# Quant Boosters - Hemant Malhotra - Set 4

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

• a. 1
b. Less than 1
c. More than 1
d. Between 1 and 2
e. None of the above

• Note -
In R = (a + b * rtc)^k,

• If k is odd then f = X
fractional part will be f = (a - b * rtc)^n
• If k is even then f + X = 1
f = 1 - (a - b * rtc)^k

here x = (9 + 4sqrt5)^48
so f = 1 - ( 9 - 4sqrt5)^48
so x * (1 - f) = (9 + 4sqrt5)^48 * (9-4sqrt5)^48 = 1

• Q2) p, q and r are roots of the equation x^3 - 7x^2 - 6x + 5 = 0. Find the value of (p + 1)(q + 1)(r + 1).

• we need (p+1) (q+1) (r+1)
so need equation whose roots are p+1 ,q+1 and r+1
so apply transformation replace x by x-1
(x-1)^3-7(x-1)^2-6(x-1)+5=0
x^3-1-3x(x-1)-7x^2-7+14x-6x+11=0
x^3-10x^2+11x+3=0
so product of roots will be -3

• Q3) p, q, r are roots of the equation x^3 - 7x^2 - 6x + 5 = 0. Find the value of (p + q)(q + r)(r + p).

• p + q + r = 7 so p + q = 7 - r
q + r = 7 - p
and p + r = 7 - q
so we need (7-p)(7-q)(7-r)
means we need equation whose roots are 7-p,7-q,7-r or
x^3-7x^2-6x+5=(x-p)((x-q))(x-r))
put x=7
so 7^3-7 * 7^2-6 * 7+5 = - 37

• Q4) If three medians of the triangle are of the lengths 9, 12 and 15, find the area of the triangle.

• Method 1 – Direct formula
Area of triangle using medians, A = 4/3 ( Sqrt(s(s-p)(s-q)(s-r))
p, q and r are medians and s = (p+q+r)/3

Method2- ABC is the triangle, AD, BE, CF are medians and G centroid
extend GD to a point M such that GD = DM, then CM = BG
so, triangle MCG; has sides 6, 8 and 10
area = 24
area of ABC = 24 * 3 = 72.

Method 3
4 * (a^2+b^2+c^2)=3 * (9^2+12^2+15^2)
so a^2+b^2+c^2=3 * (81+225+144)/4
now we have a^2+b^2+c^2

• Q5) Given 3x^2 + x = 1, find the value of 6x^3 - x^2 - 3x + 2015.

• 6x^3+2x^2 = (2x)(3x^2+x) = 2x
and -(3x^2+x) = -1
so 6x^3-x^2-3x+2015 = 2x-1-2x+2015 = 2014.

• Q6) Let a, b and c be the roots of x^3 - 2x^2 + 2x + 4 = 0.
Find the value of 1/(a^2 + b^2) + 1/(b^2 + c^2) + 1/(c^2 + a^2) + 1/4

• a + b + c = 2
ab + bc + ac = 2
abc = -4
here a+b+c=ab+bc+ca so a^2+b^2+c^2=0
question reduces to
-(1/a^2 + 1/b^2 + 1/c^2) + 1/4
now we can easily find the value by solving this

Another method : x^3-2x^2+2x+4=0
roots are a,b,c equation whose roots are a^2,b^2,c^2
will be (x^3/2)-2x+2sqrtx+4=0
x*sqrtx-2x+2sqrtx+4=
x^3+4x+4x^2=4x^2+16-16x
so x^3+20x-12=0
now roots of this equation are a^2,b^2,c^2
so equation whose roots are 1/a^2,1/b^2,1/c^2 will be reverse of this
so 1/x^3+20/x-12=0
-12x^3+20x^2+1=0
so 12x^3-20x^2-1=0
so 1/a^2 + 1/b^2 + 1/c^2 = 5/4
so -5/4 + 1/4 = -1

• Q7) From 100 litres of milk, 10 litres of water is added and then 20 litres of solution is removed. Next 30 litres of the water is added and 20 litres of solution is removed. Find the amount of milk, in litres, in the solution now.

• Method to tackle These Type Of Questions
always take initial as those part which is not added,
here milk as it is never added
here in first step 10 litre of water is added so solution=100+10=110
in the 1st time 20 is removed out off 110 so
left-out part is 90/110 so our multiplying factor is 90/110
no 30 is added to 90 so 120 solution
in the 2nd case 20 is removed from 120 so left out is 100/120
so amount of milk= 100 * 90/110 * 100/120=750/11
PS- Always remember Our concentration should be on total solution and removed part .... Added part will only increase the value of solution

• Q8) A shopkeeper claims to sell his items at a loss of 10% but instead he secretly makes a profit of 20% by using a false weight. what is the effective profit percentage he gets on every 100gm he sells?

• Method1- Approach= He claims to sell 120 when he actually sells 100 (using false weights)
Now, he claims to sell them at a loss of 10%
He makes 120*0.9 = 108
so 8% profit

Method2- = Multiplying factor
100 * 9/10 * 12/10=108 so 8%

Method3- 20-10+(20*(-10)/100=8%

• Q9) A dishonest trader, at the time of selling and purchasing, weighs 10% less and 20% more per kilogram respectively. Find the percentage profit earned by treachery. (Assuming he sells at Cost Price)
a) 30%
b) 20%
c) 25%
d) 33.33%

• Method1- Let's say original weight is 100 now he is taking 20% extra so total 120 and selling 10% less so 90
So (120-90/90) * 100
= 30/90 * 100= 33.33%

Method2-
multiplying factor
1 * 10/9 * 6/5=1.33 so 33%

• Q10) A beaker contains milk and water in the ratio 5 : 3. If a person removes 6 L of this mixture and replaces it with pure water, the ratio of milk to water becomes 1 : 1. What is the amount of milk present in beaker initially

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