(a) Find the equation of the horizontal asymptote to the graph of y=f(x).

Question 1. [Maximum mark: 6] The relation sin x +cos  y =0.5 is graphed alongside for . P lies on the y-axis, and the tangents at P and Q are perpendicular. The coordinates of Q is __________.

Question 2. [Maximum mark: 6]  Jack and John have decided to play a game. One roll of a die is considered as one round of the game. On each round, John agrees to pay Jack $4 if 1 or 2 is rolled, Jack agrees to pay John $2 if 3, 4, 5 or 6 is rolled, and who is paid wins the round. In the end, who earns money wins the game. Now Jack and John roll a die 12 times. Let A denote the number of rounds Jack wins. Find the expected value of A, E[A]= __________.

(b) Find f (x') for x ∈ ,x≠0. 

(c) Find the expression for , and state the domain of .

(a) Write down the value of n and the value of p. 

(b) State an assumption that is being made for X to be considered as following a binomial distribution. 

(c) Show that the variance of the proportion of marked fish in the sample, Var(X/300), is 0.000425. 

The estate manager feels confident that the proportion of marked fish in the lake will be within 1.5 standard deviations of the proportion of marked fish in the sample and decides these will form the upper and lower bounds of his estimate.(d) Taking the value for the variance given in (c) as a good approximation for the true variance, find upper and lower bounds for the number of fish in the lake when t=8.

(e) Given this result, comment on the validity of the linear model used initially.

(a) Find an expression for the volume of water V, in m3 , in the left part of the water tank in terms of β?.  

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