Quant Boosters by Hemant Malhotra - Set 17


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    A person takes a loan of Rs.200 at 5% simple interest . He returns 100 at the end of one year . In order to clear his dues at the end of 2 years , he would pay :
    a) 125.50
    b) 110
    c) 115.50
    d) NOT

    after 1 year SI=10 rs
    so amount =210 rs
    he returned 100 rs so 110 rs remaining
    now SI=110 * 5/100=5.5
    so amount =110+5.5=115.5

    For what value of x would y assume the maximum value where y= min (x+3,2x-1,2-x). Also find the maximum value of y.

    just check those are increasing or decreasing
    x+3 = increasing
    2x-1 = increasing
    2-x decreasing
    so compare
    x+3=2-x so x=-1/2
    2x-1=2-x
    so x=1
    now check for these two values
    at x=1 y=min(4,1,1)=1
    and x=-1/2 , y=min((5/3,-2,5/2)=-2
    so max of min will be 1

    At what time between 4 to 5 PM Minute hand and hour hand will make angle of 60

    90 degree = 15 div
    so 60 degree =10 div
    At 4 there is a gap of 20
    but we need a gap of 10
    so either minute hand will be 10 before or after the hour hand
    so (20-10 )=10 and (20+10 ) = 30
    so 4 past ((12 * 10/11 ) or 4 past ((12 * 30/11)

    In an alloy, zinc and copper are in ratio 1:2. In second alloy the same elements are in ratio 2:3. In what ratio should these two be mixed to form a new alloy in which zinc and copper are in ratio 5:8

    zinc in alloy 1 =1/3
    zinc in alloy 2 = 2/5
    in new alloy = 5/13
    1/3 --------------------- 2/5
    ............ 5/13..............
    A ............................B

    A/B= 2/5-5/13 / 5/13 -1/3
    = so 3:10

    Shoelace Theorem to find area of polygons with given vertices

    Find area of triangle (2,4) (3,-8) (1,2)
    2 .... 4
    3 .... -8
    1.... 2
    2 .... 4 (this is repeated from first point)
    then 2 * -8 + 3 * 2 + (1 * 4 ) = -6
    and then 4 * 3 + (-8) * 1 + (2 * 2) = 8
    so take difference of these two
    -6 - 8 = -14 take mod 14
    so area will be 1/2 * 14 = 7

    Another example, if we want to find area of (1,1) (2,3) (3,4) (5,6) (-1,-1)
    1……..1
    2 …….3
    3……..4
    5……...6
    -1…….-1
    1……...1 (this is repeated from first point)
    1 * 3 + 2 * 4 + 3 * 6 - 5 * 1 - 1 = 23
    now
    1 * 2 + 3 * 3 + 4 * 5 - 6 - 1 = 24
    now take difference = 24 - 23 = 1
    so area = 1/2 * 1 = 1/2

    Find range of a for which one negative and two positive roots of x^3-3x+a=0 are possible

    Approach x^3-3x=-a
    let f(x)=x^3-3x and g(x)=-a
    For 2 positive and 1 negative value g(x) will cut positive x axis 2 times and negative x axis one time
    this is possible when -2

    Kangna decides to participate in a beauty contest in her office.There are 123 contestants in all,including her. As all the participants enter a room and sit around a round table,she observes that all the other girls are prettier than her. Getting jealous,she devises a divide and rule policy.Soon a fight breaks out such that the contestants start killing each other,in the order : 1st kills 2nd,hands over the weapon to 3rd, who kills 4th and so on..Find the sum of the number on the seats she should sit if she wants to be among the last 2 alive. Numbering is 1 to 123 clockwise.

    For last survivor represent number in form of 2^m +n
    where m is as maximum possible here 123 so 2^m +n=123 so 2^6 +59=123
    so m=6 and n=59
    then last survivor will be 2 * n+1
    so last survivor=2 * 59+1=119th number
    for 2nd last represent number in form of 3*2^m +n
    so 3 * 2^m +n=123
    3 * 2^5 +27 =123
    then 2nd last survivor will be 2 * n+1=2 * 27+1=55th so sum=119+55=174

    If you want to read Extension of this Concept Follow this link http://www.scribd.com/doc/97328978/Josephus-Problem

    If a,b are integers and a-b is even then which of the following must always be even?
    (A) ab
    (B) a^2 + b^2 + 1
    (C) a^2 + b + 1
    (D) ab - b

    a-b=2m
    so a & b both odd or a & b both even
    a) when both odd then product odd so rejected
    b) when both even then even+even+1=odd
    c) when both odd then odd+odd+1=odd so rejected
    d) when both odd then odd * odd - odd = even or even * even - even = even so D ans

    Let F(x) =x^5+x^4+x^3+x^2+x+1 . Find remainder when F(x^12) is divided by F(x)

    x^n-a^n=(x-a)(x^(n-1)+x^(n-2) + ... a^(n-1)
    f(x^12)=1+(x^12)+(x^12)^2+(x^12)^3+(x^12)^4+(x^12)^5
    f(x^12)=1+(((x^12 -1)+1+(x^12)^2 -1 +1 +((x^12)^3 -1 +1 +(x^12)^4 -1 +1 +(x^12)^5 -1+1
    f(x^12)=6+((x^12-1)+(x^12)^2-1+(x^12)^3 -1+(x^12)^4 -1 +(x^12)^5-1
    now x^12-1=(x^6-1)(x^6+1)
    x^6-1=(x-1)(1+x+x^2+x^3+x^4+x^5)
    so every term will contain x^6-1 and will be divisible by 1+x+x^2+x^3+x^4+x^5
    so OA=6

    Two different positive numbers x and y such that each differ from their reciprocals by 1 . Find x + y

    x and y will be roots of |a-1/a|=1
    so a-1/a=1 or a-1/a=-1
    a^2-a-1=0 or a^2+a-1=0
    a=(1+-√(5))/2 or a=(-1+-√(5))/2
    so 4 values and sum of x and y will be (1+√5)/2 + (-1+√5)/2= √5


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