TMUA Practice Question — Worked Solution

P2·Differentiation·6.5Mock 0 · P2 · Q10

tangentquarticintersection countcrossing vs touchingsufficient not necessary

The curve

C

has equation

y = x^{4} - 2 x^{2}

and

L

is the line

y = m x + c

, where

m

and

c

are real constants.

L

is a tangent to

C

if the gradient of

C

equals

m

at one or more of the points where

L

meets

C

.Consider the following two conditions on

L

:Condition I:

L

meets

C

at exactly one point.Condition II:

L

meets

C

at exactly three points.

For

L

to be a tangent to

C

, which one of the following correctly describes the two conditions?

	Condition I	Condition II
A	necessary and sufficient	not necessary and not sufficient
B	necessary but not sufficient	not necessary and not sufficient
C	sufficient but not necessary	sufficient but not necessary
D	sufficient but not necessary	not necessary and not sufficient
E	not necessary and not sufficient	sufficient but not necessary
F	not necessary and not sufficient	not necessary and not sufficient

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