ZIMSEC A Level Additional Mathematics Paper 1 November 2010

Question 101

[2 marks]vectors

The position vectors of the points

A

B

and

C

relative to the origin

O

are

a = 3i - j

b = 2i + j + 3k

and

c = i - j + 6k

. Find the value of

a \cdot (b \times c)

Answer this when you sit the paper.

Question 102

[2 marks]vectors

For three non-zero vectors

a

b

and

c

, the scalar triple product

a \cdot (b \times c)

measures which quantity?

Athe area of the parallelogram on b and c
Bthe volume of the parallelepiped on a, b, c
Cthe volume of the tetrahedron on a, b, c
Dthe length of the projection of a onto b x c

Question 201

[2 marks]hyperbolic functions

Let

y = \sinh^{-1} 2x

. Writing

\sinh y

in exponential form and clearing the fraction gives which equation in

e^y

A $e^{2y} + 4xe^y - 1 = 0$
B $e^{2y} - 4xe^y - 1 = 0$
C $e^{2y} - 4xe^y + 1 = 0$
D $e^{2y} - 2xe^y - 1 = 0$

Question 202

[2 marks]hyperbolic functions

Given that

y = \sinh^{-1} x

, express

y

in logarithmic form.

Answer this when you sit the paper.

Question 301

[2 marks]series

The standard result is

\sum_{r=1}^{m} r^3 = \left[\frac{m(m+1)}{2}\right]^2

. Which expression gives

\sum_{r=k+1}^{3k} r^3

A $\left[\frac{3k(3k+1)}{2}\right]^2-\left[\frac{(k+1)(k+2)}{2}\right]^2$
B $\left[\frac{3k(3k+1)}{2}-\frac{k(k+1)}{2}\right]^2$
C $\left[\frac{3k(3k+1)}{2}\right]^2-\left[\frac{k(k+1)}{2}\right]^2$
D $\left[\frac{3k(3k+1)}{2}\right]^2+\left[\frac{k(k+1)}{2}\right]^2$

Question 302

[2 marks]series

Using

\sum_{r=1}^{m} r^3 = \left[\frac{m(m+1)}{2}\right]^2

, write

\sum_{r=1}^{3k} r^3

in terms of

k

in its simplest form.

Answer this when you sit the paper.

Question 303

[2 marks]series

Given that

\sum_{r=k+1}^{3k} r^3 = k^2(4k+1)(5k+2)

for

k \geq 1

, evaluate

21^3 + 22^3 + 23^3 + \ldots + 60^3

Answer this when you sit the paper.

Question 401

[2 marks]groups

The set of positive rational numbers forms a group under the operation

a * b = \frac{ab}{10}

. Find the identity element.

Answer this when you sit the paper.

Question 402

[2 marks]groups

Under the operation

a * b = \frac{ab}{10}

on the positive rational numbers, write the inverse of an element

a

in terms of

a

Answer this when you sit the paper.

Question 403

[2 marks]groups

The positive rational numbers form a group under

a * b = \frac{ab}{10}

. Which group axiom fails if the set is narrowed to the positive integers?

Aclosure, since ab/10 need not be whole
Bidentity, since 10 is not a positive integer
Cassociativity, since division is not associative
Dcommutativity, since ab is not always ba

Question 501

[1 marks]groups

The set

G = \{0, 1, 2\}

forms a group under addition modulo 3. State the identity element.

Answer this when you sit the paper.

Question 502

[2 marks]groups

In the group

G = \{0, 1, 2\}

under addition modulo 3, find the inverse of the element 2.

Answer this when you sit the paper.

Question 503

[3 marks]groups

The set

G = \{0, 1, 2\}

under addition modulo 3 is shown to be abelian. Which observation establishes that?

Aevery element of G is its own inverse under +
Bthe identity element 0 is a member of the set
Cthe operation is closed and also associative
Dthe Cayley table is symmetric in the diagonal

Question 601

[2 marks]proof by induction

Find the value of

10^n + 12\left(4^{n+1}\right) + 5

when

n = 1

Answer this when you sit the paper.

Question 602

[3 marks]proof by induction

For

f(n) = 10^n + 12\left(4^{n+1}\right) + 5

, the difference

f(n+1) - f(n)

simplifies to which expression?

A $10^n + 36\left(4^{n+1}\right)$
B $9\left(10^n + 4^{n+2}\right)$
C $9\left(10^n + 4^{n+1}\right)$
D $9\times10^n + 12\left(4^{n+1}\right)$

Question 603

[1 marks]proof by induction

An induction proof shows that

10^n + 12\left(4^{n+1}\right) + 5

is divisible by 9 for every positive integer

n

. What does the inductive step assume?

Athe result holds for every positive integer n
Bthe result holds for n = 1 and for n = 2
C9 is a factor of 10^k for that value of k
Dthe result holds for one value n = k

Question 701

[3 marks]reduction formulae

For

I_n = \int \sin^n ax\, dx

, where

n

is a positive integer and

a

is a constant, which reduction formula is correct?

A $I_n=\frac{1}{na}\sin^{n-1}ax\cos ax+\frac{n-1}{n}I_{n-2}$
B $I_n=-\frac{1}{n}\sin^{n-1}ax\cos ax+\frac{n-1}{na}I_{n-2}$
C $I_n=-\frac{1}{na}\sin^{n-1}ax\cos ax+\frac{n-1}{n}I_{n-1}$
D $I_n=-\frac{1}{na}\sin^{n-1}ax\cos ax+\frac{n-1}{n}I_{n-2}$

Question 702

[1 marks]reduction formulae

Write

\sin^2 \theta

in terms of

\cos 2\theta

Answer this when you sit the paper.

Question 703

[2 marks]reduction formulae

Find

\int \sin^2 3x\, dx

Answer this when you sit the paper.

Question 704

[3 marks]reduction formulae

Find

\int \sin^4 3x\, dx

A $-\frac{1}{4}\sin^3 3x\cos 3x+\frac{3x}{8}-\frac{1}{16}\sin 6x+c$
B $\frac{1}{12}\sin^3 3x\cos 3x+\frac{3x}{8}-\frac{1}{16}\sin 6x+c$
C $-\frac{1}{12}\sin^3 3x\cos 3x+\frac{3x}{8}-\frac{1}{16}\sin 6x+c$
D $-\frac{1}{12}\sin^3 3x\cos 3x+\frac{3x}{8}-\frac{1}{12}\sin 6x+c$

Question 801

[2 marks]partial fractions

Write

y(y + 1)\left(4y^2 - 1\right)

as a product of linear factors.

Answer this when you sit the paper.

Question 802

[3 marks]partial fractions

Express

\frac{10y^2 + 2y - 2}{y(y + 1)\left(4y^2 - 1\right)}

in partial fractions.

A $\frac{2}{y}-\frac{2}{y+1}+\frac{1}{2y-1}-\frac{1}{2y+1}$
B $\frac{2}{y}-\frac{2}{y+1}-\frac{1}{2y-1}+\frac{1}{2y+1}$
C $\frac{2}{y}+\frac{2}{y+1}+\frac{1}{2y-1}-\frac{1}{2y+1}$
D $\frac{1}{y}-\frac{2}{y+1}+\frac{2}{2y-1}-\frac{1}{2y+1}$

Question 803

[3 marks]partial fractions

The expression

S_y = \frac{10y^2 + 2y - 2}{y(y + 1)\left(4y^2 - 1\right)}

has partial fractions

\frac{2}{y} - \frac{2}{y+1} + \frac{1}{2y-1} - \frac{1}{2y+1}

. Find

\sum_{y=1}^{n} S_y

in terms of

n

Answer this when you sit the paper.

Question 804

[1 marks]partial fractions

A series has

\sum_{y=1}^{n} S_y = \frac{2n}{n+1} + \frac{2n}{2n+1}

. Deduce the value of

\sum_{y=1}^{\infty} S_y

Answer this when you sit the paper.

Question 901

[3 marks]matrices and transformations

The transformation

\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} X \\ Y \end{pmatrix}

acts on the line

2x + y - 3 = 0

. Find the equation of the image.

Answer this when you sit the paper.

Question 902

[2 marks]matrices and transformations

Describe fully the transformation with matrix

\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}

Areflection in the line y = -x
Brotation of 90 degrees about the origin
Creflection in the line y = x
Drotation of 180 degrees about the origin

Question 903

[2 marks]matrices and transformations

Find the invariant points of the transformation with matrix

\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}

Aevery point on the line y = x
Bevery point on the line y = -x
Cthe origin and no other point
Devery point on the x-axis and the y-axis

Question 904

[2 marks]matrices and transformations

Find the image of the point

(4, 1)

under the transformation

\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} X \\ Y \end{pmatrix}

Answer this when you sit the paper.

Question 1001

[2 marks]matrices

Find the determinant of the matrix

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

Answer this when you sit the paper.

Question 1002

[3 marks]matrices

Find the inverse of the matrix

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

A $\frac{1}{2}\begin{pmatrix} -1 & -3 & 5 \\ 0 & -2 & 4 \\ -1 & 1 & -1 \end{pmatrix}$
B $\frac{1}{2}\begin{pmatrix} -1 & 0 & -1 \\ -3 & 2 & 1 \\ 5 & 4 & -1 \end{pmatrix}$
C $\frac{1}{2}\begin{pmatrix} -1 & 0 & -1 \\ -3 & -2 & 1 \\ 5 & 4 & -1 \end{pmatrix}$
D $-\frac{1}{2}\begin{pmatrix} -1 & 0 & -1 \\ -3 & -2 & 1 \\ 5 & 4 & -1 \end{pmatrix}$

Question 1003

[2 marks]matrices

The equations

-x - 2y - z + 4 = 0

x + 3y + 2z + 11 = 0

and

-x + 2y + z - 10 = 0

are written as

A\mathbf{v} = \mathbf{b}

, where

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

and

\mathbf{v} = (x, y, z)

. Find

\mathbf{b}

A(4, 11, -10)
B(-4, -11, -10)
C(-4, -11, 10)
D(4, -11, 10)

Question 1004

[3 marks]matrices

Solve the simultaneous equations

-x - 2y - z + 4 = 0

x + 3y + 2z + 11 = 0

and

-x + 2y + z - 10 = 0

Answer this when you sit the paper.

Question 1101

[3 marks]differential equations

Given that

y = Ae^{4x} + Be^{-4x}

, where

A

and

B

are constants, express

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

in terms of

y

Answer this when you sit the paper.

Question 1102

[2 marks]differential equations

Find the roots of the auxiliary equation of the differential equation

\frac{d^2y}{dx^2} + 20\frac{dy}{dx} + 101y = 0

Answer this when you sit the paper.

Question 1103

[2 marks]differential equations

Find the general solution of the differential equation

\frac{d^2y}{dx^2} + 20\frac{dy}{dx} + 101y = 0

A $y=Ae^{-10x}+Be^{-101x}$
B $y=e^{10x}(A\cos x+B\sin x)$
C $y=e^{-10x}(A\cos x+B\sin x)$
D $y=e^{-10x}(A\cos 10x+B\sin 10x)$

Question 1104

[2 marks]differential equations

The differential equation

\frac{d^2y}{dx^2} + 20\frac{dy}{dx} + 101y = 0

has

y = 10

when

x = 0

and

y = 15

when

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

. Find the particular solution.

A $y=e^{-10x}\left(10\cos x-15e^{15\pi}\sin x\right)$
B $y=e^{10x}\left(10\cos x-15e^{15\pi}\sin x\right)$
C $y=e^{-10x}\left(15\cos x-10e^{15\pi}\sin x\right)$
D $y=e^{-10x}\left(10\cos x+15e^{15\pi}\sin x\right)$

Question 1201

[2 marks]series

Find the sum of all the multiples of 7 from 0 to 1 000 inclusive.

Answer this when you sit the paper.

Question 1202

[3 marks]series

Find the sum of all the integers from 0 to 1 000 inclusive which leave a remainder of 0, 1 or 2 when divided by 7.

Answer this when you sit the paper.

Question 1203

[3 marks]series

A circle is divided into

n

arcs whose lengths are in the ratio

1^3 : 2^3 : 3^3 : \ldots : n^3

. Write down, in terms of

n

, the angle in degrees subtended at the centre by the largest arc.

Answer this when you sit the paper.

Question 1204

[2 marks]series

The largest of

n

arcs of a circle divided in the ratio

1^3 : 2^3 : \ldots : n^3

subtends

\frac{1440n}{(n+1)^2}

degrees at the centre. Which inequality must be solved to find the values of

n

for which this angle is less than

48^\circ

A $n^2 - 28n + 1 > 0$
B $n^2 + 28n + 1 > 0$
C $n^2 - 30n + 1 > 0$
D $n^2 - 28n + 1 < 0$

Question 1205

[2 marks]series

The largest of

n

arcs of a circle divided in the ratio

1^3 : 2^3 : \ldots : n^3

subtends

\frac{1440n}{(n+1)^2}

degrees at the centre. Find the smallest value of

n

for which this angle is less than

48^\circ

Answer this when you sit the paper.

Question 1301

[2 marks]integration

Given that

y = \sinh^{-1} x

, find

\frac{dy}{dx}

Answer this when you sit the paper.

Question 1302

[2 marks]integration

Express

4x^2 - 4x + 5

in the form

(ax + b)^2 + c

, where

a

b

and

c

are integers.

Answer this when you sit the paper.

Question 1303

[3 marks]integration

Find

\int \frac{1}{4x^2 - 4x + 5}\, dx

A $\frac{1}{4}\tan^{-1}\left(\frac{2x-1}{4}\right)+c$
B $\frac{1}{4}\tan^{-1}\left(\frac{2x-1}{2}\right)+c$
C $\frac{1}{2}\tan^{-1}\left(\frac{2x-1}{2}\right)+c$
D $\frac{1}{8}\tan^{-1}\left(\frac{2x-1}{2}\right)+c$

Question 1304

[3 marks]integration

Find

\int \frac{1}{\sqrt{4x^2 - 4x + 5}}\, dx

A $\cosh^{-1}\left(\frac{2x-1}{2}\right)+c$
B $\frac{1}{2}\sinh^{-1}\left(\frac{2x-1}{2}\right)+c$
C $\frac{1}{4}\sinh^{-1}\left(\frac{2x-1}{2}\right)+c$
D $\frac{1}{2}\sinh^{-1}\left(\frac{2x-1}{4}\right)+c$

Question 1305

[3 marks]integration

Evaluate

\int_{\frac{1}{2}}^{\frac{3}{2}} \frac{1}{4x^2 - 4x + 5}\, dx

, giving the exact answer.

Answer this when you sit the paper.

Question 1401

[3 marks]vectors

A line has direction

d = 3i - k

and passes through the point

(8, 1, -1)

. A plane has equation

r \cdot n = 5

, where

n = -i + 10j - 3k

. Which pair of checks shows that the line lies in the plane?

Ad.n = 0 and the plane passes through the origin
Bd x n = 0 and the point (8, 1, -1) is in the plane
Cthe point (8, 1, -1) is in the plane, on its own
Dd.n = 0 and the point (8, 1, -1) is in the plane

Question 1402

[2 marks]vectors

Find the angle between the planes

r \cdot (-i + 10j - 3k) = 5

and

r \cdot (i + j + 3k) = 6

A $90^\circ$
B $60^\circ$
C $0^\circ$
D $45^\circ$

Question 1403

[1 marks]vectors

Find the magnitude of the vector

-i + 10j - 3k

, leaving your answer in surd form.

Answer this when you sit the paper.

Question 1404

[3 marks]vectors

Find the perpendicular distance of the point

P(3, 2, -1)

from the line

r = 8i + j - k + t(3i - k)

Answer this when you sit the paper.

Question 1405

[3 marks]vectors

Find the perpendicular distance of the point

P(3, 2, -1)

from the plane

r \cdot (-i + 10j - 3k) = 5

Answer this when you sit the paper.

Question 1406

[3 marks]vectors

A point

P

is at a distance

\frac{1}{2}\sqrt{14}

from a line

l

and at a distance

\frac{3}{22}\sqrt{110}

from a plane that contains

l

. The foot of the perpendicular from

P

l

A

and the foot of the perpendicular from

P

to the plane is

B

. Find the area of triangle

ABP

A $\frac{6\sqrt{10}}{11}$
B $\frac{3\sqrt{10}}{22}$
C $\frac{4\sqrt{11}}{11}$
D $\frac{3\sqrt{10}}{11}$

Additional Mathematics Paper 1 November 2010

Sit this paper online

The questions

Question 101

Question 102

Question 201

Question 202

Question 301

Question 302

Question 303

Question 401

Question 402

Question 403

Question 501

Question 502

Question 503

Question 601

Question 602

Question 603

Question 701

Question 702

Question 703

Question 704

Question 801

Question 802

Question 803

Question 804

Question 901

Question 902

Question 903

Question 904

Question 1001

Question 1002

Question 1003

Question 1004

Question 1101

Question 1102

Question 1103

Question 1104

Question 1201

Question 1202

Question 1203

Question 1204

Question 1205

Question 1301

Question 1302

Question 1303

Question 1304

Question 1305

Question 1401

Question 1402

Question 1403

Question 1404

Question 1405

Question 1406

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