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, $x < y$

$\Rightarrow$ 5x + 3y = 60 $\Rightarrow$ x=3, y=15, x=6, y = 10, or x=0, y = 20 $\Rightarrow$ 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

.

A. 3:4

B. 3:8

C. 3:7

D. 1:2

SOLUTION

Solution : C

Option (c)

Each triangle formed by diagonals = $\frac{1}{4}$ $^{t h}$ area of rectangle. Thus area of shaded triangle= $\frac{1}{8}$ $^{t h}$ area of rectangle, remaining area= $\frac{7}{8}$ $^{t h}$ of the rectangle. Thus ratio of costs: 3x $(\frac{1}{8}) : 1 x (\frac{8}{4})$ ; = 3:7.

Question 3

The roots of the equation are?

C. -a ,-b

D. a,b

E. (a-b), (a+b)

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 x²+2x+1=0solving for x, we get x=-1,-1
Which 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?

A. 138

B. 104

C. 78

D. none of these

SOLUTION

Solution : A

$S = \frac{120}{100} \times 200 = 240$

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.

A. 2246

B. 2824

C. 3452

D. 2102

E. 1857

SOLUTION

Solution : D

Option (d)

If 1 is the smallest element of the set, all or none of the other elements can be selected $- 2^{11}$ 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 = $2^{11}$

Similarly, If 2 is the smallest element in the set, 4,6,8,10 and 12 can be selected in $2^{5}$ ways

If 3 is the smallest element in the set 6,9,12 $2^{3}$ different sets
$. . . . . . 2^{11} + 2^{5} + 2^{3} + 2^{2} + 2^{1} + 6 = 2102$

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 ?

A. 1

B. 8

C. infinity

D. 2

E. 4

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.

A. 2

B. 3

C. 4

D. 1

E. none of these

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! $\times$ 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?

A. 105

B. 150

C. 135

D. 120

SOLUTION

Solution : D

Total steps are constant. Taking speed of Katrina to Priyanka to escalator as 3:1:x
$\Rightarrow 150 - (\frac{150}{3}) \times x = 75 + (\frac{75}{1}) \times x \Rightarrow 150 - 50 x = 75 + 75 x \Rightarrow 75 = 125 x \Rightarrow x = \frac{3}{5}$
So steps $= 15 - \frac{150}{3} \times \frac{3}{5} = 120$

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.