Problem Solving Select Many GRE Sample Questions 4

GRE Problem Solving Select Many Sample Questions

1. Question:

Which of the following is true for the equations 11/u-7/v=1 and 9/u-4/v=6? Indicate all correct statements.

• A. The equations have a unique solutions
• B. The equations represent parallel lines
• C. u<v
• D. v<u
• E. u+v=5/6
Correct Answer: A, D and E

Explanation:

11/u-7/v=1 and 9/u-4/v=6

Put x = 1/u and y = 1/v

11x - 7y = 1 ...(1)

9x - 4y = 6 ...(2)

Multiply (1) by 4 and (2) by 7 and subtract one from the other

44x - 28y - 63x + 28y = 4 - 42

- 19x = -38

x = 38/19=2

Putting in (1), we get

11(2) - 7y = 1

7y = 22-1

y = 21/7 = 3

x = 2, y = 3

u = 1/x = 1/2 and v = 1/y = 1/3

The equations have a unique solution.

Option A is true and B is false.

v

u+v = 1/2+1/3 = (3+2)/6 = 5/6

Option E is true.

2. Question:

If x/a+y/b=2 and ax-by = a^2-b^2, then which of the following statements is true? Indicate all correct statements.

• A. ax+by=2
• B. bx-ay=0
• C. bx+ay = 2
• D. x/b+y/a = (a^2-b^2)
• E. x/b+y/a = (a^2+b^2)/ab

[a^2=a*a]

Explanation:

x/a+y/b=2 ...(1)

ax-by=a^2-b^2 ...(2)

Multiply (1) by a^2 and subtract (2) from it

ax+a^2y/b- ax + by = 2a^2 -a^2 + b^2

a^2y/b+by = a^2+b^2

(a^2+b^2)y/b = (a^2+b^2)

y = b

Put y = b in (1)

x/a+b/b=2

x/a = 2-1 = 1

x = a

x=a and y = b

bx-ay=ba-ab=0

x/b+y/a = a/b+ b/a

= (a^2+b^2)/ab

Options B and E are true and A, C and D are false.

3. Question:

The difference between two numbers is 5 and the difference between their squares is 65. Which of the following is true? Indicate all correct options.

• A. The sum of the numbers is 13
• B. The sum of their squares is 100
• C. The product of the numbers is 40
• D. The difference of their cubes is 665
• E. The product of their squares is 1296
Correct Answer: A, D and E

Explanation:

Let the two numbers be x and y, x>y

x-y = 5 ...(1)

x^2-y^2=65 ...(2)

Put x = y+5 in (2)

(y+5)^2-y^2=65

y^2+25+10y-y^2=65

10y = 65 - 25 = 40

y = 4

x = y+5 = 4+5 = 9

The sum of the numbers = x+y = 9+4= 13

The sum of their squares = x^2+y^2

= 9^2+4^2 = 81+16 = 97

The product of the numbers = xy=9*4=36

The difference of their cubes = x^3-y^3

= 9^3-4^3 = 729 - 64 = 665

The product of their squares = x^2*y^2

= 9^2*4^2 = 81*16 = 1296

Options A, D and E are true.