Solved Questions On Finding Number Of Integral Solutions Concept - Vikas Saini



  • Q.1) How many integral values of x satisfy the equation x = |2x - |120-3x|| ?

    Case 1
    x = 2x - |120 – 3x|
    x = |120 – 3x|
    case a) x = 120 – 3x
    x = 30.
    Case b) x = 3x – 120
    x = 60.

    Case 2
    -x = 2x - |120 – 3x|
    3x = |120 – 3x|
    3x = 120 – 3x
    x = 20.
    x = 20,30,60.
    3 values.

    Q. 2) How many integral values of (x,y) satisfy the equation x^2 – y^2 = 627 ?

    x^2 – y^2 = 627.
    627 = 3 x 11 x 19.
    Total number of factors = 8.
    This can be done in negative too.
    Total integral values = 2 x 8 = 16.

    Q.3) Find the number of positive integral positive solution for a+b+c = 20.

    By direct formula,
    x1 + x2 ... xK = N.
    Positive solution = (N – 1)C(K-1)
    = (20 – 1)C(3-1)
    = 19C2
    = 171.

    Q.4) Find the number of positive integral solutions of x1+x2+x3+x4 = 20, where x1 > x2 ?

    x1 + x2 + x3 + x4 = 20.
    If a + b + c ....... r = k.
    Then by total positive solution = (k-1)C(r-1)
    Total positive solution = (20-1)C(4-1) = 19C3 = 969.
    But we need to remove case when x1 = x2.
    x1 + x1 + x3 + x4 = 20.
    2x1+x3+x4 = 20.
    Total solutions = 17+15+13+11+9+7+5+3+1 = 81.
    This will be removed from total solutions = 969 – 81 = 888.
    now only possibility in above equation is that x1 < x2 and x1 > x2.
    So we need to take only x1 > x2 cases.
    Hence total positive solution = 888 / 2 = 444.

    Q.5) Find the number of positive integral for a,b,c and d such that their sum is not more than 15.

    a + b + c + d < 15.
    a + b + c + d = 14,13,12,11,10,9,8,7,6,5,4.
    Total no of positive solution = 13C3 + 12C3 ... 1
    = 286+220+165+120+84+56+35+20+10+4+1
    =1001.

    Q.6) How many integral values of x wil satisfy |x-3|+|2x+4|+|x| < = 11.

    X = 0,1,2,-1,-2,-3.
    Total solution = 6 values.

    Q.7) Number of integral solutions of x^2 – y^2 = 287 ?

    287 = 7 x 41.
    Total no of factors = 2 x 2 = 4.
    It can be written in negative form also.
    Total integral solution = 2 x 4 = 8.

    Q.8) Total number of integral solution of a^3 + b^3 + c^3 = 43655.

    Not any number can be written as 9n+-4.
    43655 mod 9 = -4 or 5.
    Hence 0.

    Q.9) Find the total no of integral solutions of w^4 + x^4 + y^4 + z^4 = 1797.

    Unit digit of N^4 = 1,6,5, where N is a natural number.
    Unit digit of Sum of w^4,x^4,y^4,z^4 never can be get as 7.
    Hence 0 solution.

    Q.10) How many integral values of x will satisfy x^2 – 2|x|-8 > = 0.

    Case 1
    X^2 -2x-8 >= 0.
    X^2 – 2x -8 = 0.
    X = 4,-2.
    x^2 – 2x – 8 = -9.
    x = 1.

    Case 2
    x^2 + 2x – 8 >=0.
    x^2 + 2x – 8 =0.
    X = -4,2.
    x^2 + 2x – 8 = -9.
    x^2 +2x +1 = 0.
    x = -1.

    Case 3
    x^2 – 2x -8 = 8.
    x= 0,2.

    For values of x = -4,-2,-1,0,1,2,4.
    Total values = 7.

    Q.11) Find the number of total integral solutions of x^2 + y^2 = 100.

    x^2 + y^2 = 100.
    (0,10)(10,0)(-10,0)(0,-10),(6,8)(6,-8),(-6,8),(-6,-8)
    Total solution =8.

    Q.12) Number of integral solution of x^2 + y^2 = 36.

    x^2 + y^2 = 36.
    (0,6)(0,-6),(-6,0),(6,0)
    Total = 4 solutions.

    Q.13) Number of integral solutions x^2 + y^2 < 36.

    Sum of square possible below 36 are 0,1,4,9,16,25,2,10,17,26,8,13,20,29,18,34,32.
    x^2 + y^2 = 0 (0,0)
    1 solution
    X^2 + y^2 = 1[(0,1),(0,-1),(-1,0),(1,0)]
    4 solution.
    X^2 + y^2 = 9. [(0,3)(0,-3),(3,0)(-3,0)]
    4 solution.
    X^2 + y^2 = 16. [(0,-4),(0,4),(4,0)(-4,0)
    4 solutions.
    X^2 + y^2 = 25. [(0,5),(0,-5),(5,0),(-5,0),(3,4),(-3,-4),(3,-4),(-3,4)
    8 solutions.
    As same as for sum = 2,8,18,32.
    Each has 2 solutions.
    And sum = 10,17,26,13,20,29,34.
    Each has 4 solutions.
    Total solutions = 1+4+4+4+8+2x4+4x7
    = 57.

    Q.14) Number of integral solutions of equation a+b+c+d = 30.
    Where a >=2, b>= 0, c>=-5, d >=8.

    (a+2)+(b+0)+(c-5)+(d+8) = 30.
    a+b+c+d = 25.
    Total integral solutions = (25+4-1)C(4-1) = 28C3 = 3276.

    Q. 15) Find number of positive integral solution of abc = 120 ?

    abc = 120.
    120 = 2^3 x 3 x 5.
    By direct formula,if N = x^a x y^b x z^c
    Then total solution = (a+2)C2 x (b+2)C2 x (c+2)C2
    Total number of solution = (3+2)C2 x (1+2)C2 c (1+2)C2
    = 10 x 3 x 3
    = 90.

    Q.16) If E = |x| + |x – 1| + |x-2| + |x-3| + |x-4|, for how many integral values of x is E less than or equal to 54 ?

    x = -8 to 12.
    Total 21 values.

    Q.17) Find number of positive integral values satisfying (x^2 + 5x + 6 )^(x-1) = 36.

    36 = 6^2, 3^2 x 2^2.
    x– 1 = 2.
    X = 3.
    X^2 + 5x +6 = (x+3) (x+2)
    (x+3)^(x-1) (x+2)^(x-1) = 3^2 x 2^2.
    x= 0.
    None of the value of x satisfy here.
    Hence 0.

    Q.18) Find number of positive integral values satisfying (x+y)^(y-1) = 100.

    100 = 10^2 , (-10)^2.
    Y - 1 =2.
    Y = 3.
    (x + y) =10, -10.
    x + 3 = 10, -10.
    x = 7, -13.
    Hence two solution.

    Q. 19) Determine the number of positive integral solutions (x,y) to x^y = y^90.

    90 = 2 x 3^2 x 5.
    No of factors =2 x 3 x 2 = 12.
    Factors are 1,2,3,5,6,9,10,15,18,30,45,90.
    (x,y)=(1,1),(2^89,2),(3^87,3),(5^85, 5),(6^84,6),(9^81,9),(10^80,10),(15^75,15),(18^72,18),(30^60,30),(45^45,45),(90,90)
    Hence 12 positive solution

    Q.20) Determine the number of positive integral solutions (x,y) to x^y = y^100.

    100 = 2^2 x 5^2.
    No of factors = 3 x 3 = 9.
    Factors = 1,2,4,5,10,20,25,50,100.
    (x,y) = (1,1),(2^98,2),(4^96, 4),(5^95,5),(10^90,10),(20^80,20),(25^75,25),(50^50,50),(100,100).

    Hence 9 positive solution.

    Q.21) Determine the number of positive integral solutions (x,y) to x^y = y^72.

    72 = 2^3 x 3^2.
    No of factors = 4 x 3 = 12.
    Factors = 1,2,3,4,6,8,9,12,18,24,36,72.
    (x,y) = (1,1),(2^70,2), (3^69,3)(4^68,4),(6^66,6),(8^64,8),(9^63,9),(12^60,12),(18^54,18),(24^48,24),(36^36),(72,72).
    Hence 12 solutions.

    Q.22) How many pairs of integers (a,b) are possible such that a^2 – b^2 = 288 ?

    288 = 2^5 x 3^2.
    No of factors = 6 x 3 = 18.
    Total no of integer solution = 18.

    Q.23)Find unordered number of integral solutions of a x b x c = 110.

    110 = 2 x 5 x 11.
    Factors = 8
    No of solution = 3 x 3 x 3 = 27.
    No of integral solution = 4 x 27 = 108.
    Unordered integral solutions = 108 / 3! = 18.
    Hence 18 solutions.

    Q.24) Find the integral solutions of |x| + |y| = 7.

    Direct formula
    |x| + |y| = n.
    Total solution = 4n.
    Total integral solution = 4 x 7 = 28.

    Q.25)Find integral solution of |x| + |y| < 7.

    |x| + |y| = 0,1,2,3,4,5,6.
    |x|+|y| = 0.
    Solution :- 1.
    |x| + |y| = 1.
    As per direct formula, total solution = 4 x 1 = 4.
    Total solutions = 1 + 4(1+2+3+4+5+6)
    Total solution = 85.

    Q.26) Find integral solutions |x| + |y| = < 7.

    |x|+|y| = 0,1,2,3,4,5,6,7.
    Total solutions = 1 + 4(1+2+3+4+5+6+7)
    = 113.

    Q.27) Find the integral solutions of |x+2| + |y| = 7.

    By direct formula,
    Total solutions = 4 x 7 = 28.

    Q.28) Find the integral solutions of |x+2|+|y| < = 7.

    By direct formula,
    Total solutions = 1 + 4(1+2+3+4+5+6+7)
    = 113.

    Q.29) Find the integral solutions of |x+2| + |y+4| = 7.

    By direct formula,
    Total solution = 4 x 7 = 28.

    Q.30) Find the integral solutions of |x+2| + |y+4| < = 7.

    By direct formula,
    Solutions = 1 + 4(1+2+3+4+5+6+7)
    = 113.

    Q.31) Number of integral solutions xy = 2x – y.

    xy = 2x – y.
    Divided by xy both sides
    1 = 2/y – 1/x.
    1 = (y-2) (x+1)
    -2 = (x+1) (y-2)
    (-1) x (2) = (x+1) (y-2)
    (2) x (-1) = (x+1) (y-2)
    (-2) x (1) = (x+1) (y-2)
    1 x (-2) = (x+1) (y-2)

    4 integral solutions

    Q.32) Total no of positive integral solutions of a + b + c + d < 14.

    Let’s take a dummy variable e
    a + b + c + d + e = 14.
    By direct formula
    Positive solutions = (14 – 1)C(5-1)
    = 13C4
    = 715.

    Q.33)How many integral possible for below given equation 3/x – 2/y= 1/12.

    36/x – 24/y = 1.
    (x – 36) (y + 24) = -36 x 24.
    36 x 24 = 2^2 x 3^2 x 2^3 x 3
    = 2^5 x 3^3.
    Total solutions = 24.

    Q.34) How many integral pairs (x,y) satisfy 2^x – 3^y = 55 ?

    2^x = 55 + 3^y.
    X = 6, y =2.
    Only 1 solution

    Q.35) Number of integral solutions of 1/x + 1/y = 1/48.

    48/x + 48/y = 1.
    (x-48) (y-48) = 48 x 48.
    48^2 = 2^8 x 3^2.
    Total no of factors = 9 x 3 = 27.
    Hence no of integral solutions = 27.

    Q.36) Number the number of positive integral solutions of the equation a(a^2 – b) = (b^3 + 61)

    a^3 – ab = b^3 + 61.
    a^3 – b^3 = ab + 61.
    a = 6, b = 5.
    Only 1 solution.

    Q.37) How many natural solutions for possible (x+1) (x-3) (x+5)(x-7)(x+9).........................(x-99) < 0.

    If we plot above equation on x-y axis then on positive side x = 3,7..........99.
    On negative side x = -1,-5...................-97.
    Total values = 75.

    Q.38) Find the number of integral solution of x^2 + y^2 = 70.

    Sum of square of two numbers can’t be in form of 4n + 2.
    Hence 0 solution.

    Q.39) How many natural solutions are possible for the equation 2^n – 7 = x^2.

    2^n = x^2 + 7.
    n = 3 then x = 1.
    n = 4 then x =3.
    n =5 then x = 5.
    n = 7 then x = 11.
    and so on.

    Q.40) If 4y – 3x = 5, what is the smallest integer value of x for which y > 100 ?

    4y – 5 = 3x.
    y = k + 100.
    4(k + 100) – 5 = 3x.
    4k + 400 – 5 =3x.
    4k + 395 = 3x.
    K =1 then x = 133.
    Smaller value of x is 133.

    Q.41) Total number of integral solutions of 13x – 3y = 1000 for 100 < x < 200.

    13x = 3y + 1000.
    y = 113, then x = 103.
    Y = 126, then x = 106.
    Y = 139, then x = 109.
    So on..
    X = 103,106,109..............199.
    Total 33 values.



  • This post is deleted!

Log in to reply
 

Looks like your connection to MBAtious was lost, please wait while we try to reconnect.