# Tutor profile: Hương T.

## Questions

### Subject: Geometry

ImageImage http://gmatclub.com/forum/download/file.php?id=26518 A sheet of paper ABDE is a 12-by-18-inch rectangle, as shown in Figure 1. The sheet is then folded along the segment CF so that points A and D coincide after the paper is folded, as shown in Figure 2 (The shaded area represents a portion of the back side of the paper, not visible in Figure 1). What is the area, in square inches, of the shaded triangle shown? A) 72 B) 78 C) 84 D) 96 E) 108

Area of the shape is Area of triangle ACF = 1/2*(ABxAF) AB = 12 (height of the paper) Problem turns into find AF Suppose EF =x. Also, points A and D coincide after the paper is folded, so CD = CA = AF = 18-x, BC = EF = x (ACDF is a parallelogram). Noticing that line AC is the hypotenuse of triangle ABC, we can calculate it with the information we have. 12^2+x^2=(18-x)^2 144+x^2=324-36x+x^2 36x=180 x=5 Thus, the base, AF = AC = 18-5=13 And the area of the triangle is 1/2*13*12=78 Answer is B

### Subject: Numerical Analysis

The last digit of 12^12+13^13–14^14×15^15 A. 0 B. 1 C. 5 D. 8 E. 9

Note A: Last digit of power of number has the last number is 2: Form of power: (4n) : last digit: 6 Form of power: (4n+1) : last digit: 2 Form of power: (4n+2) : last digit:4 Form of power: (4n+3) : last digit:8 => last digit of 12^12 is 6 Note B: Last digit of power of number has the last number is 3 Form of power: (4n) : last digit: 1 Form of power: (4n+1) : last digit: 3 Form of power: (4n+2) : last digit:9 Form of power: (4n+3) : last digit:7 => last digit of 13^13 is 3 Note C: Last digit of power of number has the last number is 4 Odd power: 2n+1: last digit 4 Even power: 2n: last digit 6 => last digit of 14^14 is 6 Note D: The last digit of power of number has the last number is 5 always is 5 => last digit of 15^15 is 5 Normally: we will create conclusion about this kind of question like: 6+3 - (last digit of 6*5) = 9 (wrong answer) However, because 14^14x15^15 much greater than (12^12+13^13) so the result must be negative and the last digit is (last digit of 6x5) - (6+3) = 1 The Answer is: B (1)

### Subject: Algebra

How many integrals value of x satisfy the inequality (1−x^2)(4−x^2)(9−x^2)>0 ? A 0 B 1 C 3 D 5 E Greater than 5

Solution 1: X is an integer so 9-x^2 > 4-x^2 > 1-x^2 (1−x^2)(4−x^2)(9−x^2)>0 ? <=> x^2<1 or 4<x^2<9 <=> x^2 = 0 <=> x = 0 Only 1 value : B Solution 2: Try: X = 0: Result is 36>0 Try: X = 1,2,3 : Result is 0 Try: X = 4 and above: Result is negative Only 1 value: B

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