The Hard Grade 9 Questions Thread 2019

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.

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.

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?

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?

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.

5/27?

@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.

Which bit of Grade 9 questions does this apply to?