Quant Boosters  Hemant Malhotra  Set 8

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  99.70% / XAT 2017  QA : 99.975
Number of Questions  30
Topic  Quant Mixed Bag
Solved ?  Yes
Source  Elite's Grid Prep Forum

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  99.70% / XAT 2017  QA : 99.975
Q1) Find number of integral solutions of 2/x + 3/y = 1/10

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  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 * F1
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

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  99.70% / XAT 2017  QA : 99.975
Q2) Find the smallest positive integer such that (n13)/(5n+6) is a nonzero reducible fraction.

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  99.70% / XAT 2017  QA : 99.975
hcf(n13,5n+6)
hcf((n13,5n+65n+65))
hcf(n13,71))
so reducible form for min n=
n  13 = 71
so n = 84

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  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.

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  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

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  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?

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  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

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  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

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  99.70% / XAT 2017  QA : 99.975
x + y + z = 15
so positive solutions = 151c31 = 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

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  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.

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  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

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  99.70% / XAT 2017  QA : 99.975
Q7) Find all primes p, q, so that p^2 − 2q^2 = 1.

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  99.70% / XAT 2017  QA : 99.975
p^22q^2=1
(p^21) = 2 * q^2
(p1)(p+1) = 2 * q^2
when p1=1
then p+1 = 2q^2
so 2q^2=3 not possible
when p1=2
then p+1=2q^2
4=2q^2 so q^2=2 no possible
when p+1=2 and p1=q^2
p=1 and 0=q^2 not possible
now when p1=2q
and p+1=q
so 2p=3q
so p=3q/2
so q=2 and p=3 will satisfy

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  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

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  99.70% / XAT 2017  QA : 99.975
a = 7 * ( ar + ar^2 + ar^3 ... )
a=7(ar/1r)
so aar=7ar
so 1r=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

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  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?

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  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

hemant_malhotra
Director at ElitesGrid  CAT 2016  QA : 99.94, LRDI  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 ?