mathsprintplus v5.0 help sheet · 2020-02-26 · mathsprintplus v5.0 help sheet how to use...

MATHSprintPLUS V5.0 HELP SHEET

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MATHSprintPLUS 5.03 Topic List

PURE: BASIC ALGEBRA Indices Evaluate ±ve indices

*3², 4⁻³, 7⁰, etc. (±ve and zero integer index) *(1/4)², (1/3)⁻³, etc. (±ve integer index) Simplify +ve indices

*a×a×a, a²×a³, a³/a², (a³)², a×a³/a²*6a²×3a, 6a²/3a, (2a²)³

*(2a²)³×3a, (2a²)³/4a *6a²×3a×4a³, 6a²×3a/4a³

*(6a²±3a)×4a³, (6a²±4a³)/2a *6a²b³×3ab², 6a²b³/3ab², (2ab²)³

Simplify -ve indices *1/(a×a×a), a²×a⁻³, a³/a⁻², (a⁻³)², a×a⁻³/a²*6a⁻²×3a, 6a⁻²/3a, (2a⁻²)³

*(2a⁻²)³×3a, (2a⁻²)³/4a *6a⁻²×3a×4a³, 6a⁻²×3a/4a³

*(6a⁻²±3a)×4a³, (6a⁻²±4a³)/2a Evaluate fractional indices

*4¹ᴵ², etc. (+ve unit fraction index)*27⁻¹ᴵ³, etc. (-ve unit fraction index)*4¹ᴵ², 27⁻¹ᴵ³, etc. (±ve unit fraction index)*8²ᴵ³, etc. (+ve non-unit fraction index)*64⁻²ᴵ³, etc. (-ve non-unit fraction index)*8²ᴵ³, 64⁻²ᴵ³, etc. (±ve non-unit fraction index)

*(1/4)¹ᴵ², etc. (+ve unit fraction index) *(1/8)⁻¹ᴵ³, etc. (-ve unit fraction index) *(1/4)¹ᴵ², (1/8)⁻¹ᴵ³, etc. (±ve unit fraction index)

Simplify fractional indices *a¹ᴵ³×a¹ᴵ², (a³)²ᴵ³, etc.

*√a×(√a)³, √a/(√a)³, etc. Expanding brackets Expand over one bracket

*±2(3x±4), ±2(3x±4)±4(7x±1) *2x(3x±4), 2xy(3x±4y), etc.*2x⁴y⁵z²(3yz⁴±4x²), 2x⁴y²(3y⁴±4x²y²±x), etc.

Expand product of 2 or 3 brackets (1 variable) *(x ± 3)(x ± 4)

*(2x ± 5)(3x ± 2) *(2x ± 5)²

*7(2x ± 5)(3x ± 2) *-7(2x ± 5)(3x ± 2) *±7(2x ± 5)(3x ± 2)

*7(2x ± 5)² *-7(2x ± 5)²

*±7(2x ± 5)² *x(x ± 2)(x ± 3)*x(2x ± 5)(3x ± 2)

*(x ± 2)(x ± 3)(x ± 4) *(2x ± 5)(3x ± 2)(4x ± 1) *(x ± 2)³ *(2x ± 5)³

Expand product of 2 or 3 brackets (2 variables) *(x ± 3y)(x ± 2y) *(x ± 3y)² *(x + 3y)(x - 3y) *(2x ± 5y)(3x ± 2y)

*(2x ± 5y)² *(2x + 5y)(2x - 5y)

*x(2x ± 5y)(3x ± 2y) *(2x ± 5y)(3x ± 2y ± 4)

*(2x ± 5y ± 3)(3x ± 2y ± 4) Factorising Factorise into 1 bracket

*±6x±8, 6x²±8x, 6x²±8xy, 6x²y±8xy², etc. *6x⁴y⁶z⁵±8x⁷y⁵z², 6x⁴y⁶±8x⁷y⁵±4x⁵y², etc.

Factorise into 2 brackets *x² ± 5x ± 6, x² - 9*9x² - 16*2x² ± 5x ± 3*x² ± 5xy ± 6y², x² - 9y²*9x² - 16y²*2x² ± 5xy ± 3y²

Surds Simplify surds

*Easy square factor, eg: √20*Harder square factor, eg: √108

*√45 ± √20 *√15 × √5, √15 / √5

*Misc basic *√3(2 ± √3) *√3(2 ± 5√3) *√3(2 ± √5) *√3(2 ± 3√5)

*Misc expanding 1 bracket *(4 ± √5)² *(3 ± √5)(4 ± √5) *(3 ± 2√5)(4 ± 3√5)

*Misc expanding 2 brackets *(3 ± 2√5)(4 ± 3√7)

*(3√7 ± 2√5)² Surds - rationalise denominator

*2/√6, etc.*1/(2 ± √5), etc.

*±4/(2 ± √5), etc. *1/(7 ± 3√5), etc.

*±4/(7 ± 3√5), etc. *1/(√5 ± √3), etc.

*1/(√5 ± 3√3), etc. *Misc 1 *(2 ± √5)/(3 ± √5), etc. *(2 ± 4√5)/(3 ± 2√5), etc. *(√5 + √3)/(√5 - √3), etc. *(√5 + 3√3)/(√5 - 3√3), etc. *Misc 2

PURE: QUADRATICS Solving quadratic equations Factorise and solve *(x + 2)(x + 3) = 0, etc. *x(x + 4) = 0, etc. *x² + 4x = 0, etc. *x² ± 5x ± 6 = 0, etc. *x² + 10x + 25 = 0, etc. *x² - 25 = 0, etc. *Misc x² ± Ax ± B = 0 *3x² - 7x + 2 = 0, etc. Rearrange, factorise and solve *(x - 1)(x + 2) = 4 *x + 6/x = 5, etc. *x - 4/(x + 2) = 1, etc. *3/(x + 7) - 3/(x + 9) = 2, etc. *Misc. equations Solve using the formula *x²+x-5=0, etc., surd form *2x²+x-5=0, etc., surd form *x²+x-5=0, etc., 3 sig figs *2x²+x-5=0, etc., 3 sig figs Complete the square and solve *(x + p)² + q *a(x + p)² + q Functions Evaluate f(a) *1 step *2 steps with x² *2 steps with √x *2 steps with x², √x *3 steps with x² *3 steps with √x *3 steps with x², √x *Misc 2 and 3 step function *Rational function 1 *Rational function 2 *Misc rational function *Misc function Solve f(x) = a *1 step *2 steps with x² *2 steps with √x *2 steps with x², √x

*3 steps with x² *3 steps with √x *3 steps with x², √x *Misc 2 and 3 step function *Rational function 1 *Rational function 2 *Misc rational function *Misc function Solve f(x) = 0 *(x + 3)(x - 4) = 0 *x² - x - 12 = 0 *x(x + 3)(x - 4) = 0 *x³ - x² - 12x = 0 *(x + 1)(x + 3)(x - 4) = 0 *Misc polynomial factorised *Misc polynomial unfactorised *Misc polynomial Solve f(x) = g(x) *x² + 2x = 3x + 12 *x² + 2x - 5 = 3x + 7 *Misc quadratic *x³ - 12x = x² *x³ + 2x² - 10x = 3x² + 2x *Misc cubic *Misc quadratic/cubic Solve quadratics with substitution *u = x² *u = x³ *u = √x *u = ³√x *u = kˣ *Misc substitution Quadratic graphs Complete the square to: *Find line of symmetry (x+p)²+q *Find line of symmetry a(x+p)²+q *Find min/max value (x+p)²+q *Find min/max value a(x+p)²+q *Find co-ordinates of vertex (x+p)²+q *Find co-ordinates of vertex a(x+p)²+q *Find a misc feature (x+p)²+q *Find a misc feature a(x+p)²+q *List co-ordinates of vertex and all axis intersections (x+p)²+q *List co-ordinates of vertex and all axis intersections a(x+p)²+q Find the quadratic equation *Given the vertex and y-intercept (x+p)²+q *Given the vertex and y-intercept a(x+p)²+q *Given x- and y-axis intersections (x+p)²+q *Given x- and y-axis intersections a(x+p)²+q *From a sketch graph y = (x+p)²+q *From a sketch graph y = a(x+p)²+q

*From a sketch graph y = x² + bx + c *From a sketch graph y = ax² + bx + c Sketch the quadratic graph *From the equation y = (x+p)²+q *From the equation y = a(x+p)²+q *From the equation y = x² + bx + c *From the equation y = ax² + bx + c The discriminant Discriminant basics *Calculate the discriminant *Find the number of roots of a quadratic Discriminant equations and inequalities *Find k in ax² + bx + k = 0 given the number of real roots *Find k in ax² + kx + c = 0 given the number of real roots

PURE: EQUATIONS and INEQUALITIES Linear simultaneous equations Linear sim eqn, integer, no multiply *2x+3y=7, 4x+3y=11 (+ve x,y) *2x+3y=7, 4x-3y=11 (+ve x,y) *2x+3y=7, 4x±3y=11 (+ve x,y) *2x+3y=7, 4x+3y=11 (±ve x,y) *2x+3y=7, 4x-3y=11 (±ve x,y) *2x+3y=7, 4x±3y=11 (±ve x,y) Linear sim eqn, integer, multiply one equation ×2 or ×3 *2x+y=5, 3x+2y=8 (+ve x,y) *2x+y=5, 3x-2y=4 (+ve x,y) *2x+y=5, 3x±2y=4 (+ve x,y) *2x+y=5, 3x+2y=8 (±ve x,y) *2x+y=5, 3x-2y=4 (±ve x,y) *2x+y=5, 3x±2y=4 (±ve x,y) Linear sim eqn, integer, multiply one equation ×4 or more *2x+y=5, 3x+4y=10 (+ve x,y) *2x+y=5, 3x-4y=2 (+ve x,y) *2x+y=5, 3x±4y=2 (+ve x,y) *2x+y=5, 3x+4y=10 (±ve x,y) *2x+y=5, 3x-4y=2 (±ve x,y) *2x+y=5, 3x±4y=2 (±ve x,y) Linear sim eqn, integer, multiply both equations *2x+3y=7, 3x+4y=10 (+ve x,y) *2x+3y=7, 3x-4y=2 (+ve x,y) *2x+3y=7, 3x±4y=2 (+ve x,y) *2x+3y=7, 3x+4y=10 (±ve x,y) *2x+3y=7, 3x-4y=2 (±ve x,y) *2x+3y=7, 3x±4y=2 (±ve x,y) Linear sim eqn, (half)-integer, no multiply

*2x+3y=7, 4x+3y=11 (+ve x,y) *2x+3y=7, 4x-3y=11 (+ve x,y) *2x+3y=7, 4x±3y=11 (+ve x,y) *2x+3y=7, 4x+3y=11 (±ve x,y) *2x+3y=7, 4x-3y=11 (±ve x,y) *2x+3y=7, 4x±3y=11 (±ve x,y) Linear sim eqn, (half)-integer, multiply one equation ×2 or ×3 *2x+y=5, 3x+2y=8 (+ve x,y) *2x+y=5, 3x-2y=4 (+ve x,y) *2x+y=5, 3x±2y=4 (+ve x,y) *2x+y=5, 3x+2y=8 (±ve x,y) *2x+y=5, 3x-2y=4 (±ve x,y) *2x+y=5, 3x±2y=4 (±ve x,y) Linear sim eqn, (half)-integer, multiply one equation ×4 or more *2x+y=5, 3x+4y=10 (+ve x,y) *2x+y=5, 3x-4y=2 (+ve x,y) *2x+y=5, 3x±4y=2 (+ve x,y) *2x+y=5, 3x+4y=10 (±ve x,y) *2x+y=5, 3x-4y=2 (±ve x,y) *2x+y=5, 3x±4y=2 (±ve x,y) Linear sim eqn, (half)-integer, multiply both equations *2x+3y=7, 3x+4y=10 (+ve x,y) *2x+3y=7, 3x-4y=2 (+ve x,y) *2x+3y=7, 3x±4y=2 (+ve x,y) *2x+3y=7, 3x+4y=10 (±ve x,y) *2x+3y=7, 3x-4y=2 (±ve x,y) *2x+3y=7, 3x±4y=2 (±ve x,y) Quadratic simultaneous equations Elimination, rational solutions *y=x², y=2x+3, etc. *y=x²+7x+9, y=2x+3, etc. Easy substitution, rational solutions *xy = k, x + ny = p *ax² + by² = k, x + ny = p *ax² - by² = k, x + ny = p *ax² ± bxy = k, x + ny = p *ax² ± bxy + cy² = k, x + ny = p *ax² ± bxy - cy² = k, x + ny = p *xy + dx + ey = k, x + ny = p *ax² + by² + dx + ey = k, x + ny = p *ax² - by² + dx + ey = k, x + ny = p *ax² ± bxy + dx + ey = k, x + ny = p *ax² ± bxy + cy² + dx + ey = k, x + ny = p *ax² ± bxy - cy² + dx + ey = k, x + ny = p Harder substitution, rational solutions *xy = k, mx + ny = p *ax² + by² = k, mx + ny = p *ax² - by² = k, mx + ny = p *ax² ± bxy = k, mx + ny = p *ax² ± bxy + cy² = k, mx + ny = p

*ax² ± bxy - cy² = k, mx + ny = p *xy + dx + ey = k, mx + ny = p *ax² + by² + dx + ey = k, mx + ny = p *ax² - by² + dx + ey = k, mx + ny = p *ax² ± bxy + dx + ey = k, mx + ny = p *ax² ± bxy + cy² + dx + ey = k, mx + ny = p *ax² ± bxy - cy² + dx + ey = k, mx + ny = p Linear inequalities Solving linear inequalities (±ve x coeff) *±x ± b < c, ±x ± b > c *±ax < c, ±ax > c *±ax ± b < c, ±ax ± b > c *±a(x ± b) < c, ±a(x ± b) > c *Misc Solving linear inequalities and plotting (±ve x coeff) *±x ± b < c, ±x ± b > c *±ax < c, ±ax > c *±ax ± b < c, ±ax ± b > c *±a(x ± b) < c, ±a(x ± b) > c *Misc Quadratic inequalities Simple x² *eg x² > 4 *eg 2x²+4 < 22 Factorised *(x - 4)(x + 5) > 0, etc. *-(x - 4)(x + 5) > 0, etc. *±(x - 4)(x + 5) > 0, etc. Factorisable *x² + 5x - 6 > 0, etc. *-x² - 5x + 6 > 0, etc. *±x² ± 5x ± 6 > 0, etc. *x² - 6 > -5x, x(x - 5) > -6, etc. Multiply through by x² *5 > 20/x, 1 + 6/x² < -5/x, etc.

PURE: GRAPHS and TRANSFORMATIONS Graph sketching - basic shapes Powers of x *y=x, x², x³ [3] *y=1/x, 1/x² [2] *y=x, x², x³, 1/x, 1/x² [5] *y=√x, ³√x [2] *Misc [7] Graph sketching - cubics Sketch a cubic *Sketch y = (x + 3)(x - 2)(x - 4), etc. *Sketch y = -(x + 3)(x - 2)(x - 4), etc. *Sketch y = (x + 3)²(x - 2), etc. *Sketch y = -(x + 3)²(x - 2), etc.

*Sketch y = (x + 3)³, etc. *Sketch y = -(x + 3)³, etc. *Sketch misc factorised *Factorise and sketch +ve cubic with 3 distinct roots *Factorise and sketch -ve cubic with 3 distinct roots *Factorise and sketch +ve cubic with double repeated root *Factorise and sketch -ve cubic with double repeated root Suggest an equation for a cubic *y = (x + 3)(x - 2)(x - 4), etc. *y = -(x + 3)(x - 2)(x - 4), etc. *y = (x + 3)²(x - 2), etc. *y = -(x + 3)²(x - 2), etc. *y = (x + 3)³, etc. *y = -(x + 3)³, etc. *Misc cubic Graph sketching - quartics Sketch a quartic *Sketch y = (x + 3)(x - 2)(x - 4)(x + 1), etc. *Sketch y = -(x + 3)(x - 2)(x - 4)(x + 1), etc. *Sketch y = (x + 3)²(x - 2)(x - 4), etc. *Sketch y = -(x + 3)²(x - 2)(x - 4), etc. *Sketch y = (x + 3)³(x - 2), etc. *Sketch y = -(x + 3)³(x - 2), etc. *Sketch misc factorised *Factorise and sketch +ve quartic with four distinct roots *Factorise and sketch -ve quartic with four distinct roots *Factorise and sketch +ve quartic with double repeated root *Factorise and sketch -ve quartic with double repeated root Suggest an equation for a quartic *y = (x + 3)(x - 2)(x - 4)(x + 1), etc. *y = -(x + 3)(x - 2)(x - 4)(x + 1), etc. *y = (x + 3)²(x - 2)(x - 4), etc. *y = -(x + 3)²(x - 2)(x - 4), etc. *y = (x + 3)³(x - 2), etc. *y = -(x + 3)³(x - 2), etc. *Misc quartic Points of intersection Number of intersections by sketching *Linear and quadratic *Linear and cubic *Linear and y=1/x *Linear and y=1/x² *Linear and misc. *Quadratic and cubic *Quadratic and y=1/x

*Quadratic and y=1/x² *Cubic and y=1/x *Cubic and y=1/x² *Quadratic/cubic and misc. Transforming graphs Basics *Describe single-step transformation of y = f(x) *Apply single-step transformation to y = f(x) *Describe two-step transformation of y = f(x) *Apply two-step transformation to y = f(x) Transform a function algebraically (1 step) *Translate: given f(x) = xⁿ, find f(x ± k) *Translate: given f(x) = xⁿ, sketch f(x ± k) *Translate f(x) = xⁿ given a translation vector *Translate: transform a point on y = f(x) to y = f(x ± k) *Stretch: given f(x) = xⁿ, find kf(x) or f(kx) *Stretch: given f(x) = xⁿ, sketch kf(x) or f(kx) *Stretch f(x) = xⁿ given a scale factor and direction *Stretch: transform a point on y = f(x) to y = kf(x) or f(kx) *Reflect: given f(x) = xⁿ, find -f(x) or f(-x) *Reflect: given f(x) = xⁿ, sketch -f(x) or f(-x) *Reflect f(x) = xⁿ given a reflection line *Reflect: transform a point on y = f(x) to y = -f(x) or f(-x) *Misc: given f(x) = xⁿ, find f(x ± k), ±kf(x) or f(±kx) *Misc: given f(x) = xⁿ, sketch f(x ± k), ±kf(x) or f(±kx) *Misc algebraic transformation of f(x) = xⁿ *Misc: transform a point on y = f(x)

PURE: STRAIGHT-LINE GRAPHS Line segments Line segments in positive quadrant *Gradient *Midpoint *Length *Gradient and mid-point *Gradient, mid-point & length Line segments in any quadrant *Gradient *Midpoint *Length *Gradient and mid-point *Gradient, mid-point & length Features of linear graphs Gradient and y-intercept from y = mx + c *Identify integer gradient

*Identify integer y-intercept *Identify integer gradient and y-intercept *Identify fractional gradient *Identify fractional y-intercept *Identify fractional gradient and y-intercept Find x-intercept from y = mx + c *Identify integer x-intercept *Identify fractional x-intercept Converting between y = mx + c and ax + by + c = 0 *Rewrite integer y=mx + c as ax + by + c = 0 *Rewrite fractional y=mx + c as ax + by + c = 0 *Rewrite ax + by + c = 0 as integer y=mx + c *Rewrite ax + by + c = 0 as fractional y=mx + c Gradient and y-intercept from ax + by + c = 0 *Identify integer gradient *Identify integer y-intercept *Identify integer gradient and y-intercept *Identify fractional gradient *Identify fractional y-intercept *Identify fractional gradient and y-intercept Find x-intercept from ax + by + c = 0 *Identify integer x-intercept *Identify fractional x-intercept Find the equation of a linear graph Find equation given gradient m and one point *Integer m; y = mx + c *Integer m; ax + by + c = 0 *Fractional m; y = mx + c *Fractional m; ax + by + c = 0 Find equation given two points *Integer m; y = mx + c *Integer m; ax + by + c = 0 *Fractional m; y = mx + c *Fractional m; ax + by + c = 0 Parallel and perpendicular lines Test if lines are parallel, perpendicular or neither *Parallel or not: y = mx + c, y = nx + d *Parallel or not: ax + by + c = 0, dx + ey + f = 0 *Parallel or not: y = mx + c, ax + by + c = 0 *Parallel or not: misc equations *Perpendicular or not: y = mx + c, y = nx + d *Perpendicular or not: ax + by + c = 0, dx + ey + f = 0 *Perpendicular or not: y = mx + c, ax + by + c = 0 *Perpendicular or not: misc equations *Parallel, perpendicular or neither: y = mx + c, y = nx + d *Parallel, perpendicular or neither: ax + by + c = 0, dx + ey + f = 0 *Parallel, perpendicular or neither: y = mx + c, ax + by + c = 0 *Parallel, perpendicular or neither: misc equations

Give the equation of a parallel or perpendicular line through a point *Parallel line as y = mx + c *Parallel line as ax + by + c = 0 *Perpendicular line as y = mx + c *Perpendicular line as ax + by + c = 0 *Parallel/perpendicular line as y = mx + c *Parallel/perpendicular line as ax + by + c = 0 Find the perpendicular bisector of a line segment *Perpendicular bisector as y = mx + c *Perpendicular bisector as ax + by + c = 0

PURE: CIRCLES Midpoints and perpendicular bisectors Find the midpoint of a line segment *Midpoint of points in +ve quadrant *Midpoint of points in any quadrant Find the perpendicular bisector of a line segment *Perpendicular bisector as y = mx + c *Perpendicular bisector as ax + by + c = 0 Circle equation Give the equation of a circle *Centre O, radius r *Centre (a, b), radius r *Centre O passing through given point *Centre (a, b) passing through given point *With given diameter Give information about a circle *Radius of a circle centre O *Radius (integer) and centre of a circle *Radius (surd) and centre of a circle *Radius (integer) and centre by completing the square *Radius (surd) and centre by completing the square Circle sketch graphs Sketch a circle from its equation *Completed square form *By completing the square Give the equation of a circle from a sketch *Completed square form *By completing the square Equation of radius and tangent Radius of a circle *Gradient of radius, centre O *Equation of radius, centre O *Gradient of radius, centre (a, b) *Equation of radius, centre (a, b) Tangent to a circle *Gradient of tangent, centre O *Equation of tangent, centre O *Gradient of tangent, centre (a, b)

*Equation of tangent, centre (a, b) Intersections of straight lines and circles Solve intersection of circle with x=k, y=k *Circle and x- or y-axis *Circle and y=k, x=k Number of intersections of circle and line *Circle and y = mx + c *Circle and ax + by + c = 0 Solve intersection of circle with line *Circle and y = mx + c, 2 solutions *Circle and ax + by + c = 0, 2 solutions *Circle and y = mx + c, 1 solution *Circle and ax + by + c = 0, 1 solution *Circle and y = mx + c, 1 or 2 solutions *Circle and ax + by + c = 0, 1 or 2 solutions Solve for K: line intersects circle 0/1/2 times *ax + by + c = 0 and (x-p)² + (y-q)² = K *ax + by + c = 0 and (x-K)² + (y-q)² = r² *ax + by + c = 0 and (x-p)² + (y-K)² = r² *ax + by + K = 0 and (x-p)² + (y-q)² = r² *y = Kx and (x-p)² + (y-q)² = r² *Misc

PURE: ALGEBRAIC METHODS Algebraic fractions Numeric common factor *2x/6, etc. *2(x+2)/6, etc. *(2x+4)/6, etc. *(2x+4)/6x, etc. *(2x+4)/(6x-2), etc. *Misc Algebraic common factor *x(x+2)/6x, etc. *(x²+2x)/6x, etc. *x(x+2)/x(x-3), etc. *(x²+2x)/(x²-3x), etc. *(2x²+4x)/(4x²-12x), etc. *Misc 1 *(x+2)(x+1)/(x+2), etc. *(x²+3x+2)/(x+2), etc. *(x²-4)/(x±2), etc. *(x²-4)/(x²±2x), etc. *(x+2)(x+1)/(x+3)(x+2), etc. *(x²+3x+2)/(x²+5x+6), etc. *Misc 2 *(2x²-5x-3)/(2x-3)(x-3), etc. *(2x²-5x-3)/(2x²-9x+9), etc. Polynomial arithmetic Polynomial basics

*Order *Add cubics *Subtract cubics Multiply polynomials *Multiply linear by quadratic *Multiply linear by cubic *Multiply quadratic by quadratic *Multiply quadratic by cubic *Multiply misc. Divide polynomials *Divide quadratic by x + 3, etc. *Divide quadratic by 3x - 2, etc. *Divide cubic by x + 3, etc. *Divide cubic by 3x - 2, etc. *Divide quartic by x + 3, etc. *Divide quartic by 3x - 2, etc. *Divide quintic by x + 3, etc. *Divide quintic by 3x - 2, etc. Factor and remainder theorems Factor theorem *Is linear a factor of cubic? *Factorise cubic into linears *Solve for one unknown *Solve for two unknowns Remainder theorem *Remainder when cubic is divided by linear Mathematical proof Algebra toolkit for proof *Specify even, odd numbers, multiples, etc. [10] *Specify consecutives, even/odd squares, etc. [12] *Specify one more than/twice a square number, etc. [10] *Find simple sums and products, etc. [11] *Find sums/differences of squares, cubes, etc. [10] *How to prove even, odd, remainder, positive, etc. [13] Find a counterexample *Primes, factors, inequalities [11] *Formulae for primes, squares, cubes [11] *Quadrilateral facts [10] Algebraic proof *Combining evens and odds [14] *Multiples and remainders [15] *Primes, factors, inequalities [11] *Formulae for primes, squares, cubes [12] *Quadrilateral facts [10] *Algebraic exhaustion, factorials [8] *Completing the square *Misc proof True or false? *Primes, factors, inequalities

*Formulae for primes, squares, cubes *Quadrilateral facts

PURE: THE BINOMIAL EXPANSION Pascal's triangle, factorials and nCr Pascal's triangle *Complete the given row [5] *Find the next two values in the given row [10] Evaluate factorial expressions *Evaluate n! *Evaluate m!/n! *Evaluate algebraic n!/(n-k)! or (n+k)!/n! Evaluate nCr *Evaluate small numerical nCr *Evaluate larger numerical nCr *Evaluate algebraic nCr [8] Binomial expansions (1 ± bx)ⁿ *Expand (1 + x)ⁿ *Expand (1 + bx)ⁿ *Expand (1 - bx)ⁿ *Expand (1 ± bx)ⁿ *Expand (1 + x²)ⁿ, etc. (a ± bx)ⁿ *Expand (a + bx)ⁿ *Expand (a - bx)ⁿ *Expand (a ± bx)ⁿ (x ± by)ⁿ *Expand (x + y)ⁿ *Expand (x + by)ⁿ *Expand (x - by)ⁿ *Expand (x ± by)ⁿ (ax ± by)ⁿ *Expand (ax + by)ⁿ *Expand (ax - by)ⁿ *Expand (ax ± by)ⁿ

PURE: TRIGONOMETRIC RATIOS Non-right-angled triangles Find a missing side or angle *Sine Rule - find Side *Sine Rule - find Angle *Sine Rule - find Misc *Cosine Rule - find Side *Cosine Rule - find Angle *Cosine Rule - find Misc

*Sine/Cosine Rule - find Side *Sine/Cosine Rule - find Angle *Sine/Cosine Rule - find Misc Find the area *Find area of triangle Trigonometric graphs Sketch a graph *y=sinx, cosx, tanx [3] *y=cosecx, secx, cotx [3] *Trig graph [6] *y=sin⁻¹x, cos⁻¹x, tan⁻¹x [3]

PURE: TRIG EQUATIONS Exact values of sin/cos/tan 0 to 90° *sin 0, 30, 45, 60, 90° [5] *cos 0, 30, 45, 60, 90° [5] *tan 0, 30, 45, 60, 90° [5] *sin/cos/tan 0, 30, 45, 60, 90° 120 to 360° *sin 120-360° [12] *cos 120-360° [12] *tan 120-360° [12] *sin/cos/tan 120-360° 0 to -360° *sin 0 to -360° [17] *cos 0 to -360° [17] *tan 0 to -360° [17] *sin/cos/tan 0 to -360° Solve trig equations: exact solutions sin/cos/tan x = c *Solve sin x = c *Solve cos x = c *Solve tan x = c *Solve sin/cos/tan x = c sin/cos/tan kx = c *Solve sin kx = c *Solve cos kx = c *Solve tan kx = c *Solve sin/cos/tan kx = c

PURE: DIFFERENTIATION Differentiation basics Definitions *Definition to term [5] *Term to definition [5] Differentiation by rule *y = xⁿ *y = Axⁿ *y = Ax⁻ⁿ *y = A/xⁿ

*y = Axⁿ, y = Ax⁻ⁿ or y = A/xⁿ *y = 5x³ - 3x² + 6, etc. *y = 5x³ - 3/x², etc. Applying differentiation Gradient at a point *y = 5x³, etc. *y = 4/x, etc. *y = 5x³, y = 4/x, etc. *y = 5x³ - 3x² + 6, etc. *y = 5x³ - 3/x², etc. Solve gradient = m *y = 5x³, etc. *y = 4/x, etc. *y = 5x³, y = 4/x, etc. Find maximum/minimum points *Quadratics *Cubics *Quadratics/cubics Kinematics *Find v given x=at³+bt²+ct+d *Find a given x=at³+bt²+ct+d *Find a given v=at³+bt²+ct+d *Find v or a given x or v *Find v at t=k given x=at³+bt²+ct+d *Find a at t=k given x=at³+bt²+ct+d *Find a at t=k given v=at³+bt²+ct+d *Find v or a at t=k given x or v Chain rule: linear inner function Polynomials *y = (2x + 7)³ *y = 4(2x + 7)³ *y = (2x + 7)⁻³ *y = 4(2x + 7)⁻³ *y = 1/(2x + 7)³ *y = 4/(2x + 7)³ Exponentials *y = e³ˣ, etc. *y = 4e³ˣ, etc. Trig functions *y = 4sin(3x), etc. *y = 4cos(3x), etc. *y = 4tan(3x), etc. *Misc trig function

PURE: INTEGRATION Indefinite integration Axⁿ Given dy/dx, find y *x³, etc. *Ax³, etc. *Ax⁻³, etc. *Ax³ or Ax⁻³, etc. *A/x³, etc.

*Ax¹ᴵ³, etc. *A³√x, etc. *Ax⁻¹ᴵ³, etc. *5x³ - 3x² + 6, etc. *5x³ - 3/x², etc. Integrate *x³, etc. *Ax³, etc. *Ax⁻³, etc. *Ax³ or Ax⁻³, etc. *A/x³, etc. *Ax¹ᴵ³, etc. *A³√x, etc. *Ax⁻¹ᴵ³, etc. *5x³ - 3x² + 6, etc. *5x³ - 3/x², etc. *x³(3x² + 4), etc. *(2x - 3)(4x + 1), etc. *(2x - x³)(3x² + 6), etc. *Misc product *(3x³ + 4x²)/x, etc. Definite integration Axⁿ Evaluate with +ve limits *x³, etc. *Ax³, etc. *Ax⁻³, etc. *Ax³ or Ax⁻³, etc. *A/x³, etc. *Ax¹ᴵ³, etc. *A³√x, etc. *Ax⁻¹ᴵ³, etc. *5x³ - 3x² + 6, etc. *5x³ - 3/x², etc. *x³(3x² + 4), etc. *(2x - 3)(4x + 1), etc. *(2x - x³)(3x² + 6), etc. *Misc product *(3x³ + 4x²)/x, etc. Evaluate with ±ve limits *x³, etc. *Ax³, etc. *Ax⁻³, etc. *Ax³ or Ax⁻³, etc. *A/x³, etc. *x³(3x² + 4), etc. *(2x - 3)(4x + 1), etc. *(2x - x³)(3x² + 6), etc. *Misc product *(3x³ + 4x²)/x, etc.

PURE: EXPONENTIALS and LOGARITHMS Basics of logarithms Rewrite logs and powers *3² = 9, etc. to log form *log₄ 64 = 3, etc. to power form Evaluate logs *log₃ 81, etc. *log₃ 1/81, etc. *log₃ √3, etc. *log₁ₗ₄ 16, etc. *Misc exact logs *logₙ n³, logₙ 1/√n, etc. *Calculator: log₃ 4, etc. Log laws: numeric combine *log₃ 6 + log₃ 8 *log₃ 2 + log₃ 4 + log₃ 5 *log₃ 8 - log₃ 4 *log₃ 8 + log₃ 5 - log₃ 2 *Misc log add/subtract *2log₃ 4 *Misc basic log laws *3log₃ 4 + log₃ 2 *3log₃ 4 - log₃ 2 *3log₃ 4 ± log₃ 2 *Misc combining logs Log laws: algebraic combine *2log₁₀ x + log₁₀ y - 3log₁₀ z *2log₁₀ x + log₁₀ y - 3log₁₀ z + 4 *2log₁₀ x + ½log₁₀ y - 3log₁₀ z *2log₁₀ x + ½log₁₀ y - 3log₁₀ z - 4 *2logₙ x + logₙ y - 3logₙ z *2logₙ x + logₙ y - 3logₙ z + 4 *2logₙ x + ½logₙ y - 3logₙ z *2logₙ x + ½logₙ y - 3logₙ z - 4 Log laws: split *log₁₀ (x²y/z³) *log₁₀ (1000x²y/z³) *log₁₀ (x²√y/z³) *log₁₀ (x²√y/1000z) *logₙ (x²y/z³) *logₙ (n³x²y/z³) *logₙ (x²√y/z³) *logₙ (x²√y/n³z) Equations with logarithms Solve log equations *log₃ x = 4, etc. *log₁ₗ₃ x = 4, etc. *log₃ x = 4, log₁ₗ₃ x = 4, etc. *logₓ 81 = 2, etc. Solve equations by taking logs

*3ˣ = 7, etc. *2³ˣ⁺⁴ = 7, etc. Exponentials Differentiate *y = e³ˣ *y = 4e³ˣ

PURE: FURTHER MATHS FM Matrices Basic operations (2x2) *A + B *A - B *kA *kA + lB *kA - lB *AB *A² *Det(A) *Is A singular / Det(A)=0 ? *A⁻¹ Solve matrix and vector equations (2x2) *Solve AX = B for matrix X *Solve XA = B for matrix X Basic operations (3x3) *A + B *A - B *kA *kA + lB *kA - lB *AB *A² *Det(A) *Is A singular / Det(A)=0 ? *Cofactors of A *Transpose of cofactors of A *A⁻¹ FM Complex Numbers Basic operations: a+bi *Real part of a complex number *Imaginary part of a complex number *Add two complex numbers *Subtract two complex numbers *Multiply two complex numbers *Square a complex number *Conjugate of a complex number *Multiply a complex number by its conjugate *Reciprocal of a complex number *Divide two complex numbers *Modulus of a complex number *Argument of a complex number Quadratics with complex roots

*Solve z² + c = 0 (integer) *Solve z² + c = 0 (surd) *Solve z² = a + bi (complex z) *Solve z² + bz + c = 0 (conjugate pair) *Solve z² + bz + c = 0 (non conjugate pair) FM Vectors Scalar (dot) product *Scalar product of two vectors *Are two vectors perpendicular? *Is angle ABC a right angle? *Magnitude of a vector *Angle between two vectors *Acute angle between two vectors *Angle ABC *Make a perpendicular vector Vector (cross) product *Vector product of two vectors *Unit normal vector Area and Volume using vectors *Area of parallelogram given 2 sides *Area of parallelogram given 4 points *Area of triangle given 2 sides *Area of triangle given 3 points *Volume of parallelepiped given 3 edges *Volume of tetrahedron given 3 edges *Volume of tetrahedron given 4 points Lines in 3D *Vector to Cartesian equation of a line (standard) *Cartesian to vector equation of a line (standard) *Vector to Cartesian equation of a line (harder) *Cartesian to vector equation of a line (harder) *Vector equation of a line through 2 points *Cartesian equation of a line through 2 points *Acute angle between two lines (vector/vector) *Acute angle between two lines (vector/Cartesian) *Acute angle between two lines (Cartesian/Cartesian) *Does a point lie on a line (vector)? *Does a point lie on a line (Cartesian)? *Do three points lie on the same line? *Perpendicular distance from a point to a line (vector) *Perpendicular distance from a point to a line (Cartesian) *Two lines (vector/vector): intersect/skew/parallel? *Two lines (vector/Cartesian): intersect/skew/parallel? *Two lines (Cartesian/Cartesian): intersect/skew/parallel? *Shortest distance between two lines (vector) *Shortest distance between two lines (Cartesian)

Planes in 3D *Vector to Cartesian equation of a plane *Vector equation of a plane through 3 points *Cartesian equation of a plane through 3 points *Vector equation of a plane containing a point and a line *Cartesian equation of a plane containing a point and a line *Acute angle between two planes (Cartesian) *Acute angle between a plane (Cartesian) and a line *Perpendicular distance of a plane (Cartesian) from the origin *Perpendicular distance of a plane (Cartesian) from a point P *Does a point lie in a plane (Cartesian)? *Do four points lie in the same plane? *Vector line of intersection of two planes (Cartesian) *Point of intersection of a line (vector) with a plane (Cartesian) *Point of intersection of a line (Cartesian) with a plane (Cartesian)

A LEVEL STATISTICS Data Collection Types of data *Qualitative and Quantitative *Discrete and Continuous Sampling Methods *Census and Sample [9] *Define sampling methods [5] *Name sampling methods [5] *Advantages and disadvantages [10] Median, Quartiles from raw data List of integers *Median *Lower Quartile *Upper Quartile *IQR *Median/quartiles List of decimals *Median *Lower Quartile *Upper Quartile *IQR *Median/quartiles Outliers *Check one value using 1.5×IQR: integer *Check one value using 1.5×IQR: decimal *Find outliers using 1.5×IQR: integer list *Find outliers using 1.5×IQR: decimal list *Find extreme outliers using 3×IQR: integer list *Find extreme outliers using 3×IQR: decimal list Box Plots *Find median *Find quartiles *Find IQR *Find range *Find misc *Draw from median, LQ, UQ, min, max *Draw from median, LQ, IQR, min, max *Draw from median, LQ, UQ, min, range *Draw from median, IQR, UQ, range, max *Draw from misc *Draw including an outlier *Compare two box plots Discrete frequency table *Median *Lower Quartile *Upper Quartile *IQR *Median/quartiles Grouped frequency table interpolate *Median

*Lower Quartile *Upper Quartile *IQR *Median/quartiles Grouped frequency interpolate with rounding *Median *Lower Quartile *Upper Quartile *IQR *Median/quartiles Mean, Variance, SD List of integers *Mean *Variance *Standard Deviation *Mean/var/SD *Variance/SD *∑ x *∑ x² List of decimals *Mean *Variance *Standard Deviation *Mean/var/SD *Variance/SD *∑ x *∑ x² Outliers *Check one value using 2×SD: integer *Check one value using 2×SD: decimal *Find outliers using 2×SD: integer list *Find outliers using 2×SD: decimal list Discrete frequency table *Mean *Variance *Standard Deviation *Mean/var/SD *Variance/SD *∑ fx *∑ fx² Grouped frequency table *Mean *Variance *Standard Deviation *Mean/var/SD *Variance/SD *∑ fx *∑ fx² Grouped frequency with rounding *Mean *Variance *Standard Deviation *Mean/var/SD

*Variance/SD *∑ fx *∑ fx² Using ∑ x, ∑ x² *Find mean *Find variance *Find SD *Find variance from mean, ∑ x² *Find SD from mean, ∑ x² *Find mean from variance, ∑ x² *Find mean from SD, ∑ x² *Find n from mean, variance, ∑ x² *Find n from variance, ∑ x, ∑ x² *Find misc quantity Using ∑ fx, ∑ fx² *Find mean *Find variance *Find SD *Find variance from mean, ∑ fx² *Find SD from mean, ∑ fx² *Find mean from variance, ∑ fx² *Find mean from SD, ∑ fx² *Find n from mean, variance, ∑ fx² *Find n from variance, ∑ fx, ∑ fx² *Find misc quantity Adding an extra data point *Find new mean using ∑ x *Find new mean using old mean *Find value to make target mean *Find new variance using ∑ x, ∑ x² *Find new variance using old mean, variance *Find new SD using ∑ x, ∑ x² *Find new SD using old mean, SD *Find misc quantity Combining ∑ fx, ∑ fx² *Find new mean using ∑ x *Find new mean using old means *Find new variance using ∑ x, ∑ x² *Find new variance using old means, variances *Find new SD using ∑ x, ∑ x² *Find new SD using old means, SDs *Find misc quantity Coding: effect of transformations *Translation and the mean *Translation and the variance *Translation and the SD *Translation and misc quantity *Scaling and the mean *Scaling and the variance *Scaling and the SD *Scaling and misc quantity *Translation & scaling and the mean *Translation & scaling and the variance

*Translation & scaling and the SD *Translation & scaling and misc quantity Coding: y=(x-a)/b: x to y *Code data values *Find coded mean *Find coded variance *Find coded standard deviation *Find misc coded quantity Coding: y=(x-a)/b: y to x *Uncode data values *Find uncoded mean *Find uncoded variance *Find uncoded standard deviation *Find misc uncoded quantity Binomial probability Rewrite using P(X = k) *P(X < k) *P(X > k) *P(X = k) *P(X </>/= k) *P(k1 < X < k2) *P(k1 = X < k2) *P(k1 < X = k2) *P(k1 = X = k2) *P(k1 </= X </= k2) *Misc Find a probability from B(N, p) *P(X = k) *P(X < k) *P(X = k) *P(X > k) *P(X = k) *P(X </=/>/= k) *P(m < X < n) *P(m = X < n) *P(m < X = n) *P(m = X = n) *P(m </= X </= n) *Misc cumulative Find a probability from N & p *Prob of k successes *Prob of < k successes *Prob of = k successes *Prob of > k successes *Prob of = k successes *Prob of </=/>/= k successes *Prob of m < X < n successes *Prob of m = X < n successes *Prob of m < X = n successes *Prob of m = X = n successes *Prob of m </= X </= n successes *Misc cumulative Find a probability from N & q

*Prob of k successes *Prob of < k successes *Prob of = k successes *Prob of > k successes *Prob of = k successes *Prob of </=/>/= k successes *Prob of m < X < n successes *Prob of m = X < n successes *Prob of m < X = n successes *Prob of m = X = n successes *Prob of m </= X </= n successes *Misc cumulative Find a probability (N biased coins) *Prob of k Heads *Prob of < k Heads *Prob of = k Heads *Prob of > k Heads *Prob of = k Heads *Prob of </=/>/= k Heads *Prob of m < X < n Heads *Prob of m = X < n Heads *Prob of m < X = n Heads *Prob of m = X = n Heads *Prob of m </= X </= n Heads *Misc cumulative Find a probability (N 6-sided dice) *Prob of k successes *Prob of < k successes *Prob of = k successes *Prob of > k successes *Prob of = k successes *Prob of </=/>/= k successes *Prob of m < X < n successes *Prob of m = X < n successes *Prob of m < X = n successes *Prob of m = X = n successes *Prob of m </= X </= n successes *Misc cumulative Find a probability (N 10-sided dice) *Prob of k successes *Prob of < k successes *Prob of = k successes *Prob of > k successes *Prob of = k successes *Prob of </=/>/= k successes *Prob of m < X < n successes *Prob of m = X < n successes *Prob of m < X = n successes *Prob of m = X = n successes *Prob of m </= X </= n successes *Misc cumulative Binomial hypothesis testing Find the critical region

*1-tailed test; lower tail *1-tailed test; upper tail *2-tailed test; below sig level *2-tailed test; closest to sig level *Misc 1- or 2-tailed test Find the acceptance region *1-tailed test; lower tail *1-tailed test; upper tail *2-tailed test; below sig level *2-tailed test; closest to sig level *Misc 1- or 2-tailed test Is value in critical region? *1-tailed test; lower tail *1-tailed test; upper tail *2-tailed test; below sig level *2-tailed test; closest to sig level *Misc 1- or 2-tailed test Do full hypothesis test *1-tailed test; lower tail *1-tailed test; upper tail *2-tailed test; below sig level *2-tailed test; closest to sig level *Misc 1- or 2-tailed test Find actual significance level *1-tailed test; lower tail *1-tailed test; upper tail *2-tailed test; below sig level *2-tailed test; closest to sig level *Misc 1- or 2-tailed test Normal Distribution Basics *Features of a Normal distribution [6] *Recall 'famous' values ±s, ±2s, ±3s Standard Normal (Z) Distribution N(0, 1) *Find P(Z = k) *Find P(Z = k) *Find P(m = Z = n) *Find misc P General Normal Distribution N(µ, s²) *Find P(Z = k) *Find P(Z = k) *Find P(m = Z = n) *Find misc P *Wordy P(Z = k) *Wordy P(Z = k) *Wordy P(m = Z = n) *Wordy P(within k of µ) *Find misc wordy P Inverse Standard Normal N(0, 1) *Solve P(Z = ?) = k *Solve P(Z = ?) = k *Solve P(m = Z = ?) = k *Solve P(? = Z = n) = k

*Solve P(-? = Z = ?) = k *Solve misc P Inverse General Normal N(µ, s²) *Solve P(Z = ?) = k *Solve P(Z = ?) = k *Solve P(m = Z = ?) = k *Solve P(? = Z = n) = k *Solve misc P *Wordy solve P(Z = ?) = k *Wordy solve P(Z = ?) = k *Wordy solve P(m = Z = ?) = k *Wordy solve P(? = Z = n) = k *Solve misc wordy P Find values of µ, s *Find µ given P(X = k) *Find µ given P(X = k) *Find s given P(X = k) *Find s given P(X = k) *Find misc µ, s *Find µ & s given P(X = m), P(X = n) *Find µ & s given P(X = m), P(X = n) *Find misc µ & s *Wordy find µ given P(X = k) *Wordy find µ given P(X = k) *Wordy find s given P(X = k) *Wordy find s given P(X = k) *Find misc wordy µ, s *Wordy find µ & s given P(X = m), P(X = n) *Wordy find µ & s given P(X = m), P(X = n) *Find misc wordy µ & s Normal hypothesis tests of a population mean Find the critical region *1-tailed test; lower tail *1-tailed test; upper tail *2-tailed test *Misc 1- or 2-tailed test Find the acceptance region *1-tailed test; lower tail *1-tailed test; upper tail *2-tailed test *Misc 1- or 2-tailed test Is value in critical region? *1-tailed test; lower tail *1-tailed test; upper tail *2-tailed test *Misc 1- or 2-tailed test Do full hypothesis test *1-tailed test; lower tail *1-tailed test; upper tail *2-tailed test *Misc 1- or 2-tailed test Normal approximation to Binomial Basics

*Is the Normal approximation valid? *Find mean and variance from B(n, p) Continuity corrections *Binomial P(X = b) to Normal *Binomial P(X = b) to Normal *Binomial P(X < b) to Normal *Binomial P(X = b) to Normal *Binomial P(X > b) to Normal *Misc Binomial to Normal (1) *Binomial P(a = X = b) to Normal *Binomial P(a = X < b) to Normal *Binomial P(a < X = b) to Normal *Binomial P(a < X < b) to Normal *Misc Binomial to Normal (2) Find probabilities *Find Binomial P(X = b) using Normal *Find Binomial P(X = b) using Normal *Find Binomial P(X < b) using Normal *Find Binomial P(X = b) using Normal *Find Binomial P(X > b) using Normal *Find misc Binomial using Normal (1) *Find Binomial P(a = X = b) using Normal *Find Binomial P(a = X < b) using Normal *Find Binomial P(a < X = b) using Normal *Find Binomial P(a < X < b) using Normal *Find misc Binomial using Normal (2) Correlation Definitions *Correlation terms [8] *Regression terms [8] Scatter diagrams & PMCC *Describe type of correlation *Estimate the PMCC *Calculate PMCC from table Hypothesis testing with PMCC *1-tailed +ve: find critical region *1-tailed -ve: find critical region *2-tailed: find critical region *Misc find critical region *1-tailed +ve: is value in critical region? *1-tailed -ve: is value in critical region? *2-tailed: is value in critical region? *Misc is value in critical region? *1-tailed +ve: hypothesis test *1-tailed -ve: hypothesis test *2-tailed: hypothesis test *Misc hypothesis test Coding exponential models *y = a xn from linear *y = k bx from linear *y = a xn or y = k bx from linear *y = a xn from gradient & y-int *y = k bx from gradient & y-int

*y = a xn or y = k bx from gradient & y-int *y = a xn to linear *y = k bx to linear *y = a xn or y = k bx to linear Probability: Venn diagrams Interpret *2-Venn: find P(A and not B), etc. [10] *3-Euler: find P(A and not B), etc. [24] *3-Venn: find P(A and not B), etc. [24] *2-Venn: find P(A ∩ B'), etc. [10] *3-Euler: find P(A ∩ B'), etc. [24] *3-Venn: find P(A ∩ B'), etc. [24] Fill in frequencies *2-Venn, eg n(A or B) *3-Euler, eg n(A or B) *3-Venn, eg n(A or B) *Misc Venn, eg n(A or B) *2-Venn, eg n(A U B) *3-Euler, eg n(A U B) *3-Venn, eg n(A U B) *Misc Venn, eg n(A U B) *2-Venn wordy *3-Euler wordy *3-Venn wordy *Misc Venn wordy Fill in probabilities *2-Venn, eg p(A or B) *3-Euler, eg p(A or B) *3-Venn, eg p(A or B) *Misc Venn, eg p(A or B) *2-Venn, eg p(A U B) *3-Euler, eg p(A U B) *3-Venn, eg p(A U B) *Misc Venn, eg p(A U B) Independence: all values given *2-Venn: are A and B independent? *2-Venn: are A' and B independent? *2-Venn: are A and B' independent? *2-Venn: are A' and B' independent? *2-Venn: are A/A' and B/B' independent? *3-Venn: are A and B independent? *3-Venn: are A' and B independent? *3-Venn: are A and B' independent? *3-Venn: are A' and B' independent? *3-Venn: are A/A' and B/B' independent? Independence: one blank value *2-Venn: are A and B independent? *2-Venn: are A' and B independent? *2-Venn: are A and B' independent? *2-Venn: are A' and B' independent? *2-Venn: are A/A' and B/B' independent? *3-Venn: are A and B independent? *3-Venn: are A' and B independent?

*3-Venn: are A and B' independent? *3-Venn: are A' and B' independent? *3-Venn: are A/A' and B/B' independent? Find x if independent *2-Venn: one solution 1 *2-Venn: one solution 2 *2-Venn: misc one solution *2-Venn: two solutions 1 *2-Venn: two solutions 2 *2-Venn: misc two solutions *2-Venn: misc 1/2 solutions Find x if mutually exclusive *2-Venn: one solution 1 *2-Venn: one solution 2 *2-Venn: misc one solution Conditional probability *2-Venn: find P(A | A U B), etc. [16] *3-Euler: find P(A | A U B), etc. [32] *3-Venn: find P(A | A U B), etc. [32] Probability: two-way tables Interpret *2-way table: find P(A and not B), etc. [10] *2-way table: find P(A ∩ B'), etc. [10] Fill in frequencies *2-way table, eg n(A or B) *2-way table, eg n(A U B) *2-way table wordy Fill in probabilities *2-way table, eg p(A or B) *2-way table, eg p(A U B) Independence: all values given *2-way table: are A and B independent? *2-way table: are A' and B independent? *2-way table: are A and B' independent? *2-way table: are A' and B' independent? *2-way table: are A/A' and B/B' independent? Independence: one blank value *2-way table: are A and B independent? *2-way table: are A' and B independent? *2-way table: are A and B' independent? *2-way table: are A' and B' independent? *2-way table: are A/A' and B/B' independent? Find x if independent *2-way table: one solution 1 *2-way table: one solution 2 *2-way table: misc one solution *2-way table: two solutions 1 *2-way table: two solutions 2 *2-way table: misc two solutions *2-way table: misc 1/2 solutions Find x if mutually exclusive *2-way table: one solution 1 *2-way table: one solution 2

*2-way table: misc one solution Conditional probability *2-way table: find P(A | A U B), etc. [16]

NETWORKS/DECISION Networks Minimum Spanning Tree *Prim *Kruskal *Matrix Prim Shortest Route *Dijkstra Route Inspection *Best pairing of 4 odd nodes *Route Inspection - 4 odd nodes Travelling Salesperson *Nearest neighbour (single) *Lower bound (single) *Nearest neighbour (all nodes) *Lower bound (all nodes) Algorithms Packing *First fit *First fit decreasing *Full bin Sorting *Bubble sort *Shuttle sort Order of Algorithms *Find order from efficiency *Calculate using order Linear Programming LP Graphical Solution *Terminology *Definitions *Solving from graph *Solving from LP formulation LP Simplex Method *2-variable simplex *3-variable simplex

c MATHSprint, 2020.Licensed to MATHSprint. Circle Equations:1

Circle EquationsMATHSprint

Answer on file paper

1: 12Work out the following:

The equation of a circlecentre (-5, 2) and radius 12.

a) The equation of a circlecentre (-3, -6) and radius 10.

b)


The equation of a circle centre (-1, 3)passing through the point (3, 9).

a) The equation of a circle centre (-4, 0)passing through the point (-3, 5).

b)


The equation of a circle withdiameter from (-5, 0) to (1, -10).

a) The equation of a circle withdiameter from (-7, 2) to (1, 6).

b)

4: 12Give an equation for each of the following in the form (x - a)2 + (y - b)2 = r2

-10 -5 0 5 10

-10

-5

0

5

10

x

ya)

-10 -5 0 5 10

-10

-5

0

5

10

x

yb)

5: 12Give an equation for each of the following in the form x2 + y2 + f x + gy + c = 0

-10 -5 0 5 10

-10

-5

0

5

10

x

ya)

-10 -5 0 5 10

-10

-5

0

5

10

x

yb)

6: 12Find the co-ordinates of the points where the following intersect:

The circle (x + 4)2 + (y + 3)2 = 13and the x-axis.

a) The circle (x + 1)2 + (y - 1)2 = 10and the y-axis.

b)

7: 12How many times do the following intersect?

The circle (x - 3)2 + (y - 1)2 = 5 and the line x + 2y - 7 = 0.a)

The circle x2 + (y - 4)2 = 20 and the line x - 2y - 2 = 0.b)

8: 12Find the coordinates of the point(s) of intersection of the following:

y = -2x + 6(x + 3)2 + (y - 2)2 = 20

a) y = x + 3(x + 2)2 + (y + 1)2 = 20

b)

9: 12Find the (range of) values of k in each case:

If x - 2y + k = 0 does not intersect (x - 1)2 + y2 = 5.a)

If x + y - 3 = 0 meets (x - k)2 + (y - 3)2 = 8 at exactly one point.b)

c MATHSprint, 2020.Licensed to MATHSprint. Circle Equations:1

Answers: Circle EquationsMATHSprint

1: (x + 5)2 + (y - 2)2 = 144a) (x + 3)2 + (y + 6)2 = 100b)

2: r2 = (3 - (-1))2 + (9 - 3)2 so(x + 1)2 + (y - 3)2 = 52

a) r2 = ((-3) - (-4))2 + (5 - 0)2 so(x + 4)2 + y2 = 26

b)

3: Centre = (1 + (-5)2 , -10 + 0

2 ) = (-2, -5)r2 = (1 - (-5)

2 )2 + (-10 - 02 )2 = 32 + (-5)2 so

(x + 2)2 + (y + 5)2 = 34

a) Centre = (1 + (-7)2 , 6 + 2

2 ) = (-3, 4)r2 = (1 - (-7)

2 )2 + (6 - 22 )2 = 42 + 22 so

(x + 3)2 + (y - 4)2 = 20

b)

4: (x + 3)2 + (y - 3)2 = 9a) (x + 2)2 + (y - 2)2 = 25b)

5: (x + 4)2 + (y - 1)2 = 1x2 + y2 + 8x - 2y + 16 = 0

a) (x - 5)2 + y2 = 4x2 + y2 - 10x + 21 = 0

b)

6: (x + 4)2 + 9 = 13Intersections at (-6, 0) and (-2, 0).

a) 1 + (y - 1)2 = 10Intersections at (0, -2) and (0, 4).

b)

7: x = -2y + 7(-2y + 4)2 + (y - 1)2 = 55y2 - 18y + 12 = 0Discriminant is(-18)2 - 4 ´ 5 ´ 12 = 842 intersections

a) x = 2y + 2(2y + 2)2 + (y - 4)2 = 205y2 = 0Discriminant is02 - 4 ´ 5 ´ 0 = 01 intersection

b)

8: y = -2x + 6(x + 3)2 + (-2x + 4)2 = 205x2 - 10x + 5 = 0x2 - 2x + 1 = 0(x - 1)2 = 0(1, 4).

a) y = x + 3(x + 2)2 + (x + 4)2 = 202x2 + 12x = 0x2 + 6x = 0x(x + 6) = 0(-6, -3) and (0, 3).

b)

9: x = 2y - k(2y - k - 1)2 + (y)2 + = 55y2 + (-4k - 4)y + (k2 + 2k - 4) = 0Discriminant is:(-4k - 4)2 - 4 ´ 5 ´ (k2 + 2k - 4) = -4k2 - 8k + 96 < 0k2 + 2k - 24 > 0(k + 6)(k - 4) > 0k < -6 or k > 4.

a) y = -x + 3(x - k)2 + (-x)2 = 82x2 + (0 - 2k)x + k2 - 8 = 0Discriminant is:(0 - 2k)2 - 4 ´ 2 ´ (k2 - 8) = -4k2 + 64 = 0k2 - 16 = 0(k + 4)(k - 4) = 0k = -4 or k = 4.

b)

c MATHSprint, 2020.Licensed to MATHSprint. Exact Trig Values:1

Exact Trig ValuesMATHSprint

Answer on file paper

1: 12Give the following as exact values:

tan 90°a) cos 45°b) sin 45°c) tan 0°d)

cos 0°e) sin 30°f) sin 60°g) cos 90°h)


sin 120°a) tan 180°b) cos 135°c) tan 135°d)

sin 225°e) cos 360°f) cos 330°g) sin 150°h)


cos(-150°)a) tan(-60°)b) sin(-225°)c) sin(-270°)d)

cos(-360°)e) tan(-30°)f) cos(-240°)g) tan(-210°)h)

4: 12Solve the following equations for q in the interval 0 £ q < 360°:

tan q = -Ö3a) cos q = 12b) sin q = 1

Ö2c) tan q = - 1Ö3d)

5: 12Solve the following equations for q in the interval 0 £ q < 360°:

tan 2q = 1a) sin 3q = 1b) cos 4q = 0c) sin 5q = Ö32d)

c MATHSprint, 2020.Licensed to MATHSprint. Exact Trig Values:1

Answers: Exact Trig ValuesMATHSprint

1: tan 90° = ¥a) cos 45° = 1Ö2b) sin 45° = 1

Ö2c) tan 0° = 0d)

cos 0° = 1e) sin 30° = 12f) sin 60° = Ö3

2g) cos 90° = 0h)

2: sin 120° = Ö32a) tan 180° = 0b) cos 135° = -1

Ö2c) tan 135° = -1d)

sin 225° = -1Ö2e) cos 360° = 1f) cos 330° = Ö3

2g) sin 150° = 12h)

3: cos(-150°) = -Ö32a) tan(-60°) = -Ö3b) sin(-225°) = 1

Ö2c) sin(-270°) = 1d)

cos(-360°) = 1e) tan(-30°) = -1Ö3f) cos(-240°) = -1

2g) tan(-210°) = -1Ö3h)

4: tan q = -Ö3tan-1(-Ö3) = -60°q = 180 - 60 = 120°q = 360 - 60 = 300°

a) cos q = 12

cos-1(12) = 60°

q = 60°q = 360 - 60 = 300°

b)

sin q = 1Ö2

sin-1( 1Ö2) = 45°

q = 45°q = 180 - 45 = 135°

c) tan q = - 1Ö3

tan-1(- 1Ö3) = -30°

q = 180 - 30 = 150°q = 360 - 30 = 330°

d)

5: tan 2q = 1, 0 £ q < 360°Let X = 2q, 0 £ X < 720°tan-1(1) = 45°X = (45+360n)°, X = (225+360n)°q = (22.5+180n)°, q = (112.5+180n)°q = 22.5°, 202.5°q = 112.5°, 292.5°

a) sin 3q = 1, 0 £ q < 360°Let X = 3q, 0 £ X < 1080°sin-1(1) = 90°X = (90+360n)°q = (30+120n)°q = 30°, 150°, 270°

b)

cos 4q = 0, 0 £ q < 360°Let X = 4q, 0 £ X < 1440°cos-1(0) = 90°X = (90+360n)°, X = (270+360n)°q = (22.5+90n)°, q = (67.5+90n)°q = 22.5°, 112.5°, 202.5°, 292.5°q = 67.5°, 157.5°, 247.5°, 337.5°

c) sin 5q = Ö32 , 0 £ q < 360°

Let X = 5q, 0 £ X < 1800°

sin-1(Ö32 ) = 60°

X = (60+360n)°, X = (120 + 360n)°q = (12+72n)°, q = (24+72n)°q = 12°, 84°, 156°, 228°, 300°q = 24°, 96°, 168°, 240°, 312°

d)

c MATHSprint, 2020.Licensed to MATHSprint. Integration:1

IntegrationMATHSprint

Please answer in your books

1: 12Find an expression for y when

dydx

is the following:

x-8a) 6x7b)

2: 12Find an expression for y when

dydx

is the following:

53Öxa) 9x2

Öxb)

3: 12Find the following integrals:

ó

õ 3x6 - 7x4 dxa) ó

õ -6x9 + 7x5 - x3 + 10x2 dxb)


ó

õ x4(x3 + 5x) dxa) ó

õ x(-5x5 + 2x2) dxb)


ó

õ 3x7 + 4x6

x4 dxa) ó

õ -3x8 - 4x5

x3 dxb)

6: 12Evaluate the following definite integrals:

ó

õ

1/2

1/440x-5 dxa) ó

õ

3

1-12x3 dxb)


ó

õ

625

1-45

4Öx dxa) ó

õ

125

6440(

3Öx)2 dxb)


ó

õ

5

0-24x5 + 10x4 - 10x + 4 dxa) ó

õ

5

16x5 - 8x3 - 24x2 + 10 dxb)


ó

õ

5

4(3x - 1)(5x + 3) dxa) ó

õ

2

1(5x + 1)(3x - 1) dxb)


ó

õ

5

2

-5x5 - 8x4

x dxa) ó

õ

5

3

15x5 + 6x3

x dxb)

c MATHSprint, 2020.Licensed to MATHSprint. Integration:1

Answers: IntegrationMATHSprint

1: y = óõ

x-8 dx = -17x

-7 + ca) y = óõ

6x7 dx = 34x

8 + cb)

2: y = óõ

5x1/3 dx = 154 x4/3 + ca) y = ó

õ 9x5/2 dx = 18

7 x7/2 + cb)

3: ó

õ 3x6 - 7x4 dx = 3

7x7 - 7

5x5 + ca)

ó

õ -6x9 + 7x5 - x3 + 10x2 dx = -3

5x10 + 7

6x6 - 1

4x4 + 10

3 x3 + cb)

4: ó

õ x7 + 5x5 dx = 1

8x8 + 5

6x6 + ca) ó

õ -5x6 + 2x3 dx = -5

7x7 + 1

2x4 + cb)

5: ó

õ 3x3 + 4x2 dx = 3

4x4 + 4

3x3 + ca) ó

õ -3x5 - 4x2 dx = -1

2x6 - 4

3x3 + cb)

6: ó

õ

1/2

1/440x-5 dx = é

ë-10x-4ù

û

1/2

1/4

= (-160) - (-2560) = 2400

a) ó

õ

3

1-12x3 dx = é

ë-3x4ù

û

3

1

= (-243) - (-3) = -240

b)

7: ó

õ

625

1-45x1/4 dx = é

ë-36x5/4ù

û

625

1

= (-112500) - (-36) = -112464

a) ó

õ

125

6440x2/3 dx = é

ë24x5/3ù

û

125

64

= (75000) - (24576) = 50424

b)

8: ó

õ

5

0-24x5 + 10x4 - 10x + 4 dx = é

ë-4x6 + 2x5 - 5x2 + 4xù

û

5

0

= (-56355) - (0) = -56355

a)

ó

õ

5

16x5 - 8x3 - 24x2 + 10 dx = é

ëx6 - 2x4 - 8x3 + 10xù

û

5

1

= (13425) - (1) = 13424

b)

9: ó

õ

5

415x2 + 4x - 3 dx = é

ë5x3 + 2x2 - 3xù

û

5

4

= (660) - (340) = 320

a) ó

õ

2

115x2 - 2x - 1 dx = é

ë5x3 - x2 - xù

û

2

1

= (34) - (3) = 31

b)

10: ó

õ

5

2-5x4 - 8x3 dx = é

ë-x5 - 2x4ù

û

5

2

= (-4375) - (-64) = -4311

a) ó

õ

5

315x4 + 6x2 dx = é

ë3x5 + 2x3ù

û

5

3

= (9625) - (783) = 8842

b)

c MATHSprint, 2020.Licensed to MATHSprint. Logarithms:2

Name: Class/Set:

LogarithmsMATHSprint

1: 12Rewrite each statement as a logarithm:

53 = 125

__________

a) 4-4 = 1256

__________

b)

2: 12Rewrite each statement using a power:

log8 164 = -2

__________

a) log7 117649 = 6

__________

b)

3: 12Without using a calculator, find the value of:

log9 9Ö9 = __________a) log1/7 7 = __________b)

log9 81 = __________c) log12 1

12 = __________d)

4: 12Without using a calculator, find the value of:

loga 1a2 = __________a) log1/a Öa = __________b)

log1/a a9 = __________c) loga a

5 = __________d)

5: 12Use a calculator to find the value of the following correct to 4 d.p.:

log11 884 = __________a) log12 1142552 = __________b)

log2 10 = __________c) log9 66 = __________d)

6: 12Write as a single logarithm:

3log3 2 + log3 3 = __________a) 2log6 4 - log6 2 = __________b)

log10 5 + log10 6 = __________c) log5 21 + log5 4 - log7 7 = __________d)


log10x - 4log10y - 5log10z + 2

____________________

a) -3log10x + 5log10y - 2log10z - 4

____________________

b)


-12logax - 1

3logay + 2logaz - 3

____________________

a) logax + 13logay - 1

2logaz + 3

____________________

b)


9: 12Write in terms of log10 x, log10 y and log10 z:

log10(Ö10 3Öxyz2)

____________________

a) log10æ

è

yz2

Ö10x3ö

ø

____________________

b)

10: 12If p = loga x, q = loga y and r = loga z, write the following in terms of p, q and r:

logaæ

è

a2

x3y5zö

ø

____________________

a) logaæ

è

x3z5

ay4ö

ø

____________________

b)

11: 12Solve the following equations:

log1/3 x = -3

__________

a) log5 x = -2

__________

b)

12: 12Solve the following equations:

logx 64 = 2

__________

a) logx 36 = 2

__________

b)

13: 12Solve the following equations correct to 3 decimal places:

11x = 96

__________

a) 7x = 45

__________

b)

14: 12Solve the following equations correct to 3 decimal places:

126x - 4 = 28

__________

a) 42x - 2 = 40

__________

b)


Answers: LogarithmsMATHSprint

1: log5 125 = 3a) log4 1

256 = -4b)

2: 8-2 = 164a) 76 = 117649b)

3: log9 9Ö9 = 32a) log1/7 7 = -1b) log9 81 = 2c) log12

112 = -1d)

4: loga 1a2 = -2a) log1/a Öa = -1

2b) log1/a a9 = -9c) loga a

5 = 5d)

5: log11 884 = 2.8293a) log12 1142552 = 5.6134b)

log2 10 = 3.3219c) log9 66 = 1.9068d)

6: 3log3 2 + log3 3 = log3 24a) 2log6 4 - log6 2 = log6 8b)

log10 5 + log10 6 = log10 30c) log5 21 + log5 4 - log7 7 = log5 12d)

7: log10x - 4log10y - 5log10z + 2 = log10æ

è

100xy4z5

ö

øa)

-3log10x + 5log10y - 2log10z - 4 = log10æ

è

y5

10000x3z2ö

øb)

8: -12logax - 1

3logay + 2logaz - 3 = logaæ

è

z2

a3Öx

3Öyö

øa) logax + 1

3logay - 12logaz + 3 = loga

æ

è

a3x 3Öy

Özö

øb)

9: log10(Ö10 3Öxyz2) = 1

3log10x + log10y + 2log10z + 12a)

log10æ

è

yz2

Ö10x3ö

ø = -3log10x + log10y + 2log10z - 1

2b)

10: logaæ

è

a2

x3y5zö

ø = -3p - 5q - r + 2a) loga

æ

è

x3z5

ay4ö

ø = 3p - 4q + 5r - 1b)

11: log1/3 x = -3x = (1

3)-3 = 27

a) log5 x = -2x = 5-2 = 1

25

b)

12: logx 64 = 2x2 = 64 so x = 8

a) logx 36 = 2x2 = 36 so x = 6

b)

13: log (11x) = xlog 11 = log 96so x = log 96

log 11 = 1.903a) log (7x) = xlog 7 = log 45

so x = log 45log 7 = 1.956

b)

14: log (126x - 4) = (6x - 4)log 12 = log 28so x = 1

6(log 28log 12 + 4) = 0.890

a) log (42x - 2) = (2x - 2)log 4 = log 40so x = 1

2(log 40log 4 + 2) = 2.330

b)

c MATHSprint, 2020.Licensed to MATHSprint. Number of solutions by sketching:1

Name: Class/Set:

Number of solutions by sketchingMATHSprint

1: 12Sketch the following pairs of graphs on the same axes. Use your sketch to state the number of points of

intersection of the two graphs:

y = -x3, y = -3x.

x

y

O

__________

a) y = -1x2, y = 2x.

x

y

O

__________

b) y = 1x, y = -x.

x

y

O

__________

c)

y = x2, y = 2.

x

y

O

__________

d) y = 1x2, y = 2x.

x

y

O

__________

e) y = -x(x - 1)(x + 1), y = -3x.

x

y

O

__________

f)


2: 12Sketch the following pairs of graphs on the same axes. Use your sketch to state the number of points of

intersection of the two graphs:

y = -1x2, y = -x(x + 1)2.

x

y

O

__________

a) y = x3, y = x(x + 1).

x

y

O

__________

b) y = 1x, y = -x(x + 1).

x

y

O

__________

c)

y = 1x2, y = -x2.

x

y

O

__________

d) y = -1x, y = -x(x - 1)(x + 1).

x

y

O

__________

e) y = 1x2, y = -x(x - 1)2.

x

y

O

__________

f)


Answers: Number of solutions by sketchingMATHSprint

1: y = -x3, y = -3x.Number of intersections = 3.

x

y

O

a) y = -1x2, y = 2x.

Number of intersections = 1.

x

y

O

b) y = 1x, y = -x.


x

y

O

c)

y = x2, y = 2.Number of intersections = 2.

x

y

O

d) y = 1x2, y = 2x.


x

y

O

e) y = -x(x - 1)(x + 1), y = -3x.Number of intersections = 3.

x

y

O

f)

2: y = -1x2, y = -x(x + 1)2.


x

y

O

a) y = x3, y = x(x + 1).Number of intersections = 3.

x

y

O

b) y = 1x, y = -x(x + 1).


x

y

O

c)


y = 1x2, y = -x2.


x

y

O

d) y = -1x, y = -x(x - 1)(x + 1).


x

y

O

e) y = 1x2, y = -x(x - 1)2.


x

y

O

f)

c MATHSprint, 2020.Licensed to MATHSprint. Algebraic Fractions and Proof:1

Algebraic Fractions and ProofMATHSprint

1: 12Simplify the following as far as possible:

4z + 412z - 8

a) 12y - 24

15b)


w2 + 2ww2 - 3w

a) 12h2 + 36h16h2 - 16h

b)


t2 - 4t - 2

a) u2 - 2u - 3u2 - u - 6

b)


8r2 + 41r + 56r2 + 31r + 5

a) 6p2 - p - 2

4p2 - 4p - 3b)

5: 12Prove the following statements:

Odd + Odd = Odd.a) The sum of two consecutive even numbersis even.

b)

Even + Even = Even.c) Even ´ Even = Even.d)


The sum of four consecutive numbers is twomore than a multiple of 4.

a) The difference between the squares ofconsecutive odd numbers is a multiple of 8.

b)

The difference between the cubes ofconsecutive odd numbers is two more than amultiple of 24.

c) The difference between the cubes ofconsecutive numbers is one more than amultiple of 6.

d)


For all x ³ 0 and y ³ 0,x + y ³ Öx2 + y2.a) No three consecutive numbers are allprimes.

b)

If a and b are both even then ab is a multipleof 4.

c) All square numbers have an odd number offactors.

d)


For n > 0,4n+2 is never a square number.a) For n > 0,n(n-1)(n+1) + n is always a cubenumber.

b)

For n > 2,n3-1 is never prime.c) For n > 2,n2

-1 is never prime.d)


Answers: Algebraic Fractions and ProofMATHSprint

1: 4(z + 1)

4(3z - 2)

= z + 1

3z - 2

a) 3 ´ 4(y - 2)

3 ´ 5

= 4(y - 2)

5

b)

2: w(w + 2)w(w - 3)

= w + 2w - 3

a) 4h ´ 3(h + 3)4h ´ 4(h - 1)

= 3(h + 3)4(h - 1)

b)

3: (t - 2)(t + 2)

t - 2 = t + 2

a) (u - 3)(u + 1)(u - 3)(u + 2)

= u + 1u + 2

b)

4: (r + 5)(8r + 1)(r + 5)(6r + 1)

= 8r + 16r + 1

a) (2p + 1)(3p - 2)(2p + 1)(2p - 3)

= 3p - 22p - 3

b)

5: (2m+1) + (2n+1) = 2[m + n + 1] \ Even.a) 2n + (2n+2) = 2[2n+1] \ Even.b)

(2m) + (2n) = 2[m + n] \ Even.c) (2m)(2n) = 2[2mn] \ Even.d)

6: (n-1) + (n) + (n+1) + (n+2) = 4n + 2.a)

(2n+1)2 - (2n-1)2 = 8n.b)

(2n+1)3 - (2n-1)3 = (8n3 + 12n2 + 6n + 1) - (8n3 - 12n2 + 6n - 1) = 24[n2] + 2.c)

(n+1)3 - n3 = 3n2 + 3n + 1 = 3[n(n+1)] + 1where either n or n+1 must be a multiple of 2.

d)

7: True: consider (x + y)2 = x2 + 2xy + y2 ³ x2 + y2

Rooting both sides gives x + y ³ Öx2 + y2.a)

True: At least one of the three numbers must be even; the only even prime is 2 so the listmust be 1,2,3 - but 1 is not prime.

b)

True: a = 2m and b = 2n so ab = 4[mn].c)

True: All factors occur in pairs (f and Nf ) except for the square root which is duplicated.d)

8: True: 4n+2 is even and all even square numbers (2n)2 = 4n2 are divisible by 4 with noremainder; 4n+2 has a remainder of 2.

a)

True: n(n-1)(n+1) + n = n3 - n + n = n3.b)


True: n3-1 = (n-1)(n2+n+1) giving at least 2 factors > 1 when n > 2.c)

True: n2-1 = (n-1)(n+1) giving at least 2 factors > 1 when n > 2.d)

c MATHSprint, 2020.Licensed to MATHSprint. Simultaneous Linear and Quadratic Equations:1

Simultaneous Linear and Quadratic EquationsMATHSprint

Answer on separate paper

1: 12Solve the following simultaneous equations:

x + y = 3xy = 2

a) 3x - y = 0xy = 3

b)


4x + y = -12x2 - xy = 1

a) x - 2y = -43xy + 2y2 = -4

b)


2x - y = -3-2x2 + 3xy + y2 = -4

a) 4x + y = -2-2x2 + xy + y2 = 4

b)


5x + y = 5xy - 5x = -5

a) 3x - y = 1xy - 2x - 3y = -6

b)


x - 3y = 2x2 - 2xy - x + 5y = -6

a) x + 2y = 4xy + y2 + 2x + y = 8

b)


x - y = -5x2 + 2xy - y2 + 4x - 4y = -43

a) 2x - y = -1-2x2 - 3xy + y2 + 3x - 5y = -2

b)


Answers: Simultaneous Linear and Quadratic Equations

MATHSprint

1: x = (3 - y)xy = 2(3 - y)y = 2-y2 + 3y - 2 = 0-(y - 2)(y - 1) = 0

x = 2, y = 1x = 1, y = 2

a) y = (3x)xy = 3x(3x) = 33x2 - 3 = 03(x - 1)(x + 1) = 0

x = -1, y = -3x = 1, y = 3

b)

2: y = (-1 - 4x)2x2 - xy = 12x2 - x(-1 - 4x) = 16x2 + x - 1 = 0(3x - 1)(2x + 1) = 0

x = -12, y = 1

x = 13, y = -7

3

a) x = (2y - 4)3xy + 2y2 = -43(2y - 4)y + 2y2 = -48y2 - 12y + 4 = 04(y - 1)(2y - 1) = 0

x = -3, y = 12

x = -2, y = 1

b)

3: y = (2x + 3)-2x2 + 3xy + y2 = -4-2x2 + 3x(2x + 3) + (2x + 3)2 = -48x2 + 21x + 13 = 0(x + 1)(8x + 13) = 0

x = -138 , y = -1

4

x = -1, y = 1

a) y = (-2 - 4x)-2x2 + xy + y2 = 4-2x2 + x(-2 - 4x) + (-2 - 4x)2 = 410x2 + 14x = 02x(5x + 7) = 0

x = -75, y = 18

5

x = 0, y = -2

b)

4: y = (5 - 5x)xy - 5x = -5x(5 - 5x) - 5x = -5-5x2 + 5 = 0-5(x - 1)(x + 1) = 0

x = 1, y = 0x = -1, y = 10

a) y = (3x - 1)xy - 2x - 3y = -6x(3x - 1) - 2x - 3(3x - 1) = -63x2 - 12x + 9 = 03(x - 3)(x - 1) = 0

x = 1, y = 2x = 3, y = 8

b)

5: x = (3y + 2)x2 - 2xy - x + 5y = -6(3y + 2)2 - 2(3y + 2)y - (3y + 2) + 5y = -63y2 + 10y + 8 = 0(3y + 4)(y + 2) = 0

x = -4, y = -2x = -2, y = -4

3

a) x = (4 - 2y)xy + y2 + 2x + y = 8(4 - 2y)y + y2 + 2(4 - 2y) + y = 8-y2 + y = 0-y(y - 1) = 0

x = 4, y = 0x = 2, y = 1

b)


6: x = (y - 5)x2 + 2xy - y2 + 4x - 4y = -43(y - 5)2 + 2(y - 5)y - y2 + 4(y - 5) - 4y = -432y2 - 20y + 48 = 02(y - 6)(y - 4) = 0

x = -1, y = 4x = 1, y = 6

a)

y = (2x + 1)-2x2 - 3xy + y2 + 3x - 5y = -2-2x2 - 3x(2x + 1) + (2x + 1)2 + 3x - 5(2x + 1) = -2-4x2 - 6x - 2 = 0-2(2x + 1)(x + 1) = 0

x = -12, y = 0

x = -1, y = -1

b)

c MATHSprint, 2020.Licensed to MATHSprint. Box Plots and Outliers:1

Name: Class/Set:

Box Plots and OutliersMATHSprint

1: 12If an outlier is defined as lying more than 1.5 ´ interquartile range beyond the lower or upper quartile, decide

whether each value is an outlier:

Q1 = 138, Q3 = 174, value = 233.

__________

a) Q1 = 82, Q3 = 154, value = -16.

__________

b)

2: 12If an extreme outlier is defined as lying more than 3 ´ interquartile range beyond the lower or upper quartile,

identify any extreme outliers in the following data sets:

-417, -352, 97, 122, 144, 178, 217, 219, 250, 252, 268, 431, 702.

____________________

a)

94, 118, 264, 277, 280, 284, 297, 313, 468, 484.

____________________

b)

3: 12Find the required information from the box plot:

0 20 40 60 80 100 120 140 160 180 200

Find the lower and upper quartiles.

__________

a)

0 50 100 150 200 250 300 350 400

Find the range.

__________

b)

4: 12Draw the box plot using the given information, given that there is one outlier more than 1.5 ´ interquartile range

beyond the lower or upper quartile:

Median = 33, LQ = 29, UQ = 35, min value = 13,max value = 43.

0 10 20 30 40 50 60

a) Median = 37, LQ = 32, UQ = 40, min value = 29,max value = 66.

0 10 20 30 40 50 60 70 80

b)

5: 12For each pair of box plots, give two differences relating to location and spread.

0 10 20 30 40 50 60 70 80 90 100

A

B

____________________

a)

0 50 100 150 200 250 300 350 400 450 500

A

B

____________________

b)

c MATHSprint, 2020.Licensed to MATHSprint. Box Plots and Outliers:1

Answers: Box Plots and OutliersMATHSprint

1: Lower boundary = 84 and upper boundary = 228, so 233 is an outlier.a)

Lower boundary = -26 and upper boundary = 262, so -16 is not an outlier.b)

2: LQ = 109.5 and UQ = 260 so limits for extreme outliers are -342 and 711.5.Extreme outliers: -417, -352.

a)

LQ = 264 and UQ = 313 so limits for extreme outliers are 117 and 460.Extreme outliers: 94, 468, 484.

b)

3: LQ = 50, UQ = 140a) Range = 350b)

4:

0 10 20 30 40 50 60

a)

0 10 20 30 40 50 60 70 80

b)

5: A has a higher median/is generally higher than B.A has a bigger range/overall spread than B.A has a bigger interquartile range than B.

a) A has a lower median/is generally lower than B.A has a smaller range/overall spread than B.A has a smaller interquartile range than B.

b)

c MATHSprint, 2020.Licensed to MATHSprint. Statistics - data collection definitions:1

Name: Class/Set:

Statistics - data collection definitionsMATHSprint

Answer in your files

1: 12Decide whether the following are examples of qualitative or quantitative data:

Favourite food

__________

a) Height of student

__________

b) Air temperature

__________

c) Exam percentage

__________

d)

2: 12Decide whether the following are examples of discrete or continuous data:

Number of children

__________

a) Time to run 100m

__________

b)

Capacity of stadium

__________

c) Money in bank account

__________

d)

3: 12Answer the following:

Give two disadvantages of taking a census.

____________________

a) What is a census?

____________________

b)

What is a sampling unit?

____________________

c) Give two advantages of taking a sample.

____________________

d)

4: 12Name the following sampling methods:

Start at a random point near the start of the sampling frame; then select items at regularlyspaced intervals.

____________________

a)

Sample the first N people or items that you come across that meet your criteria.

____________________

b)

Split the sampling frame into groups and take a random sample from each group so that thesample size is proportional to the group size.

____________________

c)

Each element has an equal chance of being selected; pick them from the sampling frame usingnon-repeating random numbers.

____________________

d)


5: 12Answer the following:

Give two disadvantages of Systematic Sampling.

____________________

a)

Give two advantages of Stratified Sampling.

____________________

b)

Give two disadvantages of Stratified Sampling.

____________________

c)

Give two advantages of Systematic Sampling.

____________________

d)


Answers: Statistics - data collection definitionsMATHSprint

1: QUALitative.a) QUANTitative.b) QUANTitative.c) QUANTitative.d)

2: Discrete.a) Continuous.b) Discrete.c) Discrete.d)

3: Census: expensive - takes a long time - can't be used if the test is destructivea)

A census observes or measures every single member of a populationb)

A sampling unit is an individual item or person in the populationc)

Sample: cheaper - quicker - more questions can be asked - okay if the test is destructived)

4: Systematic Samplinga) Opportunity (or Convenience) Samplingb)

Stratified Samplingc) Simple Random Samplingd)

5: Systematic: errors if there is an underlying periodic pattern - a sampling frame is neededa)

Stratified: reflects the population more accurately - subgroups are fairly represented - cancompare different subgroups

b)

Stratified: sampling frame must be sorted into strata - clustering can still happen within eachstratum

c)

Systematic: easy & cheap - avoids picking clusters - suitable for large populations - only onerandom number needed

d)

c MATHSprint, 2020.Licensed to MATHSprint. Mean, Variance & SD calculations:1

Name: Class/Set:

Mean, Variance & SD calculationsMATHSprint

1: 12Calculate the following (round to 4 decimal places if necessary):

If x = 67, variance s2 = 58 and Sf x2 = 272820, find Sf .

____________________

a)

If variance s2 = 41, Sf x = 4680 and Sf x2 = 307152, find Sf .

____________________

b)

If Sf = 50, variance s2 = 62 and Sf x2 = 171300, find the mean x.

____________________

c)

If Sf = 27, Sf x = 729 and Sf x2 = 21654, find the variance s2.

____________________

d)


If variance s2 = 88, Sx = 3240 and Sx2 = 294768, find n.

____________________

a)

If n = 16, Sx = 208 and Sx2 = 2880, find the standard deviation s.

____________________

b)

If n = 54, variance s2 = 87 and Sx2 = 8154, find the mean x.

____________________

c)

If x = 45, variance s2 = 34 and Sx2 = 61770, find n.

____________________

d)



If a data set has n = 14, mean = 54 and variance = 7, find what value must be added to change the mean to64.

____________________

a)

If a data set has n = 7, Sx = 651 and Sx2 = 60767, find the standard deviation when an extra value 102 isadded.

____________________

b)

If a data set has n = 39, Sx = 2808 and Sx2 = 204867, find the variance when an extra value 74 is added.

____________________

c)

If a data set has n = 4, mean = 61 and standard deviation = Ö23, find the standard deviation when an extravalue 59 is added.

____________________

d)


If one data set has n = 45, mean = 96 and variance = 2 and another has n = 64, mean = 78 and variance =21, find the variance of the combined data set.

____________________

a)

If one data set has n = 20, Sx = 200 and Sx2 = 2760 and another has n = 18, Sx = 1476 and Sx2 = 122652,find the standard deviation of the combined data set.

____________________

b)

If one data set has n = 9, mean = 68 and variance = 27 and another has n = 24, mean = 46 and variance =1, find the mean of the combined data set.

____________________

c)

If one data set has n = 25, Sx = 550 and Sx2 = 13025 and another has n = 48, Sx = 144 and Sx2 = 624, findthe variance of the combined data set.

____________________

d)


Answers: Mean, Variance & SD calculationsMATHSprint

1: Sf = 27282058 + 672 = 60a) Sf = 72b)

Mean x = Ö171300

50 - 62 = 58c) Variance s2 =

2165427

- æè

72927

ö

ø

2

= 73d)

2: n = 36a) SD s = Ö2880

16 - æè

20816

ö

ø

2

= Ö11 = 3.3166 [4 dp].b)

Mean x = Ö8154

54 - 87 = 8c) n =

6177034 + 452 = 30d)

3: Value = 15 ´ 64 - 54 ´ 14 = 204a)

SD = Ö60767 + 1022

8 - æè

651 + 1028

ö

ø

2

= 6.0712b)

Variance = 204867 + 742

40 - æè

2808 + 7440

ö

ø

2

= 67.3725c)

SD = Ö(23 + 612) ´ 4 + 592

5 - æè

61 ´ 4 + 595

ö

ø

2

= 4.3635d)

4: Variance = (2 + 962) ´ 45 + (21 + 782) ´ 64

45 + 64 - æè

96 ´ 45 + 78 ´ 6445 + 64

ö

ø

2

= 91.6948a)

SD = Ö2760 + 122652

20 + 18 - æè

200 + 147620 + 18

ö

ø

2

= 36.8109b)

Mean = 68 ´ 9 + 46 ´ 24

9 + 24 = 52.0000c)

Variance = 13025 + 624

25 + 48 - æè

550 + 14425 + 48

ö

ø

2

= 96.5924d)

c MATHSprint, 2020.Licensed to MATHSprint. Normal Distribution:7

Name: Class/Set:

Normal DistributionMATHSprint


The random variable X ~ N(m, 102) and P(X £ -7) = 0.0808. Find the value of m.

____________________

a)

The random variable X ~ N(3, s2) and P(X > 2) = 0.5362. Find the value of s.

____________________

b)


The time taken to brush one's teeth is modelled by a normal distribution with a mean of m seconds and astandard deviation of 30 seconds. If a random person is selected, the probability of the time being greater than122.3 seconds is 0.1903. Find the value of m.

__________

a)

The length of a mature blue whale is modelled by a normal distribution with a mean of 23 metres and astandard deviation of s metres. If a random blue whale is selected, the probability of the length being less than15.2 metres is 0.0256. Find the value of s.

__________

b)



The random variable X ~ N(m, s2). If P(X < -5) = 0.0256 and P(X < 60) = 0.9032, find the values of m and s.

____________________

a)

The random variable X ~ N(m, s2). If P(X £ 81) = 0.7475 and P(X ³ 91) = 0.1108, find the values of m and s.

____________________

b)


The mass of a lettuce is modelled by a normal distribution with a mean of m grams and a standard deviation of sgrams. If a random lettuce is selected, the probability of the mass being less than 476.4 grams is 0.1177 and theprobability of it being less than 477 grams is 0.1217. Find the values of m and s.

__________

a)

The mass of a bag of sweets is modelled by a normal distribution with a mean of m grams and a standarddeviation of s grams. If a random bag of sweets is selected, the probability of the mass being less than 108grams is 0.9772 and the probability of it being greater than 107.8 grams is 0.0287. Find the values of m and s.

__________

b)


Answers: Normal DistributionMATHSprint

1: -7 - m

10 = -1.400 so m = 7.a)

2 - 3s

= -0.091 so s = 11.b)

2: 122.3 - m

30 = 0.877 so m = 96 seconds.a)

15.2 - 23s

= -1.950 so s = 4 metres.b)

3: -5 - ms

= -1.950 and 60 - ms

= 1.300 so m = 34 and s = 20.a)

81 - ms

= 0.667 and 91 - ms

= 1.222 so m = 69 and s = 18.b)

4: 476.4 - m

s = -1.187 and

477 - ms

= -1.167 so m = 512 grams and s = 30 grams.a)

108 - ms

= 2.000 and 107.8 - m

s = 1.900 so m = 104 grams and s = 2 grams.b)

c MATHSprint, 2020.Licensed to MATHSprint. PMCC Hypothesis Testing:1

Name: Class/Set:

PMCC Hypothesis TestingMATHSprint

1: 12Work out the following, giving any numbers correct to 4 decimal places if necessary:

Wilf thinks that there is a negative correlation between the amount of exercise per week andbody mass index and takes a sample of 5 data pairs, getting a PMCC of -0.7905.Testing Wilf's claim at the 2.5% level, is this PMCC value in the critical region?

____________________

a)

Vijay reckons that there is a correlation between the cost of a car and the reliability of a carand takes a sample of 16 data pairs, getting a PMCC of -0.4706.Testing Vijay's claim at the 20% level, is this PMCC value in the critical region?

____________________

b)

2: 12Work out the following, giving any numbers correct to 4 decimal places if necessary:

Sophie is pretty sure that there is a positive correlation between air pressure and temperatureand takes a sample of 7 data pairs, getting a PMCC of 0.7496.Carry out a hypothesis test at the 1% significance level to investigate Sophie's claim.

____________________

a)

Georgia conjectures that there is a correlation between hand span and number of T-shirtsowned and takes a sample of 10 data pairs, getting a PMCC of 0.8117.Carry out a hypothesis test at the 1% significance level to investigate Georgia's claim.

____________________

b)

c MATHSprint, 2020.Licensed to MATHSprint. PMCC Hypothesis Testing:1

Answers: PMCC Hypothesis TestingMATHSprint

1: 1-tailed test at 2.5% level, n = 5: critical region is r £ -0.8783,so -0.7905 is not in the critical region.

a)

2-tailed test at 20% level, n = 16: critical region is r £ -0.3383 or r ³ 0.3383,so -0.4706 is in the critical region.

b)

2: H0: r = 0 and H1: r ³ 0.

1-tailed test at 1% level, n = 7: critical region is r ³ 0.8329,so 0.7496 is not in the critical region.There is insufficient evidence at the 1% level to reject H0, the hypothesis that air pressure

and temperature are not correlated.

a)

H0: r = 0 and H1: r ¹ 0.

2-tailed test at 1% level, n = 10: critical region is r £ -0.7646 or r ³ 0.7646,so 0.8117 is in the critical region.There is evidence at the 1% level to reject H0, supporting the alternative hypothesis that

there is a correlation between hand span and number of T-shirts owned.

b)

c MATHSprint, 2020.Licensed to MATHSprint. Venn Diagrams:1

Name: Class/Set:

Venn DiagramsMATHSprint

1: 12Find the following probabilities.

M N

0.42 0.28 0.22

0.08

P(N ¢)

__________

a) P(M Ç N)

__________

b) P(M)

__________

c) P(N)

__________

d)


S T U

0.16 0.44 0.21 0.15 0.03

0.01

P(T Ç U)

__________

a) P(S Ç U)

__________

b) P(S Ç T)

__________

c) P(T ¢)

__________

d)


L M

N

0.260.04

0.16

0.040.06 0.06

0.280.10

P(M È N)

__________

a) P((L È M È N) ¢)

__________

b) P((L È N) ¢)

__________

c)

P(N Ç M ¢)

__________

d) P((L Ç M ¢) È (L ¢ Ç M))

__________

e) P(L È M È N)

__________

f)

P((M È N) ¢)

__________

g) P((L È M) ¢)

__________

h)


4: 12Complete the Venn diagram using the information provided.

A survey of 51 musicians finds that 17 play clarinet, 14 play guitar, 8 play bassoon, 9 play bothclarinet and guitar and 18 play guitar or bassoon.

C G B

a)

Out of 51 children, 27 own fish, 18 own gerbils, 28 own mice, 6 own both fish and gerbils, 41own fish or mice, 38 own gerbils or mice and 1 own fish, gerbils and mice.

F G

M

b)

5: 12Complete the Venn diagram using the information provided.

For events D, E and F, P(D) = 0.20, P(E) = 0.10, P(F) = 0.26, P(D Ç E) = 0.07 andP(E Ç F) = 0.02.

D E F

a)

For events T and U, P(T ¢) = 0.10,P(U ¢) = 0.40 and P(T Ç U) = 0.53.

T U

b)


Answers: Venn DiagramsMATHSprint

1: P(N ¢) = 0.50a) P(M Ç N) = 0.28b) P(M) = 0.70c) P(N) = 0.50d)

2: P(T Ç U) = 0.15a) P(S Ç U) = 0b) P(S Ç T) = 0.44c) P(T ¢) = 0.20d)

3: P(M È N) = 0.64a) P((L È M È N) ¢) = 0.10b)

P((L È N) ¢) = 0.26c) P(N Ç M ¢) = 0.34d)

P((L Ç M ¢) È (L ¢ Ç M)) = 0.54e) P(L È M È N) = 0.90f)

P((M È N) ¢) = 0.36g) P((L È M) ¢) = 0.38h)

4: C G B

8 9 1 4 4

25

a) F G

M

85

5

113 7

75

b)

5: D E F

0.13 0.07 0.01 0.02 0.24

0.53

a) T U

0.37 0.53 0.07

0.03

b)

c MATHSprint, 2020.Licensed to MATHSprint. Lines in 3 Dimensions:1

Lines in 3 DimensionsMATHSprint


Convert r = æ

ç

è

-3-40

ö

÷

ø

+ læ

ç

è

32-1

ö

÷

ø

to Cartesian form.a) Convert r = æ

ç

è

-25-4

ö

÷

ø

+ læ

ç

è

-253

ö

÷

ø

to Cartesian form.b)


Convert x + 2

2 =

y - 5-3

= z - 2

2 to vector form r = a + lb.a)

Convert x - 2

5 =

y - 24

= z - 5

2 to vector form r = a + lb.b)


Convert r = æ

ç

è

303

ö

÷

ø

+ læ

ç

è

010

ö

÷

ø

to Cartesian form.a) Convert r = æ

ç

è

34-5

ö

÷

ø

+ læ

ç

è

10-3

ö

÷

ø

to Cartesian form.b)


Convert x = 5, y - 2 = z + 1 to vector form r = a + lb.a)

Convert z = -2, x = y + 3-4

to vector form r = a + lb.b)


A vector equation for the line through (2, -1, -4) and (3, -2, -3).a)

A vector equation for the line through (0, 2, -5) and (4, 5, 0).b)


The Cartesian equation for the line through (-2, 1, -2) and (3, 5, 1).a)

The Cartesian equation for the line through (-3, -4, -4) and (-2, -9, -3).b)


The acute angle between these lines:

r = æ

ç

è

35-4

ö

÷

ø

+ læ

ç

è

125

ö

÷

ø

, r = æ

ç

è

0-2-1

ö

÷

ø

+ mæ

ç

è

25-2

ö

÷

ø

a) The acute angle between these lines:

r = æ

ç

è

443

ö

÷

ø

+ læ

ç

è

55-4

ö

÷

ø

, r = æ

ç

è

-340

ö

÷

ø

+ mæ

ç

è

122

ö

÷

ø

b)


The acute angle between these lines:

r = æ

ç

è

-10-1

ö

÷

ø

+ læ

ç

è

-125

ö

÷

ø

, x + 4-3

= y + 3

4 =

z + 14

a) The acute angle between these lines:

r = æ

ç

è

45-2

ö

÷

ø

+ læ

ç

è

524

ö

÷

ø

, x - 2-2

= y - 5

4 =

z5

b)



The acute angle between these lines:x + 5

3 =

y + 43

= z + 5

5,

x - 55

= y-4

= z - 4

4

a) The acute angle between these lines:x - 2

2 =

y - 53

= z + 5-3

, x + 4-2

= y + 5

3 =

z - 25

b)


Does the point (3, -1, -7) lie on the line r = æ

ç

è

1-3-3

ö

÷

ø

+ læ

ç

è

11-2

ö

÷

ø

?a)

Does the point (6, 9, -4) lie on the line r = æ

ç

è

-454

ö

÷

ø

+ læ

ç

è

52-4

ö

÷

ø

?b)


Does the point (1, 13, -3) lie on the line x + 2 = y - 4

3 =

z - 4-3

?a)

Does the point (0, -2, -1) lie on the line x + 1-1

= y - 3

5 =

z - 32

?b)


Do the points (5, 4, -2), (9, 6, -3), (1, 2, -1) lie on the same line?a)

Do the points (0, -2, -1), (-2, -1, 0), (-4, 0, 1) lie on the same line?b)


The perpendicular distance from the point (-2, 1, 4) to the line r = æ

ç

è

4-35

ö

÷

ø

+ læ

ç

è

-210

ö

÷

ø

.a)

The perpendicular distance from the point (5, 1, -1) to the line r = æ

ç

è

-4-40

ö

÷

ø

+ læ

ç

è

31-1

ö

÷

ø

.b)


The perpendicular distance from the point (9, -4, 1) to the line x - 5-1

= y + 2

2 = z - 5.a)

The perpendicular distance from the point (7, 9, 6) to the line x - 2 = y - 4 = z + 2

2.b)


Find the point of intersection (if any) of these lines;otherwise state if they are skew or parallel:

r = æ

ç

è

31-4

ö

÷

ø

+ læ

ç

è

-1-21

ö

÷

ø

and r = æ

ç

è

-1513-25

ö

÷

ø

+ mæ

ç

è

4-46

ö

÷

ø

.

a) Find the point of intersection (if any) of these lines;otherwise state if they are skew or parallel:

r = æ

ç

è

-96-12

ö

÷

ø

+ læ

ç

è

-11-3

ö

÷

ø

and r = æ

ç

è

-2938-12

ö

÷

ø

+ mæ

ç

è

-69-3

ö

÷

ø

.

b)



Find the point of intersection (if any) of these lines;otherwise state if they are skew or parallel:

r = æ

ç

è

-1037

ö

÷

ø

+ læ

ç

è

-211

ö

÷

ø

and x + 1

2 =

y - 2-1

= z - 2-1

.

a) Find the point of intersection (if any) of these lines;otherwise state if they are skew or parallel:

r = æ

ç

è

-4-5-2

ö

÷

ø

+ læ

ç

è

1-2-1

ö

÷

ø

and x - 22

9 =

y - 156

= z + 9-3

.

b)


Find the point of intersection (if any) of these lines;otherwise state if they are skew or parallel:x - 9-3

= y2

= z - 6-2

and x + 5-6

= y - 12

4 =

z - 18-4

.

a) Find the point of intersection (if any) of these lines;otherwise state if they are skew or parallel:x + 7-1

= y - 10

3 = z - 3 and

x + 6-1

= y - 5

2 =

z + 6-3

.

b)


The shortest distance between the lines

r = æ

ç

è

2-24

ö

÷

ø

+ læ

ç

è

1-63

ö

÷

ø

and r = æ

ç

è

-135

ö

÷

ø

+ mæ

ç

è

2-11

5

ö

÷

ø

.

a) The shortest distance between the lines

r = æ

ç

è

1-5-4

ö

÷

ø

+ læ

ç

è

43-5

ö

÷

ø

and r = æ

ç

è

-1-3-3

ö

÷

ø

+ mæ

ç

è

33-4

ö

÷

ø

.

b)


The shortest distance between the linesx + 5-9

= y + 1

2 =

z - 35

and x - 2-6

= y - 1 = z + 3

3.

a) The shortest distance between the linesx - 1

5 =

y + 15

= z + 2

2 and

x + 32

= y + 2

2 = z - 4.

b)


Answers: Lines in 3 DimensionsMATHSprint

1: x + 3

3 =

y + 42

= z-1

a) x + 2-2

= y - 5

5 =

z + 43

b)

2: r = æ

ç

è

-252

ö

÷

ø

+ læ

ç

è

2-32

ö

÷

ø

a) r = æ

ç

è

225

ö

÷

ø

+ læ

ç

è

542

ö

÷

ø

b)

3: x = 3, z = 3a) y = 4, x - 3 = z + 5-3

b)

4: r = æ

ç

è

52-1

ö

÷

ø

+ læ

ç

è

011

ö

÷

ø

a) r = æ

ç

è

0-3-2

ö

÷

ø

+ læ

ç

è

1-40

ö

÷

ø

b)

5: r = æ

ç

è

2-1-4

ö

÷

ø

+ læ

ç

è

1-11

ö

÷

ø

or r = æ

ç

è

3-2-3

ö

÷

ø

+ læ

ç

è

-11-1

ö

÷

ø

a) r = æ

ç

è

02-5

ö

÷

ø

+ læ

ç

è

435

ö

÷

ø

or r = æ

ç

è

450

ö

÷

ø

+ læ

ç

è

-4-3-5

ö

÷

ø

b)

6: x + 2

5 =

y - 14

= z + 2

3a) x + 3 =

y + 4-5

= z + 4b)

7: Use b × d = bdcosq on the direction vectors æ

ç

è

125

ö

÷

ø

and æ

ç

è

25-2

ö

÷

ø

so the acute angle is cos-1æ

è

2Ö30 ´ Ö33

ö

ø = 86.4°

a)

Use b × d = bdcosq on the direction vectors æ

ç

è

55-4

ö

÷

ø

and æ

ç

è

122

ö

÷

ø


è

7Ö66 ´ 3

ö

ø = 73.3°

b)


ç

è

-125

ö

÷

ø

and æ

ç

è

-344

ö

÷

ø


è

31Ö30 ´ Ö41

ö

ø = 27.9°

a)


ç

è

524

ö

÷

ø

and æ

ç

è

-245

ö

÷

ø


è

183Ö5 ´ 3Ö5

ö

ø = 66.4°

b)


ç

è

335

ö

÷

ø

and æ

ç

è

5-44

ö

÷

ø


è

23Ö43 ´ Ö57

ö

ø = 62.3°

a)



ç

è

23-3

ö

÷

ø

and æ

ç

è

-235

ö

÷

ø

so the acute angle is 180° - cos-1æ

è

-10Ö22 ´ Ö38

ö

ø = 69.8°

b)

10: æ

ç

è

11-2

ö

÷

ø

´

æ

ç

è

22-4

ö

÷

ø

= æ

ç

è

-4 - -4-(-4 - -4)

2 - 2

ö

÷

ø

= æ

ç

è

000

ö

÷

ø

\ YES: point lies on the line.a)

æ

ç

è

52-4

ö

÷

ø

´

æ

ç

è

104-8

ö

÷

ø

= æ

ç

è

-16 - -16-(-40 - -40)

20 - 20

ö

÷

ø

= æ

ç

è

000

ö

÷

ø

\ YES: point lies on the line.b)

11: æ

ç

è

13-3

ö

÷

ø

´

æ

ç

è

39-7

ö

÷

ø

= æ

ç

è

-21 - -27-(-7 - -9)

9 - 9

ö

÷

ø

= æ

ç

è

6-20

ö

÷

ø

\ NO: point does not lie on the line.a)

æ

ç

è

-152

ö

÷

ø

´

æ

ç

è

1-5-4

ö

÷

ø

= æ

ç

è

-20 - -10-(4 - 2)

5 - 5

ö

÷

ø

= æ

ç

è

-10-20

ö

÷

ø

\ NO: point does not lie on the line.b)

12: æ

ç

è

42-1

ö

÷

ø

´

æ

ç

è

-4-21

ö

÷

ø

= æ

ç

è

2 - 2-(4 - 4)-8 - -8

ö

÷

ø

= æ

ç

è

000

ö

÷

ø

\ YES: points lie on the same line.a)

æ

ç

è

-211

ö

÷

ø

´

æ

ç

è

-422

ö

÷

ø

= æ

ç

è

2 - 2-(-4 - -4)-4 - -4

ö

÷

ø

= æ

ç

è

000

ö

÷

ø

\ YES: points lie on the same line.b)

13: Perpendicular distance = ½(a - p) ´ b̂½ = 1Ö5

½

½

½

é

ê

ë

æ

ç

è

4-35

ö

÷

ø

-

æ

ç

è

-214

ö

÷

ø

ù

ú

û

´

æ

ç

è

-210

ö

÷

ø

½

½

½

= 1Ö5

½

½

½

æ

ç

è

6-41

ö

÷

ø

´

æ

ç

è

-210

ö

÷

ø

½

½

½

= 1Ö5

½

½

½

æ

ç

è

-1-2-2

ö

÷

ø

½

½

½

= 3Ö5

= 35Ö5 = 1.34 (2 d.p.).

a)

Perpendicular distance = ½(a - p) ´ b̂½ = 1Ö11

½

½

½

é

ê

ë

æ

ç

è

-4-40

ö

÷

ø

-

æ

ç

è

51-1

ö

÷

ø

ù

ú

û

´

æ

ç

è

31-1

ö

÷

ø

½

½

½

= 1Ö11

½

½

½

æ

ç

è

-9-51

ö

÷

ø

´

æ

ç

è

31-1

ö

÷

ø

½

½

½

= 1Ö11

½

½

½

æ

ç

è

4-66

ö

÷

ø

½

½

½

= 2Ö22Ö11

= 2Ö2 = 2.83 (2 d.p.).

b)

14: Perpendicular distance = ½(a - p) ´ b̂½ = 1Ö6

½

½

½

é

ê

ë

æ

ç

è

5-25

ö

÷

ø

-

æ

ç

è

9-41

ö

÷

ø

ù

ú

û

´

æ

ç

è

-121

ö

÷

ø

½

½

½

= 1Ö6

½

½

½

æ

ç

è

-424

ö

÷

ø

´

æ

ç

è

-121

ö

÷

ø

½

½

½

= 1Ö6

½

½

½

æ

ç

è

-60-6

ö

÷

ø

½

½

½

= 6Ö2Ö6

= 2Ö3 = 3.46 (2 d.p.).

a)

Perpendicular distance = ½(a - p) ´ b̂½ = 1Ö6

½

½

½

é

ê

ë

æ

ç

è

24-2

ö

÷

ø

-

æ

ç

è

796

ö

÷

ø

ù

ú

û

´

æ

ç

è

112

ö

÷

ø

½

½

½

= 1Ö6

½

½

½

æ

ç

è

-5-5-8

ö

÷

ø

´

æ

ç

è

112

ö

÷

ø

½

½

½

= 1Ö6

½

½

½

æ

ç

è

-220

ö

÷

ø

½

½

½

= 2Ö2Ö6

= 23Ö3 = 1.15 (2 d.p.).

b)


15: Lines are skew.a)

Point of intersection is (-5, 2, 0). [l = -4 and m = -4]b)

16: Lines are parallel: direction vectors æ

ç

è

-211

ö

÷

ø

and æ

ç

è

2-1-1

ö

÷

ø

are scalar multiples of one another.a)

Lines are skew.b)

17: Lines are parallel: direction vectors æ

ç

è

-32-2

ö

÷

ø

and æ

ç

è

-64-4

ö

÷

ø

are scalar multiples of one another.a)

Point of intersection is (-4, 1, 0). [l = -3 and m = -2]b)

18: Normal to both lines n = æ

ç

è

1-63

ö

÷

ø

´

æ

ç

è

2-11

5

ö

÷

ø

= æ

ç

è

311

ö

÷

ø

and n̂ = 1Ö11

æ

ç

è

311

ö

÷

ø

so shortest distance = (a - c) × n̂ = 1Ö11

½

½

½

é

ê

ë

æ

ç

è

2-24

ö

÷

ø

-

æ

ç

è

-135

ö

÷

ø

ù

ú

û

×

æ

ç

è

311

ö

÷

ø

½

½

½

= 1Ö11

½

½

½

æ

ç

è

3-5-1

ö

÷

ø

×

æ

ç

è

311

ö

÷

ø

½

½

½

= ½9 - 5 - 1½

Ö11 = 3

11Ö11 = 0.90 (2 d.p.).

a)

Normal to both lines n = æ

ç

è

43-5

ö

÷

ø

´

æ

ç

è

33-4

ö

÷

ø

= æ

ç

è

313

ö

÷

ø

and n̂ = 1Ö19

æ

ç

è

313

ö

÷

ø


½

½

½

é

ê

ë

æ

ç

è

1-5-4

ö

÷

ø

-

æ

ç

è

-1-3-3

ö

÷

ø

ù

ú

û

×

æ

ç

è

313

ö

÷

ø

½

½

½

= 1Ö19

½

½

½

æ

ç

è

2-2-1

ö

÷

ø

×

æ

ç

è

313

ö

÷

ø

½

½

½

= ½6 - 2 - 3½

Ö19 = 1

19Ö19 = 0.23 (2 d.p.).

b)

19: Normal to both lines n = æ

ç

è

-925

ö

÷

ø

´

æ

ç

è

-613

ö

÷

ø

= æ

ç

è

1-33

ö

÷

ø

and n̂ = 1Ö19

æ

ç

è

1-33

ö

÷

ø


½

½

½

é

ê

ë

æ

ç

è

-5-13

ö

÷

ø

-

æ

ç

è

21-3

ö

÷

ø

ù

ú

û

×

æ

ç

è

1-33

ö

÷

ø

½

½

½

= 1Ö19

½

½

½

æ

ç

è

-7-26

ö

÷

ø

×

æ

ç

è

1-33

ö

÷

ø

½

½

½

= ½-7 + 6 + 18½

Ö19 = 17

19Ö19 = 3.90 (2 d.p.).

a)


Normal to both lines n = æ

ç

è

552

ö

÷

ø

´

æ

ç

è

221

ö

÷

ø

= æ

ç

è

1-10

ö

÷

ø

and n̂ = 1Ö2

æ

ç

è

1-10

ö

÷

ø


½

½

½

é

ê

ë

æ

ç

è

1-1-2

ö

÷

ø

-

æ

ç

è

-3-24

ö

÷

ø

ù

ú

û

×

æ

ç

è

1-10

ö

÷

ø

½

½

½

= 1Ö2

½

½

½

æ

ç

è

41-6

ö

÷

ø

×

æ

ç

è

1-10

ö

÷

ø

½

½

½

= ½4 - 1 + 0½

Ö2 = 3

2Ö2 = 2.12 (2 d.p.).

b)

c MATHSprint, 2020.Licensed to MATHSprint. 3x3 Matrices:1

3x3 MatricesMATHSprint

Please answer on file paper


æ

ç

è

3-3-1

42-4

10-2

ö

÷

ø

2

a) æ

ç

è

4-1-4

-33-2

201

ö

÷

ø

2

b)


Detæ

ç

è

-3-41

24-2

-103

ö

÷

ø

a) Detæ

ç

è

2-40

-3-11

43-2

ö

÷

ø

b)


Is æ

ç

è

-4-2-3

243

000

ö

÷

ø

singular?a) Is æ

ç

è

-313

-424

0-1-2

ö

÷

ø

singular?b)

c MATHSprint, 2020.Licensed to MATHSprint. 3x3 Matrices:1

Answers: 3x3 MatricesMATHSprint

1: æ

ç

è

3-3-1

42-4

10-2

ö

÷

ø

2

= æ

ç

è

9 + -12 + -1-9 + -6 + 0-3 + 12 + 2

12 + 8 + -4-12 + 4 + 0-4 + -8 + 8

3 + 0 + -2-3 + 0 + 0-1 + 0 + 4

ö

÷

ø

= æ

ç

è

-4-1511

16-8-4

1-33

ö

÷

ø

a)

æ

ç

è

4-1-4

-33-2

201

ö

÷

ø

2

= æ

ç

è

16 + 3 + -8-4 + -3 + 0-16 + 2 + -4

-12 + -9 + -4

3 + 9 + 012 + -6 + -2

8 + 0 + 2-2 + 0 + 0-8 + 0 + 1

ö

÷

ø

= æ

ç

è

11-7-18

-25124

10-2-7

ö

÷

ø

b)

2: Detæ

ç

è

-3-41

24-2

-103

ö

÷

ø

= [-3(4 ´ 3 - 0 ´ -2)] - [2(-4 ´ 3 - 0 ´ 1)] + [-1(-4 ´ -2 - 4 ´ 1)] = [-36] - [-24] + [-4] = -16

a)

Detæ

ç

è

2-40

-3-11

43-2

ö

÷

ø

= [2(-1 ´ -2 - 3 ´ 1)] - [-3(-4 ´ -2 - 3 ´ 0)] + [4(-4 ´ 1 - -1 ´ 0)] = [-2] - [-24] + [-16] = 6

b)

3: Detæ

ç

è

-4-2-3

243

000

ö

÷

ø

= [-4(4 ´ 0 - 0 ´ 3)] - [2(-2 ´ 0 - 0 ´ -3)] + [0(-2 ´ 3 - 4 ´ -3)] = [0] - [0] + [0] = 0 so the matrix is singular.

a)

Detæ

ç

è

-313

-424

0-1-2

ö

÷

ø

= [-3(2 ´ -2 - -1 ´ 4)] - [-4(1 ´ -2 - -1 ´ 3)] + [0(1 ´ 4 - 2 ´ 3)] = [0] - [-4] + [0] = 4 so the matrix is NOT singular.

b)

c MATHSprint, 2020.Licensed to MATHSprint. Networks:1

Name: Class/Set:

NetworksMATHSprint

1: 12Find the Minimum Spanning Tree using Prim's Algorithm starting from vertex A:

15

10

19

7

17

16

14

12

6

3

20

13

9

5

4

1

11

2

18

A

B

C

D

E

F

G

H

I

J

Arcs/Length:

a)

8

5

8

20

17

9

19

4

3

2

13

1

15

12

7

14

A

B

C

D

E

F

G

H

I

Arcs/Length:

b)

2: 12Find the Minimum Spanning Tree using Prim's Algorithm starting from vertex A:

-

29

26

15

28

7

A

A

29

-

12

18

13

21

B

B

26

12

-

11

22

24

C

C

15

18

11

-

1

5

D

D

28

13

22

1

-

19

E

E

7

21

24

5

19

-F

F

Arcs:Total length=

a)

-

2

17

14

8

25

30

A

A

2

-

4

3

6

20

23

B

B

17

4

-

27

9

16

10

C

C

14

3

27

-

19

18

21

D

D

8

6

9

19

-

24

12

E

E

25

20

16

18

24

-

29

F

F

30

23

10

21

12

29

-G

G

Arcs:Total length=

b)


3: 12Find the shortest route from A to the ringed vertex using Dijkstra's Algorithm:

10

3

11

15

13

9

16

5

1

8

17

14

2

18

6

A

B

C

D

E

F

G

H

I

Route/Length:

a)

12

6

16

10

20

4

19

11

18

7

5

13

15

3

1

17

20

8

A

B

C

D

E

F

G

H

I

J

Route/Length:

b)

4: 12Find the shortest route starting and finishing at A which includes every edge.

14

12

9

2

11

16

7

1019

18

6

A B

CD

E F

Pairings/Best solution:

____________________

a)

4

2014

1517

12

A B

C

D

Pairings/Best solution:

____________________

b)


Answers: NetworksMATHSprint

1:

15

10

19

7

17

16

14

12

6

3

20

13

9

5

4

1

11

2

181

2 34

56

7

8 9A

B

C

D

E

F

G

H

I

J

Arcs: AC, CF, FI, IG, GD, GH, HJ, DB, DE. Total length=58

a)

8

5

8

20

17

9

19

4

3

2

13

1

15

12

7

14

12

3

45

6

78

A

B

C

D

E

F

G

H

I

Arcs: AD, DH, HE, DG, AB, BF, FI, FC. Total length=59

b)

2:

-

29

26

15

28

7

A

A

29

-

12

18

13

21

B

B

26

12

-

11

22

24

C

C

15

18

11

-

1

5

D

D

28

13

22

1

-

19

E

E

7

21

24

5

19

-F

F1 23 456

Arcs: AF, FD, DE, DC, CB.Total length=36

a)

-

2

17

14

8

25

30

A

A

2

-

4

3

6

20

23

B

B

17

4

-

27

9

16

10

C

C

14

3

27

-

19

18

21

D

D

8

6

9

19

-

24

12

E

E

25

20

16

18

24

-

29

F

F

30

23

10

21

12

29

-G

G1 2 34 5 67

Arcs: AB, BD, BC, BE, CG, CF.Total length=41

b)


3:

10

3

11

15

13

9

16

5

1

8

17

14

2

18

6

1 0

5 1010

8 1717

2 33

4 88

7 1325.13

3 44

6 1111

9 1717

A

B

C

D

E

F

G

H

I

Route: ADHI. Length=17

a)

12

6

16

10

20

4

19

11

18

7

5

13

15

3

1

17

20

8

1 0

4 1212

2 66

5 1616

9 3035.30

3 1010

6 2323

8 2626

7 2425.24

10 3240.32

A

B

C

D

E

F

G

H

I

J

Route: ACFGIJ. Length=32

b)

4:

14

12

9

2

11

16

7

1019

18

6

A B

CD

E F

Pairings: BC:EF:Tot:

121931

BE:CF:Tot:

251035

BF:CE:Tot:

72229

Poss. route: Length:

ABCDADCFAEDFBFEA 124 + 29 = 153

a)


4

2014

1517

12

A B

C

D

Pairings: AB:CD:Tot:

41216

AC:BD:Tot:

141731

AD:BC:Tot:

151833

Poss. route: Length:

ABACBDCDA 82 + 16 = 98

b)

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