Edexcel GCSE Mathematics (Higher)

Here are the first five terms of a quadratic sequence:

$$-3,0,5,12,21$$

Find an expression, in terms of $n$, for the nth term of this sequence. [4 marks]

method A:

• first difference = $3,5,7,9$

• second difference = $2$

• $$2\div2=1\to n^2$$

• $$n^2=1,4,9$$

  • linear sequence = $4$ →$-4$

$n^2-4$

method B:

• first difference = $3,5,7,9$

• second difference = $2$

• $$2a=2$$

  • $a = 1$

• $$3a + b = 3$$

  • $$3(1) + b = 3$$

  • $$3 + b = 3$$

  • $b = 0$

• $$a + b + c = -3$$

  • $$1 + 0 + c = -3$$

  • $c = -4$

• general formula for a quadratic sequence — $an^2+bn+c$

  • $n² - 4$

A quadratic sequence starts:

$$6,10,16,24$$

a) Show that the nth term is $n^2+n+4$. [4 marks]

b) Hence find the term that have value $136$ [2 marks]

a)

• method A:

  • first difference = $4,6,8$

  • second difference = $2$

  • $$2\div2=1\to n^2$$

  • $$n^2=1,4,9$$

    • linear sequence = $5,6,7$ →$n$

    • $$5 - 1 = 4$$

  $n^2+n+4$

  method B:

  • first difference = $4,6,8$

  • second difference = $2$

  • $$2a=2$$

    • $a = 1$

  • $$3a+b=4$$

    • $$3(1)+b=4$$

    • $$3+b=4$$

    • $b=1$

  • $$a+b+c=6$$

    • $$1+1+c=6$$

    • $c=4$

  • general formula for a quadratic sequence — $an^2+bn+c$

    • $n^2{}+n+4$

b)

• $$n^2+n+4=136$$

• $$n^2+n-132=0$$

• $$\left(n-11\right)\left(n+12\right)=0$$

  • $$n=11$$

  • $$n = -12$$

• $136$ is the $11$ th term

Factorise $12x^2-31xy+20y^2$ [2 marks]

• $$ac=240$$

  • $$PF=2⁴, 3, 5$$

• $$\frac{\left(12x-16y\right)\left(12x-15y\right)}{12}$$

  • $$\left(x-\frac43y\right)\left(12x-15y\right)$$

    • multiply first bracket by 3 to remove fraction

    • divide second bracket by 3 as you have multiplied the first bracket by 3

      • $\left(3x-4y\right)\left(4x-5y\right)$

Factorise fully $3x^2-12$ [2 marks]

• $$3\left(x^2-4\right)$$

  • $3(x+2)\left(x-2\right)$

$$f\left(x\right)+n$$

• What is the vector?

• What does $\left(x,y\right)$ become?

• $$\begin{pmatrix}0\\ n\end{pmatrix}$$

• $$\left(x,y+n\right)$$

$$f\left(x\right)-n$$

• What is the vector?

• What does $\left(x,y\right)$ become?

• $$\begin{pmatrix}0\\ -n\end{pmatrix}$$

• $$\left(x,y-n\right)$$

$$f\left(x+n\right)$$

• What is the vector?

• What does $\left(x,y\right)$ become?

• $$\begin{pmatrix}-n\\ 0\end{pmatrix}$$

• $$\left(x-n,y\right)$$

$$f\left(x-n\right)$$

• What is the vector?

• What does $\left(x,y\right)$ become?

• $$\begin{pmatrix}n\\ 0\end{pmatrix}$$

• $$\left(x+n,y\right)$$

For the function $g(x)$ — what does $-g(x)$ do?

• What does $\left(x,y\right)$ become?

• Reflection — x-axis

• $$(x,-y)$$

For the function $g(x)$ — what does $g(-x)$ do?

• What does $\left(x,y\right)$ become?

• Reflection — y-axis

• $$(-x,y)$$

For the function $g(x)$ — what does $-g(-x)$ do?

• What does $\left(x,y\right)$ become?

• Reflection — x-axis and y-axis

• $$(-x,-y)$$

The second term of a geometric sequence is $6$.

The sixth term of the geometric sequence is $\frac{3}{8}$.

Find all possible values for the first term of the geometric sequence. [4 marks]

• $$\frac38\div x^4=6$$

  • $$\frac38\div6=\frac{1}{16}$$

  • $$x^4=\frac{1}{16}$$

  • $x=\frac12$ or $-\frac12$

• $$6\div\frac12=12$$

• $$6\div-\frac12=-12$$

first term = $12$ or $-12$

S is a geometric sequence.

The first three terms of S are $(x+12)$ , $\left(x-2\right)$ , and $(x-9)$.

Find the value of $x$. [4 marks]

• $$\frac{x-9}{x-2}=\frac{x-2}{x+12}$$

• $$\left(x-9\right)\left(x+12\right)=\left(x-2_{}\right)\left(x-2\right)$$

• $$x^2+3x-108=x^2-4x+4$$

• $$7x-112=0$$

  • $$7x = 112$$

    • $x=16$

What is the equation for the surface area of a cylinder?

$$SA=2\pi r^2+2\pi rh$$

When do you use:

$$\frac12ab\sin C$$

2 sides — angle between them

Which graph has the equation:

$y = -x^2 + 4$

A

Which graph has the equation:

$$y = \frac{1}{x}$$

B

Which graph has the equation:

$y = x(x-2)(x+2)$

C

Which graph has the equation:

$y = x + 1$

D

Which graph has the equation:

$y=x³$

E

Which graph has the equation:

$y=-x³$

F

Which graph has the equation:

$$y=-x$$

G

Which graph has the equation:

$y=x²-3$

H

Which graph has the equation:

$$y=-\frac{1}{x}$$

J

Which graph has the equation:

$$y = sinx$$

C

Which graph has the equation:

$$y = cosx$$

D

Which graph has the equation:

$$y=x^3+4x$$

F

When do you use each version of the sine rule:

• $$\frac{a}{\sin A}=\frac{b}{\sin B}$$

• $$\frac{\sin A}{a}=\frac{\sin B}{b}$$

• finding a side

• finding an angle

Alternate angles are ___________.

_____ shape

• equal

• Z

Corresponding angles are ____________.

_____ shape

• equal

• F

Co-interior angles _____________________________.

_____ shape

• add to 180°

• C

What is the formula for the area of a trapezium?

$$A=\frac12\left(a+b\right)h$$

When do you use each version of the cosine rule:

• $$a^2=b^2+c^2-2bc\cos A$$

• $$\cos A=\frac{b^2+c^2-a^2}{2bc}$$

• finding a side — SAS

• finding an angle — SSS

What is the cosine rule? (2)

• $$a^2=b^2+c^2-2bc\cos A$$

• $$\cos A=\frac{b^2+c^2-a^2}{2bc}$$

What is this circle theorem?

Angle at the centre is twice the angle at the circumference

What is this circle theorem?

Radius meets a tangent at 90°

What is this circle theorem?

Tangents from a common point are equal in length

What is this circle theorem?

Angle inside a semi-circle is always 90°

What is this circle theorem?

Opposite angles in a cyclic quadrilateral sum to 180°

What is this circle theorem?

Angles in the same segment are equal

What is this circle theorem?

Alternate segment theorem

What is the value of $n$?

$$27^{\left(n-1\right)}=\frac{\sqrt3}{9}$$

• $$\left(3^3\right)^{\left(n-1\right)}=\frac{3^{\frac12}}{3^2}$$

  • $$\frac12-2=\frac12-\frac42=-\frac32$$

• $$3^{3\left(n-1\right)}=3^{-\frac32}$$

• $$3\left(n-1\right)=-\frac32$$

  • $$3n-3=-\frac32$$

  • $$3n=\frac32$$

  • $n=\frac12$

Shapes A and B are similar.

Work out the perimeter of shape B. [4 marks]

• $$20 ÷ 8 = 2.5$$

• $$4 × 2.5 = 10$$

• $$7 × 2.5 = 17.5$$

• $$6 × 2.5 = 15$$

• $$3 × 2.5 = 7.5$$

• $$2 ×2.5 = 5$$

• $$20 + 10 + 17.5 + 15 + 7.5 + 5 = 75$$

  • $75$ cm

ACEG and BCDF are straight lines.

AB, DE and FG are parallel lines.

AC = 8 cm BC = 6.4 cm DE = 7.5 cm FG = 11.25 cm EG = 2.5 cm

Work out the length of DF. [5 marks]

CE:

• Let CE = $x$

• $$\frac{x+2.5}{x}=\frac{11.25}{7.5}$$

  • $$\frac{x+2.5}{x}=\frac32$$

  • $$2(x+2.5)=3x$$

  • $$x=5$$

CD:

• Let CD = $y$

• $$\frac{y}{6.4}=\frac58$$

  • $$8y=32$$

  • $$y=4$$

DF:

• Let DF = $z$

• $$\frac{4+z}{4}=\frac32$$

  • $$2\left(4+z\right)=12$$

  • $z=2$

ABC is a triangle.

CDEF is a parallelogram such that:

D is the midpoint of AC

E is the midpoint of AB

F is the midpoint of BC

Prove that triangle ADE is congruent to triangle BEF. [4 marks]

• EB = AE — E is the midpoint of AB (given)

• AD = EF

  • D is the midpoint of AC (given)

    • AD = CD

  • CD = EF — opposite sides of a parallelogram are parallel
    equal

• DE = FB

  • F is the midpoint of CB (given)

    • CF = FB

  • CF = DE — opposite sides of a parallelogram are equal

• Congruent by SSS

Name this angle fact.

An exterior angle of a triangle is equal to the sum of the interior opposite angles.

What is the formula for the volume of a pyramid?

$\frac13$ area of base × perpendicular height

What is the formula for the area of a parallelogram?

$$b×h$$

How do you find the midpoint between two coordinates:

$\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$

$$Midpoint=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$$

How do you find the distance between two coordinates:

$\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$

$$d=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}$$

Write terms two to six of this sequence:

• first term — $T(1) = 2$

• sequence — $T(n) = 2T(n - 1)$

4, 8, 16, 32, 64

There are some red counters and some blue counters in a bag.

The ratio of red counters to blue counters is $4:1$

Two counters are removed at random.

The probability that both the counters taken are red is $\frac{22}{35}$

Work how many blue counters are in the bag.

• red counters = $4x$

• blue counters = $x$

$P(RR)$:

• $$\frac{4x}{5x}\cdot\frac{4x-1}{5x-1}=\frac{16x^2-4x}{25x^2-5x}$$

• $$\frac{16x^2-4x}{25x^2-5x}=\frac{22}{35}$$

  • $$35\left(16x^2-4x\right)=22\left(25x^2-5x\right)$$

  • $$560x^2-140x=550x^2-110x$$

  • $$10x^2-30x$$

  • $$10x=30$$

• $x=3$

Parallel lines have the same ______________.

gradient

Perpendicular lines have gradients that are ______________ _____________ of each other.

negative reciprocals

What is the equation for sine symmetry?

• $\sin\left(30\degree\right)=0.5$ and $\sin\left(150\degree\right)=0.5$

$$sin(x) = sin(180 - x)$$

What is the equation for cosine symmetry?

• $\cos\left(60\degree\right)=0.5$ and $\cos\left(300\degree\right)=0.5$

$$\cos(x)=\cos(360-x)$$

What is the equation for tan symmetry?

• $\tan\left(45\degree\right)=1$ and $\tan\left(225\degree\right)=1$

$$\tan(x)=\tan(180+x)$$

When do you use the ambiguous case of the sine rule?

• SSA — not angle between sides

• finding an angle

• side opposite known angle is shorter than other side given

Draw the line:

$y = -x$ or $-y = x$

Draw the line:

$y = x$ or $x = y$

The floor plan of an office building is drawn using a scale of $1:20$

On the plan, a conference room has a floor area of $115cm^2$

Work out the real area of the floor of this conference room.

Give your answer in $m^2$.

• Area scale factor = $1:400$

• $$400 × 115 = 46000$$

• $$46000 ÷ 10000 = 4.6$$

• $4.6m^2$

$1m^2$ = _______ $cm^2$

$10000$ or $100^2$

Describe an exponential graph.

• $$y = k^x$$

  • k > 0

• k >1 — graph shows increasing curve

• $k = 1$ — graph is constant + equal to 1

• 0 < k < 1 ph is a decreasing curve (gets closer to 0 but not = to 0)

A pattern is made from four identical squares.

The sides of the squares are parallel to the axes.

Point A has coordinates $(6, 7)$

Point B has coordinates $(38, 36)$

Point C is marked on the diagram.

Work out the coordinates of C.

• $$4x = 32$$

  • $$x = 8$$

• $$3x + y = 29$$

  • $$y = 5$$

• $$8 + 5 + 7 = 20$$

• $$8 + 8 + 6 = 22$$

$(22, 20)$

C is a circle with equation $x² + y² = 32.4$

Point P has coordinates $(a, b)$ where $a$and $b$ are positive.

The equation of the tangent to C at the point P is parallel to the line with equation $y = 9 – 3x$

Find the values of $a$ and $b$.