ZIMSEC A Level Pure Mathematics Paper 1 November 2007

Question 101

[1 marks]algebra / polynomial identity

In the identity

2x^4+6x^3+7x^2+15x+5 \equiv (x^2+3x+1)(ax^2+bx+c)

, what is the value of

b

A $b=6$
B $b=2$
C $b=-6$
D $b=0$

Question 102

[1 marks]algebra / polynomial identity

In the identity

2x^4+6x^3+7x^2+15x+5 \equiv (x^2+3x+1)(ax^2+bx+c)

, what are the values of

a

b

c

respectively?

A $a=1,\ b=3,\ c=1$
B $a=2,\ b=6,\ c=5$
C $a=2,\ b=0,\ c=5$
D $a=2,\ b=0,\ c=-5$

Question 103

[1 marks]algebra / polynomial identity

Matching the coefficient of

x^2

2x^4+6x^3+7x^2+15x+5 \equiv (x^2+3x+1)(ax^2+bx+c)

gives the equation

c+3b+a=k

. State

k

Answer this when you sit the paper.

Question 201

[1 marks]differential equations

Solving

x^2\dfrac{dy}{dx}=y

by separating variables gives

\ln y = -\dfrac{1}{x} + C

. Given

y=e^2

when

x=1

, what is the value of

C

A $C=2$
B $C=3$
C $C=-3$
D $C=1$

Question 202

[1 marks]differential equations

What is the particular solution

y

, in terms of

x

, of

x^2\dfrac{dy}{dx}=y

given that

y=e^2

when

x=1

A $y=e^{-1/x}+3$
B $y=3e^{-1/x}$
C $y=e^{3-1/x}$
D $y=e^{3+1/x}$

Question 203

[2 marks]differential equations

Separating variables in

x^2\dfrac{dy}{dx}=y

gives

\dfrac{dy}{y}=\dfrac{dx}{x^2}

. Integrate the right-hand side,

\int \dfrac{dx}{x^2}

(omit the constant of integration).

Answer this when you sit the paper.

Question 301

[1 marks]inequalities / absolute value

What is the solution set of

|2x-3| < x+1

A $\frac{2}{3} < x < \frac{3}{2}$
B $\frac{2}{3} < x < 4$
C $\frac{3}{2} < x < 4$
D $-4 < x < \frac{2}{3}$

Question 302

[1 marks]inequalities / absolute value

When solving

|2x-3|<x+1

for the branch

x<\frac32

, the inequality becomes

-(2x-3)<x+1

. At what value of

x

do the two sides become equal (the boundary of this branch)?

A $x=\frac32$
B $x=4$
C $x=\frac23$
D $x=-\frac23$

Question 303

[2 marks]inequalities / absolute value

For the branch

x\geq\frac32

|2x-3|<x+1

, solve

2x-3=x+1

to find the boundary value of

x

for this branch.

Answer this when you sit the paper.

Question 401

[1 marks]complex numbers

Given

\dfrac{3+2i}{2+ai}=x+iy

with

x,y

real and

x=y

, what is the value of

a

A $a=\frac{2}{5}$
B $a=\frac{4}{3}$
C $a=-\frac{2}{3}$
D $a=-\frac{2}{5}$

Question 402

[2 marks]complex numbers

Expanding

(3+2i)(2-ai)

, state the real part of the result, in terms of

a

Answer this when you sit the paper.

Question 403

[2 marks]complex numbers

Expanding

(3+2i)(2-ai)

, state the coefficient of

i

(the imaginary part), in terms of

a

Answer this when you sit the paper.

Question 501

[1 marks]functions / inverse functions

For

f:x\mapsto x^2+4x+1

x\geq -2

, what is the range of

f

A $f(x)\geq -2$
B $f(x)\leq -3$
C $f(x)\geq -3$
D $f(x)\geq 1$

Question 502

[1 marks]functions / inverse functions

For the same function

f:x\mapsto x^2+4x+1

x\geq -2

, which correctly gives

f^{-1}(x)

together with its domain?

A $f^{-1}(x)=\sqrt{x+3}-2$ , domain $x\geq -3$
B $f^{-1}(x)=\sqrt{x+3}-2$ , domain $x\geq -2$
C $f^{-1}(x)=\sqrt{x-3}-2$ , domain $x\geq -3$
D $f^{-1}(x)=-\sqrt{x+3}-2$ , domain $x\geq -3$

Question 503

[2 marks]functions / inverse functions

Write

f(x)=x^2+4x+1

in completed-square form

(x+p)^2+q

. State

p

and

q

as 'p, q'.

Answer this when you sit the paper.

Question 504

[2 marks]functions / inverse functions

Given

y=(x+2)^2-3

, make

x+2

the subject (taking the positive square root since

x\geq-2

). State the resulting expression for

x+2

in terms of

y

Answer this when you sit the paper.

Question 601

[1 marks]calculus / volumes of revolution

The region bounded by

x^2=y-1

x=0

y=0

and

x=1

is rotated completely about the

x

-axis. Which integral gives the volume of the solid formed?

A $\pi\int_0^1 (x^2-1)^2\,dx$
B $\pi\int_0^1 (x^2+1)\,dx$
C $\pi\int_0^1 (x^2+1)^2\,dx$
D $2\pi\int_0^1 x(x^2+1)\,dx$

Question 602

[1 marks]calculus / volumes of revolution

What is the volume of the solid generated when the region bounded by

x^2=y-1

x=0

y=0

and

x=1

is rotated completely about the

x

-axis?

A $\frac{56\pi}{15}$
B $\frac{13\pi}{15}$
C $\frac{4\pi}{3}$
D $\frac{28\pi}{15}$

Question 603

[2 marks]calculus / volumes of revolution

Expand

(x^2+1)^2

as a polynomial in

x

Answer this when you sit the paper.

Question 604

[2 marks]calculus / volumes of revolution

Evaluate

\int_0^1 (x^4+2x^2+1)\,dx

as a single fraction (before multiplying by

\pi

Answer this when you sit the paper.

Question 701

[1 marks]calculus / small changes and errors

For a cylindrical wire, an error

\delta x

in the measured diameter produces an error

\delta V

in the calculated volume. Approximately how many times the relative error in the diameter is the relative error in the volume?

Answer this when you sit the paper.

Question 702

[1 marks]calculus / small changes and errors

A cylindrical wire of fixed length

l

and diameter

x

has volume

V=\dfrac{\pi l x^2}{4}

. Which of these correctly relates the error

\delta V

to an error

\delta x

in the diameter?

A $\delta V \approx \dfrac{\pi l x}{2}\delta x$
B $\delta V \approx \dfrac{\pi l x^2}{4}\delta x$
C $\delta V \approx \dfrac{\pi l x}{4}\delta x$
D $\delta V \approx \pi l x\,\delta x$

Question 703

[2 marks]calculus / small changes and errors

For

V=\dfrac{\pi l x^2}{4}

(volume of a cylindrical wire in terms of diameter

x

), find

\dfrac{dV}{dx}

in terms of

l

and

x

Answer this when you sit the paper.

Question 704

[2 marks]calculus / small changes and errors

Simplify the ratio

\dfrac{\pi l x/2}{\pi l x^2/4}

to an expression in

x

only.

Answer this when you sit the paper.

Question 1301

[1 marks]arithmetic series / sequences

Ten oranges

O_1,\ldots,O_{10}

are placed in a line 6 m apart, with

O_1

15 m from the starting line. A contestant starts at the line, runs to fetch each orange one at a time in order, returning it to a box at the start before fetching the next. What is the total distance covered to collect all 10 oranges?

A540 m
B840 m
C420 m
D960 m

Question 1302

[1 marks]arithmetic series / sequences

Using the same set-up (oranges 6 m apart,

O_1

15 m from the start, each fetched and returned to the box before the next), what round-trip distance, in metres, does the contestant cover to fetch

O_5

alone?

Answer this when you sit the paper.

Question 1303

[2 marks]arithmetic series / sequences

Ten oranges are placed 6 m apart with

O_1

15 m from the start. The round-trip distance to collect orange

O_n

alone is

2(15+6(n-1))

m. Evaluate this for

n=10

(the last orange).

Answer this when you sit the paper.

Question 1401

[1 marks]differential equations / exponential decay

The mass

m

of mealie-meal satisfies

\dfrac{dm}{dt}=-\dfrac{k}{5}m

, with

m=m_0

t=0

. What is the general solution?

A $m=m_0e^{-kt/5}$
B $m=m_0e^{-kt}$
C $m=m_0-\dfrac{k}{5}t$
D $m=m_0e^{-5kt}$

Question 1402

[1 marks]differential equations / exponential decay

Modelling mealie-meal loss by

m=m_0e^{-kt/5}

, if 10% is lost in the first hour (so

m=0.9m_0

t=1

), what percentage of the mealie-meal remains after the girl's full 2-hour journey?

A72.9%
B80%
C81%
D90%

Question 1403

[2 marks]differential equations / exponential decay

Separating variables in

\dfrac{dm}{dt}=-\dfrac{k}{5}m

gives

\dfrac{dm}{m}=-\dfrac{k}{5}dt

. Integrate the left-hand side,

\int\dfrac{dm}{m}

(omit the constant of integration).

Answer this when you sit the paper.

Question 1404

[2 marks]differential equations / exponential decay

Given that after

t=1

hour, 10% of the mealie-meal is lost (so

m=0.9m_0

), and

m=m_0e^{-kt/5}

, state the value of

e^{-k/5}

Answer this when you sit the paper.

Question 1405

[2 marks]differential equations / exponential decay

Given

e^{-k/5}=0.9

and that after 2 hours the mass is

m_0(e^{-k/5})^2

, what percentage of the mealie-meal is LOST during the full 2-hour journey (not the percentage remaining)?

Answer this when you sit the paper.

Question 1406

[2 marks]differential equations / exponential decay

Using

\ln m = -\dfrac{k}{5}t + C

and

m=m_0

t=0

, find

C

in terms of

m_0

Answer this when you sit the paper.

Question 1407

[2 marks]differential equations / exponential decay

Given

e^{-k/5}=0.9

, taking natural logs gives

-\dfrac{k}{5}=\ln0.9

, so

k=m\ln0.9

for some constant

m

. State

m

Answer this when you sit the paper.

Question 1501

[1 marks]logarithms / series expansion

Which expression is equal to

\left(1-\dfrac{1}{n^2}\right)^{-1}

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

Question 1502

[1 marks]logarithms / series expansion

What are the first three terms of the series expansion of

\ln(1-x)

in ascending powers of

x

A $-x-\dfrac{x^2}{2}-\dfrac{x^3}{3}$
B $x-\dfrac{x^2}{2}+\dfrac{x^3}{3}$
C $-x-\dfrac{x^2}{2}-\dfrac{x^3}{6}$
D $-x+\dfrac{x^2}{2}-\dfrac{x^3}{3}$

Question 1503

[1 marks]logarithms / series expansion

Using

\ln 10=2.3025851

\ln 3=1.0986123

, and

n=10

\ln\left(\dfrac{n^2}{n^2-1}\right)\approx \dfrac{1}{n^2}+\dfrac{1}{2n^4}+\dfrac{1}{3n^6}

together with

\ln\left(\dfrac{n^2}{n^2-1}\right)=2\ln n-\ln(n+1)-\ln(n-1)

, what is

\ln 11

to six decimal places?

A2.397895
B2.397946
C2.407946
D2.417996

Question 1504

[2 marks]logarithms / series expansion

Substituting

x=1/n^2

into

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

, state the first (leading) term of the expansion of

\ln\left(1-\dfrac1{n^2}\right)

, in terms of

n

Answer this when you sit the paper.

Question 1505

[2 marks]logarithms / series expansion

The expansion of

\ln\left(1-\frac1{n^2}\right)^{-1}

is the negative of the expansion of

\ln\left(1-\frac1{n^2}\right)

. State its leading (first) term, in terms of

n

Answer this when you sit the paper.

Question 1506

[2 marks]logarithms / series expansion

Using

\ln 10=2.3025851

, evaluate

2\ln 10

, correct to 7 decimal places.

Answer this when you sit the paper.

Question 1507

[2 marks]logarithms / series expansion

Using

\ln 3=1.0986123

, evaluate

2\ln 3

(needed since

\ln 9 = 2\ln 3

), correct to 7 decimal places.

Answer this when you sit the paper.

Question 1508

[1 marks]logarithms / series expansion

Using

n=10

, evaluate

\ln\left(\dfrac{100}{99}\right)

approximately using

\dfrac1{n^2}+\dfrac1{2n^4}+\dfrac1{3n^6}

, correct to 6 decimal places.

Answer this when you sit the paper.

Question 1601

[1 marks]coordinate geometry / circles

The circle through

(-2,-4)

(3,1)

and

(-2,0)

has centre

(1,-2)

. Using the chord joining

(-2,-4)

and

(-2,0)

, which is vertical, what is the equation of its perpendicular bisector (the locus on which the centre must lie)?

A $y=-4$
B $y=-2$
C $x=1$
D $x=-2$

Question 1602

[1 marks]coordinate geometry / circles

The circle

(x-1)^2+(y+2)^2=13

has centre

(1,-2)

. What is the equation, in the form

ax+by+c=0

, of the diameter through the point

(-2,0)

A $3x+2y+4=0$
B $2x+3y-4=0$
C $2x-3y+4=0$
D $2x+3y+4=0$

Question 1603

[1 marks]coordinate geometry / circles

What is the gradient of the tangent to the circle

(x-1)^2+(y+2)^2=13

at the point

(3,1)

A $-\dfrac32$
B $-\dfrac23$
C $\dfrac32$
D $\dfrac23$

Question 1604

[2 marks]coordinate geometry / circles

The circle

(x-1)^2+(y+2)^2=13

has centre

(1,-2)

and passes through

(3,1)

. State the gradient of the radius from the centre to

(3,1)

Answer this when you sit the paper.

Question 1605

[2 marks]coordinate geometry / circles

The diameter of the circle

(x-1)^2+(y+2)^2=13

(centre

(1,-2)

) passes through the point

(-2,0)

. State its gradient.

Answer this when you sit the paper.

Question 1606

[2 marks]coordinate geometry / circles

Find the distance between the two given points

(-2,-4)

and

(-2,0)

(a chord of the circle

(x-1)^2+(y+2)^2=13

Answer this when you sit the paper.

Question 1607

[2 marks]coordinate geometry / circles

State the gradient of the chord joining the two given points

(-2,-4)

and

(3,1)

on the circle (not a diameter).

Answer this when you sit the paper.

Question 1608

[2 marks]coordinate geometry / circles

State the midpoint of the chord joining

(-2,-4)

and

(3,1)

, as 'x, y'.

Answer this when you sit the paper.

Pure Mathematics Paper 1 November 2007

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 504

Question 601

Question 602

Question 603

Question 604

Question 701

Question 702

Question 703

Question 704

Question 1301

Question 1302

Question 1303

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

Question 1602

Question 1603

Question 1604

Question 1605

Question 1606

Question 1607

Question 1608

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