Practice Paper A3 Ms

EDEXCEL CORE MATHEMATICS C3 Question Number 1.

(a)

PRACTICE PAPER A3 MARK SCHEME Scheme

( )

4x = 2x

2

Marks

= u 2 or 2 ( x + 1) = 2.2 x = 2u , → u 2 − 2u − 15 (= 0 )

M1, A1 c.s.o (2)

(b)

u 2 − 2u − 15, = (u − 5)(u + 3) u = 5 ⇒ 2x = 5 ⇒ x =

M1, A1

log 5 , = 2.32 log 2

M1, A1

[Ignore any other solution]

2.

(6 marks)

(a) f`(x) = 0.5ex − 2x

M1

f`(0) = 0.5

A1 c.s.o

Equation of tangent at A is: y = f `(0)x + f(0), i.e y = 0.5x + 0.5

M1, A1

⇒

(b) f`(x) = 0

(4)

2x =

1 2

ex

M1

4x = ex

i.e ⇒

x = ln(4x)

(4)

M1 A1 c.s.o

*

(c) x1 = ln 8.6 = 2.1517622

(3)

M1

x2

= 2.1525814

x3

= 2.152962… = 2.1530

(4dp) only

A1 c.a.o (2)

(9 marks) 3.

(a)

V graph with ‘vertex’ on x-axis M1

y

{− 12 a , (0)} and {(0), a} seen A1

(2)

Correct graph (could be separate) B1

(1)

a − 12 a

O

x

(b)

(c) Meet where

1 = |2x + a | ⇒ x|2x + a | − 1 = 0; only one meet x

(d) 2x2 + x – 1 Attempt to solve; x =

B1

(1)

B1 1 2

(no other value)

M1; A1

(3)

(7 marks)

EDEXCEL CORE MATHEMATICS C3

PRACTICE PAPER A3 MARK SCHEME

Question Number

4.

Scheme

Marks

y

(a)

( 13 , 1)

Translation in ← or →

B1

Points correct

B2, 1, 0 (–1eeoo) (3)

−1

1

(b)

x

y ( 43 , 1)

2

O (− , 1) 4 3

4 3

( , 1)

−2

2

O

B1

x > 2 correct reflection

B1

cusp at (2, 0) (not ∪)

B1

correct shape x ≥ 0

B1

symmetry in y-axis correct maxima

B1 B1

correct x intercepts

B1

(3)

x

y

(c)

x < 2 including points

x

(4)

(10 marks) 5.

(i)

A correct form of cos 2x used 2

2

2

M1 2

4 3 3 4 1 – 2   or   −   or 2  − 1 5 5 5 5 sec 2 x = (ii) (a)

1 cos 2 x

;

=

25

or 3

7

cos 2 x 1 + sin 2 x sin 2 x

or

7    25 

4 7

A1 M1A1

(b)

1 1 + tan 2 x sin 2 x

(4)

M1

Forming single fraction (or multiplying both sides by sin2x)

M1

Use of correct trig. formulae throughout and producing expression in

M1

terms of sin x and cos x Completion (cso) e.g.

2 cos 2 x cos x = = cot x 2 sin x cos x sin x

(*)

A1

(4)

(8 marks)

EDEXCEL CORE MATHEMATICS C3

PRACTICE PAPER A3 MARK SCHEME

Question Number

6.

(a)

Scheme

y=

3x − 1 x−3

⇒

Marks

y(x – 3) = 3x – 1

M1

yx – 3x = 3y – 1 x(y – 3) = 3y – 1

x=

M1

3 y −1 3 x −1 ∴ f−1(x) = = f(x) x −3 y −3

A1 cso

(3) (2)

(b)

ff(k) = f−1f(k), = k

M1 A1

(c)

g(−2) = −5

B1

f(−5) =

− 15 − 1 − 16 ,= =2 −8 −8

M1, A1

(d)

(3)

shape B1 y

g−1(x)

2 −5

2 −1 −2

(e)

6

(0, −1) and (2, 0) B1 Domain: −5 ≤ x ≤ 6 B1

(3)

x

Translation +1 →

6

g(x − 1)

−1

(lines join at (0,0)) B1

x

3 −5

Stretch ×2↕ B1

y

h(x)

12

Range: −10 ≤ h(x) ≤ 12 B1 −1

(3)

x

3 −10

(14 marks)

EDEXCEL CORE MATHEMATICS C3

PRACTICE PAPER A3 MARK SCHEME

Question Number

7.

(a)

Scheme

Marks

12 cos θ − 5 sin θ = R cos θ cosσ − R sin θ sin σ R2 = 52 + 122, ⇒ R = 13 R

5

tan σ =

5 , ⇒ σ = 22.6° (awrt 22.6) M1, A1 12

4 13

M1

θ + 22.6 = 72.1,

M1

θ = 49.5 8 tan θ

(ii) i.e.

(4)

(AWRT or 0.39c (AWRT 0.39c)

12

(b) cos (θ + 22.6) =

M1 A1

(only) − 3 tan θ = 2

A1

(3)

M1

0 = 3tan2 θ + 2tan θ − 8

M1

0 = (3 tan θ − 4)(tan θ + 2)

M1

tan θ =

4 or − 2 3

tan θ =

4 3

⇒

θ = 53.1

[ignoreθ not in range e.g . θ = 116.6 ]

A1

A1

(5)

(12 marks)

EDEXCEL CORE MATHEMATICS C3 Question Number

8.

PRACTICE PAPER A3 MARK SCHEME Scheme

f ' (x ) =

(a)

Marks

3 1 − x x2

M1 A1

3 1 1 − 2 = 0 ⇒ 3x 2 − x = 0 ⇒ x = x x 3 (b)

1 1 y = 31n   + = 3 − 31n 3  3 1    3

(c)

x=1

⇒

f(1) = 2 ⇒ m = − y−1=− (d) (i)

−

(k = 3)

1 2

M1 A1

(2)

M1 x 3   y=− +  2 2 

M1 A1

x 3 1 + = 31n x + 2 2 x

leading to 61n x + x +

(4)

B1

y=1

1 (x − 1) 2

M1 A1

(4)

M1 2 −3= 0 x

*

A1

both, except 1 d.p

M1

c.s.o

(ii) g(0.13) = 0.273… g(0.14) = -0.370…

Sign change (and continuity) ⇒ root ∈ (0.13, 0.14)

A1

(4)

(14 marks)