Sunday, September 20, 2009

A Practice PSAT/SAT Quiz with Strategies!!

UPDATE #2: Answers to the quiz are now provided at the bottom. If you disagree with any answers or would like clarification, don't hesitate to post a comment or send an email to dmarain "at gmail dot com".

UPDATE: No comments from my faithful readers yet -- I suspect they are giving students a chance to try these! I will post answers on Friday 9-25. However, students or any readers who would like to check their answers against mine need only email me at dmarain "at" gmail "dot" com and I will let them know how they did!

With the SAT/PSAT coming in a few weeks, I thought it would be helpful to your students to try a challenging "quiz". Most of these questions represent the high end level of difficulty and some are intentionally above the level of these tests. Then again, difficulty is very subjective. A student taking Honors Precalculus would have a very different perspective from the student starting Algebra 2!

Also, these questions can also be used to prepare for some math contests such as the THIRD MATHNOTATIONS FREE ONLINE MATH CONTEST! Yes, another shameless plug, but time is running out for your registration...

A Few Reminders For Students

(1) Do not worry about the time these take although I would suggest about 30 minutes. The idea is to try these, then correct mistakes and/or learn methods/strategies. It's what you do after this quiz that will be of most benefit!

(2) I added strategies and comments after the quiz. I suggest trying as many as you can without looking at these. Then go back, read the comments and re-try some. I will not provide answers yet!

(3) Don't forget these problems are copyrighted and cannot be reproduced for commercial use. See the Creative Commons License in the sidebar. Thank you...


1. If n is an even positive integer, how many digits of 1002n - 1002n-2 will be equal to 9 when the expression is expanded?

(A) 2 (B) 4 (C) 8 (E) 2n (E) 2n - 4

2. The sides of a triangle have lengths a, b and c. Let S represent (a+b+c)/2. Which of the following could be true?

I. S is less than c
II. S > c
III. S = c

(A) I only (B) II only (C) I and II only (D) I and III only (E) I, II and III

3. The mean, median and mode of 3 numbers are x, x+1 and x+1 respectively. Which of the following represents the least of the 3 numbers?

(A) x (B) x - 1 (C) x - 2 (D) x-3 (E) 2x - 2

4. (10/√5)500 (1/(2√5))500 = _________

5. A point P(x,y) lies on the graph of the equation x2y2 = 64. If x and y are both integers, how many such points are there?

(A) 4 (B) 8 (C) 16 (D) 32 (E 64

6. Each side of a parallelogram is increased by 50% while the shape is preserved. By what percent is the area of the parallelogram increased? __________


AB is parallel to CD , AB = 3, CD = 5, AD = BC = 4. If segments AD and BC are extended to form a triangle ABE (not shown), what would be the length of AE?

Figure not drawn to scale


1. Most students learn to substitute numbers for n here although it can be done algebraically by factoring. However, the real issue here is figuring out what the question is asking. Reading interpretation - ugh!!

2. When you are not given any information about what type of triangle it is, just choose a few special cases and draw a conclusion. O course, if one recalls a key inequality theorem from geometry, this problem can be done in short order.

3. If you don't feel comfortable setting this up algebraically (preferred method), PLUG IN A VALUE FOR x...

4. Your calculator may not be able to handle the exponent so skills are needed. The large exponent also suggests a Make it Simpler strategy. This is a "Grid-In" question so if one is guessing remember that most answers are simple whole numbers! Finally, if one knows their basic exponent rules and basic radical simplification, none of the above strategies are needed!

5. Possibilities should be listed carefully. It is possible to count these efficiently by recognizing the effect of reversals and signs. Easy to get this one wrong if not careful.

6. For those who do not remember or want to apply a key geometry concept about ratios in similar figures, there are a couple of essential test-taking strategies which all students should be aware of of:
(a) Consider a special case of a parallelogram
(b) choose particular values for the sides.
In the end, even strong students often make a different error, however. That darn ol' percent increase idea!

7. Should you skip this if you have no idea how to start? Absolutely not! Draw a complete diagram and even if you don't recognize the similar triangles, make an educated guess! It's a grid-in and there's no penalty for guessing. Further, answers tend to be positivc integers!!


1. B

2. B

3. C

4. 1

5. C

6. 125

7. 6

1 comment:

Eric Jablow said...

About problem 2:

You should also remind your students of a famous formula in which s=(a + b + c)/2 appears. Why would that formula tend to give evidence for the solution of problem 2? You should also ask your students how practical that formula is.

As a point of style, I always try to use capital letters only for the vertices of triangles and for their angles. I reserve small letters for their sides.