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The next set of medium/hard PS questions. I'll post OA's with detailed explanations on Friday. Please, post your solutions along with the answers.

1. The distance from the Y-axis to point K is 1/3 of the distance from the X-axis to point K. If the coordinates of K are (-3, y), what is the distance between point K and X-axis?

3. Three workers, A, B, and C, can complete a certain task in 10, 5 and x hours respectively. A starts working alone and 2 hours later B joins. After another 2 hours joins C. After that A, B, and C together complete the task in 15 minutes. What is the value of x?

4. A draining pipe can empty a pool in 4 hours. On a rainy day, when the pool is full, the draining pipe is opened and the pool is emptied in 6 hours. If rain inflow into the pool is 3 liters per hour, what is the capacity of the pool?

A. 9 liters B. 18 liters C. 27 liters D. 36 liters E. 45 iters

5. For a certain set of numbers, if x is in the set, then both -x^2 and -x^3 are also in the set. If the number 1/2 is in the set , which of the following must also be in the set ?

I. -1/64 II. 1/64 III. 1/2^(1/3)

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

6. A team contributes total of $399 from its members. If each member contributed at least $10, and no one contributed $19, what is the greatest number of members the club could have?

7. Mary spent 64 percent of her salary on food (including meat) and 16% of her salary on meat. What percent of the salary spent on food were not spent on meat?

8. Usually Holly leaves home to school at 9:00, however today she left home 20 minutes later. In order to be at school on time she increased her usual speed by 20% and still was at school 15 minutes later than usual. What is her usual time from home to school?

A. 15 minutes B. 20 minutes C. 25 minutes D. 30 minutes E. 210 minutes

BONUS QUESTION: 11. Certain bowl contains 5 red marbles and 3 blue marbles only. One by one, every marble is drawn at random and without replacement. What is the probability that the seventh marble drawn is NOT blue?

6. A team contributes total of $399 from its members. If each member contributed at least $10, and no one contributed $19, what is the greatest number of members the club could have? ributed A. 37 B. 38 C. 39 D. 40 E. 41

Sol: Since we need to find the greatest number of members, we should minimise the contribution of each member. Min Contribution : $ 10 We can start by assuming no of members from any of the given options starting with C we have, 39*10+ y= 399--> y = $9. This $9 can be shared by any contributed by any of the members

Lets take no. of members to be D then we have 400*10 > 399 and hence the case is not possible as Total contribution will exceed $ 399.

Assuming B as no of members we have 38*10+ y= 399----> y =$ 19 which can be given by eny of the remaining members (shared by 3 or more)

Hence max no possible be 39.

Ans should be C
_________________

“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.”

7. Mary spent 64 percent of her salary on food (including meat) and 16% of her salary on meat. What percent of the salary spent on food were not spent on meat?

A. 16% B. 25% C. 32% D. 48% E. 75%

Sol: Let Mary's Salary be US $ 100 Spent on food : $ 64 Spent on meat : $ 16 Salary spent on food other than meet : 64-16 = 48$

percent of the salary spent on food were not spent on meat : (64-16)/64*100 = 75% Ans should be E
_________________

“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.”

4. A draining pipe can empty a pool in 4 hours. On a rainy day, when the pool is full, the draining pipe is opened and the pool is emptied in 6 hours. If rain inflow into the pool is 3 liters per hour, what is the capacity of the pool?

A. 9 liters B. 18 liters C. 27 liters D. 36 liters E. 45 iters

Sol: The best method will be pluggin in Ans choices. Also Total water drained (Sum of Exisiting water in the pool + Water added by Rain flow) = Rate of Draining pipe * No. of hours

Let us take volume to be C i.e 27 Llt Rate of Draining pipe : 27/4 ---> 6.75 Litres Amount water drained in 6 hours = 6.75*6 = 40.50 Amount of water in the pool to be drained = 27+ 3*6 (Rain water flow rate* no. of hours) = 45

Clearly LHS is not equal to RHS and hence Volume cannot be 27 Lts

Let us take Volume to be 36 lts Rate of Draining pipe : 36/4 ---> 9 Litres Amount water drained in 6 hours = 9*6 = 54 Amount of water in the pool to be drained = 36+ 3*6 (Rain water flow rate* no. of hours) = 54

Clearly LHS= RHS and hence ans should be D = 36 Lts
_________________

“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.”

5. For a certain set of numbers, if x is in the set, then both -x^2 and -x^3 are also in the set. If the number 1/2 is in the set , which of the following must also be in the set ?

I. -1/64 II. 1/64 III. 1/2^(1/3)

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

Sol: Given if x is in the set, then both -x^2 and -x^3 are also in the set. If the number 1/2 is in the set

The we have other members in the set as - (1/2)^2 = - 1/4 - (1/2)^3 = -1/8

Now if -1/4 is in the set then we have - (-1/4)^2 = -1/16 and -(-1/4)^3= 1/64 in the set Also If -1/8 is in the set then we have - (1/8)^2 = -1/64 and -(-1/8)^3 = 1/512 in the set

Since 1/2 is in the set then surely either -x^2 = 1/2 or x^2 = -1/2 but since Sqaure of no cannot be negative therefore -x^2= 1/2 -X^3= 1/2 -----> X^3= -1/2 and x = -1/2^(1/3)

Hence Ans should Choice D
_________________

“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.”

8. Usually Holly leaves home to school at 9:00, however today she left home 20 minutes later. In order to be at school on time she increased her usual speed by 20% and still was at school 15 minutes later than usual. What is her usual time from home to school?

A. 15 minutes B. 20 minutes C. 25 minutes D. 30 minutes E. 210 minutes

Sol: Let her usual speed of Holly be S1 km/min and t1 be the time usual time taken by Holly to reach school In the case when she leaves for school at 0920 hrs and reaches school 15 minutes later than usual So we have S2 = 1.2 S1 where S2 is the speed of Holly in case when she is late and time taken by her is T1- 20 +15= T1-5 = T2 Now Distance in both cases is same so we have S1*T1= S2*T2 S1*T1= 1.2S1*(T1-5) 0.2 S1T1= 6 S1 T1 = 6/0.2 = 30 minutes

Hence ans should be D
_________________

“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.”

1. The distance from the Y-axis to point K is 1/3 of the distance from the X-axis to point K. If the coordinates of K are (-3, y), what is the distance between point K and X-axis?

A. 1/2 B. 1 C. 3 D. 4.5 E. 9.

Sol: Any Co-ordinates on X axis and Y axis indicate the distance of that point from both X axis and y axis Co-ordinates of Pt K (-3, y) means Pt K is to the left of Y axis at distance of 3 units

Also, y is the distance of Pt K from the X axis. Now, The distance from the Y-axis to point K is 1/3 of the distance from the X-axis to point K Let distance from X axis to the Point K is "3a" then we have Distance from Y axis to the point K is a Given a =-3 then 3a =-9 and hence point K lies in 3rd quadrant where x and y both are negative

Ans should be E
_________________

“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.”

2. What is the area of a region enclosed by |x/3|+|y/9|=10?

A. 675 B. 1350 C. 2700 D. 5400 E. 10800

Sol: |x/3|+|y/9|=10?

Possible equations will be

x/3+ y/9 = 10 -x/3+ y/9 = 10 x/3 -y/9 = 10 -x/3 -y/9 = 10 For each Equation we can find the values of x and y when they intersect with x and y axis Eq 1 , we have x= 30 , y= 0 and y=90, x=0 Eq 2, We have x=-30, y=0 and y=90, x=0 Eq 3, we have x=30, y=0 and y= -90, x=0 Eq 4, we have x= -30, y=0 and y=-90 when x =0

If we draw a graph using these points we will have a Rhombus.

Area of Rhombus will be 1/2 d1 * d2 = 1/2 *180*60 = 5400 should be the ans

5. For a certain set of numbers, if x is in the set, then both -x^2 and -x^3 are also in the set. If the number 1/2 is in the set , which of the following must also be in the set ?

I. -1/64 II. 1/64 III. 1/2^(1/3)

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

1/2 is in the set -> - (1/2)^2 and - (1/2)^3 are also in the set -> -1/4 and -1/8 are in the set -1/4 is in the set -> - (-1/4)^2 and - (-1/4)^3 are also in the set -> -1/16 and 1/64 are in the set -1/8 is in the set -> - (-1/8)^2 and - (-1/8)^3 are also in the set -> -1/64 and 1/512 are in the set

Answer : D
_________________

KUDOS is the good manner to help the entire community.

"If you don't change your life, your life will change you"

9. If x and y are integers and x + y = -12, which of the following must be true?

A. Both x and y are negative B. xy > 0 C. If y < 0, then x > 0 D. If y > 0, then x < 0 E. x - y > 0

Sol: Note that sum of 2 negative nos will be negative and also sum of positive no and negative no can be negative as well

So option A - Possible but not 100% Option B xy>0 Possible but no 100% (Ex let say x=-36 and y=24, xy=-864 <0) Option C : y<0, then x>0 x+y= -12 and y= -2 , x=-10, X<0 hence need not be true option D : if y>0, then x< 0 y=2, then x = -14 less than 0 y= 12, then x =-24 which is not less than 0 Hence as Y increases X becomes more and more negative and thefore option D must be true Option E x - y > 0 Let x-y =2 x+y =-12 So we get 2x= -10 and x =-5 and y = -3 ----> -5+3= -2<0

Ans D
_________________

“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.”

2. What is the area of a region enclosed by |x/3|+|y/9|=10?

We can analize the function x/3 + y/9 = 10 first. y=90-3x This intersect the Y-axis is 90 and the X-axis in 30. The original function ( with the absolute values) has the same graph as y=90-3x, only symmetric for value of x<0 and values of y <0. So the answer is: it defines a triangle of Area=\(\frac{90*30}{2}=1350\)

IMO B

As we have absolute values, don't we need to multiply 1350*4 to cover all the possibilities (x=30,y=90) , (x=-30,y=90), (x=-30,y=-90), (x=30,y=-90)
_________________

Do not dwell in the past, do not dream of the future, concentrate the mind on the present moment.

6. A team contributes total of $399 from its members. If each member contributed at least $10, and no one contributed $19, what is the greatest number of members the club could have?

A. 37 B. 38 C. 39 D. 40 E. 41

\(399 = 39*10 + 9\) --> \(399 = 37*10 + 29\) --> Thus, 37 members each gave $10 and 2 members gave a combination of 29 ( 16,13 or 12,17 or ...)

So, the greatest number of members the club could have : 39

Answer : C
_________________

KUDOS is the good manner to help the entire community.

"If you don't change your life, your life will change you"

BONUS QUESTION: 11. Certain bowl contains 5 red marbles and 3 blue marbles only. One by one, every marble is drawn at random and without replacement. What is the probability that the seventh marble drawn is NOT blue?

A. 7/8 B. 3/4 C. 2/3 D. 5/8 E. 3/8

SOLUTION: A marble can be drawn from the bowl in 8 ways. A red marble can drawn in 5 ways. So the probability that a marble is NOT blue is 5/8
_________________

Do not dwell in the past, do not dream of the future, concentrate the mind on the present moment.

7. Mary spent 64 percent of her salary on food (including meat) and 16% of her salary on meat. What percent of the salary spent on food were not spent on meat?

A. 16% B. 25% C. 32% D. 48% E. 75%

Let $100 be her salary ;

So, she spent $64 of her salary on food (including meat) and $16 of her salary on meat. Hence, she spent 64-16, which is $48 on food not spent on meat.

Percent of the salary spent on food were not spend on meat = [Amount spent on food and not spent on meat] / [Total amount spent on food] = 48/64 = 0.75

Answer : E
_________________

KUDOS is the good manner to help the entire community.

"If you don't change your life, your life will change you"

8. Usually Holly leaves home to school at 9:00, however today she left home 20 minutes later. In order to be at school on time she increased her usual speed by 20% and still was at school 15 minutes later than usual. What is her usual time from home to school?

A. 15 minutes B. 20 minutes C. 25 minutes D. 30 minutes E. 210 minutes

The data are organized in the figure below;

Notice that the distance that takes Holly to arrive from home to school as usual is the same as today; Hence, \((1.2x-6)=x\) -->\(x=30minutes\)

Answer: D

Attachments

Rock3.jpg [ 18.22 KiB | Viewed 2541 times ]

_________________

KUDOS is the good manner to help the entire community.

"If you don't change your life, your life will change you"

9. If x and y are integers and x + y = -12, which of the following must be true?

A. Both x and y are negative B. xy > 0 C. If y < 0, then x > 0 D. If y > 0, then x < 0 E. x - y > 0

Lets consider this two pairs that match with the equation above : (x,y) = (-3,-9) (x,y) = (-13,1)

A- Not always true ; -3 and -9 both are negative BUT -13 is negative whereas 1 is positive B- Not always true; (-3)*(-9) = 27>0 BUT (-13)*(1)<0 C- Not always true; If y = -9 <0 then x = -3<0 so x is not positive D- True; If y>0 then -12-x>0 then x<-12 and hence x must be negative E- Not always true; x-y = -3-9 = -12 <0

Answer : D
_________________

KUDOS is the good manner to help the entire community.

"If you don't change your life, your life will change you"

5. For a certain set of numbers, if x is in the set, then both -x^2 and -x^3 are also in the set. If the number 1/2 is in the set , which of the following must also be in the set ?

I. -1/64 II. 1/64 III. 1/2^(1/3)

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

5. For a certain set of numbers, if x is in the set, then both -x^2 and -x^3 are also in the set. If the number 1/2 is in the set , which of the following must also be in the set ?

I. -1/64 II. 1/64 III. 1/2^(1/3)

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

Answers B and C are same

Thank you, edited. C should read: III only.
_________________

3. Three workers, A, B, and C, can complete a certain task in 10, 5 and x hours respectively. A starts working alone and 2 hours later B joins. After another 2 hours joins C. After that A, B, and C together complete the task in 15 minutes. What is the value of x?

A. 1 B. 1.25 C. 2 D. 2.5 E. 4

A, B, and C, can complete a certain task in 10, 5 and x hours respectively A completes 1/10 of the total work in 1 hr B Completes 1/5 of the total work in 1 hr C completes 1/x of the total work in 1 hr A starts working alone and 2 hours later B joins. After another 2 hours joins C. A completes 2/10 of the total work in 2 hrs later B joins A A&B together can complete 1/10 + 1/5 of the total work in 1 hr = 3/10 A&B work together for 2 hour and they complete 2 * 3/10 = 6/10 of the total work

By the time C joins A&B 8/10 of the total work is complete

To complete 2/10 of the work A, B & C together take 15 minutes (1/4 hr) 2/10 = ¼( 1/10 + 1/5+ 1/x) 8/10 = 3/10 + 1/X 5/10 = 1/X X = 2 Answer = C _________________

-------------------------------------

Last edited by g106 on 03 Apr 2013, 07:48, edited 1 time in total.

2. What is the area of a region enclosed by |x/3|+|y/9|=10?

We can analize the function x/3 + y/9 = 10 first. y=90-3x This intersect the Y-axis is 90 and the X-axis in 30. The original function ( with the absolute values) has the same graph as y=90-3x, only symmetric for value of x<0 and values of y <0. So the answer is: it defines a triangle of Area=\(\frac{90*30}{2}=1350\)

IMO B

Sound reasoning but IMO you missed the last step. What you have calculated is only the part of the figure that lies when x>0 and y>0. Thus only the first quadrant. Since the expression is symmetrical in terms of -x and -y (Substituting -x for x and/or -y for y will not change the expression) the figure will be symmetrical about the x and y axes. You just need to multiply 1350 by 4 to get the area enclosed by the figure.

3. Three workers, A, B, and C, can complete a certain task in 10, 5 and x hours respectively. A starts working alone and 2 hours later B joins. After another 2 hours joins C. After that A, B, and C together complete the task in 15 minutes. What is the value of x?

A. 1 B. 1.25 C. 2 D. 2.5 E. 4

A, B, and C, can complete a certain task in 10, 5 and x hours respectively A completes 1/10 of the total work in 1 hr B Completes 1/5 of the total work in 1 hr C completes 1/x of the total work in 1 hr A starts working alone and 2 hours later B joins. After another 2 hours joins C. A completes 2/10 of the total work in 2 hrs later B joins A A&B together can complete 1/10 + 1/5 of the total work in 1 hr = 3/10 A&B work together for 2 hour and they complete 2 * 3/10 = 6/10 of the total work

By the time C joins A&B 8/10 of the total work is complete

To complete 2/10 of the work A, B & C together take 15 minutes (1/4 hr) 2/10 = ¼( 1/10 + 1/5+ 1/x) 8/10 = 3/10 + 1/X 5/10 = 1/X X = 2 Answer = C

4. A draining pipe can empty a pool in 4 hours. On a rainy day, when the pool is full, the draining pipe is opened and the pool is emptied in 6 hours. If rain inflow into the pool is 3 liters per hour, what is the capacity of the pool?

A. 9 liters B. 18 liters C. 27 liters D. 36 liters E. 45 iters

Draining pipe takes 4 hr to empty the pool Rain inflow = 3lt/hr Time taken to drain the water from the pool on rainy day = 6

It takes 2 hr to drain the additional water added due to rain Water accumulation in six hrs of rain = 3*6 = 18lts

It takes 2 hrs to drain 18 lts => in 4 hours 36 lts of water will be drained from the pool and the capacity of the pool is 36 liters

Answer:D 36 liters

5. For a certain set of numbers, if x is in the set, then both -x^2 and -x^3 are also in the set. If the number 1/2 is in the set , which of the following must also be in the set ?

I. -1/64 II. 1/64 III. 1/2^(1/3)

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

X = ½ -X^2 = ¼ = X1 -X^3 = -1/8 = X2

-X1^2 = 1/16 -X1^3 = -1/64 (I)

-X2^2 = 1/64 (II)

Ans D. I and II only

6. A team contributes total of $399 from its members. If each member contributed at least $10, and no one contributed $19, what is the greatest number of members the club could have?

A. 37 B. 38 C. 39 D. 40 E. 41

Answer. C. 39

To identify the maximum number each member of the team should have contributed closer or equal to minimum number (~$10)

37 members contributing $10 each will make $370. Since no one contributes $19, 2 members share the $29 between them which makes $399 from 39 members

7. Mary spent 64 percent of her salary on food (including meat) and 16% of her salary on meat. What percent of the salary spent on food were not spent on meat?

A. 16% B. 25% C. 32% D. 48% E. 75%

Say mar yeans $100 and spends $16 for meat and $64 for food.

Percentage spent on meat in total food expenditure is 16/64 = 25% which makes 75% not spent meat

Answer E. 75% 8. Usually Holly leaves home to school at 9:00, however today she left home 20 minutes later. In order to be at school on time she increased her usual speed by 20% and still was at school 15 minutes later than usual. What is her usual time from home to school?

A. 15 minutes B. 20 minutes C. 25 minutes D. 30 minutes E. 210 minutes

Mary left 20 Minutes later and reached school 15 minutes later. Because of the 20% speed increase she has reached the distance 5 minutes earlier

Substituting the value in each of the answer options, Speed difference between today and usual should be 20% ~Du/Tu – Dt/Tt = 20/100 = .2 Where, Du – Distance usual; Dt – Distance today; Tu – Time Usual; Tt – Time today Lets take mary’s speed is 1 unit/ minute

9. If x and y are integers and x + y = -12, which of the following must be true?

A. Both x and y are negative B. xy > 0 C. If y < 0, then x > 0 D. If y > 0, then x < 0 E. x - y > 0

X and Y are integers X+Y = -12

A. Both x and y are negative => not true as there is a possibility for one of the number is higher negative (e.g., X = -14; Y =2) B. xy > 0 => not true; one number can be positive and one number can be negative to make the condition true C. If y < 0, then x > 0 => It is not necessary that only one number should be negative we need at least one to be negative to make the statement true; In this scenario x can be negative. D. If y > 0, then x < 0 => True as at least one number should be negative to get the negative summation E. x - y > 0 => not true in the case of x = -6 and y = -6 and so many other scenarios

Answers D

BANG ON!!!!!! Just some issue with Q-5 5. For a certain set of numbers, if x is in the set, then both -x^2 and -x^3 are also in the set. If the number 1/2 is in the set , which of the following must also be in the set ?

I. -1/64 II. 1/64 III. 1/2^(1/3)

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

X = ½ -X^2 = ¼ = X1 -X^3 = -1/8 = X2

-X1^2 = 1/16 -X1^3 = -1/64 (I)

-X2^2 = 1/64 (II)

Ans D. I and II only

IMO -x^2 should be \(\frac{-1}{4}\) Likewise -X2^2 will be \(\frac{-1}{64}\) And we will never get +ve values. Thus only possible soln is A