Find the first three terms in the expansion of:
\((3a - 2b)^4\)
\(=81a^4 - 216a^3b \\+216a^2b^2 ...\)
If £180 is invested with an interest rate of 1% compounded monthly, find the value of the investment after 5 years. £189.22
Here are the coordinates of 3 vertices of a square, what are the coordinates of the 4th?
\((4,3),(9,8),(-1,8)\)
(4,13)
\( X \sim N(9.03, 0.89^2)\)
Find
\( P(7.15\lt X \lt9.01) \)
\(0.474\)
Factorise:
\(x^2+x-6\)
\((x+3)(x-2)\)
Factorise:
\(3x^2+11x-4\)
\((x+4)(3x-1)\)
Draw a rough sketch of the graph of:
\(y=x\)
Gradient 1
y intercept 0
What is the value of:
\(3^{1}\)
\(= 3\)
Find angle BCA if AC = 6m and BC = 7.7m. 38.8o
Find AC if angle BCA = 59o and AB = 5.5m. 3.30m
Describe the red region.
\(y = 8x^3 - 9x^2 + 4x\)
Find \( \dfrac{dy}{dx}\)
\(24x^2 - 18x + 4\)
\(y = \dfrac{6}{x^7} - 8\sqrt[9]{x}\)
Find \( \frac{dy}{dx}\)
\(-\frac{42}{x^8} - \frac{8}{9}x^{-\frac{8}{9}}\)
\(y=(3x+7)^4\)
Find \( \dfrac{dy}{dx}\)
\(12(3x+7)^3\)
\(y=x^3 \ln x\)
Find \( \dfrac{dy}{dx}\)
\(3x^2lnx+x^2\)
\(y=\frac{x}{\sin x}\)
Find \( \dfrac{dy}{dx}\)
\(\frac{(sinx-xcosx)}{sin^2x}\)
Find the equation of the tangent to the curve:
\(y = x^2 + 6x + 9\)
where \(x = -3\)
\(y = 0\)
Find the equation of the normal to the curve:
\(y = -4x^2 + 8x - 5\)
where \(x = -1\)
\(y = -\frac{x}{16} - \frac{273}{16}\)
\(y =15x^2 - 4x + 6\)
Find \( \int y \quad dx\)
\(5x^3 - 2x^2 + 6x+c\)
A game is played 16 times and the probability of winning is 0.5. Calculate the probability of winning exactly 7 times. 0.175
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:
\(u_{9} = 17\)
\(u_{19} = 27\)
Find the sum of the first 32 terms.784
Find the equations of the asymptotes of:
\(y=3\left(\dfrac{2x+3}{7-x}\right)\)
\(x=7,y=-6\)
In the triangle ABC,
AB = 8.9cm.
BC = 9.3cm.
CÂB = 31.1°.
Find angle BĈA.
29.7°
Evaluate:
$$\sum_{n=2}^{5} 3n+1$$
46
\(f(x)=8x^2-9x+5\)
What is the value of the discriminant and what does it indicate?
-79, No real roots
\(f(x)=x^2+7x-2\)
By completing the square find the coordinates of the vertex.
(-3.5, -14.25)
Simplify \(\log_{10}10^5\)
5
Find the integral:
\(\int \dfrac{5x}{x^2-3} \;dx\)
\(\frac{5}{2} \ln(x^2-3)+c\)
Find the equation of the straight line that passes through:
(-2, -9) and (9, 13)
\(y=2x-5\)
Find the inverse of the function \(f\):
\(f(x)=\frac{\sqrt{x}-15}{15}\)
\((15x+15)²\)
\(f(x)=2x+3 \\ g(x)=2x^2 \\[1cm] \text{Find }fgf(x)\)
\(16x^2+48x+39\)
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
\(y=\sin(x)\)
Sketch a height-time graph as this jar is filled.
Without a calculator find the exact value of
$$\cos{0°} + \sin{\frac{\pi}{6}} + \cos{60°}$$\(2\)
Without a calculator find the exact value of
$$\cos{7\pi}$$\(-1\)
Solve:
\(2d+3e-4f = 3 \\ d-e-f= -7\\ 9d+2e-2f=20\)
d = 2, e = 5, f = 4
Find the perimeter of a sector with radius 4.7cm and angle \( \frac{\pi}{3}\)
🍕
14.3cm
In how many ways can 8 different books be arranged on a shelf if 2 of them must be together?
10080
Find the equations of the asymptotes of:
$$y=\dfrac{-6x^2-4x-1}{3x+2}$$x=-2/3, y=-2x
Evaluate:
$$ \sum_{k=1}^{15} 5 \times (-2)^k $$
-109230
Find the first 4 terms in the expansion of:
\(\dfrac{1}{(1+3x)^3}\)
\(1-9x+54x^2-270x^3\)
Evaluate:
\(\int^{\pi/3}_{\pi/6} \sin{x} \; dx\)
\(\dfrac{\sqrt{3}-1}{2}\)
Every family in Happyland has either has a car or a motor scooter or both. 75% of the families have a car. 85% of the families have a scooter. A family is selected at random and it is found that they have a car. Find the probability they also have a scooter.
\(\dfrac{4}{5}\)
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
$$ \dfrac{i(2-i)}{3-2i}$$
\(-\frac{1}{13}+\frac{8}{13}i\)
Evaluate:
\(\int \ln{x}\; dx\)
\(x\ln|x|-x+c\)
Simplify:
$$\dfrac{\sin^2{x}-1}{\cos{x}}$$\(-\cos{x}\)
$$ \DeclareMathOperator{cosec}{cosec} $$Find the volume of revolution when \(y=x^2\) is rotated about the y-axis for \(0 \le y \le 4\)
\(8\pi\) cubic units
What is the difference between a permutation and a combination?
Permutations consider order; combinations do not.
Show how the first four terms of the Maclaurin series are obtained for
\(f(x) = \sqrt{x+1}\)
\(1 + \frac{x}{2} - \frac{x^2}{8} + \frac{x^3}{16}\)
Solve for \(z\)
$$ z^4 = \sqrt{3}+i $$
\(\sqrt[4]{2} cis \frac{\pi}{24},\sqrt[4]{2} cis \frac{13\pi}{24} \\ \sqrt[4]{2} cis \frac{-11\pi}{24}, \sqrt[4]{2} cis \frac{-23\pi}{24}\)
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 the sum of the cubes of the first \( n \) natural numbers is \( \left(\frac{n(n + 1)}{2}\right)^2 \)
Show true for n=1, assume true for n=k, prove for n=k+1
Simplify:
$$\sqrt{18}$$
\(3\sqrt{2}\)
Simplify:
$$\dfrac{7}{\sqrt{5}}$$\(\frac{7\sqrt{5}}{5}\)
Simplify
\((2 - 2\sqrt{2})^2\)
\(12 - 8\sqrt{2}\)
Simplify:
$$\dfrac{7}{4 + \sqrt{3}}$$\(\frac{28 - 7\sqrt{3}}{13}\)
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