Further Maths Quiz (Entire course revision)

Dividing by a fraction is the same as multiplying by its reciprocal.

To find a percentage of an amount, convert the percentage to a decimal and multiply by the amount.

To increase or decrease an amount by a percentage, multiply by the appropriate decimal multiplier.

To reverse a percentage change, divide by the decimal multiplier used in the original calculation.

To apply compound interest, multiply the initial amount by the decimal multiplier raised to the power of the number of years.

To share an amount in a ratio, find the total number of parts and then scale up to the required total.

Ratios can be combined by making the parts equal, which can be achieved by forming equivalent ratios.

A surd is an irrational number involving a root that cannot be written as a fraction with whole numbers as the numerator and denominator.

An algebraic expression involving terms that are added together. Like terms have the same variables with the same exponents and can be combined by adding their coefficients.

To simplify an expression with brackets, multiply each term in the first bracket by each term in the second bracket.

To factorize an expression, find a common factor that exists in all the terms of the expression. This factor is placed outside the parenthesis, while the remaining factors are placed inside.

A quadratic function is a polynomial function of degree two, expressed as y = ax^2 + bx + c.

A parabola is the shape of the graph of a Quadratic function. It is symmetrical and resembles either ∪ (for a > 0) or ∩ (for a < 0).

When factoring a quadratic expression of the form ax^2 + bx + c, find pairs of numbers that multiply to give ac and add to give b. These numbers represent the constants in the two brackets.

If a quadratic function's graph intersects the x-axis at two distinct points, the corresponding quadratic equation has two distinct solutions.

If a quadratic function's graph touches the x-axis at only one point, this point represents the maximum or minimum of the function, and the equation has one (repeated) solution.

The square root of a product is equal to the product of the square roots.

To simplify a surd, find the largest perfect square factor of the radicand and take its square root outside the radical.

To rationalize the denominator of a fraction with a surd, multiply both the numerator and denominator by the surd in the denominator.

To rationalize the denominator of a fraction with a sum or difference involving a surd, multiply both numerator and denominator by the conjugate of the denominator.

Multiplying out brackets involves multiplying each term inside the bracket by the term outside the bracket.

The Pythagorean Theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

If a fraction has a root (surd) in the denominator, it is often useful to rewrite the fraction with only rational numbers (whole numbers or fractions) in the denominator. This is called rationalizing the denominator.

A surd can be brought outside of a radical if the radicand can be expressed as a product of two factors, one of which is a perfect square.

Addition of surds does not follow the same rules as multiplication. We can only add surds with the same radicand.

The factor theorem states that (x - a) is a factor of polynomial P(x) if and only if P(a) = 0. This means that if we know a factor of a polynomial, we can find the corresponding root. Conversely, if we know a root, we can determine the corresponding factor.

A root is a value of x that makes the polynomial P(x) equal to zero. It represents the x-coordinates where the graph of the function crosses the x-axis.

To factorize a cubic polynomial, start by finding a root through trial and error. Then, apply the factor theorem to generate a linear factor. After that, use inspection to determine the remaining quadratic factor and factorize it further if possible.

A positive quadratic function has a minimum point, while a negative quadratic function has a maximum point. The graph is symmetrical about its vertex, which is the minimum or maximum point. It is a smooth curve that gets steeper as it increases but never becomes vertical.

A positive quadratic function will have a single turning point, which is a minimum point.

A negative quadratic function will have a single turning point, which is a maximum point.

The y-intercept is the point where the graph of the quadratic function crosses the y-axis. It is always at (0, c), where c is the constant term in the quadratic equation.

Finding the gradient of a curve at a given point. It represents the rate of change of y with respect to x.

A straight line segment connecting two points on a curve. Its gradient approximates the curve's gradient at those points.

The limit of the gradient of chords as the points they connect get closer together. This gives the exact gradient of the curve at a point.

The gradient of the tangent to a curve at a specific point. It represents the instantaneous rate of change of the curve at that point.

The limit of the ratio of the change in y to the change in x as both approach zero. It represents the instantaneous rate of change of y with respect to x.

A rule used to differentiate a function of the form y = xn. It involves bringing the power down and reducing it by one.

A method used to find the equation of the tangent line to a curve at a given point, using the derivative to determine the gradient of the tangent.

The process of reversing differentiation to find the original function from its derivative.

The distance between two points is found by using the Pythagorean theorem on a right-angled triangle formed by the coordinates.

The midpoint of a line segment is calculated by averaging the x-coordinates and the y-coordinates of the two endpoints.

The gradient of a line represents the steepness or slope of the line. It is calculated by dividing the change in y (vertical change) by the change in x (horizontal change).

The equation of a straight line can be written in the form y = mx + c, where m is the gradient (slope) and c is the y-intercept (the point where the line crosses the y-axis).

To find a point that's a proportion of the way between two points, add a fraction of the 'journey' from the starting point to the ending point.

The limit of a sequence is the value that the terms of the sequence approach as the index (n) gets larger and larger. The limit may exist and be a finite number, or it may not exist.

To find the limit of a rational function, where both the numerator and denominator go to infinity, divide both by the highest power of n in the denominator. This simplifies the expression and allows you to determine the behaviour as n approaches infinity.

The limit of a rational function as n approaches infinity can be found by using the fact that 1/n approaches 0 as n approaches infinity.