Find the first three terms in the expansion of:
\((3a - 4b)^5\)
\(=243a^5 - 1620a^4b \\+4320a^3b^2 ...\)
If £140 is invested with an interest rate of 5% compounded monthly, find the value of the investment after 5 years. £179.67
Here are the coordinates of 3 vertices of a square, what are the coordinates of the 4th?
\((4,5),(9,10),(-1,10)\)
(4,15)
\( X \sim N(9.03, 0.89^2)\)
Find
\( P(7.15\lt X \lt9.01) \)
\(0.474\)
Factorise:
\(x^2-2x-8\)
\((x+2)(x-4)\)
Factorise:
\(4x^2+7x-2\)
\((x+2)(4x-1)\)
Draw a rough sketch of the graph of:
\(y=x-2\)
Gradient 1
y intercept -2
What is the value of:
\(1^{\frac{1}{3}}\)
\(= 1\)
Find angle BCA if AB = 3.5m and AC = 4.7m. 36.7o
Find BC if angle BCA = 69o and AC = 4.7m. 13.1m
Describe the red region.
\(y = 8x^3 - 7x^2 + 8x\)
Find \( \dfrac{dy}{dx}\)
\(24x^2 - 14x + 8\)
\(y = \dfrac{3}{x^5} - 6\sqrt[7]{x}\)
Find \( \frac{dy}{dx}\)
\(-\frac{15}{x^6} - \frac{6}{7}x^{-\frac{6}{7}}\)
\(y=e^{4x+5}\)
Find \( \dfrac{dy}{dx}\)
\(4e^{4x+5}\)
\(y=x^8 \sin x\)
Find \( \dfrac{dy}{dx}\)
\(8x^7sinx+x^8cosx\)
\(y=\frac{ \ln x}{x^2}\)
Find \( \dfrac{dy}{dx}\)
\(\frac{(1-2lnx)}{x^3}\)
Find the equation of the tangent to the curve:
\(y = 5x^2 + 7x + 3\)
where \(x = 1\)
\(y = 17x - 2\)
Find the equation of the normal to the curve:
\(y = 3x^2 - 6x + 9\)
where \(x = 2\)
\(y = 9\frac{1}{3} - \frac{x}{6}\)
\(y =6x^2 - 8x + 8\)
Find \( \int y \quad dx\)
\(2x^3 - 4x^2 + 8x+c\)
A game is played 11 times and the probability of winning is 0.2. Calculate the probability of winning exactly 7 times. 0.00173
Make up a maths question using this:
\(^nC_r=\dfrac{n!}{r!(n-r)!}\)
Combinations
(from n choose r)
What letter is this?
Two terms of an arithmetic sequence:
\(u_{9} = -80\)
\(u_{14} = -125\)
Find the sum of the first 33 terms.-5016
Find the equations of the asymptotes of:
\(y=\dfrac{2x-7}{6-2x}-5\)
\(x=3,y=-6\)
In the triangle ABC,
BC = 7.9cm.
CA = 7.9cm.
BĈA = 73.9°
Find AB to 1 dp.
9.5cm
Evaluate:
$$\sum_{n=3}^{6} 2^n$$
120
\(f(x)=-4x^2+8x-8\)
What is the value of the discriminant and what does it indicate?
-64, No real roots
\(f(x)=x^2-4x+7\)
By completing the square find the coordinates of the vertex.
(2, 3)
Solve for x:
\( \log(x) + \log(29-x) = 2\)
\(x = 4 \text{ or } x = 25 \)
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:
(-4, 4) and (9, -22)
\(y=-2x-4\)
Find the inverse of the function \(f\):
\(f(x)= \sqrt{x-4}\)
\(x²+4\)
\(f(x)=7x-3 \\ g(x)=3x^2 \\[1cm] \text{Find }g \circ f(x)\)
\(147x^2-126x+27\)
Write in standard form:
\(a \times 10^3 \times b\times 10^5\)
where \(a \times b \) is a single digit number \((1 \le ab \lt 10)\)
\(ab\times10^8\)
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
$$\tan{45°} - \sin{\frac{\pi}{6}} - \cos{\frac{\pi}{3}}$$\(0\)
Without a calculator find the exact value of
$$\tan{\dfrac{19\pi}{3}}$$\(\sqrt{3}\)
Solve:
\( j+k+l= 26 \\ 2j-3k+9l= 70\\ -j+k-3l=-26\)
j = 8, k = 9, l = 9
Find the area of a sector with radius 4.6cm and angle \( \frac{\pi}{3}\)
🍕
11.1cm2
In how many ways can 7 different books be arranged on a shelf if 2 of them must be together?
1440
Find the equations of the asymptotes of:
$$y=\dfrac{8x^2-19x-15}{1-2x}$$x=1/2,y=15/2-4x
Evaluate:
$$ \sum_{k=1}^{8} \left( \dfrac12 \right)^{k-3} $$
7.97
Find the first 4 terms in the expansion of:
\((1-4x)^{-3}\)
\(1+12x+96x^2+640x^3\)
Evaluate:
\(\int^{4}_{0} e^x dx\)
\(e^{4}- 1 \approx 53.6\)
What is the probability that it was machine B that broke down if at least one of the independent machines broke down today, given machine A has a 9% chance and machine B has a 13% chance of breaking down on any given day?
\(0.624\)
Find the vector equation of the line:
\( \dfrac{x-8}{3} = \dfrac{2-y}{3} = \dfrac{z}{8} \)
\( \mathbf{r} = \begin{pmatrix} 8 \\ 2 \\ 0 \end{pmatrix} \quad + \quad t \begin{pmatrix} 3 \\ -3 \\ 8 \end{pmatrix} \)
Simplify
$$ (6-3i)(2-2i) $$
\(6-18i\)
Evaluate:
\(\int x^2 \ln{x}\; dx\)
\(\frac{x^3}{9}(3\ln x-1)+c\)
Simplify:
$$\cosec{x}\tan{x}$$\(\sec{x}\)
$$ \DeclareMathOperator{cosec}{cosec} $$Find the volume of revolution when \(y=x^3\) is rotated about the y-axis for \(1 \le y \le 2\)
\(\approx 4.10\) cubic units
What is the binomial theorem?
Clue: Expand \( (a + b)^n \)
Show how the first four terms of the Maclaurin series are obtained for
\(f(x) = \ln(1 + x)\)
\(x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4}\)
Expand and simplify:
$$ (i-\sqrt{3})^5 $$
\(16\sqrt{3} + 16i \\ \text{or } 16(\sqrt{3} + i)\)
5 alphabet blocks A, E, P, R and S are placed at random in a row. What is the likelihood that they spell out either SPEAR or PARSE?
1/60 or 1.67%
Prove by mathematical induction that the sum of the first \( n \) odd numbers is \( n^2 \)
Show true for n=1, assume true for n=k, prove for n=k+1
Simplify:
$$\sqrt{75}$$
\(5\sqrt{3}\)
Simplify:
$$\dfrac{7}{2\sqrt{5}}$$\(\frac{7\sqrt{5}}{10}\)
Simplify
\((1 + \sqrt{2})(2 + \sqrt{2})\)
\(4 + 3\sqrt{2}\)
Simplify:
$$\dfrac{4}{6 - \sqrt{5}}$$\(\frac{24 + 4\sqrt{5}}{31}\)
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