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Year 11 Advanced task 2 paper
math advanced
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NESA Number: Teacher:
Meriden School
Year 11 Mathematics Advanced
Task 2, 2023
Time allowed: 55 minutes
Instructions to all students:
- Attempt all questions.
- Answer each question in the space provided.
- All necessary working must be shown.
- Use blue or black pen.
- NESA-approved calculators may be used.
- A reference sheet is provided.
- Diagrams are not necessarily drawn to scale.
Question 1 / Question 2 / Question 3 /
Total /
Question 1 (16 marks)
(a) Given a function f (x) = x 4 + 2x 3 − 6 , find the value of f (2). 1 ......................................................................................................... .........................................................................................................
(b) Classify the following relation as one-to-one, one-to-many, many-to-one or many-to-many. 1
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(c) Two fair 6-sided dice are rolled and the numbers on the uppermost faces are observed. Find the probability that:
(i) both numbers are the same, 1 ........................................................................................................ ........................................................................................................ ........................................................................................................ ........................................................................................................ (ii) the sum of the two numbers is four. 1 ........................................................................................................ ........................................................................................................ ........................................................................................................ ........................................................................................................
(h) Complete the square to find the centre and radius of the circle x 2 + y 2 − 6 x + 4y − 3 = 0. 3
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(i) Show that f (x) = x 2 + 6 + 1 x 2 is an even function. 2 ......................................................................................................... ......................................................................................................... ......................................................................................................... ......................................................................................................... ......................................................................................................... ......................................................................................................... ......................................................................................................... .........................................................................................................
End of Question 1
Spare writing page: Clearly indicate which part of Question 1 is being answered here.
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(c) The graph below shows part of an odd function. 1
On the graph above, draw the missing part of the function.
(d) Consider the functions f (x) = √x − 1 and g(x) = 3x 2 + 4x. Find:
(i) the expression of g(f (x)), 1 ........................................................................................................ ........................................................................................................ ........................................................................................................ ........................................................................................................ (ii) the domain for g(f (x)). 1 ........................................................................................................ ........................................................................................................ ........................................................................................................ ........................................................................................................
(e) A game involves two people, Tom and Gareth. The spinner is made of five sections of equal areas. There are two red sections, two yellow sections and one green sections. Tom pays 50 cents to play the game. If the spinner lands on red he gets nothing, if the spinner lands on yellow he gets his money back, if the spinner lands on green, Gareth will pay Tom $1.
- Tom loses 50 cents for red
- Tom gets his money back for yellow
- Tom wins 90 cents for green (i) Let the random variable X represent the payout in cents for Tom when the game 1 is played once. Complete the following table.
x − 50 0 90 P (X = x)
(ii) Calculate the expected value for Tom to play this game once. 2 ........................................................................................................ ........................................................................................................ ........................................................................................................ (iii) How much would Tom be expected to lose if they played the game 500 times? 1 ........................................................................................................ ........................................................................................................ ........................................................................................................
(f) A bag contains 3 blue and 2 red marbles. Marbles are drawn at random, one by one 3 without replacement, until two red marbles have been drawn. Find the probability that exactly 3 draws will be required. ......................................................................................................... ......................................................................................................... ......................................................................................................... ......................................................................................................... ......................................................................................................... .........................................................................................................
End of Question 2
NESA Number: Teacher:
Question 3 (16 marks)
(a) Consider the functions f (x) = x 2 − 2 x − 5 with domain [− 1 , 4] and g(x) = ax 2 − 10 with domain (−∞, ∞), where a is a constant.
(i) Find the range of f (x). 2 ........................................................................................................ ........................................................................................................ ........................................................................................................ ........................................................................................................ ........................................................................................................ (ii) Determine the value of a, if g(f (1)) = 8. 2 ........................................................................................................ ........................................................................................................ ........................................................................................................ ........................................................................................................ ........................................................................................................
(b) The events A and B are such that 3 P (A) = 4 9 , P (A|B) = 3 8 , P (B|A) = 1 4. Determine the value of P (A ∪ B). ......................................................................................................... ......................................................................................................... ......................................................................................................... ......................................................................................................... ......................................................................................................... ......................................................................................................... ......................................................................................................... .........................................................................................................
(c) For the discrete random variable X the probability distribution is given by
P (X = x) =
{
kx for x = 1, 2 , 3 , 4 k(8 − x) for x = 5, 6 , 7.
(i) Show that k = 161. 2 ........................................................................................................ ........................................................................................................ ........................................................................................................ ........................................................................................................ ........................................................................................................ (ii) Find E (X). 1 ........................................................................................................ ........................................................................................................ ........................................................................................................ ........................................................................................................ (iii) Find Var (X). 2 ........................................................................................................ ........................................................................................................ ........................................................................................................ ........................................................................................................ ........................................................................................................ ........................................................................................................ ........................................................................................................
Spare writing page: Clearly indicate which part of Question 3 is being answered here.
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Year 11 Advanced task 2 paper
Subject: math advanced
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