ZIMSEC A Level Pure Mathematics Paper 1 June 2009

Question 101

[1 marks]algebra / factor theorem

Given that

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

, find the value of

k

for which

(x + 2)

is a factor of

f(x)

A $k = 1$
B $k = 2$
C $k = \frac{1}{2}$
D $k = -\frac{1}{2}$

Question 102

[1 marks]algebra / factor theorem

With

k=\frac12

, so that

f(x) = x^3 + 2x^2 - \frac12 x - 1

and

(x+2)

is a factor, which of the following is the complete factorisation of

f(x)

A $(x+2)\left(x-\frac{1}{\sqrt2}\right)\left(x+\frac{1}{\sqrt2}\right)$
B $(x+2)\left(x-\frac12\right)\left(x+\frac12\right)$
C $(x+2)\left(x^2+\frac12\right)$
D $(x-2)\left(x-\frac{1}{\sqrt2}\right)\left(x+\frac{1}{\sqrt2}\right)$

Question 103

[3 marks]algebra / factor theorem

Dividing

f(x)=x^3+2x^2-\frac12x-1

(x+2)

gives a quadratic quotient

x^2+bx+c

. State

b

and

c

as 'b, c'.

Answer this when you sit the paper.

Question 201

[1 marks]coordinate geometry / simultaneous equations

The line

y = 2x + 1

is substituted into the curve

y^2 - xy = 3

. Which quadratic equation in

x

results?

A $4x^2+3x-2=0$
B $2x^2+3x-2=0$
C $2x^2+5x-2=0$
D $2x^2+3x+1=0$

Question 202

[1 marks]coordinate geometry / simultaneous equations

The line

y = 2x + 1

intersects the curve

y^2 - xy = 3

at points

A

and

B

. Find the coordinates of the mid-point of

AB

A $\left(-\frac34,-\frac12\right)$
B $\left(\frac34,\frac12\right)$
C $\left(-\frac34,\frac12\right)$
D $\left(-\frac12,-\frac34\right)$

Question 203

[3 marks]coordinate geometry / simultaneous equations

Solving

2x^2+3x-2=0

by factorising as

(2x-1)(x+2)=0

, state both

x

-values as 'x1, x2'.

Answer this when you sit the paper.

Question 301

[1 marks]functions / inverse functions

The function

f

is defined by

f: x \mapsto x^2 - 4x

x \in \mathbb{R}

|x| \leq 1

. Which statement correctly explains why

f

is one-one on this domain?

AThe vertex of $y=x^2-4x$ is at $x=-2$ , which lies outside $[-1,1]$ ; since $[-1,1]$ lies to the right of this vertex, $f$ is strictly increasing there, so no two $x$ -values share an image.
BThe vertex of $y=x^2-4x$ is at $x=0$ , inside $[-1,1]$ , but the two branches on either side never take equal values within this domain.
CThe vertex of $y=x^2-4x$ is at $x=2$ , which lies outside $[-1,1]$ ; since $[-1,1]$ lies to the left of the vertex, $f$ is strictly decreasing there, so no two $x$ -values share an image.
DThe vertex of $y=x^2-4x$ is at $x=2$ , and since $[-1,1]$ lies to the right of the vertex, $f$ is strictly increasing there, so no two $x$ -values share an image.

Question 302

[1 marks]functions / inverse functions

For

f: x \mapsto x^2 - 4x

with domain

-1 \le x \le 1

, find an expression for

f^{-1}(x)

A $f^{-1}(x)=\sqrt{x+4}-2$
B $f^{-1}(x)=2-\sqrt{x-4}$
C $f^{-1}(x)=2+\sqrt{x+4}$
D $f^{-1}(x)=2-\sqrt{x+4}$

Question 303

[3 marks]functions / inverse functions

Complete the square: write

f(x)=x^2-4x

(x-p)^2+q

. State

p

and

q

as 'p, q'.

Answer this when you sit the paper.

Question 401

[1 marks]calculus / parametric differentiation

Given that

y = \ln(t + 2)

and

x = t^2

for

t > -2

, find

\dfrac{dy}{dx}

in terms of

t

A $2t(t+2)$
B $\dfrac{2t}{t+2}$
C $\dfrac{1}{2t(t+2)}$
D $\dfrac{1}{2(t+2)}$

Question 402

[1 marks]calculus / parametric differentiation

Given that

y = \ln(t + 2)

and

x = t^2

for

t > -2

, for which values of

t

does

y

increase as

x

increases?

A $-2<t<0$
B $t>0$
C $t<-2$
D $t>-2$

Question 403

[2 marks]calculus / parametric differentiation

For

y=\ln(t+2)

, find

\dfrac{dy}{dt}

in terms of

t

Answer this when you sit the paper.

Question 404

[2 marks]calculus / parametric differentiation

For

x=t^2

, find

\dfrac{dx}{dt}

in terms of

t

Answer this when you sit the paper.

Question 501

[1 marks]exponential equations / surds

Solve the inequality

(0.2)^{2x-3} > 10^6

A $x>-2.79$
B $x<2.79$
C $x<-2.79$
D $x<-1.5$

Question 502

[1 marks]exponential equations / surds

Solve the equation

3\sqrt{x} - 11 = \dfrac{4}{\sqrt{x}}

for

x>0

A $x=8$
B $x=4$
C $x=16$
D $x=121$

Question 503

[2 marks]exponential equations / surds

Since

0.2=\dfrac15=5^{-1}

, rewrite the left side of

(0.2)^{2x-3}>10^6

5^{k}

for some expression

k

in terms of

x

. State

k

Answer this when you sit the paper.

Question 504

[2 marks]exponential equations / surds

Evaluate

\log_5 10

, correct to 4 decimal places (needed to solve

(0.2)^{2x-3}>10^6

using base 5).

Answer this when you sit the paper.

Question 505

[1 marks]exponential equations / surds

Solving

3\sqrt{x}-11=\dfrac{4}{\sqrt{x}}

: with

u=\sqrt{x}

, this leads to

3u^2-11u-4=0

, which factorises as

(3u+1)(u-c)=0

. State

c

Answer this when you sit the paper.

Question 601

[1 marks]integration / inequalities

Find

\int x\ln(2x)\,dx

A $\dfrac{x^2}{2}\ln(2x)+\dfrac{x^2}{4}+C$
B $\dfrac{x^2}{2}\ln(2x)-\dfrac{x^2}{4}+C$
C $\dfrac{x^2}{2}\ln(2x)-\dfrac{x^2}{8}+C$
D $\dfrac{x^2}{2}\ln(2x)-\dfrac{x^2}{2}+C$

Question 602

[1 marks]integration / inequalities

Solve the inequality

|4x - 1| \leq |2x + 7|

A $-4\le x\le1$
B $x\le-1 \text{ or } x\ge4$
C $-7\le x\le4$
D $-1\le x\le4$

Question 603

[2 marks]integration / inequalities

Integrating

\int x\ln(2x)\,dx

by parts with

u=\ln(2x)

dv=x\,dx

, state

v

(the antiderivative of

dv

Answer this when you sit the paper.

Question 604

[1 marks]integration / inequalities

Using

u=\ln(2x)

, state

\dfrac{du}{dx}

Answer this when you sit the paper.

Question 605

[2 marks]integration / inequalities

Squaring both sides of

|4x-1|\leq|2x+7|

gives

16x^2-8x+1\leq4x^2+28x+49

. Simplify this to the form

x^2-3x-c\leq0

. State

c

Answer this when you sit the paper.

Question 701

[1 marks]complex numbers

The complex number

p = 3 - 5i

and

q = 4ip

. Which of the following correctly states the relationship between

|p|

and

|q|

, and between

\arg p

and

\arg q

A $|q|=4|p|$ and $\arg q=\arg p+\dfrac{\pi}{2}$
B $|q|=4|p|$ and $\arg q=\arg p-\dfrac{\pi}{2}$
C $|q|=\dfrac14|p|$ and $\arg q=\arg p+\dfrac{\pi}{2}$
D $|q|=4|p|$ and $\arg q=\arg p+\pi$

Question 702

[1 marks]complex numbers

Given

p = 3 - 5i

and

q = 4ip

, find

r = p + q

in the form

a + bi

A $r=17+7i$
B $r=23-7i$
C $r=23+7i$
D $r=-17+7i$

Question 703

[1 marks]complex numbers

Points

P

Q

R

in an Argand diagram represent

p=3-5i

q=20+12i

(where

q=4ip

) and

r=p+q=23+7i

respectively, with

O

the origin. What kind of quadrilateral is

OPRQ

, and what is its area?

Arectangle, area $=172$
Brhombus, area $=136$
Csquare, area $=136$
Drectangle, area $=136$

Question 704

[2 marks]complex numbers

Given

p=3-5i

and

q=4ip

, compute

q

in the form

a+bi

Answer this when you sit the paper.

Question 705

[2 marks]complex numbers

Given

\bar p=3+5i

(the conjugate of

p=3-5i

) and

q=20+12i

, compute

\bar p\,q

in the form

a+bi

(used to find the area of

OPRQ

Answer this when you sit the paper.

Question 801

[1 marks]calculus / differentiation of trig functions

Given that

y = \dfrac{1}{\sin 3x} = \csc 3x

, find

\dfrac{dy}{dx}

A $-3\csc 3x\cot 3x$
B $-3\csc^2 3x$
C $3\csc 3x\cot 3x$
D $-\csc 3x\cot 3x$

Question 802

[1 marks]calculus / differentiation of trig functions

Given that

y = \csc 3x

, express

\dfrac{d^2y}{dx^2} + 9y

in terms of

y

A $0$
B $18y^3$
C $9y^3$
D $18y^3-18y$

Question 803

[2 marks]calculus / differentiation of trig functions

For

y=\csc3x

, evaluate

y

x=\pi/18

(i.e.

\csc(3\times\pi/18)=\csc(\pi/6)

Answer this when you sit the paper.

Question 804

[2 marks]calculus / differentiation of trig functions

For

\dfrac{dy}{dx}=-3\cot3x\csc3x

, evaluate

\dfrac{dy}{dx}

x=\pi/18

(using

\cot(\pi/6)=\sqrt3

and

\csc(\pi/6)=2

Answer this when you sit the paper.

Question 805

[2 marks]calculus / differentiation of trig functions

Given

y=\csc3x

satisfies

\dfrac{d^2y}{dx^2}+9y=18y^3

, evaluate

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

x=\pi/18

, using

y=2

there.

Answer this when you sit the paper.

Question 901

[1 marks]trigonometry / sequences and series

Triangle

ABC

is right-angled at

A

, with

AC=2a

and angle

ACB=\theta

radians. A circular arc

AD

, with centre

C

and radius

2a

, divides the triangle into a sector

P

(bounded by the arc, of angle

\theta

) and the remaining region

Q

. Given that the area of

P

is 5 times the area of

Q

, which equation must

\theta

satisfy?

A $6\theta=5\theta\tan\theta$
B $5\theta=6\tan\theta$
C $6\theta=5\tan\theta$
D $6\theta=5\cot\theta$

Question 902

[1 marks]trigonometry / sequences and series

The first two terms of a geometric progression are 4 and 6 respectively. Calculate the sum of the first 20 terms, giving your answer correct to the nearest integer.

A $12$
B $17727$
C $26594$
D $39895$

Question 903

[1 marks]trigonometry / sequences and series

The first term of an arithmetic progression is 6 and the sum of the first 36 terms is 90. Find the common difference.

A $d=\frac15$
B $d=-\frac16$
C $d=-\frac{7}{18}$
D $d=-\frac15$

Question 904

[2 marks]trigonometry / sequences and series

Triangle

ABC

is right-angled at

A

with

AC=2a

, angle

ACB=\theta

. State the area of triangle

ABC

in terms of

a

and

\theta

(using

AB=2a\tan\theta

Answer this when you sit the paper.

Question 905

[2 marks]trigonometry / sequences and series

State the area of sector

P

(radius

2a

, angle

\theta

radians) cut from the triangle in the arc/sector problem.

Answer this when you sit the paper.

Question 906

[2 marks]trigonometry / sequences and series

A geometric progression has first two terms 4 and 6. State the common ratio

r

as a fraction.

Answer this when you sit the paper.

Question 1001

[1 marks]differential equations / data analysis

Satellite pictures give the area

A

^2

of a lake at times

t=0,1,2,3,4

weeks as

76, 70, 62, 51, 31

. What do the successive decreases in area (

6, 8, 11, 20

) show about the rate of decrease of the area?

AThe rate of decrease is constant, since $6, 8, 11$ and $20$ average to about $11$ km $^2$ per week.
BThe rate of decrease is not constant: it grows smaller each week, so the graph flattens out over time.
CThe rate of decrease is constant at exactly $20$ km $^2$ per week, matching the largest recorded drop.
DThe rate of decrease is not constant: it grows larger each week, so the graph of $A$ against $t$ is a curve, not a straight line.

Question 1002

[1 marks]differential equations / data analysis

In a second lake, the rate of decrease of the area

A

^2

is inversely proportional to the area at time

t

weeks. Which differential equation correctly models this, where

k>0

is a constant?

A $\dfrac{dA}{dt}=-\dfrac{k}{A}$
B $\dfrac{dA}{dt}=-kA$
C $\dfrac{dA}{dt}=-\dfrac{k}{A^2}$
D $\dfrac{dA}{dt}=\dfrac{k}{A}$

Question 1003

[1 marks]differential equations / data analysis

Solving

\dfrac{dA}{dt}=-\dfrac{k}{A}

gives

A^2=-2kt+C

. Given that

A=8

when

t=1

and

A=6

when

t=5

, find the area when

t=0

, correct to 2 decimal places.

A $A=8.37$
B $A=8.43$
C $A=8.00$
D $A=71$

Question 1004

[2 marks]differential equations / data analysis

Solving

\dfrac{dA}{dt}=-\dfrac{k}{A}

by separating variables gives

A\,dA=-k\,dt

. Integrate the left-hand side,

\int A\,dA

(omit the constant of integration).

Answer this when you sit the paper.

Question 1005

[2 marks]differential equations / data analysis

Using

A^2=-2kt+C

with

A=8

t=1

and

A=6

t=5

, forming two equations and subtracting to eliminate

C

gives

28=nk

. State

n

Answer this when you sit the paper.

Question 1006

[2 marks]differential equations / data analysis

Using

28=8k

, state

k

as a decimal.

Answer this when you sit the paper.

Question 1007

[1 marks]differential equations / data analysis

Using

k=3.5

and

64=-2k+C

, state

C

Answer this when you sit the paper.

Question 1101

[1 marks]vectors

Points

A

B

C

have position vectors

a=2i+j-k

b=3i+4j-2k

and

c=5i-j+2k

relative to the origin

O

. Find the size of angle

A\hat{B}C

, giving your answer to the nearest

0.1°

A $20.1°$
B $40.2°$
C $139.8°$
D $49.8°$

Question 1102

[1 marks]vectors

Points

A

B

C

have position vectors

a=2i+j-k

b=3i+4j-2k

and

c=5i-j+2k

. Given that

ABCD

is a parallelogram, find the position vector of

D

A $6i+2j+k$
B $0i+6j-5k$
C $-4i+4j-3k$
D $4i-4j+3k$

Question 1103

[1 marks]vectors

Points

A

B

have position vectors

a=2i+j-k

b=3i+4j-2k

, and

D=4i-4j+3k

is the fourth vertex of parallelogram

ABCD

. Find the area of

ABCD

, giving your answer in exact form.

A $\sqrt{206}/2$
B $\sqrt{206}$
C $206$
D $\sqrt{166}$

Question 1104

[2 marks]vectors

For

A(2,1,-1)

B(3,4,-2)

C(5,-1,2)

(position vectors

a,b,c

), compute

a-b

as a vector, in the form 'x,y,z'.

Answer this when you sit the paper.

Question 1105

[2 marks]vectors

Compute

c-b

as a vector, in the form 'x,y,z'.

Answer this when you sit the paper.

Question 1106

[2 marks]vectors

Evaluate the scalar product

(a-b)\cdot(c-b)

using

a-b=(-1,-3,1)

and

c-b=(2,-5,4)

Answer this when you sit the paper.

Question 1107

[2 marks]vectors

State

|a-b|

as an exact surd, using

a-b=(-1,-3,1)

Answer this when you sit the paper.

Question 1201

[1 marks]logarithms / exponential growth

Express

e^{\ln x + \ln y}

in its simplest form, for

x,y>0

A $xy$
B $x^y$
C $x+y$
D $\ln(xy)$

Question 1202

[1 marks]logarithms / exponential growth

Express

\ln e^{2x}

in its simplest form.

A $e^{2x}$
B $2\ln x$
C $x^2$
D $2x$

Question 1203

[1 marks]logarithms / exponential growth

The number

N

of bacteria in a culture at time

t

hours is modelled by

N=600e^{ct}

. Given that

N=600

when

t=0

and

N=1800

when

t=2

, find the exact value of

c

A $c=\frac12\ln3$
B $c=\ln3$
C $c=\frac12\ln1800$
D $c=2\ln3$

Question 1204

[2 marks]logarithms / exponential growth

For

N=600e^{ct}

, differentiate to find

\dfrac{dN}{dt}

in terms of

t

and

c

(before rewriting in terms of

N

Answer this when you sit the paper.

Question 1205

[2 marks]logarithms / exponential growth

Using

600e^{2c}=1800

, divide both sides by 600 to isolate the exponential; state the resulting equation in the form

e^{2c}=m

. State

m

Answer this when you sit the paper.

Question 1206

[2 marks]logarithms / exponential growth

From

e^{2c}=3

, take natural logs of both sides; state the resulting expression for

2c

Answer this when you sit the paper.

Question 1207

[2 marks]logarithms / exponential growth

Using

N=600e^{ct}

with

c=\frac12\ln3

, find

N

t=4

hours (as an exact integer, using

e^{2c}=3

Answer this when you sit the paper.

Question 1208

[1 marks]logarithms / exponential growth

For

N=600e^{ct}

, state

N

when

t=0

by direct substitution.

Answer this when you sit the paper.

Pure Mathematics Paper 1 June 2009

Sit this paper online

The questions

Question 101

Question 102

Question 103

Question 201

Question 202

Question 203

Question 301

Question 302

Question 303

Question 401

Question 402

Question 403

Question 404

Question 501

Question 502

Question 503

Question 504

Question 505

Question 601

Question 602

Question 603

Question 604

Question 605

Question 701

Question 702

Question 703

Question 704

Question 705

Question 801

Question 802

Question 803

Question 804

Question 805

Question 901

Question 902

Question 903

Question 904

Question 905

Question 906

Question 1001

Question 1002

Question 1003

Question 1004

Question 1005

Question 1006

Question 1007

Question 1101

Question 1102

Question 1103

Question 1104

Question 1105

Question 1106

Question 1107

Question 1201

Question 1202

Question 1203

Question 1204

Question 1205

Question 1206

Question 1207

Question 1208

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