Quant Boosters - Sagar Gupta - Set 2


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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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    m = 1/n
    mn = 1
    m^3 + n^3 +3mn [ m+n ] = (m+n)^3
    65/8 + 3 x = x^3
    5/2 = x


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    Q16) The median of five positive integers is 7. If the only mode is greater than the median and the mean is greater than 9, what is the lowest possible value of the mode of these 5 integers?


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    a < b < c < d < e
    c = 7
    d and e are same
    a + b + c + d + e > 45
    5 = a
    b = 6
    c = 7
    18 + 2d > 45
    2d > 27
    d > 13.5
    d = 14
    e = 14
    Mode =14


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    Q17) The number of factors of the number N = 4^6 + 6^8 is
    a. 18
    b. 36
    c. 54
    d. 72


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    2^12 + 2^8 * 3^8
    2^8 [ 16 + 3^8 ]
    2^8 [ 16 + 81 * 81 ]
    2^8 [ 16 + 6561 ]
    2^8 [6577]
    9 * 2 = 18 factors


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    Q18) When a natural number, N is divided by D, the remainder is 35. When 50N is divided by D, the remainder is 11. Find D
    a. 1739
    b. 43
    c. 47
    d. Cannot be determined


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    n=ad +35
    50n=bd +11
    50n=50ad+1750
    50n=bd+11
    50ad+1750 = bd +11
    1739= (b-50a)d
    d can be 1739 or 47
    CBD


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    Q19) In a row at a bus stop, A is 9th from the right and B is 7th from the left. They both interchange their positions. If there are 20 people in the row, what will be the new position of B from the left'?
    (1) 11th
    (2) 12th
    (3) 13th
    (4) 10th


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    1,2,3,4,5,6,B,8,9,10,11,A,13,14,15,16,17,18,19,20
    So, 12'th from left


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    Q20) Find the sum of all positive two-digit integers that are divisible by each of their digits


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    {11,22...99}
    10a+b mod a=0 -> b mod a=0
    10a+b mod b=0 ->10a mod b=0
    when a=b , the above 2 conditions will be satisfied -> {11,22...99}
    also : {12,15,24,36,48 } will satisfy the above conditions
    sum =630


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    Q21) If a + b + c + d + e = 8 & a^2 + b^2 + c^2 + d^2 + e^2 = 16, where a,b,c,d& e are real numbers then
    maximum (a,b,c,d,e) = ?
    a. 4
    b. 2
    c. 16/5
    d. 6/5
    e. None of these


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    Application of cauchy-schwarz inequality
    a+b+c+d = 8-e
    a^2+b^2+c^2+d^2 = 16-e^2
    now
    (a^2+b^2+c^2+d^2)(1+1+1+1) >= (a+b+c+d)^2
    or 4*(16-e^2) >= (8-e)^2
    or 64-4e^2 >= 64+e^2-16e
    or 5e^2 < = 16e
    or e(5e-16) < = 0
    so 0 < = e < = 16/5


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    Q22) Let f(x) be a function such that f(x).f(y) - f(xy) = 3(x+y+2). Then f(4)=?
    (1) can not be determined
    (2) 7
    (3) -8
    (4) either 7 or -8
    (5) none of these


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    f(0)^2 - f(0) =6
    f(0) =3 or -2
    :
    f(0) f(2) - f(0) = 3 (0+2+2)
    f(0) [ f(2) -1 ] = 12
    if f(0) = 3 , f(2) =5
    if f(0) =-2 , f(2) =-5
    :
    f(2)^2 - f(4)= 18
    25 -18 = f(4) = 7 ( -(5)^2 = 5^2 = 25 )
    Hence only 7


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    Q23) Number of non negative solutions for x + y = x^2 - xy + y^2


 

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