ZIMSEC A Level Pure Mathematics Paper 1 November 2020

Question 101

[1 marks]Coordinate geometry

Find the gradient of the line

3x-4y+2=0

Answer this when you sit the paper.

Question 102

[2 marks]Coordinate geometry

Find, in the form

ax+by+c=0

, the equation of the line passing through

(2;-3)

and parallel to the line

3x-4y+2=0

Answer this when you sit the paper.

Question 201

[1 marks]Indices and logarithms

Taking logarithms in the equation

2^{3x-2}=6

gives

3x-2=\\log_2 6

. Find the value of

\\log_2 6

, correct to 3 significant figures.

Answer this when you sit the paper.

Question 202

[2 marks]Indices and logarithms

Solve the equation

2^{3x-2}=6

, leaving the answer correct to three significant figures.

Answer this when you sit the paper.

Question 301

[1 marks]Parametric differentiation

A curve has parametric equations

x=\\cos 2t

and

y=\\sin 2t

. Find

\\dfrac{dx}{dt}

Answer this when you sit the paper.

Question 302

[2 marks]Parametric differentiation

A curve has parametric equations

x=\\cos 2t

and

y=\\sin 2t

. Find the gradient function

\\dfrac{dy}{dx}

of the curve in terms of

t

Answer this when you sit the paper.

Question 401

[1 marks]Series and sigma notation

Given that

\\displaystyle\\sum_{r=1}^{n} r^2=\\dfrac16 n(n+1)(2n+1)

, evaluate

\\displaystyle\\sum_{r=1}^{50} r^2

Answer this when you sit the paper.

Question 402

[2 marks]Series and sigma notation

Given that

\\displaystyle\\sum_{r=1}^{n} r^2=\\dfrac16 n(n+1)(2n+1)

, evaluate

\\displaystyle\\sum_{r=10}^{50} r^2

Answer this when you sit the paper.

Question 501

[1 marks]Indices

Simplify

\\dfrac{a^{-\\frac32}\\times a^{\\frac34}}{a^{-\\frac34}}

Answer this when you sit the paper.

Question 502

[2 marks]Indices

Simplify

\\left(\\dfrac{125a^3}{27b^6}\\right)^{-\\frac13}

Answer this when you sit the paper.

Question 601

[1 marks]Circles and implicit differentiation

The equation of a circle is

x^2+y^2-2x-6y+1=0

. Find the coordinates of its centre.

Answer this when you sit the paper.

Question 602

[1 marks]Circles and implicit differentiation

The equation of a circle is

x^2+y^2-2x-6y+1=0

. Find its radius.

Answer this when you sit the paper.

Question 603

[3 marks]Circles and implicit differentiation

The equation of a circle is

x^2+y^2-2x-6y+1=0

. Find the gradient of the circle at the point

(1;0)

Answer this when you sit the paper.

Question 701

[2 marks]Trigonometry, R-form

f(x)=\\sqrt3\\sin x+\\cos x

is to be written in the form

R\\cos(x-\\alpha)

, where

R

is a positive constant and

0\\le\\alpha<\\pi

. Find the value of

R

Answer this when you sit the paper.

Question 702

[2 marks]Trigonometry, R-form

f(x)=\\sqrt3\\sin x+\\cos x

is written as

R\\cos(x-\\alpha)

, where

R>0

and

0\\le\\alpha<\\pi

. Find the exact value of

\\alpha

in radians.

Answer this when you sit the paper.

Question 703

[1 marks]Trigonometry, R-form

The function

f(x)=\\sqrt3\\sin x+\\cos x

can be written as

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

. State the maximum value of

f(x)

Answer this when you sit the paper.

Question 801

[2 marks]Trigonometric equations

Using

\\cos2\\theta=2\\cos^2\\theta-1

, the equation

\\cos\\theta=2\\cos2\\theta+1

reduces to

4\\cos^2\\theta-\\cos\\theta-1=0

. Find the positive value of

\\cos\\theta

, correct to 3 significant figures.

Answer this when you sit the paper.

Question 802

[2 marks]Trigonometric equations

Using

\\cos2\\theta=2\\cos^2\\theta-1

, the equation

\\cos\\theta=2\\cos2\\theta+1

reduces to

4\\cos^2\\theta-\\cos\\theta-1=0

. Find the negative value of

\\cos\\theta

, correct to 3 significant figures.

Answer this when you sit the paper.

Question 803

[2 marks]Trigonometric equations

Solve the equation

\\cos\\theta=2\\cos2\\theta+1

and give the smallest solution in the interval

0^\\circ\\le\\theta\\le360^\\circ

, to the nearest

0,1^\\circ

Answer this when you sit the paper.

Question 901

[2 marks]Graph transformations

The diagram shows the graph of

y=f(x)

, with a maximum turning point at

B(-1;8)

. State the coordinates of the image of

B

on the graph of

y=2f(x)

Answer this when you sit the paper.

Question 902

[2 marks]Graph transformations

The diagram shows the graph of

y=f(x)

, with a minimum turning point at

D(1;2)

. State the coordinates of the image of

D

on the graph of

y=f(x-3)

Answer this when you sit the paper.

Question 903

[2 marks]Graph transformations

The diagram shows the graph of

y=f(x)

, which cuts the

x

-axis at

A(-2;0)

. State the coordinates of the image of

A

on the graph of

y=f(-x)

Answer this when you sit the paper.

Question 1001

[2 marks]Circular measure

The diagram shows a circle centre O of radius 9 cm in which

P\\hat{O}Q=\\dfrac{\\pi}{6}

and PQ is a chord. Find the length of the minor arc PQ in terms of

\\pi

Answer this when you sit the paper.

Question 1002

[2 marks]Circular measure

The diagram shows a circle centre O of radius 9 cm in which

P\\hat{O}Q=\\dfrac{\\pi}{6}

and PQ is a chord. Find the area of the triangle POQ, in cm

^2

Answer this when you sit the paper.

Question 1003

[2 marks]Circular measure

The diagram shows a circle centre O of radius 9 cm in which

P\\hat{O}Q=\\dfrac{\\pi}{6}

and PQ is a chord. The minor segment cut off by PQ is shaded. Find the area of the shaded segment in terms of

\\pi

, in cm

^2

Answer this when you sit the paper.

Question 1101

[1 marks]Functions and inverses

The function

f

is defined by

f:x\\to x^2-4x

, where

x\\in\\mathbb{R}

. State the coordinates of the turning point of the graph of

f

Answer this when you sit the paper.

Question 1102

[1 marks]Functions and inverses

The function

f

is defined by

f:x\\to x^2-4x

, where

x\\in\\mathbb{R}

. State the values of

x

at which the graph of

f

cuts the

x

-axis.

Answer this when you sit the paper.

Question 1103

[1 marks]Functions and inverses

The function

f

is defined by

f:x\\to x^2-4x

, where

x\\in\\mathbb{R}

. State the range of

f

Answer this when you sit the paper.

Question 1104

[1 marks]Functions and inverses

The function

f

is defined by

f:x\\to x^2-4x

, where

x\\in\\mathbb{R}

. If

x\\ge k

, then

f

is a one to one function. State the value of

k

Answer this when you sit the paper.

Question 1105

[3 marks]Functions and inverses

The function

f

is defined by

f:x\\to x^2-4x

for

x\\ge2

, where it is one to one. Find

f^{-1}(x)

Answer this when you sit the paper.

Question 1106

[1 marks]Functions and inverses

The function

f:x\\to x^2-4x

x\\ge2

, has inverse

f^{-1}(x)=2+\\sqrt{x+4}

. State the domain of

f^{-1}

Answer this when you sit the paper.

Question 1201

[3 marks]Polynomials, remainder and factor theorem

The polynomial

p(x)=6x^3-11x^2+ax+b

, where

a

and

b

are constants, leaves a remainder of

-24

when divided by

(x+1)

, and

(x-1)

is a factor of

p(x)

. Find the value of

a

Answer this when you sit the paper.

Question 1202

[1 marks]Polynomials, remainder and factor theorem

The polynomial

p(x)=6x^3-11x^2+ax+b

, where

a

and

b

are constants, leaves a remainder of

-24

when divided by

(x+1)

, and

(x-1)

is a factor of

p(x)

. Find the value of

b

Answer this when you sit the paper.

Question 1203

[3 marks]Polynomials, remainder and factor theorem

Factorise

p(x)=6x^3-11x^2+6x-1

completely, given that

(x-1)

is a factor.

Answer this when you sit the paper.

Question 1204

[3 marks]Polynomials, remainder and factor theorem

Find the roots of

6x^3-11x^2+6x-1=0

Answer this when you sit the paper.

Question 1301

[2 marks]Modulus equations and inequalities

Squaring both sides of

|x^2-2|=|x|

produces a quartic equation of the form

x^4+px^2+q=0

. State the values of

p

and

q

Answer this when you sit the paper.

Question 1302

[3 marks]Modulus equations and inequalities

Solve the equation

|x^2-2|=|x|

Answer this when you sit the paper.

Question 1303

[2 marks]Modulus equations and inequalities

The graphs of

y=|x^2-2|

and

y=|x|

are drawn on the same axes. State the number of points at which they intersect.

Answer this when you sit the paper.

Question 1304

[2 marks]Modulus equations and inequalities

State the range of values of

x

for which

|x^2-2|<|x|

Answer this when you sit the paper.

Question 1401

[3 marks]Quadratic inequalities and composite functions

The function

f

is defined by

f:x\\to x^2-2x+2

, where

x\\in\\mathbb{R}

. Find the set of values of

x

for which

f(x)>10

Answer this when you sit the paper.

Question 1402

[2 marks]Quadratic inequalities and composite functions

The function

f

is defined by

f:x\\to x^2-2x+2

, where

x\\in\\mathbb{R}

. Find the range of

f

Answer this when you sit the paper.

Question 1403

[1 marks]Quadratic inequalities and composite functions

The function

f:x\\to x^2-2x+2

is defined for all

x\\in\\mathbb{R}

. Does

f

have an inverse over this domain? Answer yes or no.

Answer this when you sit the paper.

Question 1404

[3 marks]Quadratic inequalities and composite functions

The functions

f

and

g

are defined by

f:x\\to x^2-2x+2

and

g:x\\to x+3

, where

x\\in\\mathbb{R}

. Find the discriminant of the equation

gf(x)=0

Answer this when you sit the paper.

Question 1501

[3 marks]Binomial expansion

Obtain the first three terms, in ascending powers of

y

, of the expansion of

\\dfrac{1-y}{\\sqrt{4-y}}

Answer this when you sit the paper.

Question 1502

[2 marks]Binomial expansion

In the expansion of

\\dfrac{1-y}{\\sqrt{4-y}}

in ascending powers of

y

, state the coefficient of

y

Answer this when you sit the paper.

Question 1503

[3 marks]Binomial expansion

Find the exact value of

\\dfrac{1-y}{\\sqrt{4-y}}

when

y=\\dfrac25

, giving the answer as a surd.

Answer this when you sit the paper.

Question 1504

[2 marks]Binomial expansion

It is given that

\\dfrac{1-y}{\\sqrt{4-y}}\\approx\\dfrac12-\\dfrac{7y}{16}-\\dfrac{13y^2}{256}

and that this expression equals

\\dfrac{\\sqrt{10}}{10}

when

y=\\dfrac25

. Use

y=\\dfrac25

in the expansion to calculate a value for

\\sqrt{10}

Answer this when you sit the paper.

Question 1601

[3 marks]Logarithms, linear laws and modulus

Solve the equation

\\ln(5+e^{-2x})=3

, giving the answer to 3 significant figures.

Answer this when you sit the paper.

Question 1602

[2 marks]Logarithms, linear laws and modulus

Two variables

x

and

t

are related by

x=mn^{-t}

, where

m

and

n

are constants. The values of

\\ln x

are plotted against the values of

t

and the points lie on a straight line with gradient

-2,3

crossing the vertical axis at

(0;3)

. Find the value of

n

, correct to 3 significant figures.

Answer this when you sit the paper.

Question 1603

[2 marks]Logarithms, linear laws and modulus

Two variables

x

and

t

are related by

x=mn^{-t}

, where

m

and

n

are constants. The values of

\\ln x

are plotted against the values of

t

and the points lie on a straight line with gradient

-2,3

crossing the vertical axis at

(0;3)

. Find the value of

m

, correct to 3 significant figures.

Answer this when you sit the paper.

Question 1604

[2 marks]Logarithms, linear laws and modulus

Solve

|x-3|<5

Answer this when you sit the paper.

Question 1701

[1 marks]Inverse functions

It is given that

f(x)=\\sqrt{9-x}

. State the domain of

f

Answer this when you sit the paper.

Question 1702

[2 marks]Inverse functions

It is given that

f(x)=\\sqrt{9-x}

. Find the inverse of the function,

f^{-1}(x)

Answer this when you sit the paper.

Question 1703

[2 marks]Inverse functions

It is given that

f(x)=\\sqrt{9-x}

, with inverse

f^{-1}(x)=9-x^2

. State the domain of

f^{-1}

Answer this when you sit the paper.

Question 1704

[2 marks]Inverse functions

State the relationship between the graphs of

y=f(x)

and

y=f^{-1}(x)

Answer this when you sit the paper.

Question 1705

[3 marks]Inverse functions

It is given that

f(x)=\\sqrt{9-x}

, with inverse

f^{-1}(x)=9-x^2

for

x\\ge0

. Find the exact

x

-coordinate of the point where the graphs of

y=f(x)

and

y=f^{-1}(x)

intersect.

Answer this when you sit the paper.

Question 1801

[3 marks]Partial fractions and series expansion

The expression

\\dfrac{3x+4}{(x-4)(x^2-8)}

is written as

\\dfrac{A}{x-4}+\\dfrac{Bx+C}{x^2-8}

. Find the value of

A

Answer this when you sit the paper.

Question 1802

[3 marks]Partial fractions and series expansion

The expression

\\dfrac{3x+4}{(x-4)(x^2-8)}

is written as

\\dfrac{A}{x-4}+\\dfrac{Bx+C}{x^2-8}

, where

A=2

. Find the values of

B

and

C

Answer this when you sit the paper.

Question 1803

[3 marks]Partial fractions and series expansion

It is given that

\\dfrac{3x+4}{(x-4)(x^2-8)}=\\dfrac{2}{x-4}-\\dfrac{2x+5}{x^2-8}

. Find the first two terms of the series expansion of this expression in ascending powers of

x

, for small

x

Answer this when you sit the paper.

Question 1804

[3 marks]Partial fractions and series expansion

It is given that

f(x)=\\dfrac{3x+4}{(x-4)(x^2-8)}=\\dfrac{2}{x-4}-\\dfrac{2x+5}{x^2-8}

. Find the series expansion of

f(x)

for small

x

, up to and including the term in

x^3

Answer this when you sit the paper.

Pure Mathematics Paper 1 November 2020

Sit this paper online

The questions

Question 101

Question 102

Question 201

Question 202

Question 301

Question 302

Question 401

Question 402

Question 501

Question 502

Question 601

Question 602

Question 603

Question 701

Question 702

Question 703

Question 801

Question 802

Question 803

Question 901

Question 902

Question 903

Question 1001

Question 1002

Question 1003

Question 1101

Question 1102

Question 1103

Question 1104

Question 1105

Question 1106

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 1601

Question 1602

Question 1603

Question 1604

Question 1701

Question 1702

Question 1703

Question 1704

Question 1705

Question 1801

Question 1802

Question 1803

Question 1804

More sittings of this paper

The answers, and why they are the answers

Sit this paper online