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Math Help - A level maths questions on inequalities and simultaneous equations

  1. #1
    Junior Member Turple's Avatar
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    Post A level maths questions on inequalities and simultaneous equations

    I have mock exams next week and have been trying to revise inequalities and sim eqs but I am finding any questions using a constant, k and the discriminant very difficult. If I post some of said questions I would appreciate some help

    Inequalities:

    'By considering the discriminant, or otherwise, find the range of values of k that give each of these equations two distinct real roots.' [I got as far as thinking that the discriminant must be greater than 0?]

    a) x+3x+k=0
    b) 3x+kx+2
    c) k(x+1)=x-k

    'Find the range of values for k that give these equations no real roots'

    a) x+6x+k=0
    b) 2x+kx+1=0
    c) (k+1)x+4kx+9=0

    'Show that x+2kx+9 is greater than or equal to 0 for all real values of x, if k is less than or equal to 9.

    Simultaneous equations:

    'Find the possible values of k if y=2x+k meets y=x-2x-7'
    a) in two distinct real points b) in just one point

    'Find the range of values of k for which kx+y=3 meets x+y=5 in two distinct points'

    'Find the range of values for qhich y=kx-2 is tangent to the curve y=x-8x+7'

    Thanks for reading through this, sorry it is so long!
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Turple View Post
    Inequalities:

    'By considering the discriminant, or otherwise, find the range of values of k that give each of these equations two distinct real roots.' [I got as far as thinking that the discriminant must be greater than 0?]

    a) x+3x+k=0
    b) 3x+kx+2
    c) k(x+1)=x-k
    Your thinking is correct. So in the first case the condition is:

    3^2-4k>0

    expand and rearrange:

    -4k>-9

    multiply through by -1 (which changes the direction of the inequality):

    4k<9

    or:

    k<9/4

    Now the others are similar, rearrange them into standard quadratics, then write out the condition on the discriminant, and simplify.

    CB
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by Turple View Post
    'Find the range of values for k that give these equations no real roots'

    a) x+6x+k=0
    b) 2x+kx+1=0
    c) (k+1)x+4kx+9=0

    'Show that x+2kx+9 is greater than or equal to 0 for all real values of x, if k is less than or equal to 9.
    No real roots requires that the discriminant be less than 0, otherwise as before.

    The discriminant of: x+2kx+9 is:

    4k^2-4\times9.

    That x+2kx+9 is greater than or equal to 0 implies that it has no real roots (as it must be positive for large absolute values of x), so this is true when:

    4k^2-36<0

    or:

    4k^2<36

    which is:

    k^2<9

    as required

    CB
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by Turple View Post
    Simultaneous equations:

    'Find the possible values of k if y=2x+k meets y=x-2x-7'
    a) in two distinct real points b) in just one point

    'Find the range of values of k for which kx+y=3 meets x+y=5 in two distinct points'

    'Find the range of values for qhich y=kx-2 is tangent to the curve y=x-8x+7'

    Thanks for reading through this, sorry it is so long!
    These all follow the same sort of pattern, you substitute y from the linear equation for the y in the other. Then simplify down to a quadratic in x.

    Then the questions all reduce to questions about the number of real roots of quadratics (tangency is equivalent to one real root).

    CB
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