Mathematics Exam Instructions and Sample Questions

A function that maps a value to another value, ensuring that each input has a unique output.

A function's inverse essentially 'undoes' the original function. If f(x) maps x to y, then f⁻¹(y) maps y back to x.

Finding the value of a variable that satisfies a given equation.

The set of all possible input values for a function. In the case f: R → R, the domain is the set of all real numbers.

The set of all possible output values that a function can produce.

The process of finding the inverse of a function.

A function that associates each element of a set (domain) with a unique element in a set (codomain).

A function where each element in the domain maps to a unique element in the codomain, and every element in the codomain is also mapped to.

A matrix is a rectangular array of numbers, symbols, or expressions. In this problem, we are given two matrices that are equal. We need to find the value of 'a' that makes the two matrices equal.

Determinant of a matrix is a scalar value that can be computed from a square matrix. In this case, we need to find the value of k that makes the determinant of the given matrix equal to 4.

A derivative of a function at a point represents the instantaneous rate of change of the function at that point. In this case, we are given a function y in terms of x, and we need to find the derivative of y with respect to x.

The process of finding the rate at which a quantity changes with respect to time is called related rates. In this case, we are given the rate at which the radius of a circle is increasing, and we need to find the rate at which the circumference is increasing.

A matrix is a rectangular array of numbers, symbols, or expressions. In this problem, we are given two matrices that are equal. We need to find the value of 'a' that makes the two matrices equal.

Determinant of a matrix is a scalar value that can be computed from a square matrix.

Finding the instantaneous rate of change of a function at a specific point

Finding the rate at which one quantity changes with respect to another quantity. Often involves time.

The area of a region bounded by a curve, y-axis, and a line can be calculated using integration. We need to find the points of intersection between the curve and the line, set up the integral, and evaluate it to determine the area.

The probability of the intersection of two events (A and B) can be calculated using the formula: P(A∩B) = P(A) * P(B|A), where P(B|A) is the conditional probability of event B occurring given that event A has already occurred.

A prime number can be divided evenly only by 1 and itself; the only even prime number is 2. The probability of an event is the number of favorable outcomes divided by the total number of outcomes.

The probability of two dice showing a total of 5 can be calculated by understanding that there are 6 ways to get a 5 (1 & 4, 2 & 3, 3 & 2, 4 & 1) and 36 possible outcomes when two dice are rolled.

Understanding the concept of a family with two children. If the family has either one boy or one girl, there are four possible outcomes: boy-boy, boy-girl, girl-boy, and girl-girl.

The definite integral of a function over an interval calculates the area between its graph and the x-axis. It measures the accumulation of the function's values over the interval.

An indefinite integral is the general form of all functions whose derivative equals the integrand. It represents the family of functions that have the same rate of change.

A differential equation is an equation that relates a function to its derivatives. It describes how the changes of a function are related to its value.

A dot product of two vectors is a scalar value that represents their projection onto each other. It captures the extent to which they point in the same direction.

A cross product of two vectors is a vector that is perpendicular to both input vectors. Its magnitude indicates how much they are not parallel.

The equation of a line describes the exact location of every point on the line in space. It's determined by a point it passes through and its direction.

A function is continuous at a point if the limit of the function as x approaches that point exists and equals the value of the function at that point.

To find the value of 'c' that makes the function continuous at x=0, we need to ensure the limit of the function as x approaches 0 from both sides matches the function's value at x=0.

The absolute value function |x| is defined as the distance of 'x' from zero, always resulting in a non-negative value.

A function is continuous at a point if the limit of the function as x approaches that point exists and equals the value of the function at that point. We need to check this separately for each point.

The derivative of tan⁻¹(u) with respect to 'x' is 1/(1+u²) multiplied by the derivative of 'u' with respect to 'x'.

The derivative of a function gives the rate of change of the function with respect to its independent variable.

To find the intervals where a function is increasing or decreasing, we need to analyze its derivative. The function is increasing where the derivative is positive and decreasing where the derivative is negative.

Finding the value of an integral by employing techniques like substitution, integration by parts, or trigonometric substitution.

A rate of change in a variable with respect to time, usually expressed in units per second (e.g., cm/s).

The volume of a sphere is determined by the formula V = (4/3)πr³, where r is the radius. Finding the derivative of this volume formula with respect to time provides the rate at which the volume changes.

Finding the shortest perpendicular distance between two lines in 3D space.

A type of integral that involves trigonometric functions (sine, cosine, tangent, etc.).

A method of solving linear problems with multiple constraints. It involves finding the optimal solution (maximum or minimum) of an objective function by considering the constraints.