ZIMSEC A Level Pure Mathematics Paper 1 June 2019

Question 101

[1 marks]Coordinate geometry

Find, in terms of

k

, the equation of the perpendicular bisector of the line joining

A(2k, k)

and

B(4k, 9k)

A $4x + y = 17k$
B $x - 4y = -17k$
C $x + 4y = 23k$
D $4x - y = 7k$

Question 102

[1 marks]Coordinate geometry

Find the midpoint of

A(2k,k)

and

B(4k,9k)

, in terms of

k

Answer this when you sit the paper.

Question 103

[1 marks]Coordinate geometry

Find the gradient of the line

AB

joining

A(2k,k)

and

B(4k,9k)

Answer this when you sit the paper.

Question 201

[1 marks]Binomial expansion

Find the first three terms, in ascending powers of

t

, of the expansion of

(1-2t)^{-4}

A $1 + 8t + 40t^2$
B $1 + 8t + 20t^2$
C $1 + 4t + 10t^2$
D $1 - 8t + 40t^2$

Question 202

[1 marks]Binomial expansion

Find the coefficient of

t

in the expansion of

(1-2t)^{-4}

Answer this when you sit the paper.

Question 203

[1 marks]Binomial expansion

Find the coefficient of

t^2

in the expansion of

(1-2t)^{-4}

Answer this when you sit the paper.

Question 301

[1 marks]Modulus inequalities

Solve the inequality

|x-1| < 2x - 4

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

Question 302

[1 marks]Modulus inequalities

Solve the boundary equation

x-1=2x-4

for

x

, the point where the two sides of

|x-1|<2x-4

are equal.

Answer this when you sit the paper.

Question 303

[1 marks]Modulus inequalities

Solving

|x-1| < 2x-4

for the case

x<1

leads to

1-x<2x-4

. Solve this inequality for

x

, stating the boundary value.

Answer this when you sit the paper.

Question 304

[1 marks]Modulus inequalities

Since

|x-1|\ge0

, the right-hand side

2x-4

must also be positive. Solve

2x-4=0

to find this boundary value of

x

Answer this when you sit the paper.

Question 401

[1 marks]Factorisation of polynomials

Factorise completely

4 + 5x - 7x^2 - 2x^3

A $(1-x)(2x+1)(x+4)$
B $(x-1)(2x+1)(x+4)$
C $(1-x)(2x-1)(x+4)$
D $(1-x)(2x+1)(x-4)$

Question 402

[1 marks]Factorisation of polynomials

For

f(x)=4+5x-7x^2-2x^3

, evaluate

f(1)

to confirm

(x-1)

is a factor.

Answer this when you sit the paper.

Question 403

[2 marks]Factorisation of polynomials

Divide

4+5x-7x^2-2x^3

(x-1)

to find the quotient, in the form

ax^2+bx+c

Answer this when you sit the paper.

Question 501

[1 marks]Exponential equations

The equation

2(2^{-x}) + 2^{x+2} = 10

becomes which quadratic in

y = 2^x

A $2y^2 - 10y + 4 = 0$
B $y^2 - 10y + 4 = 0$
C $4y^2 + 10y - 2 = 0$
D $4y^2 - 10y + 2 = 0$

Question 502

[1 marks]Exponential equations

Solve

2(2^{-x}) + 2^{x+2} = 10

, giving your answers correct to 2 significant figures.

A $x = 1.2$ or $x = 2.2$
B $x = 1.2$ or $x = -2.2$
C $x = -1.2$ or $x = 2.2$
D $x = 0.6$ or $x = -1.1$

Question 503

[2 marks]Exponential equations

Solve

4y^2-10y+2=0

for

y

using the quadratic formula. State both values, correct to 2 decimal places.

Answer this when you sit the paper.

Question 504

[1 marks]Exponential equations

State the discriminant of the quadratic

4y^2-10y+2=0

Answer this when you sit the paper.

Question 601

[1 marks]Differentiation (quotient rule)

Given

y = \dfrac{\sin\theta}{2-\cos\theta}

for

0 \le \theta \le 2\pi

, find the values of

\theta

for which

y

is stationary.

A $\theta = \tfrac{\pi}{3}$ and $\theta = \tfrac{5\pi}{3}$
B $\theta = \tfrac{2\pi}{3}$ and $\theta = \tfrac{4\pi}{3}$
C $\theta = \tfrac{\pi}{6}$ and $\theta = \tfrac{11\pi}{6}$
D $\theta = 0$ and $\theta = \pi$

Question 602

[2 marks]Differentiation (quotient rule)

Given

y=\dfrac{\sin\theta}{2-\cos\theta}

, use the quotient rule to find

\dfrac{dy}{d\theta}

, in terms of

\theta

Answer this when you sit the paper.

Question 603

[1 marks]Differentiation (quotient rule)

Setting the numerator of

\dfrac{dy}{d\theta}

to zero, solve

2\cos\theta-1=0

for

\cos\theta

Answer this when you sit the paper.

Question 604

[1 marks]Differentiation (quotient rule)

Evaluate

(2-\cos\theta)^2

when

\cos\theta=\tfrac12

, the denominator of

\dfrac{dy}{d\theta}

at the stationary point.

Answer this when you sit the paper.

Question 701

[1 marks]Functions and inverses

Given

f: x \mapsto e^x

and

h: x \mapsto x + 2

, describe the graph of

fh(x)

A $y = e^x$ translated 2 units in the negative $x$ -direction
B $y = e^x$ translated 2 units in the positive $x$ -direction
C $y = e^x$ translated 2 units upwards
D $y = e^x$ stretched vertically by factor 2

Question 702

[1 marks]Functions and inverses

The function

g: x \mapsto x^2 - 2x

is defined for

0 \le x \le 1

. Find

g^{-1}(x)

A $g^{-1}(x) = \sqrt{x+1} - 1$
B $g^{-1}(x) = 1 - \sqrt{x+1}$
C $g^{-1}(x) = 1 + \sqrt{x+1}$
D $g^{-1}(x) = \sqrt{x} + 1$

Question 703

[1 marks]Functions and inverses

Write

g(x)=x^2-2x

in completed-square form

a(x-h)^2+k

Answer this when you sit the paper.

Question 704

[1 marks]Functions and inverses

The inverse

g^{-1}(x)=1-\sqrt{x+1}

has domain

x\ge

what value?

Answer this when you sit the paper.

Question 705

[1 marks]Functions and inverses

Given

f:x\mapsto e^x

and

h:x\mapsto x+2

, find

fh(x)

as a single expression.

Answer this when you sit the paper.

Question 706

[1 marks]Functions and inverses

Find the value of

fh(x)=e^{x+2}

x=-2

Answer this when you sit the paper.

Question 801

[1 marks]Arithmetic and geometric progressions

Given that

\sin^2 x

\cos^2 x

and

5\cos^2 x - 3

are consecutive terms of an arithmetic progression, find

\cos^2 x

A $\tfrac{1}{2}$
B $0$
C $\tfrac{3}{4}$
D $1$

Question 802

[1 marks]Arithmetic and geometric progressions

A geometric progression has sum to infinity 500 and common ratio

0.8

. Find the first term.

A $80$
B $100$
C $400$
D $625$

Question 803

[1 marks]Arithmetic and geometric progressions

A geometric progression has first term 100 and common ratio

0.8

. Find the least number of terms whose sum exceeds 499.

A $27$
B $28$
C $29$
D $30$

Question 804

[1 marks]Arithmetic and geometric progressions

Given

\cos^2x=1

, evaluate the third AP term

5\cos^2x-3

Answer this when you sit the paper.

Question 805

[1 marks]Arithmetic and geometric progressions

Given

\cos^2x=1

(so

\sin^2x=0

), find the common difference

d

of the arithmetic progression

\sin^2x,\ \cos^2x,\ 5\cos^2x-3

Answer this when you sit the paper.

Question 806

[1 marks]Arithmetic and geometric progressions

State the formula for the sum to infinity

S_\infty

of a geometric progression with first term

a

and common ratio

r

(where

|r|<1

Answer this when you sit the paper.

Question 807

[2 marks]Arithmetic and geometric progressions

For the geometric progression with first term 100 and common ratio 0,8, find the sum of the first 10 terms, correct to the nearest whole number.

Answer this when you sit the paper.

Question 901

[1 marks]Integration by parts and the trapezium rule

Evaluate

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

A $8\ln 4 - 4$
B $4\ln 4 - \tfrac{15}{4}$
C $8\ln 4 - \tfrac{15}{4}$
D $8\ln 4 + \tfrac{15}{4}$

Question 902

[1 marks]Integration by parts and the trapezium rule

Use the trapezium rule with 4 ordinates to approximate

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

, correct to 2 significant figures.

A $6.3$
B $7.3$
C $7.5$
D $8.3$

Question 903

[2 marks]Integration by parts and the trapezium rule

Integrating by parts with

u=\ln x

v'=x

, find the antiderivative of

x\ln x

(ignore the constant of integration).

Answer this when you sit the paper.

Question 904

[2 marks]Integration by parts and the trapezium rule

For the trapezium rule with

h=1

over

x=1,2,3,4

applied to

x\ln x

, find

y_1+y_2

(the sum of the two middle ordinates

2\ln2+3\ln3

), correct to 4 decimal places.

Answer this when you sit the paper.

Question 905

[1 marks]Integration by parts and the trapezium rule

State the ordinate

y_4=4\ln4

x=4

for the trapezium rule applied to

x\ln x

, correct to 4 decimal places.

Answer this when you sit the paper.

Question 906

[2 marks]Integration by parts and the trapezium rule

The trapezium rule gives

\approx7.4547

for

\int_1^4x\ln x\,dx

, whose exact value is

8\ln4-\tfrac{15}{4}\approx7.3404

. Find the relative error, correct to 3 decimal places.

Answer this when you sit the paper.

Question 1001

[1 marks]Partial fractions and integration

Express

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

in the form

A + \dfrac{B}{x-1} + \dfrac{Cx+D}{x^2+1}

A $A = 0$ , $B = -1$ , $C = 2$ , $D = 1$
B $A = 1$ , $B = -1$ , $C = 2$ , $D = 0$
C $A = 1$ , $B = 1$ , $C = 2$ , $D = 0$
D $A = 1$ , $B = -1$ , $C = -2$ , $D = 0$

Question 1002

[1 marks]Partial fractions and integration

Evaluate

\displaystyle\int_2^3 \left(1 - \dfrac{1}{x-1} + \dfrac{2x}{x^2+1}\right)dx

A $1$
B $1 - \ln 2$
C $\ln\tfrac{10}{5}$
D $1 + \ln 2$

Question 1003

[2 marks]Partial fractions and integration

Find the indefinite integral of

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

with respect to

x

(ignore the constant of integration).

Answer this when you sit the paper.

Question 1004

[2 marks]Partial fractions and integration

Evaluate the antiderivative

x-\ln(x-1)+\ln(x^2+1)

x=3

, correct to 4 decimal places.

Answer this when you sit the paper.

Question 1005

[2 marks]Partial fractions and integration

Evaluate the antiderivative

x-\ln(x-1)+\ln(x^2+1)

x=2

, correct to 4 decimal places.

Answer this when you sit the paper.

Question 1006

[1 marks]Partial fractions and integration

State the value of

\ln10

(used when evaluating the antiderivative at

x=3

), correct to 4 decimal places.

Answer this when you sit the paper.

Question 1101

[1 marks]Numerical methods (iteration)

The iterative formula

x_{n+1} = \cos^{-1}\!\left(\dfrac{1}{3 - x_n^2}\right)

converges to a root of which equation?

A $\sec x = 3 - x^2$
B $\sec x = 3 + x^2$
C $\tan x = 3 - x^2$
D $\cos x = 3 - x^2$

Question 1102

[1 marks]Numerical methods (iteration)

Using

x_0 = 1.024627

in the iteration

x_{n+1} = \cos^{-1}\!\left(\dfrac{1}{3 - x_n^2}\right)

, determine the root correct to 2 decimal places.

A $1.00$
B $1.02$
C $1.03$
D $1.05$

Question 1103

[1 marks]Numerical methods (iteration)

Starting from

x=\cos^{-1}\!\left(\dfrac{1}{3-x^2}\right)

, take the cosine of both sides. State the resulting expression for

\cos x

, in terms of

x

Answer this when you sit the paper.

Question 1104

[2 marks]Numerical methods (iteration)

Using the iteration

x_{n+1}=\cos^{-1}\!\left(\dfrac{1}{3-x_n^2}\right)

with

x_0=1{,}024627

, find

x_1

, correct to 6 decimal places.

Answer this when you sit the paper.

Question 1105

[2 marks]Numerical methods (iteration)

Continuing the iteration

x_{n+1}=\cos^{-1}\!\left(\dfrac{1}{3-x_n^2}\right)

from

x_1=1{,}032372

, find

x_2

, correct to 6 decimal places.

Answer this when you sit the paper.

Question 1106

[2 marks]Numerical methods (iteration)

Continuing the iteration

x_{n+1}=\cos^{-1}\!\left(\dfrac{1}{3-x_n^2}\right)

from

x_2=1{,}027445

, find

x_3

, correct to 6 decimal places.

Answer this when you sit the paper.

Question 1201

[1 marks]Differential equations

The rate at which a quantity

x

decreases is proportional to the product of

x

and

1 - x

. Which differential equation models this?

A $\dfrac{dx}{dt} = kx(1-x)$
B $\dfrac{dx}{dt} = -k(1-x)$
C $\dfrac{dx}{dt} = -\dfrac{k}{x(1-x)}$
D $\dfrac{dx}{dt} = -kx(1-x)$

Question 1202

[1 marks]Differential equations

Solve

\dfrac{dx}{dt} = -kx(1-x)

given that

x = 0.2

initially and the rate of decrease is then

1.6

units per unit time.

A $x = \dfrac{4}{e^{10t} + 1}$
B $x = \dfrac{1}{e^{10t} + 4}$
C $x = \dfrac{1}{4e^{10t} + 1}$
D $x = \dfrac{1}{4e^{-10t} + 1}$

Question 1203

[2 marks]Differential equations

Separate variables in

\dfrac{dx}{dt}=-kx(1-x)

and integrate the left side

\int\dfrac{dx}{x(1-x)}

using

\dfrac1{x(1-x)}=\dfrac1x+\dfrac1{1-x}

. State the result (ignore the constant of integration).

Answer this when you sit the paper.

Question 1204

[2 marks]Differential equations

Given

x=0,2

t=0

, find the constant of integration

c

\ln\dfrac{x}{1-x}=-kt+c

, correct to 4 decimal places.

Answer this when you sit the paper.

Question 1205

[2 marks]Differential equations

Given that

x=0,2

initially and the rate of decrease is then

1,6

units per unit time, use

\dfrac{dx}{dt}=-kx(1-x)

to find

k

Answer this when you sit the paper.

Question 1206

[1 marks]Differential equations

State the value of

x(1-x)

x=0,2

, used to find

k

from the initial rate.

Answer this when you sit the paper.

Question 1301

[1 marks]R-form, trigonometric equations and series expansion

Express

2\sin x - 4\cos x

in the form

R\sin(x - \alpha)

, where

R > 0

and

\alpha

is acute.

A $\sqrt{20}\sin(x - 63.4°)$
B $6\sin(x - 63.4°)$
C $2\sqrt{5}\sin(x + 63.4°)$
D $2\sqrt{5}\sin(x - 26.6°)$

Question 1302

[1 marks]R-form, trigonometric equations and series expansion

Solve

2\sin x - 4\cos x = \sqrt{5}

for

0° \le x \le 360°

, giving answers to the nearest degree.

A $x = 63°$ and $x = 243°$
B $x = 93°$ and $x = 273°$
C $x = 30°$ and $x = 150°$
D $x = 93°$ and $x = 213°$

Question 1303

[1 marks]R-form, trigonometric equations and series expansion

In the series expansion of

y = e^{a-x}

, the coefficient of

x

-4

. Find the exact value of

a

A $a = -\ln 4$
B $a = \ln 4$
C $a = e^4$
D $a = 4$

Question 1304

[1 marks]R-form, trigonometric equations and series expansion

For

2\sin x-4\cos x=R\sin(x-\alpha)

, state the value of

\tan\alpha

Answer this when you sit the paper.

Question 1305

[2 marks]R-form, trigonometric equations and series expansion

Given

\tan\alpha=2

, find

\alpha

correct to 1 decimal place (degrees).

Answer this when you sit the paper.

Question 1306

[1 marks]R-form, trigonometric equations and series expansion

State the exact value of

R

for

2\sin x-4\cos x=R\sin(x-\alpha)

Answer this when you sit the paper.

Question 1307

[2 marks]R-form, trigonometric equations and series expansion

Find the first three terms, in ascending powers of

x

, of the series expansion of

y=e^{a-x}=e^ae^{-x}

, in terms of

e^a

Answer this when you sit the paper.

Question 1308

[1 marks]R-form, trigonometric equations and series expansion

The coefficient of

x

in the series expansion of

e^{a-x}

-e^a

. Given this coefficient is

-4

, find

e^a

Answer this when you sit the paper.

Question 1309

[1 marks]R-form, trigonometric equations and series expansion

Given

2\sqrt5\sin(x-63{,}4^\circ)=\sqrt5

, find

\sin(x-63{,}4^\circ)

Answer this when you sit the paper.

Question 1401

[1 marks]Parametric differentiation and circular measure

A curve has parametric equations

x = \cos^2\theta

and

y = 1 - 2\sin^2\theta

. Find

\dfrac{dy}{dx}

A $-2$
B $\tfrac{1}{2}$
C $2\cos 2\theta$
D $2$

Question 1402

[1 marks]Parametric differentiation and circular measure

A sector

AOB

of a circle of radius

r

has arc

AB

of length

\tfrac{1}{2}r

. Find angle

AOB

in degrees, correct to the nearest degree.

A $14°$
B $29°$
C $30°$
D $57°$

Question 1403

[1 marks]Parametric differentiation and circular measure

In a sector of angle 1 radian, the segment cut off by the chord is removed. Express the segment area as a percentage of the remaining triangular area, correct to 2 significant figures.

A $27\%$
B $16\%$
C $19\%$
D $84\%$

Question 1404

[1 marks]Parametric differentiation and circular measure

For

x=\cos^2\theta

, find

\dfrac{dx}{d\theta}

Answer this when you sit the paper.

Question 1405

[1 marks]Parametric differentiation and circular measure

For

y=1-2\sin^2\theta

, find

\dfrac{dy}{d\theta}

Answer this when you sit the paper.

Question 1406

[2 marks]Parametric differentiation and circular measure

Using

\cos3\theta=\cos2\theta\cos\theta-\sin2\theta\sin\theta

, substitute

\cos2\theta=2\cos^2\theta-1

and

\sin2\theta=2\sin\theta\cos\theta

, and simplify to express

\cos3\theta

in terms of

\sin\theta

and

\cos\theta

only (as

\cos\theta

minus a term).

Answer this when you sit the paper.

Question 1407

[1 marks]Parametric differentiation and circular measure

Arc

AB

of the sector has length

\tfrac12r

. Using arc length

=r\theta

, find angle

AOB

in radians.

Answer this when you sit the paper.

Question 1408

[2 marks]Parametric differentiation and circular measure

B

is the midpoint of arc

AC

, so the full sector angle

AOC=2\times0,5=1

radian. Find the area of triangle

AOC

, in terms of

r

(using

\tfrac12r^2\sin\theta

Answer this when you sit the paper.

Question 1409

[1 marks]Parametric differentiation and circular measure

Find the area of the circular sector

AOC

(angle 1 radian), in terms of

r

(using

\tfrac12r^2\theta

Answer this when you sit the paper.

Question 1501

[1 marks]Areas and volumes of revolution

Find the

x

-coordinate of the point of intersection

P

of the curves

y = e^{2x}

and

y = e^{3-x}

A $x = 3$
B $x = \tfrac{3}{2}$
C $x = 1$
D $x = \ln 3$

Question 1502

[1 marks]Areas and volumes of revolution

Find the exact area of the region bounded by

y = e^{2x}

y = e^{3-x}

and the

y

-axis.

A $e^3 + \tfrac{3}{2}e^2 - \tfrac{1}{2}$
B $\tfrac{3}{2}e^2 - e^3 + \tfrac{1}{2}$
C $e^3 - e^2 + \tfrac{1}{2}$
D $e^3 - \tfrac{3}{2}e^2 + \tfrac{1}{2}$

Question 1503

[1 marks]Areas and volumes of revolution

The region bounded by

y = e^{2x}

y = e^{3-x}

and the

y

-axis is rotated through

360°

about the

x

-axis. Find the exact volume generated.

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

Question 1504

[2 marks]Areas and volumes of revolution

Find the antiderivative of

e^{3-x}-e^{2x}

with respect to

x

(ignore the constant of integration).

Answer this when you sit the paper.

Question 1505

[2 marks]Areas and volumes of revolution

Evaluate the antiderivative

-e^{3-x}-\tfrac12e^{2x}

x=1

, in terms of

e

Answer this when you sit the paper.

Question 1506

[2 marks]Areas and volumes of revolution

Evaluate the antiderivative

-e^{3-x}-\tfrac12e^{2x}

x=0

, in terms of

e

Answer this when you sit the paper.

Question 1507

[2 marks]Areas and volumes of revolution

For the volume of revolution

V=\pi\displaystyle\int_0^1(e^{6-2x}-e^{4x})\,dx

, find the antiderivative of

e^{6-2x}-e^{4x}

with respect to

x

(ignore

\pi

and the constant of integration).

Answer this when you sit the paper.

Question 1601

[1 marks]Vectors and complex numbers

A building has floor

OABC

with

OA = 4

m and

OC = 8

m, and apex

V

vertically

5

m above

M

, the centre of the floor. Taking

O

as origin, write down

\overrightarrow{OB}

and

\overrightarrow{OV}

A $\overrightarrow{OB} = 4\mathbf{i} + 8\mathbf{j}$ , $\overrightarrow{OV} = 4\mathbf{i} + 8\mathbf{j} + 5\mathbf{k}$
B $\overrightarrow{OB} = 8\mathbf{i} + 4\mathbf{j}$ , $\overrightarrow{OV} = 4\mathbf{i} + 2\mathbf{j} + 5\mathbf{k}$
C $\overrightarrow{OB} = 4\mathbf{i} + 8\mathbf{j}$ , $\overrightarrow{OV} = 2\mathbf{i} + 4\mathbf{j} + 5\mathbf{k}$
D $\overrightarrow{OB} = 2\mathbf{i} + 4\mathbf{j}$ , $\overrightarrow{OV} = 2\mathbf{i} + 4\mathbf{j} + 5\mathbf{k}$

Question 1602

[1 marks]Vectors and complex numbers

Given

\overrightarrow{OV} = 2\mathbf{i} + 4\mathbf{j} + 5\mathbf{k}

and

\overrightarrow{OB} = 4\mathbf{i} + 8\mathbf{j}

, find the exact value of

\cos\theta

, where

\theta

is the angle between them.

A $\tfrac{1}{2}$
B $\tfrac{3}{4}$
C $\tfrac{2}{3}$
D $\tfrac{40}{45}$

Question 1603

[1 marks]Vectors and complex numbers

Express

u = \dfrac{4-8i}{i}

in the form

x + iy

and state its modulus.

A $u = 8 + 4i$ , $|u| = 4\sqrt{5}$
B $u = -8 + 4i$ , $|u| = 4\sqrt{5}$
C $u = -8 - 4i$ , $|u| = 80$
D $u = -8 - 4i$ , $|u| = 4\sqrt{5}$

Question 1604

[2 marks]Vectors and complex numbers

Find the argument of

u=-8-4i

, correct to 2 decimal places (radians).

Answer this when you sit the paper.

Question 1605

[1 marks]Vectors and complex numbers

Find

|\overrightarrow{OB}|

for

\overrightarrow{OB}=4\mathbf{i}+8\mathbf{j}

, in surd form.

Answer this when you sit the paper.

Question 1606

[1 marks]Vectors and complex numbers

Find

|\overrightarrow{OV}|

for

\overrightarrow{OV}=2\mathbf{i}+4\mathbf{j}+5\mathbf{k}

, in surd form.

Answer this when you sit the paper.

Question 1607

[1 marks]Vectors and complex numbers

Find

\overrightarrow{OV}\cdot\overrightarrow{OB}

for

\overrightarrow{OV}=2\mathbf{i}+4\mathbf{j}+5\mathbf{k}

and

\overrightarrow{OB}=4\mathbf{i}+8\mathbf{j}

Answer this when you sit the paper.

Question 1608

[3 marks]Vectors and complex numbers

To rationalise

u=\dfrac{4-8i}{i}

, multiply numerator and denominator by

i

. Simplify the numerator

(4-8i)(i)

, using

i^2=-1

, in the form

p+qi

Answer this when you sit the paper.

Question 1609

[2 marks]Vectors and complex numbers

Given

u=-8-4i

, find

u^2

, in the form

p+qi

Answer this when you sit the paper.

Pure Mathematics Paper 1 June 2019

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 501

Question 502

Question 503

Question 504

Question 601

Question 602

Question 603

Question 604

Question 701

Question 702

Question 703

Question 704

Question 705

Question 706

Question 801

Question 802

Question 803

Question 804

Question 805

Question 806

Question 807

Question 901

Question 902

Question 903

Question 904

Question 905

Question 906

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 1201

Question 1202

Question 1203

Question 1204

Question 1205

Question 1206

Question 1301

Question 1302

Question 1303

Question 1304

Question 1305

Question 1306

Question 1307

Question 1308

Question 1309

Question 1401

Question 1402

Question 1403

Question 1404

Question 1405

Question 1406

Question 1407

Question 1408

Question 1409

Question 1501