NCEA Level 2 Mathematics and Statistics flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Quadratic formula

The quadratic formula gives the roots of any quadratic equation ax²+bx+c=0, provided a is not zero: x=(−b±√(b²−4ac))/(2a).

Discriminant

For a quadratic ax²+bx+c=0, the discriminant is b²−4ac. Positive means two real roots, zero means one repeated root, negative means no real roots.

Index product rule

When multiplying powers with the same base, add the exponents: a^m×a^n=a^(m+n). This rule applies for any real base a and integer exponents m and n.

Index power rule

When raising a power to another power, multiply the exponents: (a^m)^n=a^(mn). The base stays unchanged; for example, (3²)⁴=3⁸.

Negative index

A negative exponent means the reciprocal of the positive power: a^(−n)=1/a^n, provided a≠0. For example, 5^(−2)=1/25.

Logarithm definition

If b^x=y with b>0 and b≠1, then log_b(y)=x. A logarithm answers the question: to what power must the base b be raised to give y?

Arithmetic sequence nth term

For an arithmetic sequence with first term a and common difference d, the nth term is a_n=a+(n−1)d.

Geometric sequence nth term

For a geometric sequence with first term a and common ratio r, the nth term is a_n=a×r^(n−1).

Compound interest formula

Compound interest: A=P(1+i)^n, where A is final amount, P is principal, i is decimal interest rate per period, and n is number of periods.

Completing the square

To complete the square for x²+bx+c, write (x+b/2)²+c−b²/4. This form reveals the vertex of the parabola and can solve quadratics.

Elimination method

To solve simultaneous equations by elimination, multiply each equation so one variable has matching coefficients, then add or subtract to remove that variable.

Pythagoras' theorem

In a right-angled triangle, the hypotenuse squared equals the sum of the squares of the other two sides: c²=a²+b².

Sine ratio

In a right-angled triangle, the sine of an angle equals the length of the opposite side divided by the hypotenuse.

Cosine ratio

In a right-angled triangle, the cosine of an angle equals the length of the adjacent side divided by the hypotenuse.

Tangent ratio

In a right-angled triangle, the tangent of an angle equals the length of the opposite side divided by the adjacent side.

Sine rule

For any triangle, a/sinA=b/sinB=c/sinC, where side a is opposite angle A, side b is opposite B, and side c is opposite C.

Cosine rule

For any triangle, c²=a²+b²−2ab×cosC, where C is the angle included between sides a and b.

Triangle area sine formula

The area of a triangle is ½×a×b×sinC, where a and b are two side lengths and C is the included angle.

Arc length

For a sector with radius r and central angle θ in degrees, arc length = (θ/360)×2πr.

Sector area

For a sector with radius r and central angle θ in degrees, sector area = (θ/360)×πr².

Similar figure scale factors

If two similar figures have length scale factor k, their areas scale by k² and their volumes scale by k³.

Perpendicular gradients

Two straight lines are perpendicular if the product of their gradients is −1; that is, m₁×m₂=−1.

Mean

The mean is the sum of all data values divided by the number of values. It uses every value and is affected by outliers.

Median

The median is the middle value when data are ordered from smallest to largest. For an even number of values, average the two middle values.

Interquartile range

The interquartile range is upper quartile minus lower quartile. It measures spread of the middle 50% of data and is resistant to outliers.

Standard deviation

Standard deviation measures the typical distance of data values from their mean. A larger value indicates greater spread or variability.

Complementary probability

For any event A, the probability of its complement is P(notA)=1−P(A). The complement contains all outcomes not in A.

Addition rule probability

For two events A and B, P(AorB)=P(A)+P(B)−P(AandB). If A and B cannot both occur, P(AandB)=0.

Independent events

Events A and B are independent if P(AandB)=P(A)×P(B). Knowing that one event occurs does not change the probability of the other.

Expected value

For a discrete random variable X, expected value E(X)=Σx×P(x). It is the long-run average outcome over many trials.

Correlation

Correlation measures the strength and direction of a linear relationship between two variables. It ranges from −1 to 1, and does not prove causation.

Z-score

A z-score is z=(x−μ)/σ. It states how many standard deviations a value x lies above or below the mean μ.