Quant Boosters - Maneesh - Set 4


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    Number of Questions - 30
    Topic - Quant Mixed Bag
    Solved ? - Yes
    Source -


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    Q1) f(xy) = f(x) + f(y)
    f(2401) = 10/3
    f(729) = 18
    then f(343(root3)) = ?
    (CAT 2013 question)


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    f(x) behaves as log function.
    2401 = f(49 * 49)
    4f(7)=10/3
    f(7)=10/12
    f(729) = 18
    f(27 * 27) = 6f(3) = 18
    f(sqrt3) = 3/2
    f(343(root3)) = 10/4 + 3/2 = 4


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    Q2) How many integral solution exists for the equation x + y = x^2 + y^2 - xy


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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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