Find the first three terms in the expansion of:
\((2a - 3b)^9\)
\(=512a^9 - 6912a^8b \\+41472a^7b^2 ...\)
If £160 is invested with an interest rate of 5% compounded quarterly, find the value of the investment after 6 years. £215.58
Here are the coordinates of 3 vertices of a square, what are the coordinates of the 4th?
\((2,2),(5,5),(-1,5)\)
(2,8)
\( X \sim N(33, 6^2)\)
Find
\( P(31\lt X \lt37) \)
\(0.378\)
Factorise:
\(x^2-1\)
\((x+1)(x-1)\)
Factorise:
\(4x^2-4x-3\)
\((2x+1)(2x-3)\)
Draw a rough sketch of the graph of:
\(y=2x-2\)
Gradient 2
y intercept -2
What is the value of:
\(4^{-1}\)
\(= \frac{1}{4}\)
Find angle BCA if AB = 5m and BC = 7m. 45.6o
Find AC if angle ABC = 31o and BC = 4.3m. 2.21m
Describe the red region.
\(y = 6x^3 - 3x^2 + 8x\)
Find \( \dfrac{dy}{dx}\)
\(18x^2 - 6x + 8\)
\(y = \dfrac{3}{x^9} - 7\sqrt[8]{x}\)
Find \( \frac{dy}{dx}\)
\(-\frac{27}{x^10} - \frac{7}{8}x^{-\frac{7}{8}}\)
\(y=\sin (2x^2+3)\)
Find \( \dfrac{dy}{dx}\)
\(4xcos(2x^2+3)\)
\(y=x^3 \sin x\)
Find \( \dfrac{dy}{dx}\)
\(3x^2sinx+x^3cosx\)
\(y=\frac{e^{3x}}{ \cos 4x}\)
Find \( \dfrac{dy}{dx}\)
\(\frac{(3e^{3x}cos4x+4e^{3x}sin4x)}{cos^24x}\)
Find the equation of the tangent to the curve:
\(y = x^2 - 2x + 1\)
where \(x = 0\)
\(y = 1 - 2x\)
Find the equation of the normal to the curve:
\(y = 4x^2 + 2x - 1\)
where \(x = -2\)
\(y = \frac{x}{14} + 11\frac{1}{7}\)
\(y =24x^2 - 14x + 9\)
Find \( \int y \quad dx\)
\(8x^3 - 7x^2 + 9x+c\)
A game is played 13 times and the probability of winning is 0.5. Calculate the probability of winning exactly 4 times. 0.0873
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:
\(u_{6} = 54\)
\(u_{11} = 104\)
Find the sum of the first 35 terms.6090
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,
BC = 5.7cm.
CA = 8.3cm.
BĈA = 81.0°
Find AB to 1 dp.
9.3cm
Evaluate:
$$\sum_{n=3}^{9} 73 - n^2$$
231
\(f(x)=6x^2+8x+6\)
What is the value of the discriminant and what does it indicate?
-80, No real roots
\(f(x)=x^2+2x-9\)
By completing the square find the coordinates of the vertex.
(-1, -10)
Simplify:
\(\log_2(4\sqrt{16})\)
4
Find the integral:
\(\int \cos(x)e^{\sin(x)} \;dx\)
\(e^{\sin(x)}+c\)
Find the equation of the straight line that passes through:
(-1, 5) and (3, -7)
\(y=-3x+2\)
Find the inverse of the function \(f\):
\(f(x)=\sqrt{\frac{x-2}{5}}\)
\(5x²+2\)
\(g(x)=5x-4 \\[1cm] \text{Find }g \circ g(x) \\\)
\(25x-24\)
Write in standard form:
\(a \times 10^p \times b\times 10^q\)
where \(a \times b \) is a single digit number \((1 \le ab \lt 10)\)
\(ab\times10^{p+q}\)
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:
\(2d+3e-4f = 12 \\ d-e-f= 3\\ 9d+2e-2f=63\)
d = 7, e = 2, f = 2
Find the perimeter of a sector with radius 4.1cm and angle \( \frac{\pi}{6}\)
🍕
10.3cm
A safe has a four-digit code. How many possibilities are there if no digit can be repeated and the code must be odd?
2520
Find the equations of the asymptotes of:
$$y=\dfrac{-6x^2-4x-1}{3x+2}$$x=-2/3, y=-2x
Evaluate:
$$ \sum_{k=1}^{15} 5 \times (-2)^k $$
-109230
Find the first 4 terms in the expansion of:
\((1+2x)^{\frac12}\)
\(1+x-\frac{x^2}{2}+\frac{x^3}{2}\)
Evaluate:
\(\int^{\pi/3}_{\pi/6} \sin{x} \; dx\)
\(\dfrac{\sqrt{3}-1}{2}\)
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 5% chance and machine B has a 7% chance of breaking down on any given day?
\(0.601\)
Find the point of intersection of these planes:
\(\Pi_1: \quad 2x + y - 3z = -5\)
\(\Pi_2: \quad x - 3y + 2z = 1\)
\(\Pi_3: \quad 3x - 2y + z = 2\)
\( (1,2,3) \)
Simplify
$$ (2-i)^{-2} $$
\(\frac{3}{25}+\frac{4}{25}i\)
Evaluate:
\(\int x^2 \ln{x}\; dx\)
\(\frac{x^3}{9}(3\ln x-1)+c\)
Simplify:
$$\sin{x}\cot{x}$$\(\cos{x}\)
$$ \DeclareMathOperator{cosec}{cosec} $$Find the volume of revolution when \(y=e^x\) is rotated about the x-axis for \(0 \le x \le 3\)
\(\frac{\pi}{2}(e^6-1)\) cubic units
How do you solve a quadratic inequality?
Factorise the quadratic, then analyze the sign of each factor over its domain.
Show how the first four terms of the Maclaurin series are obtained for
\(f(x) = \sqrt{x+1}\)
\(1 + \frac{x}{2} - \frac{x^2}{8} + \frac{x^3}{16}\)
Expand and simplify:
$$ (\sqrt{3}+i)^8 $$
\(-128-128\sqrt{3}i \\ \text{or } 128(-1-\sqrt{3}i)\)
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 squares of the first \( n \) natural numbers is \( \frac{n(n + 1)(2n + 1)}{6} \)
Show true for n=1, assume true for n=k, prove for n=k+1
Simplify:
$$\sqrt{50}$$
\(5\sqrt{2}\)
Simplify:
$$\dfrac{6}{\sqrt{11}}$$\(\frac{6\sqrt{11}}{11}\)
Simplify
\((3 + \sqrt{5})(3 - \sqrt{5})\)
\(4\)
Simplify:
$$\dfrac{8}{3 + \sqrt{6}}$$\(\frac{24 - 8\sqrt{6}}{3}\)
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