Oxford University MAT 2021 Solved Papers

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Question : 3 of 10

Marks: +1, -0

Tangents to

y=e^x

are drawn at

(p, e^p)

and

(q, e^q)

. These tangents cross the

x

-axis at

a

and

b

respectively. It follows that, for all

p

and

q

,

$p a=q b$ ,
$p-a< q-b$ ,
$p-a=q-b$ ,
$p-a>q-b$
$p+q=a+b$ .

Solution:

The gradient at

p

is

e^p

and so the tangent is

y=e^p(x-p)+e^p

. This crosses the

x

-axis when

e^p(a-p)+e^p=0

which happens if

a=p-1

. Similarly

q=1-b

so

p-a=q-b

(they're both 1 ).

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