2 10 pts very carefully sketch the graph of the

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2. (10 pts.) Very carefully sketch the graph of the equation x 2 = -4y below. [ See tr-t4g2.pdf. ] y x
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TEST-04/MAC1114 Page 2 of 4 3. (15 pts.) Sketch the given curve in polar coordinates. Do this as follows: (a) Carefully sketch the auxiliary curve, a rectangular graph on the coordinate system provided. (b) Then translate this graph to the polar one. [ See tr-t4g3.pdf. ] Equation: r = 2 cos(2 θ ) (a) r θ (b) y x 4. (10 pts.) Write each expression in the standard form a + bi. (a) (8 - 5i) + (-9 +2i) = -1 - 3i (b) 13/(4 - 3i) = (52/25) + (39/25)i (c) 4i 5 - 6i 7 = 4i -(6)(-i) = 10i (d) (2 - 4i) (-3 + 5i) = 14 + 22i (e) [2(cos 30° + i sin 30°)] 5 = 2 5 [cos(150°) + i sin(150°)] = -16 3 1/2 + 16i
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TEST-04/MAC1114 Page 3 of 4 5. (10 pts.) Very carefully sketch the graph of the equation (1/9)x 2 + (1/4)y 2 = 1 below. [ See tr-t4g5.pdf. ] y x 6. (5 pts.) Solve the following equation in the complex number system: x 4 - 4 x 2 - 5 = 0 By factoring, we have x 4 - 4 x 2 - 5 = 0 (x 2 - 5)(x 2 + 1) = 0 (x - 5 1/2 )(x + 5 1/2 )(x + i)(x - i) = 0 Thus, x = 5 1/2 or x = -5 1/2 or x = -i or x = i. If necessary, you can use the quadratic formula to get the roots of x 2 - 5 = 0 and x 2 + 1 = 0. Keep firmly in mind, however, the quadratic formula provides a factorization of quadratics.
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