ZIMSEC A Level Pure Mathematics Paper 1 June 2006

Question 101

[1 marks]partial fractions

In the partial fraction decomposition

\dfrac{5}{(x+1)(x^2+4)} = \dfrac{A}{x+1} + \dfrac{Bx+C}{x^2+4}

, what is the value of

A

A $\frac{1}{5}$
B $1$
C $5$
D $-1$

Question 102

[1 marks]partial fractions

Express

\dfrac{5}{(x+1)(x^2+4)}

in partial fractions.

A $-\dfrac{1}{x+1} + \dfrac{1+x}{x^2+4}$
B $\dfrac{1}{x+1} + \dfrac{x-1}{x^2+4}$
C $\dfrac{1}{x+1} + \dfrac{1-x}{x^2+4}$
D $\dfrac{1}{x+1} - \dfrac{x+1}{x^2+4}$

Question 103

[2 marks]partial fractions

5=A(x^2+4)+(Bx+C)(x+1)

with

A=1

: comparing

x^2

coefficients gives

0=A+B

, and comparing constant terms gives

5=4A+C

. State the values of

B

and

C

Answer this when you sit the paper.

Question 201

[1 marks]absolute value equations

Solve the equation

|4x + 3| = |x - 5|

for all real values of

x

A $x = \frac{8}{3}$ or $x = -\frac{2}{5}$
B $x = -\frac{8}{3}$ or $x = -\frac{2}{5}$
C $x = -\frac{8}{3}$ or $x = \frac{2}{5}$
D $x = \frac{2}{3}$ or $x = -\frac{5}{2}$

Question 202

[2 marks]absolute value equations

Case 1 of

|4x+3|=|x-5|

assumes

4x+3=x-5

. Solve for

x

Answer this when you sit the paper.

Question 203

[1 marks]absolute value equations

Case 2 of

|4x+3|=|x-5|

assumes

4x+3=-(x-5)

. Solve for

x

Answer this when you sit the paper.

Question 301

[1 marks]differential equations

Find the particular solution of the differential equation

\dfrac{dy}{dx} = -\dfrac{x}{y}

that passes through the point

(5, 12)

A $x^2+y^2=17$
B $y^2-x^2=119$
C $x^2+y^2=169$
D $x^2-y^2=169$

Question 302

[1 marks]differential equations

The general solution of

\dfrac{dy}{dx}=-\dfrac{x}{y}

passing through

(5,12)

x^2+y^2=169

. What shape does this curve describe, and what is its radius?

AA straight line through the origin
BA circle of radius $15$
CA circle of radius $13$
DA circle of radius $169$

Question 303

[1 marks]differential equations

Using the boundary point

(5,12)

on the solution curve of

\dfrac{dy}{dx}=-\dfrac{x}{y}

, evaluate

5^2+12^2

Answer this when you sit the paper.

Question 304

[1 marks]differential equations

The general solution of

\dfrac{dy}{dx}=-\dfrac{x}{y}

, of the form

x^2+y^2=k

, describes which family of curves (for

k>0

AParabolas with vertex at the origin
BCircles centred at the origin
CHyperbolas centred at the origin
DStraight lines through the origin

Question 305

[1 marks]differential equations

For the general solution

x^2+y^2=k

\dfrac{dy}{dx}=-\dfrac{x}{y}

, what must be true of

k

for the solution to be a genuine circle, rather than just the single point at the origin?

A $k<0$
B $k$ can be any real number
C $k=0$
D $k>0$

Question 401

[1 marks]circles and sectors

A circle has centre

O

and radius

OA=2

cm. Radii

OA

and

OB

are produced to

C

and

D

with

AC=BD=10

cm, so

OC=OD=12

cm. The sector

OCD

has area six times the area of the circle of radius

OA

. Find angle

AOB

A $\frac{2\pi}{3}$
B $\frac{\pi}{6}$
C $\frac{\pi}{3}$
D $\frac{\pi}{4}$

Question 402

[1 marks]circles and sectors

Using the figure

ACDB

where

AC=BD=10

cm, arc

CD

has radius

12

cm, arc

AB

has radius

2

cm, and angle

AOB=\frac{\pi}{3}

, find the exact perimeter of

ACDB

A $\frac{14\pi}{3}+20$
B $\frac{2\pi}{3}+20$
C $\frac{14\pi}{3}+10$
D $8\pi+20$

Question 403

[1 marks]circles and sectors

In the same figure,

OB=2

cm,

OC=12

cm, and angle

BOC=\frac{\pi}{3}

(since

C

lies on ray

OA

produced). Find the exact length of

BC

A $10$
B $2\sqrt{43}$
C $2\sqrt{29}$
D $2\sqrt{31}$

Question 404

[1 marks]circles and sectors

In the figure, arc

CD

has radius

12

cm and subtends angle

\dfrac{\pi}{3}

O

. Find the exact length of arc

CD

Answer this when you sit the paper.

Question 405

[1 marks]circles and sectors

In the figure, arc

BA

has radius

2

cm and subtends angle

\dfrac{\pi}{3}

O

. Find the exact length of arc

BA

Answer this when you sit the paper.

Question 406

[1 marks]circles and sectors

Find the exact area of sector

OCD

(radius

12

cm, angle

\dfrac{\pi}{3}

Answer this when you sit the paper.

Question 501

[1 marks]calculus / Maclaurin series

Differentiate

y=\sec x

with respect to

x

A $\sec x\tan x$
B $-\sec x\tan x$
C $\sec^2 x$
D $\tan x\sec^2 x$

Question 502

[1 marks]calculus / Maclaurin series

Given

y=\sec x

and

\dfrac{dy}{dx}=\sec x\tan x

, express

\dfrac{d^2y}{dx^2}

in terms of

\sec x

and

\tan x

A $\sec x\tan^2 x$
B $\sec^2 x\tan x$
C $\sec^3 x\tan x$
D $\sec x(\tan^2 x+\sec^2 x)$

Question 503

[1 marks]calculus / Maclaurin series

Given

y=\sec x

with

y(0)=1

y'(0)=0

and

y''(0)=1

, find the Maclaurin expansion of

\sec x

up to and including the term in

x^2

A $1+\frac{x^2}{2}$
B $x+\frac{x^2}{2}$
C $1+x^2$
D $1+\frac{x}{2}+\frac{x^2}{2}$

Question 504

[1 marks]calculus / Maclaurin series

For

y=\sec x

with

\dfrac{dy}{dx}=\sec x\tan x

, find

y'(0)

Answer this when you sit the paper.

Question 505

[1 marks]calculus / Maclaurin series

For

\dfrac{d^2y}{dx^2}=\sec x(\tan^2x+\sec^2x)

, find

y''(0)

Answer this when you sit the paper.

Question 506

[1 marks]calculus / Maclaurin series

State the value of

y(0)=\sec(0)

Answer this when you sit the paper.

Question 601

[1 marks]complex numbers

Express the complex number

z=\dfrac{6+4i}{1+5i}

in the form

a+bi

A $1-i$
B $1+i$
C $-1-i$
D $\frac{7}{12}+\frac{13}{12}i$

Question 602

[1 marks]complex numbers

Given

z=1-i

, find

|z|

and

\arg z

A $|z|=\sqrt2,\ \arg z=\frac{\pi}{4}$
B $|z|=\sqrt2,\ \arg z=\frac{3\pi}{4}$
C $|z|=\sqrt2,\ \arg z=-\frac{\pi}{4}$
D $|z|=2,\ \arg z=-\frac{\pi}{4}$

Question 603

[1 marks]complex numbers

Evaluate

w^2-4w+13

w=2-3i

, to confirm whether

w=2-3i

is a root of

w^2-4w+13=0

A $12i$
B $-24i$
C $0$ , so $w=2-3i$ is a root
D $-13$

Question 604

[2 marks]complex numbers

Multiplying numerator and denominator of

\dfrac{6+4i}{1+5i}

by the conjugate

1-5i

, the denominator becomes a real number. What is that real number?

Answer this when you sit the paper.

Question 605

[2 marks]complex numbers

For

w=2-3i

, find

w^2

in the form

p+qi

Answer this when you sit the paper.

Question 701

[1 marks]sequences and series

A sequence satisfies

U_{n+2}=U_{n+1}+2U_n

with

U_1=2

and

U_2=3

. Find

U_5

A $-5$
B $13$
C $20$
D $27$

Question 702

[1 marks]sequences and series

The sequence

U_n

satisfies

U_{n+2}=U_{n+1}+2U_n

with

U_1=2

U_2=3

, giving terms

2, 3, 7, 13, 27,\dots

. State whether the sequence oscillates, converges, or diverges.

AIt converges to a finite limit
BIt oscillates between positive and negative values
CIt is periodic, repeating every 3 terms
DIt diverges, increasing without bound

Question 703

[1 marks]sequences and series

A geometric progression has first term

a

and common ratio

r

. The sum of the first two terms is

90

and the sum to infinity is

91\frac{3}{7}

. Find the possible values of

r

A $r=\pm\frac14$
B $r=\pm\frac19$
C $r=\pm\frac{1}{8}$
D $r=\frac{1}{64}$

Question 704

[1 marks]sequences and series

For

U_{n+2}=U_{n+1}+2U_n

with

U_1=2

U_2=3

, find

U_3

Answer this when you sit the paper.

Question 705

[1 marks]sequences and series

For the same sequence, find

U_4

Answer this when you sit the paper.

Question 706

[2 marks]sequences and series

In the geometric progression,

a(1+r)=90

and

\dfrac{a}{1-r}=\dfrac{640}{7}

. Combining these (solving for

a

from the second equation and substituting into the first) leads to

1-r^2=\dfrac{63}{64}

. What is the value of

r^2

Answer this when you sit the paper.

Question 801

[1 marks]vectors / 3D geometry

In a triangular prism,

O

is the midpoint of

AB

where

AB=10

units, and

C

is positioned so that

BC=8

units in the

\mathbf{j}

direction. Taking

O

as the origin with

\mathbf i

along

AB

and

\mathbf j

along

BC

, find

\overrightarrow{OC}

A $5\mathbf i-8\mathbf j$
B $8\mathbf i+5\mathbf j$
C $10\mathbf i+8\mathbf j$
D $5\mathbf i+8\mathbf j$

Question 802

[1 marks]vectors / 3D geometry

In the same prism,

E=-5\mathbf i+8\mathbf j+6\mathbf k

and

P=8\mathbf j

is the midpoint of

DC

. Find

\overrightarrow{EP}

A $-5\mathbf i-6\mathbf k$
B $-5\mathbf i+6\mathbf k$
C $5\mathbf i+6\mathbf k$
D $5\mathbf i-6\mathbf k$

Question 803

[1 marks]vectors / 3D geometry

In the same prism,

\overrightarrow{AF}=10\mathbf i+8\mathbf j+6\mathbf k

and

\overrightarrow{AP}=5\mathbf i+8\mathbf j

. Find angle

FAP

, correct to 1 decimal place.

A $58.7°$
B $62.6°$
C $31.3°$
D $31.1°$

Question 804

[2 marks]vectors / 3D geometry

In the prism,

A=-5\mathbf i

and

F=5\mathbf i+8\mathbf j+6\mathbf k

. Find

\overrightarrow{AF}

Answer this when you sit the paper.

Question 805

[2 marks]vectors / 3D geometry

In the prism,

A=-5\mathbf i

and

P=8\mathbf j

. Find

\overrightarrow{AP}

Answer this when you sit the paper.

Question 901

[1 marks]logarithms and exponentials

Express

\ln\left(\dfrac{2x^2-3x-2}{\sqrt{1-x}}\right)

in the form

\ln(ax+b)+\ln(cx+d)+g\ln(1-x)

A $\ln(2x+1)+\ln(x-2)-\frac12\ln(1-x)$
B $\ln(2x+1)+\ln(x-2)+\frac12\ln(1-x)$
C $\ln(2x-1)+\ln(x+2)-\frac12\ln(1-x)$
D $\ln(2x+1)+\ln(x-2)-2\ln(1-x)$

Question 902

[1 marks]logarithms and exponentials

Solve

\dfrac{e^{3x}-e^{-3x}}{2}=4

, giving your answer correct to 3 significant figures.

A $-0.698$
B $0.693$
C $0.698$
D $1.05$

Question 903

[2 marks]logarithms and exponentials

Factorise

2x^2-3x-2

Answer this when you sit the paper.

Question 904

[1 marks]logarithms and exponentials

\ln\left(\dfrac{2x^2-3x-2}{\sqrt{1-x}}\right)=\ln(ax+b)+\ln(cx+d)+g\ln(1-x)

, state the value of

g

Answer this when you sit the paper.

Question 905

[1 marks]logarithms and exponentials

In the same expansion, state the value of

d

Answer this when you sit the paper.

Question 906

[2 marks]logarithms and exponentials

Solving

\dfrac{e^{3x}-e^{-3x}}{2}=4

can be rearranged (multiplying through by

2e^{3x}

) into a quadratic in

e^{3x}

. Using the quadratic formula, find the positive value of

e^{3x}

, correct to 3 significant figures.

Answer this when you sit the paper.

Question 1001

[1 marks]coordinate geometry

Points

A(-5,3)

and

B(7,-5)

are given. Find the equation of line

AB

in the form

ax+by+c=0

A $2x-3y+1=0$
B $2x+3y-1=0$
C $3x+2y+1=0$
D $2x+3y+1=0$

Question 1002

[1 marks]coordinate geometry

D

is the midpoint of

A(-5,3)

and

B(7,-5)

. Find the gradient of line

CD

, where

C(5,5)

A $\frac32$
B $-\frac32$
C $\frac23$
D $\frac15$

Question 1003

[1 marks]coordinate geometry

In triangle

ABC

|AB|=4\sqrt{13}

units and

CD

(perpendicular to

AB

, meeting it at the midpoint

D

) has length

2\sqrt{13}

units. Find the area of triangle

ABC

A $26$ units $^2$
B $52$ units $^2$
C $104$ units $^2$
D $4\sqrt{13}$ units $^2$

Question 1004

[1 marks]coordinate geometry

Find the gradient of line

AB

through

A(-5,3)

and

B(7,-5)

Answer this when you sit the paper.

Question 1005

[1 marks]coordinate geometry

Find the coordinates of

D

, the midpoint of

A(-5,3)

and

B(7,-5)

Answer this when you sit the paper.

Question 1006

[2 marks]coordinate geometry

Find the exact length

|AB|

for

A(-5,3)

and

B(7,-5)

, in simplified surd form.

Answer this when you sit the paper.

Question 1007

[1 marks]coordinate geometry

Find the exact length

|CD|

for

C(5,5)

and

D(1,-1)

, in simplified surd form.

Answer this when you sit the paper.

Question 1101

[1 marks]numerical methods / functions

By considering the graphs of

y=\ln(1+x)

and

y=x-6

, how many real roots does

\ln(1+x)=x-6

have?

A $0$
B $1$
C $2$
D $3$

Question 1102

[1 marks]numerical methods / functions

Using the iteration

x_{n+1}=6+\ln(x_n+1)

with

x_1=6

, find the root of

\ln(1+x)=x-6

that the iteration converges to, correct to 3 significant figures.

A $7.95$
B $8.19$
C $8.22$
D $8.31$

Question 1103

[1 marks]numerical methods / functions

Show that the equation

\ln(1+x)=x-6

may be rearranged into which of the following equivalent forms?

A $e^{x-6}-x+1=0$
B $e^{x-6}-x-1=0$
C $e^{x-6}+x-1=0$
D $e^{x+6}-x-1=0$

Question 1104

[1 marks]numerical methods / functions

Using the Newton-Raphson method on

h(x)=e^{x-6}-x-1

, starting with

x_1=0

, apply the iteration twice to approximate the smaller root of

\ln(1+x)=x-6

, correct to 4 significant figures.

A $-1.000$
B $-0.9991$
C $-0.9981$
D $0.9991$

Question 1105

[2 marks]numerical methods / functions

Using

x_{n+1}=6+\ln(x_n+1)

with

x_1=6

, find

x_2

correct to 3 significant figures.

Answer this when you sit the paper.

Question 1106

[2 marks]numerical methods / functions

For

h(x)=e^{x-6}-x-1

, find

h(0)

correct to 4 significant figures.

Answer this when you sit the paper.

Question 1107

[2 marks]numerical methods / functions

For

h(x)=e^{x-6}-x-1

, find

h'(x)

Answer this when you sit the paper.

Question 1201

[1 marks]parametric equations / calculus

Given

x=3-2\cos^2\theta

and

y=-1+5\sin^2\theta

, find

\dfrac{dy}{dx}

A $\frac25$
B $\frac52$
C $\frac54$
D $5$

Question 1202

[1 marks]parametric equations / calculus

For

x=3-2\cos^2\theta

y=-1+5\sin^2\theta

0\leq\theta\leq2\pi

, with

\dfrac{dy}{dx}=\dfrac52

constant, describe the graph of

y

against

x

AA parabola with vertex at $(1,-1)$
BA straight line segment from $(1,-1)$ to $(3,4)$ with gradient $\frac52$
CA straight line segment from $(-1,1)$ to $(4,3)$ with gradient $\frac52$
DA straight line segment from $(1,4)$ to $(3,-1)$ with gradient $-\frac52$

Question 1203

[2 marks]parametric equations / calculus

Given

x=3-2\cos^2\theta

, find

\dfrac{dx}{d\theta}

Answer this when you sit the paper.

Question 1204

[2 marks]parametric equations / calculus

Given

y=-1+5\sin^2\theta

, find

\dfrac{dy}{d\theta}

Answer this when you sit the paper.

Question 1205

[2 marks]parametric equations / calculus

Find the maximum value of

x=3-2\cos^2\theta

for

0\leq\theta\leq2\pi

Answer this when you sit the paper.

Question 1206

[1 marks]parametric equations / calculus

Find the minimum value of

x=3-2\cos^2\theta

for

0\leq\theta\leq2\pi

Answer this when you sit the paper.

Question 1301

[1 marks]algebra / polynomials

The line

y=x-2m

meets the curve

xy=8

where

x^2-2mx-8=0

. For which values of

m\in\mathbb R

does the line meet the curve?

AFor every real value of $m$ , since $4m^2+32$ is positive whatever value $m$ takes
BFor $m>0$ , since the discriminant is positive precisely in that range
CFor $m=0$ , since that is the unique value making $x^2-8=0$ solvable
DFor no real $m$ , since $4m^2+32$ is negative regardless of $m$

Question 1302

[1 marks]algebra / polynomials

The function

f(x)=2x^3+ax^2-11x+6

is divisible by

(x+2)

. Find the value of

a

A $-3$
B $-2$
C $3$
D $8$

Question 1303

[1 marks]algebra / polynomials

With

a=-3

f(x)=2x^3-3x^2-11x+6=(x+2)(2x-1)(x-3)

. Find

f(x-2)

in factored form.

A $x(2x-3)(x-5)$
B $x(2x-5)(x-5)$
C $x(2x-5)(x+1)$
D $(x+4)(2x+3)(x-1)$

Question 1304

[2 marks]algebra / polynomials

Substituting

y=x-2m

into

xy=8

gives

x^2-2mx-8=0

. Find the discriminant of this quadratic, in terms of

m

Answer this when you sit the paper.

Question 1305

[2 marks]algebra / polynomials

Given

f(x)=(x+2)(2x-1)(x-3)

, expanding

(2x-1)(x-3)

gives

2x^2+px+q

. Find the value of

q

, the constant term.

Answer this when you sit the paper.

Question 1306

[2 marks]algebra / polynomials

Given

f(x-2)=x(2x-5)(x-5)

, expanding

x(2x-5)

gives

2x^2+px

. Find the value of

p

Answer this when you sit the paper.

Question 1401

[1 marks]trigonometry

Given

\tan\alpha=\frac14

and

\tan\beta=\frac35

, where

\alpha

and

\beta

are acute, find

\alpha+\beta

A $45°$
B $40.4°$
C $90°$
D $60°$

Question 1402

[1 marks]trigonometry

Express

7\cos\theta-24\sin\theta

in the form

R\cos(\theta+\alpha)

, where

R>0

and

\alpha

is acute. Find

R

and

\alpha

, correct to the nearest 0.1°.

A $R=25,\ \alpha=106.3°$
B $R=25,\ \alpha=73.7°$
C $R=25,\ \alpha=16.3°$
D $R=31,\ \alpha=73.7°$

Question 1403

[1 marks]trigonometry

Using

7\cos\theta-24\sin\theta=25\cos(\theta+73.7°)

, solve

7\cos\theta-24\sin\theta=16

for

0\leq\theta\leq360°

, correct to the nearest 0.1°.

A $\theta=236.1°$ or $56.5°$
B $\theta=123.9°$ or $263.5°$
C $\theta=50.2°$ or $309.8°$
D $\theta=236.1°$ or $336.5°$

Question 1404

[1 marks]trigonometry

Given that

7\cos\theta-24\sin\theta

has a maximum value of

25

and a minimum value of

-25

, find the greatest and least values of

\dfrac{1}{27+7\cos\theta-24\sin\theta}

Agreatest $=\frac12$ , least $=\frac{1}{27}$
Bgreatest $=\frac14$ , least $=\frac{1}{58}$
Cgreatest $=\frac{1}{52}$ , least $=\frac12$
Dgreatest $=\frac12$ , least $=\frac{1}{52}$

Question 1405

[2 marks]trigonometry

\tan\alpha=\frac14

\tan\beta=\frac35

. Using

\tan(\alpha+\beta)=\dfrac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}

, find the numerator

\tan\alpha+\tan\beta

as a single fraction.

Answer this when you sit the paper.

Question 1406

[2 marks]trigonometry

For the same identity, find the denominator

1-\tan\alpha\tan\beta

as a single fraction.

Answer this when you sit the paper.

Question 1407

[2 marks]trigonometry

Given

7\cos\theta-24\sin\theta=R\cos(\theta+\alpha)

, find the exact value of

R

Answer this when you sit the paper.

Question 1501

[1 marks]functions and transformations

The graph of

y=x^2

is stretched in the positive

x

-direction by scale factor

4

, then translated

3

units in the negative

x

-direction. Find the equation of the resulting graph.

A $y=16(x+3)^2$
B $y=\frac{(x+3)^2}{16}$
C $y=\frac{(x+3)^2}{4}$
D $y=\frac{(x-3)^2}{16}$

Question 1502

[1 marks]functions and transformations

Given

f(x)=\dfrac{1}{x+1}

for

x>-1

, find

f^{-1}(x)

A $f^{-1}(x)=x-1$
B $f^{-1}(x)=\frac{1}{x+1}$
C $f^{-1}(x)=\frac1x-1$
D $f^{-1}(x)=\frac{1}{x-1}$

Question 1503

[1 marks]functions and transformations

State the range of

y=f(x)=\dfrac{1}{x+1}

for

x>-1

A $y>0$
B $y\geq0$
C $y<0$
D $-1<y<1$

Question 1504

[1 marks]functions and transformations

Describe how the graph of

y=f^{-1}(x)

relates to the graph of

y=f(x)

for

f(x)=\dfrac{1}{x+1}

AIt is the reflection of $y=f(x)$ in the line $y=x$
BIt is a translation of $y=f(x)$ by 1 unit
CIt is identical to the graph of $y=f(x)$ , since $f$ equals its own inverse
DIt is the reflection of $y=f(x)$ in the $x$ -axis

Question 1505

[2 marks]functions and transformations

The graph of

y=x^2

is stretched in the positive

x

-direction by scale factor 4. What is the equation of the resulting graph, before any translation is applied?

Answer this when you sit the paper.

Question 1506

[2 marks]functions and transformations

For

y=f(x)+2=\dfrac{1}{x+1}+2

, state the equation of the vertical asymptote.

Answer this when you sit the paper.

Question 1507

[2 marks]functions and transformations

For

y=f(x)+2=\dfrac{1}{x+1}+2

, state the equation of the horizontal asymptote.

Answer this when you sit the paper.

Question 1601

[1 marks]integration

Using the substitution

u=2x+1

, evaluate

\displaystyle\int_{-1}^{0}x(2x+1)^7\,dx

A $\frac19$
B $\frac{1}{18}$
C $\frac{1}{36}$
D $-\frac{1}{18}$

Question 1602

[1 marks]integration

The curve

y=x(2x+1)^7

crosses the

x

-axis at

(0,0)

and

\left(-\frac12,0\right)

. Why is the area between the curve, the

x

-axis, and the lines

x=-1

and

x=0

not equal to

\frac{1}{18}

ABecause the curve changes sign at $x=-\frac12$ within $[-1,0]$ , so the definite integral gives the signed (net) area rather than the total area
BBecause the substitution $u=2x+1$ is not valid over the whole interval $[-1,0]$
CBecause the integrand is not continuous at $x=-\frac12$
DBecause the limits of integration should be swapped

Question 1603

[1 marks]integration

Find

\displaystyle\int\ln3x\,dx

A $x\ln3x + C$
B $\frac{x^2}{2}\ln3x - \frac{x^2}{4}+C$
C $x\ln3x - 3x + C$
D $x\ln3x - x + C$

Question 1604

[2 marks]integration

Using

u=2x+1

\int_{-1}^{0}x(2x+1)^7\,dx

, the limits

x=-1

and

x=0

transform to which values of

u

Answer this when you sit the paper.

Question 1605

[2 marks]integration

The transformed integral gives an antiderivative

\dfrac{u^9}{9}-\dfrac{u^8}{8}

. Evaluate this expression at

u=1

Answer this when you sit the paper.

Question 1606

[2 marks]integration

Evaluate

\dfrac{u^9}{9}-\dfrac{u^8}{8}

u=-1

Answer this when you sit the paper.

Question 1607

[1 marks]integration

Using integration by parts on

\int\ln3x\,dx

with

u=\ln3x

and

dv=dx

, find

du

(the derivative of

\ln3x

with respect to

x

Answer this when you sit the paper.

Pure Mathematics Paper 1 June 2006

Sit this paper online

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