Maths Extension 1 Q&A by dot point

Model unrestricted growth and decay with $\frac{dN}{dt} = k N$ and solve the resulting separable differential equation

Apply integration by substitution to evaluate definite and indefinite integrals, including reverse chain rule cases

Integrate functions whose antiderivative involves $\arcsin$, $\arccos$ or $\arctan$

Differentiate an inverse function using (f^-1)'(x) = 1/f'(f^-1(x)) (equivalently dy/dx = 1/(dx/dy)), including the second derivative

Differentiate inverse trigonometric functions and integrate functions that involve them

Describe motion in the plane using a position vector r(t), differentiate to get velocity v(t)=r'(t) and acceleration a(t)=r''(t), and use the dot product to analyse the motion (speed, perpendicular velocities, angle between v and a)

Model projectile motion in two dimensions using parametric equations and find range, maximum height, time of flight and trajectory equation

Solve related-rates problems by linking two changing quantities via an equation and differentiating with respect to time

Solve separable first-order differential equations of the form $\frac{dy}{dx} = f(x) g(y)$ by separating variables and integrating both sides

Calculate volumes of revolution about the x-axis and y-axis using the disc method

State and use the binomial theorem, identify general and specific terms, and relate it to Pascal's triangle

Use the combination formula $\binom{n}{r}$ to count unordered selections, including with restrictions and complementary counting

Use the multiplication principle and the permutation formula to count ordered arrangements, including restrictions and repeated elements

State and apply the pigeonhole principle in counting and existence problems

Sketch curves given parametrically, eliminate the parameter to obtain Cartesian equations, and use parametric form for circles, parabolas and lines

Apply the division algorithm for polynomials and use the remainder and factor theorems to identify and verify factors and roots

Sketch polynomial functions using leading-term behaviour, intercepts and the multiplicity of each root

Solve polynomial and rational inequalities by factoring and analysing the sign of each factor across critical values

Use the relationships between roots and coefficients (Vieta's formulas) for polynomials of degree two, three and four

Prove divisibility statements involving an integer parameter $n$ using mathematical induction

Apply mathematical induction to prove general statements about a recursive sequence, a property of a formula, or a recursive procedure

Prove inequalities involving an integer parameter $n$ using mathematical induction

Prove identities for sums of series using the principle of mathematical induction

Define a Bernoulli random variable, compute its mean and variance, and recognise scenarios that fit the model

Define the binomial distribution $B(n, p)$, state its probability mass function, and find its mean $n p$ and variance $n p (1 - p)$

Compute exact probabilities for the binomial distribution including $P(X = k)$, $P(X \le k)$, $P(X \ge k)$, and use complementary counting

Use the normal approximation $X \sim N(n p, n p (1 - p))$ to approximate binomial probabilities for large $n$

Use the sample proportion $\hat{p} = X/n$ as a random variable, with mean $p$, variance $pq/n$ and standard deviation $\sqrt{pq/n}$, and apply its normal approximation $\hat{p} \sim N(p,\, pq/n)$

Express $a \sin x + b \cos x$ in the form $R \sin(x + \alpha)$ or $R \cos(x - \alpha)$ and use this to solve equations and find extreme values

Write general solutions to trigonometric equations using the period and the symmetries of $\sin$, $\cos$ and $\tan$

Define and sketch the inverse trigonometric functions $\arcsin$, $\arccos$ and $\arctan$, including their domains and ranges

Use the product-to-sum and sum-to-product identities to simplify trigonometric expressions and integrals

Use the sum and difference identities for sine, cosine and tangent to expand or simplify trigonometric expressions

Use the t-formula (Weierstrass substitution) to express $\sin \theta$, $\cos \theta$ and $\tan \theta$ as rational functions of $t = \tan \frac{\theta}{2}$

Use vector methods to prove geometric properties, including parallelism, perpendicularity, midpoint and ratio division

Write parametric vector equations of lines and convert between vector and Cartesian forms

Compute the scalar product of two vectors in component or geometric form and use it to find the angle between vectors and test orthogonality

Perform vector arithmetic with vectors in the plane, including component and column-vector notation, and find the magnitude and unit vector

Compute the scalar and vector projection of one vector onto another and interpret it geometrically

Count arrangements of n distinct objects around a circle as (n-1)! by treating rotations as identical, extend this to groups and blocks around a circle, alternating patterns and 'not together' restrictions handled by the complement, and recognise that for a necklace or bracelet, where a reflection also coincides, the count is (n-1)!/2

Count the distinct arrangements of n objects when some are identical: divide n! by the factorial of each repeat count to get n! over r1! r2! ... rk!, and handle the two-type special case where every object is one of two kinds (2^n arrangements in all, or choose the positions of one kind with a combination)

Count unordered selections (combinations) using ^{n}C_{r} = ^{n}P_{r}/r! = n!/(r!(n-r)!), recognise that an n-element set has 2^n subsets and that the r-element subsets number ^{n}C_{r} with the symmetry ^{n}C_{r} = ^{n}C_{n-r} and row sum 2^n, and apply the at-least, complement, conditional and geometric counting techniques to selection problems

Compute probabilities of equally likely outcomes as P = (favourable outcomes)/(total outcomes) where both counts come from the counting methods (multiplication principle, ^{n}P_{r} and ^{n}C_{r}), and apply the complement, separate-pool multiplication, cases and inclusion-exclusion to find the favourable count for selection, card-hand, digit-string and with/without-replacement problems

Define and evaluate factorials, use the recursive relation n! = n(n-1)!, and simplify factorial expressions and fractions by unrolling and cancelling

Apply three reusable counting principles to ordered arrangements with restrictions: keep tied items together by grouping them into a block and ordering inside it, count an unwanted condition and subtract it from the total (the complement), and split an 'or' count into non-overlapping cases, subtracting the overlap by inclusion-exclusion when the cases meet

Use the multiplication principle to count ordered selections across stages, count ordered selections with repetition as n^r and without repetition as nPr = n!/(n-r)!, and apply restriction techniques: deal with the difficulty first, fix a position, keep items together as a block, and count 'at least one' by the complement

Solve absolute value inequations of the form |ax + b| < k, and inequations with the unknown in the denominator, by multiplying through by the square of the denominator

Sketch the graph of $y = \dfrac{1}{f(x)}$ from the graph of $y = f(x)$: turn zeroes into vertical asymptotes, send large values to small ones, fix the points where $f = \pm 1$, and flip a maximum to a minimum where $f$ keeps its sign

Sketch the graph of $y = f(x) \pm g(x)$ by adding or subtracting ordinates: at a zero of one function the sum meets the other, opposite ordinates give a zero, equal ordinates double, and an oblique asymptote can emerge

Sketch $y = |f(x)|$ by reflecting the part of $y = f(x)$ below the $x$-axis upward, and sketch $y = f(|x|)$ by keeping the part right of the $y$-axis and mirroring it across the $y$-axis, recognising that $y = f(|x|)$ is always even and that the two transformations coincide only in special cases

Form the inverse relation by reflecting $y = f(x)$ in the line $y = x$, use the horizontal line test to decide whether the inverse is a function, find the rule for $f^{-1}(x)$ and verify it by showing $f(f^{-1}(x)) = x$, swap the domain and range, and restrict a domain so that a many-to-one function gains an inverse

Define a curve parametrically by $x = f(t)$ and $y = g(t)$, eliminate the parameter $t$ by algebra or by a Pythagorean identity to obtain the Cartesian equation, parametrise standard curves (lines, parabolas, circles, ellipses and the rectangular hyperbola), and determine the direction of travel (orientation) and any excluded points

Examine the sign of a function by building a table of test values that dodge around its zeroes and discontinuities, and use the resulting sign pattern to solve inequations and to start a curve sketch

Develop the structural consequences of the factor theorem: distinct zeroes give distinct linear factors and a degree-n polynomial has at most n zeroes; a polynomial agreeing with another at n + 1 points is identical, so a degree-n graph is fixed by n + 1 points; the number of intersections of two curves is bounded by degree via F(x) = P(x) - Q(x); and re-express a polynomial in powers of (x - a)

Apply the factor theorem, multiplicity and the sum and product of roots to the geometry of curves: a line is tangent to a curve exactly when the equation formed by solving them simultaneously has a double root; the x-coordinate of the midpoint of a chord is the average of the roots; and the number and nature of the intersections of two curves are read from the roots of the difference of the two polynomials

Sketch the graph of a polynomial in factored form using the behaviour of the leading term for large x and the multiplicity of each zero, deciding where the curve crosses, touches or has a horizontal inflection, and locate a zero between integers from a table of values

Divide one polynomial by another using long division, expressing the result in the form P(x) = D(x)Q(x) + R(x) where the remainder has degree less than the divisor, handling missing terms, and writing the result in the rational form P/D = Q + R/D

Use the remainder theorem (the remainder on division by x - a is P(a)) and the factor theorem ((x - a) is a factor if and only if P(a) = 0), test divisors of the constant term to locate integer zeroes, and combine these with division to find unknown coefficients and to fully factor a polynomial

Relate the coefficients of a polynomial to the elementary symmetric functions of its zeroes (with alternating signs) for quadratics, cubics and quartics, and use these relations to find a missing zero, to evaluate symmetric expressions of the zeroes such as the sum of squares and the sum of reciprocals, to find unknown coefficients from a condition on the zeroes, and to handle zeroes in arithmetic or geometric progression or of a special form

Define a polynomial and use the language of degree, leading term, leading coefficient, monic and the zero polynomial, including the degree of sums and products and equating coefficients of identically equal polynomials