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U and T can produce 10,000 units in x hours when working together at their constant rates. If U can produce 10,000 units in 4x/3 hours alone at the constant rate, in how many hours can T produce 10,000 units alone at the constant rate, in terms of x? A. 5x/2 B. 4x C. 2x D. x E. x/2

==> In case of work rate questions, if it is “together and alone”, you solve it reciprocally. In other words, if you assume the time it takes for T to produce 10,000 units alone as t hrs, you get t=4x from 1/(4x/3)+1/t=1/x. Therefore, B is the answer. Answer: B
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If n is the product of 3 consecutive integers, which of the following must be true?

I. a multiple of 2 II. a multiple of 3 III. a multiple of 4

A. I only B. II only C. III only D. I and II E. II and III

==> The product of 3 consecutive integers always become the multiple of 6, because the product always contains 3 and 2. Thus, in this question, it always becomes the multiple of 6 that contain 3 and 2, I and II are the answer. Therefore, the answer is D. III does not work because it becomes 1*2*3*=6, hence it cannot be the multiple of 4.

Which of the following inequalities satisfy (x-4) (x+1) >0? A. -1<x<4 B. -4<x<1 C. x<-1, 4<x D. x<-4, 1<x E. -1<x<1

==> From (x-4)(x+1)>0, you get x-4>0, x+1>0 or x-4<0, and x+1<0. Then, x-4>0 and x+1>0 is x>4 and x>-1, which becomes x>4, and x-4<0 and x+1<0 becomes x<-1 from x<4 and x<-1, hence the answer is x<-1 or 4<X. Therefore, the answer is C. Answer: C
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A triangle’s area is S=\(\sqrt{( ((p-) )(p-) )(p-c))}\), where a, b and c are the three side lengths of the triangle and p=(a+b+c)/2. When a=5, b=6 and c=7, what is the area of the triangle? A. 4√6 B. 5√6 C. 6√6 D. 7√6 E. 8√6

==> If you substitute p=(5+6+7)/2=9 into S=\(\sqrt{√(p(p-a)(p-b)(p-c))}\), it becomes S=\(\sqrt{(9*4*3*2)}\)=6√6. Therefore, the answer is C.

What is the measurement of one inner angle of the regular octagon?

A.\(115^o\) B. \(120^o\) C. \(125^o\) D. \(130^o\) E. \(135^o\)

==> An octagon’s total number of triangle is (8-2), so the total inner angle is 6*\(180^o\)=\(1,480^o\), and since it is an octagon, if you divide by 8, it becomes \(1,480^o\)/8=\(135^o\)r. Therefore, the answer is E.

The standard deviation of which of the following is equivalent to 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. {1/m, 1/r, 1/p, 1/n} E. \({m^2, r^2, p^2, n^2}\)

==> Parallel translation doesn’t change the standard deviation. Therefore, the answer is B. In other words, each elements of B moves +2 in parallel, so the standard deviation doesn’t change.

There are 5 females and 3 males. If 3 people are selected randomly, what is the probability of at least one male getting selected from the people?

A. 2/3 B. 3/31 C. 23/28 D. 25/28 E. 27/28

==> 1-probability of not selecting one male=1-(probability of 3 selected people being all females)=1-(5C3/8C3)=1-(5/28)=23/28. Therefore, the answer is C. Answer: C
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G(x) is the greatest integer less than or equal to x and L(x) is the smallest integer greater than or equal to x. Which of the following can be the value of L(1.1)-G(1.1)?

A. -2 B. -1 C. 0 D. 1 E. 2

==> You get L(1.1)-G(1.1)=2-1=1, and therefore the answer is D.

What is the remainder when \(3^5^0\) is divided by 4?

A. 0 B. 1 C. 2 D. 3 E. -1

==> The units digit of \(3^n\) is the repetition of \(3-->9-->7--->1-->3-->9-->7-->1\), so you get \(50=4*12+2\). Thus, from \(3^5^0=3^4^*^1^2^+^2\) --> \(~3^2=~9\), the units digit becomes 9, and if you divide it by 4, from \(9=4*2+1\), the remainder becomes 1. Therefore, the answer is B. Answer: B
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If x-1, x+6, x+7 are 3 side lengths of a right triangle, what is the value of x?

A. 4 B. 5 C. 6 D. 7 E. 8

==> According to the Pythagorean Theorem, if you expand \((x-1)^2+(x+6)^2=(x+7)^2\), you get \(x^2-2x+1+x^2+12x+36=x^2+14x+49\), and then when you simplify, you get \(x^2-4x-12=0\). From \((x-6)(x+2)=0\), \(x=6\), and therefore the answer is C.

m and n are positive integers. If an equation \(x^2+(m-2)x-(n-2)=0\) has only one root, what is the value of mn?

==> For ax^2+bx+c=0 to have only one root, it needs to become discriminant \((D)=b^2-4ac=0\). Therefore, you get \(D=(m-2)^2-4*1*(-(n-2))=0\), and \((m-2)^2+8(n-2)=0\). Only m=n=2 satisfies this because m and n are positive integers.