Topic 2: Algebra and graphs

Write one step a line, doing the same to both sides. Check a solution by substituting it back into the original equation.

A lesson for this topic

Extended algebra, marked line by line: the algebra repair pack comes with the course, six modules and 72 tasks that mark each step of your working.

Quick checks: multiple choice

Question 1Core · no calculator

v

= u + at

. Find

v

when

u

= 3

a

= -2

and

t

= 5

Question 2Core · calculator

E

= \frac{1}{2}mv^2

. Find

E

when

m

= 0.8

and

v

= 12.5

Question 3Core · no calculator

Find the value of

3x^2 - 2y

when

x

= -2

and

y

= -5

Question 4Core · calculator

T

= 2\pi\sqrt{\frac{L}{9.8}}

. Find

T

when

L

= 1.5

. Give your answer correct to

3

significant figures.

Question 5Core · no calculator

Simplify

5a - 3b + 2a + 7b

Question 6Core · no calculator

Expand and simplify

(2x + 3)(x - 5)

Question 7Core · no calculator

Factorise fully

12x^2y - 18xy^2

Question 8Extended · no calculator

Factorise

9x^2 - 25y^2

Question 9Extended · no calculator

Factorise

3x^2 + 10x - 8

Question 10Extended · no calculator

Write

x^2 - 6x + 11

in the form

(x + p)^2 + q

Question 11Extended · no calculator

Expand and simplify

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

Question 12Extended · no calculator

Factorise fully

6ax - 3ay + 4bx - 2by

Question 13Extended · no calculator

Simplify

\frac{x^2 - 9}{x^2 + x - 6}

Question 14Extended · no calculator

Write

\frac{2}{x + 1}

{}+ \frac{3}{x - 2}

as a single fraction.

Question 15Extended · no calculator

Simplify

\frac{5x}{6} \div \frac{10x^2}{9}

Question 16Extended · no calculator

Write

\frac{x + 4}{3}

{}- \frac{x - 2}{5}

as a single fraction in its simplest form.

Question 17Core · no calculator

3^x

= 81

. Find the value of

x

Question 18Core · no calculator

Simplify

(4x^3)^2

Question 19Core · no calculator

Simplify

15a^6 \div 5a^{-2}

Question 20Core · no calculator

Simplify

4x^5y^{-2} \times 3x^{-2}y^6

Question 21Extended · no calculator

Solve

4^{x + 1}

= 8^x

Question 22Extended · no calculator

Simplify

(27x^6)^{\frac{2}{3}}

Question 23Extended · no calculator

Simplify

(16x^8)^{-\frac{1}{2}}

Question 24Core · no calculator

Solve

4x - 7

= 2x + 9

Question 25Core · no calculator

Solve

3(2x - 5)

= x + 10

Question 26Core · no calculator

The angles of a triangle are

x^\circ

(2x + 10)^\circ

and

(3x - 40)^\circ

. Find

x

Question 27Core · no calculator

Solve the simultaneous equations

5x + 2y

= 21

and

3x - 2y

= 11

Question 28Core · calculator

Solve

2.4x + 1.7

= 9.86

Question 29Core · no calculator

Make

t

the subject of

v

= u + at

Question 30Extended · no calculator

Solve

\frac{3}{x - 1}

= \frac{2}{x + 2}

Question 31Extended · calculator

Solve

2x^2 - 5x - 1

= 0

. Give your answers correct to

3

significant figures.

Question 32Extended · no calculator

Write

x^2 + 4x - 3

in the form

(x + a)^2 - b

. Hence solve

x^2 + 4x - 3

= 0

, giving your answers in surd form.

Question 33Extended · no calculator

Make

x

the subject of

y

= \frac{x + 3}{x - 2}

Question 34Extended · no calculator

Solve the simultaneous equations

y

= x + 1

and

x^2 + y^2

= 13

Question 35Extended · no calculator

The product of two consecutive positive even numbers is

168

. Using

x

for the smaller number,

x(x + 2)

= 168

. Find the smaller number.

Question 36Extended · calculator

The area of a circle is

A

= \pi r^2

. Make

r

the subject, and find

r

when

A

= 50

, correct to

3

significant figures.

Question 37Core · no calculator

List the integers

n

such that

-3

< n \leqslant 2

Question 38Core · no calculator

A number line shows an open circle at

-1

and a filled circle at

4

, joined by a line. Which inequality does it show?

Question 39Extended · no calculator

Solve

5 - 2x

< 11

Question 40Extended · no calculator

Solve

-4 \leqslant 3x + 2

< 11

Question 41Extended · no calculator

Which point lies in the region defined by

y \geqslant 1

x + y

< 5

and

y \leqslant 2x

Question 42Core · no calculator

Find the

n

th term of the sequence

7, 11, 15, 19, \ldots

Question 43Core · no calculator

Find the

n

th term of the sequence

3, 6, 11, 18, \ldots

Question 44Core · no calculator

Find the

n

th term of the sequence

2, 9, 28, 65, \ldots

Question 45Extended · no calculator

Find the

n

th term of the sequence

4, 11, 22, 37, \ldots

Question 46Extended · no calculator

Find the

n

th term of the sequence

5, 15, 45, 135, \ldots

Question 47Core · no calculator

The

n

th term of a sequence is

5n - 3

. Which term is equal to

147

Question 48Extended · no calculator

y

is directly proportional to

x^2

. When

x

= 2

y

= 12

. Find

y

when

x

= 5

Question 49Extended · no calculator

y

is inversely proportional to

\sqrt{x}

. When

x

= 4

y

= 6

. Find

y

when

x

= 9

Question 50Extended · calculator

The brightness

B

of a lamp is inversely proportional to the square of the distance

d

from it. When

d

= 3

B

= 50

. Find

B

when

d

= 4.5

, correct to

3

significant figures.

Question 51Extended · calculator

y

is directly proportional to

x^2

. When

x

= 2.5

y

= 20

. Find

y

when

x

= 4.2

, correct to

3

significant figures.

Question 52Core · no calculator

The travel graph shows a cycle ride from home and back. Find the speed on the journey home.

Question 53Core · calculator

A straight-line conversion graph from degrees Celsius to degrees Fahrenheit passes through

(0, 32)

and

(100, 212)

. Convert

25^\circ

C to degrees Fahrenheit.

Question 54Extended · no calculator

The speed-time graph shows a tram moving between two stops. Find the total distance travelled.

Question 55Extended · calculator

A car's speed rises steadily from

3.5

m/s to

17.9

m/s in

4.8

seconds. Find its acceleration.

Question 56Extended · no calculator

The graph shows the distance travelled by a ball. A tangent is drawn to the curve at

t

= 3

; it passes through

(1, 0)

and

(5, 24)

. From the tangent, estimate the speed of the ball at

t

= 3

Question 57Core · no calculator

A table of values is made for

y

= x^2 - 2x - 3

. Find

y

when

x

= -3

Question 58Core · no calculator

The graph of

y

= x^2 - x - 6

is drawn. Read the solutions of

x^2 - x - 6

= 0

from the graph.

Question 59Core · no calculator

Which point lies on the graph of

y

= \frac{12}{x}

Question 60Extended · no calculator

The graph of

y

= 2x^2 - 5x + 1

is drawn. Which straight line should be drawn on it to solve

2x^2 - 3x - 4

= 0

Question 61Extended · calculator

Find the value of

y

= 3 \times 1.5^x

when

x

= 4

, correct to

3

significant figures.

Question 62Core · no calculator

The curve

y

= x^2 - 6x + 5

crosses the

x

-axis at

x

= 1

and

x

= 5

. What is the equation of its line of symmetry?

Question 63Core · no calculator

Where does the graph of

y

= (x - 2)(x + 4)

cross the

x

-axis?

Question 64Extended · no calculator

By writing

x^2 + 8x + 10

in the form

(x + p)^2 + q

, find the turning point of

y

= x^2 + 8x + 10

Question 65Extended · no calculator

What are the asymptotes of the graph of

y

= \frac{3}{x} + 2

Question 66Extended · no calculator

Which statement about the graph of

y

= 2^x

is true?

Question 67Extended · no calculator

y

= 4x^3 - 5x^2 + 7

. Find

\frac{dy}{dx}

Question 68Extended · no calculator

Find the gradient of the curve

y

= x^3 - 4x

at the point where

x

= 2

Question 69Extended · no calculator

Find the coordinates of the stationary point of

y

= x^2 - 6x + 2

Question 70Extended · no calculator

The curve

y

= x^3 - 12x

has turning points at

x

= -2

and

x

= 2

. Which statement is true?

Question 71Extended · calculator

Find the gradient of the curve

y

= 0.5x^3 - 2.4x

at the point where

x

= 1.8

Question 72Extended · no calculator

h(x)

= 2x^2 + 3

. Find

h(-3)

Question 73Extended · no calculator

f(x)

= 3x - 5

. Find

f^{-1}(x)

Question 74Extended · no calculator

f(x)

= 2x + 1

and

g(x)

= x^2

. Find

fg(3)

Question 75Extended · no calculator

f(x)

= x - 4

and

g(x)

= 3x^2

. Find

gf(x)

Question 76Core · calculator

s

= ut + \frac{1}{2}at^2

. Find

s

when

u

= 2.5

t

= 3.2

and

a

= 9.8

. Give your answer correct to

3

significant figures.

Question 77Core · calculator

Solve the simultaneous equations

1.5x + 2y

= 9.5

and

1.5x - y

= 2

Question 78Core · calculator

A straight-line conversion graph from dollars to pesos passes through

(0, 0)

and

(50, 920)

. Convert

37

dollars to pesos.

Question 79Extended · calculator

The sequence

2.4, 3.6, 5.4, 8.1, \ldots

is geometric. Find its

5

th term.

Question 80Extended · calculator

A table of values is made for

y

= x^3 - 2x + 1

. Find

y

when

x

= -1.5

Question 81Extended · calculator

g(x)

= 2^x - 3

. Find

g(3.5)

, correct to

3

significant figures.

Paper-style questions: Core, then Extended

Ticket pricesCore · Paper 1 style, no calculator · 8 marks

At a theatre, an adult ticket costs

x

dollars and a child ticket costs

y

dollars. The Lee family buys

2

adult and

3

child tickets for

39

dollars. The Rao family buys

1

adult and

4

child tickets for

37

dollars.

(a) Write an equation in

x

and

y

for the Lee family's tickets. [1]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(b) Solve the simultaneous equations

2x + 3y

= 39

and

x + 4y

= 37

to find the cost of each ticket. You must show your working. [3]

Show your working on paper; type the final answer here. Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

n

adults and

2n

children. Write an expression, in terms of

n

, for the total cost of their tickets in dollars. Simplify your answer. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(d) The group can spend at most

200

dollars. Find the greatest possible value of

n

. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

Turning points of a cubicExtended · Paper 2 style, no calculator · 9 marks

A curve has equation

y

= x^3 - 6x^2 + 9x + 1

(a) Find

\frac{dy}{dx}

. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(b) Solve

\frac{dy}{dx}

= 0

. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

x

= 1

. [1]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(d) Explain how you know that the turning point where

x

= 1

is a maximum. [1]

0 words · a sentence

(e) Find the gradient of the curve at the point where

x

= 4

. [1]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).

(f) Find the equation of the tangent to the curve at the point where

x

= 4

, in the form

y

= mx + c

. [2]

Type fractions as 3/8, powers as x^2 and roots as sqrt(5).