C897 Mathematics: Content Knowledge

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Free C897 Mathematics: Content Knowledge Questions

1.

A teacher graphs y = log2(x + 3). What is the domain?

  • x > -3

  • x ≥ 0

  • All real x

  • x > 0

Explanation



Correct Answer:

x > -3


2.

Evaluate the expression 23 × 32 ÷ (2 × 3).

  • 24

  • 12

  • 6

  • 36

Explanation

Explanation:

Step 1: Simplify powers: 23 = 8 and 32 = 9. Substitute:

8 × 9 ÷ (2 × 3)

Step 2: Multiply the numerator: 8 × 9 = 72

Multiply the denominator: 2 × 3 = 6

Step 3: Divide numerator by denominator: 72 ÷ 6 = 12

Step 4: The simplified value of the expression is 12.

Correct Answer:

12


3.

Simplify the expression: (2⁻¹ × 3²) / (6⁻¹).

  • 9

  • 3

  • 27

  • 1

Explanation

Explanation:

Step 1: Rewrite using the negative exponent rule:

Correct Answer

27


4.

Factor: 2x² + 7x + 3.

  • (2x + 1)(x + 3)

  • (2x + 3)(x + 1)

  • (x + 1)(2x + 3)

  • Prime

Explanation


5.

A student researching the continuum hypothesis discovers Cohen’s forcing technique. Which independence result did Cohen establish relative to ZFC?

  • CH is independent of ZFC

  • ~CH is independent of ZFC

  • Both A and B

  • GCH is provable in ZFC

Explanation

Explanation:

Step 1: The continuum hypothesis (CH) posits that there is no set whose cardinality is strictly between that of the integers and the real numbers.

Step 2: Kurt Gödel showed in 1940 that CH cannot be disproven from ZFC (it is consistent with ZFC if ZFC itself is consistent).

Step 3: In 1963, Paul Cohen developed forcing, which allowed him to show that CH cannot be proven from ZFC either — it is independent. His result also showed that the negation of CH (~CH) is consistent with ZFC if ZFC itself is consistent.

Step 4: Therefore, Cohen’s key contribution is that both CH and ~CH are independent of ZFC.

Correct Answer:

Both A and B


6.

A teacher models exponential decay with carbon dating. If half-life is 5730 years, what fraction remains after 11460 years?

  • 1/4

  • 1/2

  • 1/8

  • 3/4

Explanation

Explanation:

Step 1: Recall that in exponential decay, the fraction remaining after n half-lives is:

​​​​​​​


7.

Solve the quadratic by completing the square: x² + 6x - 7 = 0.

  • x = -3 ± √16

  • x = -3 ± √11

  • x = 3 ± √16

  • x = -6 ± √43

Explanation


8.

For a lesson on polynomials, factor x2 + 6x + 8.

  • (x + 4)(x + 2)

  • (x + 4)(x - 2)

  • (x - 4)(x + 2)

  • (x - 4)(x - 2)

Explanation

Explanation:

Step 1: Identify two numbers that multiply to 8 (constant term) and add to 6 (coefficient of x). These numbers are 2 and 4.

Step 2: Write the factored form using these numbers:

    x2+6x+8=(x+4)(x+2)

Step 3: Optionally, verify by expanding:

      x+4x+2=x2+2x+4x+8=x2+6x+8

Correct Answer

x+4x+2


9.

What is the amplitude of y = -3cos(2x)?

  • -3

  • 2

  • 3

  • 1/2

Explanation


10.

Simplify the rational expression (x² - 9)/(x² - 6x + 9).

  • (x - 3)/(x - 3)

  • (x + 3)/(x - 3)

  • 1

  • (x + 3)/(x + 3)

Explanation


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