Substitution into formulae

Use the formula

v = u + at

to find the final velocity, when:

• the initial velocity is 8 m/s

• the acceleration is 6.5 m/s²

• the time is 14 seconds.

Use the formula

to find the value of q when p = 39 and r = 6.

A straight line has equation y = 5 x + 2.

Write down the gradient of the line.

Find the value of y when x = 7.

Make x the subject of y = 5 x + 2.

The population of fish, P, in a pond is predicted by the formula

P = 640 × 1.025ⁿ

where n is the number of years after 1st January 2020.

Write down the population of fish on 1st January 2020.

Write down the percentage increase per year used in the formula.

Show that the formula predicts that the population of fish will reach 1000 between

1st January 2038 and 1st January 2039.

Give one reason why this might not happen.

You may use these kinematics formulae to answer this question.

v = u + at

where a is the constant acceleration, u is the initial velocity, v is the final velocity, s is the displacement and t is the time taken.

A particle has an initial velocity of 2 m/s. After 6 seconds the particle has a velocity of 11 m/s. The particle had constant acceleration.

Work out the distance the particle has travelled in the 6 seconds.

Show that the equation x⁴ + 3x² - 2x - 5 = 0 has a solution between x = 1 and x = 2.

Find this solution correct to 1 decimal place. You must show calculations to support your answer.

You may use these kinematics formulae to answer these questions.

v = u + at

v² = u² + 2as

A moving particle accelerates at 2 m/s² for 8 seconds. The particle’s final velocity after the 8 seconds is 21 m/s.

Show that the velocity of the particle at the start of the 8 seconds is 5 m/s.

Work out the distance travelled by the particle during the 8 seconds.

Here is a formula.

a = b (c + d)

Find the value of a when b = 3, c = 7 and d = 5.

Find the value of c when a = 30, b = 4 and d = 6.

Use the formula

v² = u² + 2as

to find the final velocity when

• the initial velocity is 6 m/s

• the acceleration is 4 m/s²

• the distance travelled is 8 m.

The following kinematics formulas may be used in this question.

v = u + at

v² = u² + 2as

A particle has an initial velocity of 0 m/s. The particle accelerates uniformly at 3 m/s² for 4 seconds.

Find the distance travelled by the particle in the 4 seconds.

Show that the formula v = u + at can be rearranged to a =

Use the formula

to find the acceleration, am/s², when a particle takes 4 seconds to increase from an initial velocity of 3 m/s to a final velocity of 9 m/s.

Part of the graph of y = 2 x + 1 is drawn on this grid.

Write down the y-intercept.

The line continues to the right.

Will the line pass above, below or through the point (40, 80)? Show how you decide.

The line y = 2 x + 1 will pass........ the point (40, 80) because........

Write down the equation of a line that is parallel to y = 2 x + 1.

Finley is asked to solve the equation 5 x + 4 = 19.

Finley’s working is shown below.

Write down the error that Finley has made.

Charlie is asked to use the formula

v = u + at

to find the initial velocity, when

• the acceleration is 5 m/s²

• the final velocity is 29 m/s

• the time is 3 seconds.

Charlie’s working is shown below.

Write down the error that Charlie has made.

Rearrange this formula to make u the subject.

A rocket accelerates at 90 m / s² and travels 270 km. The rocket’s final velocity is 8000 m / s.

Using part (a), or otherwise, calculate the rocket’s initial velocity in m / s.

The diagram shows the graph of y = kx - x² + 2, where k is an integer.

Show that k = 3.

Use the graph to solve 3x - x² + 2 = 3. Give your answers to 1 decimal place.

The price of a seat on a flight, £ P, is given by

P = 49 × 1.009ⁿ

where n is the number of seats already sold on this flight.

Write down the percentage increase in price of the second seat sold compared to the first seat sold.

Show that the price of the 40th seat sold is less than £70.

Rearrange this formula to make d the subject.

f = 5 d + 4

Use the formula

v = u + at

to find the final velocity, when

• the initial velocity is 5 m/s

• the acceleration is 7.5 m/s²

• the time is 6 seconds.

Find the value of 4 x + 5 y when x = 3 and y = - 2.

The graph shows the solution to this pair of simultaneous equations.

2 x + y = 14 x + 2 y = 13

Use the solution to work out the value of 3 x + y. You must show how you work out your answer.

You may use these kinematics formulae to answer this question.

v = u + at

A particle has an initial velocity of 3 m/s. After 20 seconds the particle has a velocity of 11 m/s.

Work out the distance the particle has travelled after 20 seconds.

An estimate for the number of seals on an island is given by the formula

P = 5200 × 1.02^t

where P is the number of seals t years after the start of year 2015.

Write down the annual percentage increase in the number of seals on the island.

Use the formula to show that there may have been about 4700 seals on the island at the start

of year 2010.

Here is a rule to work out the time, in minutes, needed to cook a turkey.

Ling’s turkey takes 150 minutes to cook.

Use the rule to work out the weight of Ling’s turkey.

James cooks a different turkey. His turkey weighs 6 kg. James wants to take his turkey out of the oven at 1:15 pm.

Use the rule to work out at what time James should put his turkey in the oven. You must show your working.

In a dance competition, four judges award marks to each dancer. Each judge can award 1, 2, 3, 4 or 5 marks.

The four judges’ median mark, m, is put into the formula

S = 10 m - 5

to get the dancer’s score, S.

Sam is awarded marks of 4, 3, 1 and 4. Work out Sam’s score.

Taylor gets a score of 40. Taylor says

The judges must have awarded marks of 4, 4, 5 and 5 because the median is 4.5 and 4.5 × 10 - 5 = 40.

Why is Taylor not correct? Show working to support your reason.

A car accelerates at 4.06 m/s² for 10.1 seconds from an initial velocity of 2.93 m/s.

Harper rounds each value to 1 significant figure. Harper uses the rounded values and the formula

to estimate the distance travelled in the 10.1 seconds. Harper’s answer is 430 metres.

Using Harper’s method, show that their answer is wrong.

Rearrange this formula to make t the subject.

The graph of y = x² + 6x - 2 is shown below. The roots of the equation x² + 6x - 2 = 0 are at p and q.

Calculate y when x = 1.

Without solving the equation, explain why q must lie between 0 and 1.

Explain why using a method of iteration is not the most appropriate way of finding a solution to this equation.

The formula

P = 6800 × 1.045ⁿ

is used to predict the population, P, of an island n years after 2018.

Write down the population of the island in 2018.

Write down the percentage growth rate used in the formula.

Work out the population predicted by the formula for the year 2030.

Give one reason why the answer to (c)(i) may not be reliable.

The following kinematics formulas may be used in this question.

v = u + at

v² = u² + 2as

The initial velocity of a particle is 20 m/s. The acceleration of the particle is -8 m/s². After t seconds, the particle has travelled 25 m.

Show that 4t² - 20t + 25 = 0.

Solve 4t² - 20t + 25 = 0.

Show that the particle is stationary when it has travelled 25 m.

A train is travelling with a velocity of 15 m / s. It then accelerates at 0.5 m / s² for 6 seconds.

Use the formula v = u + at to calculate the velocity of the train after the 6 seconds.

Rearrange the formula v = u + at to make a the subject.

The base of a cone is fixed to the top of a cylinder to make a decoration.

The radius of the base of the cone and of the cylinder is r cm. The cone’s height is 5 r cm. The total height of the decoration is 6 r cm. The total volume of the decoration is 225 cm³.

Calculate the value of r. Show your working.

In this question, all lengths are in centimetres.

Here is a square.

Find the length of one side of the square when b = 4.

Use the formula F = to find the value of F when

Give your answer in standard form, correct to 2 significant figures.

The value of a house, £ H, is given by the formula

H = 165 000 × 1.03^t

where t is the number of years after 1st January 2010.

Write down the value of the house on 1st January 2010.

Write down the annual percentage increase in the value of the house.

Show that the value of the house is over £200 000 on 1st January 2017.

Martina has answered some questions on algebra. In each question, she has made an error.

Describe her error and give the correct answer to each problem.

Question 1 Simplify. 2a × a × a

Martina’s answer 4a

Martina’s error is........

Martina’s error is........

s = 225

Martina’s error is........

Find the value of 3 g - h when g = 4 and h = 5.

At the start of 2018, the population of a town was 17 150. At the start of 2019, the population of the town was 16 807.

It is assumed that the population of the town is given by the formula

where P is the population of the town t years after the start of 2018.

Write down the value of a.

Show that r = 0.98.

Show that the population is predicted to be less than 16 000 at the start of 2022.

Use the formula to work out what the population might have been at the start of 2017.

Gill uses the formula

h = 2 fg

Find the value of h when f = 1 and g = 3.

Find the value of g when h = 18 and f = 6.

The diagram shows a cylinder and a sphere.

The cylinder has radius 12 cm and height 30 cm. The cylinder and the sphere have the same volume.

Work out the radius r cm of the sphere.

Simplify.

5 x – 6 y – x + 3 y

w⁸ ÷ w²

5c²d × 3c

Work out the value of

4 x – 7 when x = 5,

You are given that 5 y = 4 x.

Find the value of y when x = 10.

Write y in terms of x.

The diagram shows a cylinder and a cone.

The cylinder has radius 2 cm and height 9 cm. The cone has radius r cm and height h cm.

The ratio r: h is 1: 4. The volume of the cone is equal to the volume of the cylinder.

Work out the value of r.

The diagram shows a straight line that passes through points A and B, and a curve that passes through points P and Q.

Find the equation of the straight line.

The equation of the curve is y = x² + kx + 8.

Find the value of k.

Diann draws line BQ. She says

Triangle ABQ is isosceles.

Is Diann correct? You must show all your working.

The graph of y = x³ - 7x - 12 is shown below. The root of the equation x³ - 7x - 12 = 0 is p.

Calculate y when x = 3.

Show that 3 < p < 4.

Find a smaller interval that contains the value of p. You must show calculations to support your answer.

Simplify.

2 p + 5 p - 3 p

6 j + 3 k - j - 5 k

Find the value of 10 h + 6 t when h = 12 and t = 4.

Rearrange this formula to make d the subject.

e = f - 7 d

This is a rule to find the time, in minutes, needed to roast lamb.

Use the rule to work out the time needed to roast a piece of lamb which weighs 4 pounds.

A different piece of lamb takes 95 minutes to roast.

Use the rule to work out the weight of this piece of lamb.

A company tests a new battery for an electric car. The distance the car travels, d km, and the charge left in the battery, C %, are measured.

Some measurements are shown in the table.

Plot these values on the grid and use them to draw a straight line.

Use your line to estimate the greatest distance the car will travel.

What assumption is made when estimating the greatest distance?

For your line in part (a), find

the gradient,

the C-axis intercept.

Use your answers to part (c) to write down the equation of your graph.

Give your equation in the form C = ad + b.

Use your equation to find the value of C when d = 210.

Comment on your answer.

Calculate s when u = 5, t = 10 and a = 3.

Make a the subject of the formula.

Find the value of 3 a + 2 b when a = 16 and b = 7.

Use the formula

v = u + at

to find the final velocity, when

• the initial velocity is 2 m/s

• the acceleration is 1.5 m/s²

• the time is 6 seconds.

Make d the subject of this formula.

c = 7 d

Find the value of s when u = 12, a = 10 and t = 4.