Quant Boosters - Sagar Gupta - Set 3


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    ab = x
    cd = y
    y + 100x = x^2 + y^2
    x^2 - 100x + y^2 - y = 0
    10000 - 4y^2 + 4y = Discriminant
    x = 100 +/- root ( 10000 - 4y^2 + 4y ) /2
    x = 50 +/- root ( 2500 - y^2+y )
    2500 - y ( y-1 ) should be a perfect square
    y = 33
    x = 88, 12
    Answer =1233 and 8833


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    Q22) Akash, Bhanu and Sam work on a project. After they complete 1/4th of the project, Sam takes a break. For 7 days only Akash and Bhanu work on the project. After that Sam relieves both of them. He completes the project in 5 days. Bhanu works 50% faster than Akash while Sam alone can finish the entire project in 20 days. How long would Bhanu take to finish the entire project?


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    bhanu = 3 units/day
    akash = 2 units/day
    sam = x units/day
    total work = 20x units
    5x + 7 ( 5 ) + 5x = 20x
    35 = 10x -> x=3.5
    total work = 70
    bhanu takes = 70/3 = 23.33 days


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    Q23) There are 3 solutions.
    Solution 1: Consists of A and B in the volume ratio 3 : 2.
    Solution 2: Consists of B and C in the volume ratio 1 : 4.
    Solution 3: Consists of B and C in the volume ratio 7 : 3.
    The 3 solutions are mixed in the volume ratio X : 3 : 2. If the percentage composition of A and B are equal in the resultant mixture, then what is the value of X?
    a) 4
    b) 5
    c) 10
    d) 11


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    solution 1 : 6ml + 4ml
    solution2 : 2 ml + 8ml
    solution 3 : 7ml + 3ml
    Resultant : X : 3 : 2
    A ( 6x ) + B ( 4x + 20 ) + C ( 30 )
    6x = 4x + 20 -> x = 10


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    Q24) How many numbers less than 1000 have sum of digits as 12?


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    a + b + c = 12
    total whole number solutions : 14C2 = 14 * 13/2 = 91
    when a=10 : b+c =2 : 3 solutions
    when a=11 : b+c =1 : 2 solutions
    when a=12 : b+c = 0 : 1 solution
    total 6
    same when b,c = 11,12
    total 6 * 3 = 18
    numbers = 91- 18 = 73


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    Q25) A number of students appear for one of the papers of the Online Cat test series. The paper consists of 100 questions. For each correct answer the candidate is awarded 1 mark and for each wrong answer 1/5th of the marks are deducted. No marks are deducted for not attempting a question. Exactly 20% of the students scored exactly 50 marks but no two attempted the same number of questions. Find the total number of students who appeared for the test?


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    Total students = 5x
    Students who scored exactly 50 marks = x
    all these x students attempted different number of questions

    Possibilities for scoring 50 marks
    50 correct , 0 wrong
    51 correct , 5 wrong
    52 correct , 10 wrong
    53 correct , 15 wrong
    54 correct , 20 wrong
    55 correct , 25 wrong
    56 correct , 30 wrong
    57 correct , 35 wrong
    58 correct , 40 wrong

    so x = 9
    5x = 45


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    Q26) Find the maximum power of 12 in C(100,60)


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    100! / 60! * 40!
    12 = 2^2 * 3
    max power of 3 in 100! = 100/3+100/9+100/27+100/81 = 33+11+3+1 = 48
    max power of 3 in 60! = 60/3 + 60/9 + 60/27 = 20 + 6 + 2 = 28
    max power of 3 in 40! = 40/3 + 40/9 + 40/27 = 13 + 4 + 1 = 18
    3^48 / 3^28 * 3^18 = ( 3^2 )
    max power of 4 in 100! = 100/4 + 100/16 + 10/64 = 32
    max power of 4 in 60! = 60/4 + 60/16 = 18
    max power of 4! in 40! = 40/4 + 40/16 =12
    2^32 / 2^18 * 2^12 = 2^2
    so highest power is 1


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    Q27) A jet is flying 2400 miles from Hawaii to San Francisco. In still air, it flies at 600 mph. There is 40 mph tailwind in the same direction. Exactly how many hours after takeoff would it becomes neutral for the plane to either go to San Francisco or to return to Hawaii in the case of an emergency?
    a. 1.25 hours
    b. 1.5 hours
    c. 1.75 hours
    d. 2 hours


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    d / 640 = 2400 - d / 560
    d = 1280 miles
    1280 = 640 * t
    t = 2 hours


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    Q28) Two motorists set out at the same time to go from A to B, a distance of 100 miles. They both followed the same route and travelled at different, though uniform speeds of an integral number of miles per hour. The difference in their speeds was a prime number of miles per hour and after they had been driving for 2 hours, the distance of the slower car from A was 5 times that of the faster car from B. At what speed did the two motorists drive?


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    A-------100miles------------B
    Sp and Sq and | Sp - Sq | = prime number
    Sp * 2 = b
    Sq *2 = 500 - 5b
    10Sp + 2 Sq = 500
    (42, 40) satisfies


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    Q29) If x/(2a+b) = y/(2b+c) = z/(2c+a) = 8 and x/2a = y/2b = z/2c = k where a + b + c not equal to 0, then k=?


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    x = 16a + 8b
    y = 16b + 8c
    z = 16c + 8a
    x/2a = 8 + 4b/a
    y/2b = 8 + 4c/b
    z/2c = 8 + 4a/c
    all are equal :
    b/a = c/b = a/c : numbers are equal
    8 + 4 = 12


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    Q30) If there are "n" students in a circular arrangement facing toward the center. It is known that their roll numbers are Integral value from 1 to "n" and they are sitting in the order of increasingroll numbers. Then what would be value of "n" if Roll number 13 is just opposite to Roll Number 29 ?


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    14 to 28 : 15 students
    30, 31, 32 and 1 to 12 : 15 students
    so total 32


 

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