Find the first three terms in the expansion of:
\((2a - 3b)^4\)
\(=16a^4 - 96a^3b \\+216a^2b^2 ...\)
If £180 is invested with an interest rate of 4% compounded monthly, find the value of the investment after 6 years. £228.73
Here are the coordinates of 3 vertices of a square, what are the coordinates of the 4th?
\((1,1),(4,5),(-3,4)\)
(0,8)
\( X \sim N(27.1, 1.8^2)\)
Find
\( P(28.1\lt X \lt29.1) \)
\(0.156\)
Factorise:
\(x^2-1\)
\((x+1)(x-1)\)
Factorise:
\(2x^2+7x-4\)
\((x+4)(2x-1)\)
Draw a rough sketch of the graph of:
\(2y=x+2\)
Gradient 0.5
y intercept 1
What is the value of:
\(5^{-3}\)
\(= \frac{1}{125}\)
Find angle ABC if AC = 4m and AB = 5.1m. 38.1o
Find AC if angle BCA = 55o and AB = 4.8m. 3.36m
Describe the red region.
\(y = 4x^3 - 3x^2 + 8x\)
Find \( \dfrac{dy}{dx}\)
\(12x^2 - 6x + 8\)
\(y = \dfrac{3}{x^3} - 8\sqrt[9]{x}\)
Find \( \frac{dy}{dx}\)
\(-\frac{9}{x^4} - \frac{8}{9}x^{-\frac{8}{9}}\)
\(y=e^{7x+8}\)
Find \( \dfrac{dy}{dx}\)
\(7e^{7x+8}\)
\(y=x \tan x\)
Find \( \dfrac{dy}{dx}\)
\(tanx+\frac{x}{cos^2x}\)
\(y=\frac{2x^2}{4x-1}\)
Find \( \dfrac{dy}{dx}\)
\(\frac{(8x^2-4x)}{(4x-1)^2}\)
Find the equation of the tangent to the curve:
\(y = x^2 + 6x + 9\)
where \(x = -3\)
\(y = 0\)
Find the equation of the normal to the curve:
\(y = -x^2 + 4x + 2\)
where \(x = 1\)
\(y = 5\frac{1}{2} - \frac{x}{2}\)
\(y =15x^2 - 14x + 4\)
Find \( \int y \quad dx\)
\(5x^3 - 7x^2 + 4x+c\)
A game is played 11 times and the probability of winning is 0.6. Calculate the probability of winning exactly 2 times. 0.00519
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:
\(u_{10} = -54\)
\(u_{20} = -124\)
Find the sum of the first 32 terms.-3184
Find the equations of the asymptotes of:
\(y=\dfrac{5x}{2-x}+3\)
\(x=2,y=-2\)
In the triangle ABC,
AB = 9.4cm.
BC = 9.9cm.
CA = 7.1cm.
Find angle CÂB.
72.2°
Evaluate:
$$\sum_{n=1}^{4} 2^n$$
30
\(f(x)=4x^2+7x+4\)
What is the value of the discriminant and what does it indicate?
-15, No real roots
\(f(x)=x^2+7x-5\)
By completing the square find the coordinates of the vertex.
(-3.5, -17.25)
Simplify \(\log_{10}10^5\)
5
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:
(-9, 1) and (9, -17)
\(y=-x-8\)
Find the inverse of the function \(f\):
\(f(x)= \sqrt{x+16}\)
\(x²-16\)
\(\text{Find }f(x) \text{ if} \\ f(1-b)=3-b \\\)
\(f(x)=x+2\)
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
\(x=\pm \sqrt{y}\)
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
$$\cos{7\pi}$$\(-1\)
Solve:
\(2x+y-3z= 2 \\ 3x+y+z= 22 \\ x-y+2z = 6\)
x = 4, y = 6, z = 4
Find the perimeter of a sector with radius 9.1cm and angle \( \frac{2\pi}{3}\)
🍕
37.3cm
How many ways can thirteen children sit in a row without the youngest being in the middle?
5748019200
Find the equations of the asymptotes of:
$$y=\dfrac{-6x^2+5x}{2x+1}$$x=-1/2,y=4-3x
The first term of a geometric sequence is 37 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:
\(\dfrac{1}{\sqrt{4+x}}\)
\(\frac{1}{2}-\frac{x}{16}+\frac{3x^2}{256}-\frac{5x^3}{2048}\)
Evaluate:
\(\int^{\pi/3}_{\pi/6} \sin{x} \; dx\)
\(\dfrac{\sqrt{3}-1}{2}\)
Tin A contains 7 red balls and 10 green balls. Tin B contains 11 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.
\(\frac{187}{271}\)
There are eight different ways that three planes can relate in three dimensions. One is where they all intersect at a point. How many of the other seven ways can you sketch or describe?
Simplify
$$ (3+i)^{-2} $$
\(\frac{2}{25}-\frac{3}{50}i\)
Evaluate:
\(\int e^x\sin{x}\; dx\)
\(\frac{e^x}{2}(sinx-cosx)+c\)
Simplify:
$$\sec{x}-\tan{x}\sin{x}$$\(\cos{x}\)
$$ \DeclareMathOperator{cosec}{cosec} $$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 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
\(f(x) = \frac{1}{x^2 + 1}\)
\(1 - x^2 + x^4 - x^6\)
Solve for \(z\)
$$ z^3 = - 8i $$
\(\sqrt{3}-i,2i,-\sqrt{3}-i\)
A committee of 5 is chosen from 15 people by random selection. Two Londoners were amongst the group from which the selection was made. Find the probability that both Londoners are chosen for the committee.
2/21 or 9.52%
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:
$$\sqrt{18}$$
\(3\sqrt{2}\)
Simplify:
$$\dfrac{5}{2\sqrt{3}}$$\(\frac{5\sqrt{3}}{6}\)
Simplify
\((2 - 2\sqrt{2})^2\)
\(12 - 8\sqrt{2}\)
Simplify:
$$\dfrac{8}{3 + \sqrt{6}}$$\(\frac{24 - 8\sqrt{6}}{3}\)
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