Find the first three terms in the expansion of:
\((3a - 4b)^6\)
\(=729a^6 - 5832a^5b \\+19440a^4b^2 ...\)
If £240 is invested with an interest rate of 3% compounded monthly, find the value of the investment after 4 years. £270.56
Here are the coordinates of 3 vertices of a square, what are the coordinates of the 4th?
\((4,5),(10,11),(-2,11)\)
(4,17)
\( X \sim N(33, 6^2)\)
Find
\( P(31\lt X \lt37) \)
\(0.378\)
Factorise:
\(x^2-1\)
\((x+1)(x-1)\)
Factorise:
\(2x^2+5x-3\)
\((x+3)(2x-1)\)
Draw a rough sketch of the graph of:
\(2y=x-2\)
Gradient 0.5
y intercept -1
What is the value of:
\(9^{\frac{1}{2}}\)
\(= 3\)
Find angle BCA if AC = 3.8m and BC = 5.3m. 44.2o
Find BC if angle BCA = 68o and AB = 5.5m. 5.93m
Describe the red region.
\(y = 4x^3 - 6x^2 + 3x\)
Find \( \dfrac{dy}{dx}\)
\(12x^2 - 12x + 3\)
\(y = \dfrac{8}{x^4} - 4\sqrt[5]{x}\)
Find \( \frac{dy}{dx}\)
\(-\frac{32}{x^5} - \frac{4}{5}x^{-\frac{4}{5}}\)
\(y=2\ln (8x^2+9)\)
Find \( \dfrac{dy}{dx}\)
\(32x(8x^2+9)^{-1}\)
\(y=\sin x \cos x\)
Find \( \dfrac{dy}{dx}\)
\(cos^2x-sin^2x\)
\(y=\frac{2x^2}{4x-1}\)
Find \( \dfrac{dy}{dx}\)
\(\frac{(8x^2-4x)}{(4x-1)^2}\)
Find the equation of the tangent to the curve:
\(y = x^2 + 6x + 9\)
where \(x = -3\)
\(y = 0\)
Find the equation of the normal to the curve:
\(y = x^2 - 2x + 1\)
where \(x = 0\)
\(y = \frac{x}{2} + 1\)
\(y =12x^2 - 12x + 7\)
Find \( \int y \quad dx\)
\(4x^3 - 6x^2 + 7x+c\)
A game is played 18 times and the probability of winning is 0.7. Calculate the probability of winning exactly 17 times. 0.0126
Make up a maths question using this:
\(\log_ax=\dfrac{\log_bx}{\log_ba}\)
Logarithm changing base formula
What letter is this?
Two terms of an arithmetic sequence:
\(u_{9} = 115\)
\(u_{12} = 154\)
Find the sum of the first 21 terms.2961
Find the equations of the asymptotes of:
\(y=\dfrac{3x-5}{6x-12}\)
\(x=2,y=\frac{1}{2}\)
In the triangle ABC,
AB = 6.3cm.
BC = 6.1cm.
CÂB = 51.0°.
Find angle BĈA.
53.4° or 126.6°
Evaluate:
$$\sum_{n=0}^{9} 5n+2$$
245
\(f(x)=-8x^2+8x+5\)
What is the value of the discriminant and what does it indicate?
224, Two distinct roots
\(f(x)=x^2-2x-2\)
By completing the square find the coordinates of the vertex.
(1, -3)
Solve for x:
\(\log_2(x) = 4\)
16
Find the integral:
\(\int \dfrac{x^2}{x^3-1} \;dx\)
\(\frac{1}{3} \ln(x^3-1)+c\)
Find the equation of the straight line that passes through:
(-7, 14) and (1, -2)
\(y=-2x+0\)
Find the inverse of the function \(f\):
\(f(x)= \sqrt{x+17}\)
\(x²-17\)
\(g(x)=5x-4 \\[1cm] \text{Find }g \circ g(x) \\\)
\(25x-24\)
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
\(y=2^x\)
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
$$\tan{4\pi}$$\(0\)
Solve:
\(2x+y-3z= 5 \\ 3x+y+z= 12 \\ x-y+2z = 3\)
x = 3, y = 2, z = 1
Find the area of a sector with radius 3.3cm and angle \( \frac{\pi}{6}\)
🍕
2.85cm2
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:
$$y=\dfrac{-x^2+3x-2}{x}$$x=0,y=3-x
The sum of the first 6 terms of a geometric sequence is 19530 and the sum of the first 7 terms is 97655. What is the first term?
5
Find the first 4 terms in the expansion of:
\(\dfrac{1}{(1+3x)^3}\)
\(1-9x+54x^2-270x^3\)
Evaluate:
\(\int^{6}_{1} (x-8)^2 \; dx\)
\(111.666666666667\)
Given equal populations of Type X and Type Y bacteria, with mutation rates of 30% and 40% respectively, if a mutated bacterium is found, what's the probability it's Type Y?
\(0.571\)
Find the vector equation of the line:
\( \dfrac{x-8}{2} = \dfrac{8-y}{9} = \dfrac{z}{7} \)
\( \mathbf{r} = \begin{pmatrix} 8 \\ 8 \\ 0 \end{pmatrix} \quad + \quad t \begin{pmatrix} 2 \\ -9 \\ 7 \end{pmatrix} \)
Simplify
$$ \dfrac{1-4i}{1+5i}$$
\(-\frac{19}{26}-\frac{9}{26}i\)
Evaluate:
\(\int x\cos{x}\; dx\)
\(xsinx+cosx+c\)
Simplify:
$$\cosec{x}\tan{x}$$\(\sec{x}\)
$$ \DeclareMathOperator{cosec}{cosec} $$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
How do you use the discriminant to determine the nature of roots?
Clue: positive, negative or zero: \( b^2 - 4ac \)
Show how the first four terms of the Maclaurin series are obtained for
\(f(x) = e^x\)
\(1 + x + \frac{x^2}{2} + \frac{x^3}{6}\)
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 \( 5^n - 1 \) 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:
$$\sqrt{27}$$
\(3\sqrt{3}\)
Simplify:
$$\dfrac{6}{\sqrt{11}}$$\(\frac{6\sqrt{11}}{11}\)
Simplify
\(4\sqrt{7} - \sqrt{63}\)
\(\sqrt{7}\)
Simplify:
$$\dfrac{2}{5 - \sqrt{7}}$$\(\frac{10 + 2\sqrt{7}}{18}\)
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