Practice Paper A1 Ms

EDEXCEL CORE MATHEMATICS C3 Question Number

PRACTICE PAPER A1 MARK SCHEME Scheme

Marks

( x − 3)( x − 5) 2 x( x + 3) × ( x − 3)( x + 3) ( x − 5) 2

1.

=

(3 × factorising)

2x x−5

B1 B1 B1

B1

(4 marks) 2.

(a)

f(x) = x + ln2x − 4;

xn + 1 = 4 − ln2xn, x0 = 2.4

x1 = 2.431…

A single sound application of

x2 = 2.418…

iteration

M1

x3 = 2.423…

At least x3 reached

M1

Root = 2.422 (A2)

(b)

2.42 or “correct” unrounded to 3 d.p. answer A1

A2, 1, 0

Choosing an appropriate interval e.g. [2.4215, 2.4225]

M1

Establishing change of sign + Conclusion

A1

(4)

(2)

(6 marks) 3.

(a)

y = f(x)

y

Fairly even , vertex on +ve x axis Only ( a2 , 0) and (0, a) on graph on in text

a

a 2

(b) y = f(2x)

Steeper, even and 1 correct intersection Only both ( a4 , 0) and (0, a) on graph or in text

a

(c)

− (2x − a) = 2x − a =

1 x 2

B1

(2)

x

y

a 4

B1

B1 [ft a2 from (a)]

B1

(2)

x

1 x 2

when x = 4, ⇒ a − 8 = 2

∴a = 10

M1, A1

when x = 4, ⇒ 8 − a = 2

∴a = 6

M1, A1

(4)

(8 marks)

EDEXCEL CORE MATHEMATICS C3

PRACTICE PAPER A1 MARK SCHEME

Question Number

Scheme

sin 2 θ 1− 1 − tan 2 θ cos 2 θ = 2 sin 2 θ 1 + tan θ 1+ cos 2 θ cos 2 θ − sin 2 θ cos 2θ = cos 2 θ + sin 2 θ 1

4.

Marks

  sin 2 θ  1 − cos 2 θ   or sec2 θ or equivalent      = cos 2θ

*

M1 M1

M1 A1

(4)

(4 marks) 3 x−4 + x( x + 2) ( x + 2)( x − 2)

5.

B1 B1

=

3( x − 2) + x( x − 4) x( x + 2)( x − 2)

M1 A1

=

( x − 3)( x + 2) x( x + 2)( x − 2)

M1 A1 A1

(7 marks) 6.

(a)

(b)

f″(x) = 2x − 2x−3 =8−

6 31 =7 (7.97) 24 32

A1

f(x) =

1 3 1 x − 2x − (+C) 3 x

M1 A1

0=9−6− (c)

M1 A1

1 +C 3

C=−

8 3

(or −2.67)

f`(x) > 0 needed, or f`(x) ≥ 0, or “as x increases, f(x) increases” f`(x) = (x −

1 2 ) , > 0 always, or ≥ 0 always x

(3)

M1 A1

(4)

B1 M1, A1

(3)

(10 marks)

EDEXCEL CORE MATHEMATICS C3 Question Number

7.

(a)

i.e

PRACTICE PAPER A1 MARK SCHEME Scheme

f(x) =

2(2 x + 1) − 6 4x − 4 ,= ( x − 1)(2 x + 1) ( x − 1)(2 x − 1)

f(x) =

4( x −1) 4 ,= ( x − 1)(2 x − 1) (2 x + 1)

Marks M1, A1

(M for attempt same denominator)

*

M1, A1 c.s.o

(M for factorising)

(4) α < f < β, α = 0 or β =

(b)

0
4 3

or 0 < y <

4 3

4 3 Both

(c)

y=

4 2 x =1

i.e x =

4− x (o.e) 2x

Range of f−1 = domain of f ∴ f−1 > 1 or y > 1 or > 1

B1

(2)

M1

4− y 2y

∴ f−1(x) = (d)

⇒ y (2x + 1) = 4

B1

M1 must be f−1(x)

A1

(3)

B1

(1)

(10 marks)

EDEXCEL CORE MATHEMATICS C3

PRACTICE PAPER A1 MARK SCHEME

Question Number

8.

Scheme (a)

y = ln(3 x − 6 ) ⇒ x=

(b)

y

+6 ; 3

{f

−1

ex + 6 ( x) } = 3

x ∈ℜ

Domain: Range:

(c)

e

⇒ 3x − 6 = e y

f

−1

( x) > 2

Attempting to find f

−1

e3 + 6 ]; (3) [ = 3

= 8.70

(d) ln curve passing through y = 0 Symmetry in x = k, k > 0 All correct and asymptote at x = 2 labelled

(e)

Meets y-axis: Meets x-axis:

(x = 0), y = ln 6 x=

5 3

, (0) ;

x=

7

, (0)

3

[May be seen on g