Find the first three terms in the expansion of:
\((4a - 3b)^6\)
\(=4096a^6 - 18432a^5b \\+34560a^4b^2 ...\)
If £160 is invested with an interest rate of 2% compounded monthly, find the value of the investment after 8 years. £187.74
Here are the coordinates of 3 vertices of a square, what are the coordinates of the 4th?
\((3,4),(9,7),(0,10)\)
(6,13)
\( X \sim N(300, 10^2)\)
Find
\( P(270\lt X \lt330) \)
\(0.997\)
Factorise:
\(x^2-2x-3\)
\((x+1)(x-3)\)
Factorise:
\(8x^2-2x-3\)
\((2x+1)(4x-3)\)
Draw a rough sketch of the graph of:
\(y=-x-1\)
Gradient -1
y intercept -1
What is the value of:
\(1^{\frac{1}{3}}\)
\(= 1\)
Find angle BCA if AB = 5.6m and BC = 7.6m. 47.5o
Find AC if angle BCA = 25o and AB = 5.9m. 12.7m
Describe the red region.
\(y = 9x^3 - 6x^2 + 2x\)
Find \( \dfrac{dy}{dx}\)
\(27x^2 - 12x + 2\)
\(y = \dfrac{2}{x^5} - 4\sqrt[5]{x}\)
Find \( \frac{dy}{dx}\)
\(-\frac{10}{x^6} - \frac{4}{5}x^{-\frac{4}{5}}\)
\(y=3\ln (8x^2+9)\)
Find \( \dfrac{dy}{dx}\)
\(48x(8x^2+9)^{-1}\)
\(y=\sin x \cos x\)
Find \( \dfrac{dy}{dx}\)
\(cos^2x-sin^2x\)
\(y=\frac{x}{\sin x}\)
Find \( \dfrac{dy}{dx}\)
\(\frac{(sinx-xcosx)}{sin^2x}\)
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:
\(y = 3x^2 - 6x + 9\)
where \(x = 2\)
\(y = 9\frac{1}{3} - \frac{x}{6}\)
\(y =15x^2 - 12x + 4\)
Find \( \int y \quad dx\)
\(5x^3 - 6x^2 + 4x+c\)
A game is played 20 times and the probability of winning is 0.9. Calculate the probability of winning exactly 18 times. 0.285
Make up a maths question using this:
\( \triangle = b^2-4ac\)
Quadratic equation discriminant
What letter is this?
Two terms of an arithmetic sequence:
\(u_{6} = 51\)
\(u_{11} = 111\)
Find the sum of the first 34 terms.6426
Find the equations of the asymptotes of:
\(y=\dfrac{10-2x}{10x}\)
\(x=0,y=-\frac{1}{5}\)
In the triangle ABC,
AB = 5.3cm.
BC = 8.2cm.
CÂB = 50.3°.
Find angle BĈA.
29.8°
Evaluate:
$$\sum_{n=2}^{6} 2^n$$
124
\(f(x)=-9x^2-7x+7\)
What is the value of the discriminant and what does it indicate?
301, Two distinct roots
\(f(x)=x^2+6x+9\)
By completing the square find the coordinates of the vertex.
(-3, 0)
Evaluate \(\log_2(32) \)
5
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:
(-1, -10) and (6, 11)
\(y=3x-7\)
Find the inverse of the function \(f\):
\(f(x)=\frac{\sqrt{x-8}}{4}\)
\(16x²+8\)
\(f(x)=2x+3 \\ g(x)=2x^2 \\[1cm] \text{Find }fgf(x)\)
\(16x^2+48x+39\)
Write in standard form:
\((a \times 10^4) \div (b\times 10^{-2})\)
where \(a \div b \) is a decimal number \((0.1 \le \frac{a}{b} \lt 1)\)
\(\frac{10a}{b}\times10^5\)
Draw a rough sketch of
\(y=x^2-8\)
Sketch a height-time graph as this jar is filled.
Without a calculator find the exact value of
$$\cos{\frac{\pi}{6}} \times \sin{45°}$$\(\dfrac{\sqrt{6}}{4}\)
Without a calculator find the exact value of
$$\tan{\dfrac{19\pi}{6}}$$\(\dfrac{1}{\sqrt{3}}\)
Solve:
\( j+k+l= 11 \\ 2j-3k+9l= -4\\ -j+k-3l=1\)
j = 1, k = 8, l = 2
Find the perimeter of a sector with radius 3.7cm and angle \( \frac{\pi}{6}\)
🍕
9.34cm
A safe has a four-digit code. How many possibilities are there if no digit can be repeated and the code must be odd?
2520
Find the equations of the asymptotes of:
$$y=\dfrac{2x^2-8x+8}{x-3}$$x=3, y=2x-2
The 6th term of a geometric sequence is 972 and the sum of the first 6 terms is 1456. Find the first term.
4
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:
\(\int^{8}_{0} e^x dx\)
\(e^{8}- 1 \approx 2980\)
Tin A contains 7 red balls and 9 green balls. Tin B contains 12 red balls and 13 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.
\(\frac{384}{559}\)
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
$$ (2-6i)(3-5i) $$
\(-24-28i\)
Evaluate:
\(\int e^x\sin{x}\; dx\)
\(\frac{e^x}{2}(sinx-cosx)+c\)
Simplify:
$$\sin{x}\cot{x}$$\(\cos{x}\)
$$ \DeclareMathOperator{cosec}{cosec} $$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
Describe the graph of an exponential function.
Clue: grow or decay rapidly, horizontal asymptote
Show how the first four terms of the Maclaurin series are obtained for
\(f(x) = \frac{1}{x^2 + 1}\)
\(1 - x^2 + x^4 - x^6\)
Solve for \(z\)
$$ z^4 = \sqrt{3}+i $$
\(\sqrt[4]{2} cis \frac{\pi}{24},\sqrt[4]{2} cis \frac{13\pi}{24} \\ \sqrt[4]{2} cis \frac{-11\pi}{24}, \sqrt[4]{2} cis \frac{-23\pi}{24}\)
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 first \( n \) even numbers is \( n(n + 1) \)
Show true for n=1, assume true for n=k, prove for n=k+1
Simplify:
$$\sqrt{45}$$
\(3\sqrt{5}\)
Simplify:
$$\dfrac{4}{5\sqrt{3}}$$\(\frac{4\sqrt{3}}{15}\)
Simplify
\(4\sqrt{7} - \sqrt{63}\)
\(\sqrt{7}\)
Simplify:
$$\dfrac{4}{6 - \sqrt{5}}$$\(\frac{24 + 4\sqrt{5}}{31}\)
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