DIGITAL SAT MATH PROBLEMS AND SOLUTIONS
(Part - 38)

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Problem 1 :

Solution :

Problem 2 :

The sum of the squares of two positive integers is 100 and the difference of their squares is 28. The sum of the two integers is

A) 12

B) 13

C) 14

D) 15

Solution :

Problem 3 :

If p and q are the roots of x² – 7x – 3 = 0, then what is the value of

p² – 7p + q² – 7q?

A) 6

B) 5

C) 3

D) 2

Solution :

Problem 4 :

The lines (m – 2)x + y = 4 and mx + 3y = k intersect at (2, 8). Find the value m + k.

Solution :

Problem 5 :

Solution :

Problem 6 :

Given that f(x) = (x – a)(x – b), where a and b are integer constants. If f(11) > 0, f(14) < 0 and f(17) > 0, then, which of the following could be the value of a + b?

A) 25

B) 28

C) 30

D) 31

Solution :

Problem 7 :

f(x) = a ⋅ b^x^/n

In the function f(x) above, a, b and n are integer constants. If f(4) = 4 and f(7) = 108, then what is the value of f(9)?

Solution :

Problem 8 :

In the figure above, m∠ABC = x°, m∠EDC = 43° and m∠BFE = 25°. If the lengths of AB and BC are equal, then, what is the value of x?

Solution :

Problem 9 :

y²(x – 6) – 9(x – 6)³

Which of the following is a factor of the above expression?

A) 3y

B) y – 6

C) y – 3x + 18

D) y + 2x – 6

Solution :

Problem 10 :

What is the solution set for the equation above?

A) {9/2}

B) {8}

C) {9/2, 8}

D) {3, 8}

Solution :

Problem 11 :

2x³– 6x²– kx + 3k = 0

In the equation above, k is a constant. For which of the following values of k, there are three different integer solutions for the above equation?

A) 0

B) 1

C) 4

D) 8

Solution :

Problem 12 :

If 4^x = 5^y = 20^z, what is the value of 1/x + 1/y + 1/z?

A) -1

B) 0

C) 1

D) 2

Solution :

Problem 13 :

If f(2x – 3) = 3 – 4x, then which of the following is equivalent to f(3x + 2)?

A) 2x + 3

B) -2x – 3

C) 6x + 7

D) -6x - 7

Solution :

Problem 14 :

In the xy-plane above, the area of triangle RST is 30. What is the value of k?

A) -2

B) 2

C) 3.5

D) 4

Solution :

Problem 15 :

In the triangle above, cos a = sin b. If the length of AB is 20 and the measure of angle b is π/3 radians, what is the area of the triangle?

A) 40√2

B) 40√3

C) 50√3

D) 100√3

Solution :

Problem 16 :

6ax² + 60bx + 6a = 0

In the equation above, a and b are constants. What is the greatest value of a/b for which the equation has at least one real solution?

Solution :

Problem 17 :

x^x = 2²⁰⁴⁸

What is the value of x?

A) 0

B) 40

C) 128

D) 256

Solution :

Problem 18 :

In the figure above, four congruent rectangles and a square are put together to form a larger square. The perimeter of each rectangle is 24, and the area of the smaller square is 36. If the dimensions of each rectangle are a and b as shown above, what is the value of b?

Solution :

Problem 19 :

In the figure above, two congruent circles are tangent to each other and are internally tangent to the larger circle. Line segment AB is the diameter of the larger circle. If the area of each smaller circle is 10, what is the area of the shaded region?

Solution :

Problem 20 :

f(x) = x² – 6x + 12

g(x) = k

In the equations above, f(x) ≥ g(x) for all real numbers x. If k is a constant, what is the value of k?

Solution :

Problem 21 :

2^{x + 2y} = 16

3^{3x + y} = 81

In the system of equations, what is the value of x?

A) 4

B) 3

C) 4/3

D) 3/4

Solution :

Problem 22 :

(x + y)² – (x – y)² = 72

In the equation above, x and y are positive integers. Which of the following CANNOT be the value of x + y?

A) 9

B) 11

C) 19

D) 24

Solution :

Problem 23 :

P = The average of a and b

Q = The average of b and c

R = The average of c and a

The various averages (arithmetic mean) of two of three numbers a, b and c are calculated and arranged as shown above. If P > Q > R, which of the following is true?

A) a = b = c

B) a > b > c

C) c > b > a

D) b > a > c

Solution :