ZIMSEC A Level Pure Mathematics Paper 1 June 2017

Question 101

[1 marks]Modulus / inequalities

Solve the inequality

|x-1| < x

A $0 < x < 1$
B $x > 1$
C $x > \tfrac{1}{2}$
D $x < \tfrac{1}{2}$

Question 102

[2 marks]Modulus / inequalities

For

x \ge 1

|x-1| = x-1

. Which statement correctly explains why every

x \ge 1

satisfies

|x-1| < x

AIt simplifies to $-1<0$ , true for all real $x$
BIt only holds when $x$ is a whole number, not all reals
CIt requires $x \neq 1$ in addition to $x \ge 1$ to hold
DIt simplifies to $x<-1$ , which never happens for $x \ge 1$

Question 201

[1 marks]Polynomials / factor theorem

Factorise completely

3x^3 + x^2 - 8x + 4

A $(x-1)(3x-2)(x+2)$
B $(x-1)(3x-4)(x+1)$
C $(x+1)(3x-2)(x-2)$
D $(x-1)(3x+2)(x-2)$

Question 202

[1 marks]Polynomials / factor theorem

Confirm

(x-1)

is a factor of

3x^3+x^2-8x+4

by evaluating the polynomial at

x=1

Answer this when you sit the paper.

Question 203

[2 marks]Polynomials / factor theorem

Divide

3x^3+x^2-8x+4

(x-1)

. State the resulting quadratic quotient in the form

3x^2+bx+c

Answer this when you sit the paper.

Question 301

[1 marks]Indices / quadratic in disguise

Solve the equation

y^{2/3} - 5y^{1/3} + 6 = 0

A $y = -8$ or $y = -27$
B $y = 2$ or $y = 3$
C $y = 8$ or $y = 27$
D $y = 4$ or $y = 9$

Question 302

[2 marks]Indices / quadratic in disguise

Expand and simplify

(x^{1/3}-2)(x^{2/3}+2x^{1/3}+1)

A $x^{2/3} - 8$
B $x - 3x^{1/3} - 2$
C $x - 2$
D $x - x^{1/3} - 2$

Question 303

[1 marks]Indices / quadratic in disguise

Solving

u^2-5u+6=0

where

u=y^{1/3}

gives two values of

u

. Which pair is correct?

A $u=6$ and $u=-1$
B $u=-2$ and $u=-3$
C $u=2$ and $u=3$
D $u=1$ and $u=6$

Question 304

[2 marks]Indices / quadratic in disguise

Using the expansion

x-3x^{1/3}-2

from part (a), find its value when

x=8

Answer this when you sit the paper.

Question 401

[1 marks]Small changes / rates

The radius of a sphere increases from 11 cm to 11.02 cm. Find the approximate percentage increase in its volume.

A $0.18\%$
B $0.36\%$
C $1.82\%$
D $0.55\%$

Question 402

[1 marks]Small changes / rates

For

V=\tfrac43\pi r^3

, find

\dfrac{dV}{dr}

A $2\pi r^2$
B $4\pi r^3$
C $\tfrac43\pi r^2$
D $4\pi r^2$

Question 403

[2 marks]Small changes / rates

Using

\delta V \approx 4\pi r^2\,\delta r

with

r=11

and

\delta r=0.02

, find

\delta V

as a multiple of

\pi

(e.g.

9.68\pi

Answer this when you sit the paper.

Question 404

[2 marks]Small changes / rates

Find

\delta V

to 3 significant figures as a decimal (take

\pi \approx 3.142

), using

\delta V \approx 4\pi r^2\,\delta r

with

r=11

\delta r=0.02

Answer this when you sit the paper.

Question 501

[1 marks]Binomial series / approximation

Expand

\dfrac{1}{\sqrt{1-2x}}

in ascending powers of

x

up to and including the term in

x^3

A $1 + x + \tfrac{1}{2}x^2 + \tfrac{1}{2}x^3$
B $1 + 2x + 3x^2 + 4x^3$
C $1 - x + \tfrac{3}{2}x^2 - \tfrac{5}{2}x^3$
D $1 + x + \tfrac{3}{2}x^2 + \tfrac{5}{2}x^3$

Question 502

[1 marks]Binomial series / approximation

State the range of values of

x

for which the expansion of

\dfrac{1}{\sqrt{1-2x}}

is valid.

A $|x| < 1$
B $|x| < \tfrac{1}{2}$
C $x > 0$
D $|x| < 2$

Question 503

[1 marks]Binomial series / approximation

By substituting

x = -\tfrac{1}{6}

into the expansion of

\dfrac{1}{\sqrt{1-2x}}

, find an approximation for

\sqrt{3}

correct to 5 decimal places.

A $0.86342$
B $1.72500$
C $1.72685$
D $1.73205$

Question 504

[2 marks]Binomial series / approximation

To approximate

\sqrt3

using the expansion of

(1-2x)^{-1/2}

, which value of

x

is substituted, and what does

1-2x

equal there?

A $x=\tfrac16$ , giving $1-2x=\tfrac23$
B $x=-\tfrac16$ , giving $1-2x=\tfrac43$
C $x=-\tfrac13$ , giving $1-2x=\tfrac53$
D $x=\tfrac12$ , giving $1-2x=0$

Question 505

[1 marks]Binomial series / approximation

In the expansion

(1-2x)^{-1/2}=1+x+\tfrac32x^2+\tfrac52x^3+\cdots

, state the coefficient of

x^2

Answer this when you sit the paper.

Question 601

[1 marks]Vectors in 3D

P(1,2,1)

Q(4,7,8)

and

R(6,4,12)

are three vertices of a parallelogram

PQRS

. Find the position vector of

S

A $-3\mathbf{i} + \mathbf{j} - 5\mathbf{k}$
B $5\mathbf{i} + 2\mathbf{j} + 11\mathbf{k}$
C $3\mathbf{i} - \mathbf{j} + 5\mathbf{k}$
D $9\mathbf{i} + 9\mathbf{j} + 19\mathbf{k}$

Question 602

[1 marks]Vectors in 3D

M

and

N

are the midpoints of

PQ

and

QR

in parallelogram

PQRS

, where

P(1,2,1)

and

R(6,4,12)

. Find the unit vector in the direction of

\overrightarrow{MN}

A $\tfrac{1}{2\sqrt{150}}(5\mathbf{i} + 2\mathbf{j} + 11\mathbf{k})$
B $\tfrac{1}{\sqrt{150}}(5\mathbf{i} + 2\mathbf{j} + 11\mathbf{k})$
C $\tfrac{1}{150}(5\mathbf{i} + 2\mathbf{j} + 11\mathbf{k})$
D $\tfrac{1}{\sqrt{150}}(-5\mathbf{i} - 2\mathbf{j} - 11\mathbf{k})$

Question 603

[2 marks]Vectors in 3D

For

P(1,2,1)

Q(4,7,8)

R(6,4,12)

, which pair correctly gives

\vec{RQ}

and

\vec{PR}

A $\vec{RQ}=(-2,3,-4)$ , $\vec{PR}=(5,2,11)$
B $\vec{RQ}=(-2,3,-4)$ , $\vec{PR}=(-5,-2,-11)$
C $\vec{RQ}=(3,-2,-4)$ , $\vec{PR}=(5,11,2)$
D $\vec{RQ}=(2,-3,4)$ , $\vec{PR}=(5,2,11)$

Question 604

[2 marks]Vectors in 3D

Find

|\vec{PR}|

, the magnitude of

\vec{PR}=5\mathbf{i}+2\mathbf{j}+11\mathbf{k}

, in simplified surd form.

Answer this when you sit the paper.

Question 701

[1 marks]Completing the square / transformations

Express

f(x) = x^2 - 8x + 10

in the form

(x+a)^2 + b

A $(x+4)^2 - 6$
B $(x-4)^2 + 6$
C $(x-4)^2 - 6$
D $(x-8)^2 - 54$

Question 702

[1 marks]Completing the square / transformations

Describe the transformations that map

y = x^2

onto

y = x^2 - 8x + 10

ATranslation 8 units in the $+x$ direction, then 10 units in the $+y$ direction
BReflection in the $y$ -axis, then translation 6 units down
CTranslation 4 units in the $+x$ direction, then 6 units in the $-y$ direction
DTranslation 4 units in the $-x$ direction, then 6 units in the $+y$ direction

Question 703

[2 marks]Completing the square / transformations

For

f(x)=x^2-8x+10=(x-4)^2-6

, which option correctly states the least value and the line of symmetry?

Aleast value $-6$ , line of symmetry $x=4$
Bleast value $4$ , line of symmetry $x=-6$
Cleast value $10$ , line of symmetry $x=8$
Dleast value $-6$ , line of symmetry $x=-4$

Question 704

[1 marks]Completing the square / transformations

Which single transformation maps

y=x^2

onto

y=(x-4)^2

(ignoring the vertical shift)?

Atranslation of 4 units in the positive $y$ -direction
Btranslation of 4 units in the positive $x$ -direction
Cstretch of scale factor 4 parallel to the $x$ -axis
Dtranslation of 4 units in the negative $x$ -direction

Question 705

[2 marks]Completing the square / transformations

Which single transformation maps

y=(x-4)^2

onto

y=(x-4)^2-6

Astretch of scale factor 6 parallel to the $y$ -axis
Btranslation of 6 units in the positive $y$ -direction
Ctranslation of 6 units in the negative $x$ -direction
Dtranslation of 6 units in the negative $y$ -direction

Question 801

[1 marks]Trigonometry / circle geometry

Solve

2\cos^2 x + 3\sin x = 3

for

0 < x < \pi

A $x = \tfrac{\pi}{6},\ \tfrac{\pi}{2},\ \tfrac{5\pi}{6}$
B $x = \tfrac{\pi}{6},\ \tfrac{5\pi}{6}$
C $x = \tfrac{\pi}{3},\ \tfrac{2\pi}{3}$
D $x = \tfrac{\pi}{2}$ only

Question 802

[1 marks]Trigonometry / circle geometry

A circle of radius

r

has diameter

BOC

, and

A

lies on the circle with angle

A\hat{O}B = \theta

. The area of the region bounded by the diameter and the arc, less triangle

AOB

, is

A $\tfrac{1}{2}r^2(\pi + \sin\theta)$
B $r^2(\pi - \sin\theta)$
C $\tfrac{1}{2}r^2(\theta - \sin\theta)$
D $\tfrac{1}{2}r^2(\pi - \sin\theta)$

Question 803

[2 marks]Trigonometry / circle geometry

Solving

2\sin^2x-3\sin x+1=0

gives two values of

\sin x

. Which pair is correct?

A $\sin x = 1$ and $\sin x = 0.5$
B $\sin x = 0.5$ and $\sin x = 0.25$
C $\sin x = 1$ and $\sin x = -0.5$
D $\sin x = -1$ and $\sin x = 0.5$

Question 804

[2 marks]Trigonometry / circle geometry

In the shaded-region diagram, a semicircle has diameter

BOC

(radius

r

). State its area in terms of

r

and

\pi

Answer this when you sit the paper.

Question 805

[2 marks]Trigonometry / circle geometry

In the shaded-region diagram, triangle

AOB

has

OA=OB=r

and angle

A\hat{O}B=\theta

. State its area in terms of

r

and

\theta

Answer this when you sit the paper.

Question 901

[1 marks]Partial fractions / integration

Express

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

in the form

x + 1 + \dfrac{A}{x} + \dfrac{B}{x+1}

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

Question 902

[1 marks]Partial fractions / integration

Evaluate

\displaystyle\int_1^2 \left(x + 1 + \dfrac{2}{x} - \dfrac{2}{x+1}\right)dx

A $\tfrac{5}{2} + 2\ln\tfrac{3}{4}$
B $\tfrac{5}{2} + 2\ln\tfrac{4}{3}$
C $\tfrac{3}{2} + 2\ln\tfrac{4}{3}$
D $\tfrac{5}{2} + 4\ln\tfrac{4}{3}$

Question 903

[2 marks]Partial fractions / integration

f(x)=x+1+\dfrac{A}{x}+\dfrac{B}{x+1}

, which pair

(A,B)

is correct for

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

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

Question 904

[2 marks]Partial fractions / integration

Find the indefinite integral

\displaystyle\int\left(x+1+\dfrac{2}{x}-\dfrac{2}{x+1}\right)dx

(omit the constant of integration).

Answer this when you sit the paper.

Question 905

[2 marks]Partial fractions / integration

Evaluate

\displaystyle\int_1^2 \dfrac{2}{x}\,dx

(just this term of the integrand) to 3 significant figures.

Answer this when you sit the paper.

Question 1001

[1 marks]Complex numbers

Given

z_1 = 2 - 3i

and

z_2 = 1 + 3i

, express

\dfrac{z_1}{z_2}

in the form

x + iy

A $-\tfrac{7}{10} + \tfrac{9}{10}i$
B $\tfrac{7}{10} - \tfrac{9}{10}i$
C $-\tfrac{7}{10} - \tfrac{9}{10}i$
D $-\tfrac{7}{5} - \tfrac{9}{5}i$

Question 1002

[1 marks]Complex numbers

Given

\dfrac{z_1}{z_2} = -\tfrac{7}{10} - \tfrac{9}{10}i

, find

\left|\dfrac{z_1}{z_2}\right|

A $\tfrac{130}{10}$
B $\tfrac{1}{10}\sqrt{32}$
C $\tfrac{1}{10}\sqrt{130}$
D $\sqrt{130}$

Question 1003

[1 marks]Complex numbers

Find

\arg\left(\dfrac{z_1}{z_2}\right)

where

\dfrac{z_1}{z_2} = -\tfrac{7}{10} - \tfrac{9}{10}i

, giving your answer in radians in the range

0

2\pi

A $0.91$
B $2.23$
C $4.05$
D $5.38$

Question 1004

[1 marks]Complex numbers

For

z_1=2-3i

and

z_2=1+3i

, find

z_1 z_2

(the product, not the quotient) in the form

x+iy

Answer this when you sit the paper.

Question 1005

[2 marks]Complex numbers

Which pair correctly gives

|z_1|

and

|z_2|

for

z_1=2-3i

z_2=1+3i

A $\sqrt{13}$ and $\sqrt{10}$
B $5$ and $10$
C $\sqrt5$ and $\sqrt{10}$
D $\sqrt{13}$ and $\sqrt8$

Question 1006

[1 marks]Complex numbers

The point

\dfrac{z_1}{z_2}=-\tfrac{7}{10}-\tfrac{9}{10}i

lies in which quadrant of the Argand diagram?

Athird quadrant
Bfirst quadrant
Csecond quadrant
Dfourth quadrant

Question 1007

[2 marks]Complex numbers

Express

\arg\left(\dfrac{z_1}{z_2}\right)

in degrees, to 1 decimal place, in the range

-180°

180°

Answer this when you sit the paper.

Question 1101

[1 marks]Numerical methods / Newton-Raphson

Let

f(x) = \ln x - \dfrac{x}{x-1}

. Which values show that a root of

\ln x = \dfrac{x}{x-1}

lies between

x = 0.3

and

x = 0.45

A $f(0.3) = 0.775$ and $f(0.45) = -0.0196$
B $f(0.3) = -0.775$ and $f(0.45) = -0.0196$
C $f(0.3) = -1.204$ and $f(0.45) = -0.798$
D $f(0.3) = -0.775$ and $f(0.45) = 0.0196$

Question 1102

[1 marks]Numerical methods / Newton-Raphson

Taking

x_0 = 0.45

, use the Newton-Raphson method twice on

\ln x = \dfrac{x}{x-1}

and give the root correct to 4 decimal places.

A $0.3000$
B $0.4464$
C $0.4465$
D $0.4500$

Question 1103

[2 marks]Numerical methods / Newton-Raphson

Find

f'(x)

for

f(x)=\ln x-\dfrac{x}{x-1}

Answer this when you sit the paper.

Question 1104

[2 marks]Numerical methods / Newton-Raphson

Taking

x_0=0.45

, perform one Newton-Raphson iteration on

f(x)=\ln x-\dfrac{x}{x-1}

to find

x_1

, correct to 6 decimal places.

Answer this when you sit the paper.

Question 1105

[2 marks]Numerical methods / Newton-Raphson

Find

f(0.45)

for

f(x)=\ln x-\dfrac{x}{x-1}

, correct to 4 decimal places.

Answer this when you sit the paper.

Question 1106

[2 marks]Numerical methods / Newton-Raphson

The values

f(0.3)<0

and

f(0.45)>0

(with

f

continuous on

[0.3,0.45]

) guarantee a root in that interval by which result?

Athe Fundamental Theorem of Calculus
Bthe Mean Value Theorem
Cthe Intermediate Value Theorem
DRolle's theorem

Question 1201

[1 marks]Coordinate geometry / circles

Find the centre and radius of the circle

x^2 + y^2 - 4x + 6y = 12

Acentre $(-2, 3)$ , radius $5$
Bcentre $(2, -3)$ , radius $25$
Ccentre $(4, -6)$ , radius $5$
Dcentre $(2, -3)$ , radius $5$

Question 1202

[1 marks]Coordinate geometry / circles

The circle

x^2 + y^2 - 4x + 6y = 12

meets the line

3y = x + 4

A

and

B

. Find the length of the chord

AB

A $\sqrt{10}$
B $10$
C $2\sqrt{10}$
D $\sqrt{5}$

Question 1203

[1 marks]Coordinate geometry / circles

A chord

AB

of length

\sqrt{10}

is drawn in a circle of radius 5 with centre

C

. Find the area of triangle

ABC

A $15$ units $^2$
B $\tfrac{15}{2}$ units $^2$
C $\tfrac{45}{2}$ units $^2$
D $\tfrac{15}{4}$ units $^2$

Question 1204

[2 marks]Coordinate geometry / circles

Solving

y^2-3y+2=0

(from substituting the line into the circle) gives two values of

y

. Which pair is correct?

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

Question 1205

[2 marks]Coordinate geometry / circles

Using

x=3y-4

, which pair of coordinates correctly gives

A

and

B

A $A(2,-2)$ and $B(1,-1)$
B $A(-2,2)$ and $B(1,1)$
C $A(2,2)$ and $B(1,-1)$
D $A(2,2)$ and $B(-1,1)$

Question 1206

[2 marks]Coordinate geometry / circles

Find the perpendicular distance

h

from centre

C(2,-3)

to chord

AB

, using

h^2+\left(\tfrac{\sqrt{10}}{2}\right)^2=5^2

, to 3 significant figures.

Answer this when you sit the paper.

Question 1207

[1 marks]Coordinate geometry / circles

The chord

AB

has length

\sqrt{10}

. Find

(AB)^2

Answer this when you sit the paper.

Question 1301

[1 marks]Series (GP and AP)

A series has

r

-th term

U_r = 2\left(\tfrac{1}{3}\right)^r

. Find

n

such that

\displaystyle\sum_{r=1}^{n}U_r = \tfrac{80}{81}

A $n = 5$
B $n = 4$
C $n = 81$
D $n = 3$

Question 1302

[1 marks]Series (GP and AP)

For an arithmetic progression the sum of the first

n

terms is

S_n = \tfrac{3}{2}n^2 - 2n

. Find the

n

-th term.

A $\dfrac{6n+7}{2}$
B $3n - 2$
C $\dfrac{3n-4}{2}$
D $\dfrac{6n-7}{2}$

Question 1303

[2 marks]Series (GP and AP)

Which option gives the first three terms

U_1,U_2,U_3

U_r=2\left(\tfrac13\right)^r

A $\tfrac13,\tfrac19,\tfrac{1}{27}$
B $2,\tfrac23,\tfrac29$
C $\tfrac23,\tfrac29,\tfrac{2}{27}$
D $\tfrac23,\tfrac19,\tfrac{1}{27}$

Question 1304

[2 marks]Series (GP and AP)

For the arithmetic progression with

S_n=\tfrac32n^2-2n

, which option gives

U_1,U_2,U_3

A $-0.5,\ 2,\ 4.5$
B $0.5,\ 2.5,\ 5.5$
C $1.5,\ -2,\ 7.5$
D $-0.5,\ 2.5,\ 5.5$

Question 1305

[2 marks]Series (GP and AP)

For

S_n=\tfrac32n^2-2n

, find

S_4

Answer this when you sit the paper.

Question 1306

[2 marks]Series (GP and AP)

Hence find

U_4=S_4-S_3

for

S_n=\tfrac32n^2-2n

Answer this when you sit the paper.

Question 1401

[1 marks]Curve sketching / areas and volumes

Solve the inequality

f(x) < 0

where

f(x) = x(x+1)(2-x)

A $-1 < x < 2$
B $x > 2$ only
C $x < -1$ or $0 < x < 2$
D $-1 < x < 0$ or $x > 2$

Question 1402

[1 marks]Curve sketching / areas and volumes

Find the area of the region bounded by

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

and the

x

-axis between

x = -1

and

x = 0

A $\tfrac{5}{12}$ units $^2$
B $-\tfrac{5}{12}$ units $^2$
C $\tfrac{7}{12}$ units $^2$
D $\tfrac{5}{6}$ units $^2$

Question 1403

[1 marks]Curve sketching / areas and volumes

The region bounded by

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

and the

x

-axis between

x = -1

and

x = 0

is rotated completely about the

x

-axis. Find the volume generated.

A $\tfrac{5}{12}\pi$
B $\tfrac{22}{105}\pi$
C $\tfrac{22}{105}$
D $\tfrac{44}{105}\pi$

Question 1404

[2 marks]Curve sketching / areas and volumes

For

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

, which option correctly gives its roots and its behaviour as

x\to\infty

Aroots $1,0,-2$ ; $y\to-\infty$
Broots $-1,0,2$ ; $y\to0$
Croots $-1,0,2$ ; $y\to-\infty$
Droots $-1,0,2$ ; $y\to+\infty$

Question 1405

[1 marks]Curve sketching / areas and volumes

Expand

x(x+1)(2-x)

as a cubic

ax^3+bx^2+cx

. State the coefficient of

x^2

Answer this when you sit the paper.

Question 1406

[2 marks]Curve sketching / areas and volumes

Let

F(x)=-\tfrac{x^4}{4}+\tfrac{x^3}{3}+x^2

be an antiderivative of

-x^3+x^2+2x

. Find

F(-1)

Answer this when you sit the paper.

Question 1407

[2 marks]Curve sketching / areas and volumes

Find

[f(x)]^2

where

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

, at

x=-0.5

, to 3 decimal places.

Answer this when you sit the paper.

Question 1408

[3 marks]Curve sketching / areas and volumes

Which definite integral correctly represents the volume generated when the region between

x=-1

and

x=0

under

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

is rotated about the

x

-axis?

A $\displaystyle\int_{-1}^{0}[f(x)]^2\,dx$
B $\pi\displaystyle\int_{-1}^{0}[f(x)]^2\,dx$
C $\pi\displaystyle\int_{-1}^{0}f(x)\,dx$
D $2\pi\displaystyle\int_{-1}^{0}xf(x)\,dx$

Question 1501

[1 marks]Differential equations

In a population of fixed size

p

x

people have disease A and

y = p - x

have disease B, and the rate of spread satisfies

\dfrac{dx}{dt} \propto xy

. Which differential equation follows?

A $\dfrac{dx}{dt} = \dfrac{c}{x(p-x)}$
B $\dfrac{dx}{dt} = cx(p - x)$
C $\dfrac{dx}{dt} = c(p - x)$
D $\dfrac{dx}{dt} = cxp$

Question 1502

[1 marks]Differential equations

To separate

\dfrac{dx}{dt} = cx(p - x)

, the fraction

\dfrac{1}{x(p-x)}

is written in partial fractions as

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

Question 1503

[1 marks]Differential equations

Given

x=\tfrac{p}{10}

t=0

, so

y=p-x=\tfrac{9p}{10}

, find

\dfrac{x}{p-x}

t=0

Answer this when you sit the paper.

Question 1504

[2 marks]Differential equations

Integrating

\dfrac1p\left(\dfrac1x+\dfrac{1}{p-x}\right)dx=c\,dt

gives

\dfrac1p\ln\left(\dfrac{x}{p-x}\right)=ct+C

. Using

\dfrac{x}{p-x}=\tfrac19

t=0

, find

C

in terms of

p

Answer this when you sit the paper.

Question 1505

[1 marks]Differential equations

Rearranging

\dfrac{x}{p-x}=\tfrac19e^{pct}

and solving for

x

gives

x=\dfrac{pe^{pct}}{9+e^{pct}}

. State the constant added to

e^{pct}

in the denominator.

Answer this when you sit the paper.

Question 1506

[2 marks]Differential equations

Separating variables in

\dfrac{dx}{dt}=cx(p-x)

gives which equation?

A $\displaystyle\int\dfrac{dx}{x(p-x)}=\displaystyle\int c\,dt$
B $\displaystyle\int x(p-x)\,dx=\displaystyle\int c\,dt$
C $\displaystyle\int\dfrac{dx}{x}=\displaystyle\int c(p-x)\,dt$
D $\displaystyle\int\dfrac{dp}{x(p-x)}=\displaystyle\int c\,dt$

Question 1507

[2 marks]Differential equations

Since

y=p-x

, and

\dfrac{dx}{dt}=cx(p-x)=cxy

, find

\dfrac{dy}{dt}

in terms of

c

x

and

y

Answer this when you sit the paper.

Question 1508

[3 marks]Differential equations

Using

x=\dfrac{pe^{pct}}{9+e^{pct}}

, find the time

t

at which

y=\tfrac{p}{10}

(i.e.

x=\tfrac{9p}{10}

), giving the answer in the form

t=\dfrac{k\ln3}{pc}

. State the value of

k

Answer this when you sit the paper.

Pure Mathematics Paper 1 June 2017

Sit this paper online

The questions

Question 101

Question 102

Question 201

Question 202

Question 203

Question 301

Question 302

Question 303

Question 304

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 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 1006

Question 1007

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 1301

Question 1302

Question 1303

Question 1304

Question 1305

Question 1306

Question 1401

Question 1402

Question 1403

Question 1404

Question 1405

Question 1406

Question 1407

Question 1408

Question 1501

Question 1502

Question 1503

Question 1504

Question 1505

Question 1506