ZIMSEC A Level Pure Mathematics Paper 1 November 2002

Question 101

[1 marks]quadratics and logarithmic graphs

For what set of values of

k

does the equation

kx^2 - 3x = k - 3

have (at least one) real root?

AAll real values of $k$
B $k \geq \tfrac{3}{2}$
C $k = \tfrac{3}{2}$ only
DAll real $k$ except $k = 0$

Question 102

[1 marks]quadratics and logarithmic graphs

Values of

x

and

y

satisfy

y = Ax^b

for constants

A

and

b

. Which pair of axes should be plotted, and what shape of graph confirms the relationship?

APlot $\ln y$ against $\ln x$ : a straight line confirms the relationship
BPlot $y$ against $\ln x$ : a straight line confirms the relationship
CPlot $y$ against $x^b$ : a straight line confirms the relationship
DPlot $\ln y$ against $x$ : a straight line confirms the relationship

Question 103

[2 marks]quadratics and logarithmic graphs

Rearranging

kx^2-3x=k-3

gives

kx^2-3x-k+3=0

. The discriminant of this quadratic simplifies to

(2k-3)^2

. For what value of

k

does this discriminant equal zero?

Answer this when you sit the paper.

Question 201

[1 marks]surds and rationalisation

Express

\dfrac{2}{3-\sqrt7}-\sqrt{63}

in the form

a+b\sqrt c

, where

a

b

and

c

are integers.

A $3+4\sqrt7$
B $3-2\sqrt7$
C $-3+2\sqrt7$
D $3-4\sqrt7$

Question 202

[1 marks]surds and rationalisation

For

\dfrac{2}{3-\sqrt7}-\sqrt{63}=a+b\sqrt c

with

a,b,c

integers, what is the value of

a+b+c

Answer this when you sit the paper.

Question 203

[1 marks]surds and rationalisation

Simplify

\sqrt{63}

in the form

k\sqrt7

. What is the value of

k

Answer this when you sit the paper.

Question 301

[1 marks]numerical integration

Use the trapezium rule with 5 equal intervals to estimate

\displaystyle\int_0^1\cos(x^2)\,dx

, giving your answer correct to 3 significant figures.

A $0.626$
B $0.770$
C $0.899$
D $1.80$

Question 302

[2 marks]numerical integration

Using the trapezium rule for

\int_0^1\cos(x^2)\,dx

with 5 equal intervals (

h=0.2

), find the last ordinate

y_5=\cos(1^2)

correct to 3 significant figures.

Answer this when you sit the paper.

Question 401

[1 marks]exponential growth and logarithms

A population increases by 3% every year. Find, to 3 significant figures, the time in years for the population to double.

A $66.7$ years
B $23.4$ years
C $23.1$ years
D $24.0$ years

Question 402

[1 marks]exponential growth and logarithms

A population increases by 3% each year. Which equation must be solved to find the number of years

n

for the population to double?

A $2^n=1.03$
B $1.03^n=2$
C $1.03n=2$
D $n^{1.03}=2$

Question 403

[2 marks]exponential growth and logarithms

Given

1.03^n=2

, taking logarithms gives

n=\dfrac{\ln 2}{\ln 1.03}

. Find

\ln 1.03

correct to 4 decimal places.

Answer this when you sit the paper.

Question 501

[1 marks]volumes of revolution

The region between the curve

y=e^x+1

, the

x

-axis, and the lines

x=-1

and

x=1

is rotated through

4

right angles about the

x

-axis. Find the volume of the solid formed, correct to 2 decimal places.

A $10.33$
B $11.39$
C $13.67$
D $32.44$

Question 502

[1 marks]volumes of revolution

Which integral correctly represents the volume generated when the region bounded by

y=e^x+1

, the

x

-axis,

x=-1

and

x=1

is rotated through 4 right angles about the

x

-axis?

A $\pi\displaystyle\int_{-1}^{1}(e^x+1)\,dx$
B $2\pi\displaystyle\int_{-1}^{1}(e^x+1)\,dx$
C $\pi\displaystyle\int_{-1}^{1}(e^x+1)^2\,dx$
D $\pi\displaystyle\int_{0}^{1}(e^x+1)^2\,dx$

Question 503

[2 marks]volumes of revolution

In evaluating

\pi\left[\dfrac{e^{2x}}{2}+2e^x+x\right]_{-1}^{1}

, find the value of

e^2-e^{-2}

correct to 3 significant figures.

Answer this when you sit the paper.

Question 601

[1 marks]modulus inequalities and graphs

Solve the inequality

2x+2>|3x-6|

A $\tfrac45<x<2$
B $x<8$
C $-8<x<\tfrac45$
D $\tfrac45<x<8$

Question 602

[1 marks]modulus inequalities and graphs

What are the coordinates of the vertex of the graph

y=|3x-6|

(the point where its two straight-line branches meet)?

A $(2,0)$
B $(0,6)$
C $(0,-6)$
D $(6,0)$

Question 603

[1 marks]modulus inequalities and graphs

Solving

2x+2>|3x-6|

for

x\geq2

(where

|3x-6|=3x-6

) leads to which inequality?

A $x<2$
B $x>-8$
C $x>8$
D $x<8$

Question 604

[2 marks]modulus inequalities and graphs

Solving

2x+2>|3x-6|

for

x<2

(where

|3x-6|=6-3x

) leads to

5x>4

. What is the resulting lower bound for

x

, as a fraction?

Answer this when you sit the paper.

Question 701

[1 marks]factor theorem and polynomials

Given that

f(x)\equiv2x^4+ax^3+bx^2-5x+6

has both

(x+2)

and

(x-3)

as factors, find the values of

a

and

b

A $a=-1,\ b=14$
B $a=-1,\ b=-14$
C $a=1,\ b=-14$
D $a=-5,\ b=-2$

Question 702

[1 marks]factor theorem and polynomials

Given

f(x)\equiv2x^4+ax^3+bx^2-5x+6

and that

(x+2)

is a factor, applying the factor theorem (

f(-2)=0

) leads to which simplified linear equation relating

a

and

b

A $-2a+b=-12$
B $-2a+b=12$
C $2a+b=-12$
D $-2a-b=-12$

Question 703

[1 marks]factor theorem and polynomials

Given

f(x)\equiv2x^4+ax^3+bx^2-5x+6

and that

(x-3)

is a factor, applying the factor theorem (

f(3)=0

) leads to which simplified linear equation relating

a

and

b

A $a+3b=-17$
B $27a+9b=153$
C $3a+b=17$
D $3a+b=-17$

Question 704

[2 marks]factor theorem and polynomials

Using

-2a+b=-12

and

3a+b=-17

together, find the value of

a

only.

Answer this when you sit the paper.

Question 801

[1 marks]sequences and series

A sequence is defined by

U_r=n-3r

. What are its first three terms

U_1,U_2,U_3

A $n-3,\ n-6,\ n-9$
B $n-3,\ n-9,\ n-27$
C $n-1,\ n-2,\ n-3$
D $n,\ n-3,\ n-6$

Question 802

[1 marks]sequences and series

For the sequence

U_r=n-3r

, find

\displaystyle\sum_{r=n}^{2n}U_r

in terms of

n

A $-\dfrac{7n(n-1)}{2}$
B $\dfrac{7n(n+1)}{2}$
C $-\dfrac{9n(n+1)}{2}$
D $-\dfrac{7n(n+1)}{2}$

Question 803

[2 marks]sequences and series

Find

\displaystyle\sum_{r=n}^{2n}n

(the constant

n

added once per term over this range) in terms of

n

, by first working out how many terms the sum has.

Answer this when you sit the paper.

Question 804

[2 marks]sequences and series

Find

\displaystyle\sum_{r=n}^{2n}r

in terms of

n

Answer this when you sit the paper.

Question 901

[1 marks]differentiation and integration by parts

Differentiate

x\sin\!\left(6x+\dfrac{\pi}{4}\right)

with respect to

x

A $\sin\!\left(6x+\tfrac{\pi}{4}\right)+x\cos\!\left(6x+\tfrac{\pi}{4}\right)$
B $6x\cos\!\left(6x+\tfrac{\pi}{4}\right)-\sin\!\left(6x+\tfrac{\pi}{4}\right)$
C $6\cos\!\left(6x+\tfrac{\pi}{4}\right)$
D $\sin\!\left(6x+\tfrac{\pi}{4}\right)+6x\cos\!\left(6x+\tfrac{\pi}{4}\right)$

Question 902

[1 marks]differentiation and integration by parts

Evaluate exactly

\displaystyle\int_{\pi/6}^{\pi/4}x\sin\!\left(6x+\dfrac{\pi}{4}\right)dx

A $-\dfrac{5\sqrt2}{144}\pi$
B $-\dfrac{\sqrt2}{144}\pi$
C $\dfrac{5\sqrt2}{144}\pi$
D $-\dfrac{5\sqrt2}{72}\pi$

Question 903

[1 marks]differentiation and integration by parts

In integrating

\int x\sin(6x+\pi/4)\,dx

by parts, the coefficient of

x\cos(6x+\pi/4)

in the antiderivative is

-\dfrac{1}{k}

. Find

k

Answer this when you sit the paper.

Question 904

[2 marks]differentiation and integration by parts

Evaluate

\cos\!\left(6\cdot\dfrac{\pi}{4}+\dfrac{\pi}{4}\right)

exactly, needed when evaluating the integral at

x=\pi/4

Answer this when you sit the paper.

Question 905

[1 marks]differentiation and integration by parts

Evaluate

\sin\!\left(6\cdot\dfrac{\pi}{6}+\dfrac{\pi}{4}\right)

exactly, needed when evaluating the integral at the lower limit

x=\pi/6

Answer this when you sit the paper.

Question 1001

[1 marks]numerical methods and error analysis

Given

x=8.7

and

y=0.3

, each correct to 1 decimal place, find the value of

xy

and estimate the relative error in this quantity.

A $xy=2.61$ , relative error $\approx0.00575$
B $xy=2.61$ , relative error $\approx0.05$
C $xy=2.61$ , relative error $\approx0.172$
D $xy=2.61$ , relative error $\approx0.0862$

Question 1002

[1 marks]numerical methods and error analysis

Given

x=8.7

and

y=0.3

, each correct to 1 decimal place, find

y-\dfrac1x

with an appropriate error bound, quoted to 1 significant figure.

A $0.19\pm0.10$
B $-0.19\pm0.05$
C $0.19\pm0.001$
D $0.19\pm0.05$

Question 1003

[1 marks]numerical methods and error analysis

Which formula estimates the relative error in the product

xy

, given errors

\delta x

x

and

\delta y

y

A $\dfrac{\delta(xy)}{xy}\approx\dfrac{\delta x}{x}\cdot\dfrac{\delta y}{y}$
B $\dfrac{\delta(xy)}{xy}\approx\dfrac{\delta x}{x}-\dfrac{\delta y}{y}$
C $\dfrac{\delta(xy)}{xy}\approx\dfrac{\delta x}{x}+\dfrac{\delta y}{y}$
D $\dfrac{\delta(xy)}{xy}\approx\dfrac{\delta x}{y}+\dfrac{\delta y}{x}$

Question 1004

[2 marks]numerical methods and error analysis

Given

x=8.7

and

y=0.3

each correct to 1 decimal place (so

\delta x=\delta y=0.05

), find the relative error

\dfrac{\delta y}{y}

correct to 3 significant figures.

Answer this when you sit the paper.

Question 1005

[1 marks]numerical methods and error analysis

Given

x=8.7

correct to 1 decimal place, find

\dfrac1x

correct to 4 decimal places.

Answer this when you sit the paper.

Question 1006

[1 marks]numerical methods and error analysis

Which formula gives the error bound in

y-\dfrac1x

, given errors

\delta x

x

and

\delta y

y

A $\delta y-\dfrac{\delta x}{x^2}$
B $\delta y+\dfrac{\delta x}{x}$
C $\delta y\cdot\dfrac{\delta x}{x^2}$
D $\delta y+\dfrac{\delta x}{x^2}$

Question 1101

[1 marks]Maclaurin series and differentiation

Given

f(x)=\ln(2-3x)

with

f''(x)=\dfrac{-9}{(2-3x)^2}

, find

f'''(x)

A $\dfrac{18}{(2-3x)^3}$
B $\dfrac{-54}{(2-3x)^3}$
C $\dfrac{54}{(2-3x)^3}$
D $\dfrac{-54}{(2-3x)^2}$

Question 1102

[1 marks]Maclaurin series and differentiation

Given

f(x)=\ln(2-3x)

, with

f(0)=\ln2

f'(0)=-\tfrac32

f''(0)=-\tfrac94

and

f'''(0)=-\tfrac{27}{4}

, write down the Maclaurin expansion of

\ln(2-3x)

up to and including the term in

x^3

A $2-\tfrac32x-\tfrac98x^2-\tfrac98x^3$
B $\ln2-\tfrac32x-\tfrac94x^2-\tfrac{27}{4}x^3$
C $\ln2+\tfrac32x-\tfrac98x^2+\tfrac98x^3$
D $\ln2-\tfrac32x-\tfrac98x^2-\tfrac98x^3$

Question 1103

[2 marks]Maclaurin series and differentiation

Given

f(x)=\ln(2-3x)

, differentiate to find

f'(x)

, then use it to find the exact value of

f'(0)

Answer this when you sit the paper.

Question 1104

[1 marks]Maclaurin series and differentiation

Given

f''(x)=\dfrac{-9}{(2-3x)^2}

, find the exact value of

f''(0)

Answer this when you sit the paper.

Question 1105

[1 marks]Maclaurin series and differentiation

Given

f'''(x)=\dfrac{-54}{(2-3x)^3}

, find the exact value of

f'''(0)

Answer this when you sit the paper.

Question 1106

[1 marks]Maclaurin series and differentiation

In a Maclaurin series expansion, what circumstance justifies neglecting terms in

x^4

and higher powers?

AWhen the function is a polynomial of degree 3
BWhen the series is evaluated at $x=0$ only
CWhen $x$ is a whole number greater than 1
DWhen $|x|$ is small, so higher powers become negligible compared to earlier terms

Question 1201

[1 marks]trigonometric functions and substitution

Given

f(x)=3\cos2x-2\sin x

, using

u=\sin x

this can be written as

f(x)=3-2u-6u^2

. Find the exact maximum value of

f(x)

A $-\dfrac{19}{6}$
B $\dfrac{17}{6}$
C $\dfrac{19}{6}$
D $3$

Question 1202

[1 marks]trigonometric functions and substitution

Given

f(x)=3\cos2x-2\sin x=3-2u-6u^2

where

u=\sin x

, find all values of

x

for

0^\circ\leq x<360^\circ

for which

f(x)=3

A $x=0^\circ,\ 180^\circ,\ 199.5^\circ,\ 340.5^\circ$
B $x=0^\circ,\ 180^\circ$ only
C $x=19.5^\circ,\ 160.5^\circ,\ 199.5^\circ,\ 340.5^\circ$
D $x=0^\circ,\ 180^\circ,\ 109.5^\circ,\ 250.5^\circ$

Question 1203

[1 marks]trigonometric functions and substitution

Which double angle identity for

\cos2x

, expressed in terms of

\sin x

, is used to rewrite

f(x)=3\cos2x-2\sin x

as a quadratic in

u=\sin x

A $\cos2x=2\cos^2x-1$
B $\cos2x=\cos^2x-\sin^2x$
C $\cos2x=1-2\sin^2x$
D $\cos2x=1-\sin^2x$

Question 1204

[2 marks]trigonometric functions and substitution

Writing

f(x)=3-2u-6u^2

in completed-square form gives a maximum at a certain value of

u

. Find that value of

u

Answer this when you sit the paper.

Question 1205

[2 marks]trigonometric functions and substitution

Solving

3-2u-6u^2=3

gives

u(2+6u)=0

. State both values of

u

that satisfy this equation.

Answer this when you sit the paper.

Question 1206

[1 marks]trigonometric functions and substitution

Given

\sin x=-\dfrac13

, find the reference angle (acute angle) correct to 2 decimal places, in degrees.

Answer this when you sit the paper.

Question 1301

[1 marks]numerical methods and Newton-Raphson

For

g(x)=e^{3x}-2x-10

, what is

g(-4)

to 3 significant figures, and what does it show together with

g(-5)=e^{-15}>0

A $g(-4)\approx-10.0$ ; the sign change shows a root lies between $x=-5$ and $x=-4$
B $g(-4)\approx-18.0$ ; the sign change shows a root lies between $x=-5$ and $x=-4$
C $g(-4)\approx2.00$ ; the sign change shows a root lies between $x=-5$ and $x=-4$
D $g(-4)\approx-2.00$ ; the sign change shows a root lies between $x=-5$ and $x=-4$

Question 1302

[1 marks]numerical methods and Newton-Raphson

For

g(x)=e^{3x}-2x-10

, besides the root between

x=-5

and

x=-4

, state the pair of consecutive integers between which another root lies.

Answer this when you sit the paper.

Question 1303

[1 marks]numerical methods and Newton-Raphson

For

g(x)=e^{3x}-2x-10

, taking

x_1=-4

and applying one step of the Newton-Raphson formula

x_2=x_1-\dfrac{g(x_1)}{g'(x_1)}

(with

g'(x)=3e^{3x}-2

), find

x_2

correct to 5 decimal places.

A $-5.00061$
B $-5.00001$
C $-4.99994$
D $-3.00001$

Question 1304

[2 marks]numerical methods and Newton-Raphson

For

g(x)=e^{3x}-2x-10

, find

g'(x)

Answer this when you sit the paper.

Question 1305

[2 marks]numerical methods and Newton-Raphson

For

g(x)=e^{3x}-2x-10

, find

g(1)

correct to 3 significant figures, confirming a root lies between

x=0

and

x=1

Answer this when you sit the paper.

Question 1306

[1 marks]numerical methods and Newton-Raphson

Which formula is used for one step of the Newton-Raphson method, starting from

x_1

A $x_2=x_1-\dfrac{g(x_1)}{g'(x_1)}$
B $x_2=x_1+\dfrac{g(x_1)}{g'(x_1)}$
C $x_2=x_1-g(x_1)g'(x_1)$
D $x_2=\dfrac{g(x_1)}{g'(x_1)}$

Question 1401

[1 marks]differential equations

Fans arrive at a stadium gate at a rate modelled by

\dfrac{dx}{dt}=-\dfrac{500}{t}

, where

x

is the number of fans present and

t

is the number of minutes remaining before the gate opens. Given

x=100

when

t=10

, find the general solution for

x

in terms of

t

A $x=500\ln\dfrac{10}{t}$
B $x=100+500\ln t$
C $x=100+500\ln\dfrac{10}{t}$
D $x=100-500\ln\dfrac{10}{t}$

Question 1402

[1 marks]differential equations

Using

x=100+500\ln\dfrac{10}{t}

(fans present

t

minutes before a gate opens, with

x=100

when

t=10

), find the number of fans present 1 minute before the gate opens, to the nearest whole number.

A $1151$
B $1251$
C $1459$
D $2403$

Question 1403

[1 marks]differential equations

Fans arrive at a stadium gate at a rate

\dfrac{dx}{dt}=-\dfrac{500}{t}

, where

x

is the number of fans present and

t

is the time remaining before the gate opens. Which further assumption is needed for this continuous model to be valid?

AThe gate opens exactly at $t=0$ minutes, with no delay
BFans arrive continuously and none leave the queue once they have joined it
CThe rate of arrival stays constant at 50 fans per minute throughout
DAll fans arrive in equally sized groups, once every minute

Question 1404

[2 marks]differential equations

The arrival rate satisfies

\dfrac{dx}{dt}=-\dfrac{k}{t}

. Given that at

t=10

the rate has size 50 per minute, find the value of

k

Answer this when you sit the paper.

Question 1405

[2 marks]differential equations

Solving

\dfrac{dx}{dt}=-\dfrac{500}{t}

gives

x=-500\ln t+C

. Given

x=100

when

t=10

, find the exact value of

C

Answer this when you sit the paper.

Question 1406

[2 marks]differential equations

Using

\dfrac{dx}{dt}=-\dfrac{500}{t}

, find the size of the arrival rate (in fans per minute) when

t=1

minute before the gate opens.

Answer this when you sit the paper.

Question 1501

[1 marks]vectors in 3D

Points A, B, C, D have coordinates

(5,7,3)

(7,8,0)

(8,10,3)

(6,9,6)

respectively. Find

\overrightarrow{AB}

and

\overrightarrow{CD}

, and state what this tells you about lines AB and CD.

A $\overrightarrow{AB}=(2,1,-3)$ , $\overrightarrow{CD}=(2,1,-3)$ ; since the vectors are identical, AB and CD are the same line
B $\overrightarrow{AB}=(-2,-1,3)$ , $\overrightarrow{CD}=(-2,-1,3)$ ; since the vectors are equal, ABCD is a parallelogram
C $\overrightarrow{AB}=(2,1,-3)$ , $\overrightarrow{CD}=(-2,-1,3)$ ; since $\overrightarrow{CD}=-\overrightarrow{AB}$ , AB and CD are parallel and equal in length
D $\overrightarrow{AB}=(2,1,-3)$ , $\overrightarrow{CD}=(-2,-1,3)$ ; since $\overrightarrow{CD}=-\overrightarrow{AB}$ , AB and CD are perpendicular

Question 1502

[1 marks]vectors in 3D

Points B and C have coordinates

(7,8,0)

and

(8,10,3)

respectively. Find the length of BC.

A $\sqrt{18}$
B $\sqrt{14}$
C $14$
D $\sqrt5$

Question 1503

[1 marks]vectors in 3D

Points A, B, C, D have coordinates

(5,7,3)

(7,8,0)

(8,10,3)

(6,9,6)

respectively. Evaluate the scalar product

\overrightarrow{AC}\cdot\overrightarrow{BD}

, and state what it shows about lines AC and BD.

A $\overrightarrow{AC}\cdot\overrightarrow{BD}=14$ ; AC and BD have equal length
B $\overrightarrow{AC}\cdot\overrightarrow{BD}=0$ ; AC and BD are perpendicular
C $\overrightarrow{AC}\cdot\overrightarrow{BD}=6$ ; AC and BD are neither parallel nor perpendicular
D $\overrightarrow{AC}\cdot\overrightarrow{BD}=-6$ ; AC and BD are parallel

Question 1504

[2 marks]vectors in 3D

Points A

(5,7,3)

and C

(8,10,3)

. Find the vector

\overrightarrow{AC}

Answer this when you sit the paper.

Question 1505

[2 marks]vectors in 3D

Points B

(7,8,0)

and D

(6,9,6)

. Find the vector

\overrightarrow{BD}

Answer this when you sit the paper.

Question 1506

[2 marks]vectors in 3D

Points A

(5,7,3)

and B

(7,8,0)

. Find the exact length of

AB

, in simplified surd form.

Answer this when you sit the paper.

Question 1601

[1 marks]logarithms, functions and inverses

Express

\ln(2x+1)-2\ln x-1

as a single logarithm.

A $\ln\dfrac{2x+1}{x^2}-1$
B $\ln\dfrac{2x+1}{ex^2}$
C $\ln\dfrac{2x+1}{ex}$
D $\ln\big(e(2x+1)x^2\big)$

Question 1602

[1 marks]logarithms, functions and inverses

For what values of

x

y=\ln(2x+1)

defined?

A $x>-1$
B $x>0$
C $x\geq-\tfrac12$
D $x>-\tfrac12$

Question 1603

[1 marks]logarithms, functions and inverses

The function

f

is defined by

f:x\mapsto(2x-5)^2-3

x\in\mathbb R

x\geq\tfrac52

. Find the range of

f

A $f(x)\geq-3$
B $f(x)\geq3$
C $f(x)\geq0$
D $f(x)\leq-3$

Question 1604

[1 marks]logarithms, functions and inverses

The function

f

is defined by

f:x\mapsto(2x-5)^2-3

x\in\mathbb R

x\geq\tfrac52

. Find

f^{-1}(x)

A $f^{-1}(x)=\dfrac{5+\sqrt{x-3}}{2}$
B $f^{-1}(x)=\dfrac{\sqrt{x+3}+5}{4}$
C $f^{-1}(x)=\dfrac{5+\sqrt{x+3}}{2}$
D $f^{-1}(x)=\dfrac{5-\sqrt{x+3}}{2}$

Question 1605

[1 marks]logarithms, functions and inverses

The point

B(2,2)

lies on the graph

y=f(x)

. What is the corresponding point on the graph

y=f\!\left(\dfrac{x}{2}\right)

A $(1,2)$
B $(4,2)$
C $(2,4)$
D $(4,4)$

Question 1606

[1 marks]logarithms, functions and inverses

The point

B(2,2)

lies on

y=f(x)

. Why does

B

also lie on the graph

y=f^{-1}(x)

A $B$ 's coordinates satisfy $x=y$ , so it lies on the line $y=x$ , and any point of $f$ on $y=x$ also lies on $f^{-1}$
B $B$ is the only point where $f$ is defined
C $f^{-1}$ is a straight line passing through every point of $f$
D $B$ lies exactly halfway between $A$ and $C$ on the curve

Question 1607

[2 marks]logarithms, functions and inverses

Under the transformation

y=f\!\left(\dfrac{x}{2}\right)

, the point A

(-4,-5)

y=f(x)

maps to which point?

Answer this when you sit the paper.

Question 1608

[2 marks]logarithms, functions and inverses

Under the transformation

y=f\!\left(\dfrac{x}{2}\right)

, the point C

(4,8)

y=f(x)

maps to which point?

Answer this when you sit the paper.

Question 1701

[1 marks]coordinate geometry and circles

The circle

x^2+y^2-2x+4y-8=0

has centre C. Find the coordinates of C and the radius of the circle.

A $C=(-1,2)$ , radius $\sqrt{13}$
B $C=(1,-2)$ , radius $13$
C $C=(1,-2)$ , radius $\sqrt{13}$
D $C=(1,-2)$ , radius $\sqrt5$

Question 1702

[1 marks]coordinate geometry and circles

The circle

x^2+y^2-2x+4y-8=0

cuts the

x

-axis at two points. Find their

x

-coordinates.

A $x=4$ only (double root)
B $x=-2$ and $x=4$
C $x=2$ and $x=-4$
D $x=-1$ and $x=8$

Question 1703

[1 marks]coordinate geometry and circles

The circle

x^2+y^2-2x+4y-8=0

has centre C

(1,-2)

and radius

\sqrt{13}

, and cuts the

x

-axis at A

(-2,0)

and B

(4,0)

. What is the exact value of

\cos(\angle ACB)

A $-\dfrac{5}{26}$
B $-\dfrac{5}{13}$
C $\dfrac{5}{13}$
D $-\dfrac{10}{13}$

Question 1704

[1 marks]coordinate geometry and circles

The circle

x^2+y^2-2x+4y-8=0

has centre C

(1,-2)

, radius

\sqrt{13}

, and cuts the

x

-axis at A and B with

\angle ACB\approx1.97

radians (3 s.f.). Since C lies below the

x

-axis, find the area of the part of the circle that lies above the

x

-axis, to 2 significant figures.

A $6.0$
B $6.8$
C $12.8$
D $34.0$

Question 1705

[2 marks]coordinate geometry and circles

The circle has centre C

(1,-2)

and cuts the

x

-axis at A

(-2,0)

and B

(4,0)

. Find the exact length of chord

AB

Answer this when you sit the paper.

Question 1706

[2 marks]coordinate geometry and circles

Using

r^2=13

and

\angle ACB\approx1.97

radians, find the area of sector

ACB

correct to 1 decimal place.

Answer this when you sit the paper.

Question 1707

[2 marks]coordinate geometry and circles

Triangle

ACB

has base

AB=6

and the perpendicular distance from C to line

AB

equal to 2 (since C is 2 units below the

x

-axis). Find the area of triangle

ACB

Answer this when you sit the paper.

Question 1801

[1 marks]differentiation and curve sketching

Given

y=x^3-x^2-5x+5

, find

\dfrac{dy}{dx}

A $3x^2-x-5$
B $x^3-2x-5$
C $3x^2-2x-5$
D $3x^2-2x+5$

Question 1802

[1 marks]differentiation and curve sketching

Find the equation of the tangent to the curve

y=x^3-x^2-5x+5

at the point where

x=2

A $y=-x+1$
B $y=3x-13$
C $y=3x-7$
D $y=3x-1$

Question 1803

[1 marks]differentiation and curve sketching

For

y=x^3-x^2-5x+5

, the turning points occur where

3x^2-2x-5=0

. Find the

x

-coordinates of the turning points.

A $x=\tfrac53$ and $x=-1$
B $x=1$ and $x=-\tfrac53$
C $x=\tfrac53$ and $x=1$
D $x=-\tfrac53$ and $x=-1$

Question 1804

[1 marks]differentiation and curve sketching

For

y=x^3-x^2-5x+5

, use the second derivative

\dfrac{d^2y}{dx^2}=6x-2

to classify the turning point at

x=-1

AMaximum, since $y''(-1)=-8<0$
BMaximum, since $y''(-1)=8>0$
CMinimum, since $y''(-1)=8>0$
DMinimum, since $y''(-1)=-8<0$

Pure Mathematics Paper 1 November 2002

Sit this paper online

The questions

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