DIGITAL SAT MATH PROBLEMS AND SOLUTIONS
(Part - 20)

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Problem 1 :

How can 25 be divided into two parts such that the sum of their reciprocals is 1/6?

Solution :

Problem 2 :

5x + 7y = 1

ax + by = 1

In the given pair of equations, a and b are constants. The graph of this pair of equations in the x y -plane is a pair of perpendicular lines. Which of the following pairs of equations also represents a pair of perpendicular lines?

A) 10x + 7y = 1 and ax - 2by = 1

B) 10x + 7y = 1 and ax + 2by = 1

C) 10x + 7y = 1 and 2ax - by = 1

D) 5x - 7y = 1 and ax + by = 1

Solution :

Problem 3 :

x² - 34x + c = 0

In the given equation, c is a constant. The equation has no real solutions if c > n. What is the least possible value of n?

Solution :

Problem 4 :

What is the positive solution to the given equation?

Solution :

Problem 5 :

A rocket contained 467,000 kilograms (kg) of propellant before launch. Exactly 21 seconds after launch, 362,105 kg of this propellant remained. On average, approximately how much propellant, in kg, did the rocket burn each second after launch?

A) 4,995

B) 17,243

C) 39,481

D) 104,895

Solution :

Problem 6 :

An object is kicked from a platform. The equation

h = -4.9t² + 7t + 9

represents this situation, where his the height of the object above the ground, in meters, t seconds after it is kicked. Which number represents the height, in meters, from which the object was kicked?

A) 0

B) 4.9

C) 7

D) 9

Solution :

Problem 7 :

The circle in the xy-plane is centered at the origin, O, and the coordinates of points A and B are also shown. What is the area of the shaded region?

A) 5π/6

B) 3π/2

C) 5π/3

D) 11π/6

Solution :

Problem 8 :

If the function f is defined by f[(x - 1)/2] = x² + 1, what is the value of f(-3)?

A) 32

B) 26

C) 10

D) -10

Solution :

Problem 9 :

In the figure above, the area of isosceles trapezoid OPQR is 128. What is the value of a?

Solution :

Problem 10 :

In the figure above, sin ∠BAD = 0.4 and sin ∠BCD = 0.5. If the length of AB is 10, what is the length of CD?

Solution :

Problem 11 :

The graph of y = (x² - 4)/(x - 3)(x - 2) has a vertical asymptote at x = a. What is the value of a?

Solution :

Problem 12 :

If √x + √y = 4√y, where x > 0 and y > 0, what is x in terms of y?

A) 16y

B) 9y

C) 6y

D) 4y

Solution :

Problem 13 :

Richard has seven times as many pienapples as Fred. Nathan has three times as many pineapples as Fred. Richard has 32 more pineapples than Nathan. How many pineapples does Fred have?

A) 7

B) 8

C) 11

D) 12

Solution :

Problem 14 :

During a high school basketball game, Michael scores 30 times for a total of 68 points. If he ended with x two-pointers and y three-pointers, solving which of the following systems of equations gives x and y?

A) x + y = 30 and 2x + 3y = 68

B) x + y = 68 and 2x + 3y = 30

C) x + y = 30 and 3x + 2y = 68

D) x + y = 68 and 3x + 2y = 30

Solution :

Problem 15 :

Given right triangle ABC above, which of the following is equal to c/b?

A) tan B

B) 1/tan B

C) cos B

D) 1/cos B

Solution :

Problem 16 :

A train traveling at an average speed of 80 miles per hour takes 8 hours to complete a given trip. How much time would it take the train to complete the same trip if it traveled at an average speed of 120 miles per hour?

A) 4 hours

B) 4 hours and 30 minutes

C) 5 hours and 20 minutes

D) 5 hours and 30 minutes

Solution :

Problem 17 :

5(x + 2)² – 15 = 45

Which of the following is a value of x that satisfies the equation above?

A) 4

B) –2 + 2√3

C) –2 – 2√2

D) 2 + 2√2

Solution :

Problem 18 :

In the xy-plane above, lines ℓ and m are perpendicular to each other at point P. If the graph of line m intersects the x-axis at point (-4, 0), what is the area of ΔPQR?

A) 84

B) 72

C) 42

D) 36

Solution :

Problem 19 :

In the figure above, circles O and P are tangent to line line ℓ and are tangent each other. If the radii of circles O and P are 4 and 9 respectively, what is the area of quadrilateral OPQR?

A) 39

B) 78

C) 108

D) 156

Solution :

Problem 20 :

If a and b are positive integers and 8(2^4a) = 2^2b/2³, what is b in terms of a?

A) a + 1

B) a + 2

C) 2a + 1

D) 2a + 3

Solution :

Problem 21 :

There are two different means, the arithmetic, A = (a + b)/2, and the harmonic, H = 2ab/(a + b). If the arithmetic mean is equal to harmonic mean, which of the following must be true?

A) a = 0 and b = 0

B) ab = 0

C) a = 1 and b = 1

D) a = b

Solution :