ZIMSEC A Level Pure Mathematics Paper 1 June 2013

Question 101

[1 marks]indices / simultaneous equations

Solve the simultaneous equations

2^x + 3^y = 5

and

2^{x+2} - 3^{y+1} = 13

A $x = 0$ , $y = 2$
B $x = 4$ , $y = 1$
C $x = 1$ , $y = 2$
D $x = 2$ , $y = 0$

Question 102

[1 marks]indices / simultaneous equations

Which substitution reduces

2^x + 3^y = 5

and

2^{x+2} - 3^{y+1} = 13

to a pair of linear equations?

A $p = 2x$ , $q = 3y$
B $p = 2^x$ , $q = 3^y$
C $p = x^2$ , $q = y^3$
D $p = \log_2 x$ , $q = \log_3 y$

Question 103

[2 marks]indices / simultaneous equations

Substituting

p = 2^x

q = 3^y

turns the equations into

p + q = 5

and

4p - 3q = 13

. Solve for

p

Answer this when you sit the paper.

Question 201

[1 marks]modulus / inequalities

Solve the inequality

|2x + 1| < 3x + 2

A $-1 < x < -\tfrac{3}{5}$
B $x > -\tfrac{2}{3}$
C $x > -\tfrac{3}{5}$
D $x < -\tfrac{3}{5}$

Question 202

[2 marks]modulus / inequalities

Squaring both sides of

|2x+1| < 3x+2

gives

5x^2+8x+3 > 0

. Factorise the left side as

(5x+3)(x+c)

and state

c

Answer this when you sit the paper.

Question 203

[1 marks]modulus / inequalities

Since

|2x+1| \ge 0

, squaring is only valid when

3x+2 > 0

. State this condition on

x

Answer this when you sit the paper.

Question 301

[1 marks]logarithms / linearising relationships

Two quantities satisfy

y = ab^x

. Plotting

\ln y

against

x

gives a straight line of gradient

0.8

with

\ln y

-intercept

3.3

. Find

a

and

b

, each correct to one decimal place.

A $a = 2.2$ , $b = 27.1$
B $a = 27.1$ , $b = 2.2$
C $a = 0.8$ , $b = 3.3$
D $a = 3.3$ , $b = 0.8$

Question 302

[2 marks]logarithms / linearising relationships

Given

y=ab^x

and

\ln b = 0.8

(the gradient), find

b

correct to one decimal place.

Answer this when you sit the paper.

Question 303

[1 marks]logarithms / linearising relationships

Given

y=ab^x

and

\ln a = 3.3

(the intercept), find

a

correct to one decimal place.

Answer this when you sit the paper.

Question 401

[1 marks]small changes / percentage error

1\%

error is made in measuring the diameter of a sphere. Find the approximate resulting percentage error in the volume.

A $2\%$
B $3\%$
C $\tfrac{1}{3}\%$
D $1\%$

Question 402

[1 marks]small changes / percentage error

1\%

error is made in measuring the diameter of a sphere. Find the approximate resulting percentage error in the surface area.

A $1\%$
B $4\%$
C $2\%$
D $3\%$

Question 403

[2 marks]small changes / percentage error

Express the volume

V

of a sphere in terms of its diameter

D

(substituting

r=D/2

into

V=\frac{4}{3}\pi r^3

Answer this when you sit the paper.

Question 501

[1 marks]polynomials / remainder and factor theorems

When

f(x) = x^3 + ax^2 + bx + c

is divided by

x^2 - 4

the remainder is

2x + 11

, and

x + 1

is a factor of

f(x)

. Find

a

b

and

c

A $a = -4$ , $b = -2$ , $c = 5$
B $a = 4$ , $b = -2$ , $c = -5$
C $a = 2$ , $b = -4$ , $c = 11$
D $a = 4$ , $b = 2$ , $c = -5$

Question 502

[2 marks]polynomials / remainder and factor theorems

Long division of

f(x)=x^3+ax^2+bx+c

x^2-4

gives remainder

(b+4)x + (4a+c)

. Matching to

2x+11

, find

b

Answer this when you sit the paper.

Question 503

[2 marks]polynomials / remainder and factor theorems

Using

b=-2

, the factor condition

f(-1)=0

gives

a-b+c=1

, and matching constants gives

4a+c=11

. Find

a

Answer this when you sit the paper.

Question 601

[1 marks]partial fractions / binomial expansion

Express

\dfrac{4x + 6}{(x+2)(x+1)(x+3)}

in partial fractions.

A $\dfrac{2}{x+2} + \dfrac{1}{x+1} - \dfrac{3}{x+3}$
B $\dfrac{2}{x+2} - \dfrac{1}{x+1} + \dfrac{3}{x+3}$
C $\dfrac{1}{x+2} + \dfrac{2}{x+1} - \dfrac{3}{x+3}$
D $\dfrac{2}{x+2} + \dfrac{1}{x+1} + \dfrac{3}{x+3}$

Question 602

[1 marks]partial fractions / binomial expansion

Using series expansions up to and including the term in

x^2

\dfrac{4x + 6}{(x+2)(x+1)(x+3)}

reduces to

A $1 - \tfrac{7}{6}x + \tfrac{41}{36}x^2$
B $1 + \tfrac{7}{6}x + \tfrac{41}{36}x^2$
C $1 - \tfrac{7}{6}x - \tfrac{41}{36}x^2$
D $\tfrac{7}{6} - x + \tfrac{41}{36}x^2$

Question 603

[2 marks]partial fractions / binomial expansion

\dfrac{4x+6}{(x+2)(x+1)(x+3)} = \dfrac{A}{x+2}+\dfrac{B}{x+1}+\dfrac{C}{x+3}

, find

B

(the coefficient for

x+1

Answer this when you sit the paper.

Question 604

[1 marks]partial fractions / binomial expansion

In the same partial-fraction decomposition, find

C

(the coefficient for

x+3

Answer this when you sit the paper.

Question 605

[2 marks]partial fractions / binomial expansion

Expand

\left(1+\frac{x}{2}\right)^{-1}

up to and including the term in

x^2

Answer this when you sit the paper.

Question 701

[1 marks]trigonometry / R-formula

Express

4\sin\theta - 3\cos\theta

in the form

R\sin(\theta - \alpha)

, where

R > 0

and

\alpha

is acute.

A $25\sin(\theta - 36.9°)$
B $7\sin(\theta - 36.9°)$
C $5\sin(\theta - 53.1°)$
D $5\sin(\theta - 36.9°)$

Question 702

[1 marks]trigonometry / R-formula

Solve

4\sin\theta - 3\cos\theta = 3

for

0° \le \theta \le 360°

A $\theta = 36.9°$ and $\theta = 143.1°$
B $\theta = 73.8°$ and $\theta = 180°$
C $\theta = 73.8°$ and $\theta = 143.1°$
D $\theta = 0°$ and $\theta = 180°$

Question 703

[1 marks]trigonometry / R-formula

For what value of

\theta

0° \le \theta \le 360°

4\sin\theta - 3\cos\theta

a maximum?

A $\theta = 126.9°$
B $\theta = 36.9°$
C $\theta = 306.9°$
D $\theta = 90°$

Question 704

[2 marks]trigonometry / R-formula

State the value of

\alpha

, in degrees correct to 1 d.p., in

4\sin\theta - 3\cos\theta = 5\sin(\theta - \alpha)

Answer this when you sit the paper.

Question 705

[2 marks]trigonometry / R-formula

For

0° \le \theta \le 360°

, find the value of

\theta

at which

4\sin\theta - 3\cos\theta

is a minimum.

Answer this when you sit the paper.

Question 801

[1 marks]parametric differentiation / turning points

A curve has parametric equations

x = 1 + t^2

and

y = 4t - t^3

t > 0

. Find

\dfrac{dy}{dx}

in terms of

t

A $\dfrac{2t}{4 - 3t^2}$
B $4 - 3t^2$
C $\dfrac{4 - t^2}{2t}$
D $\dfrac{4 - 3t^2}{2t}$

Question 802

[1 marks]parametric differentiation / turning points

For the curve

x = 1 + t^2

y = 4t - t^3

with

t > 0

, find the coordinates of the turning point.

A $\left(\tfrac{7}{3},\ \tfrac{16}{3\sqrt{3}}\right)$
B $\left(1,\ 0\right)$
C $\left(\tfrac{4}{3},\ \tfrac{16}{3\sqrt{3}}\right)$
D $\left(\tfrac{7}{3},\ \tfrac{8}{\sqrt{3}}\right)$

Question 803

[2 marks]parametric differentiation / turning points

Setting

\frac{dy}{dx}=0

for

x=1+t^2

y=4t-t^3

(

t>0

), find the exact value of

t

at the turning point.

Answer this when you sit the paper.

Question 804

[1 marks]parametric differentiation / turning points

For

y=4t-t^3

, find

\dfrac{dy}{dt}

Answer this when you sit the paper.

Question 805

[1 marks]parametric differentiation / turning points

For

x=1+t^2

, find

\dfrac{dx}{dt}

Answer this when you sit the paper.

Question 901

[1 marks]circular measure / sectors

In a school logo,

O

is the centre of sector

OACB

with angle

AOB = \tfrac{\pi}{3}

, and

X

is a point with

OX = OA = OB = r

. Find angle

AXB

A $\tfrac{\pi}{6}$
B $\tfrac{2\pi}{3}$
C $\tfrac{\pi}{12}$
D $\tfrac{\pi}{3}$

Question 902

[1 marks]circular measure / sectors

In the school-logo figure with

OX = OA = OB = r

and angle

AOB = \tfrac{\pi}{3}

, the exact area of the shaded region is

A $\dfrac{r^2}{2} + \dfrac{\pi r^2}{6}$
B $\dfrac{\sqrt{3}\pi r^2}{12}$
C $\dfrac{r^2}{12}\left(6 - \sqrt{3}\pi\right)$
D $\dfrac{r^2}{12}\left(6 + \sqrt{3}\pi\right)$

Question 903

[2 marks]circular measure / sectors

In triangle

AOX

, angle

AOX = \frac{5\pi}{6}

and

OA=OX=r

. Find the area of triangle

AOX

in terms of

r

(exact form).

Answer this when you sit the paper.

Question 904

[2 marks]circular measure / sectors

Using the cosine rule in triangle

AOX

(angle

\frac{5\pi}{6}

, sides

r

and

r

), find

AX^2

in terms of

r

(exact form).

Answer this when you sit the paper.

Question 905

[2 marks]circular measure / sectors

Find the exact area of sector

OACB

(angle

\frac{\pi}{3}

, radius

r

Answer this when you sit the paper.

Question 1001

[1 marks]vectors in 3D

Points

A

and

B

have position vectors

4\mathbf{i} - 9\mathbf{j} - \mathbf{k}

and

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

. Find the unit vector parallel to

\overrightarrow{AB}

A $\tfrac{1}{\sqrt{189}}(3\mathbf{i} - 12\mathbf{j} - 6\mathbf{k})$
B $\tfrac{1}{\sqrt{98}}(-3\mathbf{i} + 12\mathbf{j} + 6\mathbf{k})$
C $\tfrac{1}{\sqrt{189}}(-3\mathbf{i} + 12\mathbf{j} + 6\mathbf{k})$
D $\tfrac{1}{189}(-3\mathbf{i} + 12\mathbf{j} + 6\mathbf{k})$

Question 1002

[1 marks]vectors in 3D

Points

A

B

C

have position vectors

4\mathbf{i} - 9\mathbf{j} - \mathbf{k}

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

and

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

. Find

\lambda

so that

A

B

and

C

are collinear.

A $\lambda = -2$
B $\lambda = 4$
C $\lambda = 1$
D $\lambda = 2$

Question 1003

[1 marks]vectors in 3D

Points

A

and

B

have position vectors

4\mathbf{i} - 9\mathbf{j} - \mathbf{k}

and

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

relative to the origin

O

. Calculate angle

AOB

correct to the nearest

0.1°

A $151.4°$
B $28.6°$
C $61.4°$
D $118.6°$

Question 1004

[2 marks]vectors in 3D

Find

\overrightarrow{OA}\cdot\overrightarrow{OB}

for

\overrightarrow{OA}=4\mathbf{i}-9\mathbf{j}-\mathbf{k}

and

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

Answer this when you sit the paper.

Question 1005

[2 marks]vectors in 3D

Find

|\overrightarrow{OA}|

, correct to 3 significant figures, for

\overrightarrow{OA}=4\mathbf{i}-9\mathbf{j}-\mathbf{k}

Answer this when you sit the paper.

Question 1006

[1 marks]vectors in 3D

Find

|\overrightarrow{OB}|

, correct to 3 significant figures, for

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

Answer this when you sit the paper.

Question 1101

[1 marks]complex numbers

Given

p = 5 + i

and

q = -2 + 3i

, find

pq

A $-10 + 3i$
B $-7 - 17i$
C $-13 + 13i$
D $-13 - 13i$

Question 1102

[1 marks]complex numbers

Given

p = 5 + i

and

q = -2 + 3i

, express

\dfrac{p}{q}

in the form

a + ib

A $\dfrac{-7 + 17i}{13}$
B $\dfrac{-13 + 13i}{13}$
C $\dfrac{-7 - 17i}{26}$
D $\dfrac{-7 - 17i}{13}$

Question 1103

[1 marks]complex numbers

Describe the geometrical transformation which maps

ip

onto

p

on an Argand diagram.

AAn enlargement of scale factor $i$ about the origin
BA rotation of $90°$ clockwise about the origin
CA rotation of $90°$ anticlockwise about the origin
DA reflection in the real axis

Question 1104

[1 marks]complex numbers

Given

p = 5+i

, find

ip

in the form

a+ib

Answer this when you sit the paper.

Question 1105

[1 marks]complex numbers

Given

p=5+i

and

q=-2+3i

, find

p+q

in the form

a+ib

Answer this when you sit the paper.

Question 1106

[2 marks]complex numbers

Find the modulus of

p=5+i

, correct to 3 significant figures.

Answer this when you sit the paper.

Question 1107

[1 marks]complex numbers

Find the argument of

p=5+i

, in degrees correct to 1 decimal place.

Answer this when you sit the paper.

Question 1201

[1 marks]differential equations

Compounds

P

and

Q

have masses

x

and

y

with

x + y = 10

, and

x

increases at a rate proportional to the product of the two masses. The differential equation satisfied by

x

A $\dfrac{dx}{dt} = k(10 - x)$
B $\dfrac{dx}{dt} = kx + 10$
C $\dfrac{dx}{dt} = 10kx$
D $\dfrac{dx}{dt} = kx(10 - x)$

Question 1202

[1 marks]differential equations

Given

\dfrac{dx}{dt} = kx(10 - x)

with

x = 2

t = 0

and

x = 5

t = \ln 2

, find

t

when

x = 9.9

, correct to 3 significant figures.

A $1.50$
B $2.99$
C $4.60$
D $5.98$

Question 1203

[2 marks]differential equations

Using

x=2

t=0

\frac{1}{10}\ln\left|\frac{x}{10-x}\right| = kt+A

, find the constant

A

, correct to 3 decimal places.

Answer this when you sit the paper.

Question 1204

[2 marks]differential equations

Using

x=5

t=\ln2

\frac{1}{10}\ln\left|\frac{x}{10-x}\right| = kt+A

, find

k

Answer this when you sit the paper.

Question 1205

[2 marks]differential equations

Using

k=\frac15

, find

\dfrac{dx}{dt}

when

x=5

Answer this when you sit the paper.

Question 1301

[1 marks]parametric curves / coordinate geometry

A curve

C

has parametric equations

y = at

and

x = \dfrac{a}{t}

t > 0

. Write down the cartesian equation of

C

A $y = \dfrac{a^2}{x}$
B $y = ax$
C $y = \dfrac{a}{x}$
D $xy = a$

Question 1302

[1 marks]parametric curves / coordinate geometry

For the curve

y = \dfrac{a^2}{x}

, find the equation of the normal at the point where

t = 2

, i.e. at

P\left(\tfrac{a}{2},\ 2a\right)

A $8y = -2x + 15a$
B $y = 4x + 15a$
C $8y = 2x + 15a$
D $2y = 8x + 15a$

Question 1303

[1 marks]parametric curves / coordinate geometry

The normal to

y = \dfrac{a^2}{x}

P\left(\tfrac{a}{2},\ 2a\right)

meets the curve again at

Q\left(-8a,\ -\tfrac{a}{8}\right)

. Find the centre of the circle having

PQ

as diameter.

A $\left(-\tfrac{15a}{8},\ \tfrac{15a}{32}\right)$
B $\left(-\tfrac{15a}{4},\ -\tfrac{15a}{16}\right)$
C $\left(\tfrac{15a}{4},\ \tfrac{15a}{16}\right)$
D $\left(-\tfrac{15a}{4},\ \tfrac{15a}{16}\right)$

Question 1304

[2 marks]parametric curves / coordinate geometry

For the curve

y=at

x=\frac{a}{t}

, find the coordinates of point

P

t=2

(in terms of

a

Answer this when you sit the paper.

Question 1305

[2 marks]parametric curves / coordinate geometry

For the curve

y=\dfrac{a^2}{x}

, find

\dfrac{dy}{dx}

in terms of

x

and

a

Answer this when you sit the paper.

Question 1306

[3 marks]parametric curves / coordinate geometry

Find

r^2

, the square of the circle's radius (with

PQ

as diameter), in terms of

a

, as an exact fraction.

Answer this when you sit the paper.

Question 1401

[1 marks]integration (area and volume)

The region in the first quadrant bounded by

y = \dfrac{x}{2-x}

, the line

x = 1

and the

x

-axis has area

A

. Find

A

in exact form.

A $2\ln 2$
B $\ln 2 - 1$
C $2\ln 2 - 1$
D $1 - 2\ln 2$

Question 1402

[1 marks]integration (area and volume)

The region bounded by

y = \dfrac{x}{2-x}

, the line

x = 1

and the

x

-axis is rotated completely about the

x

-axis. Find the exact volume generated.

A $\pi(3 - 4\ln 2)$
B $\pi(2\ln 2 - 1)$
C $\pi(5 - 4\ln 2)$
D $\pi(4\ln 2 - 3)$

Question 1403

[2 marks]integration (area and volume)

Rewrite

\dfrac{x}{2-x}

in the form

-1+\dfrac{k}{2-x}

by algebraic division. State

k

Answer this when you sit the paper.

Question 1404

[2 marks]integration (area and volume)

Evaluate the antiderivative

-x-2\ln|2-x|

x=1

(the upper limit, before subtracting the lower-limit value).

Answer this when you sit the paper.

Question 1405

[2 marks]integration (area and volume)

Evaluate the antiderivative

-x-2\ln|2-x|

x=0

(the lower limit).

Answer this when you sit the paper.

Question 1406

[1 marks]integration (area and volume)

Evaluate

x+4\ln|2-x|+\dfrac{4}{2-x}

x=1

(part of the volume antiderivative, upper limit).

Answer this when you sit the paper.

Question 1501

[1 marks]numerical methods

Use the trapezium rule with 4 ordinates to find an approximate value of

\displaystyle\int_0^3 xe^{-x}\,dx

, correct to 3 decimal places.

A $0.356$
B $0.713$
C $0.801$
D $1.426$

Question 1502

[1 marks]numerical methods

How many roots does the equation

e^{x/2} = x + 2

have?

Anone
Bexactly one
Cexactly two
Dinfinitely many

Question 1503

[1 marks]numerical methods

Taking

4

as the first approximation to a root of

e^{x/2} = x + 2

, use the Newton-Raphson method to find the second approximation, correct to 3 significant figures.

A $3.00$
B $3.48$
C $3.62$
D $4.28$

Question 1504

[2 marks]numerical methods

Evaluate

f(3)=e^{1.5}-5

, correct to 3 significant figures (used to show a root lies between 3 and 4).

Answer this when you sit the paper.

Question 1505

[2 marks]numerical methods

Evaluate

f(4)=e^2-6

, correct to 3 significant figures.

Answer this when you sit the paper.

Question 1506

[2 marks]numerical methods

Find

f'(4) = \frac12 e^2-1

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

Answer this when you sit the paper.

Question 1507

[1 marks]numerical methods

State the interval width

h

used in the trapezium rule with 4 ordinates over

[0,3]

Answer this when you sit the paper.

Question 1601

[1 marks]geometric series / summation of series

The sum of the first 5 terms of a geometric series is 5 and the sum of the fifth to the ninth terms is 80. Find the possible values of the first term

a

A $a = \tfrac{5}{31}$ or $a = \tfrac{5}{11}$
B $a = 5$ or $a = 80$
C $a = \tfrac{5}{16}$ or $a = \tfrac{5}{11}$
D $a = \tfrac{5}{31}$ only

Question 1602

[1 marks]geometric series / summation of series

Express

4x^3 - 6x^2 + 2x

in the form

Ax(x+1)(x+2) + Bx(x+1) + Cx

A $A = 1$ , $B = -18$ , $C = 12$
B $A = 4$ , $B = -18$ , $C = 12$
C $A = 4$ , $B = -6$ , $C = 2$
D $A = 4$ , $B = 18$ , $C = -12$

Question 1603

[1 marks]geometric series / summation of series

Given

\sum_{x=1}^{n} x(x+1)(x+2) = \tfrac{n}{4}(n+1)(n+2)(n+3)

and

\sum_{x=1}^{n} x(x+1) = \tfrac{n}{3}(n+1)(n+2)

, evaluate

\sum_{x=1}^{n}\left(4x^3 - 6x^2 + 2x\right)

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

Question 1604

[2 marks]geometric series / summation of series

Dividing

ar^4(1+r+r^2+r^3+r^4)=80

a(1+r+r^2+r^3+r^4)=5

, find

r^4

Answer this when you sit the paper.

Question 1605

[2 marks]geometric series / summation of series

For

r=2

, evaluate

1+r+r^2+r^3+r^4

(used to find

a

Answer this when you sit the paper.

Question 1606

[2 marks]geometric series / summation of series

For

r=-2

, evaluate

1+r+r^2+r^3+r^4

(used to find the other value of

a

Answer this when you sit the paper.

Question 1607

[2 marks]geometric series / summation of series

Verify the identity

(n+2)(n+3)-6(n+2)+6 = n^2-n

by evaluating the left side at

n=2

Answer this when you sit the paper.

Pure Mathematics Paper 1 June 2013

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 401

Question 402

Question 403

Question 501

Question 502

Question 503

Question 601

Question 602

Question 603

Question 604

Question 605

Question 701

Question 702

Question 703

Question 704

Question 705

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 1006

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 1301

Question 1302

Question 1303

Question 1304

Question 1305

Question 1306

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