fp1 practice paper c

Paper Reference(s)

6667/01

Edexcel GCEFurther Pure Mathematics FP1Advanced Level

Practice Paper C

Time: 1 hour 30 minutes

Materials required for examination Items included with question papersMathematical Formulae Nil

Candidates may use any calculator allowed by the regulations of the JointCouncil for Qualifications. Calculators must not have the facility for

symbolicalgebra manipulation, differentiation and integration, or have retrievablemathematical formulas stored in them.

Instructions to Candidates

Write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Further Pure Mathematics FP1), the paper reference (6667), 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 9 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.

Unofficial FP1 practice paper C and mark scheme

1. A parallelogram has vertices (1, 0), (–1, 6), (2, 3) and (4, –3). It is transformed by the matrix:

.

(a) Find the positions of the vertices of its image.(2)

(b) Find the scale factor of the transformation and hence the area of the original parallelogram.

(3)

2. (a) By factorisation, show that two of the roots of the equation x3 – 27 = 0 satisfy the quadratic equation x2 + 3x + 9 = 0.

(2)

(b) Hence or otherwise, find the three cube roots of 27, giving your answers in the form a + ib, where a, b ℝ.

(3)

(c) Show these roots on an Argand diagram.(2)

3. (a) Prove by induction that, for all integers n, n 1: .

(7)

(b) Prove by induction or otherwise that for all integral values of n, 2n+2 + 32n+1 is exactly divisible by 7.

(7)

4. f(x) = 3x – x – 6.

(a) Show that f(x) = 0 has a root between x = 1 and x = 2.(2)

(b) Starting with the interval (1, 2), use interval bisection three times to find an interval of width 0.125 which contains .

(2)

(c) Taking 2 as a first approximation to , apply the Newton-Raphson procedure once to f(x) to obtain a second approximation to . Give your answer to 3 decimal places.

(4)


5. The matrices A and B are defined by:

A = B = .

(a) Find A–1, the inverse of A.(2)

(b) Find the matrix C such that (B + C) –1 = A.(5)

(c) Describe geometrically the transformation represented by C.(1)

6. (a) Show that

(4)

(b) Hence calculate the value of: .

(2)

7. The complex numbers z and w satisfy the simultaneous equations

2z + iw = –1,

z – w = 3 + 3i.

Use algebra to find z, giving your answers in the form a + ib, where a and b are real.(4)

(b) Calculate arg z, giving your answer in radians to 2 decimal places.(2)


8. (a) Show that the normal to the rectangular hyperbola xy = c2, at the point P , t 0

has equation y = t2x + ct3.

(5)

The normal to the hyperbola at P meets the hyperbola again at the point Q.

(b) Find, in terms of t, the coordinates of the point Q.(5)

Given that the mid-point of PQ is (X, Y) and that t 1,

(c) show that ,

(2)

(d) show that, as t varies, the locus of the mid-point of PQ is given by the equation

4xy + c2 = 0.

(2)

9. The temperature oC of a room t hours after a heating system has been turned on is given by:

The heating system switches off when = 20. The time t = , when the heating system switches off, is the solution of the equation – 20 = 0.

(a) Show that lies in the interval [1.8, 2].(2)

(b) Using the end points of this interval find, by linear interpolation, an approximation to . Give your answer to 2 decimal places.

(4)

(c) Use your answer to part b to find, to the nearest minute, the time for which the heating system was on.

(1)

END


Solutions:

1) a) = M1A1

b) New area: 2 x 4 = 8 sq units M1

Det M=1- = 8 = 12 sq units M1A1 5

2) a)(x2+3x+9)(x-3)=x3-27 M1A1

b) x=3 x=- A3

c) correct labelling and at least one root shown. M1equal spacing for all three A1 7

3)a) For n=1, LHS = 1, RHS = 0.5.So true for n=1. M1A1

Assume true for n=k.

Try for n=k+1:

M2A1

Truth for n=k implies truth for n=k+1, therefore true for all n. A2b) When n=1, 2 3 +3 3=35=7 x 5 so true. M1A1Assume truth for n=k 2 k+2 + 3 2k+1 = 7pTry for n=k+1: 2 k+3 + 3 2k+3=2(2 k+2)+9(3 2k+1)=2(2 k+2+3 2k+1) + 7(3 2k+1)=2 x 7p+7 x 3 2k+1 M2A1Truth for n=k implies truth for n=k+1, therefore true for all n. A2 14

4) a) f(1) = 3-1-6 = -4f(2) = 9-2-6 = 1f(1)<0 f(2)>0 so between x=1 and x=2 function crosses x axis. M1A1

b) f(1.5) = -2.3038….f(1.75) = -0.911478…..f(1.875) = -0.02983…..so root lies in interval (1.875, 2) M1A1

c) f ’(x)= 3xln3-1 A2

2- =1.887 (to 3dpl) M1A1 8

5) a) A-1 = M1A1


b) (B+C)(B+C) -1 = (B+C) A M1

so: M1A1

therefore b=0, a=-1, d=-1, c=0 M1

C= A1

c) Rotation 180o about origin. A1 8

6) a)

LHS= M1A1

b) Substitute into expression above: =3008 M1A1 6

7) (a) Adding Eliminating either variable M1

A1

M1

= A1

(b) arctan 2 M1 A1 6

8. (a)

M1 A1

B1


M2

(b)

The normal to the curve has gradient .

The equation of the normal is

The equation may be written

Let Q be the point (cq, c/q)

Then and so

Attempt to find q, e.g. or quadratic formula

So Q has coordinates

Alternatives

Eliminate x or y between and .

So .or

Then solve using formula to obtain

x = or

So Q has coordinates

M1

A1 (5)

M1 A1

M1

A1

A1(5)

M1

A1

M1

A1

A1(5)

8. (c)

(d)

M1

A1 (2)

M1

A1

(2)


9. (a) f(1.8) = 19.6686… 20 = 0.3313… Allow awrt 0.33

f(2) = 20.6424… 20 = 0.6424… Allow awrt 0.64

f(1.8)<0 f(2)>0 so between x=1.8 and x=2 function crosses x axis. M1A1

b) 1.87 M3A1

(c) 112 (min) (1 h 52 m) B1 7


fp1 practice paper c

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