Real and Complex Numbers: Algebra 2 Math Test Answer Key – PDF Download

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.

⬇ Download the Algebra 2 Real and Complex Numbers test answer key (PDF, 102 KB)

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