Question Bank - Number of Integral/Positive/Non-Negative Solutions - Set 1


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    Q 20) How many pair of integers are possible for x^2 - y^2 = 287?

    OA : 8


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    Q 21) How many pair of positive integers are possible for a^2 = 9555^2 + b^2

    OA: 67


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    Q 22) No of integral solution for (x-3)^3 (x-6)^4 (x-9)^9 < 0 ?

    OA : 4


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    Q 23) How many integral solutions does the equation mn – 2m = 3n + 105 have?

    OA : 8


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    Q 24) Find the number of positive integral solutions of the equation 2x + 3y = 763?

    OA : 127


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    Q 25) How many positive integral solutions are possible for 1/x - 1/y = 1/12?

    OA : 7


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    Q 26) How many positive integral solutions does equation a + b + c + d = a × b × c × d has

    OA : 12


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    Q 27) Find the sum of integral roots of the following equation: 12x^4 − 91x^3 + 194x^2 − 91x + 12 = 0

    OA : 7


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    Q 28 ) If m^3 - n^3 = 9^k + 123. Find the number of non-negative integral triplet(s) (m, n, k) satisfying the given equation.

    OA : 1


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    Q 29) Find the total number of positive integral solution of 1/x + 1/y + 1/z = 1

    OA : 10


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    Q 30) Number of non-negative integral solutions of x^(2/3) + x^(1/3 ) – 42 ≤ 0

    OA : 217


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    a + b + c ≤ 10 can be written as a + b + c + d = 10, where d is a dummy variable.

    Why we need a dummy variable here ?

    It is like we are dividing 10 items into 4 baskets and the fourth dummy basked (d) will ensure x, y and z would together have ≤ 10 always. If d = 0, a + b + c = 10 and if d = 10, a + b + c = 0 --> covering all the possible values for a + b + c which are ≤ 10.

    so the answer here is (10 + 4 - 1) C (4 - 1) = 13C3 = 286 ways



  • @zabeer
    2x+3y=763
    for y=1, x=380
    from here x will change by co-efficient of y i.e.(3) while y will change by co-efficient of x i.e (2) in opposite or one will increase while another will decrease.
    now, lets get back to there
    y=1, x=380
    y=3, x=377
    y=5, x=374
    and so on till
    y=253 , x=2
    total value= (380-2)/3 +1=127



  • @zabeer
    For x^2 - y^2 = n
    Where n is odd and not a perfect square, total number of positive integral solution is given by
    =(Number of factors of n)/2
    Now, x^2 - y^2 = 287
    287 = 7 * 41
    Number of factors = 2 * 2 = 4
    Number of positive solution = 4/2 = 2
    Taking +/- x and +/- y into consideration
    2 * 4 = 8



  • @zabeer
    For equation, x1 + x2 + x3 + ... + xn = s
    Number of positive integral solution where s >= n is given by : (s-1)C(n-1)
    So, for a + b + c = 20
    = (20-1)C(3-1)
    = 19c2
    = 171



  • @zabeer

    x1 + x2 + x3 + x4 = 20

    Total positive solutions of is (20 - 1)C(4 - 1) = 19C3 = 969

    Let number of positive integral solutions when x1 > x2 = m
    Let number of positive integral solutions when x2 > x1 = m
    Let number of positive integral solutions when x1 = x2 = n

    So, 2m + n = 969

    If x1 = x2 = p then 2p + x3 + x4 = 20
    p could be 1 - 9, so 2p would be 2 - 18
    2p = 2, x3 + x4 = 18
    2p = 4, x3 + x4 = 16
    and so on till 2p = 18, x3 + x4 = 2

    Number of solutions = 17C1 + 15C1 + 13C1 + ... + 1C1
    = 1 + 3 + 5 + ... + 17
    = 9^2
    = 81

    2m + 81 = 969
    m = (969 - 81)/2 = 444



  • @zabeer

    x^(2/3) + x^(1/3) - 42 ≤ 0
    Let x^(1/3) = z
    z^2 + z - 42 ≤ 0
    (z + 7)(z - 6) ≤ 0
    for non negative solutions, 0 ≤ z ≤ 6
    z = x^(1/3)
    so 0 ≤ x^(1/3) ≤ 6
    0 ≤ x ≤ 6^3
    0 ≤ x ≤ 216
    so we have 217 values from 0 to 216.


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    @aamin_afzal Perfect!
    Just to add, for non negative solutions it would be (s + n - 1) C (n - 1)



  • @zabeer When I do 14C4 after adding dummy variable, why I don't get answer.


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    @vkabdwal

    a + b + c + d ≤ 15
    a + b + c + d + e = 15 ( e is a dummy variable and will be zero in case of a + b + c + d = 15)
    Now as we have both natural numbers (a, b, c and d) and whole numbers (e) in the same equation, let take e = E - 1, where E is natural)
    a + b + c + d + E - 1 = 15
    a + b + c + d + E = 16
    number of positive solutions = (n - 1) C (k - 1)= 15C4 = 1365
    (trap here is that when we do a + b + c + ... we should ensure that all of them be either positive or non negative (which includes 0). You can't have both in RHS)


 

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