Find the first three terms in the expansion of:
\((2a - 3b)^7\)
\(=128a^7 - 1344a^6b \\+6048a^5b^2 ...\)
If £140 is invested with an interest rate of 6% compounded quarterly, find the value of the investment after 4 years. £177.66
Here are the coordinates of 3 vertices of a square, what are the coordinates of the 4th?
\((5,1),(8,7),(-1,4)\)
(2,10)
\( X \sim N(27.1, 1.8^2)\)
Find
\( P(28.1\lt X \lt29.1) \)
\(0.156\)
Factorise:
\(x^2-x-2\)
\((x+1)(x-2)\)
Factorise:
\(6x^2-x-2\)
\((2x+1)(3x-2)\)
Draw a rough sketch of the graph of:
\(y=-x+1\)
Gradient -1
y intercept 1
What is the value of:
\(2^{1}\)
\(= 2\)
Find angle BCA if AC = 3.6m and BC = 5.2m. 46.2o
Find AC if angle ABC = 24o and BC = 4.8m. 1.95m
Describe the red region.
\(y = 8x^3 - 9x^2 + 2x\)
Find \( \dfrac{dy}{dx}\)
\(24x^2 - 18x + 2\)
\(y = \dfrac{6}{x^6} - 3\sqrt[4]{x}\)
Find \( \frac{dy}{dx}\)
\(-\frac{36}{x^7} - \frac{3}{4}x^{-\frac{3}{4}}\)
\(y=\sin (2x^2+3)\)
Find \( \dfrac{dy}{dx}\)
\(4xcos(2x^2+3)\)
\(y=x(5x^2+6)^5\)
Find \( \dfrac{dy}{dx}\)
\((5x^2+6)^5+50x^2(5x^2+6)^4\)
\(y=\frac{2x^2}{4x-1}\)
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:
\(y = x^2 - 2x + 1\)
where \(x = 0\)
\(y = \frac{x}{2} + 1\)
\(y =15x^2 - 16x + 5\)
Find \( \int y \quad dx\)
\(5x^3 - 8x^2 + 5x+c\)
A game is played 19 times and the probability of winning is 0.4. Calculate the probability of winning exactly 6 times. 0.145
Make up a maths question using this:
\( \tan \theta \equiv \dfrac{ \sin \theta}{ \cos \theta}\)
Trigonometric identity
What letter is this?
Two terms of an arithmetic sequence:
\(u_{8} = -4\)
\(u_{16} = -20\)
Find the sum of the first 45 terms.-1530
Find the equations of the asymptotes of:
\(y=\dfrac{2x-7}{6-2x}-5\)
\(x=3,y=-6\)
In the triangle ABC,
AB = 5.3cm.
BC = 7.1cm.
CA = 6.1cm.
Find angle CÂB.
76.7°
Evaluate:
$$\sum_{n=3}^{6} 2^n$$
120
\(f(x)=-4x^2-3x-2\)
What is the value of the discriminant and what does it indicate?
-23, No real roots
\(f(x)=x^2+5x-3\)
By completing the square find the coordinates of the vertex.
(-2.5, -9.25)
Simplify \(\log_{10}10^5\)
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:
(-3, -15) and (8, 7)
\(y=2x-9\)
Find the inverse of the function \(f\):
\(f(x)=\frac{\sqrt{x}-18}{15}\)
\((15x+18)²\)
\(f(x)=5x+1 \\ g(x)=x^2 \\[1cm] \text{Find }gf(3x)\)
\(225x^2+30x+1\)
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
\(x=\pm \sqrt{y}\)
Sketch a height-time graph as this jar is filled.
Without a calculator find the exact value of
$$\sin{\frac{\pi}{2}} \div \cos{45°}$$\(\sqrt{2}\)
Without a calculator find the exact value of
$$\sin{\dfrac{13\pi}{6}}$$\(\dfrac{1}{2}\)
Solve:
\(2x+y-3z= 8 \\ 3x+y+z= 32 \\ x-y+2z = 12\)
x = 8, y = 4, z = 4
Find the area of a sector with radius 7.2cm and angle \( \frac{\pi}{3}\)
🍕
27.1cm2
How many ways can five children sit in a row without the youngest being in the middle?
96
Find the equations of the asymptotes of:
$$y=\dfrac{-x^2+3x-2}{x}$$x=0,y=3-x
The first term of a geometric sequence is 24 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:
\((1-\dfrac{x}{2})^{\frac13}\)
\(1-\frac{x}{6}-\frac{x^2}{36}-\frac{5x^3}{648}\)
Evaluate:
\(\int^{3}_{0} (x-8)^2 \; dx\)
\(129\)
Given equal populations of Type X and Type Y bacteria, with mutation rates of 70% and 80% respectively, if a mutated bacterium is found, what's the probability it's Type Y?
\(0.533\)
Find the area of the triangle with sides:
\( \begin{pmatrix} 2 \\ 8 \\ 0 \end{pmatrix}, \; \begin{pmatrix} 2 \\ -2 \\ 9 \end{pmatrix} \; \text{and} \; \begin{pmatrix} 0 \\ -10 \\ 9 \end{pmatrix} \)
38.4 square units
Simplify
$$ (1+i)^{4} $$
\(-4\)
Evaluate:
\(\int x^2 \ln{x}\; dx\)
\(\frac{x^3}{9}(3\ln x-1)+c\)
Simplify:
$$\dfrac{\tan{x}}{\sec{x}}$$\(\sin{x}\)
$$ \DeclareMathOperator{cosec}{cosec} $$Find the volume of revolution when \(y=x\) is rotated about the x-axis for \(0 \le x \le 1\)
\(\frac{\pi}{3}\) 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
\(f(x) = \ln(1 + x)\)
\(x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4}\)
Find the four 4th roots of 1
\(1, i, -1, -i\)
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 \( n \) natural numbers is \( \frac{n(n + 1)}{2} \)
Show true for n=1, assume true for n=k, prove for n=k+1
Simplify:
$$\sqrt{32}$$
\(4\sqrt{2}\)
Simplify:
$$\dfrac{3}{4\sqrt{2}}$$\(\frac{3\sqrt{2}}{8}\)
Simplify
\((1 + \sqrt{2})(2 + \sqrt{2})\)
\(4 + 3\sqrt{2}\)
Simplify:
$$\dfrac{2}{5 - \sqrt{7}}$$\(\frac{10 + 2\sqrt{7}}{18}\)
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