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What is the difference between the hypotenuse’s length of the right triangle with 2 shorter sides of 10 and 24 and the hypotenuse’s length of the right triangle with 2 shorter sides of 7 and 24?

A. 1 B. 2 C. 3 D. 4 E. 5

==>For Pythagorean Theorem, 5:12:13=10:24:26 and 7:24:25 appear most frequently. Thus, the length of hypotenuse each becomes 26 and 25, and the difference becomes 26-25=1.

In the x- y plane, there are 4 points (0,0), (0,4), (6,4), and (6,0). If these 4 points makes a rectangle, what is the probability that x+y<4? A. 1/2 B. 1/3 C. 1/4 D. 1/5 E. 2/5

ANS: As you can see from the diagram below, the correct answer is 1/3. Hence, B is our answer choice.

There is a circle inscribed in a square, shown on the above figure. If the length of the square is 2, what is the area of one of the 4 regions shaded?

A. \(1-π\) B. \(2-π\) C. \(4-π\) D. \(1-(\frac{π}{2})\) E. \(1-(\frac{π}{4})\)

==> The area of the square-area of the \(circle=2^2- π=4- π\), and since it asks for one of the 4 regions shaded, you get \(\frac{4- π}{4}=1-(\frac{π}{4})\).

What is the scope including 1/11+1/12+1/13+......+1/20?

A 1/6~1/5 B. 1/5~1/4 C. 1/4~1/3 D. 1/3~1/2 E. 1/2~1

==>The sum of consecutive reciprocal number sequence is decided by the first number and the last number. Thus, from 10/20=1/11+1/12+.....+1/20<1/11+1/12+.....+1/20<1/11+1/12+.....+1/20=10/11, you get 1/2=10/20<1/11+1/12+.....+1/20<10/11<1, which becomes 1/2~1.

Therefore, the answer is E. Answer: E
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In the x- y plane, there are 4 points (0,0), (0,4), (6,4), and (6,0). If these 4 points makes a rectangle, what is the probability that x+y<4? A. 1/2 B. 1/3 C. 1/4 D. 1/5 E. 2/5

ANS: As you can see from the diagram below, the correct answer is 1/3. Hence, B is our answer choice.

As you can see from the diagram above, the area of the red rectangle is 24 and the area of the triangle below green line is 8. The ratio of the area of the triangle to that of the red rectangle is 1/3. Hence, B is our answer choice.
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The area of the red rectangle is 24 and the area of the triangle below green line is 8. The ratio of the area of the triangle to that of the red rectangle is 1/3.
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As you can see from the diagram above, the area of the red rectangle is 24 and the area of the triangle below the green line is 8. The ratio of the triangle area to the red rectangle area is 8/24 or 1/3. Hence, B is our answer choice.
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As you can see from the diagram below, the area of the red rectangle is 24 and the area of the triangle below green line is 8. The ratio of the area of the triangle to that of the red rectangle is 1/3. Hence, B is our answer choice.

The standard deviation of which of the following is equivalent with that of \({m, r, p, n}\)?

A. {\({2m, 2r, 2p, 2n}\)} B. {\({m+2, r+2, p+2, n+2}\)} C. {\({|m|, |r|, |p|, |n|}\)} D. {\({\frac{1}{m}, \frac{1}{r}, \frac{1}{p}, \frac{1}{n}}\)} E. {\({m^2, r^2, p^2, n^2}\)}

==> For standard deviation, if the elements move in parallel, they have the same standard deviation. Thus, the standard deviation of {m, r, p, n} is equal to the standard deviation of {m+2, r+2, p+2, n+2}.

Therefore, the answer is B. Answer: B
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