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BHASVIC MαTHS

“Mathematics seems to endow one with something like a new sense” Charles Darwin

The mean marks for a stats exam were worked out for 3 classes.

Class 1 had 12 students with a mean of 78% and standard deviation of 8



(a) Find the mean mark for all 46 students, give your ans to the nearest whole

number

A new student joins class 1 who has a mark of 83%.

(b) Explain with reasons whether the mean for this class would increase,

decrease or remain the same when the new student is added to the class, and

whether the standard deviation would increase, decrease or remain the same when

the new student is added to the class.

(c) The teacher of class 3 claims that her class’ results are not actually worse than

class 1 or class 2. The head of department looks at her results and agrees. What

could explain the lower average score?

BHASVIC MαTHS


2

Rupal throws a ball upwards at 2 ms-1 from a window which is 4 m above ground

level.

(a) Draw an st, a vt and an at graph for this motion. Explain the links between

them.

(b) Use suvat to find an equation for the height h m of the ball above the ground

after t seconds (while it is still in the air)

(c) Use your answer to part (b) to find the time the ball hits the ground

(d) How fast is the ball moving just before it hits the ground?

(e) In what way would you expect your answers to (c) and (d) to change if you

were to take air resistance into account?

BHASVIC MαTHS


3

A car travelling on a straight road slows down with constant deceleration. The

car passes a road sign with speed 40km h-1 and then a post box with speed

24 km-1. The distance between the road sign and the post box is 240m.

(a) Draw an st, a vt and an at graph for this motion. Explain the links between

them.

Find, in ms-2, the deceleration of the car

(b) using your vt graph

(c) using suvat equations

BHASVIC MαTHS


4

(a) A car is joined to a caravan with a light rigid towbar and is driving along a

level road. The car has mass 1200 kg and the caravan has mass 800 kg.

Resistance forces of 600 N and 800 N act on the car and caravan respectively.

The acceleration of the car and caravan is 0.2 ms-2.

Draw a force diagram to model this situation.

(i) How have you used the fact that the towbar is light in your model?

(ii) How have you used the fact that the towbar is rigid in your model?

(iii) It would also be possible to model the car-caravan combination as one

particle. Draw this new force-acceleration diagram.

BHASVIC MαTHS


4

(b) Particles A and B each of mass 20kg are joined by a light inextensible string

which passes over a smooth pulley so that the string hangs vertically on both

sides

Draw a force diagram to model this situation.

(i) How have you used the fact that the string is light in your model?

(ii) How have you used the fact that the string is inextensible in your model?

(iii) How have you used the fact that the pulley is smooth in your model?

(iv) It is not possible to model the mass A - mass B combination as one particle

(as we did in part a). Explain why not.

BHASVIC MαTHS


5

a) Sketch the following curves of y = f (x), stating the equations of the

asymptotes and the coordinates of any axis intercepts:

(i) 𝑓 𝑥 = 2 −1

𝑥

(ii) 𝑓 𝑥 =1

𝑥−3+ 1

(b) Sketch the following curves of y = f (x), stating the coordinates of any axis

intercepts

(i) 𝑓 𝑥 = 2(𝑥 − 4)2 + 1

(ii) 𝑓 𝑥 = 5 − 𝑥 2𝑥 + 1

(iii) 𝑓 𝑥 = 𝑥 + 2 𝑥 − 3 2𝑥 − 5

(iv) 𝑓 𝑥 = (𝑥 + 2)2 𝑥 − 3 (v) 𝑓 𝑥 = 𝑥 + 2 𝑥 + 3 1 − 𝑥 3𝑥 − 1

BHASVIC MαTHS


6

(a) Given that 2𝑥+𝑦 = 1 and 103𝑥−𝑦 = 100, find x and y

(b) Given that 𝑦 =1

8𝑥3, express each of the following in the form 𝑘𝑥𝑛 where k

and n are constants.

(i) 𝑦1

3

(ii) 1

2𝑦−2

(c) Given that 4𝑥3+ 𝑥5

𝑥 can be written as 4𝑥𝑎 + 𝑥𝑏, find a and b

(d) Solve the equation 8 + 𝑥 12 =8𝑥

3

BHASVIC MαTHS


7

Use desmos to look at the graph of 𝑓 𝑥 = 4𝑥2 − 4𝑘𝑥 − 𝑘

(when you type the 4kx, click ‘add slider’).

(a) Using the slider,

(i) find the values of k for which the quadratic 𝑓 𝑥 has a repeated real root

(ii) Write in set notation and in interval notation, the set of values of k for which

the quadratic 𝑓 𝑥 has no real roots

(iii) Write in set notation and in interval notation, the set of values of k for which

the quadratic 𝑓 𝑥 has two distinct real roots

(b) By completing the square, find in terms of the constant k the roots of the

equation 𝑓 𝑥 = 4𝑥2 − 4𝑘𝑥 − 𝑘

(c) Explain the connection between what you have found in part (b) and what

you found in part (a)

BHASVIC MαTHS


8

The points A (-1, -2), B (7, 2) and C (k, 4), where k is a constant, are the vertices

of the triangle ABC. Angle ABC is a right angle.

Find the area of the triangle ABC

*lines/circles/triangles are geometry questions so always draw a diagram*

BHASVIC MαTHS


9

The line 𝑙1 has equation 2𝑥 − 𝑦 + 4 = 0

The line 𝑙2 has equation 6𝑥 − 3𝑦 − 9 = 0

(a) Prove that 𝑙1 and 𝑙2 are parallel

(b) Prove that the point A (3, 10) lies on 𝑙1

The 𝑙3line is perpendicular to 𝑙1 and passes through the point A

(c) Find the point of intersection of 𝑙3 and 𝑙2

(d) Find the shortest distance between 𝑙1 and 𝑙2

*lines/circles/triangles are geometry questions so always draw a diagram*

BHASVIC MαTHS


10

The diagram shows a section of a suspension bridge carrying a road over water.

The height of the cables above water level in metres can be modelled by the

function ℎ 𝑥 = 0.00012𝑥2 + 200, where 𝑥 is the displacement in metres from

the centre of the bridge.

(a) Interpret the meaning of the constant term 200 in the model

(b) Use the model to find the two values of 𝑥 at which the height is 346m.

(c) Given that the towers at each end are 346m tall, use your answer to part b to

calculate the length of the bridge to the nearest metre.

BHASVIC MαTHS


1 - Answers

(a) 71%

(b) Answer not given

(c) Answer not given

BHASVIC MαTHS


2 - Answers

(a) Answer not given

(b) ℎ = 4 + 2𝑡 − 4.9𝑡2

(c) 1.13 s

(d) 9.08 ms-1

(e) Answer not given

BHASVIC MαTHS


3 - Answers

(a) Answer not given

(b) Answer not given

(c) 0.165 ms-2

BHASVIC MαTHS


4 - Answers

Answers not given

BHASVIC MαTHS


5 - Answers

In the library computers you can sketch the graphs on ‘autograph’. On your

phone you could use the free app ‘desmos’ or use wolfram alpha. Or, use your

graphical calculator to check. It is important you try these yourself first, don’t go

straight to the answers!

BHASVIC MαTHS


6 - Answers

(a) Show your method! 𝑥 =1

2, 𝑦 = −

1

2

(b) Show your method!

(i) 1

2𝑥

(ii) 32𝑥−6

(c) Show your method! 𝑎 =5

2, 𝑏 = 2

(d) 𝑥 = 4 3

BHASVIC MαTHS


7 - Answers

(ai) 𝑘 = 0

(aii) Answer not given

(aiii) Answer not given

(b) 𝑥 = 2𝑘 ± 4𝑘2 + 𝑘

(c) Answer not given

BHASVIC MαTHS


8 - Answers

(a) 6

(b) 2𝑥 + 𝑦 − 16 = 0

(c) 10

BHASVIC MαTHS


9 - Answers

(a) make sure to conclude your proof with a statement of what you have found.

(b) don’t sub in both 3 and 10 then say 0 = 0, this isn’t a proof! Sub in x = 3 and

‘find’ that y = 10 or vice versa. Conclude your proof with a statement of what

you have proved.

(c) 29

5,

43

5

(d) 7

55

BHASVIC MαTHS


10 - Answers

(a) Ans not given

(b) 𝑥 = ±1103

(c) 2206 m

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