Troy School District · Mathematics · Algebra 2 · Form B
Standards Drill · 120 minutes · 100 points

Algebra 2 — Standards-Aligned Skills Examination

A calculation-focused, standards-walkthrough exam covering every CCSS-M Algebra 2 conceptual category (N-CN, A-SSE, A-APR, A-CED, A-REI, F-IF, F-BF, F-LE, F-TF, S-ID). Each problem header cites the standard(s) the items address. Form B complements Form A (which is more applied / problem-solving).

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Instructions

Part I · No Calculator · Problems 1–5 · 50 pts · ~50 min
1. N-CN.A · Complex number arithmetic 8 points
  1. Simplify each: \(i^9,\ i^{20},\ i^{-3}\).
  2. Simplify and write in \(a + bi\) form:

    (i) \((4 + 2i) + (3 - 5i)\)    (ii) \((4 + 2i)(3 - 5i)\)    (iii) \((1 - i)^2\)    (iv) \( \dfrac{3 + i}{1 + 2i} \)

  3. Compute \(|3 - 4i|\) and \(|{-5 + 12i}|\).
  4. If \(z = 2 + 3i\), find \(\bar z\) and verify \(z \cdot \bar z = |z|^2\).
2. A-APR.A,B · Polynomial arithmetic, division, factoring 10 points
  1. Multiply: \((x - 4)(2x^2 + 3x - 1)\). Show partial products.
  2. Divide using long division: \( (x^3 - 2x^2 + 5x - 6) \div (x - 1) \). State quotient and remainder.
  3. Use synthetic division to compute \( (2x^3 - 7x^2 + 4x + 3) \div (x - 3) \). State quotient and remainder.
  4. Factor each completely:

    (i) \( x^4 - 81 \)    (ii) \( x^3 + 8 \) (sum of cubes)    (iii) \( x^4 - 5x^2 + 4 \) (treat as quadratic in \(x^2\))    (iv) \( 3x^2 - 11x - 4 \)

  5. Use the Remainder Theorem: find \(P(-2)\) where \(P(x) = x^3 + 4x^2 - 7x + 3\).
3. A-REI · Equation solving — quadratic, rational, radical 12 points
  1. Solve each quadratic by factoring:

    (i) \( x^2 - 5x - 14 = 0 \)    (ii) \( 6x^2 + 11x - 10 = 0 \)    (iii) \( 4x^2 - 25 = 0 \)

  2. Solve each quadratic by the quadratic formula. Express exactly:

    (i) \( 2x^2 + 3x - 1 = 0 \)    (ii) \( x^2 - 4x + 7 = 0 \)

  3. Solve by completing the square: \( x^2 + 8x - 5 = 0 \). Show every step.
  4. Solve each rational equation. Check for extraneous roots:

    (i) \( \dfrac{2}{x - 1} + \dfrac{3}{x + 2} = 1 \)    (ii) \( \dfrac{x}{x - 3} = \dfrac{6}{x - 3} + 2 \)

  5. Solve each radical equation. Check for extraneous roots:

    (i) \( \sqrt{x + 7} = x - 5 \)    (ii) \( \sqrt{2x - 3} = 5 \)

4. F-IF · Function key features 10 points
  1. For \(f(x) = x^2 - 4x - 5\):

    (i) Find the vertex (use \(x = -b/(2a)\) and substitution).
    (ii) Find the \(x\)-intercepts (factor or formula).
    (iii) State the axis of symmetry.
    (iv) Sketch the parabola; label vertex and intercepts.

  2. For \( g(x) = \dfrac{x + 1}{x - 4} \):

    (i) Find the vertical and horizontal asymptotes.
    (ii) Find the \(x\)- and \(y\)-intercepts.
    (iii) State the domain.

  3. Find the average rate of change of \( h(x) = x^2 + 3x \) on \([1, 5]\).
  4. Determine if each is even, odd, or neither:

    (i) \( f(x) = 3x^4 - x^2 \)    (ii) \( g(x) = x^3 - x \)    (iii) \( h(x) = x + \cos x \)

5. F-TF · Trigonometric values and equations 10 points
  1. Find each exact value (no calculator):

    (i) \( \sin(\pi/3) \)    (ii) \( \cos(2\pi/3) \)    (iii) \( \tan(3\pi/4) \)    (iv) \( \sin(7\pi/6) \)    (v) \( \cos(5\pi/4) \)

  2. Convert to radian measure: \(120°,\ 270°,\ -60°\). Convert to degree measure: \(\pi/4,\ 5\pi/6\).
  3. Solve each on \([0, 2\pi)\):

    (i) \( 2 \cos\theta + 1 = 0 \)    (ii) \( \tan\theta = 1 \)    (iii) \( \sin(2\theta) = \tfrac{1}{2} \) (find both solutions in the interval)

  4. Verify the identity \( \tan x \cdot \cos x = \sin x \) by writing each side in terms of \(\sin\) and \(\cos\).
Part II · Calculator Permitted · Problems 6–10 · 50 pts · ~70 min
6. F-LE · Exponential and logarithmic equations 12 points
  1. Evaluate each (no calculator):

    (i) \( \log_3 81 \)    (ii) \( \log_{1/2} 16 \)    (iii) \( \ln e^{3} \)    (iv) \( \log 0.01 \)    (v) \( \log_4 \sqrt{8} \)

  2. Expand using log properties:

    (i) \( \log\!\left(\dfrac{x^2 y}{z}\right) \)    (ii) \( \ln\!\sqrt{\dfrac{x}{y}} \)

  3. Condense to a single log:

    (i) \( 3 \log x - \log y \)    (ii) \( \tfrac{1}{2}(\log a + \log b) - 2 \log c \)

  4. Solve each:

    (i) \( 4^{x} = 32 \)    (ii) \( 2 \cdot 5^{x - 1} = 50 \)    (iii) \( \log_2(x) + \log_2(x - 2) = 3 \)    (iv) \( \ln(x + 1) = 2 \) (round to 4 decimals)    (v) \( e^{2x} = 18 \) (round to 4 decimals)

  5. Compute \( \log_5 17 \) using natural log (change-of-base). Round to four decimals.
7. F-BF · Inverse, composition, transformations 10 points
  1. Let \(f(x) = 2x + 1\), \(g(x) = x^2 - 3\).

    (i) Compute \((f \circ g)(2)\).    (ii) Compute \((g \circ f)(2)\).    (iii) Find \((f \circ g)(x)\) and \((g \circ f)(x)\) symbolically.

  2. Find the inverse \(f^{-1}(x)\) for each:

    (i) \( f(x) = \dfrac{x - 5}{3} \)    (ii) \( f(x) = (x + 2)^3 \)    (iii) \( f(x) = \dfrac{1}{x - 4} \)

  3. Verify that the functions \(f(x) = 4x - 7\) and \(g(x) = \dfrac{x + 7}{4}\) are inverses by computing \(f(g(x))\) and \(g(f(x))\). Show your work.
  4. Describe the transformation taking \(y = f(x)\) to:

    (i) \( y = f(x - 2) + 5 \)    (ii) \( y = -3 f(x) \)    (iii) \( y = f(2x) - 1 \)

8. A-SSE.B.4 · Sequences and series 8 points
  1. Write the first 5 terms of each:

    (i) arithmetic with \(a_1 = 4\), \(d = 3\)    (ii) geometric with \(b_1 = 6\), \(r = -2\)    (iii) defined by \(a_n = n^2 - n + 1\)

  2. Find a closed form for the \(n\)th term:

    (i) \(2, 7, 12, 17, \ldots\)    (ii) \(3, 6, 12, 24, \ldots\)

  3. Compute each finite sum using a formula:

    (i) \( \sum_{n = 1}^{30} (2n + 5) \)    (ii) \( \sum_{n = 0}^{8} 3 \cdot 2^n \)

  4. Compute the infinite geometric sum, or state divergence:

    (i) \( \sum_{n = 0}^{\infty} 5 \cdot \left(\tfrac{1}{3}\right)^n \)    (ii) \( \sum_{n = 0}^{\infty} 2 \cdot \left(\tfrac{4}{3}\right)^n \)

9. G-GPE · Conic sections 8 points
  1. Identify each conic and write in standard form by completing the square:

    (i) \( x^2 + y^2 - 4x + 6y - 12 = 0 \) (circle: center, radius)
    (ii) \( y = x^2 - 6x + 5 \) (parabola: vertex)
    (iii) \( \dfrac{x^2}{36} + \dfrac{y^2}{16} = 1 \) (ellipse: vertices, foci)
    (iv) \( \dfrac{x^2}{4} - \dfrac{y^2}{9} = 1 \) (hyperbola: vertices, asymptotes)

  2. Write the equation of a circle with center \((-1, 4)\) and radius 5.
  3. Write the equation of a parabola with vertex \((2, -3)\) opening upward and passing through \((4, 5)\).
10. S-ID · Statistics — distributions and confidence interval 12 points
  1. The heights of adult women in a population are normally distributed with \(\mu = 64\) inches and \(\sigma = 3\) inches. Compute (using a calculator or Z-table; round to 4 decimals):

    (i) \(P(X < 60)\)    (ii) \(P(X > 70)\)    (iii) \(P(60 < X < 67)\)

  2. Apply the empirical (68-95-99.7) rule to the same distribution: estimate the proportion of women between 58 and 70 inches.
  3. Apply the empirical rule again: estimate the proportion of women between 61 and 67 inches.
  4. What height marks the 95th percentile of this distribution? Use the calculator's inverse-normal feature or a Z-table (\(z \approx 1.645\)). Round to one decimal.
  5. Compute the standard score (z-score) of an observation \(x = 70\) in the population from (a).
  6. The standard normal table gives \(P(Z < 1.5) \approx 0.9332\). What is \(P(-1.5 < Z < 1.5)\)? Justify by symmetry.