Quick!!!! F.4 Mathematics

1. If p + qi = (1 - 4i)(q - i) where p and q are real numbers, find the values of p and q.

p + qi = (1 - 4i)(q - i) = q - i -4qi - 4 = (q-4) - (4q+1)i

By comparing real part and imaginary part,
{ p = q - 4
{ q = -(4q+1)

-q = 4q + 1
5q = -1
q = -0.2
p = -4.2

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2ai. The figure shows the graph of y = x² + bx + c. Express in terms of b and c, find the axis of symmetry of the graph.

y = x² + 2(b/2)x + c
= (x + b/2)² + c - b²/4 ...(*)

The axis of symmetry is x = -b/2.

2aii. the minimum value of y.

From (*), the minimum value of y is c - b²/4.

2b. If the vertex of the graph is (1,-4), find the values b and c.

From (a),
{ -b/2 = 1
{ c - b²/4 = -4

Therefore, b = -2
c = b²/4 - 4 = -3

2c. Using the results of (2b), find the x-intercepts and y-intercept of the graph.

y = x² - 2x - 3

When y = 0,
x² - 2x - 3 = 0
(x - 3)(x + 1) = 0
x = 3 or x = -1
The x-intercepts are 3 and -1.

When x = 0, y = -3.
The y-intercept is -3.

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