XAT 2018

Let the cost price of the article be Rs.100.
The shopkeeper raises the price by x% and then decreases it by y%.
As a result, he reaches the cost price of the article. 
Also, it has been given that y is the difference between y% and x%.
$$y = x-y$$
$$2y = x$$
We know that $$(1+2y)(1-y)*100 = 100$$
$$(1+2y)(1-y) = 1$$
$$1-y+2y-2y^2 = 1$$
$$2y^2-y=0$$
$$2y = 1$$
$$y = 1/2$$ or $$0.5$$
$$x = 2y$$ 
=> $$x = 1$$ or $$x = 100$$%
Therefore, option D is the right answer. 

Liquids A and B are in the ratio 5:3. The volume of water is one-third the total mixture. 
Let us assume the volume of the total mixture to be 24x.

Volume of liquid A = 10x
Volume of liquid B = 6x
Volume of water = 8x

The mixture is passed through some medium that absorbs water completely and some quantity of liquid A. 
Water absorbed = 8x
Let the amount of liquid A absorbed be y.

8x+y = 200
=> y = 200-8x -----(1)
It has been given that (10x-y)/6x  = 7/9

Substituting (1), we get,
(10x-200+8x)/6x = 7/9
(18x-200)/6x = 7/9
162x-1800=42x
120x = 1800
=> x = 1800/120
Amount of water absorbed = 8*1800/120 = 120 ml.
Therefore, option E is the right answer. 

The radius of the coin is 3 cm. 
So, if the coin should not cross the edge, the centre of the coin should at least be 3 cm away from the edge of the tile. 

In the given diagram the center of the circle should lie in the blue region. As we can see, the area in which the centre of the coin can fall is a square of side 10-3-3 = 4 cm. 
Therefore, the area in which the centre of the coin can fall is 16 square cm. 
Area of the tile = 100 square cm.
Required probability = 16/100 = 0.16.
Therefore, option E is the right answer. 

2 litres are required to paint the surface of a solid sphere. 
Surface area of the solid sphere = $$4\pi*r^2$$
Now, the solid sphere is cut into 4 identical parts. This is possible only when the sphere is cut into 4 quarter spheres. 
After making the first cut, 2 hemispheres will be formed. 2 circles of area $$\pi*r^2$$ will get exposed in addition to the surface area of the sphere (the base surface of the bottom hemisphere and the base of the top hemisphere).
Now, another perpendicular cut will be made along the diameter of the sphere. 2 additional surfaces will get exposed again (one on left hemisphere and the another on the right hemisphere).

Area exposed after making 4 identical pieces = $$4*\pi*r^2$$ + $$2*\pi*r^2$$ + $$2*\pi*r^2$$
                                                                   = $$8*\pi*r^2$$
2 litres of paint is required to paint an area of $$4\pi*r^2$$
=> 4 litres of paint will be required to paint an area of $$8*\pi*r^2$$.

Therefore, option D is the right answer.

The man walks with a speed of V1 for 30 minutes. 
=> Distance covered = 30*V1.
On a particular day, he walks for 10 minutes at V1, takes a rest of 5 minutes, and then covers the distance by walking at V2 for 30 minutes. 
=> 10*V1 + 30*V2 = 30*V1
20*V1 = 30*V2
V1 = 1.5V2---------(1)
=> Total distance = 30*1.5*V2 = 45V2.
Now, we know that the person took 10 + 5 + 30 = 45 minutes to cover the entire distance. 
=> Average speed = 45V2/45 = V2.
Ratio of V2 and the average speed = 1:1.
Therefore, option A is the right answer. 

One boat is stationed to the North of the light house and the other boat is stationed to the East of the light house. 

The boat stationed to the East subtends an angle of 45 degrees and the boat stationed to the North subtends an angle of 30 degrees. 
Now, distance between the boat stationed to the East and the light house,d1 = tan 45
$$300/d1 = 1$$
=> $$d1 = 300$$ feet

Distance between the boat stationed to the North and the light house,d2 = tan 30
$$300/d2 = 1/\sqrt{3}$$
=> $$d2=300\sqrt{3}$$

Shortest distance between the 2 boats = $$\sqrt{300^2+(300\sqrt{3})^2}$$
                                                           = $$300*\sqrt{4}$$
                                                           =  $$600$$ feet.
Therefore, option C is the right answer.

It has been given that A+B+C = 41.
Let the common root be B. 
All the roots are positive integers. 
The product of the roots of one of the equations is 35. 
35 can be obtained only in 2 ways - either as 5*7 or 35*1. 
A+B+C = 41.
If A and B are 5 and 7 in any order, then C = 41 - 5 - 7 = 29.
If A and B are 35 and 1 in any order, then C = 41 - 35 - 1 = 5. 
As we can see, in either case, 5 is one of the 3 roots. 
Therefore, option C is the right answer. 

Let us find out the difference between the times given to figure out the pattern. 
The times given are 1:55 pm, 2:03 pm, 2:11 pm, 2:24 pm, 2:45 pm, 3:19 pm and 4:14 pm.
The difference between 2 consecutive times given are  8 minutes, 8 minutes, 13 minutes, 21 minutes, 34 minutes, and 55 minutes.
We can observe that the difference between the times are in the Fibonacci series. 
8 + 13 = 21
21 + 13 = 34
34 + 21 = 55

The Fibonacci series is as follows:
1,1,2,3,5,8,13,21,34,55.
But the first difference in the times given is 8. 
Therefore, the missing time must be such that it divides the interval of 8 minutes into 3 minutes and 5 minutes. 
The missing time should be 1:58 pm and hence, option B is the right answer. 

Let the number of girls in the school be G. 
=> Number of boys = G+30.
Some girls joined the class and the number of boys and girls became 3:5. 
Let the number of girls who joined the class be 'X'.
It has been given that (G+30)/(G+X) = 3/5
5G + 150 = 3G + 3X
2G + 150 = 3X
=> X = (2G/3) + 50.
2G has to be divisible by 3. 
Therefore, the least value that G can take is 3. 
When G = 3, X = 2 + 50
X = 52. 
The least number of girls who could have joined is 52. 
Therefore, option E is the right answer.

We draw BD perpendiculat to AC.
In right angled triangle BDC,  BD / BC = sin 30°
or, BD = (V2 * T)/2 ......(i)
In right angled triangle BDA, BD / BA = sin 45°
Or, BD = (V1 * T)/$$\sqrt{2}$$ ......(ii)
From (i) and (ii), we get
V2 / V1 = $$\sqrt{2}$$
Total distance to be travelled from C to A = CD + DA  = $$\sqrt{3}$$BD + BD
= BD($$1 + \sqrt{3}$$)
Replacing BD = (V2 * T)/2 in the avove equation,
CA = $$\dfrac{\text{(V2 * T)}}{2} (1 + \sqrt{3}$$)
Time taken at speed V2 = $$0.5(\sqrt{3}+1)$$T
Hence, option C is the correct answer.