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What is a surd?
A surd is an irrational number that cannot be simplified to remove the square root (or cube root etc.). For example, √2 is a surd.
What is a rational number?
A rational number is any number that can be written as a fraction of two integers. It includes integers, finite decimals, and repeating decimals.
What is an irrational number?
An irrational number cannot be expressed as a fraction and has a non-repeating, non-terminating decimal. Examples include π and √2.
What is a proof by contradiction?
Proof by contradiction involves assuming the opposite of what you want to prove, then showing that this leads to a contradiction.
What is a composite number?
A composite number is a natural number greater than 1 that is not prime. It has more than two factors.
What is meant by a periodic function?
A periodic function repeats its values at regular intervals. The length of one full cycle is called the period.
What is the unit circle?
The unit circle is a circle with radius 1, centred at the origin. It is used to define trigonometric functions.
What does it mean for a function to be even?
An even function is symmetric about the y-axis. For all x, f(x) = f(–x).
What does it mean for a function to be odd?
An odd function has rotational symmetry about the origin. For all x, f(–x) = –f(x).
What is the domain of a function?
The domain is the set of all possible input values (x-values) for which the function is defined.
What is the range of a function?
The range is the set of all output values (y-values) that a function can produce.
What is the modulus of a complex number?
The modulus is the distance of the complex number from the origin in the complex plane. For z = a + bi, |z| = √(a² + b²).
What is the argument of a complex number?
The argument is the angle the complex number makes with the positive real axis, measured in radians.
What is De Moivre’s Theorem?
De Moivre’s Theorem states that (cosθ + i sinθ)^n = cos(nθ) + i sin(nθ), useful for finding powers of complex numbers.
What is a conjugate of a complex number?
The conjugate of a complex number a + bi is a – bi. Multiplying a complex number by its conjugate gives a real number.
What is an identity in algebra?
An identity is an equation that is true for all values of the variable. For example, (a + b)² ≡ a² + 2ab + b².
What is a parametric equation?
A parametric equation expresses coordinates (x, y) in terms of a third variable, usually t. Ex: x = cos(t), y = sin(t).
What does orthogonal mean?
Orthogonal means perpendicular, especially when referring to vectors or lines in space.
What is the scalar (dot) product of two vectors?
The scalar product is a · b = |a||b|cosθ, where θ is the angle between them. It gives a scalar value.
What is the vector (cross) product?
The vector product of two 3D vectors gives a vector perpendicular to both. Not on the LC course, but useful to know.
What is the unit vector?
A unit vector has a magnitude of 1 and indicates direction. It is found by dividing a vector by its magnitude.
What is a convex polygon?
A convex polygon is a polygon where all interior angles are less than 180°, and no line segment between two points in the polygon goes outside.
What is a concave polygon?
A concave polygon has at least one interior angle greater than 180°, and a line between two points in it can go outside.
What is a cyclic quadrilateral?
A cyclic quadrilateral is a four-sided figure with all its vertices on the circumference of a circle. Opposite angles add to 180°.
What is a transversal line?
A transversal is a line that intersects two or more other lines at distinct points.
What is the sine rule?
In any triangle, a/sinA = b/sinB = c/sinC. Used when dealing with non-right triangles.
What is the cosine rule?
In any triangle, c² = a² + b² – 2ab cos(C). Used for sides and angles in non-right triangles.
What is an asymptote?
An asymptote is a line that a graph approaches but never touches. Can be vertical, horizontal, or slanted.
What is a discontinuity in a function?
A discontinuity is a point where a function is not defined or jumps in value, breaking its continuity.
What is a transformation matrix?
A transformation matrix changes the position, size, or orientation of a shape. Common in coordinate geometry and linear algebra.
What is a bijective function?
A bijective function is both injective (one-to-one) and surjective (onto), meaning each input maps to a unique output and all outputs are covered.
What is an injective function?
An injective function (one-to-one) maps each element of the domain to a unique element in the range.
What is a surjective function?
A surjective function (onto) covers all elements of the range — every possible output value is mapped by some input.
What is the difference between a relation and a function?
A relation pairs inputs with outputs, but a function must pair each input with exactly one output.
What is the image of a point under a transformation?
The image is the resulting point after a transformation has been applied to the original point.
What is the centroid of a triangle?
The centroid is the point where all medians intersect. It divides each median in a 2:1 ratio.
What is the incenter of a triangle?
The incenter is the point where the angle bisectors meet. It is the centre of the incircle.
What is the circumcenter of a triangle?
The circumcenter is where perpendicular bisectors meet. It is the centre of the circle that passes through all three vertices.
What is the orthocenter of a triangle?
The orthocenter is the intersection of the triangle’s altitudes.
What is the exponential function?
An exponential function has the form f(x) = a^x, where the base a > 0 and a ≠ 1.
What is a logarithmic function?
A logarithmic function is the inverse of an exponential function: if y = logₐ(x), then a^y = x.
What is a recursive sequence?
A recursive sequence defines each term based on previous term(s). E.g., Tₙ = Tₙ₋₁ + 3.
What is an arithmetic sequence?
An arithmetic sequence increases or decreases by a constant difference. E.g., 2, 5, 8, 11...
What is a geometric sequence?
A geometric sequence multiplies by a constant ratio. E.g., 2, 4, 8, 16...
What is an inflection point?
An inflection point is where the curve changes concavity — from concave up to concave down or vice versa.
What is a local maximum?
A local maximum is a point where the function reaches a peak in a small region. The derivative changes from positive to negative.
What is a local minimum?
A local minimum is a point where the function reaches a valley in a small region. The derivative changes from negative to positive.
What does it mean to differentiate implicitly?
Implicit differentiation is used when functions are not given explicitly as y=f(x). You differentiate both sides of the equation with respect to x.
What is a limit?
A limit describes the value a function approaches as the input approaches a certain point. Key to calculus.