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Please consider the five (5) questions below, having 3 subparts each. Each question subpart is worth 4 points. Answer each question by solving each subpart and writing your answers in the space provided.
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……………………………………………………………………………Question – 1: Consider the arithmetic sequence, , such that, ,
(a) Write down the closed form of this sequence, and hence calculate the 15th term of the sequence.
(b)Describe the long term behaviour of the sequence, .
(c)If , find the first four terms of
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Question – 2: Consider the circle, C, and the line, L, described by the equations: , & , respectively.
(a) Is the line L tangent to the circle C?. If so, find the point of tangency.
(b) Find the equation of the line passes through the center of C, and perpendicular to the line L.
(c) Find the points where the circle C intersects the y-axis, and where circle C intersects the x-axis, if any.
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Question–3: Consider the points, A (–1, – 6), and B (1, 6).
(a) Find the equation of the circle, C, the diameter of which is the segment AB.
(b) Find the area of the largest square, S, that can be inscribed inside the circle, C.
(c) Find the area of the smallest square, T, that can contain the circle, C.
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Question – 4: A piece of wire 100 cm long is cut into two pieces of length x and 100-x, respectively. the first piece is bent into the shape of a square, and the second is bent into the shape of a circle.
(a) Express as a function of x, the sum A of the area of the square and the circle.
(b) Find the value of x, if any, that makes the area of the square equal to the area of the circle.
(c) Specify the domain and the range of the function A(x)
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Question - 5: Consider the functions, f and g, given by:
, and .
(a) Find the domain of the function .
(b) Show that f is not one-one, and determine (the largest) sub-domains on which f is one-one. Find the inverse of f on a sub-domain containing, the value, x = 2.
(c) Find the function and specify the range and the domain of this function
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