3. Pure Mathematics A2 3

For $|f(x)|=g(x)$ require $g(x)\ge0$, then solve both sign branches and check the original. For variable-side inequalities, split at zeros of both sides and intersect each result with its sign interval.

Write descending powers with zero coefficients for gaps. Repeatedly divide leading terms, multiply the whole divisor, subtract, and stop only when the remaining degree is lower than the divisor degree.

Handle dividends of degree at most $4$ and either linear or quadratic divisors. Synthetic division is only a shortcut for suitable linear divisors.

Turn every factor or remainder statement into an equation. Solve simultaneous equations for unknown coefficients where needed; divide out confirmed factors before solving the lower-degree polynomial.

If division by $x+2$ leaves remainder $5$, use $P(-2)=5$, not zero. Use full polynomial division when the quotient is also required.

Factor the denominator, write the complete matching template, multiply through by the original denominator, then substitute convenient roots and/or equate coefficients to solve all constants.

Substitute the solved coefficients back and recombine as a check before using the decomposition in later algebra or integration.

Rewrite the expression as a constant multiple of $(1+u)^n$, substitute $u$ into the displayed initial terms, expand and collect powers. Do not seek a general term; it is excluded.

Transform the convergence condition with the same substitution: $|u|<1$. In the example, $|-2x|<1$, so $|x|<1/2$. For a shifted/scaled $u(x)$, solve the resulting inequality and report the full set.

Record the positivity conditions before combining or expanding logs. Move coefficients into powers when useful, combine to one logarithm, then translate back to index form.

For $k>0$, $e^{kx}$ increases; for $k<0$, it decreases. Both stay positive and approach, but never cross, the $x$-axis in one direction.

Rearrange until each exponential expression is positive, take $\ln$ of both sides, use $\ln(a^{g(x)})=g(x)\ln a$, then solve the resulting algebraic equation and check in the original.

If terms such as $a^{2x}$ and $a^x$ occur together, set $t=a^x$ with $t>0$, solve the polynomial in $t$, reject non-positive roots, then take logs to recover $x$.

Transform measured uncertainties and units consistently, then interpret the gradient and intercept in the original parameters. A straight plot supports the model but does not prove causation.

If ln y=0.7x+1.2, then a=e^{1.2} and b=e^{0.7} for y=ab^x.

Start with the corresponding cosine, sine or tangent graph over the required angles. Its denominator zeros become reciprocal asymptotes; where the denominator is $\pm1$, the reciprocal is also $\pm1$; use the denominator sign between asymptotes to choose each branch.

$\sec(\pi/3)=2$ because $\cos(\pi/3)=1/2$. At $x=\pi/2$, cosine is zero, so secant is undefined and has a vertical asymptote.

Write $a\sin heta+b\cos heta=R\sin( heta+\alpha)$ by matching $R\cos\alpha=a$ and $R\sin\alpha=b$. Thus $R=\sqrt{a^2+b^2}$ and choose the quadrant of $\alpha$ from both signs. A cosine form is equally valid when coefficient matching is adjusted.

For exact values, expand a compound angle built from known angles. For equations, rewrite to one function or $R$-form, use its range to decide existence, find all solutions in the stated interval, and check any denominator restrictions.

Identify the outer function, write its base derivative with the inner expression unchanged, multiply by the inner derivative, then combine constant multiples, sums and differences.

Label $u$ and $v$, compute $u'$ and $v'$ separately using chain rules where needed, substitute without changing the quotient numerator order, then factor or simplify.

Find the parameter or point coordinates first, evaluate the tangent gradient $m$, then use $y-y_0=m(x-x_0)$. A non-vertical normal has gradient $-1/m$; handle horizontal/vertical tangent cases geometrically.

Match the whole integrand to one row, keep the linear inner expression unchanged, divide by its gradient $a$, carry any outside constant, and include $+C$ for an indefinite integral.

Differentiate the answer: the chain factor $a$ must cancel the inserted $1/a$. For definite integrals, use an interval that does not cross a point where the integrand is undefined.

First rewrite the squared sine or cosine as a constant plus/minus a double-angle cosine. Then integrate term by term using the linear-inner rule, including the factor created by the doubled angle.

Recombine the decomposition before integration, then differentiate the final antiderivative. Preserve intervals that do not cross denominator zeros.

Differentiate the denominator or inner function, compare it with the numerator, factor out the required constant, then apply the rule. If a leftover remains, split it and use another approved standard form.

Identify the product (write $\ln x=1\cdot\ln x$ if needed), state $u,dv,du,v$, substitute into the formula, evaluate the remaining simpler integral and add $C$.

Use the substitution supplied by the question. Find $du$, rewrite every factor and $dx$ in $u$, and do not mix variables. For a definite integral, convert both limits immediately or substitute back before using the original limits.

For definite limits $x=a,b$, replace them by $u(a),u(b)$ and finish entirely in $u$. If the substitution is not one-to-one over the interval, follow the question structure carefully and verify the transformed bounds.

Differentiate an indefinite answer or compare a definite result numerically/sign-wise with the original integrand.

For a continuous $f$, $f(1)<0$ and $f(2)>0$ locates at least one root between $1$ and $2$. State the function values or their signs, not just the interval.

A graph gives visual approximate evidence; a sign-change bracket gives endpoint evidence. A narrower bracket gives a tighter location but remains an interval, not the exact root.

Choose a valid starting value, retain guard digits, and stop when successive values or the residual meet the tolerance. Verify the final value in f(x)=0.

For x=cosx, x₀=1 gives iterates that approach approximately 0.739; reporting the last iterate without a tolerance is incomplete.

Use the stated starting value, keep guard digits, tabulate $n$ and $x_n$, apply the same formula repeatedly, watch for settling/divergence/cycling, and continue until successive values justify the prescribed rounded answer.

For a requested number of decimal places, obtain successive values that round consistently at that precision and substitute the reported approximation into the original equation as a residual sense-check when practical.

$\overrightarrow{AB}$ is the directed vector from $A$ to $B$; a bold/lowercase symbol such as $\mathbf a$ can name the same free vector. Equality means equal corresponding components.

A point $P(x,y,z)$ names a location. Its position vector $\overrightarrow{OP}$ has the same numerical entries only because the origin and direction from $O$ are specified.

If $\overrightarrow{OA}=(2,1)$ and $\overrightarrow{OB}=(8,5)$, midpoint $M$ has position $(5,3)$. Check by equal displacements $\overrightarrow{AM}=\overrightarrow{MB}=(3,2)$.

Keep one origin for position vectors, subtract final minus initial for displacement, calculate its magnitude, then divide every component by that magnitude if a unit direction is required. The same workflow works in 2D or 3D.

A unit vector must have magnitude $1$. Reversing direction negates the unit vector but keeps the same magnitude.

Given point $A$ and direction $\mathbf b$, use $\mathbf r=\overrightarrow{OA}+t\mathbf b$. Given points $A,B$, use direction $\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}$.

A different point on the same line or any non-zero scalar multiple of the direction produces an equivalent equation.

Use different parameters for the two lines. Equate position components, solve two equations, then verify the same parameter pair in the remaining component. If consistent, substitute to find the intersection point.

If $\mathbf d=k\mathbf b$, test whether $\mathbf c-\mathbf a$ is also parallel to $\mathbf b$. If yes the equations describe the same line; otherwise they are distinct parallel lines.

For line directions $\mathbf b,\mathbf d$, compute $\cos\theta=(\mathbf b\cdot\mathbf d)/(|\mathbf b||\mathbf d|)$. If the smaller angle between lines is requested, use the acute value, equivalently $|\mathbf b\cdot\mathbf d|$ in the numerator.

This foot method works in 2D or 3D and in cuboid/tetrahedron contexts: the joining vector from the foot to the point must be perpendicular to the line direction.

Name dependent/independent variables, convert “rate of change” to a derivative, make the right side depend only on stated quantities, introduce $k$ with sign/units, then use any given value and rate at the same instant to evaluate $k$.

Check dimensions: in $dy/dt=ky$, $k$ has units inverse to the independent variable. State an initial relation separately if supplied.

Check constant solutions from $g(y)=0$ before dividing. Separate completely, use any required technique from 3.5, include one arbitrary constant, then rearrange explicitly or leave a valid implicit relation as appropriate.

Differentiate the general solution and substitute it into the original equation. Record intervals/domain restrictions created by logarithms or division.

Solve generally first, apply the actual initial point (not automatically $x=0$), solve the constant exactly, and check both the differential equation and the initial condition.

Choose the branch/interval containing the initial point when logs, square roots or implicit relations create multiple possibilities.

If $M(t)=50e^{-0.06t}$ kg for $t\ge0$, then $M(0)=50$ kg, the model predicts continuous decay, $M(t)>0$, and $M(t)\to0$ as $t\to\infty$. The constant $0.06$ has units h$^{-1}$ if $t$ is hours.

Also verify the formula satisfies the ODE and initial data, but algebraic validity is only the start: state the contextual conclusion in words and sensible accuracy.

A non-zero complex number has arguments differing by $2\pi$. Unless specified, use a consistent principal interval such as $-\pi<\theta\le\pi$; $0\le\theta<2\pi$ may also be convenient. $\arg0$ is undefined.

For $z=-1+i\sqrt3$, $|z|=2$ and a principal argument is $2\pi/3$; its conjugate is $-1-i\sqrt3$.

After simplifying, write the answer in a+bi form. Equality of complex numbers requires equality of both components.

(1+2i)/(3−i) =[(1+2i)(3+i)]/10=(1+7i)/10.

Write the conjugate root immediately, multiply the pair to form a real factor, divide the cubic/quartic polynomial by it, then solve the remaining lower-degree factor and verify the total degree.

$-2+3i$ is plotted at $(-2,3)$ in quadrant II; its modulus is $\sqrt{13}$ and its argument must lie in the chosen quadrant-II branch.

Label axes, scale and reference points. Read a plotted point back as real coordinate plus $i$ times imaginary coordinate.

Operate moduli and arguments separately, reduce the final argument to the requested interval, and convert to Cartesian form only if asked.

Square both answers and show full Cartesian working. The two roots are always opposites for a non-zero complex number.

Multiplication by $i=e^{i\pi/2}$ rotates every point $90^\circ$ anticlockwise about the origin without changing modulus. Multiplication by $-2$ scales by $2$ and rotates by $\pi$.

Addition/subtraction use parallelogram/displacement geometry; multiplication/division use origin-centred scale and rotation. Conjugates keep modulus and negate the argument within branch conventions.

Predict the geometric result, then confirm with Cartesian or polar arithmetic.

Plot reference points first, draw each boundary, decide included/excluded style from equality, shade the correct side/interior/exterior, then intersect all regions.

$|z-(2+i)|<3$ is the open disc centred at $(2,1)$ with radius $3$. Adding $\arg(z-(2+i))=\pi/4$ restricts to points on the corresponding ray that also lie inside the disc.

For equal distances, do not draw a circle: the locus is a straight perpendicular bisector. Test one simple point to choose a side for an inequality.