Quant Boosters  Maneesh  Set 4

x + y = x² + y²  xy
=> 2x + 2y = 2x² + 2y²  2xy
=> (x  y)² + (x  1)² + (y  1)² = 2
So, two of these three terms have to be 1 and third one to be 0
Hence, only solutions are (0, 0), (2, 2), (1, 0), (1, 2), (0, 1), (2, 1)
(solved by Hemant Yadav sir)

Q3) How many integer solution exists for the equation xy = x + y + 15

OA  10
solution by shashank srivastava
xy= x+y+15
Now take x and y to rhs we get xyxy=15
Add 1 to both lhs nd rhs for factorising we get xyxy+1=16
Or (x1)(y1)=16
Now it means (x1) and (y1) should be factors of 16 as their multiplication gives result as 16
no of factors of 16 is 4 + 1 = 5
but that is for positive only considering negatives we get another 5 so total sums upto 2 * 5 = 10.

Q4) An ant crawls from one corner of a room to the diametrically opposite corner along the shortest possible path. If the dimensions of the room are a * b * c then what distance does the ant cover? (Most famous question of CAT  2013)

It depends on a, b, c
answer will be smallest of √{a² + (b + c)²}, √{b² + (a + c)²} and √{c² + (a + b)²}
(solution by hemant yadav sir)

Q5) Two consecutive numbers are removed from a list of first ‘n’ natural numbers. The average of the
remaining numbers is 21 (1/3). What is the product of the two numbers that have been removed?

Average of first n natural numbers used to be a value near to middle number
considrng 21 as middle number > so n can be (39 + 2), (42 + 2), (45 + 2)....
when n = 41 > sum = 832 + x + x + 1 = 833 + 2x
sum of first 41 natural numbers = 41 * 21 = 861
2x = 28
x = 14
two missd numbers 14 and 15
product = 14 * 15 = 210

Q6) A train after travelling 50 KM from A meets with an accident and proceeds at (4/5)th of the former speed and reaches B 45 min late. Had the accident happened 20 KM further on it would have arrived 12 min sooner. Find the original speed and distance.

Let original speed of train is S
20/(4S/5) – 20/S = 12/60 = 1/5
10080 = 4S/5
4S = 100
S = 25 KM/HR
Let the original distance is D
D50/20  D50/25 = 3/4
5D2504D+200 = 75
D = 125 KM

Q7) I have six identical oranges and six distinct apples. In how many ways can I have a basket of five fruits containing at least one apple and at least one orange?

1 orange 4 apple > 6c4
2 orange 3 apple > 6c3
3 orange 2 apple > 6c2
4 orange 1 apple > 6c1
15 + 20 + 15 + 6 = 56

Q8) A survey on a sample of 25 new cars being sold at a local auto dealer was conducted to see which of the three popular options, air conditioning, radio and power windows were already installed.
The survey found:
15 had air conditioning
2 had air conditioning and power windows but no radios
12 had radio
6 had air conditioning and radio but no power windows
11 had power windows
4 had radio and power windows
3 had all three options.What is the number of cars that had none of the options?
a. 4
b. 3
c. 1
d. 2

N(AURUP) = N(A)+N(R)+N(P)N(AnR)  N(AnP)  N(PnR) + N(AnPnR)  car with no options
25 = 15 + 12 + 11  6  2  4 + 3  car with no options
25 = 29  car with no options
car with no options = 4

Q9) A Father is willing his estates like this. If a boy is born, wife has 1/3 part and remaining for boy. If a girl is born, Wife has 2/3 and remaining for the girl. But twins (Boy + Girl) are born. What is the share that the daughter would get?

M : B = 1 : 2
M : G = 2 : 1
M : B : G = 2 : 4 : 1
daughter's share = 1/7

Q10) 70% of the students who joined XLRI last year play football, 75% play cricket, 80% play basketball and 85% play carrom. The minimum percentage of people who play all games is:
a) 5%
b) 10%
c) 15%
d) 20%
e) None of the above

100  (30 + 25 + 20 + 15) = 10

Q11) N is a fourdigit number in which each of the digits used appears at least two times. The number of different values that N can assume is

let first digit is 1 then other 3 digits can be 1 itself > 1 way
one of them can be 1 and other two any of (0,2,3....9) > 3 * 9 = 27
total 27 + 1 = 28similarly when first digit is 2,3,4...9 > 9 * 28 = 252

Q12) There are 2 bags, one consists of 3 red and 4 black balls and the other consists of 4 red and 2 black balls. One bag is selected at random and from a selected bag a ball is drawn. Let E be an event that first bag is selected, F be an event that second bag is selected, G be an event that a ball drawn is red. Find P(G/E).