Find the first three terms in the expansion of:
\((3a - 4b)^4\)
\(=81a^4 - 432a^3b \\+864a^2b^2 ...\)
If £200 is invested with an interest rate of 1% compounded monthly, find the value of the investment after 4 years. £208.16
Here are the coordinates of 3 vertices of a square, what are the coordinates of the 4th?
\((4,4),(10,9),(-1,10)\)
(5,15)
\( X \sim N(33, 6^2)\)
Find
\( P(31\lt X \lt37) \)
\(0.378\)
Factorise:
\(x^2-2x-8\)
\((x+2)(x-4)\)
Factorise:
\(3x^2+8x-16\)
\((x+4)(3x-4)\)
Draw a rough sketch of the graph of:
\(y=-x\)
Gradient -1
y intercept 0
What is the value of:
\(1^{-3}\)
\(= 1\)
Find angle ABC if AB = 5.3m and BC = 6.3m. 32.7o
Find AC if angle BCA = 21o and AB = 5.4m. 14.1m
Describe the red region.
\(y = 9x^3 - 5x^2 + 5x\)
Find \( \dfrac{dy}{dx}\)
\(27x^2 - 10x + 5\)
\(y = \dfrac{4}{x^9} - 7\sqrt[8]{x}\)
Find \( \frac{dy}{dx}\)
\(-\frac{36}{x^10} - \frac{7}{8}x^{-\frac{7}{8}}\)
\(y=e^{6x+7}\)
Find \( \dfrac{dy}{dx}\)
\(6e^{6x+7}\)
\(y=x \tan x\)
Find \( \dfrac{dy}{dx}\)
\(tanx+\frac{x}{cos^2x}\)
\(y=\frac{e^{3x}}{ \cos 4x}\)
Find \( \dfrac{dy}{dx}\)
\(\frac{(3e^{3x}cos4x+4e^{3x}sin4x)}{cos^24x}\)
Find the equation of the tangent to the curve:
\(y = -2x^2 - 4x + 6\)
where \(x = 3\)
\(y = 24 - 16x\)
Find the equation of the normal to the curve:
\(y = 2x^2 - x + 3\)
where \(x = -1\)
\(y = \frac{x}{5} + 6\frac{1}{5}\)
\(y =21x^2 - 6x + 7\)
Find \( \int y \quad dx\)
\(7x^3 - 3x^2 + 7x+c\)
A game is played 11 times and the probability of winning is 0.8. Calculate the probability of winning exactly 7 times. 0.111
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:
\(u_{5} = 34\)
\(u_{11} = 94\)
Find the sum of the first 49 terms.11466
Find the equations of the asymptotes of:
\(y=\dfrac{4-7x}{3-14x}\)
\(x=\frac{3}{14},y=\frac{1}{2}\)
In the triangle ABC,
BĈA = 71.7°.
BC = 6.9cm.
AB̂C = 45.95°.
Find CA to 1 dp.
5.6cm
Evaluate:
$$\sum_{n=1}^{6} 3n+1$$
69
\(f(x)=6x^2+7x-9\)
What is the value of the discriminent and what does it indicate?
265, Two distinct roots
\(f(x)=x^2+2x-1\)
By completing the square find the coordinates of the vertex.
(-1, -2)
Solve for x:
\(\log_3x = 2\)
9
Find the integral:
\(\int \dfrac{5x}{x^2-3} \;dx\)
\(\frac{5}{2} \ln(x^2-3)+c\)
Find the equation of the straight line that passes through:
(-8, -19) and (7, 11)
\(y=2x-3\)
Find the inverse of the function \(f\):
\(f(x)=\frac{x+3}{7}\)
\(7x-3\)
\(\text{Find }f(x) \text{ if} \\ f(a^3)=2a^6 \\\)
\(f(x)=2x^2\)
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
\(y+x=2\)
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
$$\cos{\dfrac{16\pi}{3}}$$\(-\dfrac{1}{2}\)
Solve:
\(2x+y-3z= 15 \\ 3x+y+z= 26 \\ x-y+2z = 5\)
x = 7, y = 4, z = 1
Find the area of a sector with radius 3.2cm and angle \( \frac{\pi}{6}\)
🍕
2.68cm2
A safe has a seven-digit code. How many possibilities are there if no digit can be repeated and the code must be odd?
302400
Find the equations of the asymptotes of:
$$y=\dfrac{2x^2+3x-9}{x+2}$$x=-2,y=2x-1
The fourth term of a geometric sequence is \(-16\) and the sum to infinity is \(32\). What is the common ratio?
-0.669
Find the first 4 terms in the expansion of:
\((1+x)^{-8}\)
\(1-8x-36x^2-120x^3\)
Evaluate:
\(\int^{\pi/3}_{\pi/6} \sin{x} \; dx\)
\(\dfrac{\sqrt{3}-1}{2}\)
Every family in Happyland has either has a car or a motor scooter or both. 64% of the families have a car. 82% of the families have a scooter. A family is selected at random and it is found that they have a car. Find the probability they also have a scooter.
\(\dfrac{23}{32}\)
Find the parametric equation of the line:
\( \dfrac{x-7}{7} = \dfrac{5-y}{6} = \dfrac{z}{7} \)
\( x=7+7\lambda \quad y = 5 -6\lambda \quad z=7 \lambda \)
Simplify
$$ \dfrac{i(2-i)}{3-2i}$$
\(-\frac{1}{13}+\frac{8}{13}i\)
Evaluate:
\(\int xe^x\; dx\)
\(xe^x-e^x+c\)
Simplify:
$$\dfrac{\cot{x}}{\cosec{x}}$$\(\cos{x}\)
$$ \DeclareMathOperator{cosec}{cosec} $$Find the volume of revolution when \(y=x(4-x)\) is rotated about the x-axis for \(0 \le x \le 4\)
\(\frac{512\pi}{15}\) cubic units
Describe the behavior of a function at its asymptote.
Clue: approaches but never reaches
Show how the first four terms of the Maclaurin series are obtained for
\(f(x) = \ln(1 + x)\)
\(x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4}\)
Given |z| = 8, find:
$$ |(3+4i)z| $$
\(40\)
6 girls sit at random on 6 seats in a row. Determine the probability that the two friends Karen and Julie sit at the ends of the row.
1/15 or 6.67%
Prove by mathematical induction that \( 2^n > n^2 \) for all integers \( n \) greater
than four
Show true for n=1, assume true for n=k, prove for n=k+1
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