Find the first three terms in the expansion of:
\((4a - 2b)^5\)
\(=1024a^5 - 2560a^4b \\+2560a^3b^2 ...\)
If £140 is invested with an interest rate of 4% compounded quarterly, find the value of the investment after 4 years. £164.16
Here are the coordinates of 3 vertices of a square, what are the coordinates of the 4th?
\((5,1),(8,4),(2,4)\)
(5,7)
\( X \sim N(9.03, 0.89^2)\)
Find
\( P(7.15\lt X \lt9.01) \)
\(0.474\)
Factorise:
\(x^2+x-12\)
\((x+4)(x-3)\)
Factorise:
\(5x^2+11x-12\)
\((x+3)(5x-4)\)
Draw a rough sketch of the graph of:
\(y=-2x+1\)
Gradient -2
y intercept 1
What is the value of:
\(5^{-3}\)
\(= \frac{1}{125}\)
Find angle BCA if AB = 3.2m and BC = 4.2m. 49.6o
Find AC if angle ABC = 61o and AB = 3m. 5.41m
Describe the red region.
\(y = 5x^3 - 3x^2 + 4x\)
Find \( \dfrac{dy}{dx}\)
\(15x^2 - 6x + 4\)
\(y = \dfrac{8}{x^{8}} - 5\sqrt[6]{x}\)
Find \( \frac{dy}{dx}\)
\(-\frac{64}{x^{9}} - \frac{5}{6}x^{-\frac{5}{6}}\)
\(y=\sin (2x^2+3)\)
Find \( \dfrac{dy}{dx}\)
\(4xcos(2x^2+3)\)
\(y=7x^2e^x\)
Find \( \dfrac{dy}{dx}\)
\(14xe^x+7x^2e^x\)
\(y=\frac{x^2+5}{3x-7}\)
Find \( \dfrac{dy}{dx}\)
\(\frac{(3x^2-14x-15)}{(3x-7)^2}\)
Find the equation of the tangent to the curve:
\(y = -4x^2 + 8x - 5\)
where \(x = -1\)
\(y = 16x - 1\)
Find the equation of the normal to the curve:
\(y = x^2 - 2x + 1\)
where \(x = 0\)
\(y = \frac{x}{2} + 1\)
\(y =21x^2 - 14x + 9\)
Find \( \int y \quad dx\)
\(7x^3 - 7x^2 + 9x+c\)
A game is played 10 times and the probability of winning is 0.9. Calculate the probability of winning exactly 3 times. 0.00000875
Make up a maths question using this:
\(z=\dfrac{x-\mu}{\sigma}\)
Standardised Normal Variable
What letter is this?
Two terms of an arithmetic sequence:
\(u_{8} = 86\)
\(u_{18} = 216\)
Find the sum of the first 19 terms.2128
Find the equations of the asymptotes of:
\(y=\dfrac{5x}{2-x}+3\)
\(x=2,y=-2\)
In the triangle ABC,
BĈA = 79.9°.
BC = 6.4cm.
AB̂C = 58.03°.
Find CA to 1 dp.
8.1cm
Evaluate:
$$\sum_{n=0}^{5} 2^n$$
63
\(f(x)=-8x^2-8x-9\)
What is the value of the discriminant and what does it indicate?
-224, No real roots
\(f(x)=x^2-3x+9\)
By completing the square find the coordinates of the vertex.
(1.5, 6.75)
Solve for x:
\( \log(x) + \log(29-x) = 2\)
\(x = 4 \text{ or } x = 25 \)
Find the integral:
\(\int \cos(x)e^{\sin(x)} \;dx\)
\(e^{\sin(x)}+c\)
Find the equation of the straight line that passes through:
(-7, -16) and (8, 14)
\(y=2x-2\)
Find the inverse of the function \(f\):
\(f(x)=\sqrt{\frac{x-5}{4}}\)
\(4x²+5\)
\(f(x)=x^2-1 \\[1cm] \text{Find }f \bullet f(x) \\\)
\(x^4-2x^2\)
Write in standard form:
\(a \times 10^{-1} \times b\times 10^{-1}\)
where \(a \times b \) is a three digit number \((100 \le ab \lt 1000)\)
\(\frac{ab}{100}\times10^0\)
Draw a rough sketch of
\(y=2^x\)
Sketch a height-time graph as this jar is filled.
Without a calculator find the exact value of
$$\cos{\frac{\pi}{6}} \times \sin{45°}$$\(\dfrac{\sqrt{6}}{4}\)
Without a calculator find the exact value of
$$\tan{\dfrac{19\pi}{3}}$$\(\sqrt{3}\)
Solve:
\(2x+y-3z= 7 \\ 3x+y+z= 26 \\ x-y+2z = 11\)
x = 7, y = 2, z = 3
Find the area of a sector with radius 3.4cm and angle \( \frac{\pi}{3}\)
🍕
6.05cm2
In how many ways can 8 different books be arranged on a shelf if 4 of them must be together?
2880
Find the equations of the asymptotes of:
$$y=\dfrac{-6x^2-4x-1}{3x+2}$$x=-2/3, y=-2x
The sum of the first 3 terms of a geometric sequence is 155 and the sum of the first 4 terms is 780. What is the first term?
5
Find the first 4 terms in the expansion of:
\(\dfrac{1}{(1+3x)^3}\)
\(1-9x+54x^2-270x^3\)
Evaluate:
\(\int^{4}_{0} (x-8)^2 \; dx\)
\(149.333333333333\)
The probability that I drop and brake my phone when I visit a coffee shop is 0.06. Today I visited two coffee shops and broke my phone in one of them. What is the probability that it was the first shop where the accident occurred?
\(0.515\)
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:
\(\int x^2 \ln{x}\; dx\)
\(\frac{x^3}{9}(3\ln x-1)+c\)
Simplify:
$$\sin{x}\cot{x}$$\(\cos{x}\)
$$ \DeclareMathOperator{cosec}{cosec} $$Find the volume of revolution when \(y=\sqrt{x}\) is rotated about the y-axis for \(1 \le y \le 4\)
\(\frac{1023\pi}{5}\) cubic units
What is the inverse of a function?
Clue: swaps the roles of x and y
Show how the first four terms of the Maclaurin series are obtained for
\(f(x) = \frac{1}{1-x}\)
\(1 + x + x^2 + x^3\)
Find the five 5th roots of 1
\(1, cis\frac{2\pi}{5}, cis\frac{4\pi}{5},\\ cis\frac{-2\pi}{5}, cis\frac{-4\pi}{5}\)
A team of 11 is randomly chosen from a squad of 18 including the club captain and vice captain. Determine the probability that both the captain and vice-captain are chosen.
55/153 or 35.9%
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
Simplify:
$$\sqrt{45}$$
\(3\sqrt{5}\)
Simplify:
$$\dfrac{7}{\sqrt{5}}$$\(\frac{7\sqrt{5}}{5}\)
Simplify
\(4\sqrt{7} - \sqrt{63}\)
\(\sqrt{7}\)
Simplify:
$$\dfrac{5}{2 - \sqrt{3}}$$\(\frac{10 + 5\sqrt{3}}{1} = 10 + 5\sqrt{3}\)
Calculate the standard deviation of the following numbers:
15, 19, 19, 21, 21, 25
3
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