Unit FP2: Further Pure Mathematics 2

The manipulation and solution of The solution of inequalities such as algebraic inequalities and 1 x inequations, including those >, | x2 − 1 | > 2(x + 1). x − a x − b involving the modulus sign.

Use fp2.1.1 - manipulation and solution to connect the rule to the data and decision in the question.

This matters because fp2.1.1 - manipulation and solution determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp2.1.1 - manipulation and solution to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP2.1.1 - Manipulation and solution is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Summation of simple finite series n 1 ∑ using the method of differences.; Students should be able to sum series such as by r(r +1) r=1 1 1 1 using partial fractions such as = −. r(r +1) r r +1.

Use fp2.2.1 - summation of simple finite series n 1 ∑ to connect the rule to the data and decision in the question.

This matters because fp2.2.1 - summation of simple finite series n 1 ∑ determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp2.2.1 - summation of simple finite series n 1 ∑ to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP2.2.1 - Summation of simple finite series n 1 ∑ is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Euler’s relation eiθ = cos θ + i sin θ.; Students should be familiar with cos θ = (eiθ + e−iθ) and sin θ = (eiθ − e−iθ). 2i.

Use fp2.3.1 - euler’s relation eiθ = cos θ + i sin θ to connect the rule to the data and decision in the question.

This matters because fp2.3.1 - euler’s relation eiθ = cos θ + i sin θ determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp2.3.1 - euler’s relation eiθ = cos θ + i sin θ to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP2.3.1 - Euler’s relation eiθ = cos θ + i sin θ is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

De Moivre’s theorem and its To include finding cos nθ and sin mθ in terms of powers of application to trigonometric sin θ and cos θ and also powers of sin θ and cos θ in terms identities and to roots of a complex of multiple angles.; Students should be able to prove number.; De Moivre’s theorem for any integer n.

Use fp2.3.2 - de moivre’s theorem and its to connect the rule to the data and decision in the question.

This matters because fp2.3.2 - de moivre’s theorem and its determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp2.3.2 - de moivre’s theorem and its to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP2.3.2 - De Moivre’s theorem and its is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Interpret and sketch complex-number loci and regions defined by modulus and argument conditions, including |z − a| = b, |z − a| = k|z − b| and arg(z − a) = β.

Use fp2.3.3 - loci and regions in the argand diagram to connect the rule to the data and decision in the question.

This matters because fp2.3.3 - loci and regions in the argand diagram determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp2.3.3 - loci and regions in the argand diagram to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP2.3.3 - Loci and regions in the Argand diagram is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Elementary transformations from az + b Transformations such as w = z2 and w =, where the z-plane to the w-plane. cz + d a, b, c, d ∈, may be set. ℂ.

Use fp2.3.4 - transformations between the z-plane and w-plane to connect the rule to the data and decision in the question.

This matters because fp2.3.4 - transformations between the z-plane and w-plane determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp2.3.4 - transformations between the z-plane and w-plane to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP2.3.4 - Transformations between the z-plane and w-plane is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Further solution of first order The formation of the differential equation may be required. differential equations with Students will be expected to obtain particular solutions and separable variables. also sketch members of the family of solution curves.

Use fp2.4.1 - further solution of first order to connect the rule to the data and decision in the question.

This matters because fp2.4.1 - further solution of first order determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp2.4.1 - further solution of first order to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP2.4.1 - Further solution of first order is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

First order linear differential The integrating factor e ∫Pdx may be quoted without proof. dy equations of the form + Py = Q dx where P and Q are functions of x.

Use fp2.4.2 - first order linear differential to connect the rule to the data and decision in the question.

This matters because fp2.4.2 - first order linear differential determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp2.4.2 - first order linear differential to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP2.4.2 - First order linear differential is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Differential equations reducible to the above types by means of a given substitution.

Use fp2.4.3 - reducible first order differential equations to connect the rule to the data and decision in the question.

This matters because fp2.4.3 - reducible first order differential equations determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp2.4.3 - reducible first order differential equations to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: use the formula and units given in the question, show the substitution and interpret the result; the calculation alone is not the conclusion.

The linear second order differential The auxiliary equation may have real distinct, equal or d2y dy complex roots. f(x) will have one of the forms equation a + b + cy = f(x) dx2 dx k epx, A + Bx, p + qx + cx2 or m cos ωx + n sin ωx. where a, b and c are real constants Students should be familiar with the terms ‘complementary and the particular integral can be function’ and ‘particular integral’. found by inspection or trial.; Students should be able to solve equations of the form d2y + 4y = sin 2x. dx2.

Use fp2.5.1 - linear second order differential to connect the rule to the data and decision in the question.

This matters because fp2.5.1 - linear second order differential determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp2.5.1 - linear second order differential to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP2.5.1 - Linear second order differential is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Differential equations reducible to the above types by means of a given substitution.

Use fp2.5.2 - reducible second order differential equations to connect the rule to the data and decision in the question.

This matters because fp2.5.2 - reducible second order differential equations determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp2.5.2 - reducible second order differential equations to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: use the formula and units given in the question, show the substitution and interpret the result; the calculation alone is not the conclusion.

Third and higher order derivatives.

Use fp2.6.1 - third and higher order derivatives to connect the rule to the data and decision in the question.

This matters because fp2.6.1 - third and higher order derivatives determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp2.6.1 - third and higher order derivatives to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP2.6.1 - Third and higher order derivatives is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Derivation and use of Maclaurin The derivation of the series expansion of ex, sin x, cos x, series. ln (1 + x) and other simple functions may be required.

Use fp2.6.2 - derivation to connect the rule to the data and decision in the question.

This matters because fp2.6.2 - derivation determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp2.6.2 - derivation to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP2.6.2 - Derivation is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Derivation and use of Taylor series.; The derivation, for example, of the expansion of sin x in ascending powers of (x − π) up to and including the term in (x − π)3.

Use fp2.6.3 - derivation to connect the rule to the data and decision in the question.

This matters because fp2.6.3 - derivation determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp2.6.3 - derivation to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP2.6.3 - Derivation is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

Use of Taylor series method for Students may, for example, be required to find the solution series solutions of differential in powers of x as far as the term in x4,of the differential equations. equation d2y dy dy + x + y = 0, such that y = 1, = 0 at x = 0. dx2 dx dx.

Use fp2.6.4 - taylor series method for differential equations to connect the rule to the data and decision in the question.

This matters because fp2.6.4 - taylor series method for differential equations determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp2.6.4 - taylor series method for differential equations to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: use the formula and units given in the question, show the substitution and interpret the result; the calculation alone is not the conclusion.

Use polar coordinates (r, θ) with r ≥ 0 and sketch standard polar curves, including lines, circles, spirals, cardioids, limacons and lemniscates.

Use fp2.7.1 - polar coordinates and polar curves to connect the rule to the data and decision in the question.

This matters because fp2.7.1 - polar coordinates and polar curves determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp2.7.1 - polar coordinates and polar curves to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP2.7.1 - Polar coordinates and polar curves is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.

β The ability to find tangents parallel to, or at right angles to, Use of the formula 1 ∫ r2 dθ 2 the initial line is expected. α for area.

Use fp2.7.2 - area and tangents in polar coordinates to connect the rule to the data and decision in the question.

This matters because fp2.7.2 - area and tangents in polar coordinates determines what can be inferred or chosen; begin with the stated conditions and keep the conclusion tied to the evidence.

Example: apply fp2.7.2 - area and tangents in polar coordinates to one small, clearly defined case, show the key step or comparison, and explain the result in words.

Boundary: FP2.7.2 - Area and tangents in polar coordinates is not a universal recommendation. Check the syllabus scope, assumptions, units and the limits of the evidence before generalising.