# A level maths questions on inequalities and simultaneous equations

• February 21st 2009, 02:11 AM
Turple
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!
• February 21st 2009, 02:32 AM
CaptainBlack
Quote:

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:

$3^2-4k>0$

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
• February 21st 2009, 02:38 AM
CaptainBlack
Quote:

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: $x²+2kx+9$ is:

$4k^2-4\times9.$

That $x²+2kx+9$ 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
• February 21st 2009, 02:41 AM
CaptainBlack
Quote:

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