ZIMSEC A Level Pure Mathematics Paper 1 June 2018

Question 101

[1 marks]Exponentials and logarithms

Solve the equation

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

, giving your answer in exact form.

A $x = \tfrac{1}{2}(2 + \ln 4)$
B $x = 2 + \ln 4$
C $x = \tfrac{1}{3}\ln 4 - 2$
D $x = \tfrac{1}{3}(2 + \ln 4)$

Question 102

[2 marks]Exponentials and logarithms

Taking natural logarithms of both sides of

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

and simplifying gives

3x=2+\ln4

. Evaluate

3x

, correct to 3 decimal places.

Answer this when you sit the paper.

Question 201

[1 marks]Completing the square

Express

f(x) = 13 + 4x - 2x^2

in the form

a + b(x+c)^2

A $15 - 2(x-1)^2$
B $15 + 2(x-1)^2$
C $13 - 2(x-1)^2$
D $11 - 2(x+1)^2$

Question 202

[1 marks]Completing the square

State the coordinates of the turning point of

y = 13 + 4x - 2x^2

A $(1, 15)$
B $(2, 15)$
C $(-1, 15)$
D $(1, 13)$

Question 203

[2 marks]Completing the square

Completing the square,

13+4x-2x^2=a+b(x+c)^2

. State the values of

b

and

c

(not

a

Answer this when you sit the paper.

Question 301

[1 marks]Modulus inequalities

Solve the inequality

|x+3| > 2|x|

A $x > 3$ only
B $x < -1$ or $x > 3$
C $-1 < x < 3$
D $-3 < x < 1$

Question 302

[2 marks]Modulus inequalities

Squaring both sides of

|x+3|>2|x|

and simplifying gives an inequality of the form

x^2+bx+c<0

(with unit coefficient of

x^2

). State the values of

b

and

c

Answer this when you sit the paper.

Question 303

[1 marks]Modulus inequalities

Factorise

x^2-2x-3

Answer this when you sit the paper.

Question 401

[1 marks]Logarithmic functions / transformations

The function

f

is defined by

f(x) = \ln(x-2)

. State the domain of

f

A $x > 0$
Ball real $x$
C $x > 2$
D $x \ge 2$

Question 402

[1 marks]Logarithmic functions / transformations

For

f(x)=\ln(x-2)

, state the domain of

y=f(-x)

Answer this when you sit the paper.

Question 403

[1 marks]Logarithmic functions / transformations

State the equation of the vertical asymptote of

y=f(-x)

, where

f(x)=\ln(x-2)

Answer this when you sit the paper.

Question 404

[2 marks]Logarithmic functions / transformations

State the interval of

x

, within the domain of

f(x)=\ln(x-2)

, for which

f(x)<0

(the region where

y=|f(x)|

differs from

y=f(x)

, needed for the sketch).

Answer this when you sit the paper.

Question 501

[1 marks]Vectors in 3D

Points

A

B

C

have position vectors

6\mathbf{i}+2\mathbf{j}+6\mathbf{k}

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

and

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

. Calculate angle

ABC

correct to the nearest

0.1°

A $27.3°$
B $45.0°$
C $62.7°$
D $152.7°$

Question 502

[1 marks]Vectors in 3D

For triangle

ABC

with

\overrightarrow{BA} = (4,-1,5)

and

\overrightarrow{BC} = (2,-3,4)

, determine the exact area of the triangle.

A $\tfrac{31}{2}$
B $\tfrac{1}{2}\sqrt{257}$
C $\sqrt{257}$
D $\tfrac{1}{2}\sqrt{1218}$

Question 503

[1 marks]Vectors in 3D

For triangle

ABC

with

\overrightarrow{BA}=4\mathbf{i}-\mathbf{j}+5\mathbf{k}

and

\overrightarrow{BC}=2\mathbf{i}-3\mathbf{j}+4\mathbf{k}

, find

\overrightarrow{BA}\cdot\overrightarrow{BC}

Answer this when you sit the paper.

Question 504

[1 marks]Vectors in 3D

Find

|\overrightarrow{BA}|

for

\overrightarrow{BA}=4\mathbf{i}-\mathbf{j}+5\mathbf{k}

, in exact surd form.

Answer this when you sit the paper.

Question 505

[2 marks]Vectors in 3D

Find

\overrightarrow{BA}\times\overrightarrow{BC}

for

\overrightarrow{BA}=4\mathbf{i}-\mathbf{j}+5\mathbf{k}

and

\overrightarrow{BC}=2\mathbf{i}-3\mathbf{j}+4\mathbf{k}

, in the form

p\mathbf{i}+q\mathbf{j}+r\mathbf{k}

(used to find the area of triangle

ABC

Answer this when you sit the paper.

Question 601

[1 marks]Complex numbers

The complex number

w = -2 + 2\sqrt{3}i

. Find

|w|

A $4$
B $2$
C $16$
D $2\sqrt{3}$

Question 602

[1 marks]Complex numbers

The complex number

w = -2 + 2\sqrt{3}i

. Find the argument of

\bar{w}

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

Question 603

[1 marks]Complex numbers

Given

w = -2 + 2\sqrt{3}i

, express

\dfrac{w+1}{w}

in the form

x + iy

A $\tfrac{7}{8} + \tfrac{\sqrt{3}}{8}i$
B $\tfrac{7}{8} - \tfrac{\sqrt{3}}{8}i$
C $1 - \tfrac{\sqrt{3}}{8}i$
D $\tfrac{9}{8} + \tfrac{\sqrt{3}}{8}i$

Question 604

[1 marks]Complex numbers

Find the argument of

w=-2+2\sqrt3i

itself (not its conjugate), giving your answer in degrees between

0°

and

360°

Answer this when you sit the paper.

Question 605

[2 marks]Complex numbers

To simplify

\dfrac{w+1}{w}

for

w=-2+2\sqrt3i

, multiply numerator and denominator by

\overline{w}

. Find the resulting numerator

(w+1)\overline{w}

, in the form

p+qi

Answer this when you sit the paper.

Question 701

[1 marks]Partial fractions

Express

\dfrac{x^3 - x^2 - x - 4}{x(x^2+2)}

in partial fractions.

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

Question 702

[1 marks]Partial fractions

Since

x^3-x^2-x-4

and

x(x^2+2)=x^3+2x

have the same degree, dividing gives quotient 1 with remainder

r(x)

. Find

r(x)

, in the form

ax^2+bx+c

Answer this when you sit the paper.

Question 703

[2 marks]Partial fractions

Using

x^2+3x+4\equiv A(x^2+2)+(Bx+C)x

, find

A

(by setting

x=0

Answer this when you sit the paper.

Question 704

[2 marks]Partial fractions

Using the same identity

x^2+3x+4\equiv A(x^2+2)+(Bx+C)x

with

A=2

, compare coefficients of

x^2

and

x

to find

B

and

C

Answer this when you sit the paper.

Question 801

[1 marks]Maclaurin series

Given

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

with

y = 1

when

x = 0

, find

\dfrac{d^2y}{dx^2}

x = 0

A $1$
B $2$
C $5$
D $10$

Question 802

[1 marks]Maclaurin series

Given

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

with

y = 1

x = 0

, obtain the Maclaurin series for

y

as far as the term in

x^3

A $1 + 2x + \tfrac{5}{2}x^2 + \tfrac{10}{3}x^3$
B $2x + \tfrac{5}{2}x^2 + \tfrac{5}{3}x^3$
C $1 + 2x + \tfrac{5}{2}x^2 + \tfrac{5}{3}x^3$
D $1 + 2x + 5x^2 + 10x^3$

Question 803

[2 marks]Maclaurin series

Given

\dfrac{d^2y}{dx^2}=5

x=0

, differentiate again to find

\dfrac{d^3y}{dx^3}=2\dfrac{d^2y}{dx^2}

, and hence state its value at

x=0

Answer this when you sit the paper.

Question 804

[1 marks]Maclaurin series

State

\dfrac{dy}{dx}

x=0

, given

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

and

y=1

x=0

Answer this when you sit the paper.

Question 805

[1 marks]Maclaurin series

State the coefficient of

x^3

in the Maclaurin series for

y

(i.e.

\dfrac{y'''(0)}{3!}

), as an exact fraction.

Answer this when you sit the paper.

Question 901

[1 marks]Trigonometric identities

Express

\cos 2A

in terms of

\tan A

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

Question 902

[1 marks]Trigonometric identities

By letting

2A = 45°

in the identity

\cos 2A = \dfrac{1 - \tan^2 A}{1 + \tan^2 A}

, deduce the exact value of

\tan^2 22.5°

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

Question 903

[2 marks]Trigonometric identities

Using

2A=45°

\cos45°=\dfrac{1-\tan^2 22.5°}{1+\tan^2 22.5°}

, and letting

t=\tan^2 22.5°

, the equation rearranges to

t=\dfrac{2-\sqrt2}{2+\sqrt2}

. Rationalising the denominator, state the resulting numerator

(2-\sqrt2)^2

, expanded in the form

p+q\sqrt2

Answer this when you sit the paper.

Question 904

[1 marks]Trigonometric identities

State the rationalised denominator

(2+\sqrt2)(2-\sqrt2)

Answer this when you sit the paper.

Question 905

[2 marks]Trigonometric identities

State the value of

\cos45°=\dfrac{\sqrt2}{2}

, correct to 4 decimal places.

Answer this when you sit the paper.

Question 1001

[1 marks]Polynomials / factor theorem

The polynomials

f(x) = 2x^3 + 7x^2 + ax + b

and

g(x) = x^3 + ax^2 - 5x + 2b

have

x+3

as a common factor. Find

a

and

b

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

Question 1002

[1 marks]Polynomials / factor theorem

Given

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

and

g(x) = x^3 + 2x^2 - 5x - 6

, both divisible by

x+3

, find their second common factor.

A $x + 1$
B $x - 1$
C $x - 2$
D $2x - 1$

Question 1003

[2 marks]Polynomials / factor theorem

Since

x+3

is a factor of

f(x)=2x^3+7x^2+ax+b

, evaluate

f(-3)=0

and simplify to an equation in the form

-3a+b=k

. Find

k

Answer this when you sit the paper.

Question 1004

[2 marks]Polynomials / factor theorem

Since

x+3

is a factor of

g(x)=x^3+ax^2-5x+2b

, evaluate

g(-3)=0

and simplify to an equation in the form

9a+2b=k

. Find

k

Answer this when you sit the paper.

Question 1005

[1 marks]Polynomials / factor theorem

Using

-3a+b=-9

and

9a+2b=12

, find

a

Answer this when you sit the paper.

Question 1101

[1 marks]Turning points / calculus

Given

y = \dfrac{2-x}{x^2-3}

, find the

x

-coordinates of the turning points.

A $x = \pm\sqrt{3}$
B $x = 1$ and $x = 3$
C $x = -1$ and $x = -3$
D $x = 2$ only

Question 1102

[1 marks]Turning points / calculus

For the curve

y = \dfrac{2-x}{x^2-3}

, state the turning points and their nature.

A $\left(1, \tfrac{1}{2}\right)$ maximum and $\left(3, \tfrac{1}{6}\right)$ minimum
Bboth are points of inflexion
C $\left(1, -\tfrac{1}{2}\right)$ minimum and $\left(3, -\tfrac{1}{6}\right)$ maximum
D $\left(1, -\tfrac{1}{2}\right)$ maximum and $\left(3, -\tfrac{1}{6}\right)$ minimum

Question 1103

[2 marks]Turning points / calculus

Using the quotient rule, find

\dfrac{dy}{dx}

for

y=\dfrac{2-x}{x^2-3}

, and simplify the numerator to the form

x^2+bx+c

(over denominator

(x^2-3)^2

). State

b

and

c

Answer this when you sit the paper.

Question 1104

[1 marks]Turning points / calculus

Factorise

x^2-4x+3

Answer this when you sit the paper.

Question 1105

[1 marks]Turning points / calculus

Find the

y

-coordinate of the turning point at

x=1

, for

y=\dfrac{2-x}{x^2-3}

Answer this when you sit the paper.

Question 1106

[1 marks]Turning points / calculus

Find the

y

-coordinate of the turning point at

x=3

, for

y=\dfrac{2-x}{x^2-3}

Answer this when you sit the paper.

Question 1201

[1 marks]Coordinate geometry

Find the equation of the line through

(4, -1)

perpendicular to

2x - 3y = 10

, giving your answer in the form

ax + by = c

A $3x - 2y = 14$
B $3x + 2y = 10$
C $3x + 2y = 14$
D $2x + 3y = 5$

Question 1202

[1 marks]Coordinate geometry

The line

y + 2x = 0

meets the curve

2x^2 + y^2 + 4x - 3y = 4

P

and

Q

. Find their coordinates.

A $P(1, -2)$ and $Q(-2, 4)$
B $P\left(\tfrac{1}{3}, \tfrac{2}{3}\right)$ and $Q(-2, -4)$
C $P\left(\tfrac{1}{3}, -\tfrac{2}{3}\right)$ and $Q(-2, 4)$
D $P\left(-\tfrac{1}{3}, \tfrac{2}{3}\right)$ and $Q(2, -4)$

Question 1203

[1 marks]Coordinate geometry

Find the length of

PQ

where

P\left(\tfrac{1}{3}, -\tfrac{2}{3}\right)

and

Q(-2, 4)

A $7\sqrt{5}$
B $\tfrac{7}{3}$
C $\tfrac{245}{9}$
D $\tfrac{7}{3}\sqrt{5}$

Question 1204

[1 marks]Coordinate geometry

State the gradient of the line perpendicular to

2x-3y=10

Answer this when you sit the paper.

Question 1205

[2 marks]Coordinate geometry

Substitute

y=-2x

into

2x^2+y^2+4x-3y=4

and simplify to the form

6x^2+bx+c=0

. State

b

and

c

Answer this when you sit the paper.

Question 1206

[2 marks]Coordinate geometry

Divide

6x^2+10x-4=0

by 2 and factorise the result, giving both linear factors.

Answer this when you sit the paper.

Question 1207

[1 marks]Coordinate geometry

Find the

x

-coordinate of the midpoint of

PQ

, where

P=\left(\tfrac13,-\tfrac23\right)

and

Q=(-2,4)

Answer this when you sit the paper.

Question 1208

[1 marks]Coordinate geometry

Find the

y

-coordinate of the midpoint of

PQ

, where

P=\left(\tfrac13,-\tfrac23\right)

and

Q=(-2,4)

Answer this when you sit the paper.

Question 1209

[1 marks]Coordinate geometry

State the gradient of the line

y+2x=0

Answer this when you sit the paper.

Question 1301

[1 marks]Series / binomial expansion

A salary scale begins at \$315 a month and rises to a maximum of \$765 by equal monthly increments of \$25. Find the number of months it takes to reach the maximum.

A $18$
B $19$
C $20$
D $31$

Question 1302

[1 marks]Series / binomial expansion

Find the term independent of

x

in the expansion of

\left(\dfrac{1}{x^2} - \dfrac{2x}{3}\right)^6

A $\tfrac{160}{243}$
B $\tfrac{80}{27}$
C $-\tfrac{80}{27}$
D $\tfrac{16}{81}$

Question 1303

[2 marks]Series / binomial expansion

The salary scale has first term \$315, common difference \$25, and 19 terms (reaching maximum \$765). Find the sum of all 19 monthly salaries,

S_{19}

Answer this when you sit the paper.

Question 1304

[1 marks]Series / binomial expansion

The series

\sum_{r=1}^{16}(\sqrt3)^r

is geometric with first term

a=\sqrt3

and common ratio

r=\sqrt3

. Evaluate

(\sqrt3)^{16}

(i.e.

3^8

Answer this when you sit the paper.

Question 1305

[1 marks]Series / binomial expansion

Using the geometric series sum formula,

S_{16}=\dfrac{\sqrt3(6561-1)}{\sqrt3-1}=\dfrac{6560\sqrt3}{\sqrt3-1}

. Rationalising by multiplying numerator and denominator by

(\sqrt3+1)

gives denominator

(\sqrt3-1)(\sqrt3+1)

. Evaluate this denominator.

Answer this when you sit the paper.

Question 1306

[2 marks]Series / binomial expansion

Continuing the rationalisation of

S_{16}

, the numerator becomes

6560\sqrt3(\sqrt3+1)=6560(3+\sqrt3)

. Expand this to the form

p+q\sqrt3

(before dividing by the denominator).

Answer this when you sit the paper.

Question 1307

[1 marks]Series / binomial expansion

In the expansion of

\left(\dfrac{1}{x^2}-\dfrac{2x}{3}\right)^6

, the general term is

\binom{6}{k}\left(\dfrac{1}{x^2}\right)^{6-k}\left(-\dfrac{2x}{3}\right)^k

. Find the power of

x

in this term, in terms of

k

, in the form

mk+c

. State

m

and

c

Answer this when you sit the paper.

Question 1308

[1 marks]Series / binomial expansion

Hence find the value of

k

for which the term is independent of

x

(power of

x

equals 0).

Answer this when you sit the paper.

Question 1309

[1 marks]Series / binomial expansion

State

\dbinom{6}{4}

Answer this when you sit the paper.

Question 1401

[1 marks]Circular measure / Newton-Raphson

OAB

is a sector of a circle of radius

r

with

A\hat{O}B = 2\theta

radians. Write down expressions for the arc

AB

and the chord

AB

Aarc $= 2r\theta$ , chord $= r\sin 2\theta$
Barc $= r\theta$ , chord $= 2r\sin 2\theta$
Carc $= r\theta$ , chord $= r\sin\theta$
Darc $= 2r\theta$ , chord $= 2r\sin\theta$

Question 1402

[1 marks]Circular measure / Newton-Raphson

In a sector with arc

AB = 6

cm and chord

AB = 5

cm, where

A\hat{O}B = 2\theta

, which equation does

\theta

satisfy?

A $5\theta = 6\sin 2\theta$
B $5\theta = 6\sin\theta$
C $\theta = \tfrac{6}{5}\cos\theta$
D $6\theta = 5\sin\theta$

Question 1403

[1 marks]Circular measure / Newton-Raphson

Taking

\theta_1 = 1

, apply the Newton-Raphson method twice to

5\theta = 6\sin\theta

and give the root correct to 3 decimal places.

A $0.900$
B $1.000$
C $1.027$
D $1.028$

Question 1404

[2 marks]Circular measure / Newton-Raphson

Let

f(\theta)=5\theta-6\sin\theta

(radians). Evaluate

f(0{,}9)

, correct to 3 decimal places.

Answer this when you sit the paper.

Question 1405

[2 marks]Circular measure / Newton-Raphson

Evaluate

f(1{,}1)=5(1{,}1)-6\sin(1{,}1)

, correct to 3 decimal places.

Answer this when you sit the paper.

Question 1406

[2 marks]Circular measure / Newton-Raphson

With

\theta_1=1

, evaluate

f(1)=5(1)-6\sin(1)

, correct to 4 decimal places (radians).

Answer this when you sit the paper.

Question 1407

[2 marks]Circular measure / Newton-Raphson

Evaluate

f'(\theta)=5-6\cos\theta

\theta=1

, correct to 4 decimal places.

Answer this when you sit the paper.

Question 1501

[1 marks]Areas and volumes of revolution

Find the coordinates of the points where the line

y = \tfrac{5}{2} - \tfrac{1}{2}x

meets the curve

y = 1 + \dfrac{1}{x}

for

x > 0

A $\left(\tfrac{1}{2}, 3\right)$ and $(2, \tfrac{3}{2})$
B $(-1, 0)$ and $(2, \tfrac{3}{2})$
C $(1, 2)$ and $\left(2, \tfrac{3}{2}\right)$
D $(1, 2)$ and $(2, 2)$

Question 1502

[1 marks]Areas and volumes of revolution

Find the exact area of the region enclosed by the line

y = \tfrac{5}{2} - \tfrac{1}{2}x

and the curve

y = 1 + \dfrac{1}{x}

between their intersections.

A $\tfrac{3}{4} - \ln 2$
B $\tfrac{3}{2} - \ln 2$
C $\tfrac{3}{4} + \ln 2$
D $\ln 2 - \tfrac{3}{4}$

Question 1503

[1 marks]Areas and volumes of revolution

The region enclosed by

y = \tfrac{5}{2} - \tfrac{1}{2}x

and

y = 1 + \dfrac{1}{x}

is rotated through

2\pi

radians about the

x

-axis. Find the exact volume generated.

A $\pi\left(\tfrac{19}{12} + \ln 4\right)$
B $\pi\left(\tfrac{3}{4} - \ln 2\right)$
C $\pi\left(\tfrac{19}{12} - \ln 4\right)$
D $\pi\left(\tfrac{19}{6} - \ln 4\right)$

Question 1504

[2 marks]Areas and volumes of revolution

Solve

\dfrac52-\dfrac{x}{2}=1+\dfrac1x

by multiplying through by

2x

, giving a quadratic in the form

x^2+bx+c=0

. State

b

and

c

Answer this when you sit the paper.

Question 1505

[2 marks]Areas and volumes of revolution

Find the antiderivative of

\dfrac32-\dfrac{x}{2}-\dfrac1x

with respect to

x

(ignore the constant of integration).

Answer this when you sit the paper.

Question 1506

[1 marks]Areas and volumes of revolution

Evaluate the antiderivative

\dfrac{3x}{2}-\dfrac{x^2}{4}-\ln x

x=2

Answer this when you sit the paper.

Question 1507

[1 marks]Areas and volumes of revolution

Evaluate the antiderivative

\dfrac{3x}{2}-\dfrac{x^2}{4}-\ln x

x=1

Answer this when you sit the paper.

Question 1508

[2 marks]Areas and volumes of revolution

For the volume of revolution, expand

\left(\dfrac52-\dfrac{x}{2}\right)^2

(i.e.

y_{line}^2

), giving the answer in the form

\dfrac14(x^2+bx+c)

. State

b

and

c

Answer this when you sit the paper.

Question 1509

[2 marks]Areas and volumes of revolution

Expand

\left(1+\dfrac1x\right)^2

(i.e.

y_{curve}^2

), giving the answer in the form

1+\dfrac{p}{x}+\dfrac{q}{x^2}

. State

p

and

q

Answer this when you sit the paper.

Question 1601

[1 marks]Differential equations

Solve the differential equation

\dfrac{dy}{dx} = y\tan x

, given that

y = 1

when

x = 0

A $y = \cos x$
B $y = \sec x$
C $y = \tan x$
D $y = e^{\tan x}$

Question 1602

[1 marks]Differential equations

A spray kills cockroaches at a rate inversely proportional to

\sqrt{N}

, the square root of the number alive. Which differential equation models this?

A $\dfrac{dN}{dt} = -\dfrac{k}{\sqrt{N}}$
B $\dfrac{dN}{dt} = -k\sqrt{N}$
C $\dfrac{dN}{dt} = -kN$
D $\dfrac{dN}{dt} = \dfrac{k}{\sqrt{N}}$

Question 1603

[1 marks]Differential equations

For

\dfrac{dN}{dt} = -\dfrac{k}{\sqrt{N}}

with 144 cockroaches initially and 81 after 36 minutes, find the number left one hour after spraying.

A $0$
B $16$
C $25$
D $36$

Question 1604

[2 marks]Differential equations

Separating variables in

\dfrac{dy}{dx}=y\tan x

gives

\int\dfrac1y\,dy=\int\tan x\,dx

. Evaluate the right-hand integral

\int\tan x\,dx

(ignore the constant of integration), in terms of

\cos x

Answer this when you sit the paper.

Question 1605

[2 marks]Differential equations

Separating variables in

\dfrac{dN}{dt}=-\dfrac{k}{\sqrt N}

and integrating gives

\dfrac23N^{3/2}=-kt+C

. Using

N=144

when

t=0

, find

C

(i.e.

\dfrac23(144)^{3/2}

Answer this when you sit the paper.

Question 1606

[2 marks]Differential equations

Using

N=81

when

t=36

\dfrac23N^{3/2}=-kt+1152

, find

k

Answer this when you sit the paper.

Question 1607

[2 marks]Differential equations

Hence express

N^{3/2}

in terms of

t

, in the form

N^{3/2}=A-Bt

. State

A

and

B

Answer this when you sit the paper.

Question 1608

[2 marks]Differential equations

Using

N^{3/2}=1728-27{,}75t

, find

N^{3/2}

t=60

minutes (one hour after spraying).

Answer this when you sit the paper.

Pure Mathematics Paper 1 June 2018

Sit this paper online

The questions

Question 101

Question 102

Question 201

Question 202

Question 203

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 601

Question 602

Question 603

Question 604

Question 605

Question 701

Question 702

Question 703

Question 704

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 1203

Question 1204

Question 1205

Question 1206

Question 1207

Question 1208

Question 1209

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 1501

Question 1502

Question 1503

Question 1504

Question 1505