Find the first three terms in the expansion of:
\((2a - 3b)^5\)
\(=32a^5 - 240a^4b \\+720a^3b^2 ...\)
If £240 is invested with an interest rate of 2% compounded monthly, find the value of the investment after 6 years. £270.57
Here are the coordinates of 3 vertices of a square, what are the coordinates of the 4th?
\((3,2),(9,7),(-2,8)\)
(4,13)
\( X \sim N(-25, 3^2)\)
Find
\( P(-20\lt X \lt-10) \)
\(0.0478\)
Factorise:
\(x^2+x-12\)
\((x+4)(x-3)\)
Factorise:
\(2x^2-3x-2\)
\((2x+1)(x-2)\)
Draw a rough sketch of the graph of:
\(y=-2x-1\)
Gradient -2
y intercept -1
What is the value of:
\(3^{-3}\)
\(= \frac{1}{27}\)
Find angle BCA if AC = 5m and BC = 6.5m. 39.7o
Find AC if angle ABC = 66o and AB = 5.2m. 11.7m
Describe the red region.
\(y = 8x^3 - 4x^2 + 2x\)
Find \( \dfrac{dy}{dx}\)
\(24x^2 - 8x + 2\)
\(y = \dfrac{7}{x^{4}} - 5\sqrt[6]{x}\)
Find \( \frac{dy}{dx}\)
\(-\frac{28}{x^{5}} - \frac{5}{6}x^{-\frac{5}{6}}\)
\(y=(8x^6+5)^8\)
Find \( \dfrac{dy}{dx}\)
\(384x^5(8x^6+5)^7\)
\(y=\sin x \cos x\)
Find \( \dfrac{dy}{dx}\)
\(cos^2x-sin^2x\)
\(y=\frac{e^{3x}}{ \cos 4x}\)
Find \( \dfrac{dy}{dx}\)
\(\frac{(3e^{3x}cos4x+4e^{3x}sin4x)}{cos^24x}\)
Find the equation of the tangent to the curve:
\(y = -5x^2 + 7x - 3\)
where \(x = 2\)
\(y = 17 - 13x\)
Find the equation of the normal to the curve:
\(y = 4x^2 + 2x - 1\)
where \(x = -2\)
\(y = \frac{x}{14} + 11\frac{1}{7}\)
\(y =6x^2 - 18x + 4\)
Find \( \int y \quad dx\)
\(2x^3 - 9x^2 + 4x+c\)
A game is played 18 times and the probability of winning is 0.4. Calculate the probability of winning exactly 6 times. 0.166
Make up a maths question using this:
\( A = 4\pi r^2 \)
Surface area of a sphere
What letter is this?
Two terms of an arithmetic sequence:
\(u_{8} = -16\)
\(u_{17} = -25\)
Find the sum of the first 21 terms.-399
Find the equations of the asymptotes of:
\(y=\dfrac{4-7x}{3-14x}\)
\(x=\frac{3}{14},y=\frac{1}{2}\)
In the triangle ABC,
AB = 8.9cm.
BC = 8.8cm.
CÂB = 35.3°.
Find angle BĈA.
35.8° or 144.2°
Evaluate:
$$\sum_{n=1}^{7} n^2 - 7n$$
-56
\(f(x)=-8x^2+9x-8\)
What is the value of the discriminant and what does it indicate?
-175, No real roots
\(f(x)=x^2-7x+4\)
By completing the square find the coordinates of the vertex.
(3.5, -8.25)
Evaluate \(\log_2(32) \)
5
Find the integral:
\(\int x\sqrt{x^2+3} \;dx\)
\(\frac{1}{3}(x^2+3)^{\frac32}+c\)
Find the equation of the straight line that passes through:
(-4, -15) and (0, -3)
\(y=3x-3\)
Find the inverse of the function \(f\):
\(f(x)=\frac{9+ x}{4}\)
\(4x-9\)
\(f(x)=5x+1 \\ g(x)=x^2 \\[1cm] \text{Find }gf(3x)\)
\(225x^2+30x+1\)
Write in standard form:
\((a \times 10^4) \div (b\times 10^{-2})\)
where \(a \div b \) is a decimal number \((0.1 \le \frac{a}{b} \lt 1)\)
\(\frac{10a}{b}\times10^5\)
Draw a rough sketch of
\(y=3-\dfrac{10}{x}\)
Sketch a height-time graph as this jar is filled.
Without a calculator find the exact value of
$$\sin{\frac{\pi}{4}} \times \cos{45°}$$\(\dfrac{1}{2}\)
Without a calculator find the exact value of
$$\sin{\dfrac{13\pi}{6}}$$\(\dfrac{1}{2}\)
Solve:
\( g-7h-7i=-36 \\ 2g-2h+i= 6\\ 5g+3h+i = 44\)
g = 6, h = 4, i = 2
Find the area of a sector with radius 8.5cm and angle \( \frac{\pi}{6}\)
🍕
18.9cm2
Ansh is with five people in a queue. How many ways can they line up without Ansh being at the back?
600
Find the equations of the asymptotes of:
$$y=\dfrac{-x^2+3x-2}{x}$$x=0,y=3-x
Evaluate:
$$ \sum_{k=1}^{8} \left( \dfrac12 \right)^{k-3} $$
7.97
Find the first 4 terms in the expansion of:
\(\dfrac{1}{\sqrt{4+x}}\)
\(\frac{1}{2}-\frac{x}{16}+\frac{3x^2}{256}-\frac{5x^3}{2048}\)
Evaluate:
\(\int^{\pi/3}_{\pi/6} \sin{x} \; dx\)
\(\dfrac{\sqrt{3}-1}{2}\)
What is the probability that it was machine B that broke down if at least one of the independent machines broke down today, given machine A has a 8% chance and machine B has a 11% chance of breaking down on any given day?
\(0.607\)
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?
Simplify
$$ \sqrt{5-12i} $$
\(3-2i \; \text{ or } -3+2i\)
Evaluate:
\(\int \ln{x}\; dx\)
\(x\ln|x|-x+c\)
Simplify:
$$\dfrac{\cot{x}}{\cosec{x}}$$\(\cos{x}\)
$$ \DeclareMathOperator{cosec}{cosec} $$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
Describe the behavior of a function at its inflection point.
The concavity of the function changes
Show how the first four terms of the Maclaurin series are obtained for
\(f(x) = \frac{1}{x^2 + 1}\)
\(1 - x^2 + x^4 - x^6\)
Solve for \(z\)
$$ z^4 = \sqrt{3}+i $$
\(\sqrt[4]{2} cis \frac{\pi}{24},\sqrt[4]{2} cis \frac{13\pi}{24} \\ \sqrt[4]{2} cis \frac{-11\pi}{24}, \sqrt[4]{2} cis \frac{-23\pi}{24}\)
Of the 22 people on boat, 3 are professional singers. If 4 people get sea sick, determine the chance that all three singers do not.
204/385 or 53.0%
Prove by mathematical induction that the sum of the squares of the first \( n \) natural numbers is \( \frac{n(n + 1)(2n + 1)}{6} \)
Show true for n=1, assume true for n=k, prove for n=k+1
Simplify:
$$\sqrt{45}$$
\(3\sqrt{5}\)
Simplify:
$$\dfrac{7}{2\sqrt{5}}$$\(\frac{7\sqrt{5}}{10}\)
Simplify
\((3 + 2\sqrt{5})(6 - 3\sqrt{5})\)
\(3\sqrt{5}-12\)
Simplify:
$$\dfrac{2}{5 - \sqrt{7}}$$\(\frac{10 + 2\sqrt{7}}{18}\)
Calculate the standard deviation of the following numbers:
27, 29, 30, 31, 33
2
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