ADVANCED

Binomial Theorem (1)

Find the first three terms in the expansion of:

$(2a - 3b)^5$

$=32a^5 - 240a^4b \\+720a^3b^2 ...$

Compound Interest

If £240 is invested with an interest rate of 4% compounded quarterly, find the value of the investment after 9 years. £343.38

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:

$x^2+x-6$

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

Factorise (Quadratic 2)

Factorise:

$6x^2+x-1$

$(2x+1)(3x-1)$

Graph (Linear)

Draw a rough sketch of the graph of:

$y=2x-2$

Gradient 2
y intercept -2

Indices

What is the value of:

$1^{1}$

$= 1$

Trigonometry (Angle)

Find angle BCA if AB = 5.2m and BC = 6.8m. 49.9^o

Trigonometry (Side)

Find AC if angle ABC = 37^o and AB = 4m. 3.01m

Venn Diagrams

Describe the red region.

Differentiation (1)

$y = 2x^3 - 6x^2 + 7x$

Find $\dfrac{dy}{dx}$

$6x^2 - 12x + 7$

Differentiation (2)

$y = \dfrac{3}{x^2} - 4\sqrt[5]{x}$

Find $\frac{dy}{dx}$

$-\frac{6}{x^3} - \frac{4}{5}x^{-\frac{4}{5}}$

Differentiation (3)

$y=9\ln (4x^2+5)$

Find $\dfrac{dy}{dx}$

$72x(4x^2+5)^{-1}$

Differentiation (4)

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

Find $\dfrac{dy}{dx}$

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

Differentiation (5)

$y=\frac{2x^2}{4x-1}$

Find $\dfrac{dy}{dx}$

$\frac{(8x^2-4x)}{(4x-1)^2}$

Differentiation (6)

Find the equation of the tangent to the curve:
$y = -2x^2 - 4x + 6$
where $x = 3$
$y = 24 - 16x$

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)

$y =9x^2 - 4x + 4$

Find $\int y \quad dx$

$3x^3 - 2x^2 + 4x+c$

Binomial Distribution

A game is played 17 times and the probability of winning is 0.9. Calculate the probability of winning exactly 2 times.   0.000000000000110

Formulas

Make up a maths question using this:

$\triangle = b^2-4ac$

Quadratic equation discriminant

Greek Letters

What letter is this?

Sequences (Arithmetic)

Two terms of an arithmetic sequence:
$u_{8} = 35$
$u_{12} = 59$
Find the sum of the first 15 terms.525

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,
AB = 7.2cm.
BC = 9.4cm.
CÂB = 76.8°.
Find angle BĈA.

48.2°

Sigma

Evaluate:

$$\sum_{n=3}^{8} n^2 - 7n$$

-32

Discriminant

$f(x)=-4x^2+9x+1$

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

Completing The Square

$f(x)=x^2+2x-4$

By completing the square find the coordinates of the vertex.
(-1, -5)

Logarithms

Solve for x:

$\log_2(x) = 4$

16

Integration (3)

Find the integral:

$\int \dfrac{\ln(x)}{x} \;dx$

$\dfrac{\ln(x)^2}{2}+c$

Graph (2 points)

Find the equation of the straight line that passes through:

(-5, 5) and (8, -8)

$y=-x+0$

Functions (Inverse)

Find the inverse of the function $f$:

$f(x)= \sqrt{x}-4$

$(x+4)²$

Functions (Composite)

$f(x)=5x+1 \\ g(x)=x^2 \\[1cm] \text{Find }gf(3x)$

$225x^2+30x+1$

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

$y=2^x$

Graph (Fill)

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

Trig (Special Angles)

Without a calculator find the exact value of

$\dfrac{\sqrt{3}}{4}$

Trig (Large Angles)

Without a calculator find the exact value of

$0$

Simultaneous Eqns (3)*

Solve:

$j+k+l= 11 \\ 2j-3k+9l= 49\\ -j+k-3l=-17$

j = 2, k = 3, l = 6

Radian Measures

Find the perimeter of a sector with radius 5.7cm and angle $\frac{\pi}{4}$

🍕

15.9cm

Combinatorics*

Ansh is with eight people in a queue. How many ways can they line up without Ansh being at the back?

322560

Asymptotes (Ob)*

Find the equations of the asymptotes of:

x=3, y=2x-2

Sequences (Geometric)

The sum of the first 6 terms of a geometric sequence is 5460 and the sum of the first 7 terms is 21844. What is the first term?

4

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:

$\int^{8}_{0} e^x dx$

$e^{8}- 1 \approx 2980$

Probability (Conditional)

Given equal populations of Type X and Type Y bacteria, with mutation rates of 50% and 30% respectively, if a mutated bacterium is found, what's the probability it's Type Y?

$0.375$

Vectors*

Find the point of intersection of these planes:

$\Pi_1: \quad 2x + y - 3z = -5$

$\Pi_2: \quad x - 3y + 2z = 1$

$\Pi_3: \quad 3x - 2y + z = 2$

$(1,2,3)$

Graph (Advanced)*

Sketch the graph of:

Graph Plotter

Complex Numbers 1*

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

$\frac{2}{25}-\frac{3}{50}i$

Integration (4)*

Evaluate:

$\int x\cos{x}\; dx$

$xsinx+cosx+c$

Trig (Identities)*

Simplify:

$8\sin{x}$

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 formula for compound interest?

$FV = PV(1 + \frac{r}{100k})^{kn}$

Maclaurin Series*

Show how the first four terms of the Maclaurin series are obtained for
$f(x) = \sec(x)$

$1 + \frac{x^2}{2} + \frac{5x^4}{24} + \frac{61x^6}{720}$

Complex Numbers 2*

Solve for $z$
$$z^4 = \sqrt{3}+i$$

$\sqrt[4]{2} cis \frac{\pi}{24},\sqrt[4]{2} cis \frac{13\pi}{24} \\ \sqrt[4]{2} cis \frac{-11\pi}{24}, \sqrt[4]{2} cis \frac{-23\pi}{24}$

Probability (Counting)*

5 alphabet blocks A, E, P, R and S are placed at random in a row. What is the likelihood that they spell out either SPEAR or PARSE?

1/60 or 1.67%

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:
$$\sqrt{27}$$
$3\sqrt{3}$

Surds (2)

Simplify:
$$\dfrac{9}{\sqrt{6}}$$ $\frac{9\sqrt{6}}{6} = \frac{3\sqrt{6}}{2}$

Surds (3)

Simplify

$8\sqrt{3}(1 - \sqrt{3})$

$8\sqrt{3} - 24$

Surds (4)

Simplify:
$$\dfrac{4}{6 - \sqrt{5}}$$ $\frac{24 + 4\sqrt{5}}{31}$

Last Lesson

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

?