ZIMSEC A Level Pure Mathematics Paper 1 June 2012

Question 101

[1 marks]modulus / inequalities

Given that

-4 < x < 2

is the solution of the inequality

|x + a| < b

, find the values of

a

and

b

A $a = 1$ , $b = 3$
B $a = 1$ , $b = 6$
C $a = -1$ , $b = 3$
D $a = 3$ , $b = 1$

Question 102

[1 marks]modulus / inequalities

The inequality

|x + a| < b

(with

b > 0

) is equivalent to which of the following?

A $-b - a < x < b - a$
B $-b + a < x < b + a$
C $x < -b - a$ or $x > b - a$
D $x < b - a$ only

Question 103

[1 marks]modulus / inequalities

Given

-4 < x < 2

is the solution of

|x + a| < b

, matching gives

-b - a = -4

and

b - a = 2

. Find the value of

a

(by adding the two equations).

Answer this when you sit the paper.

Question 201

[1 marks]numerical integration / trapezium rule

Use the trapezium rule with 4 equal intervals to estimate

\displaystyle\int_0^{0.2} \dfrac{\cos x^2}{e^{x^2}}\,dx

, correct to 4 decimal places.

A $0.1231$
B $0.1960$
C $0.1972$
D $0.3945$

Question 202

[3 marks]numerical integration / trapezium rule

Using the trapezium rule with 4 equal intervals over

[0, 0.2]

for

\int_0^{0.2}\frac{\cos x^2}{e^{x^2}}\,dx

, the ordinates are

y_0=1

y_1=0.9975

y_2=0.9904

y_3=0.97775

y_4=0.9229

. Evaluate

y_0 + y_4 + 2(y_1+y_2+y_3)

Answer this when you sit the paper.

Question 301

[1 marks]small-angle approximations / trigonometry

In triangle

ABD

with height

AD = h

and angle

ABD = \tfrac{1}{6}\pi

, find the length

BD

A $h\sqrt{3}$
B $\dfrac{h}{\sqrt{3}}$
C $2h$
D $\dfrac{h}{2}$

Question 302

[1 marks]small-angle approximations / trigonometry

Given that

x

is small enough for powers above

x^2

to be neglected, the approximation for

\tan\left(\tfrac{1}{4}\pi + x\right)

obtained from

\dfrac{1 + \tan x}{1 - \tan x}

leads to

DC \approx

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

Question 303

[1 marks]small-angle approximations / trigonometry

State the exact value of

\tan\frac{1}{6}\pi

(used to find

BD

in the triangle).

Answer this when you sit the paper.

Question 304

[2 marks]small-angle approximations / trigonometry

Using the binomial expansion, write

(1+x)^{-1}

up to and including the term in

x^2

Answer this when you sit the paper.

Question 401

[1 marks]small changes / implicit differentiation

Given that

x^3 y^2 = 1

, find the approximate percentage change in

y

when

x

increases by

0.5\%

Aa decrease of $0.75\%$
Ban increase of $0.75\%$
Ca decrease of $1.5\%$
Da decrease of $0.33\%$

Question 402

[3 marks]small changes / implicit differentiation

Differentiate

x^3y^2 = 1

implicitly with respect to

x

to find

\frac{dy}{dx}

in terms of

x

and

y

Answer this when you sit the paper.

Question 403

[1 marks]small changes / implicit differentiation

x

increases by

0.5\%

, express

\delta x

in terms of

x

Answer this when you sit the paper.

Question 501

[1 marks]partial fractions

Express

\dfrac{x^4}{x^4 - 1}

in partial fractions.

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

Question 502

[2 marks]partial fractions

Factorise

x^4 - 1

completely over the real numbers.

Answer this when you sit the paper.

Question 503

[2 marks]partial fractions

Express

\dfrac{1}{(x^2-1)(x^2+1)}

in partial fractions with denominators

(x^2-1)

and

(x^2+1)

Answer this when you sit the paper.

Question 601

[1 marks]binomial expansion

The expansions of

(1+x)^{1/5}

and

\dfrac{1 + px}{1 + qx}

in ascending powers of

x

are identical up to and including the term in

x^2

. Find

p

and

q

A $p = \tfrac{1}{5}$ , $q = -\tfrac{2}{25}$
B $p = \tfrac{3}{5}$ , $q = \tfrac{1}{5}$
C $p = \tfrac{2}{5}$ , $q = \tfrac{3}{5}$
D $p = \tfrac{3}{5}$ , $q = \tfrac{2}{5}$

Question 602

[3 marks]binomial expansion

Use the binomial series to expand

(1+x)^{1/5}

up to and including the term in

x^2

Answer this when you sit the paper.

Question 603

[3 marks]binomial expansion

Expand

(1+px)(1+qx)^{-1}

up to and including the term in

x^2

, in terms of

p

and

q

Answer this when you sit the paper.

Question 701

[1 marks]arithmetic and geometric progressions

An arithmetic progression has first term

a

and common difference

d > 0

. Given that its first, second and fifth terms are consecutive terms of a geometric progression, express

a

in terms of

d

A $a = d$
B $a = \tfrac{d}{4}$
C $a = \tfrac{d}{2}$
D $a = 2d$

Question 702

[1 marks]arithmetic and geometric progressions

An arithmetic progression of 10 terms has

a = \tfrac{d}{2}

and the difference between its first and last terms is 36. Find the sum of all its terms.

A $180$
B $200$
C $220$
D $400$

Question 703

[3 marks]arithmetic and geometric progressions

The terms

a

a+d

a+4d

are consecutive terms of a geometric progression, so

(a+d)^2 = a(a+4d)

. Expand and simplify this to a relation of the form

d^2 = \ldots

in terms of

a

and

d

Answer this when you sit the paper.

Question 704

[2 marks]arithmetic and geometric progressions

The arithmetic progression has 10 terms with

d>0

, and the difference between its first and last (10th) terms is 36. Find the value of

d

Answer this when you sit the paper.

Question 801

[1 marks]complex numbers

Express

w = \dfrac{4 + 3i}{3 - 2i}

in the form

x + iy

A $\tfrac{6}{13} + \tfrac{17}{13}i$
B $\tfrac{6}{13} - \tfrac{17}{13}i$
C $\tfrac{18}{13} + \tfrac{1}{13}i$
D $\tfrac{6}{5} + \tfrac{17}{5}i$

Question 802

[1 marks]complex numbers

Find the argument of

w = \dfrac{4 + 3i}{3 - 2i}

, correct to the nearest

0.1°

A $19.4°$
B $70.6°$
C $109.4°$
D $-70.6°$

Question 803

[3 marks]complex numbers

Find the modulus of

w = \dfrac{4+3i}{3-2i}

Answer this when you sit the paper.

Question 804

[2 marks]complex numbers

Multiply out

(4+3i)(3+2i)

to find the numerator of

w = \dfrac{4+3i}{3-2i}

after rationalising, in the form

p + qi

Answer this when you sit the paper.

Question 901

[1 marks]coordinate geometry / circles

Find the equation of the circle which passes through the origin and touches the line

4x - 3y = 5

at the point

(2, 1)

A $(x - 2)^2 + (y - 1)^2 = \tfrac{25}{4}$
B $\left(x - \tfrac{5}{2}\right)^2 + y^2 = \tfrac{25}{4}$
C $x^2 + \left(y - \tfrac{5}{2}\right)^2 = 25$
D $x^2 + \left(y - \tfrac{5}{2}\right)^2 = \tfrac{25}{4}$

Question 902

[3 marks]coordinate geometry / circles

Find the equation of the normal to the line

4x - 3y = 5

at the point

(2, 1)

, in the form

ay = bx + c

Answer this when you sit the paper.

Question 903

[3 marks]coordinate geometry / circles

Find the equation of the perpendicular bisector of the chord from

(0,0)

(2,1)

, in the form

y = mx + c

Answer this when you sit the paper.

Question 1001

[1 marks]exponential and logarithmic equations

Solve the equation

3(7^x) - 2(7^{-x}) = 5

, giving your answer correct to 2 decimal places.

A $x = 0.36$
B $x = 2.00$
C $x = 0.56$
D $x = -0.56$

Question 1002

[1 marks]exponential and logarithmic equations

Solve

\log_{10}(2x^3 + 1) - 3\log_{10} x = 1

A $x = \tfrac{1}{8}$
B $x = 10$
C $x = 2$
D $x = \tfrac{1}{2}$

Question 1003

[3 marks]exponential and logarithmic equations

Let

y = 7^x

. Rewrite

3(7^x) - 2(7^{-x}) = 5

as a quadratic equation in

y

Answer this when you sit the paper.

Question 1004

[3 marks]exponential and logarithmic equations

Rewrite

\log_{10}(2x^3+1) - 3\log_{10}x = 1

without logarithms and hence find the value of

x^3

Answer this when you sit the paper.

Question 1101

[1 marks]vectors in 3D

In a cube of side 8 units with

O

at the origin,

M(0, 4, 4)

is the centre of a face and

P(4, 8, 8)

is the midpoint of an edge. Find a unit vector parallel to

\overrightarrow{MP}

A $\tfrac{1}{4}(\mathbf{i} + \mathbf{j} + \mathbf{k})$
B $\tfrac{1}{\sqrt{3}}(\mathbf{i} + \mathbf{j} + \mathbf{k})$
C $\tfrac{1}{\sqrt{3}}(\mathbf{i} - \mathbf{j} + \mathbf{k})$
D $\tfrac{1}{12}(4\mathbf{i} + 4\mathbf{j} + 4\mathbf{k})$

Question 1102

[1 marks]vectors in 3D

In a cube of side 8 units,

\overrightarrow{MP} = (4, 4, 4)

and

\overrightarrow{MD} = (8, -4, 4)

. Find angle

DMP

correct to the nearest

0.1°

A $28.1°$
B $45.0°$
C $61.9°$
D $70.5°$

Question 1103

[3 marks]vectors in 3D

In the cube with

O

at the origin,

A=(8,0,0)

B=(0,8,0)

C=(0,0,8)

F=B+C=(0,8,8)

, find the coordinates of

M

, the point of intersection of

\overrightarrow{OF}

and

\overline{BC}

Answer this when you sit the paper.

Question 1104

[3 marks]vectors in 3D

Given

\overrightarrow{MP} = (4,4,4)

, find

|\overrightarrow{MP}|

, leaving your answer in surd form.

Answer this when you sit the paper.

Question 1201

[1 marks]trigonometric identities and equations

Simplify

\csc 2x - \cot 2x

A $2\tan x$
B $\tan x$
C $\cot x$
D $\sec x$

Question 1202

[1 marks]trigonometric identities and equations

Using

\csc 2x - \cot 2x \equiv \tan x

, solve

\csc 2x - \cot 2x + 5 = 3\sec^2 x

for

0 < x \le 360°

A $x = 45°,\ 225°$ only
B $x = 45°,\ 146.3°,\ 225°,\ 326.3°$
C $x = 33.7°,\ 135°,\ 213.7°,\ 315°$
D $x = 45°,\ 135°,\ 225°,\ 315°$

Question 1203

[3 marks]trigonometric identities and equations

Substitute

\sec^2x = 1+\tan^2x

into

\csc 2x - \cot 2x + 5 = 3\sec^2x

(using

\csc2x-\cot2x=\tan x

) and rearrange fully into the form

3\tan^2x - \tan x - 2 = 0

. State this quadratic equation.

Answer this when you sit the paper.

Question 1204

[3 marks]trigonometric identities and equations

Solve

3\tan^2x - \tan x - 2 = 0

using the quadratic formula. State both values of

\tan x

Answer this when you sit the paper.

Question 1301

[1 marks]differentiation / optimisation

An open rectangular box has length

\dfrac{32}{x^2}

cm, width

x

cm and volume

128

^3

. Find its depth

d

and total surface area

A

in terms of

x

A $d = 4x$ , $A = 4x^2 + \dfrac{256}{x}$
B $d = 4x$ , $A = 8x^2 + \dfrac{288}{x}$
C $d = \dfrac{4}{x}$ , $A = 8x^2 + \dfrac{288}{x}$
D $d = 4x$ , $A = 8x^2 + \dfrac{32}{x}$

Question 1302

[1 marks]differentiation / optimisation

The surface area of an open box is

A = 8x^2 + \dfrac{288}{x}

^2

. Find the value of

x

that minimises

A

, correct to 3 decimal places.

A $x = 3.000$
B $x = 4.659$
C $x = 10.483$
D $x = 2.621$

Question 1303

[3 marks]differentiation / optimisation

The open box has length

\dfrac{32}{x^2}

cm, width

x

cm and depth

d = 4x

cm. Find the combined area of the two pairs of side faces (excluding the base), in terms of

x

Answer this when you sit the paper.

Question 1304

[3 marks]differentiation / optimisation

Differentiate

A = 8x^2 + \dfrac{288}{x}

with respect to

x

Answer this when you sit the paper.

Question 1305

[2 marks]differentiation / optimisation

Find

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

for

A = 8x^2 + \dfrac{288}{x}

, used to confirm the stationary point is a minimum.

Answer this when you sit the paper.

Question 1401

[1 marks]curve sketching / Newton-Raphson

Find the coordinates and nature of the turning points on the graph of

y = x^3 - 12x - 12

A $(-2, -28)$ minimum and $(2, 4)$ maximum
B $(-2, 4)$ minimum and $(2, -28)$ maximum
C $(0, -12)$ maximum only
D $(-2, 4)$ maximum and $(2, -28)$ minimum

Question 1402

[1 marks]curve sketching / Newton-Raphson

For which set of values of

k

does the equation

x^3 - 12x - 12 - k = 0

have more than one real root?

A $k > 4$
B $k < -28$
C $-28 \le k \le 4$
D $-12 \le k \le 4$

Question 1403

[1 marks]curve sketching / Newton-Raphson

Taking

3.9

as the first approximation, apply the Newton-Raphson method to

x^3 - 12x - 12 = 0

and give the root correct to 3 decimal places.

A $3.864$
B $3.884$
C $3.885$
D $3.900$

Question 1404

[3 marks]curve sketching / Newton-Raphson

Find

\dfrac{dy}{dx}

for

y = x^3 - 12x - 12

and solve

\dfrac{dy}{dx}=0

for

x

Answer this when you sit the paper.

Question 1405

[2 marks]curve sketching / Newton-Raphson

State the

y

-intercept of the graph

y = x^3 - 12x - 12

Answer this when you sit the paper.

Question 1406

[3 marks]curve sketching / Newton-Raphson

Using the Newton-Raphson method with

f(x) = x^3 - 12x - 12

and first approximation

x_1 = 3.9

, find

x_2

, correct to 3 decimal places.

Answer this when you sit the paper.

Question 1501

[1 marks]quadratics / integration (area and volume)

Express

9x - x^2

in the form

a + b(x + c)^2

and hence state the coordinates of the turning point of

y = 9x - x^2

A $\tfrac{81}{2} - \left(x - \tfrac{9}{2}\right)^2$ ; turning point $\left(\tfrac{9}{2},\ \tfrac{81}{2}\right)$
B $9 - \left(x - \tfrac{9}{2}\right)^2$ ; turning point $\left(\tfrac{9}{2},\ 9\right)$
C $\tfrac{81}{4} - \left(x - \tfrac{9}{2}\right)^2$ ; turning point $\left(\tfrac{9}{2},\ \tfrac{81}{4}\right)$
D $\tfrac{81}{4} + \left(x - \tfrac{9}{2}\right)^2$ ; turning point $\left(\tfrac{9}{2},\ \tfrac{81}{4}\right)$

Question 1502

[1 marks]quadratics / integration (area and volume)

The region

R

is bounded by the curve

y = 9x - x^2

and the

x

-axis. Calculate the area of

R

A $40.5$ units $^2$
B $81.0$ units $^2$
C $121.5$ units $^2$
D $243.0$ units $^2$

Question 1503

[1 marks]quadratics / integration (area and volume)

The region bounded by

y = 9x - x^2

and the

x

-axis is rotated through four right angles about the

x

-axis. Find the volume generated.

A $1968.3\pi$ units $^3$
B $984.15\pi$ units $^3$
C $14762.25\pi$ units $^3$
D $121.5\pi$ units $^3$

Question 1504

[1 marks]quadratics / integration (area and volume)

Find the roots of

y = 9x - x^2

(the

x

-intercepts of the graph).

Answer this when you sit the paper.

Question 1505

[2 marks]quadratics / integration (area and volume)

Evaluate the indefinite integral

\int(9x - x^2)\,dx

Answer this when you sit the paper.

Question 1506

[3 marks]quadratics / integration (area and volume)

Expand

(9x - x^2)^2

fully, in powers of

x

Answer this when you sit the paper.

Question 1507

[3 marks]quadratics / integration (area and volume)

Evaluate

\int_0^9 (81x^2 - 18x^3 + x^4)\,dx

(the integral used before multiplying by

\pi

to find the volume).

Answer this when you sit the paper.

Question 1601

[1 marks]differential equations

Solve the differential equation

(4 - x)\dfrac{dy}{dx} = y

given that

y = 4

when

x = 1

A $y = \dfrac{4}{4 - x}$
B $y = 4e^{x-1}$
C $y = \dfrac{12}{4 - x}$
D $y = 12(4 - x)$

Question 1602

[1 marks]differential equations

A liquid cools so that its rate of temperature decrease is proportional to

\theta

, its temperature in

°

C. Given

\theta = 60

t = 0

and

\theta = 40

t = 5

minutes, find

\theta

in terms of

t

A $\theta = 40\left(\tfrac{2}{3}\right)^{t/5}$
B $\theta = 60\left(\tfrac{2}{3}\right)^{t/5}$
C $\theta = 60\left(\tfrac{3}{2}\right)^{t/5}$
D $\theta = 60e^{-t/5}$

Question 1603

[1 marks]differential equations

A liquid cools according to

\theta = 60\left(\tfrac{2}{3}\right)^{t/5}

, where

t

is in minutes. Find its temperature 5 minutes after it has reached

40°

A $30.0°$ C
B $13.3°$ C
C $20.0°$ C
D $26.7°$ C

Question 1604

[3 marks]differential equations

Separate variables in

(4-x)\dfrac{dy}{dx} = y

and integrate both sides to express

\ln y

in terms of

x

and an arbitrary constant

c

Answer this when you sit the paper.

Question 1605

[2 marks]differential equations

Using

y = 4

when

x = 1

\ln y = -\ln(4-x) + c

, find the value of

c

Answer this when you sit the paper.

Question 1606

[2 marks]differential equations

Write the differential equation relating

\theta

and

t

, given that the rate of decrease of temperature is proportional to

\theta

(let

k>0

be the constant of proportionality).

Answer this when you sit the paper.

Question 1607

[3 marks]differential equations

Given

\theta = 60

t=0

and

\theta = 40

t=5

, find the value of

5k

Answer this when you sit the paper.

Pure Mathematics Paper 1 June 2012

Sit this paper online

The questions

Question 101

Question 102

Question 103

Question 201

Question 202

Question 301

Question 302

Question 303

Question 304

Question 401

Question 402

Question 403

Question 501

Question 502

Question 503

Question 601

Question 602

Question 603

Question 701

Question 702

Question 703

Question 704

Question 801

Question 802

Question 803

Question 804

Question 901

Question 902

Question 903

Question 1001

Question 1002

Question 1003

Question 1004

Question 1101

Question 1102

Question 1103

Question 1104

Question 1201

Question 1202

Question 1203

Question 1204

Question 1301

Question 1302

Question 1303

Question 1304

Question 1305

Question 1401

Question 1402

Question 1403

Question 1404

Question 1405

Question 1406

Question 1501

Question 1502

Question 1503

Question 1504

Question 1505

Question 1506

Question 1507

Question 1601

Question 1602

Question 1603

Question 1604

Question 1605

Question 1606

Question 1607

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