ZIMSEC A Level Pure Mathematics Paper 1 June 2011

Question 101

[1 marks]inequalities / indices

Solve the inequality

|3^{-3x} - 54| < 27

A $1 < x < \tfrac{4}{3}$
B $-\tfrac{4}{3} < x < \tfrac{4}{3}$
C $x < -\tfrac{4}{3}$ or $x > -1$
D $-\tfrac{4}{3} < x < -1$

Question 102

[1 marks]inequalities / indices

Removing the modulus from

|3^{-3x} - 54| < 27

reduces it to which inequality in

3^{-3x}

A $27 < 3^{-3x} < 81$
B $0 < 3^{-3x} < 27$
C $-27 < 3^{-3x} < 27$
D $3^{-3x} > 81$

Question 103

[2 marks]inequalities / indices

Rewrite

27 < 3^{-3x}

using base 3 (i.e.

3^3 < 3^{-3x}

), then solve for

x

, stating only this one bound as an inequality in

x

Answer this when you sit the paper.

Question 201

[1 marks]binomial expansion

Find the expansion of

\sqrt{4 - 3x^2}

up to and including the term in

x^4

A $2 - \tfrac{3}{4}x^2 - \tfrac{9}{64}x^4$
B $2 - \tfrac{3}{2}x^2 - \tfrac{9}{16}x^4$
C $4 - \tfrac{3}{2}x^2 - \tfrac{9}{32}x^4$
D $2 - \tfrac{3}{4}x^2 + \tfrac{9}{64}x^4$

Question 202

[1 marks]binomial expansion

In the expansion of

\sqrt{4 - 3x^2}

in ascending powers of

x

, what is the coefficient of

x^4

A $-\tfrac{9}{16}$
B $-\tfrac{3}{4}$
C $-\tfrac{9}{64}$
D $\tfrac{9}{64}$

Question 203

[2 marks]binomial expansion

Write

\sqrt{4-3x^2}

2(1+u)^{1/2}

, stating

u

in terms of

x

Answer this when you sit the paper.

Question 301

[1 marks]coordinate geometry

Point

Q

lies on the line

y - 3x = 0

and on the perpendicular bisector of the line joining

M(6, 3)

and

N(2, 1)

. Find the coordinates of

Q

A $(6, 18)$
B $(2, 6)$
C $(-2, -6)$
D $(4, 2)$

Question 302

[1 marks]coordinate geometry

What is the gradient of the perpendicular bisector of the line joining

M(6, 3)

and

N(2, 1)

A $-\tfrac12$
B $-2$
C $\tfrac12$
D $2$

Question 303

[1 marks]coordinate geometry

Find the midpoint of

M(6,3)

and

N(2,1)

Answer this when you sit the paper.

Question 304

[2 marks]coordinate geometry

Hence write the equation of the perpendicular bisector of MN, in the form

y=mx+c

Answer this when you sit the paper.

Question 401

[1 marks]indices / equations

Find the value of

x

which satisfies

3(2^{2x+1}) - 2^{1-2x} = -5

A $x = 1$
B $x = -2$
C $x = \tfrac14$
D $x = -1$

Question 402

[1 marks]indices / equations

Which substitution turns

3(2^{2x+1}) - 2^{1-2x} = -5

into a quadratic equation?

A $u = \log_2 x$
B $u = 2^{x}$
C $u = 2^{2x}$
D $u = 2x + 1$

Question 403

[2 marks]indices / equations

Solve

(4u-1)(3u+2)=0

and state the positive root

u

Answer this when you sit the paper.

Question 404

[1 marks]indices / equations

Why is

u=-\frac23

rejected as a solution, given

u=2^{2x}

A $u$ must be an integer since $2x$ is an exponent.
B $u=-\frac23$ gives a non-real value of $x$ when substituted back.
C $u=-\frac23$ makes $(4u-1)(3u+2)=0$ false, so it isn't an algebraic root.
D $2^{2x}$ is always positive, so $u$ cannot be negative.

Question 501

[1 marks]numerical integration / trapezium rule

Using the trapezium rule with 5 trapezia, find the approximate value of

\displaystyle\int_0^1 x^2 e^x \, dx

, correct to 4 decimal places.

A $0.6487$
B $0.7183$
C $0.7454$
D $1.4908$

Question 502

[1 marks]numerical integration / trapezium rule

The trapezium rule with 5 trapezia estimates

\displaystyle\int_0^1 x^2 e^x \, dx

0.7454

. Given that the exact value is

e - 2

, the percentage error in the approximation is closest to

A $0.3\%$
B $3.8\%$
C $7.5\%$
D $26.6\%$

Question 503

[1 marks]numerical integration / trapezium rule

For the trapezium rule with 5 strips over

[0,1]

, what is the strip width

h

Answer this when you sit the paper.

Question 504

[2 marks]numerical integration / trapezium rule

Calculate

y_3=(0.6)^2e^{0.6}

, to 4 decimal places.

Answer this when you sit the paper.

Question 505

[1 marks]numerical integration / trapezium rule

State the exact value of the integral,

e-2

, to 4 decimal places.

Answer this when you sit the paper.

Question 601

[1 marks]trigonometry

In triangle

ABC

the sides

BC

CA

and

AB

are

a

b

and

c

respectively, and

\cot C = \dfrac{a}{c}\operatorname{cosec} B - \cot B

. Given

a = 14.7

cm,

c = 17.3

cm and

B = 64.2°

, find angle

C

correct to the nearest

0.1°

A $41.5°$
B $50.5°$
C $65.3°$
D $24.7°$

Question 602

[1 marks]trigonometry

In triangle

ABC

a = 14.7

cm,

c = 17.3

cm,

B = 64.2°

and

C = 65.3°

. Find angle

A

correct to the nearest

0.1°

A $50.5°$
B $65.3°$
C $115.5°$
D $25.8°$

Question 603

[1 marks]trigonometry

In right triangle ABD (D the foot of the perpendicular from A), express

AD=x

in terms of

c

and angle

B

Answer this when you sit the paper.

Question 604

[1 marks]trigonometry

In right triangle ABD, express

BD

in terms of

c

and angle

B

Answer this when you sit the paper.

Question 605

[2 marks]trigonometry

Using

a=14.7

c=17.3

B=64.2°

, evaluate

\cot C=\dfrac{a}{c}\operatorname{cosec}B-\cot B

to 4 decimal places.

Answer this when you sit the paper.

Question 701

[1 marks]graph transformations

The diagram shows the graph of

y = f(x)

, passing through

A(-1, 0)

B(0, 1)

and

C(1, 0)

. On the graph of

y = f(-x) + 3

, the image of

B

A $(0, 3)$
B $(0, -2)$
C $(0, 4)$
D $(3, 1)$

Question 702

[1 marks]graph transformations

The diagram shows the graph of

y = f(x)

, passing through

A(-1, 0)

B(0, 1)

and

C(1, 0)

. On the graph of

y = 2f(x + 1)

, the image of

C

A $(0, 2)$
B $(0, 0)$
C $(2, 0)$
D $(1, 0)$

Question 703

[1 marks]graph transformations

Under

y=f(2x)

(horizontal stretch factor

\frac12

), find the image of

A(-1,0)

Answer this when you sit the paper.

Question 704

[1 marks]graph transformations

Under

y=f(2x)

, find the image of

C(1,0)

Answer this when you sit the paper.

Question 705

[1 marks]graph transformations

Under

y=f(-x)+3

, find the image of

A(-1,0)

Answer this when you sit the paper.

Question 706

[1 marks]graph transformations

Under

y=2f(x+1)

, find the image of

B(0,1)

Answer this when you sit the paper.

Question 801

[1 marks]differentiation

Given that

y = e^{2x}\sin 5x

, which differential equation does

y

satisfy?

A $\dfrac{d^2y}{dx^2} - 4\dfrac{dy}{dx} - 21y = 0$
B $\dfrac{d^2y}{dx^2} - 2\dfrac{dy}{dx} + 25y = 0$
C $\dfrac{d^2y}{dx^2} - 4\dfrac{dy}{dx} + 29y = 0$
D $\dfrac{d^2y}{dx^2} + 4\dfrac{dy}{dx} + 29y = 0$

Question 802

[1 marks]differentiation

y = e^{2x}\sin 5x

, then

\dfrac{dy}{dx} =

A $2e^{2x}\sin 5x - 5e^{2x}\cos 5x$
B $10e^{2x}\sin 5x\cos 5x$
C $2e^{2x}\cos 5x$
D $2e^{2x}\sin 5x + 5e^{2x}\cos 5x$

Question 803

[2 marks]differentiation

Differentiate

5e^{2x}\cos5x

(one term of

dy/dx

) using the product rule; give the result as a sum of two terms before simplifying.

Answer this when you sit the paper.

Question 804

[1 marks]differentiation

Differentiate

2e^{2x}\sin5x

(the other term of

dy/dx

); give the result as a sum of two terms.

Answer this when you sit the paper.

Question 805

[1 marks]differentiation

From

\frac{dy}{dx}=5e^{2x}\cos5x+2y

, make

5e^{2x}\cos5x

the subject.

Answer this when you sit the paper.

Question 901

[1 marks]complex numbers

Given

z_1 = 2 - 4i

and

z_2 = 6 - 2i

, express

z_1 z_2

in the form

a + ib

A $4 - 28i$
B $20 - 28i$
C $12 - 28i$
D $12 + 8i$

Question 902

[1 marks]complex numbers

Given

z_1 = 2 - 4i

and

w = \dfrac{1}{z_1}

, find the exact value of

|w|

A $\tfrac{\sqrt{5}}{10}$
B $\sqrt{20}$
C $\tfrac{1}{10}$
D $\tfrac{\sqrt{5}}{5}$

Question 903

[1 marks]complex numbers

Find

z_1-z_2

for

z_1=2-4i

z_2=6-2i

, in the form

a+bi

Answer this when you sit the paper.

Question 904

[2 marks]complex numbers

Find

w=\dfrac{1}{z_1}

for

z_1=2-4i

, in the form

a+bi

(rationalise using the conjugate).

Answer this when you sit the paper.

Question 905

[2 marks]complex numbers

Find

\arg(w)

for

w=\dfrac{1}{10}+\dfrac{1}{5}i

, to 1 decimal place (in degrees).

Answer this when you sit the paper.

Question 1001

[1 marks]parametric differentiation

A curve has parametric equations

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

and

y = t + \ln(2t+1)

t > -0.5

. Find the equation of the tangent to the curve at

t = 1

A $y = \tfrac15 x + 1 + \ln 3$
B $y = 5x + 4 - 6\ln 3$
C $y = 5x - 4 - 6\ln 3$
D $y = 5x - 4 + 6\ln 3$

Question 1002

[1 marks]parametric differentiation

For the curve

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

y = t + \ln(2t+1)

with

t > -0.5

, it can be shown that

\dfrac{dy}{dx} = \dfrac{2t+3}{2t-1}

. What is the gradient of the curve at

t = 1

A $5$
B $\tfrac15$
C $\tfrac53$
D $-5$

Question 1003

[1 marks]parametric differentiation

Find

\dfrac{dx}{dt}

for

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

Answer this when you sit the paper.

Question 1004

[1 marks]parametric differentiation

Find

\dfrac{dy}{dt}

for

y=t+\ln(2t+1)

Answer this when you sit the paper.

Question 1005

[3 marks]parametric differentiation

Find the coordinates

(x,y)

of the point on the curve at

t=1

, giving exact values involving

\ln3

Answer this when you sit the paper.

Question 1101

[1 marks]trigonometric equations / calculus

Solve the equation

\cos 2x = \cos x

for

0 \le x \le 2\pi

A $x = \tfrac{\pi}{3}, \tfrac{5\pi}{3}$ only
B $x = 0, \tfrac{2\pi}{3}, \tfrac{4\pi}{3}, 2\pi$
C $x = 0, \tfrac{\pi}{3}, \tfrac{5\pi}{3}, 2\pi$
D $x = 0, \pi, 2\pi$

Question 1102

[1 marks]trigonometric equations / calculus

Given

f(x) = \cos 2x - \cos x

, the derivative

f'(x)

factorises as

A $-2\sin 2x \cos x$
B $\sin x(4\cos x - 1)$
C $\cos x(1 - 4\sin x)$
D $\sin x(1 - 4\cos x)$

Question 1103

[2 marks]trigonometric equations / calculus

Solve

\cos x=\frac14

for

0\le x\le2\pi

, giving the first (smaller) solution in radians, to 3 d.p.

Answer this when you sit the paper.

Question 1104

[2 marks]trigonometric equations / calculus

Solve

\cos x=\frac14

for

0\le x\le2\pi

, giving the second (larger) solution in radians, to 3 d.p.

Answer this when you sit the paper.

Question 1105

[2 marks]trigonometric equations / calculus

Using

\cos2x=2\cos^2x-1

, rewrite

\cos2x=\cos x

as a quadratic in

\cos x

. State the quadratic expression, in the form

a\cos^2x+b\cos x+c

Answer this when you sit the paper.

Question 1201

[1 marks]arithmetic series

A teacher earns

\$x

in his first year and his salary increases each year by 10% of his first year's salary. His total salary after

n

years is

A $\dfrac{nx(n+19)}{20}$
B $\dfrac{nx(n+9)}{10}$
C $x(1.1)^{n}$
D $\dfrac{nx(n-1)}{20}$

Question 1202

[1 marks]arithmetic series

A teacher's total salary after

n

years is

\dfrac{nx(n+19)}{20}

, where

\$x

is his first year's salary. Find the least value of

n

for which his total salary exceeds 100 times his first salary.

A $n = 45$
B $n = 100$
C $n = 37$
D $n = 36$

Question 1203

[1 marks]arithmetic series

What is the common difference of the salary arithmetic progression, in terms of

x

Answer this when you sit the paper.

Question 1204

[2 marks]arithmetic series

From

\dfrac{nx(n+19)}{20}>100x

, cancel

x

and simplify to a quadratic inequality in

n

, in the form

n^2+bn+c>0

Answer this when you sit the paper.

Question 1205

[3 marks]arithmetic series

Using the quadratic formula on

n^2+19n-2000=0

, evaluate the positive root, to 1 decimal place.

Answer this when you sit the paper.

Question 1301

[1 marks]vectors

PQRS

is a square with

\overrightarrow{PR} = \begin{pmatrix} -3 \\ 3 \\ 12 \end{pmatrix}

and

\overrightarrow{SQ} = \begin{pmatrix} 5 \\ -11 \\ 4 \end{pmatrix}

. Find

\overrightarrow{PQ}

A $\begin{pmatrix} 1 \\ -4 \\ 8 \end{pmatrix}$
B $\begin{pmatrix} -4 \\ 7 \\ 4 \end{pmatrix}$
C $\begin{pmatrix} -1 \\ 4 \\ -8 \end{pmatrix}$
D $\begin{pmatrix} 2 \\ -8 \\ 16 \end{pmatrix}$

Question 1302

[1 marks]vectors

PQRS

is a square in which

\overrightarrow{PQ} = \begin{pmatrix} 1 \\ -4 \\ 8 \end{pmatrix}

. Calculate the area of the square.

A $9$ square units
B $18$ square units
C $40.5$ square units
D $81$ square units

Question 1303

[2 marks]vectors

Find

\overrightarrow{OP}=\overrightarrow{OQ}-\overrightarrow{PQ}

, given

\overrightarrow{OQ}=(3,-1,2)

and

\overrightarrow{PQ}=(1,-4,8)

Answer this when you sit the paper.

Question 1304

[2 marks]vectors

Find

\overrightarrow{OR}=\overrightarrow{OP}+\overrightarrow{PR}

, given

\overrightarrow{OP}=(2,3,-6)

and

\overrightarrow{PR}=(-3,3,12)

Answer this when you sit the paper.

Question 1305

[2 marks]vectors

Find

\overrightarrow{OS}=\overrightarrow{OQ}-\overrightarrow{SQ}

, given

\overrightarrow{OQ}=(3,-1,2)

and

\overrightarrow{SQ}=(5,-11,4)

Answer this when you sit the paper.

Question 1306

[1 marks]vectors

Find

|\overrightarrow{PQ}|

, the side length of the square, given

\overrightarrow{PQ}=(1,-4,8)

Answer this when you sit the paper.

Question 1401

[1 marks]integration (substitution and by parts)

Using the substitution

y = \sqrt{x}

, find the exact value of

\displaystyle\int_0^4 \dfrac{1}{\sqrt{x}+1}\,dx

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

Question 1402

[1 marks]integration (substitution and by parts)

Given that

\displaystyle\int_0^1 \dfrac{dx}{\sqrt{x}+1} = 2 - 2\ln 2

, find the exact value of

\displaystyle\int_0^1 \dfrac{\ln(\sqrt{x}+1)}{\sqrt{x}}\,dx

A $2\ln 2 - 2$
B $2 - 2\ln 2$
C $4\ln 2 + 2$
D $4\ln 2 - 2$

Question 1403

[1 marks]integration (substitution and by parts)

With

y=\sqrt{x}

, find

dx

in terms of

y

and

dy

Answer this when you sit the paper.

Question 1404

[2 marks]integration (substitution and by parts)

Rewrite

\dfrac{2y}{y+1}

by polynomial division, in the form

2-\dfrac{2}{y+1}

. State the simplified integrand.

Answer this when you sit the paper.

Question 1405

[2 marks]integration (substitution and by parts)

Evaluate

\displaystyle\int_0^1\dfrac{dx}{\sqrt x+1}

(needed in part (ii)), exact value.

Answer this when you sit the paper.

Question 1406

[2 marks]integration (substitution and by parts)

In the integration by parts for

\displaystyle\int_0^1\dfrac{\ln(\sqrt x+1)}{\sqrt x}dx

, with

u=\ln(\sqrt x+1)

and

dv=\dfrac{dx}{\sqrt x}

, find

v

Answer this when you sit the paper.

Question 1407

[3 marks]integration (substitution and by parts)

Evaluate

[2\sqrt x\ln(\sqrt x+1)]_0^1

, the boundary term from integration by parts.

Answer this when you sit the paper.

Question 1501

[1 marks]exponentials, area and volume of revolution

The region

R

is bounded by the curves

y = e^x

and

y = e^{-x}

and the line

x = 1

. Find the exact area of

R

A $e + \dfrac{1}{e}$
B $e + \dfrac{1}{e} - 2$
C $e - \dfrac{1}{e} - 2$
D $e - \dfrac{1}{e}$

Question 1502

[1 marks]exponentials, area and volume of revolution

The region

R

bounded by

y = e^x

y = e^{-x}

and

x = 1

is rotated through

2\pi

radians about the

x

-axis. Find the exact volume generated.

A $\pi\left(e + \dfrac{1}{e} - 2\right)^2$
B $\pi\left(e^2 + \dfrac{1}{e^2} - 2\right)$
C $\dfrac{\pi}{2}\left(e^2 + \dfrac{1}{e^2} - 2\right)$
D $\dfrac{\pi}{2}\left(e^2 - \dfrac{1}{e^2}\right)$

Question 1503

[1 marks]exponentials, area and volume of revolution

Solve

e^x-e^{-x}=0

for

x

Answer this when you sit the paper.

Question 1504

[1 marks]exponentials, area and volume of revolution

Find the point of intersection of

y=e^x

and

y=e^{-x}

Answer this when you sit the paper.

Question 1505

[2 marks]exponentials, area and volume of revolution

Find the antiderivative of

e^x-e^{-x}

(ignore the constant of integration).

Answer this when you sit the paper.

Question 1506

[2 marks]exponentials, area and volume of revolution

Expand

(e^x)^2-(e^{-x})^2

in terms of

e^{2x}

and

e^{-2x}

Answer this when you sit the paper.

Question 1507

[3 marks]exponentials, area and volume of revolution

Find the antiderivative of

e^{2x}-e^{-2x}

(used for the volume integral), ignoring the constant.

Answer this when you sit the paper.

Question 1508

[2 marks]exponentials, area and volume of revolution

Evaluate

e+\dfrac1e

to 4 decimal places (used in the area calculation).

Answer this when you sit the paper.

Question 1601

[1 marks]differential equations (Newton's law of cooling)

Bread cools in a box at

20°

C so that

\dfrac{d\theta}{dt} = -k(\theta - 20)

, where

\theta\,°

C is its temperature at time

t

minutes. Given that

\theta = 100

when

t = 0

and

\theta = 80

when

t = 10

, the particular solution is

A $\theta = 20\left(\tfrac34\right)^{t/10} + 80$
B $\theta = 80\left(\tfrac34\right)^{t/10} + 20$
C $\theta = 100\left(\tfrac45\right)^{t/10}$
D $\theta = 80e^{-t/10} + 20$

Question 1602

[1 marks]differential equations (Newton's law of cooling)

The temperature of cooling bread is

\theta = 80\left(\tfrac34\right)^{t/10} + 20

, where

t

is in minutes. Find its temperature when

t = 20

A $45°$ C
B $60°$ C
C $65°$ C
D $80°$ C

Question 1603

[1 marks]differential equations (Newton's law of cooling)

The temperature of cooling bread is

\theta = 80\left(\tfrac34\right)^{t/10} + 20

, where

t

is in minutes. To the nearest minute, when has the temperature dropped by

40°

A $24$ minutes
B $29$ minutes
C $14$ minutes
D $20$ minutes

Question 1604

[1 marks]differential equations (Newton's law of cooling)

Write the right-hand side of the differential equation

\dfrac{d\theta}{dt}=\ldots

, given the cooling rate is proportional to the difference between

\theta

and

20°C

, with constant

k

Answer this when you sit the paper.

Question 1605

[1 marks]differential equations (Newton's law of cooling)

Using

\theta=100

t=0

\theta=Be^{-kt}+20

, find

B

Answer this when you sit the paper.

Question 1606

[2 marks]differential equations (Newton's law of cooling)

Using

\theta=80

t=10

, write and solve

60=80e^{-10k}

for

e^{-10k}

Answer this when you sit the paper.

Question 1607

[2 marks]differential equations (Newton's law of cooling)

Hence evaluate

k=\dfrac{1}{10}\ln\left(\dfrac{4}{3}\right)

, to 4 decimal places.

Answer this when you sit the paper.

Question 1608

[2 marks]differential equations (Newton's law of cooling)

Solve

\left(\frac34\right)^{t/10}=0.5

for

\dfrac{t}{10}=\dfrac{\ln0.5}{\ln0.75}

, to 3 decimal places.

Answer this when you sit the paper.

Question 1609

[3 marks]differential equations (Newton's law of cooling)

Separate variables in

\dfrac{d\theta}{dt}=-k(\theta-20)

to write

\int\frac{d\theta}{\theta-20}=-k\int dt

, then integrate the left side (state the antiderivative, ignoring the constant).

Answer this when you sit the paper.

Pure Mathematics Paper 1 June 2011

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 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 605

Question 701

Question 702

Question 703

Question 704

Question 705

Question 706

Question 801

Question 802

Question 803

Question 804

Question 805

Question 901

Question 902

Question 903

Question 904

Question 905

Question 1001

Question 1002

Question 1003

Question 1004

Question 1005

Question 1101

Question 1102

Question 1103

Question 1104

Question 1105

Question 1201

Question 1202

Question 1203

Question 1204

Question 1205

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

Question 1501

Question 1502

Question 1503

Question 1504

Question 1505

Question 1506

Question 1507

Question 1508

Question 1601