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

Algebra 2 Honors — Standards-Aligned Skills Examination

A calculation-focused, standards-walkthrough exam covering every CCSS-M Algebra 2 conceptual category at the honors level (N-CN, A-SSE, A-APR, A-CED, A-REI, F-IF, F-BF, F-LE, F-TF, S-IC). 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 1–6 · 50 pts · ~50 min
1. N-CN.A · Complex number arithmetic 8 points
  1. Simplify each power of \(i\): \(i^{15},\ i^{42},\ i^{-7},\ i^{100}\).
  2. Simplify and write in the form \(a + bi\):

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

  3. Compute the modulus \(|z|\) for each: \(z_1 = 3 - 4i,\ z_2 = -5 + 12i,\ z_3 = -7\).
  4. Find the conjugate \(\bar z\) and verify that \(z \cdot \bar z = |z|^2\) for \(z = 6 - 5i\).
2. N-CN.B–C · Quadratics with complex roots; FTA 6 points
  1. Solve each quadratic equation. Express solutions in \(a + bi\) form when needed.

    (i) \(x^2 + 25 = 0\)    (ii) \(x^2 - 4x + 13 = 0\)    (iii) \(2x^2 + 3x + 5 = 0\)

  2. Construct a polynomial of least degree with real coefficients having zeros \(2 + i\) and \(-3\). Express in expanded form.
  3. State the Fundamental Theorem of Algebra in one sentence. Then state how many complex zeros (counting multiplicity) a degree-7 polynomial must have.
3. A-APR.A,B · Polynomial arithmetic, division, factoring 10 points
  1. Multiply and simplify: \((2x - 3)(x^2 + 4x - 5)\). Show partial products.
  2. Divide using polynomial long division: \( (2x^3 - 5x^2 + 3x - 4) \div (x - 2) \). State quotient and remainder.
  3. Use synthetic division to compute \( (3x^4 - x^3 + 6x - 8) \div (x + 2) \). State quotient and remainder.
  4. Factor each completely over the integers (and where indicated, over the reals):

    (i) \( x^4 - 16 \)    (ii) \( x^3 - 27 \)    (iii) \( 2x^3 + x^2 - 8x - 4 \) (group)    (iv) \(6x^2 + 11x - 35\)

  5. Apply the Remainder Theorem: find \(P(3)\) where \(P(x) = 2x^4 - 5x^3 + x - 7\). Show your work using either substitution or synthetic division.
  6. List all possible rational zeros of \( Q(x) = 2x^3 - 5x^2 + x + 6 \) per the Rational Root Theorem. Then determine which (if any) are actual zeros, and factor \(Q(x)\) completely.
4. A-SSE · Structure in expressions 6 points
  1. Rewrite each expression by recognizing structure:

    (i) factor \(9x^4 - 25\) as a difference of squares    (ii) factor \(x^6 - 64\) as both a difference of squares and a difference of cubes; check that the two factorings are consistent    (iii) factor \(4^{x} - 4\) as \(4(4^{x-1} - 1)\) and verify by expansion

  2. Complete the square: \(2x^2 - 12x + 5 = 2(x - h)^2 + k\). Find \(h\) and \(k\).
  3. Express \( \dfrac{2x^2 - 5x + 4}{x - 1} \) as a polynomial plus a remainder fraction. Show the long division.
5. A-CED, A-REI · Equation solving — rational and radical 10 points
  1. Solve each rational equation. Check for extraneous solutions:

    (i) \( \dfrac{1}{x} + \dfrac{1}{x - 3} = \dfrac{5}{x(x-3)} \)    (ii) \( \dfrac{x}{x + 2} - \dfrac{4}{x - 2} = \dfrac{16}{x^2 - 4} \)

  2. Solve each radical equation. Check for extraneous solutions:

    (i) \( \sqrt{x + 5} = x - 1 \)    (ii) \( \sqrt{2x + 3} - \sqrt{x - 2} = 2 \)    (iii) \( \sqrt[3]{2x - 1} = 3 \)

  3. Solve the absolute-value equation \( |2x - 5| = |x + 4| \). Show both cases.
  4. Solve the polynomial inequality \( x^3 - 4x^2 - 5x \ge 0 \) by factoring and using a sign chart.
6. F-TF.A–B · Unit circle, identities, exact values 10 points
  1. Find the exact value of each (no calculator):

    (i) \( \sin(7\pi/6) \)    (ii) \( \cos(11\pi/6) \)    (iii) \( \tan(5\pi/4) \)    (iv) \( \sec(2\pi/3) \)    (v) \( \csc(-\pi/3) \)    (vi) \( \cot(7\pi/4) \)

  2. Convert to radian measure: \(150°,\ 240°,\ -45°\). Convert to degree measure: \(7\pi/12,\ -3\pi/4\).
  3. Simplify using identities:

    (i) \( \sin^2 \theta + \cos^2 \theta + \tan^2 \theta - \sec^2 \theta \)    (ii) \( \dfrac{\sin(2\theta)}{1 - \cos(2\theta)} \)

  4. Solve each trig equation on \([0, 2\pi)\). Give all solutions exactly:

    (i) \( 2\sin\theta - 1 = 0 \)    (ii) \( \cos^2\theta = \tfrac{1}{4} \)    (iii) \( \tan\theta = -\sqrt{3} \)

  5. Verify the identity \( \sec^2 x - \tan^2 x = 1 \) by writing each side in terms of sine and cosine.
Part II · Calculator Permitted · Problems 7–12 · 50 pts · ~70 min
7. F-IF.B–C · Function key features and graphs 8 points
  1. For each function, identify domain, range, intercepts, asymptotes, and end behavior:

    (i) \( f(x) = \dfrac{2x - 1}{x + 3} \)    (ii) \( g(x) = (x - 2)^2 (x + 1) \)    (iii) \( h(x) = 2 - \log_3(x + 1) \)

  2. Find the average rate of change of \( f(x) = x^2 - 3x + 5 \) on the interval \([1, 4]\).
  3. Sketch \( y = -2|x - 3| + 4 \). Label the vertex, both intercepts, and end behavior.
  4. Determine whether each function is even, odd, or neither — algebraically:

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

8. F-BF.A–B · Building functions, inverse, composition, transformations 8 points
  1. Let \(f(x) = 3x - 5\) and \(g(x) = x^2 + 1\). Compute \((f \circ g)(2)\), \((g \circ f)(2)\), \((f \circ g)(x)\), and \((g \circ f)(x)\).
  2. Find \(f^{-1}(x)\) for each function. State its domain.

    (i) \( f(x) = \dfrac{2x - 7}{4} \)    (ii) \( f(x) = \sqrt{x + 3} \)    (iii) \( f(x) = e^{2x} - 1 \)

  3. Describe the transformations from \(y = f(x)\) to:

    (i) \( y = -2 f(x - 3) + 1 \)    (ii) \( y = f(-x) + 4 \)    (iii) \( y = \tfrac{1}{3} f(2x) \)

  4. If \(f(2) = 5\), \(f(3) = -1\), and \(f^{-1}(0) = 7\), compute \(f(f^{-1}(0))\) and \(f^{-1}(f(2))\). State the property of inverse functions used.
9. F-LE · Exponential and logarithmic equations 10 points
  1. Evaluate each (no calculator):

    (i) \( \log_2 64 \)    (ii) \( \log_{1/3} 27 \)    (iii) \( \ln e^{5} \)    (iv) \( \log_5 \sqrt{125} \)    (v) \( \log_{10} 0.001 \)

  2. Expand each into a sum/difference using log properties:

    (i) \( \log\!\left(\dfrac{x^3 \sqrt{y}}{z^2}\right) \)    (ii) \( \ln\!\left(\dfrac{e^{2x}}{x + 1}\right) \)

  3. Condense into a single logarithm:

    (i) \( 2 \log x + 3 \log y - \tfrac{1}{2} \log z \)    (ii) \( \log_2 12 - \log_2 4 + \tfrac{1}{2} \log_2 9 \)

  4. Solve each equation:

    (i) \( 5^{x+1} = 125 \)    (ii) \( 3^{2x-1} = 27 \)    (iii) \( \log_2(x) + \log_2(x+2) = 3 \)    (iv) \( \ln(2x - 1) = 4 \) (round to 4 decimals)    (v) \( e^{2x} - 5 e^x + 6 = 0 \) (factor as a quadratic in \(e^x\))

  5. Apply the change-of-base formula to compute \( \log_4 50 \) using natural log. Round to four decimals.
10. A-SSE.B.4, F-LE · Sequences and series 8 points
  1. Identify each sequence as arithmetic, geometric, or neither. Then give a closed-form for the \(n\)th term.

    (i) \(7, 11, 15, 19, \ldots\)    (ii) \(3, 6, 12, 24, \ldots\)    (iii) \(1, 4, 9, 16, \ldots\)    (iv) \(20, 12, 4, -4, \ldots\)

  2. Compute each finite sum using a formula:

    (i) \( \sum_{n = 1}^{50} (3n - 2) \)    (ii) \( \sum_{n = 0}^{9} 4 \cdot \left(\tfrac{1}{2}\right)^n \)    (iii) \( \sum_{n = 1}^{20} 5 \)

  3. Compute each infinite geometric series, or state that it diverges:

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

11. G-GPE · Conic sections (Algebra 2 honors extension) 8 points
  1. Identify the conic and put each into standard form by completing the square; state the relevant features (center, vertices, foci, directrix, asymptotes — as appropriate):

    (i) \( x^2 + y^2 - 6x + 8y - 11 = 0 \) (circle: center, radius)
    (ii) \( y^2 = 12 x \) (parabola: vertex, focus, directrix)
    (iii) \( \dfrac{(x - 1)^2}{25} + \dfrac{(y + 3)^2}{9} = 1 \) (ellipse: center, vertices, foci)
    (iv) \( \dfrac{x^2}{16} - \dfrac{y^2}{9} = 1 \) (hyperbola: vertices, foci, asymptotes)

  2. Write the equation in standard form for the ellipse with foci \((\pm 4, 0)\) and major-axis length 10. Show \(a, b, c\) calculations.
  3. Write the equation in standard form for a hyperbola with vertices \((0, \pm 3)\) and asymptotes \(y = \pm \tfrac{3}{4} x\).
12. S-IC.A–B · Inferential statistics 8 points
  1. A simple random sample of \(n = 64\) adults yields a sample mean \(\bar x = 78\) with sample standard deviation \(s = 12\). Construct a 95% confidence interval for the population mean. Use \(t^* \approx 2.00\). Round endpoints to two decimals.
  2. A school board member surveys a random sample of \(n = 200\) students; \(\hat p = 0.36\) of them report walking to school.

    (i) Compute the standard error of \(\hat p\) using \( \text{SE} = \sqrt{\hat p (1 - \hat p)/n} \). Round to four decimals.
    (ii) Construct a 95% confidence interval for the true proportion using \(\hat p \pm 1.96 \cdot \text{SE}\). Round to four decimals.
    (iii) Interpret the interval in one sentence in context.

  3. The sampling distribution of \(\bar X\) for a sample of size \(n\) drawn from a population with mean \(\mu = 50\) and standard deviation \(\sigma = 12\) has what mean and standard deviation, when \(n = 36\)? When \(n = 100\)? Comment in one sentence on how the standard error changes as \(n\) grows.
  4. State the difference between an experiment, an observational study, and a survey. Give one situation each in which only one of the three is appropriate; explain why in one sentence.