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# FMSQ Maths

## Calculus

### Integration

Integration is the reverse of differentiation.

Be able to integrate kxⁿ where n is a positive integer or 0, and the sum of such functions

Be able to find a constant of integration.

Be able to find the equation of a curve, given its gradient function and one point.

### Differentiation

Be able to differentiate kxⁿ where n is a positive integer or 0, and the sum of such functions

The gradient function dy/dx gives the gradient of the curve and measures the rate of change of y with x

The gradient of the fuction is the gradient of the tangent at that point.

Be able to find the equation of a tangent and normal at any point of a curve.

Be able to use differentiation to find stationary points on a curve.

Be able to determine the nature of a stationary point.

Be able to sketch a curve with known stationary points.

### Kinematic applications

Be able to use differentiation and integration with repect to time to solve simple problems involving variable acceleration.

Be able to recognise the special case where the use of constant acceleration of formulae is appropriate.

Be able to solve problems using the formulae.

### Definite Integrals

Definition of definite integrals

Definition of indefinite integrals

Be able to evaluate definite integrals

Be able to find the area between a curve, two ordinates and the x axis.

Be able to find the area between two curves.

## Algebra

### Manipulations of equations

Simplifying expressions including: algebraic fractions, square roots and polynomials, Algebraic Fractions, x/n=y, x=ny, Square roots., √54, √(9x6), 3√6, Polynomial, 5x²+25x+15, 5(x²+5x+3), Take out a factor of 5, and thus divide every part of the orginal polynomial by 5.

### Remanider Theorem

Remainders of a polynomial up to order 3 when divided by a linear factor

### Factor Theorem

Linear factors of a polynomial up to order 3

### Solution of equations

Use of brackets

Solving linear equations with one unknown, x+4=6 x=2

Solving quadratic equations by factorisation, the use of formula and by completing the square, Factorisation, x²+6x+8, (x+4) (x+2), Two factors of 8 that add together to make 6., Quadratic Formula, x=-b±√(b²-4ac) ÷ 2a, ax²+bx+c=0, Completing the square., (x-(b/2))²-(b/2)²+c, ax²+bx+c=0

Solving a cubic equation by factorisation, Use factor theorum to find one factor, and then factorise the quadratic you are left with.

Solving two linear simultaneous equations in 2 unknowns, 6y+4x=-22 y+3x=8, y=8-3x, Subsitute the rearranged equation into the first one., 6(8-3x)+4x=-22, 48-18x+4x=-22, 48-14x=-22, -14x=-70, 14x=70, x=5

Solving two simultaneous equations where one equation is linear and the other is quadratic., x²+3x+2=0 x+y=5, x=5-y, Subsitute the rearranged equation into the first one., (5-y)²+3(5-y)+2=0, 15-3y+2+y²-10y+25=0, 42-13y+y²=0, (y-6)(y-7)=0, y=6 or y=7, ஃ x=-1 or x=-2

Set up and solve problems leading to linear, quadratic and cubic equations in one unknown, and to simultaneious linear equations in two unknowns.

### Inequalities

Ability to manipulate inequalities, x+4>2x+7, -3>x

### Binomial Expansion

Understand and be able to apply the binomial expansion (a+b)^n where n is a positive integer

### Application to probability

Recognise probability situations which give rise to the binomial distribution

## Co-ordinate Geometry (2D)

### Straight lines

Know the definition of the gradient of a line

Know the relationship between the gradients of parallel and perpendicular lines

Be able to calculate the distance between two points

Be able to find the midpoint of a line segment

Be able to form the equation of a straight line

Be able to solve simultaneous equations graphically

### Co-ordinate geometry of circles

Know that the equation of a circle, centre (0,0) radius r is x²+y²=r²

Know that (x-a)² + (y-b)² = r², is the equation of a circle with centre (a,b) and radius r

### Inequalities

Be able to illustrate linear inequalities in two variables.

### Applications to linear programming

Be able to express real situations in terms of linear inequalities

Be able to use graphs of linear inequalities to solve 2D maximisation and minimisation problems.

Know the definition of objective function and be able to find it in 2D cases.

## Trigonometry

### Ratios of an angle and their graphs

Able to use the definitions of sinθ, cosθ and tanθ for any angle.

Apply trig to right angled triangles

Know the sine and cosine rules and be able to apply them

Be able to apply trig to triangles with any angles

Know and be able to use the identity tanθ = sinθ /cosθ

Know and be able to use the identity sin²θ + cos²θ = 1

Be able to solve simple trig. equations in given intervals

Be able to apply tig. to 2 and 3D problems.