SAT Math: Challenging Problems #1

In my work as an SAT tutor, I spend time with some extremely bright students.  They keep me on my toes, for sure.  One of my current students has asked me to periodically come up with “some hard problems … like the ones you’d see as the last problem in a section.” These are the problems we’ll being doing as part of our work this week.  Give them a try and see what you think.  The answers appear at the bottom of this post.

1)  In the diagram below, the circle with center at D has an area of 36 pi .  Triangles ABC and DEF are equilateral.  If segment BC is tangent to the circle at G, what is the area of the yellow shaded region?

Challenge 1 Number 1

(A)   9\sqrt{3}
(B)  12\sqrt{3}
(C)  24\sqrt{3}
(D)  36\sqrt{3}
(E)  48\sqrt{3}

2)  On a recent physics test, the x juniors in the class had an average (arithmetic mean) score of 85 and the y seniors had an average (arithmetic mean) score of 94.  If the average (arithmetic mean) for the entire class was 90, and the class consists only of juniors and seniors, what is the ratio of x to y?

(A)  2:3
(B)  3:4
(C)  4:5
(D)  5:4
(E)  3: 2

3)  In the diagram below, the function f(x) is defined to be f(x) = x2 – 4x + 3.  If g(x) is the function defined by g(x) = f(x + h) + k.  What is the value of h + k?

Challenge 1 Number 3

(A)  – 9
(B)  – 2
(C)  – 1
(D)  1
(E)   6

4)  In a deck of cards, each card has a different number greater than 10 on it.  Five cards are dealt and the average (arithmetic mean) of the numbers on those cards is 22.  If Larry has the card with the largest number of the five on it, what is the largest possible number that could appear on Larry’s card?

(A)  22
(B)  42
(C)  60
(D)  66
(E)  106

5)  A cube is inscribed inside a sphere so that each vertex of the cube touches one point on the sphere.  If the sphere has a radius of 6 inches, what is the surface area, in square inches, of the cube?

(A)   28\sqrt{3}
(B)  48\sqrt{3}
(C)  192\sqrt{3}
(D)  288
(E)  432

If you have questions about these problems or anything else to do with the SAT, leave a comment below or send me an email at .

1)  (D)
2)  (C)
3)  (A)
4)  (C)
5)  (D)

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