Find the first three terms in the expansion of:
\((4a - 2b)^8\)
\(=65536a^8 - 262144a^7b \\+458752a^6b^2 ...\)
If £120 is invested with an interest rate of 5% compounded monthly, find the value of the investment after 6 years. £161.88
Here are the coordinates of 3 vertices of a square, what are the coordinates of the 4th?
\((2,3),(8,9),(-4,9)\)
(2,15)
\( X \sim N(27.1, 1.8^2)\)
Find
\( P(28.1\lt X \lt29.1) \)
\(0.156\)
Factorise:
\(x^2+2x-8\)
\((x+4)(x-2)\)
Factorise:
\(5x^2+16x-16\)
\((x+4)(5x-4)\)
Draw a rough sketch of the graph of:
\(y=-x-1\)
Gradient -1
y intercept -1
What is the value of:
\(1^{\frac{1}{2}}\)
\(= 1\)
Find angle ABC if AB = 3.2m and BC = 4.7m. 47.1o
Find AC if angle BCA = 68o and AB = 3.3m. 1.33m
Describe the red region.
\(y = 7x^3 - 7x^2 + 2x\)
Find \( \dfrac{dy}{dx}\)
\(21x^2 - 14x + 2\)
\(y = \dfrac{4}{x^8} - 2\sqrt[3]{x}\)
Find \( \frac{dy}{dx}\)
\(-\frac{32}{x^9} - \frac{2}{3}x^{-\frac{2}{3}}\)
\(y=9\ln (4x^2+5)\)
Find \( \dfrac{dy}{dx}\)
\(72x(4x^2+5)^{-1}\)
\(y=x(4x^2+5)^6\)
Find \( \dfrac{dy}{dx}\)
\((4x^2+5)^6+48x^2(4x^2+5)^5\)
\(y=\frac{x^2+5}{3x-7}\)
Find \( \dfrac{dy}{dx}\)
\(\frac{(3x^2-14x-15)}{(3x-7)^2}\)
Find the equation of the tangent to the curve:
\(y = x^2 + 6x + 9\)
where \(x = -3\)
\(y = 0\)
Find the equation of the normal to the curve:
\(y = -4x^2 + 8x - 5\)
where \(x = -1\)
\(y = -\frac{x}{16} - \frac{273}{16}\)
\(y =6x^2 - 10x + 3\)
Find \( \int y \quad dx\)
\(2x^3 - 5x^2 + 3x+c\)
A game is played 17 times and the probability of winning is 0.8. Calculate the probability of winning exactly 10 times. 0.0267
Make up a maths question using this:
\( \int \dfrac{1}{x} = \ln |x| + c\)
Reciprocal Integral formula
What letter is this?
Two terms of an arithmetic sequence:
\(u_{7} = 89\)
\(u_{17} = 219\)
Find the sum of the first 42 terms.11655
Find the equations of the asymptotes of:
\(y=\dfrac{2x+5}{2x+3}\)
\(x=-\frac{3}{2},y=1\)
In the triangle ABC,
BĈA = 31.7°.
BC = 7.5cm.
AB̂C = 123.16°.
Find CA to 1 dp.
14.8cm
Evaluate:
$$\sum_{n=3}^{5} 116 - n^2$$
298
\(f(x)=6x^2+9x+1\)
What is the value of the discriminant and what does it indicate?
57, Two distinct roots
\(f(x)=x^2+8x+1\)
By completing the square find the coordinates of the vertex.
(-4, -15)
Express \(\log_2(32)\) in terms of a log to base 4.
\( 10\log_4(2) \text{ or } \log_4(1024) \)
Find the integral:
\(\int (x+3)\sqrt{x^2+6x+8} \;dx\)
\(\frac{1}{3}(x^2+6x+8)^{\frac32}+c\)
Find the equation of the straight line that passes through:
(-4, -5) and (2, 1)
\(y=x-1\)
Find the inverse of the function \(f\):
\(f(x)= \sqrt{x-9}\)
\(x²+9\)
\(f(x)=5x+1 \\ g(x)=x^2 \\[1cm] \text{Find }gf(3x)\)
\(225x^2+30x+1\)
Write in standard form:
\(a \times 10^2 \times b\times 10^3\)
where \(a \times b \) is a two digit number \((10 \le ab \lt 100)\)
\(\frac{ab}{10}\times10^6\)
Draw a rough sketch of
\(x=\pm \sqrt{y}\)
Sketch a height-time graph as this jar is filled.
Without a calculator find the exact value of
$$\cos{30°} \div \sin{\frac{\pi}{3}}$$\(1\)
Without a calculator find the exact value of
$$\sin{5\pi}$$\(0\)
Solve:
\(2x+y-3z= 11 \\ 3x+y+z= 27 \\ x-y+2z = 11\)
x = 8, y = 1, z = 2
Find the area of a sector with radius 3.5cm and angle \( \frac{2\pi}{3}\)
🍕
12.8cm2
How many ways can fifteen children sit in a row without the youngest being in the middle?
1220496076800
Find the equations of the asymptotes of:
$$y=\dfrac{2x^2+3x-9}{x+2}$$x=-2,y=2x-1
The first term of a geometric sequence is 30 and the fourth term is twice this value. What is the common ratio?
\( \sqrt[3]{2} \)
Find the first 4 terms in the expansion of:
\((1+x)^{-8}\)
\(1-8x-36x^2-120x^3\)
Evaluate:
\(\int^{3}_{0} e^x dx\)
\(e^{3}- 1 \approx 19.1\)
Each afternoon the probability my cat sleeps is 0.5 and the probability that my dog sleeps is 0.4. The probability that the dog sleeps given that the cat is sleeping is 0.9. Find the probability that both sleep in the afternoon.
\(0.45\)
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:
\(\int e^x\sin{x}\; dx\)
\(\frac{e^x}{2}(sinx-cosx)+c\)
Simplify:
$$\tan{x}\cot{x}$$\(1\)
$$ \DeclareMathOperator{cosec}{cosec} $$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
How do you determine the domain of a function?
The domain of a function is the set of all possible input values (x-values) for which the function is defined.
Show how the first four terms of the Maclaurin series are obtained for
\(f(x) = \cos(x)\)
\(1 - \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720}\)
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 first \( n \) natural numbers is \( \frac{n(n + 1)}{2} \)
Show true for n=1, assume true for n=k, prove for n=k+1
Simplify:
$$\sqrt{8}$$
\(2\sqrt{2}\)
Simplify:
$$\dfrac{2}{\sqrt{10}}$$\(\frac{2\sqrt{10}}{10} = \frac{\sqrt{10}}{5}\)
Simplify
\((9 + 2\sqrt{2})(9 - 2\sqrt{2})\)
\(73\)
Simplify:
$$\dfrac{5}{2 - \sqrt{3}}$$\(\frac{10 + 5\sqrt{3}}{1} = 10 + 5\sqrt{3}\)
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