ZIMSEC A Level Pure Mathematics Paper 1 November 2015

Question 101

[1 marks]remainder theorem

A polynomial

f(x)

leaves remainder

-9

when divided by

(x-3)

and remainder

-6

when divided by

(2x-1)

. Find the remainder when

f(x)

is divided by

(x-3)(2x-1)

A $-15$
B $-\tfrac{6}{5}x - \tfrac{27}{5}$
C $\tfrac{6}{5}x - \tfrac{27}{5}$
D $-\tfrac{27}{5}x - \tfrac{6}{5}$

Question 102

[1 marks]remainder theorem

A polynomial

f(x)

is divided by the quadratic

(x-3)(2x-1)

. Since the divisor has degree 2, what form must the remainder take?

Azero, since it divides exactly
B $ax+b$ , a linear expression
Ca constant $b$
D $ax^2+bx+c$ , a quadratic

Question 103

[2 marks]remainder theorem

Writing

f(x)=(x-3)(2x-1)Q(x)+ax+b

and using

f(3)=-9

and

f\left(\tfrac12\right)=-6

, which pair of simultaneous equations results?

A $3a+b=9$ and $a+2b=12$
B $3a+b=-9$ and $a+2b=-12$
C $3a+b=-9$ and $\tfrac12 a+b=-6$
D $3a-b=-9$ and $a+2b=-12$

Question 201

[1 marks]small changes / percentage change

Given

y = 2x^3 + \dfrac{8}{x}

, find the percentage increase in

y

x

increases from 2 to 2.003.

A $0.33\%$
B $3.30\%$
C $0.66\%$
D $0.15\%$

Question 202

[1 marks]small changes / percentage change

Given

y=2x^3+\dfrac{8}{x}

, what is

\dfrac{dy}{dx}

A $6x^2-\dfrac{8}{x}$
B $2x^2-\dfrac{8}{x^2}$
C $6x^2+\dfrac{8}{x^2}$
D $6x^2-\dfrac{8}{x^2}$

Question 203

[2 marks]small changes / percentage change

x=2

\dfrac{dy}{dx}=6(2)^2-\dfrac{8}{2^2}=22

. Using

\delta y\approx\dfrac{dy}{dx}\delta x

with

\delta x=0.003

, what is

\delta y

, correct to 3 decimal places?

Answer this when you sit the paper.

Question 301

[1 marks]circle geometry / segments and sectors

In a circle of radius

r

, a chord

PQ

subtends a right angle at the centre

O

. Find the area of the minor segment cut off by

PQ

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

Question 302

[1 marks]circle geometry / segments and sectors

In a circle centre

O

radius

r

, chord

PQ

subtends a right angle at

O

and chord

QR

makes an inscribed angle

PQR = \tfrac{\pi}{12}

. The shaded area bounded by chords

PQ

QR

and arc

PR

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

Question 303

[1 marks]circle geometry / segments and sectors

Chord

PQ

subtends a right angle at centre

O

, and inscribed angle

PQR=\dfrac{\pi}{12}

subtends arc

PR

. By the inscribed angle theorem (central angle = twice the inscribed angle), what is the central angle

QOR

A $\dfrac{\pi}{3}$
B $\dfrac{\pi}{12}$
C $\dfrac{\pi}{24}$
D $\dfrac{\pi}{6}$

Question 304

[2 marks]circle geometry / segments and sectors

What is the area of the segment cut off by chord

QR

, given central angle

QOR=\dfrac{\pi}{3}

A $\dfrac12 r^2\left(\dfrac{\pi}{3}+\dfrac{\sqrt3}{2}\right)$
B $\dfrac12 r^2\left(\dfrac{\pi}{6}-\dfrac{\sqrt3}{2}\right)$
C $r^2\left(\dfrac{\pi}{3}-\dfrac{\sqrt3}{2}\right)$
D $\dfrac12 r^2\left(\dfrac{\pi}{3}-\dfrac{\sqrt3}{2}\right)$

Question 401

[1 marks]modulus functions / inequalities

Solve the inequality

|3x+4| < |2x+1|

A $x < -3$ or $x > -1$
B $-\tfrac{4}{3} < x < -\tfrac{1}{2}$
C $-3 < x < -1$
D $-1 < x < 3$

Question 402

[2 marks]modulus functions / inequalities

Solving

3x+4=2x+1

and

3x+4=-(2x+1)

, what are the two

x

-values where

y=|3x+4|

and

y=|2x+1|

intersect?

A $x=-\tfrac43$ and $x=-\tfrac12$
B $x=-3$ and $x=-1$
C $x=-3$ and $x=1$
D $x=3$ and $x=-1$

Question 403

[2 marks]modulus functions / inequalities

x=0

|3x+4|=4

and

|2x+1|=1

, so

|3x+4|>|2x+1|

there. Since

x=0

lies outside

(-3,-1)

, what does this confirm about the solution of

|3x+4|<|2x+1|

Athe solution set is $x>0$ only
Bthere is no solution to the inequality
Cthe solution set is exactly the interval between the two intersection points, $-3<x<-1$
Dthe solution set is everywhere except $-3<x<-1$

Question 501

[1 marks]partial fractions

Express

\dfrac{5x^2 - 5x + 11}{(x-2)(x^2+3)}

in partial fractions.

A $\dfrac{3}{x-2} + \dfrac{2x+1}{x^2+3}$
B $\dfrac{3}{x-2} + \dfrac{2x-1}{x^2+3}$
C $\dfrac{7}{x-2} + \dfrac{2x-1}{x^2+3}$
D $\dfrac{3}{x-2} - \dfrac{2x-1}{x^2+3}$

Question 502

[1 marks]partial fractions

Writing

\dfrac{5x^2-5x+11}{(x-2)(x^2+3)}=\dfrac{A}{x-2}+\dfrac{Bx+C}{x^2+3}

and substituting

x=2

, what is

A

Answer this when you sit the paper.

Question 503

[1 marks]partial fractions

With

A=3

, substituting

x=0

into

5x^2-5x+11=A(x^2+3)+(Bx+C)(x-2)

gives

11=3A-2C

. What is

C

Answer this when you sit the paper.

Question 504

[2 marks]partial fractions

With

A=3

and

C=-1

, substituting

x=1

into

5x^2-5x+11=A(x^2+3)+(Bx+C)(x-2)

gives

11=4A-(B+C)

. What is

B

A $B=2$
B $B=-2$
C $B=1$
D $B=5$

Question 601

[1 marks]binomial expansion

Expand

(3-2x)^{-3}

up to and including the term in

x^3

A $\tfrac{1}{27} - \tfrac{2}{27}x + \tfrac{8}{81}x^2 - \tfrac{80}{729}x^3$
B $\tfrac{1}{27} + \tfrac{2}{9}x + \tfrac{8}{27}x^2 + \tfrac{80}{243}x^3$
C $1 + 2x + \tfrac{8}{3}x^2 + \tfrac{80}{27}x^3$
D $\tfrac{1}{27} + \tfrac{2}{27}x + \tfrac{8}{81}x^2 + \tfrac{80}{729}x^3$

Question 602

[1 marks]binomial expansion

State the set of values of

x

for which the expansion of

(3-2x)^{-3}

is valid.

A $-\tfrac{2}{3} < x < \tfrac{2}{3}$
B $-\tfrac{3}{2} < x < \tfrac{3}{2}$
C $|x| < 1$
D $|x| < 3$

Question 603

[1 marks]binomial expansion

To expand

(3-2x)^{-3}

by the binomial series, it is first written in the form

3^{-3}(1+kx)^{-3}

. What is

k

A $-\dfrac23$
B $\dfrac23$
C $-2$
D $-3$

Question 604

[2 marks]binomial expansion

In the expansion of

(3-2x)^{-3}

, what is the coefficient of

x^2

A $\dfrac{4}{81}$
B $\dfrac{2}{27}$
C $\dfrac{8}{81}$
D $\dfrac{8}{27}$

Question 701

[1 marks]linearising relationships / graphs

Variables

x

and

y

satisfy

y = ax^2 + bx

. Which quantity should be plotted against

x

to obtain a straight line, and what do its gradient and intercept represent?

APlot $\ln y$ against $\ln x$ : gradient $a$ , intercept $\ln b$
BPlot $\tfrac{y}{x}$ against $x$ : gradient $b$ , intercept $a$
CPlot $\tfrac{y}{x}$ against $x$ : gradient $a$ , intercept $b$
DPlot $y$ against $x^2$ : gradient $a$ , intercept $b$

Question 702

[1 marks]linearising relationships / graphs

For the table with

x=3

y=22.3

, what is

\dfrac{y}{x}

, correct to 1 decimal place (needed to plot

\dfrac{y}{x}

against

x

Answer this when you sit the paper.

Question 703

[2 marks]linearising relationships / graphs

Using the plotted points

(3,7.4)

and

(5,13.2)

on the graph of

\dfrac{y}{x}

against

x

, what is the gradient, correct to 1 decimal place?

Answer this when you sit the paper.

Question 704

[1 marks]linearising relationships / graphs

The graph of

\dfrac{y}{x}

against

x

for

y=ax^2+bx

is a straight line of gradient

a=2,9

and

y

-intercept

b=-2

. Use these values to estimate

y

when

x=4

Answer this when you sit the paper.

Question 801

[1 marks]vectors / scalar product

Points

A

and

B

have position vectors

\begin{pmatrix} n-2 \\ n \\ 3-n \end{pmatrix}

and

\begin{pmatrix} n+1 \\ n-4 \\ 2n \end{pmatrix}

. Given

|\overrightarrow{AB}| = 13

and

n > 0

, find

n

A $n = -3$
B $n = 5$
C $n = 13$
D $n = 3$

Question 802

[1 marks]vectors / scalar product

With

n = 5

A = (3, 5, -2)

\overrightarrow{AB} = \begin{pmatrix} 3 \\ -4 \\ 12 \end{pmatrix}

and

\overrightarrow{AC} = \begin{pmatrix} 1 \\ -2 \\ 7 \end{pmatrix}

. Find angle

B\hat{A}C

correct to the nearest

0.1°

A $83.9°$
B $174.0°$
C $6.0°$
D $45.0°$

Question 803

[2 marks]vectors / scalar product

Position vectors give

\overrightarrow{AB}=\begin{pmatrix}3\\-4\\3n-3\end{pmatrix}

. Setting

|\overrightarrow{AB}|=13

, which quadratic in

n

results after simplifying

9+16+(3n-3)^2=169

A $n^2-2n+15=0$
B $n^2-2n-15=0$
C $n^2+2n-15=0$
D $9n^2-18n-144=0$

Question 804

[1 marks]vectors / scalar product

With

n=5

A=(3,5,-2)

and

C=(4,3,5)

. What is

\overrightarrow{AC}

as a column vector, written as three comma-separated numbers?

Answer this when you sit the paper.

Question 805

[1 marks]vectors / scalar product

With

\overrightarrow{AB}=\begin{pmatrix}3\\-4\\12\end{pmatrix}

and

\overrightarrow{AC}=\begin{pmatrix}1\\-2\\7\end{pmatrix}

, what is the dot product

\overrightarrow{AB}\cdot\overrightarrow{AC}

Answer this when you sit the paper.

Question 901

[1 marks]coordinate geometry / circles

Find the equation of the circle with centre

(5, 4)

which touches the line joining

(0, 5)

and

(4, 1)

A $(x-5)^2 + (y-4)^2 = 8$
B $(x-5)^2 + (y-4)^2 = 64$
C $(x-3)^2 + (y-2)^2 = 8$
D $(x-5)^2 + (y-4)^2 = 2\sqrt{2}$

Question 902

[2 marks]coordinate geometry / circles

Solving

y=x-1

together with

y=-x+5

, what is the point of contact between the circle and the line?

A $(4,1)$
B $(3,2)$
C $(2,3)$
D $(1,4)$

Question 903

[1 marks]coordinate geometry / circles

The centre is

(5,4)

and the point of contact is

(3,2)

. What is the square of the radius (that is,

(5-3)^2+(4-2)^2

Answer this when you sit the paper.

Question 904

[3 marks]coordinate geometry / circles

The line joining

(0,5)

and

(4,1)

has gradient

-1

, so its equation is

y=-x+5

. The radius from centre

(5,4)

to the point of tangency is perpendicular to this line. What is the equation of this radius?

A $y=x+1$
B $y=-x-1$
C $y=x-1$
D $y=-x+9$

Question 1001

[1 marks]harmonic form / trigonometric equations

Express

6\cos\theta - 8\sin\theta

in the form

R\cos(\theta + \alpha)

, where

R > 0

and

\alpha

is acute.

A $10\cos(\theta + 53.1°)$
B $14\cos(\theta + 53.1°)$
C $10\cos(\theta + 36.9°)$
D $10\cos(\theta - 53.1°)$

Question 1002

[1 marks]harmonic form / trigonometric equations

Solve the equation

6\cos\theta - 8\sin\theta = -2.5

for

-180° \le \theta \le 180°

A $\theta = 104.5°$ or $\theta = -104.5°$
B $\theta = 51.4°$ or $\theta = 157.6°$
C $\theta = -51.4°$ or $\theta = 157.6°$
D $\theta = 51.4°$ or $\theta = -157.6°$

Question 1003

[1 marks]harmonic form / trigonometric equations

State the minimum and maximum values of

\dfrac{1}{6\cos\theta - 8\sin\theta + 13}

Aminimum $\tfrac{1}{13}$ , maximum $\tfrac{1}{3}$
Bminimum $\tfrac{1}{3}$ , maximum $\tfrac{1}{23}$
Cminimum $\tfrac{1}{23}$ , maximum $\tfrac{1}{3}$
Dminimum $-\tfrac{1}{10}$ , maximum $\tfrac{1}{10}$

Question 1004

[1 marks]harmonic form / trigonometric equations

Writing

6\cos\theta-8\sin\theta=R\cos(\theta+\alpha)

with

R=10

, what is

\alpha

, correct to 1 decimal place (in degrees)?

Answer this when you sit the paper.

Question 1005

[2 marks]harmonic form / trigonometric equations

To solve

10\cos(\theta+53.1°)=-2.5

, the equation

\cos(\theta+53.1°)=-0.25

is formed. What angle (in degrees, correct to 1 decimal place, taking the principal value) has cosine

-0.25

Answer this when you sit the paper.

Question 1006

[1 marks]harmonic form / trigonometric equations

What is the range of the denominator

6\cos\theta-8\sin\theta+13

\theta

varies, given

6\cos\theta-8\sin\theta

ranges from

-10

10

A $3$ to $23$
B $-10$ to $10$
C $0$ to $26$
D $6$ to $20$

Question 1101

[1 marks]parametric differentiation / normals

A curve has parametric equations

x = \csc\Phi

and

y = \cot\Phi

. Show that

\dfrac{dy}{dx}

equals

A $\cos\Phi$
B $\sec\Phi$
C $-\sec\Phi$
D $\csc\Phi$

Question 1102

[1 marks]parametric differentiation / normals

For the curve

x = \csc\Phi

y = \cot\Phi

, find the equation of the normal at the point where

\Phi = \tfrac{\pi}{6}

A $y = -\tfrac{2}{\sqrt{3}}x + 2\sqrt{3}$
B $y = \tfrac{\sqrt{3}}{2}x$
C $y = \tfrac{2}{\sqrt{3}}x - \tfrac{\sqrt{3}}{3}$
D $y = -\tfrac{\sqrt{3}}{2}x + 2\sqrt{3}$

Question 1103

[1 marks]parametric differentiation / normals

\Phi=\dfrac{\pi}{6}

x=\csc\Phi

and

y=\cot\Phi

. What are

x

and

y

, written as \"x,y\"?

Answer this when you sit the paper.

Question 1104

[1 marks]parametric differentiation / normals

\Phi=\dfrac{\pi}{6}

, using

\dfrac{dy}{dx}=\sec\Phi

, what is

\dfrac{dy}{dx}

(as a fraction over

\sqrt3

Answer this when you sit the paper.

Question 1105

[1 marks]parametric differentiation / normals

Given the tangent gradient at

\Phi=\dfrac{\pi}{6}

\dfrac{2}{\sqrt3}

, what is the gradient of the normal there?

Answer this when you sit the paper.

Question 1106

[2 marks]parametric differentiation / normals

For the parametric curve

x=\csc\Phi

y=\cot\Phi

, what are

\dfrac{dx}{d\Phi}

and

\dfrac{dy}{d\Phi}

A $\dfrac{dx}{d\Phi}=-\csc\Phi\cot\Phi,\ \dfrac{dy}{d\Phi}=-\csc^2\Phi$
B $\dfrac{dx}{d\Phi}=-\csc^2\Phi,\ \dfrac{dy}{d\Phi}=-\csc\Phi\cot\Phi$
C $\dfrac{dx}{d\Phi}=\csc\Phi\cot\Phi,\ \dfrac{dy}{d\Phi}=\csc^2\Phi$
D $\dfrac{dx}{d\Phi}=-\csc\Phi\cot\Phi,\ \dfrac{dy}{d\Phi}=\csc^2\Phi$

Question 1201

[1 marks]implicit differentiation / Maclaurin expansion

Given that

\ln y = xy

with

y > 0

, show that

\dfrac{dy}{dx}

equals

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

Question 1202

[1 marks]implicit differentiation / Maclaurin expansion

Given

\ln y = xy

with

y > 0

, find the Maclaurin expansion of

y

up to and including the term in

x^2

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

Question 1203

[2 marks]implicit differentiation / Maclaurin expansion

Differentiating

\ln y=xy

implicitly with respect to

x

(product rule on the right), which equation results before solving for

\dfrac{dy}{dx}

A $\dfrac1y\dfrac{dy}{dx}=xy$
B $\dfrac1y\dfrac{dy}{dx}=x+y\dfrac{dy}{dx}$
C $y\dfrac{dy}{dx}=y+x\dfrac{dy}{dx}$
D $\dfrac1y\dfrac{dy}{dx}=y+x\dfrac{dy}{dx}$

Question 1204

[2 marks]implicit differentiation / Maclaurin expansion

x=0

(where

y=1

and

\dfrac{dy}{dx}=1

), using

\dfrac{d^2y}{dx^2}=\dfrac{2y\frac{dy}{dx}-xy^2\frac{dy}{dx}+y^3}{(1-xy)^2}

, what is

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

x=0

Answer this when you sit the paper.

Question 1205

[1 marks]implicit differentiation / Maclaurin expansion

The Maclaurin series is

y=y(0)+y'(0)x+\left[?\right]x^2+\dots

. What fills the bracket, the coefficient of

x^2

in general?

A $2y''(0)$
B $\dfrac{y''(0)}{2!}$
C $y''(0)$
D $\dfrac{y''(0)}{3!}$

Question 1206

[2 marks]implicit differentiation / Maclaurin expansion

x=0

\ln y=xy=0

gives

y=1

, and using

\dfrac{dy}{dx}=\dfrac{y^2}{1-xy}

then gives

\dfrac{dy}{dx}=1

there. What are

y(0)

and

y'(0)

, written as \"y(0),y'(0)\"?

Answer this when you sit the paper.

Question 1301

[1 marks]integration / areas and volumes of revolution

Find the area of the region bounded by the

x

-axis and the curve

y = \sin x + \sqrt{3}\cos x

between

x = -\tfrac{\pi}{3}

and

x = \tfrac{2\pi}{3}

A $4$ square units
B $2\sqrt{3}$ square units
C $\pi$ square units
D $2$ square units

Question 1302

[1 marks]integration / areas and volumes of revolution

Express

\sin x + \sqrt{3}\cos x

in the form

R\sin(x + \alpha)

A $4\sin\left(x + \tfrac{\pi}{3}\right)$
B $\sqrt{3}\sin\left(x + \tfrac{\pi}{3}\right)$
C $2\sin\left(x + \tfrac{\pi}{6}\right)$
D $2\sin\left(x + \tfrac{\pi}{3}\right)$

Question 1303

[1 marks]integration / areas and volumes of revolution

The region bounded by the

x

-axis and

y = \sin x + \sqrt{3}\cos x

from

x = -\tfrac{\pi}{3}

x = \tfrac{2\pi}{3}

is rotated completely about the

x

-axis. Find the volume generated.

A $2\pi^2$ cubic units
B $4\pi^2$ cubic units
C $4\pi$ cubic units
D $\pi^2$ cubic units

Question 1304

[2 marks]integration / areas and volumes of revolution

What is

\displaystyle\int(\sin x+\sqrt3\cos x)\,dx

A $-\sin x+\sqrt3\cos x+C$
B $-\cos x+\sqrt3\sin x+C$
C $\cos x+\sqrt3\sin x+C$
D $-\cos x-\sqrt3\sin x+C$

Question 1305

[1 marks]integration / areas and volumes of revolution

What is the value of

-\cos x+\sqrt3\sin x

x=\dfrac{2\pi}{3}

Answer this when you sit the paper.

Question 1306

[2 marks]integration / areas and volumes of revolution

Using

\sin^2\theta=\dfrac{1-\cos2\theta}{2}

, what does

4\sin^2\left(x+\dfrac{\pi}{3}\right)

equal, in terms of

\cos\left(2x+\dfrac{2\pi}{3}\right)

A $2-2\cos\left(2x+\dfrac{2\pi}{3}\right)$
B $2+2\cos\left(2x+\dfrac{2\pi}{3}\right)$
C $4-4\cos\left(2x+\dfrac{2\pi}{3}\right)$
D $1-\cos\left(2x+\dfrac{2\pi}{3}\right)$

Question 1307

[1 marks]integration / areas and volumes of revolution

In the volume integral, the term

\sin\left(2x+\dfrac{2\pi}{3}\right)

is evaluated at both

x=-\dfrac{\pi}{3}

and

x=\dfrac{2\pi}{3}

. What value does it take at both limits?

A $\dfrac{\sqrt3}{2}$
B $0$
C $1$
D $-1$

Question 1401

[1 marks]complex numbers / simultaneous equations

Complex numbers

z

and

w

satisfy

z + iw = 2

and

iz + w = 2 + 3i

. Find

z

and

w

A $z = \tfrac{5}{2} + i$ , $w = 1 - \tfrac{1}{2}i$
B $z = \tfrac{5}{2} - i$ , $w = 1 + \tfrac{1}{2}i$
C $z = 2 - i$ , $w = 1 + 3i$
D $z = 1 + \tfrac{1}{2}i$ , $w = \tfrac{5}{2} - i$

Question 1402

[1 marks]complex numbers / simultaneous equations

Given

z = \tfrac{5}{2} - i

and

w = 1 + \tfrac{1}{2}i

, find

|zw|

correct to 2 decimal places.

A $2.69$
B $3.00$
C $3.01$
D $9.06$

Question 1403

[1 marks]complex numbers / simultaneous equations

Given

z = \tfrac{5}{2} - i

and

w = 1 + \tfrac{1}{2}i

, find

\arg\dfrac{z}{w}

correct to the nearest

0.1°

A $48.4°$
B $131.6°$
C $-48.4°$
D $-41.6°$

Question 1404

[2 marks]complex numbers / simultaneous equations

Equating real and imaginary parts of

z+iw=2

and

iz+w=2+3i

(with

z=x+iy

w=a+ib

), which system of 4 equations results?

A $x-b=2,\ a+y=0,\ a-y=2,\ x+b=2$
B $x-b=2,\ a-y=0,\ a+y=2,\ x+b=3$
C $x-b=2,\ a+y=0,\ a-y=2,\ x+b=3$
D $x+b=2,\ a-y=0,\ a+y=2,\ x-b=3$

Question 1405

[1 marks]complex numbers / simultaneous equations

Solving

a+y=0

and

a-y=2

together, what is the pair

(a,y)

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

Question 1406

[1 marks]complex numbers / simultaneous equations

Before taking its modulus, what does

zw=\left(\dfrac52-i\right)\left(1+\dfrac12 i\right)

simplify to?

A $3+\dfrac14 i$
B $3-\dfrac14 i$
C $\dfrac52+\dfrac14 i$
D $2+\dfrac14 i$

Question 1407

[2 marks]complex numbers / simultaneous equations

Before taking its argument, rationalising

\dfrac{z}{w}=\dfrac{\frac52-i}{1+\frac12 i}

by the conjugate of

w

gives which complex number?

A $\dfrac85-\dfrac95 i$
B $\dfrac85+\dfrac95 i$
C $\dfrac95-\dfrac85 i$
D $\dfrac25-\dfrac35 i$

Question 1501

[1 marks]trapezium rule / integration

Use the trapezium rule with 4 equal intervals to estimate

\displaystyle\int_{\pi/6}^{\pi/2} \csc^2 x\,dx

, correct to 3 decimal places.

A $0.904$
B $1.732$
C $1.808$
D $1.904$

Question 1502

[1 marks]trapezium rule / integration

Evaluate

\displaystyle\int_{\pi/6}^{\pi/2} \csc^2 x\,dx

exactly.

A $\tfrac{1}{\sqrt{3}}$
B $2$
C $\tfrac{\pi}{3}$
D $\sqrt{3}$

Question 1503

[1 marks]trapezium rule / integration

The trapezium rule gives

1.808

as an estimate of

\displaystyle\int_{\pi/6}^{\pi/2} \csc^2 x\,dx = \sqrt{3}

. Find the percentage error.

A $8.8\%$
B $4.4\%$
C $1.8\%$
D $0.44\%$

Question 1504

[1 marks]trapezium rule / integration

Using the trapezium rule with 4 equal intervals on

\displaystyle\int_{\pi/6}^{\pi/2}\csc^2x\,dx

, what is the strip width

h

Answer this when you sit the paper.

Question 1505

[1 marks]trapezium rule / integration

What is

\csc^2\left(\dfrac{\pi}{4}\right)

Answer this when you sit the paper.

Question 1506

[1 marks]trapezium rule / integration

What is

\csc^2\left(\dfrac{5\pi}{12}\right)

, correct to 6 decimal places?

Answer this when you sit the paper.

Question 1507

[1 marks]trapezium rule / integration

What is

\displaystyle\int\csc^2x\,dx

A $-\tan x+C$
B $-\csc x\cot x+C$
C $-\cot x+C$
D $\cot x+C$

Question 1508

[1 marks]trapezium rule / integration

The exact integral is

\sqrt3\approx1.732

and the trapezium estimate is

1.808

. What is

|\sqrt3-1.808|

, correct to 3 decimal places?

Answer this when you sit the paper.

Question 1509

[2 marks]trapezium rule / integration

In evaluating

\left[-\cot x\right]_{\pi/6}^{\pi/2}

, what are

\cot\left(\dfrac{\pi}{6}\right)

and

\cot\left(\dfrac{\pi}{2}\right)

, written as \"cot(pi/6),cot(pi/2)\"?

Answer this when you sit the paper.

Question 1601

[1 marks]graphical solution / Newton-Raphson method

Let

f(x) = (x-1)e^x - x

. Which calculation shows that a root of

(x-1)e^x = x

lies between

x = 1.2

and

x = 1.5

A $f(1.2) = -0.53977$ and $f(1.5) = -0.74084$
B $f(1.2) = 1.2$ and $f(1.5) = 1.5$
C $f(1.2) = 0.53977$ and $f(1.5) = 0.74084$
D $f(1.2) = -0.53977$ and $f(1.5) = 0.74084$

Question 1602

[1 marks]graphical solution / Newton-Raphson method

Taking

x_1 = 1.2

, apply the Newton-Raphson method twice to

(x-1)e^x = x

and give the root correct to 4 decimal places.

A $1.2000$
B $1.3508$
C $1.3796$
D $1.5000$

Question 1603

[1 marks]graphical solution / Newton-Raphson method

Dividing both sides of

(x-1)e^x=x

(x-1)

, what does

e^x

equal?

A $x(x-1)$
B $x+1$
C $\dfrac{x}{x-1}$
D $\dfrac{x-1}{x}$

Question 1604

[2 marks]graphical solution / Newton-Raphson method

To show

(x-1)e^x=x

has two real roots by sketching, which two graphs are drawn on the same axes?

A $y=e^{x-1}$ and $y=x$
B $y=x-1$ and $y=x$
C $y=e^x$ and $y=\dfrac{x}{x-1}$
D $y=e^x$ and $y=x-1$

Question 1605

[1 marks]graphical solution / Newton-Raphson method

For

f(x)=(x-1)e^x-x

, what is

f'(x)

A $e^x-1$
B $(x-1)e^x-1$
C $xe^x$
D $xe^x-1$

Question 1606

[2 marks]graphical solution / Newton-Raphson method

Taking

x_1=1.2

, one Newton-Raphson iteration

x_2=x_1-\dfrac{f(x_1)}{f'(x_1)}

gives which value, correct to 4 decimal places?

Answer this when you sit the paper.

Question 1607

[2 marks]graphical solution / Newton-Raphson method

For

f(x)=(x-1)e^x-x

, what are

f(1.2)

and

f(1.5)

, each correct to 5 decimal places, written as \"f(1.2),f(1.5)\"?

Answer this when you sit the paper.

Question 1701

[1 marks]differential equations / related rates

A tank with a

3

m square base and height

5

m leaks at a rate proportional to the wetted area. Given that the level falls at

0.3

m/h when the depth is 3 m, the depth

y

after

t

hours satisfies

A $\dfrac{dy}{dt} = \dfrac{-3}{10}(4y + 3)$
B $\dfrac{dy}{dt} = \dfrac{-3}{50}y$
C $\dfrac{dy}{dt} = \dfrac{-3}{50}\left(\dfrac{4y}{3} + 1\right)$
D $\dfrac{dy}{dt} = \dfrac{-3}{50}(12y + 9)$

Question 1702

[1 marks]differential equations / related rates

Solve

\dfrac{dy}{dt} = \dfrac{-3}{50}\left(\dfrac{4y}{3} + 1\right)

given that the tank is initially full to a depth of 5 m.

A $y = \tfrac{1}{4}\left(23e^{-2t/25} + 3\right)$
B $y = 5e^{-2t/25}$
C $y = \tfrac{1}{4}\left(23e^{-2t/25} - 3\right)$
D $y = \tfrac{1}{4}\left(23e^{-3t/50} - 3\right)$

Question 1703

[1 marks]differential equations / related rates

For a tank in which the depth is

y = \tfrac{1}{4}\left(23e^{-2t/25} - 3\right)

metres and the full depth is 5 m, how long does it take to become half full?

A $5.71$ hours
B $7.13$ hours
C $8.94$ hours
D $12.5$ hours

Question 1704

[2 marks]differential equations / related rates

The tank has a

3

m square base and height

5

m. What is the total area in contact with water (base plus the 4 sides) when the depth is

y

A $12y+9$
B $9y+12$
C $3y+9$
D $12y+3$

Question 1705

[1 marks]differential equations / related rates

Since the base area is

9

^2

, the volume of water at depth

y

V=9y

. What is

\dfrac{dV}{dt}

in terms of

\dfrac{dy}{dt}

A $\dfrac{9}{dy/dt}$
B $y\dfrac{dy}{dt}$
C $9\dfrac{dy}{dt}$
D $\dfrac{dy}{dt}$

Question 1706

[1 marks]differential equations / related rates

Using

9\dfrac{dy}{dt}=-k(12y+9)

with

y=3

and

\dfrac{dy}{dt}=-0.3

, what is

k

Answer this when you sit the paper.

Question 1707

[2 marks]differential equations / related rates

Separating variables in

\dfrac{dy}{dt}=-\dfrac{3}{50}\left(\dfrac{4y}{3}+1\right)

, which equation results (before integrating)?

A $\dfrac{dy}{4y+3}=-\dfrac{3}{50}\,dt$
B $\dfrac{3\,dy}{4y+3}=\dfrac{3}{50}\,dt$
C $(4y+3)\,dy=-\dfrac{3}{50}\,dt$
D $\dfrac{3\,dy}{4y+3}=-\dfrac{3}{50}\,dt$

Question 1708

[2 marks]differential equations / related rates

Integrating gives

\dfrac34\ln(4y+3)=-\dfrac{3}{50}t+C

. Using the initial condition

y=5

t=0

, what is

C

A $\ln23$
B $\dfrac34\ln5$
C $\dfrac34\ln23$
D $\dfrac34\ln20$

Question 1709

[1 marks]differential equations / related rates

From

\dfrac34\ln(4y+3)=-\dfrac{3}{50}t+\dfrac34\ln23

, dividing by

\dfrac34

and rearranging gives

\ln\left(\dfrac{4y+3}{23}\right)

equal to which expression in

t

A $-\dfrac{3}{50}t$
B $-\dfrac{4}{25}t$
C $\dfrac{2}{25}t$
D $-\dfrac{2}{25}t$

Question 1710

[1 marks]differential equations / related rates

The tank is 'half full' when the depth

y

equals what value (half of the full depth of 5 m)?

Answer this when you sit the paper.

Pure Mathematics Paper 1 November 2015

Sit this paper online

The questions

Question 101

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Question 601

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Question 1001

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Question 1201

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Question 1501

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Question 1601

Question 1602