Quant Boosters - Vikas Saini - Set 5



  • Some concepts before the solution

    Relative speed of 2 clock hands
    12 H = 60 div
    1 H = 5 div.
    SH = 60 M = 50 div.
    SH = 5/60 div/min.
    SH = 1/12 div/min.
    SM = 60 div / 60 min = 1div/1min
    Relative speed = 1 - 1/12 = 11/12 div/min.
    T = 60 / (11/12)
    T = 65 + (5/11)
    T = 1:05:5/11
    Both hand will coincide after every 1:05:5/11.

    Examples

    By what time both hands will coincide in between 5-6 PM ?
    (1:05:5/11) x 5
    = 5:25:25/11
    = 5:27:3/11

    By what time both hand will coincide in between 9 - 10 PM ?
    (1:05:5/11) x 9
    = 9:45:45/11
    = 9:49:1/11.

    By what time in between 2-3 PM both hand will make angle of 30°.
    angle = | 5.5 M - 30 H |
    30° = | 5.5 x M - 30 x 2 |
    30° = | 5.5 M - 60° |
    90° = 5.5 M
    M = 900 / 55 = 180/11 = 16 (4/11).
    Time = 2:16:4/11.

    By what time in between 5-6 PM both hands will make angle of 60°.
    angle = | 5.5 M - 30 H |
    60° = | 5.5 M - 30 x 5 |
    60° = 5.5 M - 150
    210° = 5.5 M
    M = 2100/55 = 420/11 = 38(2/11)
    Time = 5:38:2/11.

    Given question

    3 Min = 5 sec
    1 min = 5/3 sec.
    60 min = 100 sec.
    1 Hour = 100 sec.
    14 Hour = 1400 sec.
    1400 sec = 23 min 20 seconds.
    Time = 10:23:20 sec.



  • Q21) If all the roots of the equation (x – m)^2 * (x – 10) + 4 = 0 are integers, find the number of distinct values that ‘m’ can have ?



  • (x – m)^2 * (x-10) = -4.
    -4 = 4 x (-1) , 1 x (-4),

    (x – m)^2 (x -10) = 4 x (-1)
    x – 10 = -1.
    x = 9.
    (9 – m)^2 = 4.
    (9 – m) = 2, -2.
    m = 7, 11.

    (x – m)^2 (x – 10) = 1 x (-4)
    x – 10 = -4.
    x = 6.
    ( 6 – m)^2 = 1.
    6 – m =1, -1.
    m = 5, 7.

    Hence m can take three values.



  • Q22) Find the number of integer solutions of the equation x^2/y = 4x – 3, where x and y are non zero real numbers



  • Suppose x^2 / y = t.
    t = 4x – 3.
    x = 1, t = 1, y = 1.
    x = 3, t = 9, y = 1.
    Only two integer solutions.



  • Q23) If a, b are integers then how many ordered pairs (a, b) satisfy the equation a^2 + ab + b^2 = 1 ?



  • a^2 + ab + ab + b^2 – ab = 1
    (a+b)^2 = 1 + ab.
    a = 0, b = 1.
    a = 1, b = 0.
    a = 0, b = -1.
    a = -1, b = 0.
    a = 1, b = -1.
    a = -1, b=1.
    6 ordered pairs.



  • Q24) x^2 * y^3 = 8, where x, y > 0. What is the minimum value of 4x + 3y ?



  • 4x + 3y = P.
    4x = 2k
    x = k/2.
    3y = 3k.
    y = k.
    (k/2)^2 k^3 = 8.
    K^5/4 = 8.
    K = 2.
    4x + 3y = 2k + 3k = 5k = 10.



  • Q25) [ log (1) + log (1+3) + log (1+3+5) ... log(1+3+5…….19) - 2 (log 1 + log 2 ... log 7) = m + nx + ay. If log 2 = x and log 3 = y, then find the value of m, n & a ?



  • (log 1 + log 4 + log 9……..log 100) – 2(log 1 + log 2…….log 7)
    = 2(log 1 + log 2 + log 3……..log 10) – 2( log 1 + log 2…………log 7)
    = 2 (log 8 +log 9 + log 10)
    = 2(3log2 + 2log3 + 1)
    = 2 + 6log 2 + 4log 3
    = 2 + 6x + 4y
    m = 2, x =6, y =4.



  • Q26) All three roots of the cubic equation x^3 - 10x^2 + 31x - k = 0 are prime numbers. What is the value of K ?



  • Let’s three roots l,m and n.
    l + m + n = 10.
    lm + mn + nl = 31.
    lmn = k.
    all roots are prime.
    Hence l = 2, m = 3, n = 5.
    lmn = 30.




  • {x} = x – [x]

    (a) 0
    (b) 1
    (c) 2
    (d) 3








  • Hence 0.



  • Q28) When ‘2’ is added to each of the three roots of x^3 - Ax^2 + Bx - C = 0, we get the roots of x^3 + Px^2 + Qx - 18 = 0. If A, B, C, P and Q are all non zero real numbers then what is the value of (4A + 2B + C).



  • Let three roots of first equation are l, m, n respectively.
    18 = (l+2)(m+2)(n+2)
    18 = (lm+2l+2m+4)(n+2)
    18 = lmn + 2lm + 2ln + 4l +2mn + 4m + 4n + 8.
    10 = lmn + 2(lm+ln+mn) + 4(l+n+m)
    10 = C+2B+4A.



  • Q29) If a, b and c are root of the equation 3x^3 + 42x + 93 = 0, then what is value of a^3 + b^3 + c^3 ?



  • Here coefficient of x^2 = 0.
    means a+b+c = 0.
    Then a^3 + b^3 + c^3 = 3abc = 93



  • Q30) ax^2 + bx + c = 0 is a quadratic equation with rational coefficients such that a + b + c = 0, then which of the following is necessarily true ?
    (a) Both the roots of this equation are less than 1.
    (b) One of the roots of the equation is c/a.
    (c) Exactly one of the root is 1.
    (d) b & c both.


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