Find the first three terms in the expansion of:
\((4a - 5b)^8\)
\(=65536a^8 - 655360a^7b \\+2867200a^6b^2 ...\)
If £240 is invested with an interest rate of 3% compounded quarterly, find the value of the investment after 8 years. £304.83
Here are the coordinates of 3 vertices of a square, what are the coordinates of the 4th?
\((1,2),(5,7),(-4,6)\)
(0,11)
\( X \sim N(65, 9^2)\)
Find
\( P(36\lt X \lt48) \)
\(0.0288\)
Factorise:
\(x^2-x-6\)
\((x+2)(x-3)\)
Factorise:
\(6x^2-x-1\)
\((3x+1)(2x-1)\)
Draw a rough sketch of the graph of:
\(y=2x+1\)
Gradient 2
y intercept 1
What is the value of:
\(5^{-1}\)
\(= \frac{1}{5}\)
Find angle BCA if AC = 4.6m and BC = 5.7m. 36.2o
Find BC if angle BCA = 68o and AB = 4.6m. 4.96m
Describe the red region.
\(y = 6x^3 - 3x^2 + 4x\)
Find \( \dfrac{dy}{dx}\)
\(18x^2 - 6x + 4\)
\(y = \dfrac{8}{x^{8}} - 3\sqrt[4]{x}\)
Find \( \frac{dy}{dx}\)
\(-\frac{64}{x^{9}} - \frac{3}{4}x^{-\frac{3}{4}}\)
\(y=3\ln (8x^2+9)\)
Find \( \dfrac{dy}{dx}\)
\(48x(8x^2+9)^{-1}\)
\(y=\sin x \sqrt{ x^2 + 5}\)
Find \( \dfrac{dy}{dx}\)
\(cosx \sqrt{x^2+5}+\frac{xsinx}{\sqrt{x^2+5}}\)
\(y=\frac{x+2}{x-4}\)
Find \( \dfrac{dy}{dx}\)
\(-\frac{6}{(x-4)^2}\)
Find the equation of the tangent to the curve:
\(y = x^2 - 2x + 1\)
where \(x = 0\)
\(y = 1 - 2x\)
Find the equation of the normal to the curve:
\(y = 4x^2 + 2x - 1\)
where \(x = -2\)
\(y = \frac{x}{14} + 11\frac{1}{7}\)
\(y =21x^2 - 6x + 4\)
Find \( \int y \quad dx\)
\(7x^3 - 3x^2 + 4x+c\)
A game is played 12 times and the probability of winning is 0.8. Calculate the probability of winning exactly 4 times. 0.000519
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:
\(u_{10} = -100\)
\(u_{17} = -170\)
Find the sum of the first 29 terms.-4350
Find the equations of the asymptotes of:
\(y=\dfrac{5x}{2-x}+3\)
\(x=2,y=-2\)
In the triangle ABC,
AB = 9.7cm.
BC = 5.1cm.
CÂB = 28.1°.
Find angle BĈA.
116° or 64°
Evaluate:
$$\sum_{n=4}^{8} 72 - n^2$$
170
\(f(x)=7x^2-3x+7\)
What is the value of the discriminant and what does it indicate?
-187, No real roots
\(f(x)=x^2-9x+9\)
By completing the square find the coordinates of the vertex.
(4.5, -11.25)
Evaluate \(\log_2(32) \)
5
Find the integral:
\(\int x\sqrt{x^2+3} \;dx\)
\(\frac{1}{3}(x^2+3)^{\frac32}+c\)
Find the equation of the straight line that passes through:
(-4, 1) and (4, 9)
\(y=x+5\)
Find the inverse of the function \(f\):
\(f(x)=\frac{x+6}{3}\)
\(3x-6\)
\(f(x)=7x-3 \\ g(x)=3x^2 \\[1cm] \text{Find }g \circ f(x)\)
\(147x^2-126x+27\)
Write in standard form:
\((a \times 10^p) \div (b\times 10^q)\)
where \(a \div b \) is a single digit number \((1 \le \frac{a}{b} \lt 10)\)
\(\frac{a}{b}\times10^{p-q}\)
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
$$\cos{\frac{\pi}{4}} + \sin{45°}$$\(\sqrt{2}\)
Without a calculator find the exact value of
$$\cos{\dfrac{16\pi}{3}}$$\(-\dfrac{1}{2}\)
Solve:
\( j+k+l= 15 \\ 2j-3k+9l= 60\\ -j+k-3l=-23\)
j = 9, k = 1, l = 5
Find the area of a sector with radius 4.7cm and angle \( \frac{\pi}{4}\)
🍕
8.67cm2
In how many ways can 12 different books be arranged on a shelf if 2 of them must be together?
79833600
Find the equations of the asymptotes of:
$$y=\dfrac{x^2+x-6}{x-1}$$x=1,y=x+2
The first term of a geometric sequence is 32 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:
\(\int^{\pi/3}_{\pi/6} \sin{x} \; dx\)
\(\dfrac{\sqrt{3}-1}{2}\)
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 0.3. Find the probability that there will be a high level of pollution on a day when it is cloudy.
\(0.429\)
Find the vector product:
\( \begin{pmatrix} 4 \\ 4 \\ 0 \end{pmatrix} \; \times \; \begin{pmatrix} 8 \\ -9 \\ 7 \end{pmatrix} \)
\( \begin{pmatrix} 28 \\ -28 \\ -68 \end{pmatrix} \)
Simplify
$$ \dfrac{2-i}{1+3i}$$
\(-\frac{1}{10}-\frac{7}{10}i\)
Evaluate:
\(\int x\cos{x}\; dx\)
\(xsinx+cosx+c\)
Simplify:
$$\dfrac{\sin^2{x}-1}{\cos{x}}$$\(-\cos{x}\)
$$ \DeclareMathOperator{cosec}{cosec} $$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
What is the difference between a permutation and a combination?
Permutations consider order; combinations do not.
Show how the first four terms of the Maclaurin series are obtained for
\(f(x) = \frac{1}{x^2 + 1}\)
\(1 - x^2 + x^4 - x^6\)
Expand and simplify:
$$ (i-\sqrt{3})^5 $$
\(16\sqrt{3} + 16i \\ \text{or } 16(\sqrt{3} + i)\)
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 \( n \) even numbers is \( n(n + 1) \)
Show true for n=1, assume true for n=k, prove for n=k+1
Simplify:
$$\sqrt{75}$$
\(5\sqrt{3}\)
Simplify:
$$\dfrac{7}{\sqrt{5}}$$\(\frac{7\sqrt{5}}{5}\)
Simplify
\((9 + 2\sqrt{2})(9 - 2\sqrt{2})\)
\(73\)
Simplify:
$$\dfrac{7}{4 + \sqrt{3}}$$\(\frac{28 - 7\sqrt{3}}{13}\)
Calculate the standard deviation of the following numbers:
30, 38, 38, 42, 42, 50
6
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