Thread: 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) x²+3x+k=0
b) 3x²+kx+2
c) k(x²+1)=x-k

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

a) x²+6x+k=0
b) 2x²+kx+1=0
c) (k+1)x²+4kx+9=0

'Show that x²+2kx+9 is greater than or equal to 0 for all real values of x, if k² is less than or equal to 9.

Simultaneous equations:

'Find the possible values of k if y=2x+k meets y=x²-2x-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 x²+y²=5 in two distinct points'

'Find the range of values for qhich y=kx-2 is tangent to the curve y=x²-8x+7'

Thanks for reading through this, sorry it is so long!

Originally Posted by Turple

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) x²+3x+k=0
b) 3x²+kx+2
c) k(x²+1)=x-k

Your thinking is correct. So in the first case the condition is:



expand and rearrange:



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



or:



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

CB

Originally Posted by Turple

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

a) x²+6x+k=0
b) 2x²+kx+1=0
c) (k+1)x²+4kx+9=0

'Show that x²+2kx+9 is greater than or equal to 0 for all real values of x, if k² is less than or equal to 9.

No real roots requires that the discriminant be less than 0, otherwise as before.

The discriminant of: is:



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



or:



which is:



as required

CB

Originally Posted by Turple

Simultaneous equations:

'Find the possible values of k if y=2x+k meets y=x²-2x-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 x²+y²=5 in two distinct points'

'Find the range of values for qhich y=kx-2 is tangent to the curve y=x²-8x+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