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Questions and Answers
What is the determinant of matrix A if it is defined as A = [[2, 3], [4, 5]]?
What is the determinant of matrix A if it is defined as A = [[2, 3], [4, 5]]?
A matrix whose determinant is zero is called a non-singular matrix.
A matrix whose determinant is zero is called a non-singular matrix.
False
What does the symbol 'det' represent in linear algebra?
What does the symbol 'det' represent in linear algebra?
Determinant
The determinant of a 1x1 matrix is simply the _______ of the matrix.
The determinant of a 1x1 matrix is simply the _______ of the matrix.
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Match the following matrices with their determinants:
Match the following matrices with their determinants:
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Which of the following matrices is singular?
Which of the following matrices is singular?
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The determinant of a matrix is applicable only to square matrices.
The determinant of a matrix is applicable only to square matrices.
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What does the leading diagonal of a matrix refer to?
What does the leading diagonal of a matrix refer to?
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What is the maximum value of the objective function $3x + 4y$ for the given inequalities?
What is the maximum value of the objective function $3x + 4y$ for the given inequalities?
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The inequality $x + y ≥ 0$ represents the shaded region in the first quadrant.
The inequality $x + y ≥ 0$ represents the shaded region in the first quadrant.
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What is the range of the function y = cosx?
What is the range of the function y = cosx?
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What are the coordinates of point A based on the given information?
What are the coordinates of point A based on the given information?
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The roots of the quadratic equation $x^2 + 3x + 2 = 0$ are __________.
The roots of the quadratic equation $x^2 + 3x + 2 = 0$ are __________.
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The period of the function y = tanx is π radians.
The period of the function y = tanx is π radians.
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Match the objective functions with their corresponding constraints.
Match the objective functions with their corresponding constraints.
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What is the first term (a) of the series if the given series is 36 + 12 + 4 + ...?
What is the first term (a) of the series if the given series is 36 + 12 + 4 + ...?
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The value of $r$ in the geometric series 36, 12, 4,... is ______.
The value of $r$ in the geometric series 36, 12, 4,... is ______.
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Which of the following represents a constraint that must be satisfied for the objective function M = x + y?
Which of the following represents a constraint that must be satisfied for the objective function M = x + y?
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Match the following functions with their characteristics:
Match the following functions with their characteristics:
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The equation $y = 2x + 3$ is a linear equation.
The equation $y = 2x + 3$ is a linear equation.
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What is the degree of the constant term in the equation $y = 2x + 3$?
What is the degree of the constant term in the equation $y = 2x + 3$?
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Which of the following angles corresponds to a cosine value of 0?
Which of the following angles corresponds to a cosine value of 0?
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The geometric series with common ratio greater than 1 will always diverge.
The geometric series with common ratio greater than 1 will always diverge.
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Identify the number of terms in a geometric progression if the first term is 'a' and the common ratio is 'r,' and if r is less than 1.
Identify the number of terms in a geometric progression if the first term is 'a' and the common ratio is 'r,' and if r is less than 1.
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At which value of x is the function f(x) = (x² - 1)/(x - 1) discontinuous?
At which value of x is the function f(x) = (x² - 1)/(x - 1) discontinuous?
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A function is continuous at a point if the limit from the left and the limit from the right both equal the function value at that point.
A function is continuous at a point if the limit from the left and the limit from the right both equal the function value at that point.
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What is the limit of f(x) as x approaches 2 from the left (f(1.99))?
What is the limit of f(x) as x approaches 2 from the left (f(1.99))?
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The function f(x) is continuous at x = _____ because there are no gaps or jumps.
The function f(x) is continuous at x = _____ because there are no gaps or jumps.
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Which of the following functions is likely to be continuous over its entire defined range?
Which of the following functions is likely to be continuous over its entire defined range?
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For any function to be continuous at a point a, it must satisfy the definition: lim f(x) as x approaches a from the left must equal lim f(x) as x approaches a from the _____ and f(a) must exist.
For any function to be continuous at a point a, it must satisfy the definition: lim f(x) as x approaches a from the left must equal lim f(x) as x approaches a from the _____ and f(a) must exist.
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What is the value of f(2) for the function f(x) = (x² - 1)/(x - 1)?
What is the value of f(2) for the function f(x) = (x² - 1)/(x - 1)?
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Match each function with its continuity characteristics:
Match each function with its continuity characteristics:
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What is the acute angle between the lines given by the equations √3𝑥 – 𝑦 + 8 = 0 and 𝑦 + 10 = 0?
What is the acute angle between the lines given by the equations √3𝑥 – 𝑦 + 8 = 0 and 𝑦 + 10 = 0?
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The slope of the line given by the equation 2x – y + 3 = 0 is negative.
The slope of the line given by the equation 2x – y + 3 = 0 is negative.
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What is the slope (m1) of the line represented by the equation x – 3y + 2 = 0?
What is the slope (m1) of the line represented by the equation x – 3y + 2 = 0?
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The obtuse angle between the lines can be found using the equation tanθ = ±(__________)
The obtuse angle between the lines can be found using the equation tanθ = ±(__________)
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Match the following lines with their corresponding slopes:
Match the following lines with their corresponding slopes:
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What value of 'a' will make the lines ax + 5y – 16 = 0 and 6x + 10y – 9 = 0 perpendicular?
What value of 'a' will make the lines ax + 5y – 16 = 0 and 6x + 10y – 9 = 0 perpendicular?
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The obtuse angle between the two lines 2x – y + 3 = 0 and x – 3y + 2 = 0 is 135°.
The obtuse angle between the two lines 2x – y + 3 = 0 and x – 3y + 2 = 0 is 135°.
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What is the angle θ if tanθ = √3?
What is the angle θ if tanθ = √3?
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What is the nature of the graph of the equation $y = 2x^2$?
What is the nature of the graph of the equation $y = 2x^2$?
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The vertex of the equation $y = x^2 - 6x$ can be found at the point (3, -9).
The vertex of the equation $y = x^2 - 6x$ can be found at the point (3, -9).
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What is the equation of the line of symmetry for the parabola $y = 4x^2 + 8x + 5$?
What is the equation of the line of symmetry for the parabola $y = 4x^2 + 8x + 5$?
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The vertex of the parabola represented by the equation $y = x^2 + 4x - 1$ is located at the point (___, ___).
The vertex of the parabola represented by the equation $y = x^2 + 4x - 1$ is located at the point (___, ___).
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Match the following equations to their types of graphs:
Match the following equations to their types of graphs:
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Which of the following equations represents a function with a maximum value?
Which of the following equations represents a function with a maximum value?
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The graph of the function $y = 3x^3$ will have symmetry about the y-axis.
The graph of the function $y = 3x^3$ will have symmetry about the y-axis.
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Identify the vertex of the parabola from the equation $y = x^2 + 2x - 5$.
Identify the vertex of the parabola from the equation $y = x^2 + 2x - 5$.
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Study Notes
Trigonometric Functions (Cosine, Tangent)
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Cosine Function (y = cosx):
- Domain: -360° < x° < 360° or -2π < x < 2π
- Range: -1 ≤ y ≤ 1
- Period: 2π
- Values at key angles:
- x = -360°, -270°, -180°, -90°, 0°, 90°, 180°, 270°, 360°
- cosx = 1, 0, -1, 0, 1, 0, -1, 0, 1
-
Tangent Function (y = tanx):
- Domain: values differing by 90°, excluding odd multiples of 90°
- Range: -∞ < y < ∞
- Asymptotes: vertical lines at odd multiples of 90° (x = ±90°, ±270°, etc.)
- Values at key angles:
- x = -360°, -270°, -180°, -90°, 0°, 90°, 180°, 270°, 360°
- tanx = 0, undefined, 0, undefined, 0, undefined, 0, undefined, 0
Geometric Series
-
Sum of a finite geometric series:
- Formula 1: Sn = a(r^n - 1) / (r - 1) where a is the first term, r is the common ratio, and n is the number of terms.
- Formula 2: Sn = (ar^n - a) / (r - 1), where l = ar^(n-1) is the last term.
- Alternative formula for |r| < 1: Sn = (a - lr) / (1 - r)
Linear Inequalities
- Linear inequalities in two variables represent regions on a coordinate plane
- Common forms: ax + by + c < 0, ax + by + c ≤ 0, ax + by + c > 0, ax + by + c ≥ 0
Quadratic Equations and Graphs
-
Parabolas:
- The graph of a quadratic equation is a parabola.
- Key features: vertex, axis of symmetry, opening direction (up or down)
-
Finding solutions graphically/algebraically:
- Use graphs of both functions to find intersection points.
- Use substitution methods to find solutions based on equal values of the variables.
-
Determining the equation of a parabola
- Use provided graph to understand vertex, direction of opening, appropriate formula
Continuity
- Continuous function: lim(x→a⁻)f(x) = lim(x→a⁺)f(x) = f(a)
- Discontinuous function: A function with 'jumps', 'holes', or 'breaks'.
-
Graphs of (some) common continuous functions:
- y = x + 2
- y = x²
- y = x³
- y = cosx
Matrices and Determinants
-
Determinant of a Matrix:
- For a 1x1 matrix [a₁₁], det(A) = a₁₁
- For a 2x2 matrix [a₁₁ a₁₂; a₂₁ a₂₂], det(A) = a₁₁a₂₂ - a₂₁a₁₂
- A matrix with a determinant of zero is called a singular matrix.
-
Example problem: Calculating the determinant of various 2x2 matrices, calculating an angle between two lines using their slopes.
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Description
This quiz covers key concepts of trigonometric functions such as cosine and tangent, including their domains, ranges, and key angle values. Additionally, it explores the sum of finite geometric series with relevant formulas. Test your knowledge on these fundamental mathematical topics!
