Find the first three terms in the expansion of:
\((2a - 4b)^5\)
\(=32a^5 - 320a^4b \\+1280a^3b^2 ...\)
If £200 is invested with an interest rate of 3% compounded quarterly, find the value of the investment after 5 years. £232.24
Here are the coordinates of 3 vertices of a square, what are the coordinates of the 4th?
\((3,1),(8,7),(-3,6)\)
(2,12)
\( X \sim N(50, 5^2)\)
Find
\( P(40\lt X \lt60) \)
\(0.955\)
Factorise:
\(x^2-x-2\)
\((x+1)(x-2)\)
Factorise:
\(4x^2-1\)
\((2x+1)(2x-1)\)
Draw a rough sketch of the graph of:
\(y=-2x-2\)
Gradient -2
y intercept -2
What is the value of:
\(3^{1}\)
\(= 3\)
Find angle ABC if AC = 4.6m and BC = 5.8m. 52.5o
Find AB if angle ABC = 54o and BC = 3.7m. 2.17m
Describe the red region.
\(y = 4x^3 - 5x^2 + 2x\)
Find \( \dfrac{dy}{dx}\)
\(12x^2 - 10x + 2\)
\(y = \dfrac{9}{x^5} - 4\sqrt[5]{x}\)
Find \( \frac{dy}{dx}\)
\(-\frac{45}{x^6} - \frac{4}{5}x^{-\frac{4}{5}}\)
\(y=9\ln (3x^2+4)\)
Find \( \dfrac{dy}{dx}\)
\(54x(3x^2+4)^{-1}\)
\(y=x^4 \ln x\)
Find \( \dfrac{dy}{dx}\)
\(4x^3lnx+x^3\)
\(y=\frac{2x^2}{4x-1}\)
Find \( \dfrac{dy}{dx}\)
\(\frac{(8x^2-4x)}{(4x-1)^2}\)
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:
\(y = -4x^2 + 8x - 5\)
where \(x = -1\)
\(y = -\frac{x}{16} - \frac{273}{16}\)
\(y =24x^2 - 6x + 6\)
Find \( \int y \quad dx\)
\(8x^3 - 3x^2 + 6x+c\)
A game is played 10 times and the probability of winning is 0.4. Calculate the probability of winning exactly 5 times. 0.201
Make up a maths question using this:
\(S_n = \dfrac{u_1(r^n-1)}{r-1}\)
The sum of a geometric sequence
What letter is this?
Two terms of an arithmetic sequence:
\(u_{8} = -17\)
\(u_{12} = -25\)
Find the sum of the first 36 terms.-1368
Find the equations of the asymptotes of:
\(y=\dfrac{2x-7}{6-2x}-5\)
\(x=3,y=-6\)
In the triangle ABC,
AB = 6.4cm.
BC = 9.3cm.
CÂB = 71.3°.
Find angle BĈA.
40.7°
Evaluate:
$$\sum_{n=3}^{7} n^2 - 3n$$
60
\(f(x)=6x^2+4x-3\)
What is the value of the discriminant and what does it indicate?
88, Two distinct roots
\(f(x)=x^2+5x+2\)
By completing the square find the coordinates of the vertex.
(-2.5, -4.25)
Express \(\log_2(32)\) in terms of a log to base 4.
\( 10\log_4(2) \text{ or } \log_4(1024) \)
Find the integral:
\(\int \dfrac{\ln(x)}{x} \;dx\)
\(\dfrac{\ln(x)^2}{2}+c\)
Find the equation of the straight line that passes through:
(-2, 0) and (4, 6)
\(y=x+2\)
Find the inverse of the function \(f\):
\(f(x)=\frac{\sqrt{x}-16}{18}\)
\((18x+16)²\)
\(f(x)=2x-3 \\[1cm] \text{Find }f \bullet f(1-\sqrt{m}) \\\)
\(-5-4\sqrt{m}\)
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
\(y=3-\dfrac{10}{x}\)
Sketch a height-time graph as this jar is filled.
Without a calculator find the exact value of
$$\sin{\frac{\pi}{4}} \times \cos{45°}$$\(\dfrac{1}{2}\)
Without a calculator find the exact value of
$$\cos{720°}$$\(1\)
Solve:
\(2d+3e-4f = -14 \\ d-e-f= -6\\ 9d+2e-2f=10\)
d = 2, e = 2, f = 6
Find the area of a sector with radius 2.3cm and angle \( \frac{\pi}{4}\)
🍕
2.08cm2
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:
$$y=\dfrac{-6x^2-4x-1}{3x+2}$$x=-2/3, y=-2x
The first term of a geometric sequence is 37 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:
\(\dfrac{1}{2-x}\)
\(\frac{1}{2}+\frac{x}{4}+\frac{x^2}{8}+\frac{x^3}{16}\)
Evaluate:
\(\int^{8}_{1} (x-8)^2 \; dx\)
\(114.333333333333\)
27 Scouts went hiking. 11 got lost, 13 got blisters, and 5 got both lost and blisters. Find the probability that a randomly selected Scout got blisters, given that they were not lost.
\(\dfrac{1}{2}\)
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:
\(\int x\sec^2x\; dx\)
\(xtanx+\ln|cosx|+c\)
Simplify:
$$\dfrac{\cot{x}}{\cosec{x}}$$\(\cos{x}\)
$$ \DeclareMathOperator{cosec}{cosec} $$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
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
\(f(x) = (1 + x)^n\)
\(1 + nx + \frac{n(n-1)x^2}{2} + \frac{n(n-1)(n-2)x^3}{6}\)
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 the sum of the squares of the first \( n \) natural numbers is \( \frac{n(n + 1)(2n + 1)}{6} \)
Show true for n=1, assume true for n=k, prove for n=k+1
Simplify:
$$\sqrt{32}$$
\(4\sqrt{2}\)
Simplify:
$$\dfrac{4}{5\sqrt{3}}$$\(\frac{4\sqrt{3}}{15}\)
Simplify
\((3 + 2\sqrt{5})(6 - 3\sqrt{5})\)
\(3\sqrt{5}-12\)
Simplify:
$$\dfrac{5}{3 - \sqrt{2}}$$\(\frac{15 + 5\sqrt{2}}{7}\)
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