Q: Which trigonometric identity is correct?

$$\sin^2$ \theta + \cos^2$ \theta = 1$$

Question 1

Which of the following best describes a function?

Accepted Answer

A relation where each input has exactly one output.

Question 2

Given the quadratic function $$f(x) = 2x^2$ - 4x + 1$$, what is the vertex of the parabola?

Accepted Answer

$$(1, -1)$$

Question 3

Which equation represents a circle with center at $$(2, -3)$$ and radius $$5$$?

Accepted Answer

$$(x-2)^2$$ + $$(y+3)^2 = 25$

Question 4

Solve the system of equations: $$2x + 3y = 7$$ and $$x - y = 1.$$

Accepted Answer

$$x = 2, y = 1$$

Question 5

Which of the following is the correct formula for the compound interest $$A$$ after $$t$$ years, with principal $$P$$, annual rate $$r$$, and $$n$$ compounding periods per year?

Accepted Answer

$$A = P \left(1 + \frac{r}{n}\$$right)^{nt}$$

Question 6

If $$f(x) = \log_2$(x-1)$$, what is the domain of $$f$$?

Accepted Answer

$$x \u003e 1$$

Question 7

Which trigonometric identity is correct?

Accepted Answer

$$\sin^2$ 	heta + \cos^2$ 	heta = 1$$

Question 8

A Ferris wheel with a radius of $$20$$ meters makes one complete revolution every $$2$$ minutes. Which trigonometric function best models the height $$h(t)$$ above the ground after $$t$$ minutes, if the lowest point is $$2$$ meters above the ground?

Accepted Answer

$$h(t) = 20 \cos(\pi t) + 22$$

Question 9

Which of the following is the solution to the equation $$3^{2x} = 81$?

Accepted Answer

$$x = 2$$

Question 10

Given the function $$f(x) = x^3$ - 6x^2$ + 9x$$, what are the $$x-$$intercepts?

Accepted Answer

$$x = 0, 3$$

Question 11

Which of the following is the correct equation for a line passing through $$(1,2)$$ with slope $$-3$$?

Accepted Answer

$$y - 2 = -3(x - 1)$$

Question 12

If $$\sin 	heta = \frac{3}{5}$$ and $$	heta is in $$the first quadrant, what is $$\cos 	heta$$?

Accepted Answer

$$\frac{4}{5}$$

Question 13

Which of the following is the solution to $$\log_{10}(x) = 2$$?

Accepted Answer

$$x = 100$$

Question 14

A projectile is launched from the ground with an initial velocity of $$v_0 = 40$ m/s at an angle of 30$$^\circ$$ above the horizontal. Which equation gives the height h(t$$)$$ after t$ seconds? (Assume g = 9.8$ m/s^2$)

Accepted Answer

$$h(t) = 40 \sin 30^\circ \cdot t - \frac{1}{2} \cdot 9.8 t^2$

Question 15

Which of the following is the correct parametric equation for a particle moving in a circle of radius $$r$$ centered at the origin?

Accepted Answer

$$x = r \cos t, \ y = r \sin t$$

Question 16

Which conic section is represented by the equation $$x^2$$ - $$y^2 = 1$?

Accepted Answer

Hyperbola

Question 17

If $$f(x) = 2x + 1$$ and $$g(x) = x^2$, what is (f$$ \circ g)(3)$$?

Accepted Answer

$$19$$

Question 18

Which of the following best describes the range of the function $$f(x) = e^x$$?

Accepted Answer

$$(0, \infty)$$

Question 19

A population of bacteria triples every 4 hours. If the initial population is $$P_0$$, which equation models the population after $$t$$ hours?

Accepted Answer

$$P(t) = P_0$ \cdot 3$$^{t/4}$$

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