Quant Boosters  Hemant Malhotra  Set 4

Proper roots will be 1001/2, 1/3 and 1/5 (for maximum a + b)
30x^3  ax^2 + bx  1001 = 30(x  1001/2)(x  1/3)(x  1/5)
now put x = 1
a + b = 1031 + 30(1003)(4)(6)/30
so a + b = 23041
for improper it should be a = 347 and b = 1142 and the roots be ( 7/5 , 11/3 and 13/2 ) ..
a + b = 1489
OA = for proper = 23041 and for improper 1489

Q21) Determine the number of positive integral solutions (x, y) to x^y = y^(90)

Factors of 90=2 * 3^2*5=12
1,2,3,5,6,9,10,15,18,30,45,90
x=y^(90/y)
now check for square
2^2 will satisfy
3^3
and 6^2 will satisfy

Q22) Determine the number of positive integral solutions (x, y) to x^y = y^{60}.

x=y^(60/y)
all factors of 60 will satisfy
1,2,3,4,5,6,10,12,15,20,30,60
now check for squares of factors
y = 16
x = 16^((60/16)=2^15
now no cubes is satisfying
so OA=13

Q23) Determine the number of positive integral solutions (x, y) to x^y = y^{72}.

x^y=y^72
now we will look for factors of 72 and their perfect square
factors 1,2,3,4,6,8,9,12,18,24,36,72
now squares of them
16,,144 will satisfy
now cubes
3^3 , 6^3 will satisfy so overall 16 solutions

Q24) Determine the number of positive integral solutions (x, y) to x^y = y^{100}

x = y^(100/y)
factors of 100
1,2,4,5,10,20,25,50,100
now check for squares of factors
at y=16 u will get x integer
rest no value satisfying
OA = 10

Q25) If x, y and z are real numbers such that x + y + z = 4 and x^2 + y^2 + z^2 = 6 then x, y and z lie in
a) [3/2,2]
b) [2/3,2]
c) [0,2/3]
d) None of these

x+y+z=4 and x^2+y^2+z^2 =6
apply Cauchy
(y * 1 + z * 1)^2 < = (y^2+z^2)(1^2+1^2)
(4x)^2 < = ((6x^2)*2
16+x^28x < = 122x^2
3x^28x+4 < = 0
2/3 < = x < = 2
so OA=B

Q26) Between 2 junctions A and B there are 12 intermediate stations. The number of ways in which a train can be made to stop at 4 of these stations, so that no two halting stations are consecutive is?

Method1 Direct formula when u want to choose r numbers out of n numbers such that no two values are consecutive = 124+1C4=9c4
Method2
A... A1 .... A2 ... A3... A4 ..... B
here A1,A2,A3 and A4 are 4 halting stations
now A1,A2,A3,A4 should not be consecutive
NOW let x1 be d number of stations before A1
so x1>=0 (( its possible that after A , next station is A1 )
A1 and A2 should not be consecutive so there must be atleast one stations
so let number of stations x2
so x2>=1
same for A2 and A3
x3>=1
same for A3 and A4
x4>=1
now station After A4 could be zero also
so x5>=0
total station=12
and 4 are halting and 8 are other stations
so x1+x2+x3+x4+x5=8
where x1>=0 letx1=a
x2>=1 so x2=1+b
x3=1+c
x4=1+d
x5=e
so a+b+c+d+e=5
so number of ways=5+51C51=9c4

Q27) Find the number of ways of distributing 27 ladoos to Swetabh, Raman and Gaurav such that number of ladoos Swetab gets are more than the number of ladoos Raman gets which in turn is more than the number of ladoos Gaurav gets?

method 1
let G=x then R=x+y+1 and S=x+y+z+2
so 3x+2y+z+3=27
so 3x+2y+z=24
now make cases and solve u will get ur ansmethod 2
S+R+G=27
now ordered=29c2
now here find unordered
case when 2 same one different
=2S+G=27
so G=272S
so S will vary from 0 to 13 so 14 values but 1 value will be there when all same
so 2 same one different case will be 13
and when all same
3S=27 so 1 case
so 6 * a + 3 * 13 + 1=29C2
so find a and that will be your answer

Q28) If numbers 320ab and 298cd are divisible by 35 and 65 respectively then find the number of divisors of [398cd – 320ab + 2]^10 where 320ab is the largest possible number satisfying the given conditions

320ab mod 35
means div by 5 and div by 7
so b could be o or 5
0ab32 mod 7=0
ab4 mod 7=0
ab mod 7=4
when b=5 then a = 9
max value of ab=95
now same process and u will find cd=45
OA: 41261 ; numbers will be 32095 and 39845.

Q29) A salesman sells two kinds of trousers: cotton and woollen. A pair of cotton trousers is sold at 30% profit and a pair of woollen trousers is sold at 50% profit. The salesman has calculated that if he sells 100% more woollen trousers than cotton trousers, his overall profit will be 45%. However he ends up selling 50% more cotton trousers than woollen trousers. What will be his overall profit?
a. 37.5%
b. 40%
c. 41%
d. 42.33%

Let Cotton CP=a
Woolen CP=b
now let Cotton = x then Woolen=2x
So SP of cotton=1.3 * x * a
and Wollen =1.5 * 2x * b=3 * x * b
now overall profit =x*((1.3a+3b))((ax+2x*b))/((ax+2bx))
((0.3a+b))/(a+2b))=45/100=9/20
6a+20b=9a+18b
so 3a=2bnow let woollen =2x then cotton=3x
so CP = 2x * a + 3x * b
and SP = 2x * 1.3a + 3x * b * 1.5
so profit=((2.6a+4.5b2a3b)/(2a+3b)
((0.6a+1.5b))/(2a+3b))
here 3a=2b
so 40% approx

Q30) 3 men and 5 women together can finish a job in 3 days. Working on the same job 3 women take 5 days more than the time required by 2 men . What is the ratio of efficiency of a man to a woman ?