ZIMSEC A Level Pure Mathematics Paper 1 November 2022

Question 101

[2 marks]Binomial expansion

Expand

(4-3x)^{-\frac{1}{2}}

in ascending powers of

x

. Give the coefficient of

x

as an exact fraction.

Answer this when you sit the paper.

Question 102

[2 marks]Binomial expansion

Expand

(4-3x)^{-\frac{1}{2}}

in ascending powers of

x

. Give the coefficient of

x^2

as an exact fraction.

Answer this when you sit the paper.

Question 103

[1 marks]Binomial expansion

The binomial expansion of

(4-3x)^{-\frac{1}{2}}

in ascending powers of

x

is valid for

|x|<a

. State the value of

a

Answer this when you sit the paper.

Question 201

[1 marks]Functions

The function

g

is defined by

g:x\rightarrow\sqrt{x}

x\geq p

. State the least value of

p

for which

g

is defined.

Answer this when you sit the paper.

Question 202

[2 marks]Functions

The functions

h

and

g

are defined by

h:x\rightarrow x^3

x\in\mathbb{R}

and

g:x\rightarrow\sqrt{x}

x\geq0

. Find

gh(x)

in the form

x^n

, where

n

is rational. State

n

Answer this when you sit the paper.

Question 203

[2 marks]Functions

The function

g

is defined by

g:x\rightarrow\sqrt{x}

x\geq0

. Find

g^2(x)

, that is

g(g(x))

, in the form

x^n

where

n

is rational. State

n

Answer this when you sit the paper.

Question 301

[2 marks]Parametric equations

A curve

C

has parametric equations

x=3+4\cos\theta

and

y=1+4\sin\theta

. Eliminate

\theta

to find the cartesian equation of

C

Answer this when you sit the paper.

Question 302

[2 marks]Parametric equations

A curve

C

has parametric equations

x=3+4\cos\theta

and

y=1+4\sin\theta

, and its cartesian equation is

(x-3)^2+(y-1)^2=16

. State the coordinates of the centre of this circle.

Answer this when you sit the paper.

Question 303

[1 marks]Parametric equations

A circle has equation

(x-3)^2+(y-1)^2=16

. State its radius.

Answer this when you sit the paper.

Question 401

[2 marks]Graph transformations

The curve

y=f(x)

, where

f(x)=\ln(x+1)

, has vertical asymptote

x=-1

and passes through the origin. The graph of

y=|f(x)|

is sketched. State the coordinates of the point where

y=|f(x)|

touches the

x

-axis.

Answer this when you sit the paper.

Question 402

[2 marks]Graph transformations

The curve

y=f(x)

, where

f(x)=\ln(x+1)

, has vertical asymptote

x=-1

. State the equation of the vertical asymptote of the curve

y=f(x-2)

Answer this when you sit the paper.

Question 403

[2 marks]Graph transformations

The curve

y=f(x)

is defined by

f(x)=\ln(x+1)

. Find the

y

-intercept of the curve

y=1-f(x)

Answer this when you sit the paper.

Question 501

[1 marks]Linear laws

In an experiment the variables

x

and

y

satisfy

y=ax^2+b

, where

a

and

b

are constants. To obtain a straight line graph,

y

is plotted on the vertical axis. State what must be plotted on the horizontal axis.

Answer this when you sit the paper.

Question 502

[1 marks]Linear laws

In an experiment the variables

x

and

y

satisfy

y=ax^2+b

, and

y

is plotted against

x^2

. One reading is

x=3.8

. Calculate the value of

x^2

that is plotted for this reading.

Answer this when you sit the paper.

Question 503

[2 marks]Linear laws

In an experiment the variables

x

and

y

satisfy

y=ax^2+b

. Two of the readings are

x=1.6

y=5.4

and

x=5.0

y=32.3

. Plotting

y

against

x^2

gives a straight line of gradient

a

. Use these two readings to calculate

a

, correct to 2 significant figures.

Answer this when you sit the paper.

Question 504

[2 marks]Linear laws

In an experiment the variables

x

and

y

satisfy

y=ax^2+b

, and the gradient of the graph of

y

against

x^2

gives

a=1.2

. One reading is

x=1.6

y=5.4

. Find the value of

b

, correct to 2 significant figures.

Answer this when you sit the paper.

Question 601

[3 marks]Differential equations

Find the particular solution of the differential equation

(x+1)^2\dfrac{dy}{dx}+2=0

, given that

y=1

when

x=0

. Express

y

in terms of

x

Answer this when you sit the paper.

Question 602

[2 marks]Differential equations

The curve

y=\dfrac{2}{x+1}-1

is sketched. State the equation of its vertical asymptote.

Answer this when you sit the paper.

Question 603

[2 marks]Differential equations

Find the

x

-coordinate of the point where the curve

y=\dfrac{2}{x+1}-1

crosses the

x

-axis.

Answer this when you sit the paper.

Question 701

[1 marks]Variation

The total daily takings

C

of a cooperative partly varies as the number of chickens sold

N

, and partly varies as the number of dozens of eggs sold

D

. Write down an expression for

C

in terms of

N

D

and the constants

h

and

k

Answer this when you sit the paper.

Question 702

[3 marks]Variation

The total daily takings of a cooperative are given by

C=hN+kD

, where

N

is the number of chickens sold and

D

the number of dozens of eggs sold. A total of

\$120

was collected when 15 chickens and 15 dozens of eggs were sold, and

\$90

was collected when 10 chickens and 20 dozens of eggs were sold. Find the value of

h

and the value of

k

Answer this when you sit the paper.

Question 703

[1 marks]Variation

The total daily takings of a cooperative are

C=hN+kD

, where

N

is the number of chickens sold,

D

the number of dozens of eggs sold, and it is found that

h=7

and

k=1

. State, in dollars, the amount taken for one dozen eggs.

Answer this when you sit the paper.

Question 704

[2 marks]Variation

The total daily takings of a cooperative are

C=7N+D

, where

N

is the number of chickens sold and

D

the number of dozens of eggs sold. Find the total daily takings, in dollars, if 20 chickens and 30 dozen eggs were sold.

Answer this when you sit the paper.

Question 801

[1 marks]Mathematical induction

The statement

\displaystyle\sum_{r=1}^{n}(8r-1)=4n^2+3n

is to be proved by induction. Evaluate the left hand side when

n=1

Answer this when you sit the paper.

Question 802

[2 marks]Mathematical induction

In proving

\displaystyle\sum_{r=1}^{n}(8r-1)=4n^2+3n

by induction, it is assumed that

\displaystyle\sum_{r=1}^{k}(8r-1)=4k^2+3k

. Write down, in terms of

k

and in its simplest form, the single term that must be added to reach

\displaystyle\sum_{r=1}^{k+1}(8r-1)

Answer this when you sit the paper.

Question 803

[2 marks]Mathematical induction

Given that

\displaystyle\sum_{r=1}^{n}(8r-1)=4n^2+3n

, evaluate

\displaystyle\sum_{r=1}^{10}(8r-1)

Answer this when you sit the paper.

Question 901

[3 marks]Partial fractions

Express

\dfrac{2}{4r^2-1}

in partial fractions.

Answer this when you sit the paper.

Question 902

[3 marks]Partial fractions

Given that

\dfrac{2}{4r^2-1}=\dfrac{1}{2r-1}-\dfrac{1}{2r+1}

, find

\displaystyle\sum_{r=1}^{n}\dfrac{2}{4r^2-1}

as a single fraction in terms of

n

Answer this when you sit the paper.

Question 903

[2 marks]Partial fractions

Given that

\displaystyle\sum_{r=1}^{n}\dfrac{2}{4r^2-1}=\dfrac{2n}{2n+1}

, evaluate

\displaystyle\sum_{r=1}^{5}\dfrac{2}{4r^2-1}

, giving the answer as a fraction in its lowest terms.

Answer this when you sit the paper.

Question 1001

[1 marks]Integration

The curve

y=x(4-x)

meets the

x

-axis at two points. State the two values of

x

at which it does so.

Answer this when you sit the paper.

Question 1002

[2 marks]Integration

Find

\displaystyle\int x(4-x)\,dx

Answer this when you sit the paper.

Question 1003

[2 marks]Integration

The region

R

is bounded by the curve

y=x(4-x)

and the

x

-axis between

x=0

and

x=4

. Calculate the area of

R

, giving the answer as an exact fraction.

Answer this when you sit the paper.

Question 1004

[3 marks]Integration

The region bounded by the curve

y=x(4-x)

and the

x

-axis between

x=0

and

x=4

is rotated through

360^\circ

about the

x

-axis. Calculate the exact volume of the solid generated, in terms of

\pi

Answer this when you sit the paper.

Question 1101

[2 marks]Quadratic and cubic equations

For the equation

x^2+(3k-2)x+k+2=0

, the condition for no real roots reduces to

9k^2-16k-4<0

. Solve the equation

9k^2-16k-4=0

Answer this when you sit the paper.

Question 1102

[3 marks]Quadratic and cubic equations

Find the range of values of

k

for which the equation

x^2+(3k-2)x+k+2=0

has no real roots.

Answer this when you sit the paper.

Question 1103

[2 marks]Quadratic and cubic equations

Show that

x=2

is a root of

2x^3-9x^2+7x+6=0

by evaluating

2x^3-9x^2+7x+6

x=2

Answer this when you sit the paper.

Question 1104

[3 marks]Quadratic and cubic equations

Solve the equation

2x^3-9x^2+7x+6=0

Answer this when you sit the paper.

Question 1201

[2 marks]Remainder and factor theorem

f(x)=ax^4-9x^3+bx^2-3x+2

is exactly divisible by

x+2

. Using the factor theorem, this reduces to an equation of the form

4a+b=c

. Find the value of

c

Answer this when you sit the paper.

Question 1202

[3 marks]Remainder and factor theorem

f(x)=ax^4-9x^3+bx^2-3x+2

is exactly divisible by

x+2

and leaves a remainder

-12

when divided by

x-1

. Find the values of

a

and

b

Answer this when you sit the paper.

Question 1203

[2 marks]Remainder and factor theorem

Verify that

1-2x

is a factor of

f(x)=-6x^4-9x^3+4x^2-3x+2

by evaluating

f\left(\dfrac{1}{2}\right)

Answer this when you sit the paper.

Question 1204

[3 marks]Remainder and factor theorem

The quartic

f(x)=-6x^4-9x^3+4x^2-3x+2

factorises as

(x+2)(1-2x)\left(3x^2+1\right)

. State all the real roots of

f(x)=0

Answer this when you sit the paper.

Question 1301

[2 marks]Numerical methods

To show that the root of

\cot x=1+x^2

lies between 0,6 and 0,9, the value of

\cot x

is needed at each end. Evaluate

\cot(0,6)

, with

x

in radians, correct to 3 decimal places.

Answer this when you sit the paper.

Question 1302

[2 marks]Numerical methods

The iterative formula

x_{n+1}=\tan^{-1}\left(\dfrac{1}{1+x_n^2}\right)

, with

x

in radians, is used to find the root of

\cot x=1+x^2

. Taking

x_0=0.6

, find

x_1

correct to 4 decimal places.

Answer this when you sit the paper.

Question 1303

[2 marks]Numerical methods

The iterative formula

x_{n+1}=\tan^{-1}\left(\dfrac{1}{1+x_n^2}\right)

, with

x

in radians, is used to find the root of

\cot x=1+x^2

. Given that

x_1=0.6340

, find

x_2

correct to 4 decimal places.

Answer this when you sit the paper.

Question 1304

[2 marks]Numerical methods

The iterative formula

x_{n+1}=\tan^{-1}\left(\dfrac{1}{1+x_n^2}\right)

, with

x

in radians, is used to find the root of

\cot x=1+x^2

. Given that

x_2=0.6196

, find

x_3

correct to 4 decimal places.

Answer this when you sit the paper.

Question 1401

[2 marks]Matrices

Find the determinant of

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

Answer this when you sit the paper.

Question 1402

[2 marks]Matrices

The simultaneous equations

2x+3y-z=0

x+2y+z=2

and

3x+3y+2z=4

are written in the matrix form

A\mathbf{X}=\mathbf{B}

, where

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

and

\mathbf{X}=\begin{pmatrix}x\\y\\z\end{pmatrix}

. Write down the column vector

\mathbf{B}

Answer this when you sit the paper.

Question 1403

[2 marks]Matrices

For

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

, whose determinant is 8, the adjugate is

\begin{pmatrix}1 & -9 & 5\\ 1 & 7 & -3\\ -3 & 3 & 1\end{pmatrix}

. Find the element of

A^{-1}

in the first row and second column.

Answer this when you sit the paper.

Question 1404

[3 marks]Matrices

Given that

A^{-1}=\dfrac18\begin{pmatrix}1 & -9 & 5\\ 1 & 7 & -3\\ -3 & 3 & 1\end{pmatrix}

, solve the simultaneous equations

2x+3y-z=0

x+2y+z=2

3x+3y+2z=4

. Give the values of

x

y

and

z

Answer this when you sit the paper.

Question 1501

[2 marks]Vectors

The points

A

and

B

have position vectors

\mathbf{a}=3\mathbf{i}+3\mathbf{j}+4\mathbf{k}

and

\mathbf{b}=2\mathbf{i}+5\mathbf{j}+3\mathbf{k}

. Find the vector

\overrightarrow{BA}

Answer this when you sit the paper.

Question 1502

[3 marks]Vectors

The points

A

B

and

C

have position vectors

\mathbf{a}=3\mathbf{i}+3\mathbf{j}+4\mathbf{k}

\mathbf{b}=2\mathbf{i}+5\mathbf{j}+3\mathbf{k}

and

\mathbf{c}=3\mathbf{i}+4\mathbf{j}+3\mathbf{k}

. Find the angle

A\hat{B}C

, in degrees.

Answer this when you sit the paper.

Question 1503

[3 marks]Vectors

The points

A

B

and

C

have position vectors

\mathbf{a}=3\mathbf{i}+3\mathbf{j}+4\mathbf{k}

\mathbf{b}=2\mathbf{i}+5\mathbf{j}+3\mathbf{k}

and

\mathbf{c}=3\mathbf{i}+4\mathbf{j}+3\mathbf{k}

, and the angle

A\hat{B}C

30^\circ

. Find the area of triangle

ABC

in exact form.

Answer this when you sit the paper.

Question 1504

[2 marks]Vectors

The points

A

and

B

have position vectors

\mathbf{a}=3\mathbf{i}+3\mathbf{j}+4\mathbf{k}

and

\mathbf{b}=2\mathbf{i}+5\mathbf{j}+3\mathbf{k}

. Find the position vector of

F

, the midpoint of

AB

Answer this when you sit the paper.

Question 1505

[2 marks]Vectors

The points

A

B

and

C

have position vectors

\mathbf{a}=3\mathbf{i}+3\mathbf{j}+4\mathbf{k}

\mathbf{b}=2\mathbf{i}+5\mathbf{j}+3\mathbf{k}

and

\mathbf{c}=3\mathbf{i}+4\mathbf{j}+3\mathbf{k}

, and

F

is the midpoint of

AB

with position vector

\dfrac52\mathbf{i}+4\mathbf{j}+\dfrac72\mathbf{k}

. Evaluate the scalar product

\overrightarrow{CF}\cdot\overrightarrow{AB}

Answer this when you sit the paper.

Pure Mathematics Paper 1 November 2022

Sit this paper online

The questions

Question 101

Question 102

Question 103

Question 201

Question 202

Question 203

Question 301

Question 302

Question 303

Question 401

Question 402

Question 403

Question 501

Question 502

Question 503

Question 504

Question 601

Question 602

Question 603

Question 701

Question 702

Question 703

Question 704

Question 801

Question 802

Question 803

Question 901

Question 902

Question 903

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 1301

Question 1302

Question 1303

Question 1304

Question 1401

Question 1402

Question 1403

Question 1404

Question 1501

Question 1502

Question 1503

Question 1504

Question 1505

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