ZIMSEC A Level Pure Mathematics Paper 1 November 2006

Question 501

[1 marks]sequences and functions

A sequence is defined by

U_{n+1} = \dfrac{1}{1-U_n}

, with

U_1 = a

, where

a \neq 1

. What is

U_3

in terms of

a

A $\dfrac{a}{a-1}$
B $\dfrac{1-a}{a}$
C $\dfrac{a-1}{a}$
D $\dfrac{1}{1-a}$

Question 502

[1 marks]sequences and functions

The function

f

is defined by

f: x \mapsto \ln x - 9

, where

x \in \mathbb{R}^+

. What is

f^{-1}(x)

A $e^{x-9}$
B $e^{x+9}$
C $e^x + 9$
D $\ln(x+9)$

Question 503

[2 marks]sequences and functions

A sequence is defined by

U_{n+1}=\dfrac1{1-U_n}

, with

U_1=a

a\neq1

, and

U_3=\dfrac{a-1}a

. Continuing the sequence, state

U_4

in terms of

a

Answer this when you sit the paper.

Question 504

[1 marks]sequences and functions

The function

f

is defined by

f:x\mapsto\ln x-9

x\in\mathbb R^+

, with inverse

f^{-1}(x)=e^{x+9}

. State the domain of

f^{-1}

Answer this when you sit the paper.

Question 505

[2 marks]sequences and functions

Given

f^{-1}(x)=e^{x+9}

(the inverse of

f:x\mapsto\ln x-9

), evaluate

f^{-1}(0)

in exact form.

Answer this when you sit the paper.

Question 601

[1 marks]trigonometry and iterative methods

The shape ABC is formed from a wire, where AB is an arc of a circle centre O radius 3 cm, and CA, CB are tangents to the circle each of length

t

cm. This gives the iterative formula

t_{n+1} = 5 - 3\tan^{-1}\!\left(\dfrac{t_n}{3}\right)

, used with

t_1 = 3

. What is

t_2

, to 3 decimal places?

A $0.328$
B $2.644$
C $4.215$
D $7.356$

Question 602

[1 marks]trigonometry and iterative methods

Continuing the iteration

t_{n+1} = 5 - 3\tan^{-1}\!\left(\dfrac{t_n}{3}\right)

from

t_1 = 3

, given that

t_2 \approx 2.644

, what is

t_3

, to 3 decimal places?

A $2.644$
B $2.833$
C $4.278$
D $7.166$

Question 603

[1 marks]trigonometry and iterative methods

Continuing the iteration

t_{n+1} = 5 - 3\tan^{-1}\!\left(\dfrac{t_n}{3}\right)

, given that

t_3 \approx 2.833

, what is

t_4

, to 3 decimal places?

A $2.644$
B $2.732$
C $4.244$
D $7.268$

Question 604

[1 marks]trigonometry and iterative methods

A wire of length 10 cm forms shape ABC, where AB is an arc of a circle centre O radius 3 cm subtending angle

2\theta

at the centre, and CA, CB are tangents to the circle, each of length

t

cm. Since arc length

=

radius

\times

angle, arc AB

=3(2\theta)=6\theta

. Using total wire length

10=

arc AB

+CA+CB

, what is

CA+CB

in terms of

\theta

A $10-3\theta$
B $10-6\theta$
C $5-6\theta$
D $5-3\theta$

Question 605

[1 marks]trigonometry and iterative methods

In the shape ABC (wire of length 10 cm, arc AB radius 3 cm subtending angle

2\theta

, tangents CA

=

=t

cm), the right-angled triangle formed by the radius (3 cm), the tangent length

t

, and the line from the centre to C has a right angle at the point of tangency. What is

\tan\theta

in terms of

t

A $6/t$
B $3/t$
C $t/6$
D $t/3$

Question 606

[2 marks]trigonometry and iterative methods

For the shape ABC (arc AB radius 3 cm, tangents CA

=

=t

cm), using

\tan\theta=t/3

with the initial estimate

t_1=3

, state

\theta

in radians, exact form, in terms of pi.

Answer this when you sit the paper.

Question 701

[1 marks]integration and volumes of revolution

Evaluate exactly

\displaystyle\int_1^4 \dfrac{2x+3}{x+1}\,dx

A $8 + \ln\!\left(\dfrac{5}{2}\right)$
B $6 + \ln 4$
C $6 - \ln\!\left(\dfrac{5}{2}\right)$
D $6 + \ln\!\left(\dfrac{5}{2}\right)$

Question 702

[1 marks]integration and volumes of revolution

The region R is bounded by the curve

y=x^2

, the

y

-axis, and the lines

y=1

and

y=\dfrac{9}{2}

. A cup is formed by rotating R through 4 right angles about the

y

-axis. What is the volume of the cup, in terms of

\pi

A $\dfrac{77\pi}{8}$
B $\dfrac{77\pi}{16}$
C $\dfrac{77\pi}{4}$
D $\dfrac{81\pi}{8}$

Question 703

[1 marks]integration and volumes of revolution

The region R is bounded by the curve

y=x^2

, the

y

-axis, and the lines

y=1

and

y=\dfrac{9}{2}

, and is rotated about the

y

-axis to form a cup. Water is poured into the cup until the upper surface of the water has radius

\sqrt3

. What is the volume of water in the cup, in terms of

\pi

A $\dfrac{9\pi}{2}$
B $\dfrac{77\pi}{8}$
C $4\pi$
D $\pi$

Question 704

[1 marks]integration and volumes of revolution

Rewriting

\dfrac{2x+3}{x+1}

by polynomial division as

A+\dfrac{B}{x+1}

(used before integrating

\displaystyle\int_1^4\dfrac{2x+3}{x+1}\,dx

), state

A

and

B

as 'A, B'.

Answer this when you sit the paper.

Question 705

[2 marks]integration and volumes of revolution

Using

\dfrac{2x+3}{x+1}=2+\dfrac1{x+1}

, evaluate

\displaystyle\int_1^4\dfrac1{x+1}\,dx

exactly (one part of the full integral).

Answer this when you sit the paper.

Question 706

[1 marks]integration and volumes of revolution

The region R, bounded by

y=x^2

, the

y

-axis, and the lines

y=1

and

y=\dfrac92

, is rotated about the

y

-axis to form a cup. Water is poured in until the upper surface has radius

\sqrt3

. Since

y=x^2

, state the value of

y

at this water level.

Answer this when you sit the paper.

Question 707

[1 marks]integration and volumes of revolution

Finding the volume of the cup (formed by rotating

y=x^2

about the

y

-axis between

y=1

and

y=\dfrac92

) needs

\left[\dfrac{y^2}2\right]_1^{4.5}

. State the value of

\dfrac{y^2}2

y=4.5

Answer this when you sit the paper.

Question 801

[1 marks]differentiation and tangents/normals

The point P

(-2, 6)

lies on the curve

y = 2 - x^2 - x^3

. What is the equation of the tangent to the curve at P, in the form

y = mx + c

A $y = -8x + 6$
B $y = \dfrac{x}{8} + \dfrac{25}{4}$
C $y = -8x - 10$
D $y = -16x - 26$

Question 802

[1 marks]differentiation and tangents/normals

The point P

(-2, 6)

lies on the curve

y = 2 - x^2 - x^3

. What is the equation of the normal to the curve at P, in the form

y = mx + c

A $y = 8x + 22$
B $y = -8x - 10$
C $y = \dfrac{x}{8} + \dfrac{25}{4}$
D $y = -\dfrac{x}{8} + \dfrac{23}{4}$

Question 803

[1 marks]differentiation and tangents/normals

The point P

(-2, 6)

lies on the curve

y = 2 - x^2 - x^3

. The tangent to the curve at P meets the

y

-axis at A, and the normal to the curve at P meets the

y

-axis at B. What is the length of AB?

A $16.25$ units
B $10.25$ units
C $3.75$ units
D $21.25$ units

Question 804

[2 marks]differentiation and tangents/normals

The point P

(-2,6)

lies on the curve

y=2-x^2-x^3

. Find

\dfrac{dy}{dx}

and evaluate it at

x=-2

(the gradient of the tangent at P).

Answer this when you sit the paper.

Question 805

[1 marks]differentiation and tangents/normals

The tangent to

y=2-x^2-x^3

at P

(-2,6)

has gradient

-8

. State the gradient of the normal at P.

Answer this when you sit the paper.

Question 806

[1 marks]differentiation and tangents/normals

The tangent to

y=2-x^2-x^3

at P

(-2,6)

has equation

y=-8x-10

. State the

y

-coordinate of point A, where this tangent meets the

y

-axis.

Answer this when you sit the paper.

Question 807

[1 marks]differentiation and tangents/normals

The normal to

y=2-x^2-x^3

at P

(-2,6)

has equation

y=\dfrac x8+\dfrac{25}4

. State the

y

-coordinate of point B, where this normal meets the

y

-axis.

Answer this when you sit the paper.

Question 901

[1 marks]complex numbers and logarithms

The complex number

z = x + iy

satisfies

\dfrac{z}{z+2} = 2 - i

. What are the values of

x

and

y

A $x=-1,\ y=-3$
B $x=3,\ y=1$
C $x=-3,\ y=1$
D $x=-3,\ y=-1$

Question 902

[1 marks]complex numbers and logarithms

Solve the simultaneous equations

2\log y = \log 2 + \log x

and

2^y = 4^x

for

x

and

y

A $x=\tfrac18,\ y=\tfrac12$
B $x=1,\ y=2$
C $x=\tfrac12,\ y=1$
D $x=0,\ y=0$

Question 903

[2 marks]complex numbers and logarithms

The complex number

z=x+iy

satisfies

\dfrac z{z+2}=2-i

. Rearranging as

z=(2-i)(z+2)

and collecting all

z

terms on one side gives

z(a+bi)=4-2i

for some complex coefficient. State this coefficient in the form 'a+bi'.

Answer this when you sit the paper.

Question 904

[1 marks]complex numbers and logarithms

Solving

2\log y=\log2+\log x

without logarithms gives an equation in the form

y^2=kx

. State

k

Answer this when you sit the paper.

Question 905

[1 marks]complex numbers and logarithms

Rewriting

2^y=4^x

2^y=2^{2x}

, the exponents give

y=cx

. State

c

Answer this when you sit the paper.

Question 906

[2 marks]complex numbers and logarithms

Substituting

y=2x

into

y^2=2x

gives

4x^2=2x

, which factorises as

2x(2x-c)=0

. State

c

Answer this when you sit the paper.

Question 907

[1 marks]complex numbers and logarithms

Solving

2x(2x-1)=0

for the simultaneous equations

2\log y=\log2+\log x

and

2^y=4^x

gives

x=0

x=\tfrac12

. Why is

x=0

rejected?

A $4^x$ is undefined when $x=0$
B $x=0$ gives an irrational value of $y$
C $x=0$ is not a solution of $2x(2x-1)=0$
D $\log x$ is undefined at $x=0$ , so it lies outside the equation's domain

Question 1001

[1 marks]differentiation and related rates

A right circular metallic cylinder has radius

r

cm and height

h

cm, with volume

200\text{ cm}^3

. What is

h

in terms of

r

A $h = \dfrac{200}{\pi r^2}$
B $h = \dfrac{200}{\pi r}$
C $h = \dfrac{\pi r^2}{200}$
D $h = 200\pi r^2$

Question 1002

[1 marks]differentiation and related rates

A right circular cylinder of radius

r

cm and height

h

cm always has volume

200\text{ cm}^3

. At the instant when

r=4

cm,

h

is increasing at

0.4

cm/minute. What is the rate of change of

r

at that instant, to 4 decimal places?

A $0.2011$ cm/min
B $-0.4021$ cm/min
C $-0.2011$ cm/min
D $-0.0040$ cm/min

Question 1003

[1 marks]differentiation and related rates

The total surface area of a cylinder of radius

r

cm is

S = 2\pi r^2 + \dfrac{400}{r}

. What is

\dfrac{dS}{dr}

A $4\pi r - \dfrac{400}{r}$
B $4\pi r - \dfrac{400}{r^2}$
C $2\pi r - \dfrac{400}{r^2}$
D $4\pi r + \dfrac{400}{r^2}$

Question 1004

[1 marks]differentiation and related rates

A cylinder's total surface area is

S = 2\pi r^2 + \dfrac{400}{r}

, where

r

is the radius in cm. At the instant when

r=4

cm and

\dfrac{dr}{dt} \approx -0.2011

cm/min, what is the rate at which

S

is changing, and is it increasing or decreasing?

A $\approx -5.08\text{ cm}^2/\text{min}$ , so $S$ is decreasing
B $\approx -10.16\text{ cm}^2/\text{min}$ , so $S$ is decreasing
C $\approx +5.08\text{ cm}^2/\text{min}$ , so $S$ is increasing
D $\approx -2.54\text{ cm}^2/\text{min}$ , so $S$ is decreasing

Question 1005

[2 marks]differentiation and related rates

A cylinder of volume

200\text{ cm}^3

has

h=\dfrac{200}{\pi r^2}

. Evaluate

h

when

r=4

cm, correct to 4 decimal places.

Answer this when you sit the paper.

Question 1006

[2 marks]differentiation and related rates

Differentiating

V=\pi r^2h=200

implicitly with respect to

t

and rearranging gives

\dfrac{dr}{dt}=-\dfrac r{2h}\dfrac{dh}{dt}

. Using

r=4

and

h\approx3.9789

, evaluate the coefficient

-\dfrac r{2h}

, correct to 4 decimal places.

Answer this when you sit the paper.

Question 1007

[2 marks]differentiation and related rates

The total surface area of a cylinder of radius

r

S=2\pi r^2+\dfrac{400}r

, with

\dfrac{dS}{dr}=4\pi r-\dfrac{400}{r^2}

. Evaluate

\dfrac{dS}{dr}

r=4

, correct to 3 decimal places.

Answer this when you sit the paper.

Question 1101

[1 marks]trigonometry and series expansions

Expanding

\cos(2x+x)

\cos2x\cos x - \sin2x\sin x

and using

\cos2x = 2\cos^2x-1

and

\sin2x = 2\sin x\cos x

, which of the following is the fully simplified result, in terms of

\cos x

only?

A $4\cos^3 x - 3\cos x$
B $3\cos x - 4\cos^3 x$
C $4\cos^3 x + 3\cos x$
D $2\cos^3 x - \cos x$

Question 1102

[1 marks]trigonometry and series expansions

Using

\cos^3x = \dfrac{\cos3x+3\cos x}{4}

together with the series

\cos x = 1-\dfrac{x^2}{2}+\dfrac{x^4}{24}-\dfrac{x^6}{720}+\ldots

, what is the coefficient of

x^4

in the resulting series for

\cos^3x

A $\dfrac{7}{8}$
B $\dfrac{5}{16}$
C $\dfrac{27}{32}$
D $\dfrac{7}{2}$

Question 1103

[1 marks]trigonometry and series expansions

Using the approximation

\cos^3x \approx 1-\dfrac{3}{2}x^2+\dfrac{7}{8}x^4-\dfrac{61}{240}x^6

, what is the relative error in this approximation for

\cos^3x

when

x=\dfrac13\pi

, to 1 significant figure?

A $0.3$
B $0.4$
C $0.5$
D $0.6$

Question 1104

[1 marks]trigonometry and series expansions

To prove

\cos3x=4\cos^3x-3\cos x

, the method begins by expanding

\cos(2x+x)

using the addition formula

\cos(A+B)=\cos A\cos B-\sin A\sin B

. What is this expansion?

A $\sin2x\cos x-\cos2x\sin x$
B $\cos2x\sin x-\sin2x\cos x$
C $\cos2x\cos x-\sin2x\sin x$
D $\cos2x\cos x+\sin2x\sin x$

Question 1105

[2 marks]trigonometry and series expansions

Expanding

\cos(3x)

using the series

\cos u=1-\dfrac{u^2}2+\dfrac{u^4}{24}-\ldots

with

u=3x

, state the coefficient of

x^2

in this series for

\cos3x

alone (before combining with

3\cos x

or dividing by 4).

Answer this when you sit the paper.

Question 1106

[2 marks]trigonometry and series expansions

Expanding

\cos(3x)

using the series

\cos u=1-\dfrac{u^2}2+\dfrac{u^4}{24}-\ldots

with

u=3x

, state the coefficient of

x^4

in this series for

\cos3x

alone (before combining with

3\cos x

or dividing by 4).

Answer this when you sit the paper.

Question 1107

[2 marks]trigonometry and series expansions

The relative error calculation for

\cos^3x

x=\dfrac13\pi

needs the true value of

\cos^3x

at that point. Using

\cos(\pi/3)=\dfrac12

, evaluate

\cos^3x

exactly at

x=\pi/3

Answer this when you sit the paper.

Question 1201

[1 marks]vectors

Points A, B and C have position vectors

6\mathbf{i}+2\mathbf{j}+6\mathbf{k}

2\mathbf{i}+3\mathbf{j}+\mathbf{k}

and

14\mathbf{i}+16\mathbf{k}

respectively. What is the exact value of

|\overrightarrow{AB}|

A $\sqrt{50}$
B $\sqrt{42}$
C $\sqrt{41}$
D $\sqrt{59}$

Question 1202

[1 marks]vectors

Points A, B and C have position vectors

6\mathbf{i}+2\mathbf{j}+6\mathbf{k}

2\mathbf{i}+3\mathbf{j}+\mathbf{k}

and

14\mathbf{i}+16\mathbf{k}

respectively, so that

\overrightarrow{AB}=-4\mathbf{i}+\mathbf{j}-5\mathbf{k}

and

\overrightarrow{AC}=8\mathbf{i}-2\mathbf{j}+10\mathbf{k}

. Which statement correctly describes the arrangement of A, B and C?

AA, B and C are collinear, with B lying between A and C
BA, B and C are collinear, with A lying between B and C
CA, B and C are collinear, with C lying between A and B
DA, B and C are not collinear, since $\overrightarrow{AC}$ is not a scalar multiple of $\overrightarrow{AB}$

Question 1203

[1 marks]vectors

Point A has position vector

6\mathbf{i}+2\mathbf{j}+6\mathbf{k}

, so that

\overrightarrow{AB}=-4\mathbf{i}+\mathbf{j}-5\mathbf{k}

. Point D has position vector

\mathbf{i}-d\mathbf{j}+9\mathbf{k}

, and

\overrightarrow{AD}

is perpendicular to

\overrightarrow{AB}

. What is the value of

d

A $d = 3$
B $d = -3$
C $d = 5$
D $d = 18$

Question 1204

[1 marks]vectors

With A at

6\mathbf{i}+2\mathbf{j}+6\mathbf{k}

\overrightarrow{AB}=-4\mathbf{i}+\mathbf{j}-5\mathbf{k}

, and D at

\mathbf{i}-3\mathbf{j}+9\mathbf{k}

(so that

\overrightarrow{AD}

is perpendicular to

\overrightarrow{AB}

), what is the exact area of triangle ABD?

A $50.5$
B $\dfrac{\sqrt{2478}}{2}$
C $\sqrt{2478}$
D $\dfrac{\sqrt{2478}}{4}$

Question 1205

[2 marks]vectors

Points A, B and C have position vectors

6\mathbf i+2\mathbf j+6\mathbf k

2\mathbf i+3\mathbf j+\mathbf k

and

14\mathbf i+16\mathbf k

respectively. State

\overrightarrow{AC}=C-A

as 'x,y,z'.

Answer this when you sit the paper.

Question 1206

[1 marks]vectors

With

\overrightarrow{AB}=-4\mathbf i+\mathbf j-5\mathbf k

and

\overrightarrow{AC}=8\mathbf i-2\mathbf j+10\mathbf k

, state the scalar

k

such that

\overrightarrow{AC}=k\cdot\overrightarrow{AB}

(confirming A, B, C are collinear).

Answer this when you sit the paper.

Question 1207

[2 marks]vectors

Point D has position vector

\mathbf i-3\mathbf j+9\mathbf k

(using

d=3

), and A has position vector

6\mathbf i+2\mathbf j+6\mathbf k

. State

\overrightarrow{AD}=D-A

as 'x,y,z'.

Answer this when you sit the paper.

Question 1208

[2 marks]vectors

Given

\overrightarrow{AD}=-5\mathbf i-5\mathbf j+3\mathbf k

, state

|\overrightarrow{AD}|

exactly.

Answer this when you sit the paper.

Question 1209

[1 marks]vectors

\overrightarrow{AD}

is perpendicular to

\overrightarrow{AB}=-4\mathbf i+\mathbf j-5\mathbf k

, where D has position vector

\mathbf i-d\mathbf j+9\mathbf k

and A has

6\mathbf i+2\mathbf j+6\mathbf k

. Before solving for

d

, state

\overrightarrow{AD}\cdot\overrightarrow{AB}

in terms of

d

Answer this when you sit the paper.

Question 1301

[1 marks]series and sequences

Given that

\displaystyle\sum_{r=1}^{n}(2r-3) = 255

, what is the value of

n

A $n = 17$
B $n = 15$
C $n = 16$
D $n = 19$

Question 1302

[1 marks]series and sequences

An athlete runs a 40 km marathon, then each day after runs 80% of the distance run the previous day, starting with 32 km on the first day after the race. What distance does the athlete run on the tenth day after the marathon, to 3 decimal places?

A $5.369$ km
B $2.749$ km
C $3.436$ km
D $4.295$ km

Question 1303

[1 marks]series and sequences

An athlete runs a 40 km marathon, then each day after runs 80% of the distance run the previous day, starting with 32 km on the first day after the race. What is the first day after the marathon on which the athlete's total distance run (since the race) exceeds 155 km?

Aday 13
Bday 15
Cday 17
Dday 16

Question 1304

[2 marks]series and sequences

Simplifying

\displaystyle\sum_{r=1}^n(2r-3)

gives a quadratic in

n

of the form

n^2+bn

(no constant term). State

b

Answer this when you sit the paper.

Question 1305

[2 marks]series and sequences

Solving

n^2-2n-255=0

(from

\displaystyle\sum_{r=1}^n(2r-3)=255

) gives two roots, one positive and one negative (the negative root is rejected since

n

must be a positive integer). State the negative root.

Answer this when you sit the paper.

Question 1306

[1 marks]series and sequences

An athlete runs a 40 km marathon, then each day after runs 80% of the distance run the previous day, starting with 32 km on the first day after the race. State the common ratio of this geometric sequence of daily distances.

Answer this when you sit the paper.

Question 1307

[2 marks]series and sequences

An athlete runs 32 km on the first day after a marathon, then 80% of the previous day's distance each day after. The total distance after

n

days is

S_n=160(1-0.8^n)

. Evaluate

S_{15}

(total distance after 15 days), correct to 3 decimal places.

Answer this when you sit the paper.

Question 1308

[2 marks]series and sequences

Using

S_n=160(1-0.8^n)

for the athlete's total distance after

n

days, evaluate

S_{16}

, correct to 3 decimal places.

Answer this when you sit the paper.

Question 1401

[1 marks]differential equations

Solve the differential equation

\dfrac{dy}{dx} = x^2y^2

given that

y=-\dfrac12

when

x=3

, expressing

y

in terms of

x

A $y = \dfrac{3}{33-x^3}$
B $y = \dfrac{3}{21-x^3}$
C $y = \dfrac{-3}{x^3}$
D $y = \dfrac{3}{x^3-21}$

Question 1402

[1 marks]differential equations

A liquid is heated in an oven kept at

200°C

. The rate of increase of the liquid's temperature

\theta°C

(

\theta<200

) at time

t

minutes is proportional to

(200-\theta)

. Which differential equation correctly models this, for some positive constant

k

A $\dfrac{d\theta}{dt} = 200-k\theta$
B $\dfrac{d\theta}{dt} = k(200-\theta)$
C $\dfrac{d\theta}{dt} = k(\theta-200)$
D $\dfrac{d\theta}{dt} = k\theta(200-\theta)$

Question 1403

[1 marks]differential equations

A liquid's temperature

\theta°C

satisfies

\theta = 200-Ae^{-kt}

, where

t

is time in minutes. The liquid starts at

0°C

and reaches

100°C

after 6 minutes. What is the temperature after 10 minutes, to the nearest degree?

A $168°C$
B $68°C$
C $100°C$
D $137°C$

Question 1404

[2 marks]differential equations

Separating variables in

\dfrac{dy}{dx}=x^2y^2

gives

\displaystyle\int\dfrac1{y^2}\,dy=\displaystyle\int x^2\,dx

. State the antiderivative of

\dfrac1{y^2}

with respect to

y

(omit the constant of integration).

Answer this when you sit the paper.

Question 1405

[2 marks]differential equations

Solving

\dfrac{dy}{dx}=x^2y^2

gives

-\dfrac1y=\dfrac{x^3}3+C

. Using

y=-\dfrac12

when

x=3

, evaluate

C

Answer this when you sit the paper.

Question 1406

[2 marks]differential equations

A liquid's temperature

\theta

satisfies

\dfrac{d\theta}{dt}=k(200-\theta)

. Separating variables, state the antiderivative of

\dfrac1{200-\theta}

with respect to

\theta

(omit the constant of integration).

Answer this when you sit the paper.

Question 1407

[1 marks]differential equations

A liquid's temperature satisfies

\theta=200-Ae^{-kt}

, starting at

\theta=0

when

t=0

. State the value of

A

Answer this when you sit the paper.

Question 1408

[2 marks]differential equations

A liquid's temperature satisfies

\theta=200-200e^{-kt}

and reaches

100°C

after 6 minutes. State

e^{-6k}

as a decimal.

Answer this when you sit the paper.

Question 1409

[2 marks]differential equations

Using

e^{-6k}=0.5

, evaluate

k

, correct to 4 decimal places.

Answer this when you sit the paper.

Question 1410

[2 marks]differential equations

A liquid's temperature satisfies

\theta=200-200e^{-kt}

with

k\approx0.1155

. Evaluate

e^{-10k}

, correct to 4 decimal places (needed to find

\theta

t=10

Answer this when you sit the paper.

Pure Mathematics Paper 1 November 2006

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The questions

Question 501

Question 502

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Question 504

Question 505

Question 601

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Question 603

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Question 605

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Question 701

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Question 704

Question 705

Question 706

Question 707

Question 801

Question 802

Question 803

Question 804

Question 805

Question 806

Question 807

Question 901

Question 902

Question 903

Question 904

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Question 906

Question 907

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

Question 1209

Question 1301

Question 1302

Question 1303

Question 1304

Question 1305

Question 1306

Question 1307

Question 1308

Question 1401

Question 1402

Question 1403

Question 1404

Question 1405

Question 1406

Question 1407

Question 1408

Question 1409

Question 1410

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