Quant Boosters - Hemant Malhotra - Set 12


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


    Number of Questions - 30
    Solved ? - Not Yet!
    Topic - Quant Mixed Bag
    Source - Elite's Grid CAT Preparation Forum


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


    Q1) A function V(a, b) is defined for positive integers a, b and satisfies V(a, a) = a, V(a, b) = V(b, a), V(a, a+b) = (1 + a/b) V(a, b). The value represented by V(66, 14) is ?
    (a) 364
    (b) 231
    (c) 455
    (d) 472
    (e) none


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


    Here we know value of V(a, a + b)
    so 1st term should be less than 2nd term, but in V(66, 14) 66 > 14
    so change this we know that V(a, b) = V(b, a)
    so V(66, 14) = V(14, 66)
    now proceed
    V(66,14 ) = V(14,66= (33/26) * V(14,52)
    = (33/26) * (26/19) * V(14, 38 )
    = (33/19) * (19/12) * V(14, 24)
    = (33/12) * (12/5) * V(14,10)
    = (33/5) * (7/2) * V(10, 4)
    =(231/10) * (5/3) * V(4, 6)
    = (77/2) * 3 * V(4, 2)
    = (231/2) * 2 * V(2, 2)
    = 231 * 2
    = 462


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


    Q2) f (x + y) = f(x) * f(y) then find f(3) if f(1) = 4


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


    If f(x + y) = f(x) * f(y) then f(x)= (f(1))^x always
    Here f(1) = 4 = > f(x) = 4^x so f(3) = 4^3 = 64


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


    Q3) Find range of the function (x^2 + x + 3)/(x^2 + x + 1)


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


    Method 1 :
    y=x^2+x+3/(x^2+x+1))
    yx^2+yx+y=x^2+x+3
    x^2(y-1)+x(y-1)+y-3=0
    x real so D > =0
    (y-1)^2-4*(y-1)(y-2) > =0
    y-1)(y-1-4y+12) > =0
    (y-1)(-3y+11) > =0
    (y-1)(3y-11)
    so (1,11/3]

    Method 2 :
    y=1+((2/(x^2+x+1))
    now x^2+x+1=x^2+x+1/4-1/4+1
    (x+1/2)^2+3/4
    so min value is 3/4 and maax infinity
    so y=1+2/inf
    y=1
    and y=1+2/(3/4))
    y=1+8/3=11/3


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


    Q4) How many integral values from 1 to 15 can the expression (x^2 + 34x -71)/(x^2 + 2x - 7) not take?
    a) 2
    b) 3
    c) 4
    d) none


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


    (x^2 +34x -71)/(x^2+2x-7)=y
    (1-y)x^2+ (34-2y)x -71+7y=0
    b^2-4ac > =0 for real values therefore
    (34-2y)^2-4*(1-y)(-71+7y) > =0
    now on simplifying
    y^2- 14y +45 > =0
    that is y < = 5 and y > = 9 so y can't take values
    6,7,8 so three integral values


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


    Q5) f(x) * f(1/x) = f(x) + f(1/x) for all real x. If f(3) = -26 find f(4)


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


    If f(x) * f(1/x) = f(x) + f(1/x)
    then f(x) = 1+-x^n
    now f(3) = -26
    so 1+-(x)^n = -26
    so +-(x)^n = -27
    so +-(3)^n = -27 (negative case will be considered bcz in positive case 3^n=-27 not possible)
    so -3^n = -27
    so n = 3
    and f(x) = 1-x^n
    so f(4) = 1-4^3
    = -63


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


    Q6) If f(x − 1) + f(x + 1) = f(x) and f(2) = 6, f(0) = 1, then what is the value of f(50) ?
    a) −7
    b) 6
    c) 1
    d) 7


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


    Since we know both f(0) and f(2), we can find f(1).
    f(1) = f(0) + f(2) = 7
    f(2) = f(1) + f(3)
    f(3) = f(2) – f(1)
    = 6 – 7
    = −1
    Also, f(3) = f(2) + f(4)
    f(4) = f(3) – f(2) = −7
    Continuing in a similar way, we can find out
    f(0) = 1
    f(1) = 7
    f(2) = 6
    f(3) = −1
    f(4) = −7
    f(5) = −6
    f(6) = 1
    f(7) = 7 and so on
    After every 6 integral values of x, f(x) repeats itself.
    f(6) = f(12) = f(18 ) = f(24) = f(30) = f(36) = f(42) = f(48 ) = 1
    f(49) = 7
    f(50) = 6


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


    Q7) What is the maximum value for the equation -2x^2 + 12x + 15


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


    Q8) For the equation x^2 + 6x + |n| to have real roots, how many values can n take ?


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


    Q9) Find the solution set of the equation x^3 - 4x^2 + x + 6 > 0
    a) (-1, 2)
    b) (-1, 2) U (3, ∞)
    c) (∞, -1) U (2, 3)
    d) (-1, 2) U (2, 3)


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


    Q10) Find the largest value of t such that x^t + 1 divides 1 + x + x^2 + x^3 + ... + x^143


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


    Q11) If x + 1/x = 3 then find the value of x^4 + 1/x^4


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


    Q12) A father and his son are waiting at a bus stop in the evening. There is a lamp post behind them. The lamp post, the father and his son stand on the same straight line. The father observes that the shadows of his
    head and his son's head are incident at the same point on the ground. If the heights of the lamp post, the
    father and his son are 6 metres, 1.8 metres and 0.9 metres respectively, and the father is standing 2.1
    metres away from the post then how far (in metres) is son standing form his father?


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


    Q13) For how many values of y, y(y + 4)(y + 6)(y + 8) < 300 is satisfied


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