ZIMSEC A Level Pure Mathematics Paper 1 June 2005

Question 101

[1 marks]coordinate geometry

The points A and B have coordinates

(m^2, 2m)

and

(3m^2, 6m)

respectively. What is the midpoint of AB, in terms of

m

A $(2m^2, 8m)$
B $(4m, 2m^2)$
C $(2m^2, 4m)$
D $(4m^2, 4m)$

Question 102

[1 marks]coordinate geometry

The points A and B have coordinates

(m^2, 2m)

and

(3m^2, 6m)

respectively, so the midpoint of AB is

(2m^2, 4m)

. Find the values of

m

for which this midpoint lies on the line

y = 2x - 8

A $m = 2$ or $m = -1$
B $m = 1$ or $m = -1$
C $m = 2$ or $m = 1$
D $m = -2$ or $m = 1$

Question 103

[1 marks]coordinate geometry

Substituting the midpoint

(2m^2,4m)

of AB into

y=2x-8

gives

4m=4m^2-8

, which simplifies to

m^2-m-c=0

. Find the value of

c

Answer this when you sit the paper.

Question 201

[1 marks]binomial expansion

In the expansion of

(1+2x)^4

in ascending powers of

x

, what is the coefficient of the

x^2

term?

A $8$
B $16$
C $24$
D $32$

Question 202

[1 marks]binomial expansion

Given that

(1+2x)^4 = 1+8x+24x^2+32x^3+16x^4

, find the coefficient of the

x^3

term in the expansion of

(2x+3)(1+2x)^4

A $48$
B $96$
C $136$
D $144$

Question 203

[1 marks]binomial expansion

In the expansion of

(1+2x)^4

in ascending powers of

x

, what is the coefficient of the

x

term (the term in

x^1

Answer this when you sit the paper.

Question 204

[1 marks]binomial expansion

In finding the

x^3

coefficient of

(2x+3)(1+2x)^4

, one contribution comes from

2x\times24x^2

. What is the coefficient of

x^3

from this contribution alone?

Answer this when you sit the paper.

Question 301

[1 marks]inequalities

To solve

|x+3| > |2x-4|

by squaring both sides, the inequality reduces to which quadratic inequality?

A $3x^2-22x+7<0$
B $3x^2-14x+7<0$
C $3x^2-22x+7>0$
D $3x^2-22x-7<0$

Question 302

[1 marks]inequalities

Solve the inequality

|x+3| > |2x-4|

A $x<\tfrac13$ or $x>7$
B $\tfrac13<x<7$
C $-7<x<-\tfrac13$
D $\tfrac13<x<3$

Question 303

[2 marks]inequalities

Solving

3x^2-22x+7<0

requires the roots of

3x^2-22x+7=0

. State both roots.

Answer this when you sit the paper.

Question 401

[1 marks]trigonometry

Express

\cos 2x + 2\sin 2x

in the form

R\cos(2x-\alpha)

, where

R

is exact and

\alpha

is acute. What are

R

and

\alpha

(to 2 dp)?

A $R=\sqrt5,\ \alpha=26.57°$
B $R=3,\ \alpha=63.43°$
C $R=\sqrt5,\ \alpha=63.43°$
D $R=\sqrt5,\ \alpha=45°$

Question 402

[1 marks]trigonometry

Using

\cos 2x + 2\sin 2x = \sqrt5\cos(2x-63.43°)

, which of the following is a value of

x

satisfying

\cos 2x + 2\sin 2x = 1

for

0° \le x \le 360°

A $90°$
B $270°$
C $180°$
D $45°$

Question 403

[2 marks]trigonometry

Using

\cos2x+2\sin2x=\sqrt5\cos(2x-63.43°)

, solve

\cos2x+2\sin2x=1

for

0°<x\leq180°

, giving the value of

x

(other than

x=0°

x=180°

) correct to 1 decimal place.

Answer this when you sit the paper.

Question 404

[2 marks]trigonometry

Using

\cos2x+2\sin2x=\sqrt5\cos(2x-63.43°)

, solve

\cos2x+2\sin2x=1

for

180°<x\leq360°

, giving the value of

x

(other than

x=180°

x=360°

) correct to 1 decimal place.

Answer this when you sit the paper.

Question 501

[1 marks]algebra and functions

Given that

\dfrac{4}{x} = 3 - \sqrt5

, find the exact value of

x

in simplified surd form.

A $\dfrac{6+2\sqrt5}{7}$
B $3+\sqrt5$
C $3-\sqrt5$
D $-3-\sqrt5$

Question 502

[1 marks]algebra and functions

Given that

f(x) = e^{2x-3}

for all real

x

, find an expression for

f^{-1}(x)

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

Question 503

[1 marks]algebra and functions

For

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

, where

f(x) = e^{2x-3}

, what is the domain of

f^{-1}(x)

A $x \in \mathbb{R}$
B $x \geq 0$
C $x > -3$
D $x > 0$

Question 504

[2 marks]algebra and functions

Given

y=e^{2x-3}

, taking natural logarithms of both sides gives

\ln y=

___, in terms of

x

Answer this when you sit the paper.

Question 505

[1 marks]algebra and functions

To find

x=\dfrac{4}{3-\sqrt5}

in surd form, the denominator is rationalised by multiplying by

\dfrac{3+\sqrt5}{3+\sqrt5}

. What does the denominator simplify to?

Answer this when you sit the paper.

Question 506

[1 marks]algebra and functions

Which transformation maps the graph of

y=e^x

onto the graph of

y=e^{2x-3}

AStretch horizontally, scale factor $\tfrac12$ , then translate left by $\tfrac32$ units
BStretch horizontally, scale factor $\tfrac12$ , then translate right by $\tfrac32$ units
CTranslate right by 3 units, then stretch vertically, scale factor 2
DStretch horizontally, scale factor 2, then translate right by 3 units

Question 601

[1 marks]polynomials

Given that

(x+2)

and

(2x-1)

are factors of

f(x) = 2x^3 + ax^2 + bx + 6

, find the value of

a

A $-11$
B $-5$
C $-3$
D $3$

Question 602

[1 marks]polynomials

Given that

(x+2)

and

(2x-1)

are factors of

f(x) = 2x^3 + ax^2 + bx + 6

, and that

a=-3

, find the value of

b

A $-11$
B $-3$
C $2$
D $11$

Question 603

[1 marks]polynomials

Given that

f(x) = 2x^3 - 3x^2 - 11x + 6

and that

(x+2)

and

(2x-1)

are two of its factors, find the third factor.

A $(x-3)$
B $(2x-3)$
C $(x-6)$
D $(x+3)$

Question 604

[1 marks]polynomials

Given

f(x)=2x^3+ax^2+bx+6

and that

(x+2)

is a factor, applying the factor theorem (

f(-2)=0

) leads to which simplified equation relating

a

and

b

A $4a-2b=-10$
B $2a+b=5$
C $-2a+b=5$
D $2a-b=5$

Question 605

[3 marks]polynomials

Given

f(x)=2x^3+ax^2+bx+6

and that

(2x-1)

is a factor, applying the factor theorem with

f\left(\tfrac12\right)=0

and clearing fractions leads to which simplified equation relating

a

and

b

A $a+2b=-25$
B $a+2b=25$
C $2a+b=-25$
D $a-2b=-25$

Question 701

[1 marks]calculus/parametric differentiation

Given that

x = a\cos\theta + b\sin\theta

and

y = a\sin\theta - b\cos\theta

, find

\dfrac{dy}{dx}

\theta = \dfrac{\pi}{2}

A $\dfrac{b}{a}$
B $-\dfrac{a}{b}$
C $-\dfrac{b}{a}$
D $\dfrac{a}{b}$

Question 702

[1 marks]calculus/parametric differentiation

Given that

x = a\cos\theta + b\sin\theta

and

y = a\sin\theta - b\cos\theta

, the expression

x^2+y^2

simplifies to which of the following, independent of

\theta

A $(a-b)^2$
B $2(a^2+b^2)$
C $a^2-b^2$
D $a^2+b^2$

Question 703

[2 marks]calculus/parametric differentiation

Given

x=a\cos\theta+b\sin\theta

, find

\dfrac{dx}{d\theta}

Answer this when you sit the paper.

Question 704

[2 marks]calculus/parametric differentiation

Given

y=a\sin\theta-b\cos\theta

, find

\dfrac{dy}{d\theta}

Answer this when you sit the paper.

Question 705

[1 marks]calculus/parametric differentiation

Which trigonometric identity is the key step used to show that

x^2+y^2=a^2+b^2

is independent of

\theta

A $\tan^2\theta+1=\sec^2\theta$
B $\sin\theta\cos\theta=\tfrac12\sin2\theta$
C $\sin^2\theta+\cos^2\theta=1$
D $\cos2\theta=1-2\sin^2\theta$

Question 801

[1 marks]sequences and series

An arithmetic progression has first term

-3

, last term

25

, and the sum of all its terms is

1837

. Find the number of terms,

n

A $157$
B $166$
C $167$
D $183$

Question 802

[1 marks]sequences and series

An arithmetic progression has

167

terms, with first term

-3

and last term

25

. Find the common difference.

A $\dfrac{14}{83}$
B $-\dfrac{14}{83}$
C $\dfrac{28}{83}$
D $\dfrac{28}{167}$

Question 803

[1 marks]sequences and series

In a geometric progression, the fifth term is

100

and the seventh term is

400

. Find the possible values of the second term.

A $\pm50$
B $\dfrac{25}{4}$
C $25$
D $\pm\dfrac{25}{2}$

Question 804

[2 marks]sequences and series

In the geometric progression, dividing

ar^6=400

ar^4=100

gives

r^2=4

. Find both possible values of

r

Answer this when you sit the paper.

Question 805

[2 marks]sequences and series

Given

r^2=4

and

ar^4=100

, find the value of the first term

a

of the geometric progression.

Answer this when you sit the paper.

Question 901

[1 marks]integration

Evaluate

\displaystyle\int_{0.4}^{0.5} xe^{3x}\,dx

, giving your answer correct to 3 significant figures.

A $0.0738$
B $0.175$
C $0.249$
D $0.323$

Question 902

[1 marks]integration

Use the trapezium rule with 4 ordinates to obtain an approximate value for

\displaystyle\int_1^4 x^3 \ln x\,dx

A $34.5$
B $62.0$
C $79.6$
D $159.1$

Question 903

[1 marks]integration

Integrating by parts,

\int xe^{3x}\,dx=\dfrac{xe^{3x}}{3}-\dfrac{e^{3x}}{k}+C

. Find the value of

k

Answer this when you sit the paper.

Question 904

[2 marks]integration

For the trapezium rule estimate of

\int_1^4 x^3\ln x\,dx

with ordinates at

x=1,2,3,4

, find the ordinate

f(3)=27\ln3

correct to 3 significant figures.

Answer this when you sit the paper.

Question 905

[2 marks]integration

For the trapezium rule estimate of

\int_1^4 x^3\ln x\,dx

with ordinates at

x=1,2,3,4

, find the ordinate

f(2)=8\ln2

correct to 3 significant figures.

Answer this when you sit the paper.

Question 1001

[1 marks]numerical methods

Let

g(x) = \ln(x+2) - 2x + 8

. What are the signs of

g(4)

and

g(5)

, confirming that a root of

\ln(x+2)=2x-8

lies between

x=4

and

x=5

A $g(4) > 0$ and $g(5) < 0$
B $g(4) > 0$ and $g(5) > 0$
C $g(4) < 0$ and $g(5) < 0$
D $g(4) < 0$ and $g(5) > 0$

Question 1002

[1 marks]numerical methods

Applying the Newton-Raphson method once to

g(x) = \ln(x+2) - 2x + 8

, starting with

x_1 = 5

, find the second approximation

x_2

, correct to 4 decimal places.

A $4.9708$
B $4.9709$
C $5.0291$
D $5.3786$

Question 1003

[2 marks]numerical methods

For

g(x)=\ln(x+2)-2x+8

, find

g'(x)

Answer this when you sit the paper.

Question 1004

[2 marks]numerical methods

For

g(x)=\ln(x+2)-2x+8

, find the exact value of

g'(5)

as a fraction.

Answer this when you sit the paper.

Question 1005

[2 marks]numerical methods

For

g(x)=\ln(x+2)-2x+8

, find

g(5)

correct to 4 decimal places.

Answer this when you sit the paper.

Question 1101

[1 marks]differential equations

Water enters a lake at a constant rate of

12

^3

/s. The volume of water is

V=kh^3

, where

h

is the depth at the dam wall. Starting from

\dfrac{dV}{dt}=3kh^2\dfrac{dh}{dt}=12

, find

\dfrac{dh}{dt}

in terms of

k

and

h

A $\dfrac{36}{kh^2}$
B $\dfrac{4}{kh^2}$
C $\dfrac{12}{kh^2}$
D $\dfrac{4}{kh}$

Question 1102

[1 marks]differential equations

Given

\dfrac{dh}{dt} = \dfrac{4}{kh^2}

, separate the variables and integrate to find the general solution (with

C

an arbitrary constant).

A $3kh^3 = 4t + C$
B $\dfrac{kh^3}{3} = 4t + C$
C $kh^3 = 4t + C$
D $\dfrac{kh^3}{3} = -4t + C$

Question 1103

[1 marks]differential equations

Using

\dfrac{kh^3}{3} = 4t + C

, given that the depth is

24

m when

t=0

, find in terms of

k

the time taken for the depth to increase to

30

A $274.5k$
B $2250k$
C $4392k$
D $1098k$

Question 1104

[2 marks]differential equations

Separating variables in

\dfrac{dh}{dt}=\dfrac{4}{kh^2}

gives

\dfrac{kh^3}{3}=4t+C

. Given that

h=24

when

t=0

, find

C

in terms of

k

Answer this when you sit the paper.

Question 1105

[2 marks]differential equations

Using

\dfrac{kh^3}{3}=4t+4608k

, find the value of

\dfrac{k(30)^3}{3}

, in terms of

k

Answer this when you sit the paper.

Question 1106

[2 marks]differential equations

Using

9000k=4t+4608k

, find

4t

, in terms of

k

Answer this when you sit the paper.

Question 1201

[1 marks]vectors

With respect to origin O, the position vectors of P and R are

13\mathbf{i}

and

5\mathbf{i}+12\mathbf{k}

respectively. The diagonal OQ bisects the diagonal PR at W. Find the position vector of W.

A $9\mathbf{i}$
B $9\mathbf{i} + 6\mathbf{k}$
C $9\mathbf{i} + 6\mathbf{j}$
D $18\mathbf{i} + 12\mathbf{k}$

Question 1202

[1 marks]vectors

With respect to origin O, W is the midpoint of PR with position vector

9\mathbf{i}+6\mathbf{k}

, and

\overrightarrow{OQ} = 3\overrightarrow{OW}

. Find the position vector of Q.

A $27\mathbf{i} + 6\mathbf{k}$
B $9\mathbf{i} + 18\mathbf{k}$
C $27\mathbf{i} + 18\mathbf{k}$
D $18\mathbf{i} + 12\mathbf{k}$

Question 1301

[1 marks]complex numbers

The complex numbers

Z_1

and

Z_2

satisfy

Z_1 Z_2 = 5 - 4i

. Find

|Z_1 Z_2|

A $\sqrt9$
B $41$
C $9$
D $\sqrt{41}$

Question 1302

[1 marks]complex numbers

Given that

Z_1 Z_2 = 5 - 4i

, find

\arg(Z_1 Z_2)

, correct to 2 decimal places.

A $141.34°$
B $-38.66°$
C $38.66°$
D $-51.34°$

Question 1303

[1 marks]complex numbers

The complex numbers

Z_1 = 1 + ai

and

Z_2 = -b - i

, where

a

and

b

are real and positive, satisfy

Z_1 Z_2 = 5 - 4i

. Find the exact values of

a

and

b

A $a=\dfrac{\sqrt{37}-5}{2},\ b=\dfrac{5+\sqrt{37}}{2}$
B $a=\dfrac{5-\sqrt{37}}{2},\ b=\dfrac{-5-\sqrt{37}}{2}$
C $a=\dfrac{5+\sqrt{37}}{2},\ b=\dfrac{\sqrt{37}-5}{2}$
D $a=\dfrac{5+\sqrt{41}}{2},\ b=\dfrac{\sqrt{41}-5}{2}$

Pure Mathematics Paper 1 June 2005

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The questions

Question 101

Question 102

Question 103

Question 201

Question 202

Question 203

Question 204

Question 301

Question 302

Question 303

Question 401

Question 402

Question 403

Question 404

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 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 1301

Question 1302

Question 1303

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