Find the first three terms in the expansion of:
\((2a - 3b)^5\)
\(=32a^5 - 240a^4b \\+720a^3b^2 ...\)
If £200 is invested with an interest rate of 6% compounded quarterly, find the value of the investment after 8 years. £322.06
Here are the coordinates of 3 vertices of a square, what are the coordinates of the 4th?
\((4,3),(7,8),(-1,6)\)
(2,11)
\( X \sim N(43, 7^2)\)
Find
\( P(32\lt X \lt45) \)
\(0.554\)
Factorise:
\(x^2-16\)
\((x+4)(x-4)\)
Factorise:
\(4x^2+9x-9\)
\((x+3)(4x-3)\)
Draw a rough sketch of the graph of:
\(y=-x-2\)
Gradient -1
y intercept -2
What is the value of:
\(4^{-3}\)
\(= \frac{1}{64}\)
Find angle BCA if AB = 5.4m and AC = 6.4m. 40.2o
Find BC if angle BCA = 49o and AB = 5m. 6.63m
Describe the red region.
\(y = 4x^3 - 3x^2 + 4x\)
Find \( \dfrac{dy}{dx}\)
\(12x^2 - 6x + 4\)
\(y = \dfrac{2}{x^7} - 8\sqrt[9]{x}\)
Find \( \frac{dy}{dx}\)
\(-\frac{14}{x^8} - \frac{8}{9}x^{-\frac{8}{9}}\)
\(y=\sqrt{2x^8+3}\)
Find \( \dfrac{dy}{dx}\)
\(8x^7(2x^8+3)^{-\frac{1}{2}}\)
\(y=9x^2e^x\)
Find \( \dfrac{dy}{dx}\)
\(18xe^x+9x^2e^x\)
\(y=\frac{e^{3x}}{ \cos 4x}\)
Find \( \dfrac{dy}{dx}\)
\(\frac{(3e^{3x}cos4x+4e^{3x}sin4x)}{cos^24x}\)
Find the equation of the tangent to the curve:
\(y = 3x^2 - 6x + 9\)
where \(x = 2\)
\(y = 6x + 3\)
Find the equation of the normal to the curve:
\(y = 2x^2 - x + 3\)
where \(x = -1\)
\(y = \frac{x}{5} + 6\frac{1}{5}\)
\(y =12x^2 - 4x + 3\)
Find \( \int y \quad dx\)
\(4x^3 - 2x^2 + 3x+c\)
A game is played 11 times and the probability of winning is 0.8. Calculate the probability of winning exactly 3 times. 0.000216
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:
\(u_{6} = 11\)
\(u_{20} = 67\)
Find the sum of the first 44 terms.3388
Find the equations of the asymptotes of:
\(y=\dfrac{2x+5}{2x+3}\)
\(x=-\frac{3}{2},y=1\)
In the triangle ABC,
AB = 8.6cm.
BC = 5.1cm.
CA = 10.1cm.
Find angle CÂB.
30.3°
Evaluate:
$$\sum_{n=2}^{8} n^2 - 2n$$
133
\(f(x)=8x^2-8x-2\)
What is the value of the discriminent and what does it indicate?
128, Two distinct roots
\(f(x)=x^2-8x-1\)
By completing the square find the coordinates of the vertex.
(4, -17)
Simplify:
\(\log_2(4\sqrt{16})\)
4
Find the integral:
\(\int (x+3)\sqrt{x^2+6x+8} \;dx\)
\(\frac{1}{3}(x^2+6x+8)^{\frac32}+c\)
Find the equation of the straight line that passes through:
(-5, 24) and (7, -12)
\(y=-3x+9\)
Find the inverse of the function \(f\):
\(f(x)=\frac{x+6}{9}\)
\(9x-6\)
\(\text{Find }f(x) \text{ if} \\ ff(2a-3)=\\18a-27 \\\)
\(f(x)=3x\)
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
\(x=\pm \sqrt{y}\)
Sketch a height-time graph as this jar is filled.
Without a calculator find the exact value of
$$\cos{0°} + \sin{\frac{\pi}{6}} + \cos{60°}$$\(2\)
Without a calculator find the exact value of
$$\sin{780°}$$\(\dfrac{\sqrt{3}}{2}\)
Solve:
\(2d+3e-4f = 8 \\ d-e-f= -7\\ 9d+2e-2f=31\)
d = 3, e = 6, f = 4
Find the area of a sector with radius 5.4cm and angle \( \frac{\pi}{3}\)
🍕
15.3cm2
In how many ways can 8 different books be arranged on a shelf if 4 of them must be together?
2880
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 48 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:
\((1+4x)^{\frac{3}{2}}\)
\(1+6x+6x^2-4x^3\)
Evaluate:
\(\int^{80}_{40} \dfrac{1}{x} dx\)
\(\ln{2} \approx 0.693\)
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 7% chance and machine B has a 12% chance of breaking down on any given day?
\(0.661\)
Find the parametric equation of the line:
\( \dfrac{x-2}{5} = \dfrac{3-y}{7} = \dfrac{z}{5} \)
\( x=2+5\lambda \quad y = 3 -7\lambda \quad z=5 \lambda \)
Simplify
$$ \dfrac{1-4i}{1+5i}$$
\(\frac{21}{26}-\frac{9}{26}i\)
Evaluate:
\(\int x\tan^{-1}x\; dx\)
\((\frac{x^2+1}{2})tan^{-1}x-\frac{x}{2}+c\)
Simplify:
$$\sec{x}-\tan{x}\sin{x}$$\(\cos{x}\)
$$ \DeclareMathOperator{cosec}{cosec} $$Find the volume of revolution when \(y=\frac{1}{x}\) is rotated about the x-axis for \(1 \le x \le 2\)
\(\frac{\pi}{2}\) cubic units
What is the difference between a rational and an irrational number?
Rational can be expressed as a fraction with integer numerator and denominator
Show how the first four terms of the Maclaurin series are obtained for
\(f(x) = \cos(x)\)
\(1 - \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720}\)
Expand and simplify:
$$ (i-\sqrt{3})^5 $$
\(16\sqrt{3} + 16i \\ \text{or } 16(\sqrt{3} + i)\)
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 cubes of the first \( n \) natural numbers is \( \left(\frac{n(n + 1)}{2}\right)^2 \)
Show true for n=1, assume true for n=k, prove for n=k+1
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