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Page | 1 Practice Test on Topic 18 Complex Numbers Test on Topic 18 Complex Numbers 1.Express the following as complex numbers a + bi (a) (b) 2 25 (c) 2 + 8 (d) 4 + 9 (e) 4 + 16 (f) 36 + 25 2.Write each of the following as a complex number in the form a + bi (a) 81 36 (b) (2i – 1)(3i – 2) 3.Write each of the following as a complex number in the form a + bi. (a) (5 + 4i) + (3 + 2i) (c) 2(3 + 2i) + 5(4 – 3i) (b) (3 + i) – (7 + 2i) (d) 4(3 + 4i) – 2(6 – 6i) 4. Write each of the following as a complex number in the form a + bi. (a) 2i(3 + 4i) (c) (2 + i)(4 – 3i) (e) (2 + 3i) 2 (b) 5i(3 + 4i) (d) (5 + 4i)(5 – 4i) (f) 5. 6. The quadratic equation x 2 – 2x + 6 = 0 has two solutions which are both complex numbers. Use the quadratic formula to find these solutions. The quadratic equation x 2 – x + 4 = 0 has two solutions which are both complex numbers. Use the quadratic formula to find these solutions.
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Practice Test on Topic 18 Complex Numbers Complex Numbers Solutions 1.Express the following as complex numbers a + bi (a)=0 + 3i (b) 2 25 = 2 – 5i (c) 2 + 8 = 2i 2 2i = 0 3 2i (d) 4 + 9 =4+ 9 1 = 4+ 9 1 =4 + 3i (e) 4 + 16 = 4 1 + 16 1 ====== 4 1 + 16 1 2i + 4i 6i (f) 36 + 25 = 6i + 5i = 0 + 11i 2.Write each of the following as a complex number in the form a + bi (b) 81 36 =9i +6i = 15i (b)(2i – 1)(3i – 2)=6i 2 – 4i – 3i + 2=– 6 – 7i + 2=– 4 – 7i 3.Write each of the following as a complex number in the form a + bi. (a) (5 + 4i) + (3 + 2i)=5 + 3 + 4i + 2i=8 + 6i (c) (3 + i) – (7 + 2i) =3 + i – 7 – 2i = 3 – 7 + i– 2i = – 4 – i (c) 2(3 + 2i) + 5(4 – 3i) (d) 4(3 + 4i) – 2(6 – 6i) ==== 6 + 6i + 20 – 15i 12 + 16i – 12 + 12i ==== 9 + 20 + 6i – 15i 12 – 12 + 16i + 12i ==== 29 – 9i 28i 4. Write each of the following as a complex number in the form a + bi. Page | 2
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Practice Test on Topic 18 Complex Numbers Page | 3 (a) 2i(3 + 4i) =6i + 8i 2 Using the distributive law ====== 6i + 8(–1) 6i – 8 – 8 + 6i Using the property that i 2 = – 1 Rearranging terms (b) 5i(3 + 4i) =15i + 20i 2 ====== Using the distributive law 15i + 20(–1) Using the property that i 2 = – 1 15i – 20 – 20 + 15i Rearranging terms Using FOIL Using the property that i 2 = – 1 Using FOIL Using the property that i 2 = – 1 (c) (2 + i)(4 – 3i) = = (d) (5 + 4i)(5 – 4i) = = 8 – 6i + 4i – 3i 2 8 – 2i – 3(–1) 8 – 2i + 3 11 – 2i 25 – 20i + 20i – 16i 2 8 – 2(–1) 8+2 10 (e)(2 + 3i) 2 =(2 + 3i)(2 + 3i) ======== 4 + 6i + 6i + 9i 2 4 + 12i + 9(–1) 4 + 12i – 9 – 5 + 12i Using the distributive law Using the property that i 2 = – 1 Adding and subtracting like terms (f) First we convert all the numbers to be in terms of i = (-) i Using the distributive law Using the property that i 2 = – 1 = = – 4 – 12i
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Practice Test on Topic 18 Complex Numbers Page | 4 5.The quadratic equation x 2 – 2x + 6 = 0 has two solutions which are both complex numbers. Use the quadratic formula to find these solutions. Solution: x 2 – 2x + 6 = 0 a=1b= – 2c=6 b 2 – 4ac =(–2) 2 – 4(1)(6) =4 – 24=– 20 So b 2 4ac = 20 = 20 i = 2 5 i x= b b 2 4ac 2a = ( 4 ) 2 5 i 2(1) = 4 2 5 i 2 The two solutions arex= 4 2 5 i 2 =2 – 5 i andx= 4 2 5 i 2 = 2+ 5 i There are 2 complex number solutions they are x = 2 – 5 i and x = 2 + 5 i 6.The quadratic equation x 2 – x + 4 = 0 has two solutions which are both complex numbers. Use the quadratic formula to find these solutions. Solution: x 2 – 2x + 2 = 0 a=1 b= – 1 c=4 b 2 – 4ac =(– 1) 2 – 4 x 1 x (4) =1 – 16 = – 15 So b 2 4ac = 15 = 15 1 = 15 1 = 15 i = 2 15i 2 x So x or x ====== b b 2 4ac 2a 2 15i 2 2 15i 2 There are two complex numbers as solutions they are x = 2 15i 2 and x = 2 15i 2
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