Find the first three terms in the expansion of:
\((2a - 3b)^5\)
\(=32a^5 - 240a^4b \\+720a^3b^2 ...\)
If £140 is invested with an interest rate of 3% compounded monthly, find the value of the investment after 9 years. £183.33
Here are the coordinates of 3 vertices of a square, what are the coordinates of the 4th?
\((1,3),(6,8),(-4,8)\)
(1,13)
\( X \sim N(33, 6^2)\)
Find
\( P(31\lt X \lt37) \)
\(0.378\)
Factorise:
\(x^2-x-2\)
\((x+1)(x-2)\)
Factorise:
\(10x^2+x-2\)
\((2x+1)(5x-2)\)
Draw a rough sketch of the graph of:
\(y=2x+1\)
Gradient 2
y intercept 1
What is the value of:
\(2^{-1}\)
\(= \frac{1}{2}\)
Find angle ABC if AC = 3.8m and BC = 5.4m. 44.7o
Find AC if angle ABC = 35o and AB = 4.5m. 3.15m
Describe the red region.
\(y = 5x^3 - 4x^2 + 7x\)
Find \( \dfrac{dy}{dx}\)
\(15x^2 - 8x + 7\)
\(y = \dfrac{9}{x^{5}} - 6\sqrt[7]{x}\)
Find \( \frac{dy}{dx}\)
\(-\frac{45}{x^{6}} - \frac{6}{7}x^{-\frac{6}{7}}\)
\(y=8\ln (5x^2+6)\)
Find \( \dfrac{dy}{dx}\)
\(80x(5x^2+6)^{-1}\)
\(y=x \tan x\)
Find \( \dfrac{dy}{dx}\)
\(tanx+\frac{x}{cos^2x}\)
\(y=\frac{x^2+5}{3x-7}\)
Find \( \dfrac{dy}{dx}\)
\(\frac{(3x^2-14x-15)}{(3x-7)^2}\)
Find the equation of the tangent to the curve:
\(y = 5x^2 + 7x + 3\)
where \(x = 1\)
\(y = 17x - 2\)
Find the equation of the normal to the curve:
\(y = -x^2 + 4x + 2\)
where \(x = 1\)
\(y = 5\frac{1}{2} - \frac{x}{2}\)
\(y =15x^2 - 8x + 6\)
Find \( \int y \quad dx\)
\(5x^3 - 4x^2 + 6x+c\)
A game is played 16 times and the probability of winning is 0.3. Calculate the probability of winning exactly 12 times. 0.000232
Make up a maths question using this:
\(u_n=u_1+(n-1)d\)
The nth term of an arithmetic sequence
What letter is this?
Two terms of an arithmetic sequence:
\(u_{6} = 40\)
\(u_{18} = 148\)
Find the sum of the first 41 terms.7175
Find the equations of the asymptotes of:
\(y=\dfrac{2x-7}{6-2x}-5\)
\(x=3,y=-6\)
In the triangle ABC,
BC = 8.5cm.
CA = 10.2cm.
BĈA = 61.7°
Find AB to 1 dp.
9.7cm
Evaluate:
$$\sum_{n=0}^{6} 2^n$$
127
\(f(x)=-4x^2+2x-2\)
What is the value of the discriminant and what does it indicate?
-28, No real roots
\(f(x)=x^2+6x-7\)
By completing the square find the coordinates of the vertex.
(-3, -16)
Solve for x:
\(\log_3x = 2\)
9
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:
(-3, -2) and (0, -5)
\(y=-x-5\)
Find the inverse of the function \(f\):
\(f(x)=\sqrt{\frac{x-5}{7}}\)
\(7x²+5\)
\(g(x)=5x-4 \\[1cm] \text{Find }g \circ g(x) \\\)
\(25x-24\)
Write in standard form:
\(a \times 10^p \times b\times 10^q\)
where \(a \times b \) is a two digit number \((10 \le ab \lt 100)\)
\(\frac{ab}{10}\times10^{p+q+1}\)
Draw a rough sketch of
\(y=x^2+7x\)
Sketch a height-time graph as this jar is filled.
Without a calculator find the exact value of
$$\tan{\frac{\pi}{6}} \times \cos{45°}$$\(\dfrac{1}{\sqrt{6}}\)
Without a calculator find the exact value of
$$\tan{\dfrac{19\pi}{3}}$$\(\sqrt{3}\)
Solve:
\( g-7h-7i=-93 \\ 2g-2h+i= 3\\ 5g+3h+i = 53\)
g = 5, h = 7, i = 7
Find the perimeter of a sector with radius 4.7cm and angle \( \frac{\pi}{3}\)
🍕
14.3cm
A safe has a seven-digit code. How many possibilities are there if no digit can be repeated and the code must be odd?
302400
Find the equations of the asymptotes of:
$$y=\dfrac{2x^2+3x-9}{x+2}$$x=-2,y=2x-1
The 7th term of a geometric sequence is 78125 and the sum of the first 7 terms is 97655. Find the first term.
5
Find the first 4 terms in the expansion of:
\((1+2x)^{\frac12}\)
\(1+x-\frac{x^2}{2}+\frac{x^3}{2}\)
Evaluate:
\(\int^{7}_{0} x^2-2x+7 \; dx\)
\(114\)
Tin A contains 3 red balls and 5 green balls. Tin B contains 7 red balls and 10 green balls. A dice is thrown and if the score is less than 3 a ball is selected from tin A; otherwise a ball is selected from tin B. Given that the ball selected was red, calculate the probability that it came from tin B.
\(\frac{112}{163}\)
Find the cartesian equation of this plane:
\( \mathbf{r} = \begin{pmatrix} -2 \\ 2 \\ 3 \end{pmatrix} \; + \; s \begin{pmatrix} 3 \\ 1 \\ 0 \end{pmatrix} \; + \; t \begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix} \)
2x-6y+5z=-1
Simplify
$$ \dfrac{2-i}{1+3i}$$
\(-\frac{1}{10}-\frac{7}{10}i\)
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 \(y=\frac{1}{x}\) is rotated about the x-axis for \(1 \le x \le 2\)
\(\frac{\pi}{2}\) 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
\(f(x) = \ln(1 + x)\)
\(x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4}\)
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 school committee of 8 is chosen at random from 12 senior students and 4 junior students. Find the probability that all four junior students are chosen.
1/26 or 3.85%
Prove by mathematical induction that \( 5^n - 1 \) is divisible by 4 for all natural numbers \( n \)
Show true for n=1, assume true for n=k, prove for n=k+1
Simplify:
$$\sqrt{20}$$
\(2\sqrt{5}\)
Simplify:
$$\dfrac{6}{\sqrt{11}}$$\(\frac{6\sqrt{11}}{11}\)
Simplify
\(7\sqrt{7} - 3\sqrt{7}\)
\(4\sqrt{7}\)
Simplify:
$$\dfrac{8}{3 + \sqrt{6}}$$\(\frac{24 - 8\sqrt{6}}{3}\)
Calculate the standard deviation of the following numbers:
4, 2, 5, 8, 6
2
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