Find the first three terms in the expansion of:
\((3a - 4b)^7\)
\(=2187a^7 - 20412a^6b \\+81648a^5b^2 ...\)
If £160 is invested with an interest rate of 2% compounded quarterly, find the value of the investment after 6 years. £180.35
Here are the coordinates of 3 vertices of a square, what are the coordinates of the 4th?
\((5,2),(9,6),(1,6)\)
(5,10)
\( X \sim N(43, 7^2)\)
Find
\( P(32\lt X \lt45) \)
\(0.554\)
Factorise:
\(x^2-x-2\)
\((x+1)(x-2)\)
Factorise:
\(3x^2-4x-4\)
\((3x+2)(x-2)\)
Draw a rough sketch of the graph of:
\(y=2x\)
Gradient 2
y intercept 0
What is the value of:
\(4^{-3}\)
\(= \frac{1}{64}\)
Find angle ABC if AB = 5.2m and BC = 7.1m. 42.9o
Find BC if angle BCA = 64o and AB = 4.2m. 4.67m
Describe the red region.
\(y = 9x^3 - 8x^2 + 2x\)
Find \( \dfrac{dy}{dx}\)
\(27x^2 - 16x + 2\)
\(y = \dfrac{3}{x^8} - 6\sqrt[7]{x}\)
Find \( \frac{dy}{dx}\)
\(-\frac{24}{x^9} - \frac{6}{7}x^{-\frac{6}{7}}\)
\(y=2\ln (2x^2+3)\)
Find \( \dfrac{dy}{dx}\)
\(8x(2x^2+3)^{-1}\)
\(y=e^{5x} \cos x\)
Find \( \dfrac{dy}{dx}\)
\(5e^{5x}cosx-e^{5x}sinx\)
\(y=\frac{x^2+5}{3x-7}\)
Find \( \dfrac{dy}{dx}\)
\(\frac{(3x^2-14x-15)}{(3x-7)^2}\)
Find the equation of the tangent to the curve:
\(y = -2x^2 - 4x + 6\)
where \(x = 3\)
\(y = 24 - 16x\)
Find the equation of the normal to the curve:
\(y = x^2 - 2x + 1\)
where \(x = 0\)
\(y = \frac{x}{2} + 1\)
\(y =21x^2 - 6x + 2\)
Find \( \int y \quad dx\)
\(7x^3 - 3x^2 + 2x+c\)
A game is played 12 times and the probability of winning is 0.1. Calculate the probability of winning exactly 10 times. 0.00000000535
Make up a maths question using this:
\( P(A|B) = \dfrac{P(A \cap B)}{P(B)} \)
Conditional probability formula
What letter is this?
Two terms of an arithmetic sequence:
\(u_{6} = -19\)
\(u_{20} = -61\)
Find the sum of the first 34 terms.-1819
Find the equations of the asymptotes of:
\(y=\dfrac{5x}{2-x}+3\)
\(x=2,y=-2\)
In the triangle ABC,
AB = 7.6cm.
BC = 9.9cm.
CA = 6.9cm.
Find angle CÂB.
86.0°
Evaluate:
$$\sum_{n=4}^{5} n^2 - 8n$$
-31
\(f(x)=7x^2+9x-5\)
What is the value of the discriminant and what does it indicate?
221, Two distinct roots
\(f(x)=x^2+7x-6\)
By completing the square find the coordinates of the vertex.
(-3.5, -18.25)
Write the following in terms of logs to base 10:
\(\log_a(z)\)
\( \dfrac{\log_{10}(z)}{\log_{10}(a)}\)
Find the integral:
\(\int 3xe^{x^2} \;dx\)
\(\frac{3}{2}e^{x^2}+c\)
Find the equation of the straight line that passes through:
(-5, -20) and (2, 1)
\(y=3x-5\)
Find the inverse of the function \(f\):
\(f(x)=\frac{\sqrt{x}-19}{14}\)
\((14x+19)²\)
\(g(x)=5x-4 \\[1cm] \text{Find }g \circ g(x) \\\)
\(25x-24\)
Write in standard form:
\(a \times 10^3 \times b\times 10^5\)
where \(a \times b \) is a single digit number \((1 \le ab \lt 10)\)
\(ab\times10^8\)
Draw a rough sketch of
\(y=x(5-x)\)
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
$$\tan{4\pi}$$\(0\)
Solve:
\( 5a+2b+c=39 \\ 3a+4b+2c= 43 \\ a+5b+c=31\)
a = 5, b = 4, c = 6
Find the perimeter of a sector with radius 4.8cm and angle \( \frac{\pi}{3}\)
🍕
14.6cm
How many ways can twelve people be divided into two equal groups?
462
Find the equations of the asymptotes of:
$$y=\dfrac{x^2+x-6}{x-1}$$x=1,y=x+2
Evaluate:
$$ \sum_{k=1}^{8} \left( \dfrac12 \right)^{k-3} $$
7.97
Find the first 4 terms in the expansion of:
\((1+2x)^{\frac12}\)
\(1+x-\frac{x^2}{2}+\frac{x^3}{2}\)
Evaluate:
\(\int^{7}_{0} x^2-2x+7 \; dx\)
\(114\)
The probability that it is cloudy on a particular day is 0.7. The probability that it is cloudy with a high level of pollution on a particular day is 0.4. Find the probability that there will be a high level of pollution on a day when it is cloudy.
\(0.571\)
Find the vector product:
\( \begin{pmatrix} 6 \\ 4 \\ 0 \end{pmatrix} \; \times \; \begin{pmatrix} 5 \\ -8 \\ 8 \end{pmatrix} \)
\( \begin{pmatrix} 32 \\ -48 \\ -68 \end{pmatrix} \)
Simplify
$$ (1+i)^{4} $$
\(-4\)
Evaluate:
\(\int x^2 \ln{x}\; dx\)
\(\frac{x^3}{9}(3\ln x-1)+c\)
Simplify:
$$\dfrac{\sin^2{x}-1}{\cos{x}}$$\(-\cos{x}\)
$$ \DeclareMathOperator{cosec}{cosec} $$Find the volume of revolution when \(y=\sqrt[3]{x^2}\) is rotated about the y-axis for \(2 \le y \le 3\)
\(\frac{65\pi}{4}\) cubic units
Describe the behavior of a function at its asymptote.
Clue: approaches but never reaches
Show how the first four terms of the Maclaurin series are obtained for
\(f(x) = \ln(1 + x)\)
\(x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4}\)
Expand and simplify:
$$ (\sqrt{3}+i)^8 $$
\(-128-128\sqrt{3}i \\ \text{or } 128(-1-\sqrt{3}i)\)
6 children, three boys and three girls, are randomly seated on a row of 6 chairs. Determine the likelihood that the three boys are seated together.
1/5 or 20%
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
\(\sqrt{50} - 3\sqrt{2}\)
\(2\sqrt{2}\)
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