FP1 practice papers A to G - · PDF fileTurn over Paper Reference(s) 6667/01 Edexcel GCE...

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Paper Reference(s)

6667/01Edexcel GCEFurther Pure Mathematics FP1Advanced SubsidiaryPractice Paper A Time: 1 hour 30 minutes

Materials required for examination Items included with question papersMathematical Formulae Answer Booklet

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 CandidatesIn the boxes on the answer book, write your centre number, candidate number, your surname, initials and signature. Check that you have the correct question paper.When a calculator is used, the answer should be given to an appropriate degree of accuracy.

Information for CandidatesA booklet ‘Mathematical Formulae and Statistical Tables’ is provided.Full marks may be obtained for answers to ALL questions.The marks for individual questions and the parts of questions are shown in round brackets: e.g. (2).There are 9 questions in this question paper. The total mark for this paper is 75. There are 8 pages in this question paper. Any blank pages are indicated.

Advice to CandidatesYou must ensure that your answers to parts of questions are clearly labelled.You should show sufficient working to make your methods clear to the Examiner.Answers without working may not gain full credit.

Printer’s Log. No.

N33681A

This publication may be reproduced only in accordance with Edexcel Limited copyright policy. ©2007 Edexcel Limited.

W850/R6667/57570 2/2/

*N33681A*

FP1 practice papers A to G

N33681A 2

1. A =

2 14 3

−⎛ ⎞⎜ ⎟⎝ ⎠

, B = 3 14 2

⎛ ⎞⎜ ⎟−⎝ ⎠

, I = 1 00 1⎛ ⎞⎜ ⎟⎝ ⎠

.

(a) Show that AB = cI, stating the value of the constant c. (2)

(b) Hence, or otherwise, find A–1. (2)

(Total 4 marks)

2. f(x) = 5 – 2x + 3–x

The equation f(x) = 0 has a root, α , between 2 and 3.

Starting with the interval (2, 3), use interval bisection twice to find an interval of width 0.25 which contains α .

(Total 4 marks)

3. f(n) = (2n + 1)7n – 1.

(a) Show that f(k + 1) – f(k) = (ak + b)7k, stating the values of the constants a and b.(3)

(b) Use induction to prove that, for all positive integers n, f(n) is divisible by 4.(4)

(Total 7 marks)

4. f(x) = x3 + x – 3.

(a) Use differentiation to find f ′(x).(2)

The equation f(x) = 0 has a root, α , between 1 and 2.

(b) Taking 1.2 as your first approximation to α , apply the Newton-Raphson procedure once to f(x) to obtain a second approximation to α . Give your answer to 3 significant figures.

(4)

(Total 6 marks)

N33681A 3 Turn over

5. Given that 3 + i is a root of the equation f(x) = 0, where

f(x) = 2x3 + ax2 + bx – 10, a, b ∈ ℝ,

(a) find the other two roots of the equation f(x) = 0,(5)

(b) find the value of a and the value of b.(3)

(Total 8 marks)

6. (a) Write down the 2 × 2 matrix which represents an enlargement with centre (0, 0) and scale factor k.

(1)

(b) Write down the 2 × 2 matrix which represents a rotation about (0, 0) through –90°. (2)

(c) Find the 2 × 2 matrix which represents a rotation about (0, 0) through –90° followed by an enlargement with centre (0, 0) and scale factor 3.

(2)

The point A has coordinates (a + 2, b) and the point B has coordinates (5a + 2, 2 – b). A is transformed onto B by a rotation about (0, 0) through –90° followed by an enlargement with centre (0, 0) and scale factor 3.

(d) Find the values of a and b.(5)

(Total 10 marks)

N33681A 4

7. Given that z = 1 + √3i and that zw = 2 + 2i, find

(a) w in the form a + ib, where a, b ∈ ℝ,(3)

(b) the argument of w,(2)

(c) the exact value for the modulus of w.(2)

On an Argand diagram, the point A represents z and the point B represents w.

(d) Draw the Argand diagram, showing the points A and B.(2)

(e) Find the distance AB, giving your answer as a simplified surd.(2)

(Total 11 marks)

8. The parabola C has equation y2 = 4ax, where a is a constant.

The point (3t2, 6t) is a general point on C.

(a) Find the value of a.(1)

(b) Show that an equation for the tangent to C at the point (3t2, 6t) is

ty = x + 3t2.(4)

The point Q has coordinates (3q2, 6q). The tangent to C at the point Q crosses the x-axis at the point R.

(c) Find, in terms of q, the coordinates of R.(3)

The directrix of C crosses the x-axis at the point D.

Given that the distance RD = 12 and q > 1,

(d) find the exact value of q.(4)

(Total 12 marks)

N33681A 5

9. (a) Prove by induction that, for all positive integers n,

2

1

1 ( 1)(2 1)6

n

rr n n n

=

= + +∑ .

(6)

(b) Show that = 61 n(n+ 7)(2n + 7).

(5)

(c) Hence calculate the value of 40

10( 1)( 5)

rr r

=

+ +∑ .

(2)

(Total 13 marks)

TOTAL FOR PAPER: 75 MARKS

END

Turn over

Paper Reference(s)

6667/01Edexcel GCEFurther Pure Mathematics FP1Advanced SubsidiaryPractice Paper BTime: 1 hour 30 minutes

Materials required for examination Items included with question papersMathematical Formulae Answer Booklet

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 CandidatesIn the boxes on the answer book, write your centre number, candidate number, your surname, initials and signature. Check that you have the correct question paper.When a calculator is used, the answer should be given to an appropriate degree of accuracy.

Information for CandidatesA booklet ‘Mathematical Formulae and Statistical Tables’ is provided.Full marks may be obtained for answers to ALL questions.The marks for individual questions and the parts of questions are shown in round brackets: e.g. (2).There are 8 questions in this question paper. The total mark for this paper is 75.There are 4 pages in this question paper. Any blank pages are indicated.

Advice to CandidatesYou must ensure that your answers to parts of questions are clearly labelled.You should show sufficient working to make your methods clear to the Examiner. Answers without working may not gain full credit.

Printer’s Log. No.

N33682A

This publication may be reproduced only in accordance with Edexcel Limited copyright policy. ©2007 Edexcel Limited.

W850/XXXX/57570 2/

*N33682A*

1.

where q is a real constant.

(a) Find det A in terms of q. (2)

(b) Show that A is non-singular for all values of q.(3)

(Total 5 marks)

2. Given that z = 22 + 4i and = 6 – 8i, find

(a) (2)

(b) w in the form a + bi, where a and b are real, (3)

(c) the argument of z, in radians to 2 decimal places. (2)

(Total 7 marks)

3. (a) Show that . (5)

(b) Hence calculate the value of (2)

(Total 7 marks)

4.

The root α of the equation f(x) = 0 lies in the interval [0.5, 0.6].

(a) Using the end points of this interval find, by linear interpolation, an approximation to α, giving your answer to 3 significant figures.

(4)

(b) Taking 0.55 as a first approximation to α, apply the Newton-Raphson procedure once to f(x) to find a second approximation to α, giving your answer to 3 significant figures.

(5)

(Total 9 marks)

A =− −

⎛⎝⎜

⎞⎠⎟

q

q

3

2 1,

z

wz

w,

( )( ) ( ) ( )r r n n nr

n

− + = − +=

∑ 1 21

31 4

1

( )( ).r rr

− +=

∑ 1 25

20

f ( )x xx

= − +2 35

5. (a) Given that 2 + i is a root of the equation

z2 + bz + c = 0, where b and c are real constants,

(i) write down the other root of the equation,

(ii) find the value of b and the value of c.(5)

(b) Given that 2 + i is a root of the equation

z3 + mz2 + nz – 5 = 0, where m and n are real constants,

find the value of m and the value of n.(5)

(Total 10 marks)

6. A, B and C are non-singular 2 × 2 matrices such that

AB = C.

(a) Show that B = A–1C.(2)

The triangle T1 has vertices at the points with coordinates (0, 0), (5, 0) and (0, 3).

Triangle T1 is mapped onto triangle T2 by the transformation given by C.

(b) Find det C.(1)

(c) Hence, or otherwise, find the area of triangle T2.(3)

Triangle T1 is mapped onto triangle T2 by the transformation given by B followed by the transformation given by A.

(d) Using part (a) or otherwise, find B.(4)

(e) Describe fully the geometrical transformation represented by B. (2)

(Total 12 marks)

A C=− −

−

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=− −

−⎛⎝⎜

⎞⎠⎟

1

2

1

21

2

1

2

1 1

1 1, .

7. (a) Show that the normal to the rectangular hyperbola xy = 4, at the general point

t ≠ 0 has equation

(5)

The normal to the hyperbola at the point A (–4, –1) meets the hyperbola again at the point B.

(b) Find the coordinates of B.(7)

(Total 12 marks)

8. (a) f(n) = n3 – 10n + 15.

Given that f(k + 1) – f(k) = ak2 + bk + c,

(i) find the values of a, b and c.(3)

(ii) Use induction to prove that, for all positive integers n, f(n) is divisible by 3.(4)

(b) Prove by induction that, for n ∈ℤ+, (6)

(Total 13 marks)

TOTAL FOR PAPER: 75 MARKS

END

P tt

22

, ,⎛⎝⎜

⎞⎠⎟

y t xt

t= + −2 322 .

r nr n

r

n

2 2 1 1 21

= + −=

∑ { ( ) }.

FP1 practice paper C - compiled from the practice questions issued by Edexcel in 2008, sometimes adapted. For each question, look in the mark scheme document for the mark scheme with the # number in square brackets. 77 marks.

1.[#1] Given that z = 22 + 4i and wz

= 6 – 8i, find

(a) w in the form a + bi, where a and b are real, (3)

(b) the argument of z, in radians to 2 decimal places. (2)

[P4 January 2002 Qn 1]

2.[#2] (a) Prove by induction that

n

rrr

1)1)(1( = 6

1 n (n – 1)(2n + 5).

(5)(b) Deduce that n(n – 1)(2n + 5) is divisible by 6 for all n > 1.

(2)


3.[#3] f(x) = x3 + x – 3.

The equation f(x) = 0 has a root, , between 1 and 2.

(a) By considering f (x), show that is the only real root of the equationf(x) = 0.

(3)(b) Taking 1.2 as your first approximation to , apply the Newton-Raphson procedure once to f(x) to obtain a second approximation to .Give your answer to 3 significant figures.

(2)(c) Prove that your answer to part (b) gives the value of correct to 3significant figures. (2)


4.[#4] Given that 2 + i is a root of the equation

z2 + bz + c = 0, where b and c are real constants,

(i) write down the other root of the equation,

(ii) find the value of b and the value of c. (5)

[*P4 January 2002 Qn 5]

FP1 questions from old P4, P5, P6 and FP1, FP2, FP3 papers L Version March 2009

5.[#5] Prove by using standard results that

n

r

r1

2 )1(6 (n 1)n(2n + 5).

(4)

[P4 June 2002 Qn 1]

6.[#6] Given that z = 3 + 4i and w = 1 + 7i,

(a) find w .(1)

The complex numbers z and w are represented by the points A and B onan Argand diagram.

(b) Show points A and B on an Argand diagram.(1)

(c) Prove that △OAB is an isosceles right-angled triangle.(5)

(d) Find the exact value of arg | zw| .

(3)

[P4 June 2002 Qn 5]

7.[#7] The point P (2p ,2p

) and the point Q (2q,2q

) , where p q, lie on

the rectangular hyperbola with equation xy = 4.

The tangents to the curve at the points P and Q meet at the point R.

Show that at the point R,

x = qppq

4 and y = qp

4.

(8)

[*P5 June 2002 Qn 7]

FP1 questions from old P4, P5, P6 and FP1, FP2, FP3 papers – Version 2 – March 2009 2

8.[#8] For n ℤ+ prove that

(a) 23n + 2 + 5n + 1 is divisible by 3,(9)

(b) (−2 −19 4 )

n

= (1−3n −n9n 3n+1 ) .

(7)

[P6 June 2002 Qn 6]

9.[#9] f(x) = 2 sin 2x + x – 2.

The root of the equation f(x) = 0 lies in the interval [2, ].

Using the end points of this interval find, by linear interpolation, anapproximation to .

(4)

[You won't have sine in linear interpolation questions in the real exam, but really having sincein this question doesn't make it harder. P4 January 2003 Qn 4]

10.[#10] Given that z = 3 – 3i express, in the form a + ib, where a and b are

real numbers,

(a) z2,(2)

(b)z1

.

(2)

(c) Find the exact value of each of z , z2 and z1

.

(2)

The complex numbers z, z2 and z1

are represented by the points A, B and

C respectively on an Argand diagram. The real number 1 is representedby the point D, and O is the origin.

(d) Show the points A, B, C and D on an Argand diagram. (2)

(e) Prove that triangle OAB is similar to triangle OCD. (3)

[P4 January 2003 Qn 6]END OF PAPER


FP1 practice paper D - compiled from the practice questions issued by Edexcel in 2008, sometimes adapted. For each question, look in the mark scheme document for the mark scheme with the # number in square brackets. 66 marks.

1.[#11] (a) Using the fact that 3 is the real root of the cubic equation x3 –27 = 0, show that the complex roots of the cubic satisfy the quadraticequation x2 + 3x + 9 = 0.

(2)

(b) Hence, or otherwise, find the three cube roots of 27, giving youranswers in the form

a + ib, where a, b ℝ.(3)

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

[#P4 June 2003 Qn 3]

2.[#12.] 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 tofind an interval of width 0.125 which contains .

(2)

[*P4 June 2003 Qn 4]

3.[#13.] z = i2i3

aa

, a ℝ.

(a) Given that a = 4, find z. (3)

(b) Show that there is only one value of a for which arg z = 4

, and find

this value. (6)

[P4 June 2003 Qn 5]


4. [#14] f(n) = (2n + 1)7n – 1.

Prove by induction that, for all positive integers n, f(n) is divisible by 4.(6)

[P6 June 2003 Qn 2]

5. [#15.] Given that z = 2 – 2i and w = –3 + i,

(a) find the modulus and argument of wz2. (6)

(b) Show on an Argand diagram the points A, B and C which represent z,w and wz2 respectively, and determine the size of angle BOC.

(4)

[It's unusual these days to be asked to find the size of an angle, but angle BOC is just 2π plusarg wz2 minus arg w. P4 January 2004 Qn 3]

6. [#16] (a) Show by induction that

n

r

rr1

)5)(1( = 61

n(n+ 7)(2n + 7).

(4)

(b) Hence calculate the value of

40

10

)5)(1(r

rr .

(2)

[P4 June 2004 Qn 1]

7. [17] f(x) = 2x + x 4.

The equation f(x) = 0 has a root in the interval [1, 2].

Use linear interpolation on the values at the end points of this interval tofind an approximation to .

(2)

[*P4 June 2004 Qn 2]

8.[#18] The complex number z = a + ib, where a and b are real numbers,satisfies the equation

z2 + 16 30i = 0.

(a) Show that ab = 15.(2)


(b) Write down a second equation in a and b and hence find the roots of

z2 + 16 30i = 0.(4)

[P4 June 2004 Qn 3]

9.[#19] Given that z = 1 + 3i and that zw

= 2 + 2i, find

(a) w in the form a + ib, where a, b ℝ,(3)

(b) the argument of w,(2)

(c) the exact value for the modulus of w.(2)

On an Argand diagram, the point A represents z and the point B represents w.

(d) Draw the Argand diagram, showing the points A and B.(2)

(e) Find the distance AB, giving your answer as a simplified surd.(2)

[P4 June 2004 Qn 5]

10.[#20] Show that the normal to the rectangular hyperbola xy = c2, at the

point P (ct , ct ) , t 0 has equation

y = t2x + tc

ct3.

(5)

[*P5 June 2004 Qn 8]


FP1 practice paper E - compiled from the practice questions issued by Edexcel in 2008, sometimes adapted. For each question, look in the mark scheme document for the mark scheme with the # number in square brackets. 70 marks.

1.[#21] Given that 2 2 2 2iz and w = 1 – i3, find

(a)wz

,

(3)

(b) arg ( zw ) .

(3)

(c) On an Argand diagram, plot points A, B, C and D representing the

complex numbers z, w, ( zw ) and 4, respectively.

(3)

(d) Show that AOC = DOB.(2)

(e) Find the area of triangle AOC.(2)

[#FP1/P4 January 2005 Qn 8]

2.[#22] Given that 2 is a root of the equation z3 + 6z + 20 = 0,

(a) find the other two roots of the equation,(3)

(b) show, on a single Argand diagram, the three points representing theroots of the equation,

(1)

(c) prove that these three points are the vertices of a right-angledtriangle.

(2)[#FP1/P4 June 2005 Qn 2]

3.[#23] f(x) = 1 – ex + 3 sin 2x

The equation f(x) = 0 has a root in the interval 1.0 < x < 1.4.

Starting with the interval (1.0, 1.4), use interval bisection three times tofind the value of to one decimal place. (3)

[You won't have ex or sine in interval bisection questions this year, but... FP1/P4 June 2005 Qn4]


4.[#24] z = –4 + 6i.

(a) Calculate arg z, giving your answer in radians to 3 decimal places.(2)

The complex number w is given by w = i2

A, where A is a positive

constant. Given that w = 20,

(b) find w in the form a + ib, where a and b are constants, (4)

(c) calculate arg zw

.

(3)

[FP1/P4 June 2005 Qn 5]

5.[#25] The point P(ap2, 2ap) lies on the parabola M with equation y2 = 4ax,where a is a positive constant.

(a) Show that an equation of the tangent to M at P is

py = x + ap2.(3)

The point Q(16ap2, 8ap) also lies on M.

(b) Write down an equation of the tangent to M at Q.(2)

[*FP2/P5 June 2005 Qn 5]

6.[#26] The sequence of real numbers u1, u2, u3, ... is such that u1 = 5.2 and

un + 1 = 6−8

un+3.

(b) Prove by induction that un > 5, for n ℤ+.

(4)

[It's an unusual induction question, but there's an outside chance you could see one like itthis year. FP3/P6 June 2005 Qn 1]


7.[#27]

Prove by using standard results that

n

r

rr1

)2)(1( = 31

(n – 1)n(n + 4).

(6)

[FP1/P4 January 2006 Qn 1]

8.[#28] Given that ii2

zz

= i, where is a positive, real constant,

(a) show that z = ( λ2

+1) + i ( λ2

−1) .

(5)

Given also that arg z = arctan 21 , calculate

(b) the value of ,(3)

(c) the value of z 2.(2)

[FP1/P4 January 2006 Qn 3]

9.[#29] The temperature C of a room t hours after a heating system has been turned on is given by

= t + 26 – 20e–0.5t, t 0.

The heating system switches off when = 20. The time t = , when theheating system switches off, is the solution of the equation – 20 = 0,where lies in the interval [1.8, 2].

(a) Using the end points of the interval [1.8, 2], find, by linearinterpolation, an approximation to . Give your answer to 2 decimalplaces.

(4)

(b) Use your answer to part (a) to estimate, giving your answer to thenearest minute, the time for which the heating system was on.

(1)

[You won't have et in questions this year, but... FP1/P4 January 2006 Qn 5]


10.[#30] The parabola C has equation y2 = 4ax, where a is a constant.

(a) Show that an equation for the normal to C at the point P(ap2, 2ap) is

y + px = 2ap + ap3.(4)

The normals to C at the points P(ap2, 2ap) and Q(aq2, 2aq), p q, meet atthe point R.

(b) Find, in terms of a, p and q, the coordinates of R.(5)

[*FP2/P5 January 2006 Qn 9]


FP1 practice paper F - compiled from the practice questions issued by Edexcel in 2008, sometimes adapted. For each question, look in the mark scheme document for the mark scheme with the # number in square brackets. 68 marks.

1.[#31] A transformation T : ℝ2 ℝ2 is represented by the matrix

A = 4 2

2 1

, where k is a constant.

Find the image under T of the line with equation y = 2x + 1.(4)

[You won't have a question this year quite like this. But you should be able to do it with just

the added hint that you're asked to find what happens to ( x2x+1) when multiplied by

A. FP3/P6 January 2006 Qn 3]

2.[#32] Prove by induction that, for n ℤ+,

n

r

rr1

2 = 2{1 + (n – 1)2n}.

(5)

[*FP3/P6 January 2006 Qn 5]

3.[#33] The complex numbers z and w satisfy the simultaneous equations

2z + iw = –1,

z – w = 3 + 3i.

(a) 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)

[FP1 June 2006 Qn 1]

4.[#34] f(x) = 0.25x – 2 + 4 sin x.

(a) Show that the equation f(x) = 0 has a root between x = 0.24 and x =0.28.

(2)


(b) Starting with the interval [0.24, 0.28], use interval bisection threetimes to find an interval of width 0.005 which contains .

(3)

[You won't have sine in questions like this, but... FP1 June 2006 Qn 6]

5.[#35] (a) Find the roots of the equation

2 2 17 0,z z

giving your answers in the form a + ib, where a and b are integers.(3)

(b) Show these roots on an Argand diagram.(1)

[FP1 January 2007 Qn 1]

6.[#36] The complex numbers 1z and 2z are given by

1

1

5 3i,

1 i,

z

z p

where p is an integer.

(a) Find 2

1

zz

, in the form a + ib, where a and b are expressed in terms of

p.(3)

Given that 2

1

arg ,4

zz

(b) find the value of p. (2)



7.[#37] f (x) = x3 + 8x – 19.

(a) Show that the equation f(x) = 0 has only one real root.(3)

(b) Show that the real root of f(x) = 0 lies between 1 and 2.(2)

(c) Obtain an approximation to the real root of f(x) = 0 by performing twoapplications of the Newton-Raphson procedure to f(x) , using x = 2 asthe first approximation. Give your answer to 3 decimal places.

(4)

(d) By considering the change of sign of f(x) over an appropriate interval,show that your answer to part (c) is accurate to 3 decimal places.

(2)


8.[#38] z = 3 – i.

z* is the complex conjugate of z.

(a) Show that z

z =

21

– 23 i.

(3)

(b) Find the value of

zz

.

(2)

(c) Verify, for z = 3 – i, that arg z

z = arg z – arg z*.

(4)

(d) Display z, z* and z

z on a single Argand diagram.

(2)

(e) Find a quadratic equation with roots z and z* in the form ax2 + bx + c= 0, where a, b and c are real constants to be found.

(2)



9.[#39] The points P(ap2, 2ap) and Q(aq2, 2aq), p q, lie on the parabola Cwith equation y2 = 4ax, where a is a constant.

(a) Show that an equation for the chord PQ is (p + q) y = 2(x + apq) .(3)

The normals to C at P and Q meet at the point R.

(b) Show that the coordinates of R are ( a(p2 + q2 + pq + 2), –apq(p + q) ).(7)

[*FP2 June 2007 Qn 8]

10.[#40] Prove by induction that

for n ℤ+,

n

r

r1

2)12( = 31 n(2n – 1)(2n + 1).

(5)



FP1 practice paper G - compiled from the practice questions issued by Edexcel in 2008, sometimes adapted, and with two questions addedfrom AQA papers. For each question, look in the mark scheme document for the mark scheme with the # number in square brackets. 74 marks.

1.[#41] Given that f(n) = 34n + 24n + 2,

(a) show that, for k ℤ+, f(k + 1) – f(k) is divisible by 15,(4)

(b) prove that, for n ℤ+, f (n) is divisible by 5,(3)

[*FP3 June 2007 Qn 6]

2.[#42.] Given that x = – 21 is the real solution of the equation

2x3 – 11x2 + 14x + 10 = 0,

find the two complex solutions of this equation. (6)

[Watch out! It's 2x3, not x3.... FP1 January 2008 Qn 2]

3.[#43] f(x) = 3x2 + x – tan ( x2 ) – 2, – < x < .

The equation f(x) = 0 has a root in the interval [0.7, 0.8].

Use linear interpolation, on the values at the end points of this interval,to obtain an approximation to . Give your answer to 3 decimalplaces.

(4)

[You won't have tan in your questions this year. Still, having tan in it doesn't make it reallyany harder. FP1 January 2008 Qn 4]

4.[#44] z = –2 + i.

(a) Express in the form a + ib

(i)z1


(ii) z2. (4)

(b) Show that z2 – z = 52.(2)

(c) Find arg (z2 – z).(2)

(d) Display z and z2 – z on a single Argand diagram. (2)


5.[#45] (a) Write down the value of the real root of the equation

x3 – 64 = 0.(1)

(b) Find the complex roots of x3 – 64 = 0 , giving your answers in theform a + ib, where a and b are real.

(4)

(c) Show the three roots of x3 – 64 = 0 on an Argand diagram.(2)


6.[#46] The complex number z is defined by

z = ii2

aa

, a ℝ, a > 0 .

Given that the real part of z is 21 , find

(a) the value of a,(4)

(b) the argument of z, giving your answer in radians to 2 decimal places.(3)



7.[#47]

A = ( k −21−k k ) , where k is constant.

A transformation T : ℝ2 → ℝ2 is represented by the matrix A.

(a) Find the value of k for which the line y = 2x is mapped onto itselfunder T.

(3)

(You won't have a question quite like this. But all you have to do is find a k where multiplying ( x2x) produces

an answer where the bottom number is twice the top number)

(b) Show that A is non-singular for all values of k.(3)

(c) Find A–1 in terms of k.(2)

A point P is mapped onto a point Q under T.

The point Q has position vector ( 4−3) relative to an origin O.

(This just means that its coordinates relative to O are x=4, y=-3. That's all!)

Given that k = 3,

(d) find the position vector of P.

(i.e. what its x-coordinate and y-coordinate are)

(3)



8.[#48]

9.[#49] (from AQA rather than Edexcel, but will test your matrix skills)


10.[#50]


FP1 practice papers A to G - · PDF fileTurn over Paper Reference(s) 6667/01 Edexcel GCE...

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