c3 gold

Paper Reference(s)

Paper Reference(s)

6665/01Edexcel GCE

Core Mathematics C3Gold Level (Hard) G1Time: 1 hour 30 minutes

Materials required for examination Items included with question papersMathematical Formulae (Green) Nil

Candidates may use any calculator allowed by the regulations of the Joint

Council for Qualifications. Calculators must not have the facility for symbolic

algebra manipulation, differentiation and integration, or have retrievable

mathematical formulas stored in them.Instructions to Candidates

Write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Core Mathematics C3), the paper reference (6665), your surname, initials and signature.Information for Candidates

A booklet Mathematical Formulae and Statistical Tables is provided.

Full marks may be obtained for answers to ALL questions.

There are 8 questions in this question paper. The total mark for this paper is 75.

Advice to Candidates

You must ensure that your answers to parts of questions are clearly labelled.

You must show sufficient working to make your methods clear to the Examiner. Answers

without working may gain no credit.

Suggested grade boundaries for this paper:

A*ABCDE

685950433731

1.The point P lies on the curve with equationy = 4e2x + 1.The y-coordinate of P is 8.

(a) Find, in terms of ln 2, the x-coordinate of P.(2)

(b) Find the equation of the tangent to the curve at the point P in the form y = ax + b, where a and b are exact constants to be found.

(4)June 2008

2.A curve C has equationy = , x .The point P on C has x-coordinate 2. Find an equation of the normal to C at P in the form ax+by+ c = 0, where a, b and c are integers.

(7)June 2010

3. A curve C has equation y = x2ex.

(a)Find , using the product rule for differentiation.

(3)

(b)Hence find the coordinates of the turning points of C.

(3)

(c)Find .

(2)

(d)Determine the nature of each turning point of the curve C.

(2)

June 2007

4.(i) Given that y = , find .(4)

(ii)Given that x = tan y, show that = .

(5)January 2010

5.

Figure 1Figure 1 shows a sketch of the curve C with the equation y = (2x2 5x + 2)ex.

(a) Find the coordinates of the point where C crosses the y-axis.

(1)

(b) Show that C crosses the x-axis at x = 2 and find the x-coordinate of the other point where C crosses the x-axis.

(3)

(c) Find .(3)

(d) Hence find the exact coordinates of the turning points of C.

(5)June 2010

6.(a)Express 3 sin x + 2 cos x in the form R sin (x + ) where R > 0 and 0 < < .

(4)

(b)Hence find the greatest value of (3 sin x + 2 cos x)4.

(2)

(c)Solve, for 0 < x < 2, the equation 3 sin x + 2 cos x = 1, giving your answers to 3 decimal places.

(5)

June 2007

7.(a) Express 4 cosec2 2 cosec2 in terms of sin and cos .

(2)

(b) Hence show that

4 cosec2 2 cosec2 = sec2 .(4)

(c) Hence or otherwise solve, for 0 < < (,

4 cosec2 2 cosec2 = 4

giving your answers in terms of (.

(3)June 2012

8.(a) Starting from the formulae for sin (A + B) and cos (A + B), prove that

tan (A + B) = .(4)

(b) Deduce that

tan = .

(3)

(c) Hence, or otherwise, solve, for 0 ( ( ,

1 + 3 tan = (3 tan ) tan ( ).

Give your answers as multiples of .

(6)January 2012

TOTAL FOR PAPER: 75 MARKSENDQuestion NumberSchemeMarks

1. (a)

M1

A1 (2)

(b)

B1

B1

M1

A1 ( 4)

(6 marks)

Question NumberSchemeMarks

2.At P,

B1

M1A1

M1

m(N) = or

M1

N:

M1

N:

A1

[7]


3. (a) M1,A1,A1 (3)

(b)If , ex(x2 + 2x) = 0 setting (a) = 0M1

[ex ( 0] x(x + 2) = 0

( x = 0 ) x = 2A1

x = 0, y = 0 and x = 2, y = 4e2 ( = 0.54)A1 (3)

(c)

M1, A1 (2)

(d)x = 0, > 0 (=2) x = 2, < 0 [ = 2e2 ( = 0.270)]

M1: Evaluate, or state sign of, candidates (c) for at least one of candidates x value(s) from (b) M1

(minimum (maximumA1 (cso) (2)

(10 marks)


Q4 (i)

M1

A1

Apply quotient rule:

M1

A1

(4)

(ii)

M1*

A1

dM1*

dM1*

Hence, (as required)A1 AG

(5)

[9]


5. (a)

M1A1

(2)

(b)

B1

M1;A1 oe.

(3)

(c)

M1;A1

(2)

(d). Hence

M1

Either or

or

B1

or

A1

(3)

[10]


6. (a)Complete method for R: e.g. , ,

M1

or 3.61 (or more accurate)A1

Complete method for [Allow

( = 0.588 (Allow 33.7)M1

A1 (4)

(b)Greatest value = = 169M1, A1 (2)

(c) ( = 0.27735) sin(x + their () =

M1

( x + 0.588) = 0.281( 03 ) or 16.1 A1

(x + 0.588) = ( 0.28103

Must be ( their 0.281 or 180 their 16.1M1

or (x + 0.588) = 2( + 0.28103

Must be 2( + their 0.281 or 360 + their 16.1 M1

x = 2.273 or x = 5.976 (awrt) Both (radians only)A1 (5)

If 0.281 or 16.1 not seen, correct answers imply this A mark(11 marks)


7. (a)

B1 B1

(2)

(b)

M1

Using

M1

M1A1*

(4)

(c)

M1

A1,A1

(3)

(9 marks)

Question numberSchemeMarks

8. (a)M1A1

(M1

A1 * (4)

(b)M1

M1

A1 * (3)

(c)M1

M1

=M1 A1

M1

=A1 (6)

(13 marks)

Statistics for C3 Practice Paper G1Mean score for students achieving grade:

QuMax scoreModal scoreMean %ALLA*ABCDE

16694.155.314.483.772.972.121.01

27765.306.726.245.745.124.222.991.55

310737.349.127.876.785.514.092.18

49605.387.706.174.833.742.421.21

510616.058.917.666.224.883.752.771.66

611626.849.407.475.743.992.440.99

79575.098.656.985.043.592.441.600.76

813516.6312.089.667.535.974.353.191.60

756246.7862.0750.5240.6830.9721.6210.96

EMBED Equation.DSMT4

y

x

EMBED Equation.DSMT4

O

Gold 1: 9/122Gold 1: 9/1210Gold 1This publication may only be reproduced in accordance with Edexcel Limited copyright policy.20072013 Edexcel Limited.

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