Mathematics Class 12 Quiz

The inverse of a matrix A, written as A⁻¹, is a matrix that, when multiplied by A, results in the identity matrix (I). In other words, A * A⁻¹ = A⁻¹ * A = I.

The determinant of a matrix is a scalar value that represents certain properties of the matrix. It's calculated using specific formulas based on the matrix's elements.

A square matrix is called singular if its determinant is zero. This implies the matrix doesn't have an inverse.

A square matrix is called non-singular if its determinant is not zero. This means the matrix has an inverse.

The adjoint of a matrix is another matrix obtained by a specific process involving the cofactors and transposing the matrix.

The product of a matrix and its adjoint results in a scalar multiple of the identity matrix. This scalar is equal to the determinant of the matrix squared.

A homogeneous function is a function where multiplying each input by a constant factor results in the output being multiplied by that factor raised to a specific power. The power is called the degree of homogeneity.

A function is said to be strictly increasing in a particular interval if, for any two distinct points within the interval, the larger input always results in a larger output. It's essentially always 'going up' in that interval.

A tangent space at a point on a curve or surface represents all possible directions that a curve can take at that point. It's essentially a flat plane touching the curve or surface at that point.

In calculus, the derivative of a function f(x) at a point represents the instantaneous rate of change of the function at that point. Think of it as the slope of the line tangent to the curve at that point.

The marginal cost of producing x items is the cost of producing one more item after x items have already been produced. It's essentially the rate of change of the total cost function.

The dot product of two vectors is a scalar value calculated by multiplying the corresponding components of the vectors and summing the results. It's related to the angle between the vectors.

The angle between two vectors can be determined using the dot product. The formula involves the magnitudes of the vectors and their dot product.

A linear programming problem (LPP) is an optimization problem where the objective function is linear, and the constraints are also linear inequalities. It's often used to find the best solution under limited resources.

Feasible region in an LPP is the set of all points that satisfy all the constraints of the problem. It represents the possible solutions within the given limitations.

An objective function in an LPP defines the quantity that needs to be maximized or minimized. It's the function you want to optimize within the feasible region.

In an LPP, the optimal solution is the point within the feasible region that results in the maximum or minimum value of the objective function. It's the best possible solution under the given constraints.

The greatest integer function, denoted by [x], gives the largest integer less than or equal to x. It's a discontinuous function that jumps at integer values.

A one-to-one function (injective function) means that each distinct input maps to a unique output. No two inputs can share the same output.

An onto function (surjective function) means that every element in the codomain (range) is mapped to by at least one element in the domain. There's no output left 'unclaimed'.

A relation from set A to set B is a set of ordered pairs (a, b) where a is in A and b is in B. It establishes a connection between the elements of the two sets.

An equivalence relation is a specific type of relation that is reflexive, symmetric, and transitive. It partitions a set into distinct equivalence classes.

A function can be expressed in different ways, including set notation, graphs, or equations. When focusing on the mathematical relationship, the concept of a function itself remains the same.

A parabola is a U-shaped curve defined by a quadratic equation. It's a symmetrical curve with a specific focus and directrix.

The area of a region bounded by a curve and the x-axis can be calculated by integrating the function representing the curve over the desired interval on the x-axis.

A system of linear equations represents a set of equations where each equation involves linear relationships between variables. Solving the system involves finding values for the variables that satisfy all equations simultaneously.

Matrix method is a technique used to solve systems of linear equations using matrices. It involves representing the equations as matrices and then applying various matrix operations to find the solution.

The second derivative of a function f(x) represents the rate of change of the first derivative (the instantaneous rate of change of the function). It tells you how the rate of change of the function is changing.

The shortest distance between two skew lines (lines that don't intersect and are not parallel) involves finding a perpendicular line that connects the two skew lines. This perpendicular line represents the shortest distance.

The image of a point reflected across a line is the point that is symmetrical to the original point with respect to the line. It's like mirroring the point across the line.