ZIMSEC A Level Pure Mathematics Paper 2 June 2012

Section a

Section a, Question 1

[2 marks]Numerical methods, Newton-Raphson

Use the Newton-Raphson method to find, correct to 3 decimal places, the root of the equation

\sin x+\cos x=e^{x}-1

that lies between

-4

and

-3

Answer this when you sit the paper.

[2 marks]Numerical methods, Newton-Raphson

The equation

\sin x+\cos x=e^{x}-1

is to be solved by the Newton-Raphson method, writing

f(x)=\sin x+\cos x-e^{x}+1

. Which of these is the correct iterative formula?

A $x_{n+1}=x_n-\dfrac{\sin x_n+\cos x_n-e^{x_n}+1}{\cos x_n+\sin x_n-e^{x_n}}$
B $x_{n+1}=x_n+\dfrac{\sin x_n+\cos x_n-e^{x_n}+1}{\cos x_n-\sin x_n-e^{x_n}}$
C $x_{n+1}=x_n-\dfrac{\sin x_n+\cos x_n-e^{x_n}+1}{\cos x_n-\sin x_n-e^{x_n}}$
D $x_{n+1}=x_n-\dfrac{\cos x_n-\sin x_n-e^{x_n}}{\sin x_n+\cos x_n-e^{x_n}+1}$

[2 marks]Numerical methods, Newton-Raphson

For

f(x)=\sin x+\cos x-e^{x}+1

it is found that

f(-4)=1,085

and

f(-3)=-0,181

. Why does this show that the equation

\sin x+\cos x=e^{x}-1

has a root between

-4

and

-3

ABecause $f$ is continuous on $[-4,-3]$ and changes sign across it, so it takes the value zero inside.
BBecause $f(-4)$ is larger than $f(-3)$ , and a decreasing function is guaranteed to have a root somewhere.
CBecause the two values differ by more than 1, which is what guarantees that a root lies between them.
DBecause $-4$ and $-3$ are consecutive integers, and every interval of length 1 contains at least one root.

Section a, Question 2

[2 marks]Mathematical induction and series

Given that

\displaystyle\sum_{r=1}^{n}(r+3)(2r+1)=\frac{n}{6}\left(4n^{2}+27n+41\right)

, evaluate

\displaystyle\sum_{r=1}^{10}(r+3)(2r+1)

Answer this when you sit the paper.

[2 marks]Mathematical induction and series

Expand and simplify the product

(r+3)(2r+1)

Answer this when you sit the paper.

[2 marks]Mathematical induction and series

In proving by induction that

\displaystyle\sum_{r=1}^{n}(r+3)(2r+1)=\frac{n}{6}\left(4n^{2}+27n+41\right)

, what must be checked in the base case

n=1

AThat the formula is a quadratic in $n$ multiplied by $n$ , which it plainly is for every value of $n$ .
BThat $\dfrac{n}{6}\left(4n^{2}+27n+41\right)$ is a whole number when $n=1$ , since a sum must be a whole number.
CThat $\dfrac{1}{6}(4+27+41)$ equals 1, which is the number of terms in the sum at that stage.
DThat both sides equal 12, since the sum is $(1+3)(2+1)$ and the formula gives $\dfrac{1}{6}(4+27+41)$ .

Section a, Question 3

[2 marks]Differentiation, stationary values

An isosceles triangle has a constant perimeter of

2k

and its two equal sides are each of length

x

cm. Find, in terms of

x

and

k

, the length of the base of the triangle.

Answer this when you sit the paper.

[2 marks]Differentiation, stationary values

An isosceles triangle has a constant perimeter of

2k

, its two equal sides are each of length

x

cm and its base is therefore

2k-2x

. Find, in terms of

x

and

k

, the perpendicular height of the triangle.

Answer this when you sit the paper.

[2 marks]Differentiation, stationary values

An isosceles triangle has a constant perimeter of

2k

, equal sides of length

x

cm each, base

2k-2x

and perpendicular height

\sqrt{2kx-k^{2}}

. Which expression gives its area

A

in terms of

x

and

k

A $A=(2k-2x)\sqrt{2kx-k^{2}}$
B $A=\dfrac{1}{2}x\sqrt{2kx-k^{2}}$
C $A=(k-x)\left(2kx-k^{2}\right)$
D $A=(k-x)\sqrt{2kx-k^{2}}$

[2 marks]Differentiation, stationary values

The area of an isosceles triangle of constant perimeter

2k

, whose equal sides are each

x

cm, is

A=(k-x)\sqrt{2kx-k^{2}}

. Find, in terms of

k

, the value of

x

at which the area is stationary.

Answer this when you sit the paper.

Section a, Question 4

[2 marks]Differential equations, exponential decay

A car is bought for

\$4\,000

. A model assumes that the value

\$V

of the car,

t

months after purchase, decreases at a rate which is proportional to

V

. Which differential equation states this model?

A $\dfrac{dV}{dt}=-kV$ , with $k>0$
B $\dfrac{dV}{dt}=kV$ , with $k>0$
C $\dfrac{dV}{dt}=-kt$ , with $k>0$
D $\dfrac{dV}{dt}=-\dfrac{k}{V}$ , with $k>0$

[3 marks]Differential equations, exponential decay

A car bought for

\$4\,000

has value

\$V

after

t

months, where

V=4\,000e^{-kt}

. After 3 years the car is valued at

\$2\,000

. Find the value of

k

, correct to 4 decimal places.

Answer this when you sit the paper.

[3 marks]Differential equations, exponential decay

A car bought for

\$4\,000

has value

\$V

after

t

months, where

V=4\,000e^{-kt}

and

k=\dfrac{\ln 2}{36}

. Calculate the value of the car when it is 15 months old, giving your answer to the nearest dollar.

Answer this when you sit the paper.

[3 marks]Differential equations, exponential decay

A car bought for

\$4\,000

has value

\$V

after

t

months, where

V=4\,000e^{-kt}

and

k=\dfrac{\ln 2}{36}

. Calculate the age of the car when its value is

\$1\,600

, giving your answer to the nearest month.

Answer this when you sit the paper.

Section a, Question 5

[3 marks]Vectors, lines and planes

The lines

L_1

and

L_2

have vector equations

\mathbf{r}=(2+\lambda)\mathbf{i}+(-3+2\lambda)\mathbf{j}+(1+2\lambda)\mathbf{k}

and

\mathbf{r}=(8+3\mu)\mathbf{i}+(5+2\mu)\mathbf{j}+(13+6\mu)\mathbf{k}

respectively. Find the coordinates of their point of intersection.

Answer this when you sit the paper.

[2 marks]Vectors, lines and planes

The line

L_1

has vector equation

\mathbf{r}=(2+\lambda)\mathbf{i}+(-3+2\lambda)\mathbf{j}+(1+2\lambda)\mathbf{k}

and the plane

\pi

2x-z=3

. Which statement verifies that

L_1

lies wholly in

\pi

AThe direction of $L_1$ is perpendicular to the normal of $\pi$ , which on its own settles the matter entirely.
BThe point $(2,-3,1)$ satisfies the equation $2x-z=3$ , which on its own settles the matter entirely.
C $2(2+\lambda)-(1+2\lambda)=3$ holds when $\lambda=0$ , so the whole of the line $L_1$ must lie inside $\pi$ .
D $2(2+\lambda)-(1+2\lambda)=3$ for every $\lambda$ , so every point of $L_1$ satisfies the plane's equation.

[3 marks]Vectors, lines and planes

The plane

\pi

2x-z=3

and A is the point with position vector

12\mathbf{i}+5\mathbf{j}+6\mathbf{k}

. B is the foot of the perpendicular from A to

\pi

. Which is a vector equation of the line AB?

A $\mathbf{r}=12\mathbf{i}+5\mathbf{j}+6\mathbf{k}+t\left(2\mathbf{i}+\mathbf{j}-\mathbf{k}\right)$
B $\mathbf{r}=12\mathbf{i}+5\mathbf{j}+6\mathbf{k}+t\left(2\mathbf{i}-\mathbf{k}\right)$
C $\mathbf{r}=2\mathbf{i}-\mathbf{k}+t\left(12\mathbf{i}+5\mathbf{j}+6\mathbf{k}\right)$
D $\mathbf{r}=12\mathbf{i}+5\mathbf{j}+6\mathbf{k}+t\left(3\mathbf{i}\right)$

[3 marks]Vectors, lines and planes

The plane

\pi

2x-z=3

and A is the point with position vector

12\mathbf{i}+5\mathbf{j}+6\mathbf{k}

. B is the foot of the perpendicular from A to

\pi

, so that the line AB is

\mathbf{r}=12\mathbf{i}+5\mathbf{j}+6\mathbf{k}+t(2\mathbf{i}-\mathbf{k})

. Find the position vector of B.

Answer this when you sit the paper.

Section a, Question 6

[2 marks]Complex numbers and polynomial roots

The complex number

a=p+iq

, with

p

and

q

real and conjugate

\bar{a}

, satisfies

4a\bar{a}+12i=8a+16

. Find the value of

q

Answer this when you sit the paper.

[3 marks]Complex numbers and polynomial roots

The complex number

a=p+iq

, with

p

and

q

real and conjugate

\bar{a}

, satisfies

4a\bar{a}+12i=8a+16

, and comparing imaginary parts gives

q=\dfrac{3}{2}

. Which equation must

p

satisfy?

A $4p^{2}-8p-7=0$
B $4p^{2}+8p+7=0$
C $4p^{2}-8p+25=0$
D $4p^{2}-8p-16=0$

[2 marks]Complex numbers and polynomial roots

The polynomial

p(x)=2x^{4}+x^{3}+17x^{2}+9x-9

. Evaluate

p(3i)

Answer this when you sit the paper.

[1 marks]Complex numbers and polynomial roots

The polynomial

p(x)=2x^{4}+x^{3}+17x^{2}+9x-9

has real coefficients and

3i

is a root of

p(x)=0

. State the other complex root.

Answer this when you sit the paper.

[2 marks]Complex numbers and polynomial roots

The polynomial

p(x)=2x^{4}+x^{3}+17x^{2}+9x-9

has

3i

and

-3i

as roots, so

x^{2}+9

is a factor. What is the other quadratic factor?

A $x^{2}+2x-9$
B $2x^{2}-x-1$
C $2x^{2}+x-1$
D $2x^{2}+x+1$

[3 marks]Complex numbers and polynomial roots

The polynomial

p(x)=2x^{4}+x^{3}+17x^{2}+9x-9

factorises as

(x^{2}+9)(2x^{2}+x-1)

. Find the two real roots of

p(x)=0

Answer this when you sit the paper.

Section a, Question 7

[3 marks]Matrices, transformations and simultaneous equations

The transformation with matrix

\begin{pmatrix}-2&1\\3&2\end{pmatrix}

maps a line onto the line

y'=6x'+8

. Find the equation of the original line, giving your answer in the form

ax+by=c

where

a

b

and

c

are integers.

Answer this when you sit the paper.

[3 marks]Matrices, transformations and simultaneous equations

Given

M=\begin{pmatrix}3&2&1\\-2&2&0\\1&0&0\end{pmatrix}

and

N=\begin{pmatrix}0&1&3\\1&4&-2\\1&0&2\end{pmatrix}

, find

MN

A $\begin{pmatrix}0&2&3\\-2&8&0\\1&0&0\end{pmatrix}$
B $\begin{pmatrix}3&2&0\\11&6&1\\7&-10&3\end{pmatrix}$
C $\begin{pmatrix}3&11&7\\2&6&-10\\0&1&3\end{pmatrix}$
D $\begin{pmatrix}3&11&7\\-2&6&10\\0&1&3\end{pmatrix}$

[2 marks]Matrices, transformations and simultaneous equations

Given

M=\begin{pmatrix}3&2&1\\-2&2&0\\1&0&0\end{pmatrix}

and

N=\begin{pmatrix}0&1&3\\1&4&-2\\1&0&2\end{pmatrix}

, find the determinant of

MN

Answer this when you sit the paper.

[3 marks]Matrices, transformations and simultaneous equations

Given that

MN=\begin{pmatrix}3&11&7\\2&6&-10\\0&1&3\end{pmatrix}

and

\det(MN)=32

, find

(MN)^{-1}

A $32\begin{pmatrix}28&-26&-152\\-6&9&44\\2&-3&-4\end{pmatrix}$
B $\dfrac{1}{32}\begin{pmatrix}28&-26&-152\\-6&9&44\\2&-3&-4\end{pmatrix}$
C $\dfrac{1}{32}\begin{pmatrix}28&-6&2\\-26&9&-3\\-152&44&-4\end{pmatrix}$
D $\dfrac{1}{32}\begin{pmatrix}3&11&7\\2&6&-10\\0&1&3\end{pmatrix}$

[3 marks]Matrices, transformations and simultaneous equations

Given that

(MN)^{-1}=\dfrac{1}{32}\begin{pmatrix}28&-26&-152\\-6&9&44\\2&-3&-4\end{pmatrix}

, find the values of

x

y

and

z

for which

MN\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}3\\-6\\3\end{pmatrix}

Answer this when you sit the paper.

Section b

Section b, Question 8

[2 marks]Kinematics, velocity-time graphs

A car passes a fixed point A at

10\ \mathrm{ms^{-1}}

and holds that speed for

t_1

seconds. It then accelerates uniformly over the next

t_2

seconds to

15\ \mathrm{ms^{-1}}

, and then decelerates uniformly to rest in a further

t_3

seconds. Given that the acceleration and the deceleration are equal in magnitude, express

t_3

in terms of

t_2

Answer this when you sit the paper.

[2 marks]Kinematics, velocity-time graphs

A car passes a fixed point A at

10\ \mathrm{ms^{-1}}

and holds that speed for

t_1

seconds, then accelerates uniformly over

t_2

seconds to

15\ \mathrm{ms^{-1}}

, then decelerates uniformly to rest in a further

t_3

seconds. What does its velocity-time graph look like?

AA horizontal segment at $v=10$ , then a segment rising to $v=15$ , then a segment falling to $v=0$ .
BA single straight line rising from $v=10$ to $v=15$ and then falling to zero.
CA curve rising from $v=10$ to $v=15$ , then a horizontal segment, then a curve to zero.
DA horizontal segment at $v=10$ , then a segment falling to zero, then a segment rising to $v=15$ .

Section b, Question 9

[2 marks]Projectile motion

A ball is projected from the top of a building 40 m high with an initial speed of

20\ \mathrm{ms^{-1}}

at an angle of

30^\circ

to the horizontal, and lands on the ground at P. Taking

g=9,81\ \mathrm{ms^{-2}}

, find the time of flight of the ball, correct to 3 significant figures.

Answer this when you sit the paper.

[3 marks]Projectile motion

A ball is projected from the top of a building 40 m high with an initial speed of

20\ \mathrm{ms^{-1}}

at an angle of

30^\circ

to the horizontal, and lands on the ground at P after 4,05 s. Taking

g=9,81\ \mathrm{ms^{-2}}

, find the angle, to the nearest degree, that the ball's direction of motion at P makes with the horizontal.

Answer this when you sit the paper.

Section b, Question 10

[3 marks]Newton's laws, connected particles

Two particles of mass

m_1

kg and

m_2

kg, where

m_1>m_2

, are connected by a light inelastic string passing over a smooth fixed pulley. Find, in terms of

g

m_1

and

m_2

, the acceleration of the system.

Answer this when you sit the paper.

[2 marks]Newton's laws, connected particles

Two particles of mass

m_1

kg and

m_2

kg, where

m_1>m_2

, hang either side of a smooth fixed pulley on a light inelastic string. Why is the tension taken to be the same throughout the string?

ABecause the two masses are equal, so the pull on one side balances the pull on the other side exactly.
BBecause the string is inelastic, and an inelastic string cannot transmit any force along its own length.
CBecause the string is light and the pulley smooth, so neither weight nor friction can change the tension.
DBecause the acceleration has the same magnitude on both sides, which forces the two tensions to be equal.

[2 marks]Newton's laws, connected particles

Two particles of mass

m_1

kg and

m_2

kg, where

m_1>m_2

, are connected by a light inelastic string over a smooth fixed pulley, and the system accelerates at

a=\dfrac{(m_1-m_2)g}{m_1+m_2}

. What is the tension in the string?

A $\dfrac{(m_1-m_2)g}{m_1+m_2}$
B $m_1g\left(\dfrac{2m_2}{m_1+m_2}\right)$
C $\dfrac{(m_1+m_2)g}{2}$
D $m_2g\left(\dfrac{2m_1}{m_1-m_2}\right)$

Section b, Question 11

[2 marks]Friction and equilibrium of forces

A particle of weight 20 N rests on a rough horizontal surface. A force

P

N at an angle

\theta

to the horizontal is applied until the particle is on the point of moving. At that instant the normal force is

R=16\mathbf{j}

and the frictional force is

F=-9\mathbf{i}

. Find the coefficient of friction.

Answer this when you sit the paper.

[2 marks]Friction and equilibrium of forces

A particle rests on a rough horizontal surface with normal force

R=16\mathbf{j}

and frictional force

F=-9\mathbf{i}

acting on it. Find the magnitude of the contact force between the particle and the surface, correct to 3 significant figures.

Answer this when you sit the paper.

[2 marks]Friction and equilibrium of forces

A particle of weight 20 N rests on a rough horizontal surface. A force

P

N at an angle

\theta

above the horizontal is applied until the particle is on the point of moving, at which instant the normal force is 16 N and the frictional force is 9 N. Find

\theta

, to the nearest degree.

Answer this when you sit the paper.

[2 marks]Friction and equilibrium of forces

A particle of weight 20 N rests on a rough horizontal surface. A force

P

N at an angle

\theta

above the horizontal is applied until the particle is on the point of moving, at which instant the normal force is 16 N and the frictional force is 9 N. Find

P

, correct to 3 significant figures.

Answer this when you sit the paper.

Section c

Section c, Question 12

[2 marks]Geometric distribution

The random variable

W

has a geometric distribution,

W\sim\mathrm{Geo}(p)

, with

\mathrm{Var}(W)=30

. Find the value of

p

Answer this when you sit the paper.

[2 marks]Geometric distribution

The random variable

W

has a geometric distribution,

W\sim\mathrm{Geo}(p)

, with

\mathrm{Var}(W)=30

. Find

\mathrm{E}(W)

Answer this when you sit the paper.

Section c, Question 13

[3 marks]Conditional probability, tree diagrams

A court passes only two verdicts, convicted or discharged, and 80% of its verdicts are convictions. When the verdict is convicted the probability that the accused is innocent is 0,07, and when the verdict is discharged the probability that the accused is innocent is 0,4. Find the probability that a person tried by this court is innocent.

Answer this when you sit the paper.

[3 marks]Conditional probability, tree diagrams

A court passes only two verdicts, convicted or discharged, and 80% of its verdicts are convictions. When the verdict is convicted the probability that the accused is innocent is 0,07, and when the verdict is discharged the probability that the accused is innocent is 0,4, so that the probability a person tried is innocent is 0,136. Find the conditional probability that an innocent person tried by this court is convicted, correct to 3 significant figures.

Answer this when you sit the paper.

Section c, Question 14

[2 marks]Continuous random variables

The continuous random variable

X

has probability density function

\mathrm{f}(x)=2-2x

for

0\le x\le1

and

\mathrm{f}(x)=0

otherwise. What is the cumulative distribution function

\mathrm{F}(x)

0\le x\le1

A $\mathrm{F}(x)=2x-2x^{2}$
B $\mathrm{F}(x)=2-2x$
C $\mathrm{F}(x)=x^{2}-2x$
D $\mathrm{F}(x)=2x-x^{2}$

[2 marks]Continuous random variables

The continuous random variable

X

has probability density function

\mathrm{f}(x)=2-2x

for

0\le x\le1

and

\mathrm{f}(x)=0

otherwise. Find

P\left(X>\dfrac{1}{3}\right)

, as an exact fraction.

Answer this when you sit the paper.

[2 marks]Continuous random variables

The continuous random variable

X

has probability density function

\mathrm{f}(x)=2-2x

for

0\le x\le1

and

\mathrm{f}(x)=0

otherwise, so its cumulative distribution function is

\mathrm{F}(x)=2x-x^{2}

on that interval. Find the value of

p

for which

P(X<p)=\dfrac{1}{5}

, correct to 3 decimal places.

Answer this when you sit the paper.

Section c, Question 15

[2 marks]Data representation, stem and leaf and box plots

Twenty four students scored the following marks in Biology: 36, 45, 40, 60, 71, 66, 53, 42, 35, 54, 35, 43, 72, 37, 39, 34, 49, 43, 75, 58, 67, 59, 36, 67. Find the median mark.

Answer this when you sit the paper.

[2 marks]Data representation, stem and leaf and box plots

Twenty four students scored the following marks in Biology: 36, 45, 40, 60, 71, 66, 53, 42, 35, 54, 35, 43, 72, 37, 39, 34, 49, 43, 75, 58, 67, 59, 36, 67. Find the interquartile range.

Answer this when you sit the paper.

[2 marks]Data representation, stem and leaf and box plots

Twenty four students scored the following marks in Integrated Science: 88, 89, 30, 34, 48, 49, 59, 65, 67, 78, 41, 70, 54, 66, 39, 49, 37, 59, 45, 63, 52, 75, 38, 38. Find the median mark.

Answer this when you sit the paper.

[2 marks]Data representation, stem and leaf and box plots

Twenty four students sat Biology and Integrated Science. For Biology the minimum is 34, the quartiles are 38, 47 and 63, and the maximum is 75. For Integrated Science the minimum is 30, the quartiles are 40, 53 and 66,5, and the maximum is 89. Which comment on the two sets of marks is correct?

AThe two subjects have the same median, but Biology marks are more spread out.
BIntegrated Science marks are higher on average and more spread out than Biology marks.
CIntegrated Science marks are higher on average but less spread out than Biology marks.
DBiology marks are higher on average and more spread out than Integrated Science marks.

Pure Mathematics Paper 2 June 2012

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The questions

Section a

Section a, Question 1

Section a, Question 2

Section a, Question 3

Section a, Question 4

Section a, Question 5

Section a, Question 6

Section a, Question 7

Section b

Section b, Question 8

Section b, Question 9

Section b, Question 10

Section b, Question 11

Section c

Section c, Question 12

Section c, Question 13

Section c, Question 14

Section c, Question 15

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The answers, and why they are the answers

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