ADVANCED

Binomial Theorem (1)

Find the first three terms in the expansion of:

$(2a - 3b)^8$

$=256a^8 - 3072a^7b \\+16128a^6b^2 ...$

Compound Interest

If £180 is invested with an interest rate of 4% compounded monthly, find the value of the investment after 8 years. £247.75

Coordinates (Square)

Here are the coordinates of 3 vertices of a square, what are the coordinates of the 4^th?

$(5,5),(10,9),(1,10)$

(6,14)

Normal Distribution

$X \sim N(50, 5^2)$

Find

$P(40\lt X \lt60)$

$0.955$

Factorise (Quadratic 1)

Factorise:

$x^2-2x-3$

$(x+1)(x-3)$

Factorise (Quadratic 2)

Factorise:

$2x^2+x-6$

$(x+2)(2x-3)$

Graph (Linear)

Draw a rough sketch of the graph of:

$2y=x+4$

Gradient 0.5
y intercept 2

Indices

What is the value of:

$16^{\frac{1}{2}}$

$= 4$

Trigonometry (Angle)

Find angle ABC if AC = 3.8m and BC = 4.9m. 50.9^o

Trigonometry (Side)

Find BC if angle BCA = 37^o and AC = 4.4m. 5.51m

Venn Diagrams

Describe the red region.

Differentiation (1)

$y = 5x^3 - 5x^2 + 6x$

Find $\dfrac{dy}{dx}$

$15x^2 - 10x + 6$

Differentiation (2)

$y = \dfrac{5}{x^6} - 7\sqrt[8]{x}$

Find $\frac{dy}{dx}$

$-\frac{30}{x^7} - \frac{7}{8}x^{-\frac{7}{8}}$

Differentiation (3)

$y=(4x^7+9)^8$

Find $\dfrac{dy}{dx}$

$224x^6(4x^7+9)^7$

Differentiation (4)

$y=x(5x+7)^3$

Find $\dfrac{dy}{dx}$

$(5x+7)^3+15x(5x+7)^2$

Differentiation (5)

$y=\frac{ \ln x}{x^2}$

Find $\dfrac{dy}{dx}$

$\frac{(1-2lnx)}{x^3}$

Differentiation (6)

Find the equation of the tangent to the curve:
$y = 5x^2 + 7x + 3$
where $x = 1$
$y = 17x - 2$

Differentiation (7)

Find the equation of the normal to the curve:
$y = -5x^2 + 7x - 3$
where $x = 2$
$y = \frac{x}{13} - \frac{119}{13}$

Integration (1)

$y =21x^2 - 18x + 8$

Find $\int y \quad dx$

$7x^3 - 9x^2 + 8x+c$

Binomial Distribution

A game is played 18 times and the probability of winning is 0.2. Calculate the probability of winning exactly 16 times.   0.000000000642

Formulas

Make up a maths question using this:

$\int \dfrac{1}{x} = \ln |x| + c$

Reciprocal Integral formula

Greek Letters

What letter is this?

Sequences (Arithmetic)

Two terms of an arithmetic sequence:
$u_{5} = 14$
$u_{11} = 44$
Find the sum of the first 18 terms.657

Asymptotes (HV)

Find the equations of the asymptotes of:

$y=3\left(\dfrac{2x+3}{7-x}\right)$

$x=7,y=-6$

Trig Advanced

In the triangle ABC,
BC = 7.8cm.
CA = 8.4cm.
BĈA = 45.6°
Find AB to 1 dp.

6.3cm

Sigma

Evaluate:

$$\sum_{n=3}^{5} 2^n$$

56

Discriminant

$f(x)=-9x^2+2x+9$

What is the value of the discriminant and what does it indicate?
328, Two distinct roots

Completing The Square

$f(x)=x^2-2x-1$

By completing the square find the coordinates of the vertex.
(1, -2)

Logarithms

Solve for x:

$\log_3x = 2$

9

Integration (3)

Find the integral:

$\int (x+3)\sqrt{x^2+6x+8} \;dx$

$\frac{1}{3}(x^2+6x+8)^{\frac32}+c$

Graph (2 points)

Find the equation of the straight line that passes through:

(-8, -15) and (7, 15)

$y=2x+1$

Functions (Inverse)

Find the inverse of the function $f$:

$f(x)=\frac{6+ x}{3}$

$3x-6$

Functions (Composite)

$f(x)=3x+2 \\ g(x)=2x^2 \\[1cm] \text{Find }gf(x)$

$18x^2+24x+8$

Standard Form

Write in standard form:
$(a \times 10^4) \div (b\times 10^{-2})$
where $a \div b$ is a decimal number $(0.1 \le \frac{a}{b} \lt 1)$

$\frac{10a}{b}\times10^5$

Graph (Mixed)

Draw a rough sketch of

$y=\sin(x)$

Graph (Fill)

Sketch a height-time graph as this jar is filled.

Trig (Special Angles)

Without a calculator find the exact value of

$\sqrt{2}$

Trig (Large Angles)

Without a calculator find the exact value of

$\sqrt{3}$

Simultaneous Eqns (3)*

Solve:

$5a+2b+c=26 \\ 3a+4b+2c= 31 \\ a+5b+c=26$

a = 3, b = 4, c = 3

Radian Measures

Find the area of a sector with radius 7.1cm and angle $\frac{2\pi}{3}$

🍕

52.8cm²

Combinatorics*

How many ways can fourteen people be divided into two equal groups?

1716

Asymptotes (Ob)*

Find the equations of the asymptotes of:

x=-2/3, y=-2x

Sequences (Geometric)

Evaluate:
$$\sum_{k=1}^{15} 5 \times (-2)^k$$

-109230

Binomial Theorem (2)*

Find the first 4 terms in the expansion of:

$(1+4x)^{\frac{3}{2}}$

$1+6x+6x^2-4x^3$

Integration (2)

Evaluate:

$\int^{9}_{2} x^2-2x+7 \; dx$

$212$

Probability (Conditional)

27 Scouts went hiking. 16 got lost, 14 got blisters, and 9 got both lost and blisters. Find the probability that a randomly selected Scout got blisters, given that they were not lost.

$\dfrac{5}{11}$

Vectors*

Find the cartesian equation of this plane:

$\mathbf{r} = \begin{pmatrix} -2 \\ 2 \\ 3 \end{pmatrix} \; + \; s \begin{pmatrix} 3 \\ 1 \\ 0 \end{pmatrix} \; + \; t \begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}$

2x-6y+5z=-1

Graph (Advanced)*

Sketch the graph of:

Graph Plotter

Complex Numbers 1*

Simplify
$$(5-4i)(4-3i)$$

$8-31i$

Integration (4)*

Evaluate:

$\int e^x\sin{x}\; dx$

$\frac{e^x}{2}(sinx-cosx)+c$

Trig (Identities)*

Simplify:

$-\cos{x}$

Integration (Volume)*

Find the volume of revolution when $y=x^3$ is rotated about the y-axis for $1 \le y \le 2$

$\approx 4.10$ cubic units

Miscellaneous

What is the inverse of a function?

Clue: swaps the roles of x and y

Maclaurin Series*

Show how the first four terms of the Maclaurin series are obtained for
$f(x) = \frac{1}{x^2 + 1}$

$1 - x^2 + x^4 - x^6$

Complex Numbers 2*

Solve for $z$
$$z^3 = - 8i$$

$\sqrt{3}-i,2i,-\sqrt{3}-i$

Probability (Counting)*

6 girls sit at random on 6 seats in a row. Determine the probability that the two friends Karen and Julie sit at the ends of the row.

1/15 or 6.67%

Proof by Induction*

Prove by mathematical induction that $2^n > n^2$ for all integers $n$ greater
than four

Show true for n=1, assume true for n=k, prove for n=k+1

Surds (1)

Simplify:
$$\sqrt{18}$$
$3\sqrt{2}$

Surds (2)

Simplify:
$$\dfrac{3}{4\sqrt{2}}$$ $\frac{3\sqrt{2}}{8}$

Surds (3)

Simplify

$(3 + \sqrt{5})(3 - \sqrt{5})$

$4$

Surds (4)

Simplify:
$$\dfrac{6}{5 + \sqrt{2}}$$ $\frac{30 - 6\sqrt{2}}{23}$

Last Lesson

Write down a summary of your last Maths lesson focussing on what you learnt.

?