Quant Boosters - Sagar Gupta - Set 2


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    15 * 37 * 63 * 51 * 97 * 17 mod 100
    15 * 37 * (-37) * 51 * (-3) * 17 mod 100
    45 * 37^2 * 17^2 * 3 mod 100
    45 * 89 * 03 * 69
    35


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    Q6) A five digit number divisible by 3 is to be formed using numbers 0, 1,2,3,4,5 without repetition . The total number of ways in which this can be done is....
    a. 216
    b. 240
    c. 600
    d. 3125


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    1 + 2 + 3 + 4 + 5 = 15
    -> 5! ways
    0 + 1 + 2 + 4 + 5 =12
    ->5! - [ 5! /5 ] =120 - 24 =96
    total 120 +96 =216


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    Q7) The total number of ways of selecting two numbers from the set {1,2,3,4,......,30}, so that their sum is divisible by 3, is
    a. 95
    b. 145
    c. 190
    d. None of the above.


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    3k, 3k + 1, 3k + 2
    divisible by 3:
    case 1 : 3k + 3k
    10 * 9 = 90 ways
    case 2 : (3k+2) + (3k+1)
    10 * 10 = 100 ways
    total = 190 ways


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    Q8) Train X started from point A at 9:00 am with a speed 72km/hr towards station Y. 2 hrs after train Y starts from point B towards X at a speed of 90 km/hr. They cross each other at 1:30 pm. But owing to signal problems at 12:00 noon the speeds of both the trains is reduced by same quantity such that they now cross each other at 4:30 pm. Calculate the new speed of train that started from point A.


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    Train X starts at 9:00 am @72 km/hr
    at 11:00 am, it is 144 kms from A
    A...144 kms...A'......d...........B
    Now Y starts from B at 11:00am @90 km/hr
    A' and B is the position at 11:00 am
    :
    They should have met at 1:30pm
    72 * 2.5 + 90 * 2.5 = d =405 kms
    :
    But they meet at 4:30 pm
    from 11:00 to 12:00 , they travel at normal speed
    Distance reduced = 72+90=162
    Distance between them at 12:00 = 243 kms
    Now they reduce their speed by k
    (72-k) * 4.5 + (90-k) * 4.5 =243
    k=54
    72-54 =18km/hr


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    Q9) abcd is a 4 digit perfect square, then if each digit is increased by 3 the number is still a perfect square. Find abcd.


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    abcd = x^2
    1000a +100b +10c +d = x^2
    1000(a+3) +100(b+3) +10(c+3) +d+3 = y^2
    y^2-x^2=3000 +300 + 30 + 3
    y^2 - x^2 = 3 [ 1000 +100 +10 +1 ]
    y^2 - x^2 =3333
    3333=101*33
    y+x =101
    y-x =33
    y = 67
    x = 34
    x^2 =1156


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    Q10) What is the remainder when 72 to the power 7202 is divided by 625?


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    E[625]=500
    7202 mod 500 = 2
    72^2 mod 625 = 5184 mod 625 = 184


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    Q11) If a^3 + b^3 = 10 and a^2 + b^2 = 60 then find a + b?


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    (a+b)^3 = 10 + 3ab (a+b)
    (a+b)^2 = 60 + 2ab
    a+b = p
    (p^3 - 10 ) / 3p = ab
    p^2 = 60 + 2 [ (p^3 - 10 ) / 3p ]
    3p^3=180p + 2p^3 - 20
    p^3 +20 = 180p
    p=13.36


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    Q12) A and B can build a wall in 4 days and 6 days respectively working alone, while C and D can destroy the wall in 8 days and 12 days respectively working alone. If three of them start working together and the wall was built in 24 days. Which of the four persons did not work?


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    Total work =36 units
    A -> +9 units/day
    B -> +6 units/day
    C- > -4.5 units/day
    D- > -3 units/day
    24 (x+y+z) = 36
    x+y+z =1.5
    9-4.5-3 =1.5
    Hence B did not work


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    Q13) Three persons A, B and C rent the grazing of a park for Rs. 570. A put 126 oxen in the park for 3 months, B puts in 162 oxen for 5 months and C puts in 216 oxen for 4 months. What part of the rent should each person pay?
    a. 105, 220, 245
    b. 125, 205, 245
    c. 105, 225, 240
    d. data inadequate
    d. None of these


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    126 * x * 3 + 162 * 5 * x + 216 * 4 * x = 570
    x = 5/18
    A= 126 * 3 * 5/18 =105
    B=162 * 5 * 5/18 =225
    C=216 * 4 * 5/18=240
    Hence option C


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    Q14) a,b and c are three positive real numbers. The minimum value that the expression [(a+b)/2 * (b+c)/2 * (c+a)/2] can take when the product of the three numbers is 3/2 is?


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    abc = 3/2
    a+b+c > 3 [ 1.5 ]^1/3
    (a+b)/2 > 1.5[1.5]^1/3 - c/2 = 1.5^4/3 - c/2
    (b+c)/2> 1.5 [1.5]^1/3 - a/2 = 1.5^4/3 - a/2
    (a+c)/2 >1.5[1.5)^1/3 -b/2 = 1.5^4/3 - b/2
    a = b = c = (1.5)^1/3
    1.5^4/3 - 0.5* 1.5^1/3
    1.5^1/3 [ 1.5 - 0.5 ] =1*1.5^1/3 =1.5^1/3
    (1.5^1/3) ^3 = 1.5 = 3/2


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    Q15) The number 'm' and 'n' are reciprocals of each other. Both m and n are positive real numbers. If both m^3 + n^3 = 65/8, determine (m+n).


 

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