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The next set of PS questions. I'll post OA's with detailed explanations after some discussion. Please, post your solutions along with the answers.

1. The length of the diagonal of square S, as well as the lengths of the diagonals of rhombus R are integers. The ratio of the lengths of the diagonals is 15:11:9, respectively. Which of the following could be the difference between the area of square S and the area of rhombus R?

I. 63 II. 126 III. 252

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

4. The functions f and g are defined for all the positive integers n by the following rule: f(n) is the number of positive perfect squares less than n and g(n) is the number of primes numbers less than n. If f(x) + g(x) = 16, then x is in the range:

A. 30 < x < 36 B. 30 < x < 37 C. 31 < x < 37 D. 31 < x < 38 E. 32 < x < 38

x + y = 350 x = 25*p, y = 25*q where p and q may not have any other factors in common

Substituting into the equation we have: p+q = 14 The only solutions for pair (p,q) in which p and q do not have common factors are: (1,13), (3,11),(5,9)

There are exactly 3 pairs: (25,325), (75, 275), (125, 225)

1. x = 1 ? Not necessary. Let x = 3, and y = 0, then x^y = 1 2. x = 1 and y = 0. Not necessary (see item 1) 3. x = 1 or y = 0. Not necessary. Let x = -1, y=2

Answer: E

Last edited by caioguima on 18 Apr 2013, 03:48, edited 2 times in total.

10. If x is not equal to 0 and x^y=1, then which of the following must be true?

I. x=1 II. x=1 and y=0 III. x=1 or y=0

A. I only B. II only C. III only D. I and III only E. None

Sol: We need to find condition which will be true under all conditions

St 1 when x =1 then X^ (any value of y) will always give us 1. So A, D are possible answers St 2 X=1 and y=0, Here and means both conditions simultaneously. But Y need not be zero as long as x= 1. Y can take any value as long as x=1 and hence B is ruled out

St 3 x=1 or y=0 here "or" means any one of the condition if true then we get x^y=1 which is true. Consider x=1 and y= 32 ----> x^y = 1, consider x= 1, y= -3, then x^y= 1 Similarly X= 2 but y=0 then 2^0 =1 Option C and D can be the possible answer

Ans Option D......

PS: I think there is a catch since St 1 covers only value of x whereas St 3 covers for all possible cases of (x,y) and I was tempted to go for option C alone as the answer. Let's see
_________________

“If you can't fly then run, if you can't run then walk, if you can't walk then crawl, but whatever you do you have to keep moving forward.”

9. What is the 101st digit after the decimal point in the decimal representation of 1/3 + 1/9 + 1/27 + 1/37?

A. 0 B. 1 C. 5 D. 7 E. 8

Sol: 1/3 + 1/9 + 1/27 + 1/37

1/3= 0.333333 1/9= 1/(3*3)= 0.3333333/3-----> 0.1111111 1/27= 1(3^3)= 0.037037037 1/37= 0.027027027 So if we add up 0.333 0.111 0.037 0.027 Sum is ( .508508508)...99th Digit will be 8,100th digit will be 5 and 101st digit will be 0...

I guessed it under timed condition......Would like to have faster way.

Ans should A
_________________

“If you can't fly then run, if you can't run then walk, if you can't walk then crawl, but whatever you do you have to keep moving forward.”

10. If x is not equal to 0 and x^y=1, then which of the following must be true?

I. x=1 II. x=1 and y=0 III. x=1 or y=0

A. I only B. II only C. III only D. I and III only E. None

All are Could be true questions, but none of them is MUST be TRUE\ 1- X can be any other value than 1 2- NoT Necessarily 3- x=2 and y =0; x=1 and and y=1 can satisfy Hence, E
_________________

Q7) The greatest common divisor of two positive integers is 25. If the sum of the integers is 350, then how many such pairs are possible?

Answer is A

Since the GCM is 25, the 2 positive integers should have 5^2 as the only common factor. The 2 potential positive integers can be found by dividing 350 / 25 and finding unique prime factors that add up to the divisor.

Q1) The length of the diagonal of square S, as well as the lengths of the diagonals of rhombus R are integers. The ratio of the lengths of the diagonals is 15:11:9, respectively. Which of the following could be the difference between the area of square S and the area of rhombus R?

I. 63 II. 126 III. 252

Answer is E

S diagonal : R diagonal 1 : R diagonal 2 = 15:11:9

Let's assume x is the unknown multiplier and the ratio of diagonals are 15x:11x:9x

\(Area of S =\frac{15x}{\sqrt{2}}\)

\(Area of R = 1/2 * 11x * 9x\)

\(Area of S - Area of R = \frac{225x - 99x}{2} = 63x\)

And I, II, II are multiples of 63 when x = 1, 2, 3
_________________

7. The greatest common divisor of two positive integers is 25. If the sum of the integers is 350, then how many such pairs are possible?

A. 1 B. 2 C. 3 D. 4 E. 5 My Answer C Let the numbers be 25a and 25b. for these numbers to have 25 as the GCD, a and b must be co-prime i.e have only 1 as the common factor. 25a+25b=350 => a+b =14 , a and b are co prime. Hence the pairs of a and b are => (1,13) , (3,11) ,(5,9).

8. The product of a positive integer x and 377,910 is divisible by 3,300, then the least value of x is:

A. 10 B. 11 C. 55 D. 110 E. 330

My answer is D (337910*x)/3300 , reduced will become (12667 *x)/110 and 12667 is not divisible by any of the factors of 110( 2 or 5 or any combinations of both). Hence X has to be 110. At least.

9. What is the 101st digit after the decimal point in the decimal representation of 1/3 + 1/9 + 1/27 + 1/37?

A. 0 B. 1 C. 5 D. 7 E. 8

My answer A 1/3 + 1/9 + 1/27 = 13/27. = .481481… 1/37=.027027… 13/27 + 1/27 = .508508.. Hence 101st digit is 0.(3*33 +2=> hence the second digit of the recurring decimal which is 0)

10. If x is not equal to 0 and x^y=1, then which of the following must be true?

I. x=1 II. x=1 and y=0 III. x=1 or y=0

A. I only B. II only C. III only D. I and III only E. None My answer C. Since the question is what MUST be true I would go with C. as X can be any integer apart form 0 and Y can be 0. It’s not a must for X=1 and y=0.

8. The product of a positive integer x and 377,910 is divisible by 3,300, then the least value of x is:

A. 10 B. 11 C. 55 D. 110 E. 330

Given: \(\frac{377,910 *x}{3,300}=integer\).

Factorize the divisor: \(3,300=2^2*3*5^2*11\).

Check 377,910 for divisibility by 2^2: 377,910 IS divisible by 2 and NOT divisible by 2^2=4 (since its last two digits, 10, is not divisible by 4). Thus x must have 2 as its factor (377,910 is divisible only by 2 so in order 377,910*x to be divisible by 2^2, x must have 2 as its factor);

Check 377,910 for divisibility by 3: 3+7+7+9+1+0=27, thus 377,910 IS divisible by 3.

Check 377,910 for divisibility by 5^2: 377,910 IS divisible by 5 and NOT divisible by 25 (in order a number to be divisible by 25 its last two digits must be 00, 25, 50, or 75, so 377,910 is NOT divisible by 25). Thus x must have 5 as its factor.

Check 377,910 for divisibility by 11: (7+9+0)-(3+7+1)=5, so 377,910 is NOT divisible by 11, thus x must have 11 as its factor.

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