ZIMSEC A Level Pure Mathematics Paper 1 November 2011

Question 101

[1 marks]modulus / inequalities

Find the set of values of

x

which satisfy

|3x + 2| < -x + 1

A $-\tfrac{1}{4} < x < 1$
B $x < -\tfrac{3}{2}$ or $x > -\tfrac{1}{4}$
C $-\tfrac{3}{2} < x < -\tfrac{1}{4}$
D $-\tfrac{2}{3} < x < \tfrac{1}{4}$

Question 102

[1 marks]modulus / inequalities

The graphs of

y = |3x + 2|

and

y = -x + 1

intersect at two points. What are the

x

-coordinates of these points?

A $x = -\tfrac{3}{2}$ and $x = -\tfrac{1}{4}$
B $x = -\tfrac{1}{4}$ and $x = \tfrac{3}{2}$
C $x = -1$ and $x = \tfrac{1}{3}$
D $x = -\tfrac{2}{3}$ and $x = 1$

Question 103

[1 marks]modulus / inequalities

For

|3x+2|<-x+1

to have any solutions, we need

-x+1>0

. Solve this for

x

Answer this when you sit the paper.

Question 104

[2 marks]modulus / inequalities

For the case

3x+2\ge0

(i.e.

x\ge-\frac23

), solve

3x+2<-x+1

for

x

Answer this when you sit the paper.

Question 201

[1 marks]polynomials / remainder theorem

The cubic

f(x) = 2x^3 + 6x^2 + kx + 12

leaves a remainder of 6 when divided by

x + 2

. Find the value of

k

A $k = 13$
B $k = 7$
C $k = 10$
D $k = -7$

Question 202

[1 marks]polynomials / remainder theorem

Given

f(x) = 2x^3 + 6x^2 + 7x + 12

, solve the equation

f(x) = 9

, which has only one real root.

A $x = 1$
B $x = 3$
C $x = -3$
D $x = -1$

Question 203

[2 marks]polynomials / remainder theorem

By the remainder theorem, write the equation obtained by substituting

x=-2

into

f(x)=2x^3+6x^2+kx+12

, set equal to

6

, simplified to an equation in

k

Answer this when you sit the paper.

Question 204

[1 marks]polynomials / remainder theorem

Evaluate

2(-1)^3+6(-1)^2+7(-1)+3

to confirm

x=-1

is a root of

2x^3+6x^2+7x+3=0

Answer this when you sit the paper.

Question 205

[2 marks]polynomials / remainder theorem

Divide

2x^3+6x^2+7x+3

(x+1)

to find the quadratic quotient.

Answer this when you sit the paper.

Question 301

[1 marks]transformations / differentiation

Which sequence of transformations maps the graph of

y = \sin x

onto the graph of

y = 6\sin(2x) - \pi

AHorizontal stretch factor $2$ , vertical stretch factor $6$ , translation $\pi$ up
BHorizontal stretch factor $\tfrac{1}{2}$ , vertical stretch factor $6$ , translation $\pi$ down
CVertical stretch factor $2$ , horizontal stretch factor $6$ , translation $\pi$ down
DHorizontal stretch factor $\tfrac{1}{2}$ , vertical stretch factor $\tfrac{1}{6}$ , translation $\pi$ down

Question 302

[1 marks]transformations / differentiation

A curve has equation

y = \dfrac{x^2 - 4}{x + 1}

. Find the equation of the normal to the curve at

P(2, 0)

, giving your answer in the form

ax + by + c = 0

with integer coefficients.

A $3x - 4y - 6 = 0$
B $3x + 4y - 6 = 0$
C $4x + 3y - 8 = 0$
D $4x - 3y - 8 = 0$

Question 303

[2 marks]transformations / differentiation

Using the quotient rule, find

\dfrac{dy}{dx}

for

y=\dfrac{x^2-4}{x+1}

, in terms of

x

Answer this when you sit the paper.

Question 304

[1 marks]transformations / differentiation

Evaluate the gradient of the curve at

x=2

Answer this when you sit the paper.

Question 305

[2 marks]transformations / differentiation

Hence find the gradient of the normal to the curve at

P(2,0)

Answer this when you sit the paper.

Question 401

[1 marks]related rates of change

t = 0

the area of a circular pond is

36\pi

^2

and its radius increases at a constant

1.1

cm s

^{-1}

. Find the rate at which the area is increasing when the area is

144

^2

A $46.8$ cm $^2$ s $^{-1}$
B $117.5$ cm $^2$ s $^{-1}$
C $13.2$ cm $^2$ s $^{-1}$
D $41.5$ cm $^2$ s $^{-1}$

Question 402

[1 marks]related rates of change

A circular pond has area

36\pi

^2

t = 0

and its radius grows at

1.1

cm s

^{-1}

. Find the rate at which the area is increasing at

t = 10

A $37.4\pi$ cm $^2$ s $^{-1}$
B $34\pi$ cm $^2$ s $^{-1}$
C $13.2\pi$ cm $^2$ s $^{-1}$
D $46.8$ cm $^2$ s $^{-1}$

Question 403

[1 marks]related rates of change

t=0

, find the radius

r

of the pond, given its area is

36\pi

cm².

Answer this when you sit the paper.

Question 404

[1 marks]related rates of change

Write

r

in terms of

t

, given

\dfrac{dr}{dt}=1.1

cm/s and

r=6

t=0

Answer this when you sit the paper.

Question 405

[3 marks]related rates of change

Find

r

(to 2 decimal places) when the area of the pond is

144

cm².

Answer this when you sit the paper.

Question 501

[1 marks]partial fractions / integration

Express

\dfrac{3}{x^2(x+3)}

in partial fractions.

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

Question 502

[1 marks]partial fractions / integration

Evaluate

\displaystyle\int_2^4 \dfrac{3}{x^2(x+3)}\, dx

, giving your answer in exact form.

A $\tfrac{3}{4} + \ln\tfrac{7}{10}$
B $\tfrac{1}{4} + \tfrac{1}{3}\ln\tfrac{7}{10}$
C $\tfrac{1}{3}\ln\tfrac{7}{10} - \tfrac{1}{4}$
D $\tfrac{1}{4} + \tfrac{1}{3}\ln\tfrac{10}{7}$

Question 503

[1 marks]partial fractions / integration

Substituting

x=0

into

3=Ax(x+3)+B(x+3)+Cx^2

, find

B

Answer this when you sit the paper.

Question 504

[1 marks]partial fractions / integration

Substituting

x=-3

into

3=Ax(x+3)+B(x+3)+Cx^2

, find

C

Answer this when you sit the paper.

Question 505

[1 marks]partial fractions / integration

Comparing coefficients of

x^2

3=Ax(x+3)+B(x+3)+Cx^2

, find

A

Answer this when you sit the paper.

Question 506

[2 marks]partial fractions / integration

Write the antiderivative of

-\dfrac{1}{3x}+\dfrac{1}{x^2}+\dfrac{1}{3(x+3)}

, combined into a single log term minus

\dfrac1x

(ignore the constant of integration).

Answer this when you sit the paper.

Question 601

[1 marks]vectors

OPQR

is a parallelogram with

O

the origin,

\mathbf{p} = \mathbf{i} + \lambda\mathbf{j} + \mathbf{k}

and

\mathbf{r} = 4\mathbf{i} + 2\mathbf{k}

, where

\lambda > 0

. Given that angle

OPR

is a right angle, find

\lambda

A $\lambda = 2$
B $\lambda = \sqrt{2}$
C $\lambda = 1$
D $\lambda = 4$

Question 602

[1 marks]vectors

OPQR

is a parallelogram with

\mathbf{p} = \mathbf{i} + 2\mathbf{j} + \mathbf{k}

and

\mathbf{r} = 4\mathbf{i} + 2\mathbf{k}

. Find the exact area of the parallelogram.

A $84$
B $4\sqrt{21}$
C $\sqrt{21}$
D $2\sqrt{21}$

Question 603

[1 marks]vectors

Find

\overrightarrow{PR}=\mathbf{r}-\mathbf{p}

in terms of

\lambda

, given

\mathbf{p}=(1,\lambda,1)

\mathbf{r}=(4,0,2)

Answer this when you sit the paper.

Question 604

[2 marks]vectors

Set up

\overrightarrow{PO}\cdot\overrightarrow{PR}=0

using

\overrightarrow{PO}=(-1,-\lambda,-1)

and

\overrightarrow{PR}=(3,-\lambda,1)

, then simplify to an equation in

\lambda^2

Answer this when you sit the paper.

Question 605

[2 marks]vectors

Find the position vector of

Q

, given

\mathbf{p}=(1,2,1)

and

\mathbf{r}=(4,0,2)

(using

\lambda=2

Answer this when you sit the paper.

Question 701

[1 marks]circle geometry / radians

In a circle of radius

r

and centre

O

, the shaded region is bounded by the chord

AC

, the diameter

AB

and the arc

BC

, with

\widehat{BAC} = \theta

radians. Find an exact expression for the perimeter of the shaded region.

A $2r(1 + \theta + \cos\theta)$
B $r(1 + 2\theta + \cos\theta)$
C $2r(1 + \theta + \sin\theta)$
D $2r(\theta + \cos\theta)$

Question 702

[1 marks]circle geometry / radians

In a circle of radius

r

, the region bounded by chord

AC

, diameter

AB

and arc

BC

has area

r^2\theta + \tfrac{1}{2}r^2\sin 2\theta

, where

\widehat{BAC} = \theta

. Find the exact area when

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

A $r^2\left(\tfrac{\pi}{6} + \tfrac{1}{2}\right)$
B $r^2\left(\tfrac{\pi}{3} + \tfrac{\sqrt{3}}{4}\right)$
C $r^2\left(\tfrac{\pi}{6} + \tfrac{\sqrt{3}}{2}\right)$
D $r^2\left(\tfrac{\pi}{6} + \tfrac{\sqrt{3}}{4}\right)$

Question 703

[1 marks]circle geometry / radians

Express

AC

in terms of

r

and

\theta

, using the fact that angle

ACB=90°

(angle in a semicircle).

Answer this when you sit the paper.

Question 704

[1 marks]circle geometry / radians

Express arc

BC

in terms of

r

and

\theta

, using the fact that arc

BC

subtends a central angle of

2\theta

Answer this when you sit the paper.

Question 705

[3 marks]circle geometry / radians

Find the area of triangle

ABC

in terms of

r

and

\theta

, using

AC=2r\cos\theta

and

BC=2r\sin\theta

Answer this when you sit the paper.

Question 801

[1 marks]functions / completing the square

Express

h(x) = 2x^2 - 6x + 11

in the form

a(x + b)^2 + c

A $2\left(x + \tfrac{3}{2}\right)^2 + \tfrac{13}{2}$
B $2\left(x - \tfrac{3}{2}\right)^2 + \tfrac{11}{2}$
C $2\left(x - \tfrac{3}{2}\right)^2 + \tfrac{13}{2}$
D $2\left(x - 3\right)^2 - 7$

Question 802

[1 marks]functions / completing the square

The function

h(x) = 2x^2 - 6x + 11

is defined for

x \in \mathbb{R}

. State its range.

A $h(x) \ge 11$
B $h(x) \ge \tfrac{3}{2}$
C $h(x) \ge \tfrac{13}{2}$
D $h(x) \le \tfrac{13}{2}$

Question 803

[1 marks]functions / completing the square

Why does

h(x) = 2x^2 - 6x + 11

x \in \mathbb{R}

, have no inverse?

AIts range does not include $0$
BIt is not defined for all real $x$
CIt is many-to-one, so not one-to-one over $\mathbb{R}$
DIt is unbounded above

Question 804

[1 marks]functions / completing the square

State the largest element of

A

for which

h(x)=2\left(x-\frac32\right)^2+\frac{13}{2}

has an inverse (the vertex

x

-value).

Answer this when you sit the paper.

Question 805

[3 marks]functions / completing the square

Find

h^{-1}(x)

for the restricted domain

x\le\frac32

, in the form

\frac32-\frac12\sqrt{2x-13}

Answer this when you sit the paper.

Question 806

[1 marks]functions / completing the square

Confirm the minimum value by evaluating

h\left(\frac32\right)

directly, using

h(x)=2x^2-6x+11

Answer this when you sit the paper.

Question 901

[1 marks]binomial expansion

Given

f(x) = \dfrac{1}{(1 + x)^2} + \sqrt{9 + x}

, expand

f(x)

up to and including the term in

x^2

A $4 - \tfrac{11}{6}x + 3x^2$
B $3 - 2x + \tfrac{647}{216}x^2$
C $4 - \tfrac{11}{6}x + \tfrac{647}{216}x^2$
D $4 + \tfrac{11}{6}x + \tfrac{647}{216}x^2$

Question 902

[1 marks]binomial expansion

For which values of

x

is the expansion of

f(x) = \dfrac{1}{(1 + x)^2} + \sqrt{9 + x}

valid?

A $|x| < 9$
B $|x| < 3$
C $|x| < 1$
Dall real $x$

Question 903

[2 marks]binomial expansion

Expand

(1+x)^{-2}

up to and including the

x^2

term.

Answer this when you sit the paper.

Question 904

[3 marks]binomial expansion

Expand

\sqrt{9+x}

up to and including the

x^2

term.

Answer this when you sit the paper.

Question 905

[1 marks]binomial expansion

State

f(0)

, the constant term of the combined expansion of

f(x)=\dfrac{1}{(1+x)^2}+\sqrt{9+x}

Answer this when you sit the paper.

Question 1001

[1 marks]complex numbers

The complex number

u

satisfies

(-4 + 3i)u = 5 - 3i

. Find

|u|

A $\tfrac{5}{\sqrt{34}}$
B $\tfrac{\sqrt{34}}{5}$
C $\sqrt{34}$
D $\tfrac{34}{25}$

Question 1002

[1 marks]complex numbers

Given that

u = -\tfrac{29}{25} - \tfrac{3}{25}i

and

w = 2i

, express

uw

in the form

a + ib

A $-\tfrac{3}{50} + \tfrac{29}{50}i$
B $-\tfrac{6}{25} + \tfrac{58}{25}i$
C $\tfrac{58}{25} - \tfrac{6}{25}i$
D $\tfrac{6}{25} - \tfrac{58}{25}i$

Question 1003

[2 marks]complex numbers

Find

u=\dfrac{5-3i}{-4+3i}

in the form

a+bi

(rationalise using the conjugate).

Answer this when you sit the paper.

Question 1004

[2 marks]complex numbers

Find

\arg(u)

for

u=-\dfrac{29}{25}-\dfrac{3}{25}i

, to 1 decimal place (in degrees,

u

is in the third quadrant).

Answer this when you sit the paper.

Question 1005

[2 marks]complex numbers

Find

\dfrac{u}{w}

for

u=-\dfrac{29}{25}-\dfrac{3}{25}i

w=2i

, in the form

a+bi

Answer this when you sit the paper.

Question 1101

[1 marks]trapezium rule / volumes of revolution

Use the trapezium rule with 4 equal intervals to find the approximate value of

\displaystyle\int_0^{\pi/4}(1 + \sin 2x)\,dx

, correct to 3 decimal places.

A $0.639$
B $1.279$
C $1.285$
D $2.558$

Question 1102

[1 marks]trapezium rule / volumes of revolution

The region

R_1

between

y = 1

and

y = 1 + \sin 2x

from

x = 0

x = \tfrac{1}{4}\pi

is rotated through

2\pi

radians about the

x

-axis. Find the volume generated.

A $2\pi + \dfrac{\pi^2}{8}$
B $\pi + \dfrac{\pi^2}{4}$
C $\pi + \dfrac{\pi^2}{8}$
D $\dfrac{\pi^2}{8}$

Question 1103

[1 marks]trapezium rule / volumes of revolution

Evaluate

y=1+\sin(2x)

x=\pi/8

, to 5 decimal places.

Answer this when you sit the paper.

Question 1104

[2 marks]trapezium rule / volumes of revolution

Evaluate

\displaystyle\int_0^{\pi/4}2\sin2x\,dx

Answer this when you sit the paper.

Question 1105

[3 marks]trapezium rule / volumes of revolution

Evaluate

\displaystyle\int_0^{\pi/4}\sin^2(2x)\,dx

, using

\sin^2(2x)=\frac{1-\cos4x}{2}

, exact value in terms of

\pi

Answer this when you sit the paper.

Question 1106

[1 marks]trapezium rule / volumes of revolution

Evaluate

y=1+\sin(2x)

x=3\pi/16

, to 5 decimal places.

Answer this when you sit the paper.

Question 1201

[1 marks]Newton-Raphson / trigonometry

Let

f(x) = 2x - \tan x

. Which pair of values shows that the smallest positive root of

2x - \tan x = 0

lies between

x = 1

and

x = 1.5

A $f(1) = 0.443$ and $f(1.5) = 1.899$
B $f(1) = 0.443$ and $f(1.5) = -11.101$
C $f(1) = 1.557$ and $f(1.5) = 14.101$
D $f(1) = -0.443$ and $f(1.5) = 11.101$

Question 1202

[1 marks]Newton-Raphson / trigonometry

Starting with

x_0 = 1

, the Newton-Raphson method is applied to

f(x) = 2x - \tan x

. The smallest positive root, correct to 3 decimal places, is

A $1.000$
B $1.166$
C $1.318$
D $1.571$

Question 1203

[2 marks]Newton-Raphson / trigonometry

Find

f'(x)

for

f(x)=2x-\tan x

Answer this when you sit the paper.

Question 1204

[3 marks]Newton-Raphson / trigonometry

Evaluate

f'(1)=2-\sec^2(1)

, to 4 decimal places.

Answer this when you sit the paper.

Question 1205

[3 marks]Newton-Raphson / trigonometry

Using

x_1=x_0-\dfrac{f(x_0)}{f'(x_0)}

with

x_0=1

f(1)\approx0.4426

f'(1)\approx-1.4255

, calculate

x_1

, to 3 decimal places.

Answer this when you sit the paper.

Question 1301

[1 marks]trigonometric equations / R-cos form

The equation

3\sin x + 5\cos x \cot x - 4 = 0

can be written in the form

a\sin^2 x + b\sin x + c = 0

. Find

a

b

and

c

A $a = 3$ , $b = -4$ , $c = 5$
B $a = 2$ , $b = -4$ , $c = -5$
C $a = 2$ , $b = -4$ , $c = 5$
D $a = 2$ , $b = 4$ , $c = -5$

Question 1302

[1 marks]trigonometric equations / R-cos form

Express

5\cos x + 6\sin x

in the form

R\cos(x - \alpha)

, where

R > 0

and

0 < \alpha < 90°

A $\sqrt{61}\cos(x - 50.2°)$
B $\sqrt{61}\cos(x - 39.8°)$
C $11\cos(x - 50.2°)$
D $\sqrt{11}\cos(x - 50.2°)$

Question 1303

[1 marks]trigonometric equations / R-cos form

State the maximum and minimum values of

5\cos x + 6\sin x

Amaximum $6$ , minimum $-6$
Bmaximum $\sqrt{61}$ , minimum $-\sqrt{61}$
Cmaximum $61$ , minimum $0$
Dmaximum $11$ , minimum $-11$

Question 1304

[2 marks]trigonometric equations / R-cos form

Solve

2\sin^2x+4\sin x-5=0

using the quadratic formula, giving the valid root for

\sin x

(the other root lies outside

[-1,1]

), to 4 decimal places.

Answer this when you sit the paper.

Question 1305

[2 marks]trigonometric equations / R-cos form

Hence solve

\sin x=0.8708

for

0\le x\le360°

, giving the first (smaller) solution, to 1 decimal place.

Answer this when you sit the paper.

Question 1306

[1 marks]trigonometric equations / R-cos form

Find

R=\sqrt{5^2+6^2}

for

5\cos x+6\sin x=R\cos(x-\alpha)

, exact value.

Answer this when you sit the paper.

Question 1307

[2 marks]trigonometric equations / R-cos form

Find

\alpha=\tan^{-1}\left(\dfrac65\right)

, to 1 decimal place (in degrees).

Answer this when you sit the paper.

Question 1308

[3 marks]trigonometric equations / R-cos form

Solve

\cos(x-50.2°)=\dfrac{4}{\sqrt{61}}

for

0\le x\le360°

, giving the second (larger) solution, to 1 decimal place.

Answer this when you sit the paper.

Question 1401

[1 marks]differential equations

Find the solution of the differential equation

x\dfrac{dy}{dx} = y + yx

given that the curve passes through

(2, 4)

A $y = 2xe^{x-2}$
B $y = 4e^{x-2}$
C $y = 2xe^{2-x}$
D $y = xe^{x} + 2$

Question 1402

[1 marks]differential equations

A cuboid tank of base area

4

^2

and height

3

m is initially empty. Water is poured in at

0.05

^3

per minute and leaks out at a constant

0.025

^3

per minute, so that

\dfrac{dh}{dt} = \dfrac{1}{160}

. When does the tank start to overflow?

A $480$ minutes
B $160$ minutes
C $240$ minutes
D $320$ minutes

Question 1403

[1 marks]differential equations

In a refined model the depth

h

metres of water in an initially empty tank satisfies

160\dfrac{dh}{dt} = 2 - h

. Express

h

in terms of

t

A $h = 2 + e^{-t/160}$
B $h = 3\left(1 - e^{-t/160}\right)$
C $h = 2e^{-t/160}$
D $h = 2\left(1 - e^{-t/160}\right)$

Question 1404

[1 marks]differential equations

Separate variables in

x\dfrac{dy}{dx}=y(1+x)

to write the right-hand side of

\dfrac{dy}{y}=\ldots\,dx

, in terms of

x

Answer this when you sit the paper.

Question 1405

[2 marks]differential equations

Integrating gives

y=Axe^x

. Using the point

(2,4)

, find the constant

A

Answer this when you sit the paper.

Question 1406

[1 marks]differential equations

For the simple leak model, find the net inflow rate (inflow minus outflow), in m³/min.

Answer this when you sit the paper.

Question 1407

[2 marks]differential equations

Using

V=4h

and net inflow rate

0.025

m³/min, write and simplify

\dfrac{dh}{dt}

Answer this when you sit the paper.

Question 1408

[2 marks]differential equations

Separate variables in

160\dfrac{dh}{dt}=2-h

to write

\int\frac{dh}{2-h}=\int\frac{dt}{160}

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

Answer this when you sit the paper.

Question 1409

[2 marks]differential equations

Using

h=0

t=0

, find the constant of integration

C

-\ln(2-h)=\dfrac{t}{160}+C

Answer this when you sit the paper.

Question 1410

[2 marks]differential equations

In the refined model, why does the tank never actually overflow (reach

h=3

m)?

ABecause the tank's cross-sectional area increases as $h$ increases, preventing $h$ from reaching $3$ m
BBecause $h=2(1-e^{-t/160})$ approaches the asymptote $h=2$ , which is below the tank's height of $3$ m
CBecause the inflow rate $0.05$ m³/min is too small to ever fill a $12$ m³ tank
DBecause the leak rate increases without bound as $h$ increases, always draining faster than the tank fills

Question 1411

[2 marks]differential equations

In the refined model, evaluate

h

t=160

minutes, to 2 decimal places.

Answer this when you sit the paper.

Pure Mathematics Paper 1 November 2011

Sit this paper online

The questions

Question 101

Question 102

Question 103

Question 104

Question 201

Question 202

Question 203

Question 204

Question 205

Question 301

Question 302

Question 303

Question 304

Question 305

Question 401

Question 402

Question 403

Question 404

Question 405

Question 501

Question 502

Question 503

Question 504

Question 505

Question 506

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 806

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 1106

Question 1201

Question 1202

Question 1203

Question 1204

Question 1205

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