Quant Boosters - Maneesh - Set 4
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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)
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Q3) How many integer solution exists for the equation xy = x + y + 15
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OA - 10
solution by shashank srivastava
xy= x+y+15
Now take x and y to rhs we get xy-x-y=15
Add 1 to both lhs nd rhs for factorising we get xy-x-y+1=16
Or (x-1)(y-1)=16
Now it means (x-1) and (y-1) 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.
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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)
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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)
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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?
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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
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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.
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Let original speed of train is S
20/(4S/5) – 20/S = 12/60 = 1/5
100-80 = 4S/5
4S = 100
S = 25 KM/HR
Let the original distance is D
D-50/20 - D-50/25 = 3/4
5D-250-4D+200 = 75
D = 125 KM
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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?
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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
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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
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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
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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?
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M : B = 1 : 2
M : G = 2 : 1
M : B : G = 2 : 4 : 1
daughter's share = 1/7
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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
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100 - (30 + 25 + 20 + 15) = 10
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Q11) N is a four-digit number in which each of the digits used appears at least two times. The number of different values that N can assume is
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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
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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).
