Mathematics Quiz: Linear Algebra and Probability

The sum of vectors EA, EB, EC, and ED is equal to twice the vector AD.

The integral of the function e^(cos(2x)) * sin(3(2n+1)x) from -π to π, where n is an integer, evaluates to 0.

The determinant of the matrix A, given as a 3x3 matrix with entries dependent on x, is equal to (2x+1)^2.

The feasible region represents the set of solutions that satisfy the constraints of a linear programming problem. The constraint x + y ≥ 2 is not represented by the given feasible region.

A linear programming problem involves finding optimal solutions within a feasible region defined by constraints. If a point lies outside the feasible region, it does not satisfy all the constraints.

The component of vector 'a' along vector 'b' is found by projecting 'a' onto 'b'. This projection is calculated using the dot product of 'a' and 'b' divided by the magnitude of 'b', then multiplied by the unit vector of 'b'.

The adjoint of a matrix (adj(A)) is calculated by finding the transpose of the matrix of cofactors. In this case, the given matrix 'A' is of order 3 and its determinant is -2. We need to find the adjoint of the matrix 2A. The determinant of 2A is 8 times the determinant of A.

The probability of an event happening when several independent events can occur is determined by multiplying the probabilities of each individual event.

A differential equation where the variables 'x' and 'y' can be separated on opposite sides of the equation. Separating variables allows for direct integration to find the general solution.

The general solution of a differential equation includes a constant of integration ('c') that represents an unknown constant value. This constant can vary depending on the initial condition of the problem.

In this case, the general solution of the given differential equation is y = cx. This means that any equation of the form y = kx, where 'k' is a positive constant, will satisfy the given equation.

A function is increasing in an interval if its derivative is positive throughout that interval.

A critical point of a function occurs where its derivative is zero or undefined.

The maximum value of a function is the largest output value the function can attain.

A definite integral with limits of integration defines the area under the curve of a function between those limits.

The probability distribution of a discrete random variable assigns probabilities to each possible value of the variable.

A differential equation relates a function to its derivatives.

A linear programming problem involves optimizing (maximizing or minimizing) an objective function subject to constraints.

The feasible region in a linear programming problem is the set of points that satisfy all the constraints.

The probability that an event will occur, given that another event has already occurred.

The error rate of a process is the proportion of items that are defective or incorrect.

The probability of an event happening is the number of favorable outcomes divided by the total number of possible outcomes.

The total probability of an event happening is the sum of the probabilities of all the ways that event can happen.

A force is a push or pull that can cause an object to accelerate.

The magnitude of a vector is its length or size.

The resultant force is the single force that has the same effect as all of the individual forces acting on an object.

The direction of a vector is the angle it makes with a reference line.

Two vectors are perpendicular if their dot product is zero. The dot product of the given vectors is (2 * 3) + (-1 * λ) + (2 * 1) = 0. Solving for λ, we get λ = 8.

A function is differentiable at a point if its derivative exists at that point. The function f(x) = x + |x| has a sharp corner at x = 0, making it non-differentiable at that point, but differentiable for all other values of x.

The sum of squares of the direction cosines of any line is always equal to 1. Therefore, (1/c)^2 + (1/c)^2 + (1/c)^2 = 1. Solving for c, we get c = ±√3.

A function has a minimum at a point if its first derivative is zero and its second derivative is positive at that point. Here, the first derivative is zero at x = 1, but the second derivative is negative at x = 1, indicating that the function has a maximum at that point, not a minimum.

A function is bijective if it is both one-to-one and onto. The given function f: {1, 2, 3, 4} → {x, y, z, p} is not onto as it does not map any element to 'p'. Hence, it is not a bijective function.

The inverse sine function, sin⁻¹(x) is defined for values of x ranging from -1 to 1. The expression inside the inverse sine function must fall within this range. So, the domain for sin⁻¹(x² - 4) is the set of x values which make the expression x² - 4 lie between -1 and 1.

The angle 33π/5 can be rewritten as 6π + 3π/5, and the cosine function has a period of 2π, so cos(33π/5) = cos(3π/5). Then, sin⁻¹(cos(3π/5)) is the angle whose sine is equal to cos(3π/5). This angle is π/2 - 3π/5 = -π/10.

The expression cos(33π/5) is equivalent to cos(6π + 3π/5), which simplifies to cos(3π/5) due to the periodicity of cosine. Therefore, sin⁻¹(cos(33π/5)) is simply sin⁻¹(cos(3π/5)). This represents the angle whose sine is equal to cos(3π/5), which is π/2 - 3π/5 or -π/10.

This relation implies that two pairs are related if the product of their corresponding components is equal.

The equivalence class of (2, 6) is the set of all pairs (c, d) in ℕ x ℕ related to (2, 6) under R, which means 2d = 6c.

The function f(x) is one-one if each element in the domain maps to a unique element in the codomain, and it is onto if every element in the codomain is mapped to by some element in the domain.

To prove that the function is one-to-one, show that f(x₁) = f(x₂) implies x₁ = x₂. To prove it is onto, show that for every y in the codomain, there exists an x in the domain such that f(x) = y.

A system of linear equations can be represented as a matrix equation Ax = b, where A is the coefficient matrix, x is the vector of unknowns, and b is the constant vector.

The image of a point with respect to a line is the point that is equidistant to the given point and the line.

The shortest distance between two skew lines is the length of the perpendicular segment connecting the two lines.

The distance of a point from the y-axis is the absolute value of the x-coordinate of the point.