Quant Boosters - Hemant Malhotra - Set 11


  • Director at ElitesGrid | CAT 2017 - QA 100 Percentile / CAT 2016 - QA : 99.94, LR-DI - 99.70% / XAT 2017 - QA : 99.975


    a * b * c = 7^2 * 11^2 * 13^2
    x+y+z=2=4c2=6
    x+y+z=2=4c2=6
    x+y+z=2=4c2=6
    so 6^3+3 * 6^3
    4 * 6^3=864
    now unordered
    2 same
    a^2 * b=1^2 * x
    7^2 * x
    11^2 * x
    13^2 * x
    (7 * 11)^2 * x
    (7 * 13)^2 * x
    (11 * 13)^2 * x
    (7 * 11 * 13)^2
    same 8 for negative so 16
    so 864-16 * 3/6 +16


  • Director at ElitesGrid | CAT 2017 - QA 100 Percentile / CAT 2016 - QA : 99.94, LR-DI - 99.70% / XAT 2017 - QA : 99.975


    Q27) In how many ways 1000 can b expressed as the product of 3 integers? (ordered and unordered)


  • Director at ElitesGrid | CAT 2017 - QA 100 Percentile / CAT 2016 - QA : 99.94, LR-DI - 99.70% / XAT 2017 - QA : 99.975


    1000=2^3 * 5^3
    so x+y+z=3
    so 5c2=10
    and x+y+z=3
    so 10
    so 10 * 10=100
    and negative 3 * 100=300
    so total =400

    now unordered
    same a^2 * b=2^3 * 5^3
    1^2 * x
    2^2 * x
    5^2 * x
    10^2 * x
    (-1)^2 * x
    (-2)^2 * x
    (-5)^2 * x
    (-10)^2 * x
    so total y=8
    but one case will be all same here so y=7
    and z=1
    so 6x + 3 * 7 + 1=400
    so x=400-22/6
    so unordered=x+y+z


  • Director at ElitesGrid | CAT 2017 - QA 100 Percentile / CAT 2016 - QA : 99.94, LR-DI - 99.70% / XAT 2017 - QA : 99.975


    Q28) Find coefficient of x^3 * y^4 * z^3 in expansion of (2x + y + 4z)^10


  • Director at ElitesGrid | CAT 2017 - QA 100 Percentile / CAT 2016 - QA : 99.94, LR-DI - 99.70% / XAT 2017 - QA : 99.975


    general term of this equation is = 10!/(r1! * r2! * r3!) * (2x)^r1 * (y)^r2 * (4z)^r3
    = 10!/(r1! * r2! * r3!) * (2)^r1 * (4)^r3 * x^r1 * y^r2 * z^r3
    now we have to find x^3 * y^4 * z^3 so r1=3 ,r2=4 and r3=3
    put tvalue of r1, r2, r3 here 10!/(r1! * r2! * r3!) * (2)^r1 * (4)^r3 * x^r1 * y^r2 * z^r3
    so coefficient is 10!/(3! * 4! * 3!) * (2)^3 * (4)^3


  • Director at ElitesGrid | CAT 2017 - QA 100 Percentile / CAT 2016 - QA : 99.94, LR-DI - 99.70% / XAT 2017 - QA : 99.975


    Q29) Find the largest coefficient in (x + y + z + k)^15


  • Director at ElitesGrid | CAT 2017 - QA 100 Percentile / CAT 2016 - QA : 99.94, LR-DI - 99.70% / XAT 2017 - QA : 99.975


    general term = 15!/(r1! * r2! * r3! * r4!) * x^r1 * y^r2 * z^r3 * k^r4
    so basically we have to maximize 15!/(r1! * r2! * r3! * r4!) term
    so we have to minimize (r1! * r2! * r3! * r4!) so for this symmetry is useful so put values as close as possible here
    r1 = r2 = r3 = 4 and r4 = 3
    so largest coefficient is 15!/(3! * (4!)^3)


  • Director at ElitesGrid | CAT 2017 - QA 100 Percentile / CAT 2016 - QA : 99.94, LR-DI - 99.70% / XAT 2017 - QA : 99.975


    Q30) Find the coefficient of x^6 in (1 + x^2 - x^3)^8


  • Director at ElitesGrid | CAT 2017 - QA 100 Percentile / CAT 2016 - QA : 99.94, LR-DI - 99.70% / XAT 2017 - QA : 99.975


    general term = 8!/r1! * r2! * r3! * (1)^r1 * (x^2)^r2 * (-x^3)^r3
    = 8!/r1! * r2! * r3! * x^(2r1) * (-1)^r3 * (x)^3r3
    = 8!/r1! * r2! * r3! * (-1)^r3 * x^(2r2+3r3)
    now we have to find coefficient of x^6 here (for any coefficient same procedure )
    so r1 + r2 + r3 = 8
    2r2 + 3r3 = 6
    find r1, r2 and r3 here
    let r3=0 so r2=3 so r1=2 and let r3=2,so r2=0 so r1=6 (r1,r2,r3 should be integer because term can't be negative)
    now coefficient will be 8!/r1! * r2! * r3! *(-1)^r3
    put both values of r1,r2 and r3 and add
    so 8!/5! * 3!+8!/2! * 6!=84



  • @hemant_malhotra as the no. of real roots is asked so it will always be 3


 

Looks like your connection to MBAtious was lost, please wait while we try to reconnect.