25 of the Hardest SAT Math Questions

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Question 1 :

x² – 12x + 7 = 0

One solution to the given equation is j – √k, where j and k are constants. What is the value of jk?

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Question 2 :

One of the two equations in a system of linear equations is given. The system has no solution. Which equation could be the second equation in this system?

7x = 105y - 133

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A) x = 15y

B) (1/7)x = 15y

C) x = 15y - 19

D) (1/7)x = 15y - 19

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Question 3 :

The measure of angle BDC, in radians, is equal to aπ/b, where a and b are constants. What is the smallest possible value of a + b?

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Question 4 :

f(x) = |8 – 5x|

The function f is defined by the given equation. For what value of k does f(k) = 6k + 3?

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Question 5 :

The given equation relates the molal boiling point elevation K, the ideal-gas constant R, the solvent boiling point T, the solvent molecular weight M, and the molal heat of vaporization H. Which equation correctly expresses the solvent boiling point in terms of the other quantities?

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Question 6 :

Given : sin (a°) = cos (b°), where a and b are constants. The number a is 70% greater than the number b. What is the value of a?

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Question 7 :

For a group of 28 or more people, an arcade charges $9 per person for the first 28 people and $15 for each additional person. How many total people were in attendance, if the group paid $447 to go to the arcade?

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Question 8 :

An artist shades in her drawing at a rate of 1.3 square inches per second. What is this rate, in square decimeters per hour? (1 inch = 0.254 decimeter). Round your answer to the nearest tenth.

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Question 9 :

In triangles PQR and XYZ, angles R and Z each have a measure of 38°. The length of side QR is equal to 11. The length of side YZ is equal to 27.5. Which additional piece of information is sufficient to prove that triangle PQR is similar to triangle XYZ?

A) The measure of angle P is 46°.

B) The measure of angle Q is 49° and the measure of angle Y is 94°.

C) The length of side PR is 14 and the length of side XZ is 35.

D) No additional information is necessary.

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Question 10 :

Point P lies on the unit circle and its position is determined by constructing an angle of 119π/6 radians with positive x-axis. What is the x-coordinate of Point P ?

A) – √3/2

B) 2√3/2

C) √3/2

D) –1/2

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Question 11 :

y = 3x² – 5x – 12

40x – 4y = a

In the given system of equations, a is a positive constant. The system has exactly one distinct real solution. What is the value of a?

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Question 12 :

If a, b and c are positive constants such that 53% of a equals 27% of b and a equals 34% of c, then what percent of b is equal to c? Round your answer to the nearest tenth. Exclude the percent sign from your answer.

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Question 13 :

If 8^{x + 1} - 2^3x = k(16^3x/4), what is the value of k?

A) 3

B) 5

C) 7

D) 8

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Question 14 :

The ratio of the speeds of two trains is 7 : 8. If the second train runs 400 miles in 5 hours, what is the speed of the first train?

A) 10 miles/hr

B) 50 miles/hr

C) 70 miles/hr

D) 80 miles/hr

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Question 15 :

Triangle ABC and PQR are similar and have the following properties.

• ∠A is the smallest angle and ∠C is the largest.

• The measure of ∠P is equal to k times the measure of ∠A, and the measure of ∠Q is equal to k times the measure ∠B.

• The perimeter of ΔPQR is 2 times the perimeter of ΔABC.

• k > 1 and PQ = 54.

Which of the following must be true?

A) k = 2

B) PQ/AB = 2

C) BC = 27

D) k = 27

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Question 16 :

If x = ²ⁿ√(4xⁿ + 21), where n is an odd positive integer constant, what is the value of xⁿ?

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Question 17 :

nx – 7k = –7x + 2n

In the equation above, n and k are constants. If the equation has no solution, what cannot be the value of n + k?

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Question 18 :

3(w – a)(w + 5) = w + 5

In the equation above, a is a conatnt. If the sum of the solutions to the equation is 34/3, what is the value of a?

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Question 19 :

The edge length, in centimeters, of cube A is 4/27 the edge length, in centimeters, of cube B. The surface area, in square centimeters, of cube B is k times the surface area, in square centimeters, of cube A. What is the value of k?

A) 64/19,683

B) 16/729

C) 729/16

D) 19,683/64

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Question 20 :

4x² + bx – 90 = (cx + m)(x – p)

Given that c, m and p are integers, which of the following must be true?

A) m is a factor of b

B) c is a factor of 90

C) c is a factor of b

D) p is a factor of 90

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Question 21 :

The results of two random samples of votes for a population are given in the table. The samples were selected from the same population, and the margins of error were calculated using the same method. Which of the following is the most appropriate reason that the margin of error for sample A is greater than the margin of error for sample B?

A) Sample A had a larger sample size.

B) Sample A had a smaller sample size.

C) Sample A had a smaller number of votes that could be recorded.

D) Sample A had a lower percent of favorable responses.

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Question 22 :

The radius and height of a cylinder are in the ratio 5 : 7 and its lateral surface area is 1750π square centimeters. What is the volume of the cyinlder, in cubic centimeters? Round your answer to the nearest hundredth.

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Question 23 :

2x² + 8x – m³ = 0

In the quadratic equation above, m is a constant. If the roots of the equation are equal, then, what is the value of m?

A) -3

B) -1

C) 1

D) -2

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Question 24 :

The sides of an equilateral triangle are shortened by 12 units, 13 units, and 14 units respectively and a right triangle is formed. Find the length of the side of equilateral triangle.

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