New Set of Mixed Questions!!!

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

Let the usual speed be \(s\) and usual time \(t\) minutes, then as the distance covered is the same we will have: \(st=1.2s*(t-20+15)\) --> \(t=30\) minutes.

Answer: D.

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

Find the x and y intercepts.

When y=0, then x=30 or x=-30.
When x=0, then y=90 or x=-90.

So, we have 4 points: (30, 0), (-30, 0) (0, 90), (0, -90). When joining these points we get the rhombus:The area of a rhombus is \(\frac{d_1*d_2}{2}\) (where \(d_1\) and \(d_2\) are the lengths of the diagonals), thus the area of the enclosed figure is 60*180/2=5,400.

Answer: D.

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

After 2 hours \(2*\frac{1}{10}=\frac{1}{5}\) of the taks will be done (as only A works);

After 4 hours \(\frac{1}{5}+2*(\frac{1}{10}+\frac{1}{5})=\frac{4}{5}\) of the task will be done and \(\frac{1}{5}\) will be left to be done;

We are told that \(\frac{1}{5}\)th of the task is done in 15 minutes (1/4th of an hour) by all three workers: \(\frac{1}{4}*(\frac{1}{10}+\frac{1}{5}+\frac{1}{x})=\frac{1}{5}\). From which we can find that \(x=2\) hours.


Answer: C

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%

64% of her salary on food;
16% of her salary on meat;

64%-16%=48% on food but not on meat --> 48/64=3/4=75% of the salary spent on food were not spent on meat.

Answer: E.

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 \(\frac{1}{2}\) is in the set, which of the following must also be in the set?

I. \(-\frac{1}{64}\)

II. \(\frac{1}{64}\)

III. \(\frac{1}{\sqrt[3]{2}}\)

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

Since \(\frac{1}{2}\) is in the set, the following must also be in the set:

\(-x^2 = -\frac{1}{4}\);

\(-x^3 = -\frac{1}{8}\).

Since \(-\frac{1}{4}\) is in the set, the following must also be in the set:

\(-x^3 =\frac{1}{64}\)

Since \(-\frac{1}{8}\) is in the set, the following must also be in the set:

\(-x^2 =-\frac{1}{64}\)

The only number we cannot obtain is \(\frac{1}{\sqrt[3]{2}}\).

Answer: D.

SOLUTIONS:

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

Point K has the coordinates (-3, y) means that it's somewhere on the line x=-3. Hence the distance from this point to the Y-axis is 3 units.

Since the distance from the Y-axis to point K is 1/3 of the distance from the X-axis to point K, then the distance from K to the X-axis is 9 units.

Answer: E.

10. If n is a non-negative integer and the remainder when 3^n is divided by 4 is a multiple of 3, then which of the following must be true?

I. n^2 divided by 4 yields the reminder of 1
II. (-2)^n is less than 0
III. n is a prime number

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

3^0=1 --> the remainder when 1 is divided by 4 is 1;
3^1=3 --> the remainder when 3 is divided by 4 is 3;
3^2=9 --> the remainder when 9 is divided by 4 is 1;
3^3=27 --> the remainder when 27 is divided by 4 is 3;
...

We can see that in order the condition to hold true n must be odd.

I. n^2 divided by 4 yields the reminder of 1 --> odd^2 divided by 4 always yields the reminder of 1. So, this statement must be true.

II. (-2)^n is less than 0 --> (-2)^odd<0. So, this statement must be true.

III. n is a prime number. Not necessarily true.

Answer: D.

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

Obviously, the team could not have 40 or more members, since $10*40 = $400 > $399. What about 39? If 37 members contribute $10 each ($10*37 = $370) and the remaining two members contributed, for example, $11 and $18, respectively, then the total number of members the team would have is 37 + 1 + 1 = 39.

Answer: C.

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

Note that the question asks which of the following statements must be true, rather than could be true.

Let's evaluate the options:

A. Both \(x\) and \(y\) are negative. This option is not always true, consider \(x = -20\) and \(y = 8\).

B. \(xy > 0\). This option is not always true either, consider \(x = -20\) and \(y = 8\).

C. If \(y < 0\), then \(x > 0\). This option is not always true, consider \(y = -2\) and \(x = -10\).

D. If \(y > 0\), then \(x < 0\). If \(y\) is positive, then \(x\) must be negative in order for the sum of \(x\) and \(y\) to be negative.

E. \(x - y > 0\). This option is not always true, consider \(x = -20\) and \(y = 8\).

Therefore, only option D is ALWAYS true: if \(y\) is positive, then \(x\) must be negative in order for the sum of \(x\) and \(y\) to be negative.


Answer: D

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?

Point k is on the line x=-3, so its distance form the y-axis (x=0) is 3. Given that "The distance from the Y-axis to point K is 1/3 of the distance from the X-axis", its distance to the y-axis is 3*3=9

IMO E

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?

rates : A=1/10 B=1/5 c=1/x

A*2h+(A+B)*2h+(A+B+C)*15min = 1 work.

(1/10)*2 +(1/10+1/5)*2+(1/10+1/5+1/x)*1/4=1

x=2 IMO 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?

To empty X liters ==> 4 h.
To empty X + 3*6 liters ==> 6 h.

\(\frac{x}{4}=\frac{x+18}{6}\)

\(x=36\)

IMO D

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.

The distance between Y axis and (-3,y) will be the distance between (0,y) and (-3,y)

which is (-3)^2=9

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

|x/3|+|y/9|=10 will touch the x-axis at (30,0) and y-axis at (0,90) in Q1. Area of this region will be .5*30*90=1350.
Hence the total area of the figure formed on all the 4 quadrants is 4*1350=5400

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

Work done by :
A in 2 hrs = 2/10.-------(1)
A and B in 2 hrs = 2(1/10 + 1/5)--------(2)
A,B and C in 15 mins = 1/4(1/10 + 1/5 + 1/x)---------(3)
(1)+(2)+(3) =1

on simplification
4/5 + 1/4(3/10 + 1/x) = 1

or 1/x= 4/5 -3/10 = 5/10 = 1/2

X= 2 hrs

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

I have actually back solved this question. Took 36 liters as capacity of the tank. Hence draining pipe rate is 36/4 = 9 l/hr.

now total out flow during rain is 9-3= 6 l/hr. multiply with 6 hrs - time taken to empty the tank- we arrive back at the original capapcity of the tank = 36 L

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

My answer is D.

Since -1/2 is there we have -1/4 and -1/8 in the set. Since -1/4 is there we will have -(-1/4)^3= 1/64.
-1/8 is there hence -(-1/8)^2 = 1/64 will be in the set.