Conquering the Hardest SAT Math Questions

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Question 1 :

If the square of a is equal to the cube of b, where a ≥ 0 and b ≥ 0, for what value of x is √a^x equal to b?

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Question 2 :

The scatterplot shows the relationship between the number of sandwiches sold on a given day, y, at a restaurant and the number of days after February 1, x. The average rate of change, in sandwiches per day, of the number of sandwiches the retaurant sold on February 4 to the number of sandwiches the retaurant sold on February 8 is equal to a. The average rate of change, in sandwiches per day, of the number of sandwiches the restaurant sold on February 7 to the number of sandwiches the restaurant sold on February 11 is equal to b. The number b is p% greater than the number a. What is the value of p, rounded to the nearest hundredth?

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Question 3 :

The function f is defined by f(x) = ax² + bx + c, where a, b, and c are constants and 2 < b < 5. The graph of y = f(x) in the xy-plane is a parabola where f(–12) = f(7). If b is an integer, what could be the value of b – a?

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Question 4 :

The value of a video game console increased by 285% from the end of 2015 to the end of 2016. The value of the console then decreased by 21% from the end of 2016 to the end of 2017. Finally, the value of the console increased by 63% from the end of 2017 to the end of 2018. What was the net percentage increase in the value of the video game console from the end of 2015 to the end of 2018?(Round to the nearest hundredth of a percent)

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Question 5 :

The functions f and g are defined by the given equations, where 2 ≤ x ≤ 8. Which of the following equations displays, as a constant or coefficient, the minimum value of the function it defines, where 2 ≤ x ≤ 8?

I. f(x) = 0.0625(0.5)(2)^x^{+ 2}

II. g(x) = 1500(0.2)^x^{- 3}

A) I only

B) II only

C) I and II

D) Neither I nor II

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Question 6 :

On data set of 26 integers is summarized in the histogram shown. For the histogram, the first interval represents the frequency of integers greater than or equal to 70, but less than 75. The second interval represents the frequency of integers greater than or equal to 75, but less than 80, and so on. What is the largest possible value of the median?

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Question 7 :

–2x² + 8x – b/a = 0

In the equation above, a, b, and c are positive constants. The product of the solutions to the given equation is 5/7. The number b is 64% less than the number c. The number a is p% less than the number c. What is the value of p?

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Question 8 :

A quadratic function models the height, in feet, of an object above the ground in terms of the time, in seconds, after the object is launched off an elevated surface. The model indicates the object hits the ground 16 seconds after being launched; the object reaches its maximum height of 1,296 feet above the ground 7 seconds after being launched. Based on the model, at what time after the object is launched, in seconds, does the object first reach a height of 972 feet?

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Question 9 :

A 30-60-90 triangle has a perimeter of 18 + 6√3 inches. What is the triangle’s area, in square inches? Round your answer to the nearest hundredth of a square inch.

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Question 10 :

A circle in the xy-plane has the equation (x + 3)² + (y – 8)² = 25. Line k is tangent to the circle (1, 5). The equation for line k is represented by the linear function, k(x). For what value of x, does k(x) = 2?

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Question 11 :

A circle in the xy-plane has its center at (3, -7) and has a radius of 12. An equation of this circle is

x² + y² + ax + by + c = 0,

where a, b and c are constants. What is the value of c?

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Question 12 :

Sphere A has a radius of 7x and sphere B has a radius of 91x. The volume of sphere B is how many times the volume of sphere A?

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Question 13 :

One gallon of resin costs $79 and will cover 48 square feet of a countertop. The countertops in a given room have a total surface area of k square feet. Which equation represents the cost c, in dollars, of resin needed to cover the countertops twice?

A) c = 48k/79

B) c = 96k/79

C) c = 79(k/24)

D) c = 79(k/96)

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Question 14 :

For each real number r, which of the following points lies on the graph of each equation in the xy-plane for the given system?

10x + 8y = 15

25x + 20y = 37.5

A) (r, -4r/5 + 15/8)

B) (r, 5r/4 + 4/5)

C) (-4r/5 + 3/2, r)

D) (2r/5 + 10, -2r/5 + 25)

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Question 15 :

A researcher selected 2 random samples of employees in a large company to estimate the percentage of employees that planned to vote in favor of a new division. Based on the first sample, the researcher estimated that 68% of the employees would vote in favor, with an associated margin of error of 9.7%. Based on the second sample, the researcher estimated that 84% of the employees would vote in favor, with an associated margin of error of 6.7%. Assuming the margins of error were calculated in the same way, which of the following best explains why the second sample obtained a smaller margin of error than the first sample?

A) The first sample contained fewer employees than the second sample.

B) The first sample contained more employees than the second sample.

C) The first sample contained a lower percentage of residents that planned to vote in favor of a new division.

D) The first sample contained a higher percentage of residents that planned to vote in favor of a new division.

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Question 16 :

The 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) has an x-intercept at (3, 0) and a y-intercept at (0, -342). What is the value of a + b?

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Question 17 :

684 is p% greater than 9. What is the value of p?

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Question 18 :

Two numbers a and b, are each greater than zero, and the fifth root of a is equal to the cube root of b. For what value of x is a^{8x - 4} equal to b?

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Question 19 :

For the given system of equations, what is the value of 7(j - k)?

(j - k) - 16(m + l) = 689

(j - k) + 12(m + l) = 1235

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Question 20 :

A model initially estimates that there are 24,000 bacteria on a petri dish. 14 days later, the model estimates that there are 96,000 bacteria on the petri dish. Assuming exponential growth, the formula B = a(2)^xdgives the estimated number of bacteria on a petri dish, where a and x are constants and B is the number of bacteria on the petri dish d days after the initial measurement. What is the value of x?

A) 1/14

B) 1/7

C) 7

D) 14

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