ZIMSEC A Level Pure Mathematics Paper 1 November 2012

Question 101

[1 marks]trigonometry / cosine rule

In triangle

ABC

AB = m

cm,

AC = 12

cm and angle

B\hat{A}C = 60°

. Given that the area of the triangle is

\sqrt{675}

^2

, find the exact value of

m

A $m = 5$
B $m = \sqrt{109}$
C $m = 15$
D $m = 5\sqrt{3}$

Question 102

[1 marks]trigonometry / cosine rule

In triangle

ABC

AB = 5

cm,

AC = 12

cm and angle

B\hat{A}C = 60°

. Find the exact length of

BC

A $\sqrt{109}$ cm
B $13$ cm
C $\sqrt{229}$ cm
D $7$ cm

Question 103

[2 marks]trigonometry / cosine rule

Using the cosine rule with

AB = 5

cm,

AC = 12

cm and angle

B\hat{A}C = 60°

, calculate

n^2 = BC^2

(before taking the square root).

Answer this when you sit the paper.

Question 201

[1 marks]numerical methods / Newton-Raphson

Let

f(x) = 6\sin(\pi + x^2) - 3

, with

x

in radians. Which values show that the equation

6\sin(\pi + x^2) = 3

has a root between

1.9

and

2

A $f(1.9) = 3$ and $f(2) = 6$
B $f(1.9) = 0.299$ and $f(2) = 1.54$
C $f(1.9) = -0.299$ and $f(2) = 1.54$
D $f(1.9) = -0.299$ and $f(2) = -1.54$

Question 202

[1 marks]numerical methods / Newton-Raphson

Starting with

x_1 = 1.9

, apply the Newton-Raphson method once to

f(x) = 6\sin(\pi + x^2) - 3

and give the second estimate to 3 significant figures.

A $1.88$
B $1.91$
C $1.95$
D $2.01$

Question 203

[2 marks]numerical methods / Newton-Raphson

Differentiate

f(x) = 6\sin(\pi + x^2) - 3

with respect to

x

to find

f'(x)

Answer this when you sit the paper.

Question 204

[2 marks]numerical methods / Newton-Raphson

Using

f'(x) = 12x\cos(\pi+x^2)

, evaluate

f'(1.9)

correct to 3 significant figures (the value used in the Newton-Raphson formula).

Answer this when you sit the paper.

Question 301

[1 marks]differentiation / Maclaurin series

Given

y = e^{\sin 3x}

, find

\dfrac{dy}{dx}

A $e^{\cos 3x}$
B $\cos 3x\, e^{\sin 3x}$
C $3e^{\sin 3x}$
D $3\cos 3x\, e^{\sin 3x}$

Question 302

[1 marks]differentiation / Maclaurin series

Find the Maclaurin series of

f(x) = e^{\sin 3x}

up to and including the term in

x^2

A $1 + 3x + 9x^2$
B $1 + 3x + \tfrac{9}{2}x^2$
C $1 + x + \tfrac{3}{2}x^2$
D $3x + \tfrac{9}{2}x^2$

Question 303

[2 marks]differentiation / Maclaurin series

For

y = e^{\sin 3x}

, evaluate

y'(0)

, the value of

\frac{dy}{dx}

x=0

(needed to build the Maclaurin series).

Answer this when you sit the paper.

Question 304

[2 marks]differentiation / Maclaurin series

For

y = e^{\sin 3x}

, evaluate

y''(0)

, the value of

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

x=0

Answer this when you sit the paper.

Question 401

[1 marks]parametric differentiation

A curve is given parametrically by

y = e^{t^2}

and

x = e^{t^2} - 1

for

1 \le t \le 2

. Find

\dfrac{dy}{dx}

A $2te^{t^2}$
B $e^{t^2}$
C $\dfrac{1}{2t}$
D $1$

Question 402

[1 marks]parametric differentiation

The curve

y = e^{t^2}

x = e^{t^2} - 1

for

1 \le t \le 2

is a straight line segment. What are the exact coordinates of its end points?

A $(e - 1, e)$ and $(e^2 - 1, e^2)$
B $(e, e - 1)$ and $(e^4, e^4 - 1)$
C $(0, 1)$ and $(e^4 - 1, e^4)$
D $(e - 1, e)$ and $(e^4 - 1, e^4)$

Question 403

[2 marks]parametric differentiation

For

x = e^{t^2} - 1

and

y = e^{t^2}

with

1 \le t \le 2

, find

\frac{dy}{dt}

in terms of

t

Answer this when you sit the paper.

Question 404

[2 marks]parametric differentiation

The graph of

y

against

x

for

1 \le t \le 2

, where

\frac{dy}{dx} = 1

throughout, is:

Aan exponential curve
Ba straight line of gradient $-1$
Ca parabola opening upward
Da straight line of gradient 1

Question 501

[1 marks]geometric progressions

The 6th term of a geometric progression is

\tfrac{1}{81}

and the 3rd term is

-\tfrac{1}{3}

. Find the common ratio and the first term.

A $r = -3$ , $a = -\tfrac{1}{27}$
B $r = -\tfrac{1}{3}$ , $a = 3$
C $r = -\tfrac{1}{3}$ , $a = -3$
D $r = \tfrac{1}{3}$ , $a = -3$

Question 502

[1 marks]geometric progressions

A geometric progression has first term

-3

and common ratio

-\tfrac{1}{3}

. Find its sum to infinity.

A $\tfrac{9}{4}$
B $-\tfrac{4}{9}$
C $-\tfrac{9}{4}$
D $-\tfrac{9}{2}$

Question 503

[1 marks]geometric progressions

Using (6th term)

\div

(3rd term)

= r^3

, evaluate

r^3

for this progression.

Answer this when you sit the paper.

Question 504

[2 marks]geometric progressions

Using

a = -3

and

r = -\frac{1}{3}

, find the exact value of the 4th term of the progression.

Answer this when you sit the paper.

Question 505

[2 marks]geometric progressions

Using

a = -3

and

r = -\frac{1}{3}

, find

S_4

, the sum of the first 4 terms, as an exact fraction.

Answer this when you sit the paper.

Question 601

[1 marks]exponential functions / tangents

The curve

y = p + e^{qx}

crosses the

y

-axis at

A\left(0, \tfrac{3}{2}\right)

and has gradient

-2

there. Find

p

and

q

A $p = -2$ , $q = \tfrac{1}{2}$
B $p = \tfrac{3}{2}$ , $q = -2$
C $p = \tfrac{1}{2}$ , $q = -2$
D $p = \tfrac{1}{2}$ , $q = 2$

Question 602

[1 marks]exponential functions / tangents

State the equation of the asymptote of the curve

y = \tfrac{1}{2} + e^{-2x}

A $x = \tfrac{1}{2}$
B $y = \tfrac{3}{2}$
C $y = \tfrac{1}{2}$
D $y = 0$

Question 603

[1 marks]exponential functions / tangents

The curve

y = p + e^{qx}

passes through

(0, \frac{3}{2})

. Using

y(0) = p + 1 = \frac{3}{2}

, find

p

Answer this when you sit the paper.

Question 604

[2 marks]exponential functions / tangents

Differentiate

y = p + e^{qx}

with respect to

x

to find

\frac{dy}{dx}

in terms of

q

and

x

Answer this when you sit the paper.

Question 701

[1 marks]complex numbers / Argand diagram

A complex number

z_1

has modulus 2 and lies

30°

below the negative real axis on an Argand diagram. Express

z_1

in the form

a + ib

with exact real

a

and

b

A $-\sqrt{3} - i$
B $-1 - \sqrt{3}i$
C $\sqrt{3} - i$
D $-\sqrt{3} + i$

Question 702

[1 marks]complex numbers / Argand diagram

Given

z_1 = -\sqrt{3} - i

, find the exact value of

w = \dfrac{-8\sqrt{3}\,i}{z_1}

in the form

a + ib

A $2\sqrt{3} + 6i$
B $-2\sqrt{3} - 6i$
C $2\sqrt{3} - 6i$
D $6 + 2\sqrt{3}i$

Question 703

[1 marks]complex numbers / Argand diagram

State the principal argument of

z_1

, in degrees, given

z_1

has modulus 2 and lies

30°

below the negative real axis.

Answer this when you sit the paper.

Question 704

[2 marks]complex numbers / Argand diagram

Find the modulus of

w = 2\sqrt{3} + 6i

Answer this when you sit the paper.

Question 705

[1 marks]complex numbers / Argand diagram

Find the argument of

w = 2\sqrt{3} + 6i

, in degrees.

Answer this when you sit the paper.

Question 801

[1 marks]polynomials / factor theorem

Given that

2x - 1

is a factor of

f(x) = 2x^3 + bx^2 + x - 2

, find the value of

b

A $b = 3$
B $b = -5$
C $b = 2$
D $b = 5$

Question 802

[1 marks]polynomials / factor theorem

Factorise

f(x) = 2x^3 + 5x^2 + x - 2

completely.

A $(2x - 1)(x - 1)(x - 2)$
B $(x - 1)(2x^2 + 7x + 2)$
C $(2x - 1)(x + 1)(x + 2)$
D $(2x + 1)(x - 1)(x - 2)$

Question 803

[1 marks]polynomials / factor theorem

Find the values of

x

for which

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

A $x < -2$ or $-1 < x < \tfrac{1}{2}$
B $x < -2$ only
C $-2 < x < \tfrac{1}{2}$
D $-2 < x < -1$ or $x > \tfrac{1}{2}$

Question 804

[2 marks]polynomials / factor theorem

Dividing

f(x) = 2x^3 + 5x^2 + x - 2

(2x-1)

gives quotient

x^2 + ax + b

. State the quotient.

Answer this when you sit the paper.

Question 805

[1 marks]polynomials / factor theorem

State the

x

-intercepts of

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

, in ascending order.

Answer this when you sit the paper.

Question 806

[1 marks]polynomials / factor theorem

Evaluate

f(0)

for

f(x) = 2x^3 + 5x^2 + x - 2

Answer this when you sit the paper.

Question 901

[1 marks]series / arc length

A string of length 220 cm is divided into

n

parts in the ratio

1 : 2 : 3 : \ldots : n

. The length of the longest segment is

A $\dfrac{220n}{n+1}$ cm
B $\dfrac{220}{n}$ cm
C $\dfrac{440}{n(n+1)}$ cm
D $\dfrac{440}{n+1}$ cm

Question 902

[1 marks]series / arc length

A 220 cm string is bent into a circle so that its ends just touch. Find the radius of the circle, correct to 3 significant figures.

A $17.5$ cm
B $35.0$ cm
C $70.0$ cm
D $110$ cm

Question 903

[1 marks]series / arc length

The largest segment of a string bent into a circle subtends an angle of approximately

\dfrac{12.57}{n+1}

radians at the centre. Find the smallest value of

n

for which this angle is less than

\dfrac{\pi}{12}

radians.

A $n = 48$
B $n = 47$
C $n = 49$
D $n = 12$

Question 904

[2 marks]series / arc length

Using

2\pi r = 220

, state the exact radius

r

of the circle as a fraction of

\pi

Answer this when you sit the paper.

Question 905

[2 marks]series / arc length

Using the longest-segment formula

\frac{440}{n+1}

, find its length in cm (2 d.p.) when

n = 48

Answer this when you sit the paper.

Question 906

[2 marks]series / arc length

For

n = 48

, use

S = r\theta

with the longest segment length

\approx 8.98

cm and

r \approx 35.0

cm to find

\theta

in radians, correct to 3 s.f.

Answer this when you sit the paper.

Question 1001

[1 marks]integration (area and volume)

The region

R

is enclosed by the curve

y = 1 + \sqrt{3x - 2}

, the

x

-axis and the lines

x = \tfrac{2}{3}

and

x = 2

. Calculate the area of

R

A $\tfrac{16}{9}$ units $^2$
B $\tfrac{4}{3}$ units $^2$
C $\tfrac{68}{9}$ units $^2$
D $\tfrac{28}{9}$ units $^2$

Question 1002

[1 marks]integration (area and volume)

The region enclosed by

y = 1 + \sqrt{3x - 2}

, the

x

-axis and the lines

x = \tfrac{2}{3}

and

x = 2

is rotated completely about the

x

-axis. Find the volume generated.

A $\tfrac{68\pi}{9}$ units $^3$
B $\tfrac{4\pi}{3}$ units $^3$
C $\tfrac{28\pi}{9}$ units $^3$
D $\tfrac{32\pi}{9}$ units $^3$

Question 1003

[2 marks]integration (area and volume)

Evaluate

x + \frac{2}{9}(3x-2)^{3/2}

x = 2

(the antiderivative used for the area, before subtracting the lower limit).

Answer this when you sit the paper.

Question 1004

[1 marks]integration (area and volume)

Evaluate

x + \frac{2}{9}(3x-2)^{3/2}

x = \frac{2}{3}

Answer this when you sit the paper.

Question 1005

[1 marks]integration (area and volume)

Evaluate

[1 + \sqrt{3x-2}]^2

x = 2

(the integrand used for the volume).

Answer this when you sit the paper.

Question 1006

[2 marks]integration (area and volume)

Expanding

[1+\sqrt{3x-2}]^2 = 1 + 2\sqrt{3x-2} + (3x-2)

, evaluate the middle term

2\sqrt{3x-2}

x = 2

Answer this when you sit the paper.

Question 1101

[1 marks]vectors in 3D

The points

A

and

C

have position vectors

8\mathbf{j} - 4\mathbf{k}

and

-2\mathbf{i} + 3\mathbf{j} + 5\mathbf{k}

. Find a unit vector parallel to

\overrightarrow{AC}

A $\tfrac{1}{\sqrt{110}}(-2\mathbf{i} - 5\mathbf{j} + 9\mathbf{k})$
B $\tfrac{1}{110}(-2\mathbf{i} - 5\mathbf{j} + 9\mathbf{k})$
C $\tfrac{1}{\sqrt{110}}(2\mathbf{i} + 5\mathbf{j} - 9\mathbf{k})$
D $\tfrac{1}{\sqrt{78}}(-2\mathbf{i} - 5\mathbf{j} + 9\mathbf{k})$

Question 1102

[1 marks]vectors in 3D

Points

A

B

C

have position vectors

8\mathbf{j} - 4\mathbf{k}

p\mathbf{i} + 3\mathbf{j} + 4\mathbf{k}

and

-2\mathbf{i} + 3\mathbf{j} + 5\mathbf{k}

. Find the positive value of

p

for which

BA

is perpendicular to

BC

A $p = -4$
B $p = 2$
C $p = 8$
D $p = 4$

Question 1103

[1 marks]vectors in 3D

In triangle

ABC

BA

is perpendicular to

BC

with

|\overrightarrow{BA}| = \sqrt{93}

and

|\overrightarrow{BC}| = \sqrt{17}

. Find the area of the triangle correct to 2 decimal places.

A $790.50$ units $^2$
B $9.94$ units $^2$
C $19.88$ units $^2$
D $39.76$ units $^2$

Question 1104

[2 marks]vectors in 3D

Solving

p^2 + 2p - 8 = 0

(from

\overrightarrow{BA}\cdot\overrightarrow{BC} = 0

) gives two roots. Find the negative root (not used in this question).

Answer this when you sit the paper.

Question 1105

[3 marks]vectors in 3D

With

p = 2

, find

|\overrightarrow{BA}|

, correct to 3 significant figures, where

\overrightarrow{BA} = -2\mathbf{i} + 5\mathbf{j} - 8\mathbf{k}

Answer this when you sit the paper.

Question 1106

[2 marks]vectors in 3D

With

p = 2

, find

|\overrightarrow{BC}|

, correct to 3 significant figures, where

\overrightarrow{BC} = -4\mathbf{i} + \mathbf{k}

Answer this when you sit the paper.

Question 1201

[1 marks]coordinate geometry / circles

A square of perimeter

12\sqrt{2}

cm has centre

E(4, 3)

. Find the equation of the circle which passes through all four corners of the square, in the form

x^2 + y^2 + ax + by + c = 0

A $x^2 + y^2 - 8x - 6y + 25 = 0$
B $x^2 + y^2 - 8x - 6y + 16 = 0$
C $x^2 + y^2 - 8x - 6y - 9 = 0$
D $x^2 + y^2 + 8x + 6y + 16 = 0$

Question 1202

[1 marks]coordinate geometry / circles

A chord

AD

of a circle of radius 3 cm subtends a right angle at the centre. Find the area of the minor segment cut off, correct to 2 decimal places.

A $7.07$ cm $^2$
B $1.29$ cm $^2$
C $2.57$ cm $^2$
D $4.50$ cm $^2$

Question 1203

[2 marks]coordinate geometry / circles

A square has perimeter

12\sqrt{2}

cm. Find the length of one side.

Answer this when you sit the paper.

Question 1204

[1 marks]coordinate geometry / circles

Find the diagonal length

d

of the square (used to find the circle's radius), given side

3\sqrt2

cm.

Answer this when you sit the paper.

Question 1205

[1 marks]coordinate geometry / circles

State the gradient of line

BC

Answer this when you sit the paper.

Question 1206

[1 marks]coordinate geometry / circles

State the gradient of line

BA

Answer this when you sit the paper.

Question 1207

[1 marks]coordinate geometry / circles

State the equation of the tangent to the circle at point

A = (4, 6)

Answer this when you sit the paper.

Question 1301

[1 marks]differential equations

Ants bring food to a hive at a constant

500

g per day and consume it at

\tfrac{1}{2}x

g per day, where

x

g is the amount present after

t

days. Write down the differential equation satisfied by

x

and

t

A $\dfrac{dx}{dt} = 500 + \tfrac{1}{2}x$
B $\dfrac{dx}{dt} = \tfrac{1}{2}x - 500$
C $\dfrac{dx}{dt} = 500x - \tfrac{1}{2}$
D $\dfrac{dx}{dt} = 500 - \tfrac{1}{2}x$

Question 1302

[1 marks]differential equations

Solve

\dfrac{dx}{dt} = 500 - \tfrac{1}{2}x

given that

x = 400

when

t = 0

, expressing

x

in terms of

t

A $x = 400e^{-t/2}$
B $x = 1000 - 600e^{-t/2}$
C $x = 1000 - 400e^{-t/2}$
D $x = 1000 + 600e^{-t/2}$

Question 1303

[1 marks]differential equations

For a hive in which

x = 1000 - 600e^{-t/2}

grammes of food is present after

t

days, the maximum amount of food the hive can hold is

A $600$ g, because that is the initial deficit
B $1000$ g, because $\tfrac{dx}{dt} \to 0$ as $x \to 1000$
C $500$ g, because that is the daily delivery rate
Dunbounded, because food arrives at a constant rate

Question 1304

[3 marks]differential equations

Find, correct to 3 significant figures, the time

t

at which

x = 980

grammes, given

x = 1000 - 600e^{-t/2}

Answer this when you sit the paper.

Question 1305

[2 marks]differential equations

Using

x = 400

when

t = 0

, find the value of

C

x = 1000 - Ce^{-t/2}

Answer this when you sit the paper.

Question 1306

[2 marks]differential equations

Find

\frac{dx}{dt}

, correct to the nearest whole number, when

x = 600

grammes.

Answer this when you sit the paper.

Question 1401

[1 marks]trigonometry / R-formula and transformations

Express

3\sqrt{2}(\cos x + \sin x)

in the form

R\cos(x - \alpha)

, where

R > 0

and

0° < \alpha < 90°

A $18\cos(x - 45°)$
B $6\cos(x - 45°)$
C $3\sqrt{2}\cos(x - 45°)$
D $6\cos(x - 30°)$

Question 1402

[1 marks]trigonometry / R-formula and transformations

Solve

\sqrt{2}(\cos x + \sin x) = \sqrt{3}

for

0° \le x \le 360°

A $x = 75°$ and $x = 345°$
B $x = 15°$ and $x = 75°$
C $x = 30°$ and $x = 60°$
D $x = 15°$ and $x = 285°$

Question 1403

[1 marks]trigonometry / R-formula and transformations

State the maximum and minimum values of

y = 6 + 3\sqrt{2}(\cos x + \sin x)

Amaximum $6 + 3\sqrt{2}$ , minimum $6 - 3\sqrt{2}$
Bmaximum $12$ , minimum $0$
Cmaximum $6$ , minimum $-6$
Dmaximum $18$ , minimum $-6$

Question 1404

[2 marks]trigonometry / R-formula and transformations

The graph of

y = 6 + 3\sqrt2(\cos x + \sin x)

is obtained from

y = \cos x

by which sequence of transformations?

Atranslate $45°$ in the positive $x$ -direction, stretch parallel to the $y$ -axis by factor $6$ , then translate $6$ units in the positive $y$ -direction
Bstretch parallel to the $y$ -axis by factor $6$ , translate $45°$ in the negative $x$ -direction, then translate $6$ units in the negative $y$ -direction
Ctranslate $45°$ in the positive $x$ -direction, stretch parallel to the $x$ -axis by factor $6$ , then translate $6$ units in the positive $y$ -direction
Dtranslate $6$ units in the positive $x$ -direction, stretch parallel to the $y$ -axis by factor $45$ , then translate $45$ units in the positive $y$ -direction

Question 1405

[2 marks]trigonometry / R-formula and transformations

Find

y

, correct to 1 decimal place, when

x = 0°

for

y = 6 + 3\sqrt2(\cos x + \sin x)

Answer this when you sit the paper.

Question 1406

[2 marks]trigonometry / R-formula and transformations

Find the value of

x

(

0° \le x \le 360°

) at which

y = 6 + 3\sqrt2(\cos x + \sin x) = 0

Answer this when you sit the paper.

Question 1407

[2 marks]trigonometry / R-formula and transformations

Find the value of

x

(

0° \le x \le 360°

) at which

y = 6 + 3\sqrt2(\cos x + \sin x)

attains its maximum value.

Answer this when you sit the paper.

Pure Mathematics Paper 1 November 2012

Sit this paper online

The questions

Question 101

Question 102

Question 103

Question 201

Question 202

Question 203

Question 204

Question 301

Question 302

Question 303

Question 304

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 701

Question 702

Question 703

Question 704

Question 705

Question 801

Question 802

Question 803

Question 804

Question 805

Question 806

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 1101

Question 1102

Question 1103

Question 1104

Question 1105

Question 1106

Question 1201

Question 1202

Question 1203

Question 1204

Question 1205

Question 1206

Question 1207

Question 1301

Question 1302

Question 1303

Question 1304

Question 1305

Question 1306

Question 1401

Question 1402

Question 1403

Question 1404

Question 1405

Question 1406

Question 1407

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