# Free Mixed Bag 10 Practice Test - CAT

### Question 1

There are two stationary boats (A and B) on a river. B is downstream from A. At 9AM, a raft starts floating downstream from A to B. The still water speed of the raft is 0 kmph. At the same time, a boat starts out from B to A and meets the raft in 5 hours and then reaches A. It immediately returns to B. Find the time that the boat and the raft reached B, if it is given that they both reached together?

#### SOLUTION

Solution :C

Option ( c)

Speed of river= S kmph

Speed of boat= B kmph

In 5 hours, the boat would have traveled 5(B-S)km and the raft would have traveled 5S km

The raft has to cover 5(B-S)km at S km/hr speed

The boat has to cover 5S km at (B-S) kmph and (5S+5B-5S) km at (B+S) kmph speed

5(B−S)S=5SB−S+5BB+S

It will take 52 hours= 7.07 hours after 2 pm.

Answer = option c

### Question 2

It is given that the compound interest on a certain sum of money for 3 years at 5% is Rs 122, what would be the SI for the same sum of money at the same rate of interest?

#### SOLUTION

Solution :A

Assume the principal amount to be Rs 8000 (you can assume anything)

CI for Rs 8000 for 3 years at 5% per annum

Rate = 5%

For a period of 3 years, consider the 3rd line of the Pascal’s triangle

1331

Amount is = 8000(1) + ( 5% of 8000=400)(3) + ( 5% of 400=20)(3) + ( 5% of 20 =1)(1)

Sum= 8000 + 1200 + 60 + 1 =9261 .

CI = 9261-8000= 1261

The CI is Rs 1261 when the principal is Rs 8000.

Finding the Principal when CI=Rs.122=122×80001261=Rs.774

SI =774×5×3100 = Rs 116. Note:- you can assume even 100 to make it much simpler

### Question 3

Find the inverse of the function f(x)=(x−2)13

#### SOLUTION

Solution :D

Option (d)

Let y=(x−2)13

y3=x−2

y3+2=x

Now, replace y to x.

f(x)−1=x3+2

Alternative Approach:

a) Assuming a value for x. let x=0. then f(x)=(−2)13

b) Substitute, x=(−2)13 in the answer options and see where you get f(x)=0

substituting, only option (d) gives f(x)=0 at x=(−2)13

### Question 4

Find the area enclosed between the x-axis and the following equations

y=−(x+4) −5<x<−2

y=x1 −2<x<2

y=4−x 2<x<4

#### SOLUTION

Solution :B

OPTION B

Substitute the values of x, to get corresponding values of y

You can plot the points to get the graph.

Area = 2×(12×4×2)=8 sq.units

### Question 5

In a certain locality, 60 of the families own TV sets, 85 own scooters, 70 own fridge and 95 own radio sets.130 of the families own exactly one of these things. What is the maximum possible number of families in that locality?

#### SOLUTION

Solution :C

S = 60 + 85 + 70 + 95 = 310.

X = I + II + III + IV

S = I + 2 II + 3 III + 4 IV

S - I = 2 II + 3 III + 4 IV

310 - 130 = 180= 2 II + 3 III + 4 IV

To maximize X, make III & IV=0

Then 2 II = 1800 II = 90

X max = 130+90 = 220

### Question 6

Two cars start towards each other at the same time. The distance between the cars is 3000 km. One travels at 60 km/hr and another at 90 km/hr. A bird starts with one of the cars and flies at 400 km/hr towards the other one. When it gets to the other car, it turns and flies back towards the first and continues this to and fro flying when the cars are moving towards each other. Distance travelled by the bird before the cars pass each other is:

#### SOLUTION

Solution :D

Time taken for 2 cars to meet each other = distance travelled / relative speed of cars

T= 3000(90+60)= 20 hours. The bird is travelling till the cars pass each other. The time of travel will be the time taken by the cars to pass each other.

Distance travelLed by bird = speed of bird × time in air = 400×20 = 8000 km

### Question 7

125 gallons of mixture of milk and water contains 20% water. What amount of water needs to be added to this milk-water mixture in order to increase the percentage of water to 25% of the new mixture?

#### SOLUTION

Solution :C

Option c

We need to find out how much of a solution of 100% water needs to be added to a solution containing 20% water to attain a dilution of 25% . This can be found using allegation out as follows

Ratio = 75:5 = 15:1

That is for 15 parts of a 20% water solution, one part of 100% water solution needs to be added. Therefore, for a solution of 125 gallons, 12515=8.33 gallons needs to be added

### Question 8

Sets A, B, C and D are all subsets of quadrilaterals. A is the set of rhombi, B is the set of rectangles, C is the set of parallelograms, and D is the set of kites. What is the set (A∩B)∪(C∩D)?

#### SOLUTION

Solution :D

Soln:

A∩B is a set of quadrilaterals that are both rhombi and rectangles - which are squares. C∩D is a set of quadrilaterals that are both parallelograms and kites - which are rhombi. Finally, the union of squares and rhombi are... rhombi. Hence option (d)

### Question 9

Wine is made by crushing grapes with the feet. Two grape crushers take 4 days to crush a certain amount of grapes. If one of them crushed half the grapes and the other crushed the other half, then they complete the job in 9 days. How many days will it take for the slower crusher to do the job alone?

#### SOLUTION

Solution :D

Option a can be eliminated directly as the number of days have to be > 9 days.

Answer is either b,c and d

Check for answer b.

If slower one takes 16 days, then faster takes 14−116=316⇒163 days

Slower takes 162=8 days and faster takes (162)3=83 days. Totally they will not take 9 days

Reverse Gear

Check for answer c

If the slower crusher takes 15 days, then the faster one takes 14−115=1160⇒6011 days.

Slower does his half work in 152=7.5 days and the faster does his work in 6022. together it does not add up to 9 days. Hence, answer is option (d)

### Question 10

y=4a−1+4−a−1. If ‘a’ is real then the least value of ‘y’ is:

#### SOLUTION

Solution :B

Use A.M≥G.M

4(a−1)+4(−a−1)2≥√4(a−1)∗4(−a−1)

4(a−1)+4(−a−1)≥2(4−2)12

y≥12