ADVANCED
Refreshing Revision

Binomial Theorem (1)

Find the first three terms in the expansion of:

\((3a - 2b)^4\)

\(=81a^4 - 216a^3b \\+216a^2b^2 ...\)

Compound Interest

If £120 is invested with an interest rate of 4% compounded quarterly, find the value of the investment after 5 years. £146.42

Coordinates (Square)

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

\((5,2),(9,8),(-1,6)\)

(3,12)

Normal Distribution

\( X \sim N(65, 9^2)\)

Find

\( P(36\lt X \lt48) \)

\(0.0288\)

Factorise (Quadratic 1)

Factorise:

\(x^2-2x-3\)

\((x+1)(x-3)\)

Factorise (Quadratic 2)

Factorise:

\(8x^2+2x-3\)

\((4x+3)(2x-1)\)

Graph (Linear)

Draw a rough sketch of the graph of:


\(y=x\)

Gradient 1
y intercept 0

Indices

What is the value of:

\(1^{1}\)

\(= 1\)

Trigonometry (Angle)

Find angle BCA if AB = 4.8m and BC = 6.1m. 51.9o

Trigonometry (Side)

Find AB if angle ABC = 33o and BC = 4.2m. 3.52m

Venn Diagrams

Describe the red region.

Circle part Circle part

Differentiation (1)

\(y = 5x^3 - 6x^2 + 5x\)

Find \( \dfrac{dy}{dx}\)

\(15x^2 - 12x + 5\)

Differentiation (2)

\(y = \dfrac{5}{x^3} - 7\sqrt[8]{x}\)

Find \( \frac{dy}{dx}\)

\(-\frac{15}{x^4} - \frac{7}{8}x^{-\frac{7}{8}}\)

Differentiation (3)

\(y=5\ln (4x^2+5)\)

Find \( \dfrac{dy}{dx}\)

\(40x(4x^2+5)^{-1}\)

Differentiation (4)

\(y=x(5x^2+6)^4\)

Find \( \dfrac{dy}{dx}\)

\((5x^2+6)^4+40x^2(5x^2+6)^3\)

Differentiation (5)

\(y=\frac{x+2}{x-5}\)

Find \( \dfrac{dy}{dx}\)

\(-\frac{7}{(x-5)^2}\)

Differentiation (6)

Find the equation of the tangent to the curve:
\(y = -5x^2 + 7x - 3\)
where \(x = 2\)
\(y = 17 - 13x\)

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)

\(y =6x^2 - 4x + 2\)

Find \( \int y \quad dx\)

\(2x^3 - 2x^2 + 2x+c\)

Binomial Distribution

A game is played 16 times and the probability of winning is 0.6. Calculate the probability of winning exactly 6 times.   0.0392

Formulas

What's this?

\( P(A|B) = \dfrac{P(A \cap B)}{P(B)} \)

Conditional probability formula

Greek Letters

What letter is this?

Greek Letter Greek Letter

Sequences (Arithmetic)

Two terms of an arithmetic sequence:
\(u_{8} = -31\)
\(u_{18} = -61\)
Find the sum of the first 47 terms.-3713

Asymptotes (HV)

Find the equations of the asymptotes of:

\(y=\dfrac{4-7x}{3-14x}\)

\(x=\frac{3}{14},y=\frac{1}{2}\)

Trig Advanced

In the triangle ABC,
AB = 9.3cm.
BC = 8.4cm.
CA = 12.5cm.
Find angle CÂB.

42.2°

Sigma

Evaluate:

$$\sum_{n=1}^{4} 2^n$$

30

Discriminant

\(f(x)=2x^2-8x-5\)

What is the value of the discriminent and what does it indicate?
104, Two distinct roots

Completing The Square

\(f(x)=x^2-3x-1\)

By completing the square find the coordinates of the vertex.
(1.5, -3.25)

Logarithms

What is the value of \(\ln{e^3}\) ?


3

Integration (3)

Find the integral:

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


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

Graph (2 points)

Find the equation of the straight line that passes through:

(-8, -18) and (2, 2)

\(y=2x-2\)

Functions (Inverse)

Find the inverse of the function \(f\):

\(f(x)=\frac{\sqrt{x}-17}{18}\)


\((18x+17)²\)

Functions (Composite)

\(f(x)=x^2-1 \\[1cm] \text{Find }f \bullet f(x) \\\)

\(x^4-2x^2\)

Standard Form

Write in standard form:
\(a \times 10^p \times b\times 10^q\)
where \(a \times b \) is a single digit number \((1 \le ab \lt 10)\)

\(ab\times10^{p+q}\)

Graph (Mixed)

Draw a rough sketch of

\(y=x^2-8\)

Sketch

Graph (Fill)

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

Jar Graph

Trig (Special Angles)

Without a calculator find the exact value of

$$\cos{30°} \div \sin{\frac{\pi}{3}}$$

\(1\)

Trig (Large Angles)

Without a calculator find the exact value of

$$\sin{\dfrac{13\pi}{6}}$$

\(\dfrac{1}{2}\)

Simultaneous Eqns (3)*

Solve:

\(2x+y-3z= 3 \\ 3x+y+z= 9 \\ x-y+2z = 2\)

x = 2, y = 2, z = 1

Radian Measures

Find the perimeter of a sector with radius 6.6cm and angle \( \frac{\pi}{4}\)

🍕

18.4cm

Combinatronics*

A safe has a nine-digit code. How many possibilities are there if no digit can be repeated and the code must be odd?

1814400

Asymptotes (Ob)*

Find the equations of the asymptotes of:

$$y=\dfrac{2x^2-8x+8}{x-3}$$

x=3, y=2x-2

Sequences (Geometric)

The sum of the first 7 terms of a geometric sequence is 39062 and the sum of the first 8 terms is 195312. What is the first term?

2

Binomial Theorem (2)*

Find the first 4 terms in the expansion of:

\(\dfrac{1}{(1+3x)^3}\)

\(1-9x+54x^2-270x^3\)

Integration (2)

Evaluate:

\(\int^{80}_{40} \dfrac{1}{x} dx\)


\(\ln{2} \approx 0.693\)

Probability (Conditional)

In a bookstore with equally sized fiction and non-fiction sections, if a hardcover book is selected (70% of fiction, 40% of non-fiction are hardcovers), what's the probability it's non-fiction?

\(0.364\)

Vectors*

There are eight different ways that three planes can relate in three dimensions. One is where they all intersect at a point. How many of the other seven ways can you sketch or describe?

Solution

Graph (Advanced)*

Sketch the graph of:

$$y=1^{\sin{x}}$$

Graph Plotter

Complex Numbers 1*

Simplify
$$ \dfrac{1-4i}{1+5i}$$

\(\frac{21}{26}-\frac{9}{26}i\)

Integration (4)

Evaluate:

\(\int 4x\sin{\left( \frac{x}{2} \right)}\; dx\)


\(-8xcos\frac{x}{2}+16sin\frac{x}{2}+c\)

Trig (Identities)*

Simplify:

$$\tan{x}\cot{x}$$

\(1\)

$$ \DeclareMathOperator{cosec}{cosec} $$

Integration (Volume)*

Find the volume of revolution when \(y=x\) is rotated about the x-axis for \(0 \le x \le 1\)


\(\frac{\pi}{3}\) cubic units

Miscellaneous

What is the binomial theorem?

Clue: Expand \( (a + b)^n \)

Maclaurin Series*

Show how the first four terms of the Maclaurin series are obtained for
\(f(x) = (1 + x)^n\)

\(1 + nx + \frac{n(n-1)x^2}{2} + \frac{n(n-1)(n-2)x^3}{6}\)

Complex Numbers 2*

Expand and simplify:
$$ (\sqrt{3}+i)^8 $$

\(-128-128\sqrt{3}i \\ \text{or } 128(-1-\sqrt{3}i)\)

Probability (Counting)*

A school committee of 8 is chosen at random from 12 senior students and 4 junior students. Find the probability that all four junior students are chosen.

1/26 or 3.85%

Proof by Induction*

Prove by mathematical induction that the sum of the first \( n \) odd numbers is \( n^2 \)

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

Last Lesson

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

?


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