WMA11 January 2020

8. WMA11/01 Edexcel IAL P1 January 2020, Q8 (Quadratics &​​ Equations & Inequalities)

The straight line l has equation y = k(2x – 1), where k is a constant.​​ 

The curve C has equation y = x2​​ + 2x + 11​​ 

Find the set of values of k for which l does not cross or touch C.​​ 

(6)

SOLUTION

To find the point of intersection​​ between the curve and line let us solve them simultaneosly.

Solving them by substitution method.

x2+2x+11=k 2x-1

x2+2x+11=2kx-k

x2+2x-2kx+11+k=0

x2+2-2kx+k+11=0

Now, since the question says that we need to find the range of values of k for which the line and curve doesn’t cross or touch each other, meaning thereby​​ there is no point of intersection.​​ 

So, in our case, there is no solution.​​ 

x2+2-2kx+k+11=0

When​​ a=1,​​ b=2-2k, &​​ c=k+11,

b2-4ac<0

2-2k2-41k+11<0

4-8k+4k2-4k-44<0

4k2-12k-40<0

k2-3k-10<0

k2-5k+2k-10<0

kk-5+2k-10<0

k-5k+2<0

Since the sign of inequality is less than, we have to choose that portion of graph which is below the x-axis.​​ 

Hence, the range of k is​​ 

-2<k<5