About cookies

The NCETM site uses cookies. Read more about our privacy policy

Please agree to accept our cookies. If you continue to use the site, we'll assume you're happy to accept them.

 

Personal Learning Login






Sign Up | Forgotten password?
 
Register with the NCETM

Proof : Key Stage 5 (AS-Level) : Mathematics Content Knowledge


Key Stage
Key Stage
Topic
Topic
Questions

Next Steps
Next Steps

Enter the Self-evaluation Tools
Self-evaluation Tools
Currently viewing
Key Stage 5 (AS-Level)
Proof
Question 1 of 1

1. How confident are you that you can give a clear and convincing argument to show:

a. the validity of a mathematical statement?


Example



Examples 1 & 2 Examples 3 & 4
   

Example 1:
Show that x2 + kx + 4 has no real roots for -4 < k < 4.

The discriminant of ax2 + bxc is b2 – 4ac. In this case b2 – 4ac = k2 – 4 × 1 × 4 = k2 – 16

For no real roots b2 – 4ac < 0
k2 – 16 < 0
k2 < 16

k < 4 or -k < 4
k < 4 or k > -4
⇒ -4 < k < 4

Example 2:
Show that (sin x + cos x)2 + (sin x – cos x)2 = 2

(sin x + cos x)2 = sin2 x + 2sin x cos x + cos2 x
(sin x – cos x)2 = sin2 x – 2sin x cos x + cos2 x

So (sin x + cos x)2 + (sin x – cos x)2
= 2sin2 x + 2cos2 x
= 2(sin2 x + cos2 x
= 2

Related information and resources from other sites

 
 
Add to your NCETM favourites
Remove from your NCETM favourites
Add a note on this item
Recommend to a friend
Comment on this item
Send to printer
Request a reminder of this item
Cancel a reminder of this item