Hard SAT Math Questions with Video Solutions

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Question 1 :

Point C is the center of the circle shown below. Points D and F both lie on the circle as shown. The perimeter of triangle CDF is 60 units. What is the area of shaded region? Round your answer to the nearest hundredth.

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Question 2 :

f(x) = ax² + 36x + c

g(x) = jx² – 57x + ℓ

The functions f(x) and g(x) are both quadratic functions, and m is a positive constant. The parabola formed by the graph of y = f(x) has a vertex with a y-coordinate of 3m. The parabola formed by the graph of y = g(x) has a vertex with a y-coordinate of m. If f(10) = f(14) and g(-3) = g(27), for how many values of x does f(x) = g(x)?

Note : a, c, j and ℓ are constants.

A) Exactly one

B) Exactly two

C) Infinitely many

D) Zero

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Question 3 :

In triangle ABC, angle B is a right angle, the measure of angle A is 34°,and the length of segment BC is 13 inches. What is the area of triangle ABC in square inches?

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Question 4 :

For two acute angles, ∠A and ∠B, cos A = sin B. The measures, in degrees of, ∠A and ∠B are 9x + 10 and 48 + 5x respectively. What is the value of x?

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Question 5 :

In triangle XYZ, the sum of the measures of angle X and angle Y is 90 degrees. The value of sin X = 5√3/12. What is the value of cos Y?

A) 5√13/12

B) 5√3/13

C) 10√5/21

D) 21/10√5

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Question 6 :

A set of five consecutive even integers is ordered from least to greatest. The sum of the first and fourth even integers is 5,860% more than the quotient of the fifth even integer and 30. The first even integer is p% less than the fifth even integer, where p is a constant. What is the value of p? (Round your answer to the nearest hundredth).

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Question 7 :

A circle is inscribed in an equilateral triangle as shown above. Point L is the center of the circle. Points A, B and C all lie on the circle and the sides of the triangle. The perimeter of the triangle is 90√3. The length of arc AC is equal to kπ. What is the value of k?

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Question 8 :

The numbers a, b and p are all positive constants where 3 < a < 20 and 8 < b < 12.If the positive difference of a and 3 is p% less than the product of 5 and b, which equation correctly expresses p in terms of a and b?

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Question 9 :

A mathematician has two right circular cones. Cone A and Cone B are similar. Cone A has a surface area of 1,225π square inches and cone B has a surface area of 1,600π square inches. The radius of Cone A is 133 inches. The height of Cone B is 345 inches. What is the slant height, in inches of Cone B?

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Question 10 :

A line intersects two parallel lines, forming four acute angles and four obtuse angles. The measure of one of these eight angles is (9x – 220)°. The sum of the four of the eight angles is s°. Which of the following could NOT be equivalent to s, for all values of x?

A) 360

B) –18x + 980

C) –36x + 1,600

D) 18x – 700

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Question 11 :

In the equation above, k is a constant. If the equation has one solution, what is the value of k?

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Question 12 :

The perimeter of a right isosceles triangle is 256 + 256√2. What is the length of the hypotenuse?

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Question 13 :

Two lines intersect at exactly one point, forming two acute angles and two obtuse angles. The measure of one of these angles is (7x − 80)°. Which of the following could NOT be the sum of the measures of any two of these angles?

A) (14x − 160)°

B) (−14x + 520)°

C) (14x – 520)°

D) 180°

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Question 14 :

A data set contains consecutive positive integers. The range of the data set is 30 and the largest integer in the data set is seven times the smallest one. If one of the integers is selected at random from the data set, what is the probability that the selected integer is a multiple of both 4 and 5?

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Question 15 :

Length of a rectangle is half of the square of its width. And also, the length of the rectangle is 8 more than 3 times the width. What is the area of rectangle?

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Question 16 :

(4x + 4)(ax – 1) – x² + 4

In the expression above, a is a constant. If the expression is equivalent to bx, where b is a constant, what is the value of b?

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Question 17 :

0 = (x – k)(7x + t) – (x – k)

In the given equation, k and t are positive constants. A solution to the equation is –22/7. What is the value of t?

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Question 18 :

If (x + y)² = 86 and xy = 12, what is the value of x² + y²?

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Question 19 :

(x – p)² + (y – q)² + (z + r)² = 0

In the equation above, p, q and r are constants. If x = 5, y = 2 and z = 3, what is the value of p + q + r?

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Question 20 :

The mass of object X is 300% of the mass of object Y, and the mass of object X is 0.05% of the mass of object Z. If the mass of object Z is q% of the mass of object Y, what is the value of q/1000?

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