The Hard Grade 9 Questions Thread 2019 Watch

Notnek
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Following on from the popular grade 9 threads in 2017 and 2018, here is a thread for 2019!

Feel free to post tough questions but aim to keep them within the limits of "questions that could be in a GCSE exam if the exam writer was being really nasty". So try not to go beyond the GCSE syllabus.
Last edited by Notnek; 2 months ago
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Notnek
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I'll kick things off with one that I've posted the last 2 years and was done badly in both threads:




The value of is 32.123 degrees correct to 3 decimal places. The value of y is 62.425 degrees correct to the nearest 0.005 degrees.

By considering bounds, find the value of to a suitable degree of accuracy. You must show all of your working.
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begbie68
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z=0.851 (3dp) ?
Not showing ALL working so that others might have this to do ...
Good question
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begbie68
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Triangle OAB
vector OA = a
vector OB = b

M is point on AB so that AM:MB = 1:2
P is point on OM so that OP:PM = 3:4

N is point on OB so that AP produced meets OB at N

ON:NB = x:y
Find smallest possible integers for values x,y.
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begbie68
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Select a counter at random from one of two bags:
black bag contains 9 counters : 2 red, 3 green & 4 blue.
white bag contains 12 counters : 5 red, 4 green & 3 blue.

P(choose black bag) = 1-P(choose white bag) = 4/9

Question :
Given that the selected counter is blue, what is the probability that it came from the white bag?
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Notnek
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Bumping during peak hours. I wonder if it will take off this year?
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RDKGames
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The line \mathcal{L} : 3y-4x=3 is tangent to parabola \Pi : 9y=(x+3)^2+9 at point A, and it crosses the x-axis at point B.
The perpendicular line to \mathcal{L} goes through point A and crosses the x-axis at point C.

Find the area of the triangle ABC, giving your answer in the form \displaystyle \frac{a^4}{a^2-1} where a is a natural number.
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RDKGames
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Diagram below shows a semicircle of radius a about origin O. Points A,B,Q,P lie on the semicircle with the point Q dividing the arc AP into the ratio 3:2. It is also known that \angle POB = \theta<90, \angle PAO = \alpha and \angle QAP = \beta

a) Determine the area of the shaded region in terms of a and \theta

b) Determine the length BP in terms of a and \theta

c) Prove that the gradient of OP is \tan(\theta) .... (HINT: \tan \theta = \dfrac{\sin \theta}{\cos \theta})

d) Prove that 2\alpha = \theta

e) Prove that 2\theta - \alpha = 90

f) Hence, find the values of \alpha and \theta

g) Thus express the shaded area and length BP in terms of a

h) Find the angle \beta to 3 decimal places.

Last edited by RDKGames; 2 months ago
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mpaprika
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I'm commenting on this so I don't lose this thread
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RDKGames
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notnek has a bag containing 3 types of sweets coloured red, green, and blue which split the total amount of sweets in the ratio 5 : 3 : 2 respectively. There are an n amount of red sweets.

notnek reaches into his bag and picks up a red sweet. He is upset that he didn't get the sweet he wanted so he angrily reaches into the bag and throws out 9 blue sweets on the ground. Then he reaches into his bag and picks up a green sweet. Again, notnek doesn't want this sweet and is so frustrated that he throws out 1 red sweet. Third time lucky, he reaches into his bag and picks up a blue sweet. notnek is satisfied. The probability of these events happening in this order is \frac{1}{48}.

a.) Show that 11n^2-535n-750=0 is an equation to be satisfied.

b.) What was the probability that notnek would pick the sweet he wanted on the first pick?

c.) How many red, blue, and green sweets were there in the bag at the start?


N.B. He does not put the sweets he doesn't want back in the bag.
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RDKGames
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The graph below shows a function f(x) that is linear in two different ways in the intervals 0\leq x \leq 4 and 4 < x \leq 6.

Find the values of a,b, where a<b, such that f^{n}(a)=a and f^n(b)=b for all integers n \geq 1.

Here f^n represents \displaystyle \underbrace{fff...f}_{n \ \text{times}}(x). E.g. f^4(x) \equiv ffff(x).

Last edited by RDKGames; 2 months ago
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ThunderBeard
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(Original post by begbie68)
Triangle OAB
vector OA = a
vector OB = b

M is point on AB so that AM:MB = 1:2
P is point on OM so that OP:PM = 3:4

N is point on OB so that AP produced meets OB at N

ON:NB = x:y
Find smallest possible integers for values x,y.
1:5?
(Original post by RDKGames)
Diagram below shows a semicircle of radius a about origin O. Points A,B,Q,P lie on the semicircle with the point Q dividing the arc AP into the ratio 3:2. It is also known that \angle POB = \theta<90, \angle PAO = \alpha and \angle QAP = \beta

a) Determine the area of the shaded region in terms of a and \theta

b) Determine the length BP in terms of a and \theta

c) Prove that the gradient of OP is \tan(\theta) (I assume at GCSE you know how to express tan in terms of cos and sine??)

d) Prove that 2\alpha = \theta

e) Prove that 2\theta - \alpha = 90

f) Hence, find the values of \alpha and \theta

g) Thus express the shaded area and length BP in terms of a

h) Find the angle \beta to 3 decimal places.

Lost after d and didn’t get c so rip.
RIP gave up on the rest cause I’m a too dumb to do them. RDKGames solutions?
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ThunderBeard
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(Original post by begbie68)
Select a counter at random from one of two bags:
black bag contains 9 counters : 2 red, 3 green & 4 blue.
white bag contains 12 counters : 5 red, 4 green & 3 blue.

P(choose black bag) = 1-P(choose white bag) = 4/9

Question :
Given that the selected counter is blue, what is the probability that it came from the white bag?
5/27?
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RDKGames
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(Original post by ThunderBeard)
1:5?


Lost after d and didn’t get c so rip.
RIP gave up on the rest cause I’m a too dumb to do them. RDKGames solutions?
For part (C) I have added a hint. Does that help?

Further hint if you need it:
Spoiler:
Show

Notice that the gradient of the line OP is given by the length of the vertical blue dashed line divided by horizontal red dashed line. [change in Y divided by change in X]


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begbie68
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Which bit of Grade 9 questions does this apply to?

(Original post by RDKGames)
The line \mathcal{L} : 3y-4x=3 is tangent to parabola \Pi : 9y=(x+3)^2+9 at point A, and it crosses the x-axis at point B.
The perpendicular line to \mathcal{L} goes through point A and crosses the x-axis at point C.

Find the area of the triangle ABC, giving your answer in the form \displaystyle \frac{a^4}{a^2-1} where a is a natural number.
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begbie68
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1:5 is possibly a good attempt, but I got a different answer ... !
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begbie68
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(Original post by ThunderBeard)
5/27?
no. again, I got a different answer ...
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begbie68
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Nice one. Is notnek related to Hannah, or is it that they both like sweets? If I remember correctly, Hannah preferred the yellow sweets?!

(Original post by RDKGames)
notnek has a bag containing 3 types of sweets coloured red, green, and blue which split the total amount of sweets in the ratio 5 : 3 : 2 respectively. There are an n amount of red sweets.

notnek reaches into his bag and picks up a red sweet. He is upset that he didn't get the sweet he wanted so he angrily reaches into the bag and throws out 9 blue sweets on the ground. Then he reaches into his bag and picks up a green sweet. Again, notnek doesn't want this sweet and is so frustrated that he throws out 1 red sweet. Third time lucky, he reaches into his bag and picks up a blue sweet. notnek is satisfied. The probability of these events happening in this order is \frac{1}{48}.

a.) Show that 11n^2-535n-750=0 is an equation to be satisfied.

b.) What was the probability that notnek would pick the sweet he wanted on the first pick?

c.) How many red, blue, and green sweets were there in the bag at the start?


N.B. He does not put the sweets he doesn't want back in the bag.
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Notnek
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(Original post by begbie68)
Which bit of Grade 9 questions does this apply to?
It’s tough but I wouldn’t say it’s beyond GCSE level. It’s the kind of question that you see in GCSE further maths papers.
Last edited by Notnek; 2 months ago
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laurawatt
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eek! might have a bash at these later
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