# Free Mixed Bag 14 Practice Test - CAT

### Question 1

Two stations A and B are 920 km apart. A train T1, which stops for 5 minutes in every town-station and for 3 minutes in every village-station started from A with a speed of 60 km/h towards B and at the same time, a train T2 with a speed of 80 km/h which does not stop in any intermediate station started from B towards A. They met at C which is 560 km away from B. If the number of town-stations between A and C is less than the number of village-stations, then the least number of stations – town or village - are there between A and C is (Assume T1 stops only at town or village-stations)

#### SOLUTION

Solution :

As the train T2 covered 560 km, both trains travelled for 7 hours, but T1 travelled only 360 kms. Therefore, halting time = 60 minutes. Let the number of town-stations be x and number of village stations be y, x<y⇒ 5x + 3y = 60 ⇒ x=3, y=15, x=6, y = 10, or x=0, y = 20⇒ least is 16

### Question 2

A rectangular pattern on the wall looks like this. The shaded triangle was painted with a special paint costing thrice the paint used for the remaining rectangle. What is the ratio of expenses in painting the shaded triangle and the remaining rectangle, Given that PQ=QS and QR=RS

.

#### SOLUTION

Solution :C

Option (c)

Each triangle formed by diagonals = 14th area of rectangle. Thus area of shaded triangle= 18th area of rectangle, remaining area= 78th of the rectangle. Thus ratio of costs: 3x (18):1x(84); = 3:7.

### Question 3

The roots of the equation are?

#### SOLUTION

Solution :C

option (c )

Take a=1 and b=1, we basically need to find the value of x. now, the equation becomes 1/(2+x) = 2+1/x x

^{2}+2x+1=0solving for x, we get x=-1,-1Which is nothing but –a,-b. option c is the answer

### Question 4

There are 200 students in a class out of which 40% opt for dance, 30% opt for music and 50% for debate as extracurricular activities. 28 of these students do not choose any activity. What is the maximum number of students who opt for exactly one of these activities?

#### SOLUTION

Solution :A

S=120100×200=240

### Question 5

Consider a Master Set S= {1,2,3,4….12} How many subsets can be formed which will contain one or more elements of S (including all S) such that the elements of the sets are integral multiples of the smallest subset of the set.

#### SOLUTION

Solution :D

Option (d)

If 1 is the smallest element of the set, all or none of the other elements can be selected −211 different sets, as for each number from 2 to 12, there are two options, of getting selected or not getting selected. There are 11 such numbers 2-12 including both, therefore 2.2.2…..11 times = 211

Similarly, If 2 is the smallest element in the set, 4,6,8,10 and 12 can be selected in 25

^{ }waysIf 3 is the smallest element in the set 6,9,12 23 different sets

......211+25+23+22+21+6=2102

### Question 6

Find the area enclosed in the first quadrant represented by the function [x] + [y] = 1 where [t] is the greatest integer less than or equal to x ?

#### SOLUTION

Solution :C

Option C

Area under the graph will be Infinity as it is a discontinuous graph ; with discontinuity at (1,1).

### Question 7

Let S be the set of all four-digit positive integers whose digits are 3, 5, 7 and 9, with no digit repeated in the same integer. Calculate the remainder when the sum of all of the integers in S is divided by 9.

#### SOLUTION

Solution :E

option (e)

There are 24 numbers formed with the four given numbers.

Six of these numbers have a 3 in the 1000s position, six have a 5 in the 1000s position, six have a 7 in the 1000s position and six have a 9 in the 1000s position.

The same can be said about the distribution of numbers in the 100s, 10s and units positions.

Therefore, the sum of the 24 numbers is

6(3 + 5 + 7 + 9)(1000) + 6(3 + 5 + 7 +9)(100) + 6(3 + 5 + 7 + 9)(10) + 6(3 + 5 + 7 + 9) = 159984

The remainder is 0 when 159984 is divided by 9.

Alternative Shortcut:-

Sum of the digits will be 24 in each case, which will leave a remainder 6 when divided by 9. thereare 4! Numbers possible. Remainder when 4! × 6 is divided by 9 is zero. Hence option e

### Question 8

Katrina walks down an up-escalator and counts 150 steps. Priyanka walks up the same escalator and counts 75 steps. Katrina takes three times as many steps in a given time as Priyanka. How many steps are visible on the escalator?

#### SOLUTION

Solution :D

Total steps are constant. Taking speed of Katrina to Priyanka to escalator as 3:1:x

⇒150−(1503)×x=75+(751)×x⇒150−50x=75+75x⇒75=125x⇒x=35

So steps =15−1503×35=120

### Question 9

In the given figure, the sides YX, XZ, ZY are extended such that MY=YZ, NX=XY, XZ=ZO. The area of XYZ = 10, what is the area of DMNO

#### SOLUTION

Solution :In the given figure, let’s join Y to O, Z to N and X to M.

Consider DMNY, since MX is a median, area ofD MXN = area of DMXY.

But, area of DMXY = area of DXZY (XY is a median). Thus area of DMXN=area of DXZY

This is continued for all the smaller triangles formed, hence DMNO = 7D XZY.

Thus area of DMNO = 70.

### Question 10

Find out the graph representing the following function f(x) = x² – 3

#### SOLUTION

Solution :D

Option (d)

At x=0, f(x) = -3. Eliminate option (a) and option (c)

At f(x) = 0, x= +1.732 and –1.732.

Hence, answer is option (d)