $36 for using his phone, how many minutes did he use?

A) 210

B) 340

C) 450

D) 500

Solution :