ADVANCED
Refreshing Revision

Binomial Theorem (1)

Find the first three terms in the expansion of:

\((2a - 3b)^6\)

\(=64a^6 - 576a^5b \\+2160a^4b^2 ...\)

Compound Interest

If £100 is invested with an interest rate of 3% compounded monthly, find the value of the investment after 7 years. £123.34

Coordinates (Square)

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

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

(5,10)

Normal Distribution

\( X \sim N(27.1, 1.8^2)\)

Find

\( P(28.1\lt X \lt29.1) \)

\(0.156\)

Factorise (Quadratic 1)

Factorise:

\(x^2+x-2\)

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

Factorise (Quadratic 2)

Factorise:

\(x^2-3x-4\)

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

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:

\(4^{-2}\)

\(= \frac{1}{16}\)

Trigonometry (Angle)

Find angle BCA if AB = 3.3m and AC = 5.1m. 32.9o

Trigonometry (Side)

Find BC if angle BCA = 24o and AC = 3.9m. 4.27m

Venn Diagrams

Describe the red region.

Circle part Circle part

Differentiation (1)

\(y = 6x^3 - 4x^2 + 3x\)

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

\(18x^2 - 8x + 3\)

Differentiation (2)

\(y = \dfrac{9}{x^4} - 7\sqrt[8]{x}\)

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

\(-\frac{36}{x^5} - \frac{7}{8}x^{-\frac{7}{8}}\)

Differentiation (3)

\(y=\sqrt{7x^6-6x}\)

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

\((21x^5-3)(7x^6-6x)^{-\frac{1}{2}}\)

Differentiation (4)

\(y=x^3 \ln x\)

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

\(3x^2lnx+x^2\)

Differentiation (5)

\(y=\frac{x}{\sin x}\)

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

\(\frac{(sinx-xcosx)}{sin^2x}\)

Differentiation (6)

Find the equation of the tangent to the curve:
\(y = 3x^2 - 6x + 9\)
where \(x = 2\)
\(y = 6x + 3\)

Differentiation (7)

Find the equation of the normal to the curve:
\(y = 2x^2 - x + 3\)
where \(x = -1\)
\(y = \frac{x}{5} + 6\frac{1}{5}\)

Integration (1)

\(y =15x^2 - 8x + 5\)

Find \( \int y \quad dx\)

\(5x^3 - 4x^2 + 5x+c\)

Binomial Distribution

A game is played 10 times and the probability of winning is 0.7. Calculate the probability of winning exactly 3 times.   0.00900

Formulas

Make up a maths question using this:

\(^nC_r=\dfrac{n!}{r!(n-r)!}\)

Combinations
(from n choose r)

Greek Letters

What letter is this?

Greek Letter Greek Letter

Sequences (Arithmetic)

Two terms of an arithmetic sequence:
\(u_{9} = 9\)
\(u_{14} = 19\)
Find the sum of the first 17 terms.153

Asymptotes (HV)

Find the equations of the asymptotes of:

\(y=\dfrac{3x-5}{6x-12}\)

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

Trig Advanced

In the triangle ABC,
BĈA = 38.6°.
BC = 8.2cm.
AB̂C = 41.34°.
Find CA to 1 dp.

5.5cm

Sigma

Evaluate:

$$\sum_{n=2}^{8} 97 - n^2$$

476

Discriminant

\(f(x)=-8x^2+8x-6\)

What is the value of the discriminant and what does it indicate?
-128, No real roots

Completing The Square

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

By completing the square find the coordinates of the vertex.
(1.5, -1.25)

Logarithms

Simplify \(\log_{10}10^5\)


5

Integration (3)

Find the integral:

\(\int \cos(x)e^{\sin(x)} \;dx\)


\(e^{\sin(x)}+c\)

Graph (2 points)

Find the equation of the straight line that passes through:

(-3, 12) and (9, 0)

\(y=-x+9\)

Functions (Inverse)

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

\(f(x)=\sqrt{\frac{x-6}{3}}\)


\(3x²+6\)

Functions (Composite)

\(\text{Find }f(x) \text{ if} \\ f(a^3)=2a^6 \\\)

\(f(x)=2x^2\)

Standard Form

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

\(\frac{a}{b}\times10^{-2}\)

Graph (Mixed)

Draw a rough sketch of

\(y=x^3-4x\)

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

$$\cos{\frac{\pi}{4}} + \sin{45°}$$

\(\sqrt{2}\)

Trig (Large Angles)

Without a calculator find the exact value of

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

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

Simultaneous Eqns (3)*

Solve:

\(2x+y-3z= -1 \\ 3x+y+z= 20 \\ x-y+2z = 12\)

x = 5, y = 1, z = 4

Radian Measures

Find the perimeter of a sector with radius 2.1cm and angle \( \frac{5\pi}{6}\)

🍕

9.70cm

Combinatorics*

In how many ways can 11 different books be arranged on a shelf if 4 of them must be together?

967680

Asymptotes (Ob)*

Find the equations of the asymptotes of:

$$y=\dfrac{8x^2-19x-15}{1-2x}$$

x=1/2,y=15/2-4x

Sequences (Geometric)

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

-109230

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:

\(\int^{2}_{0} x^2-2x+7 \; dx\)


\(12.7\)

Probability (Conditional)

Each afternoon the probability my cat sleeps is 0.5 and the probability that my dog sleeps is 0.8. The probability that the dog sleeps given that the cat is sleeping is 0.9. Find the probability that both sleep in the afternoon.

\(0.45\)

Vectors*

Find the angle between the plane and the line:

\(\Pi: \quad 4x+4y-2z=7\)

\(L: \quad x+1= \dfrac{4-y}{2} = 3-z \)

\( \approx 7.82^o \)

Graph (Advanced)*

Sketch the graph of:

$$y=|\sqrt{x}-3|$$

Graph Plotter

Complex Numbers 1*

Simplify
$$ (3+i)^{-2} $$

\(\frac{2}{25}-\frac{3}{50}i\)

Integration (4)*

Evaluate:

\(\int x^2 \ln{x}\; dx\)


\(\frac{x^3}{9}(3\ln x-1)+c\)

Trig (Identities)*

Simplify:

$$\sec{x}-\tan{x}\sin{x}$$

\(\cos{x}\)

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

Integration (Volume)*

Find the volume of revolution when \(y=\sqrt[3]{x^2}\) is rotated about the y-axis for \(2 \le y \le 3\)


\(\frac{65\pi}{4}\) cubic units

Miscellaneous

Describe the behavior of a function at its asymptote.

Clue: approaches but never reaches

Maclaurin Series*

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

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

Complex Numbers 2*


Solve for \(z\)
$$ z^4 = - 16 $$

\(\sqrt{2}+i\sqrt{2},-\sqrt{2}+i\sqrt{2} \\ \sqrt{2}-i\sqrt{2},-\sqrt{2}-i\sqrt{2}\)

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 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

Surds

Simplify

\(7\sqrt{13} - 9\sqrt{13}\)


\(-2\sqrt{13}\)

Last Lesson

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

?


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