Worked solutions with a scoring guide for the 20-point test. Award the points listed in each scoring note and allow partial credit for correct steps with a later arithmetic slip. Graphs are shown where a figure helps check the reading of the plane.
Question 1: Classify marked numbers (2 points)
a) \(-4\) is rational. \(\sqrt{10}\approx 3.16\) is not a perfect square root, so it is irrational.
b) \(-4\) is an integer (not natural, because it is negative). \(\sqrt{10}\) is irrational.
a) \(-4\) rational, \(\sqrt{10}\) irrational b) integer; irrational
Scoring: 1 point for part a (both correct), 1 point for part b (both correct).
Question 2: Points in the complex plane (2 points)
Read across first, then up or down.
a) \(3+4i\) b) \(-4-2i\)
Scoring: 1 point for each correct number. Reversed parts earn no point.
Question 3: Imaginary unit (2 points)
a) \(\sqrt{-72}=i\sqrt{36\cdot 2}=6i\sqrt{2}\).
b) \(3x^{2}=-48\), so \(x^{2}=-16\) and \(x=\pm 4i\).
a) \(6i\sqrt{2}\) b) \(x=\pm 4i\)
Scoring: 1 point for each part.
Question 4: Add and subtract (2 points)
a) \(5-8=-3\), \(-3i+7i=4i\). b) \(6-9=-3\), \(2i+5i=7i\).
a) \(-3+4i\) b) \(-3+7i\)
Scoring: 1 point for each correct result.
Question 5: Multiply (3 points)
a) \(8+12i-2i-3i^{2}=8+10i+3\). b) \(9+15i+15i+25i^{2}=9+30i-25\). c) \(49-4i^{2}=49+4\).
a) \(11+10i\) b) \(-16+30i\) c) \(53\)
Scoring: 1 point for each correct product.
Question 6: Powers of i (2 points)
a) 38 = 4(9) + 2, so \(i^{38}=i^{2}=-1\). b) 83 = 4(20) + 3, so \(i^{83}=i^{3}=-i\).
a) \(-1\) b) \(-i\)
Scoring: 1 point for each correct power.
Question 7: Arrows in the plane (3 points)
a) \(z_1=2+4i\) and \(z_2=3-2i\).
b) \((2+4i)+(3-2i)=5+2i\).
c) \((2+4i)-(3-2i)=-1+6i\).
a) \(2+4i\), \(3-2i\) b) \(5+2i\) c) \(-1+6i\)
Scoring: 1 point for each part.
Question 8: Check a solution (2 points)
\((2+i)^{2}=4+4i+i^{2}=3+4i\) and \(-4(2+i)=-8-4i\).
Sum: \((3+4i)+(-8-4i)+5=0\).
The left side is 0, so \(2+i\) is a solution.
Scoring: 1 point for expanding the square correctly; 1 point for combining to get 0.
Question 9: Find the error (2 points)
Liam replaced \(i^{2}\) with \(1\) instead of \(-1\). Correctly, \(4-9i^{2}=4-9(-1)=4+9=13\).
The correct product is 13.
Scoring: 1 point for identifying the wrong value of \(i^{2}\); 1 point for 13.
20 points
See every chapter on the Algebra 2 math page.