C897 Mathematics: Content Knowledge

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Free C897 Mathematics: Content Knowledge Questions

Find the vertex of y = x² - 4x + 3.

(2, -1)
(2, 3)
(-2, -1)
(4, 3)

Explanation

Explanation:

A classroom uses matrices. If A = [[1,2],[3,4]], what is det(A)?

2
-2
-10
10

Explanation

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

x = -3 ± √16
x = -3 ± √11
x = 3 ± √16
x = -6 ± √43

Explanation

In a proof of the Poincaré conjecture in dimension 3, which topological invariant played a central role in Perelman’s Ricci flow approach?

Fundamental group
Homology groups
Ricci curvature
Betti numbers

Explanation

Explanation:

Step 1: Grigori Perelman proved the Poincaré conjecture for 3-manifolds using Richard Hamilton’s Ricci flow with surgery.

Step 2: Ricci flow is an analytic tool that evolves the metric of a manifold according to its Ricci curvature. The curvature determines how the shape of the manifold deforms over time.

Step 3: Controlling and understanding the Ricci curvature was crucial to showing that any simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

Step 4: Therefore, the central topological invariant in Perelman’s approach is the Ricci curvature.

Correct Answer:

Ricci curvature

A classroom activity involves solving literal equations for geometry formulas. Solve A = (1/2)bh for h.

h = A/(2b)
h = 2Ab
h = 2A/b
h = A/b

Explanation

Explanation:

Step 1: Start with the equation:

$A = \frac{1}{2} b h$

Step 2: Multiply both sides by 2 to eliminate the fraction:

2A = bh

Step 3: Divide both sides by b to isolate h:

$h = \frac{2 A}{b}$

Correct Answer:

h = 2A/b

What is the value of cos⁻¹(√3/2)?

π/6
π/3
π/4
π/2

Explanation

Explanation:

Step 1: Recall that cos⁻¹(x) gives the angle whose cosine is x.

Step 2: Solve cos θ = √3/2. From standard trigonometric values

$\cos (\frac{π}{6}) = \frac{\sqrt{3}}{2}$

Step 3: Therefore, the inverse cosine of $\sqrt{(3 / 2)}$ :

$θ = \frac{π}{6}$

Correct Answer:

$\frac{π}{6}$

Solve the system: 2x + y = 7, x - y = 2.

(1, 5)
(2, 3)
(3, 1)
(4, -1)

Explanation

Explanation:

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

-3
2
3
1/2

Explanation

Factor completely: 4x² - 12x + 9.

(4x - 3)(x - 3)
(2x + 3)²
(2x - 3)²
Prime

Explanation

Explanation:

10.

In a classroom proof, which axiom allows a = a?

Symmetric
Transitive
Reflexive
Additive identity

Explanation

Explanation:

Step 1: The reflexive property states that any quantity is equal to itself.

Step 2: This is a fundamental axiom in algebra used in proofs to justify that a = a.

Step 3: Therefore, the axiom that allows a = a is the reflexive property.

Correct Answer:

Reflexive

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