Quant Boosters - Hemant Malhotra - Set 8


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


    Number of Questions - 30
    Topic - Quant Mixed Bag
    Solved ? - Yes
    Source - Elite's Grid Prep Forum


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


    Q1) Find number of integral solutions of 2/x + 3/y = 1/10


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


    Concept:
    a/x + b/y=1/k
    where a, b and k are positive integers and we want number of value of (x,y) satisfying this equation
    Approach - first find number of factors of a * b * k^2 let number of factors=F
    a) total number of positive integral solutions=F
    b) total integral solutions = 2 * F-1
    c)total number of negative solution= zero ( bcz if both x and y will be negative than lhs will be negative but rhs is positive so not possible )

    solution:

    First find total number of factors of 2 * 3 * 10^2
    2^3 * 3 * 5^2 so (4 * 2 * 3) = 24 total number of factors
    so positive integral solutions = 24
    total number of integral solution = 2 * 24 - 1 = 47


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


    Q2) Find the smallest positive integer such that (n-13)/(5n+6) is a non-zero reducible fraction.


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


    hcf(n-13,5n+6)
    hcf((n-13,5n+6-5n+65))
    hcf(n-13,71))
    so reducible form for min n=
    n - 13 = 71
    so n = 84


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


    Q3) Find the minimum possible value of a + b + c if abc + bc + c = 2014, where a, b, c are positive integers.


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


    c * (ab+b+1)=2014
    c * (ab+b+1) = 2 * 19 * 53
    so c=19
    ab+b+1=106
    ab+b=105
    b * (a+1)=7*15
    so b=7 and a=14
    so sum =40


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


    Q4) If x = 1! + 2! +3! +4! + ... + n!, how many value of n, x is a perfect square?


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


    3,5,6 any perfect square will not leave this remainder when div by 7
    so 1!+2!+3!+4!=33 mod 7=5
    or 1!+2!+3!+4!+5!=153 mod 7=6
    or 1!+2!+3!+4!+5!+6! mod 7=153+620 mod7=6+4=10 mod 7=3
    so no value of n>=4 will give perfect square
    so check for n=1,2,3
    when n=1 then 1!=1 so perfect square
    when n=2 then 1!+2!=3 not a perfect square
    when n=3 then 1!+2!+3!=9=perfect square
    so n=1 and 3 only two values


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


    Q5) Find the number of integral solutions to |x| + |y| + |z| = 15
    a. 902
    b. 728
    c. 734
    d. 904


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


    x + y + z = 15
    so positive solutions = 15-1c3-1 = 14c2 = 7 * 13 = 91
    now
    when all negative =91
    when 2 positive one negative 3c2 * 91
    same for 2 negative one positive
    91 + 91 + 3 * 91 + 3 * 91
    8 * 91= 728
    now case when one is zero
    |y|+|z|=15
    then
    14 cases
    again 14 * 4=56
    so 3c2 * 56 = 3 * 56= 168
    now when 2 are zero
    |z|=15
    so 2 solutions
    so 3c2 * 2=6 solutions
    so total 728+174 =902


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


    Q6) If x^2 + xy + y^2 = 41, y^2 + yz + z^2 = 73 and z^2 + xz + x^2 = 9, find (x+y)/z.


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


    (x^3 - y^3) = 41 * (x - y)
    y^3 - z^3 = 73 * (y - z)
    z^3 - x^3 = 9 * (z - x)
    add all three
    41x - 41y + 73y - 73z + 9z - 9x = 0
    32x + 32y = 64z
    so (x+y)/z = 2


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


    Q7) Find all primes p, q, so that p^2 − 2q^2 = 1.


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


    p^2-2q^2=1
    (p^2-1) = 2 * q^2
    (p-1)(p+1) = 2 * q^2
    when p-1=1
    then p+1 = 2q^2
    so 2q^2=3 not possible
    when p-1=2
    then p+1=2q^2
    4=2q^2 so q^2=2 no possible
    when p+1=2 and p-1=q^2
    p=1 and 0=q^2 not possible
    now when p-1=2q
    and p+1=q
    so 2p=3q
    so p=3q/2
    so q=2 and p=3 will satisfy


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


    Q8) The sum of first 2 terms of an infinite GP is 18. Also, each term in the series is seven times the sum of the terms that follow. Find the first term and common ratio of the GP


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


    a = 7 * ( ar + ar^2 + ar^3 ... )
    a=7(ar/1-r)
    so a-ar=7ar
    so 1-r=7r so 8r=1 so r=1/8
    now a+ar=18
    a(1+1/8)=18
    so a * 9/8=18
    so a=16


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


    Q9) Thirty two men can complete a work in 16 days and 48 women can complete the same work in 12 days. Sixteen men and 36 women started working together and worked for 8 days. If the remaining work has to be completed in 2 days, how many additional men would be required?


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


    let total work=512 unit
    32 men per day work=512/16=32 unit
    so 1 men per day=1 unit
    in same way per day work of women=8/9 units
    now 16 M+ 36 W = 8 day work=16 * 1 * 8 + 36 * 8 * 8/9 = 384 units
    so remaining work=128 units
    let x more men needed
    so (16+x) * 2 + (36 * 8/9 * 2) = 128
    so x = 16


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


    Q10) The roots of x^3 - ax^2 + bx - c = 0 are p, q and r while the roots of x^3 + dx^2 + ex - 90 = 0 are p+3, q+3 and r+3. what is the value of 9a + 3b + c ?


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