Mathematics · May / June 2021 · Paper 41 · Question 8

y

3d

y

=

6 e

x

d

x

2

+∫

3

−

2

x

−

2

x

y

3

= −

3 e

x

3e

d

x

2

2

3

3

−

2

x

−

2

x

y

3

= −

3 e

x

−

e

+

c

2

2

Substitutes (

)

0,1

⇒

c

=

3

(

)

3

2

−

2

x

−

2

x

y

= −

2 e

x

−

e

+

2

3

2

x

2

,e

−

0

( )

3

2

As

x

→∞

→

and

y

=

2

⇒

y

=

2

(a)

1

1

∫

∫

∫

1

∫

−

−

3d

−

2

x

3

−

2

x

B1: Separates the variables either

y

y

=

6 e

x

d

x

or

y

d

y

=

x

e

d

x

6

Condone with missing integral signs but the dx and dy must be present and in the correct positions

1

2

−

M1: For integrating the lhs

y

3

→

y

3

∫

±∫

−

2

x

−

2

x

M1: For integrating the rhs by parts the right way around. Look for

x

e

d

x

→

.. e

x

...e

−

2

x

−

2

x

dM1: For fully integrating the rhs to obtain

.. e

x

±

...e

. Depending upon the previous M

A1: Correct integration with or without '+ c'.

2

2

3

3

3

−

2

x

−

2

x

3

−

2

x

−

2

x

Look for

y

= −

3 e

x

−

e

+ or equivalent such as

c

3

y

= −

6 e

x

−

3e

+

a

2

2

dM1: Must have a ''+c '' now. Substitutes (

)

0,1

⇒

c

=

...

It is dependent upon having a correct attempt to integrate one side so M1 or M2 must have been awarded.

)

(

3

2

−

2

x

−

2

x

A1: CSO

y

= −

2 e

x

−

e

+

2

(b)

−

2

x

2

(

)

3

M1: For

e

→

0

y =

''2''

2

− with g(0)

kx

Follow through on their g( )x in

y

=

g( )

x

but g( )x must be a function of e

≠

2

Implied by a correct decimal answer for their

y

=

g( )

x

3

3

A1:

y =

2

2

o.e such as

y =

8

cso. ISW after sight of this. Condone

y = ±

2

2

o.e.