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Trigonometric Equations and Inequalities Quiz

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Trigonometric Equations and Inequalities Quiz

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Questions and Answers

What is the slope of line AC?

$rac{3}{2}$

$rac{1}{3}$

$2$

$rac{1}{2}$ (correct)

What is the slope of line PD?

$1$

$2$

$rac{1}{2}$

$0$ (correct)

What is the slope of line BC?

$rac{1}{2}$

$1$

$2$

$-rac{1}{2}$ (correct)

What is the slope of line EP?

$rac{1}{2}$ Signup and view all the answers

What is the equation of line AP?

$y = 2x - 3$ Signup and view all the answers

What is the equation of line BC?

$3x - y - 8 = 0$ Signup and view all the answers

What is the value of $k$ in the given matrix equation?

$k = 4$ Signup and view all the answers

What is the value of $p$ and $q$ in the given equation?

$p = \cos\frac{\pi}{7}$, $q = \sin\frac{\pi}{7}$ Signup and view all the answers

What is the value of $27 \sec 2 - 8 \csc 2$?

$250$ Signup and view all the answers

What is the value of $5 \sin 2 - 2$?

$3$ Signup and view all the answers

What is the value of $4 \cos 2 - 1$?

$3$ Signup and view all the answers

What is the intersection of the sets $S_1$ and $S_2$?

$[1, 2], [1, 2]$ Signup and view all the answers

What is the value of $x$ in the equation $x^2 - 3x + 2 = 0$?

$x = 1, x = 2$ Signup and view all the answers

Study Notes

Analytical Geometry

Slope of AC is infinity, and slope of PD is 0.
Point D is (2, 2), and point C is (-3, -1).
Equation of AP is x + 2y = 3, and equation of BC is 3x - y + 8 = 0.

Algebra

The number of real solutions of the equation e^4x + 4e^3x - 58e^2x + 4ex + 1 = 0 is 2.
Equation 25sin^4α + 10(1 - sin^2α) = 6 is equivalent to 25sin^2α - 10sin^2α - 10sin^2α + 4 = 0.
The equation 2^15sin4α + 10(1 - sin^2α) = 6 has solution sin^2α = 4/5.

Inequalities

Let S1 = {x ∈ R - {1, 2} : (x + 2)(x^2 + 3x + 5) ≥ 0}, and S2 = {x ∈ R : 32x - 3x + 1 - 3x + 2 + 27 ≤ 0}.
S1 ∪ S2 is equal to (-∞, -2] ∪ [1, 2].

Matrices

If PQ = kI, then P is an invertible matrix, and Q = kP^(-1).
adj(P) = k, and Q = kP^(-1)I.

Complex Numbers

If α is a 7th root of unity, then 1 + α + α^2 + ... + α^6 = 0.
The equation x^2 + x + 2 = 0 has solutions p and q, where p = α + α^2 + α^3, and q = α^4 + α^5 + α^6.

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Description

This quiz covers solving trigonometric equations and inequalities, including manipulating trigonometric functions, finding values of sine, cosine, secant, and cosecant, and determining ranges of variables. It includes intermediate level questions that test understanding of trigonometry concepts and techniques.