Quant Boosters - Vikas Saini - Set 2



  • x = 1 / (4 - 3√3)
    using rationalisation
    x = 4 + 2√3 / 4
    x = 1 + √3/2.
    x^2 = 1 + 3/4 + √3 = 7/4 + √3
    x^3 = 7/4 + √3 + 7√3/8 + 3/2
    = 13/4 + √3 + 7√3 /8.
    x^4 = 49/16 + 3 + 7√3/2.
    x^4 - 4x^3 + 7x^2 -6x + 7/4
    = 97/3 + 7√3/2 -13 - 4√3 - 7√3/2 + 49/4 + 7√3 - 6 -3√3 + 7/4
    = 17/16.



  • Q11) What is the number of real solution of x^6 + 4x^3 - 12 ?



  • suppose x^3 = t & x^6 = t^2.
    t^2 + 4t - 12 = 0.
    = > t^2 +6t -2t -12 = 0.
    = > (t + 6) (t - 2) = 0.
    t = -6, 2.
    x^3 = -6
    x = -6, -6w , -6w^2.
    x^3 = 2.
    x = 2, 2w, 2w^2.
    only 2 real solution



  • Q12) In a 500 m race Dishu beats Ashu by 100 m or 5 seconds. In another race on the same track at the same speeds. Ashu and Prashant start at one end while Dishu starts at the opposite end. How many metres would Ashu have covered, by the time Dishu meets Prashant given that Dishu's speed is 10 m/sec more than that of Prashant.



  • Speed ratio of Ashu & Dishu = 400 : 500 = 4:5.
    Ashu’s speed = 100/5 = 20 m/s.
    Dishu’s speed = 25 m/s.
    Prashant’s speed = 15m/s.
    Relative speed = 25+15 = 40 m/s.
    Time when Dishu and Prashant meet = 500 / 40 = 12.5 seconds.
    Distance travelled by Ashu in given time = 12.5 x 20 = 250 meter.



  • Q13) Ram covers a part of the journey at 20 kmph and the balance at 70 kmph taking total of 8 hours to cover the distance of 400 km. How many hours has he been driving at 20 kmph?



  • 8 = D / 20 + (400 – D)/70
    8 = 7D + 800 – 2D / 140
    D = 64.
    Time = 64/20 = 3.2 = 3 hr 12 min.



  • Q14) Rohit and Virat walk from X and Y, a distance of 27 km at 5kmph and 7kmph respectively. Virat reaches Y and immediately turns back meeting Rohit at T. What is the distance of A to T ?



  • Suppose T is p km away from Y.
    27 + p / 7 = 27 – p / 5
    5(27 + p) = 7 (27 – p)
    135 + 5p = 189 – 7p
    12p = 54.
    p = 4.5
    Distance from A = 27 – 4.5 = 22.5



  • Q15) Donald and Trump leave points x and y towards y and x respectively simultaneously and travel in the same route. After meeting each other on the way, Donald takes 4 hours to reach her destination, while Trump takes 9 hours to reach his destination. If the speed of Donald is 48 km/hr, what is the speed of Trump?



  • Suppose speed of Donald is S1 and time is T1 after meeting each other.
    Speed of Trump is S2 and time T2 after meeting each other.
    After meeting, S1 / S2 = root (T2 / T1)
    48 / S2 = root (9 / 4)
    48 / S2 = 3 / 2
    S2 = 32.



  • Q16) There is an escalator is moving upwards. You take 8 seconds to take up & 10 seconds when escalator is not moving. Speed of escalator is 2 step/sec.



  • If my speed is ‘a’ and speed of escalator is ‘b’.

    When escalator is moving upwards

    1. If you are moving upwards then total speed = a+b.
    2. If you are moving downwards then total speed = |a-b|.

    When escalator is moving downwards

    1. If you are moving upwards then total speed = |a-b|.
    2. If you are moving downwards then total speed = a+b.

    formula d = st.
    8 (a+2) = 10a.
    8a+16 = 10a
    a = 8.
    Total steps = 8(a+2) = 10a = 80 steps.



  • Q17) Raju takes 40 seconds to walk up on an escalator which is moving upwards but he takes 60 seconds to walk up on an escalator which is moving downwards. How much time will he take to walk up if the escalator is not moving?



  • Suppose Ravi’s speed = a.
    Escalator’s speed = b.
    By formula d = st.
    (a+b)40 = (a-b)60
    2a + 2b = 3a – 3b.
    a = 5b.
    Total steps = (5b+b)40 = 240b.
    Time he will take when escalator is stationary = 240b / 5b = 48 seconds.



  • Q18) Rohit takes 60 seconds on an escalator which is moving down when he walks down but takes 40 seconds when he runs down. He takes 20 steps when he walking whereas he takes 30 steps when he is running. What is the total number of steps in the escalator?



  • Speed of escalator is a.
    Rohit whether runs or walks, distance must be same.
    20 + 60a = 30+40a
    20a = 10.
    a = 1/2.
    Steps = 20 + 60 x (1/2) = 50



  • Q19) Ravi and Rakesh are climbing on a moving escalator that is going up. Ravi takes 10 seconds to reach the top but Rakesh takes 8 seconds to reach the top. This happens because Rakesh is faster than Ravi. Rakesh takes 4 steps whereas Ravi can take only 3 steps in one second. What is the total number of steps in the escalator?



  • Suppose speed of escalator =a.
    (a+4)8 = (a+3)10.
    a = 1.
    Total steps = (1+4)8 = 40.



  • Q20) A and B are climbing on a moving escalator that is going up. A takes 30 steps to reach the top whereas B takes 32 steps for the same. This happens because B is faster than A. For every 4 steps that B takes, A takes only 3 steps. What is the total number of steps in the escalator?


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