Find the first three terms in the expansion of:
\((4a - 5b)^7\)
\(=16384a^7 - 143360a^6b \\+537600a^5b^2 ...\)
If £180 is invested with an interest rate of 6% compounded monthly, find the value of the investment after 7 years. £273.67
Here are the coordinates of 3 vertices of a square, what are the coordinates of the 4th?
\((4,2),(7,8),(-2,5)\)
(1,11)
\( X \sim N(50, 5^2)\)
Find
\( P(40\lt X \lt60) \)
\(0.955\)
Factorise:
\(x^2-16\)
\((x+4)(x-4)\)
Factorise:
\(2x^2+3x-2\)
\((x+2)(2x-1)\)
Draw a rough sketch of the graph of:
\(y=-x+1\)
Gradient -1
y intercept 1
What is the value of:
\(3^{-3}\)
\(= \frac{1}{27}\)
Find angle BCA if AB = 5.3m and BC = 6.3m. 57.3o
Find BC if angle BCA = 62o and AC = 5.1m. 10.9m
Describe the red region.
\(y = 9x^3 - 4x^2 + 8x\)
Find \( \dfrac{dy}{dx}\)
\(27x^2 - 8x + 8\)
\(y = \dfrac{2}{x^{2}} - 4\sqrt[5]{x}\)
Find \( \frac{dy}{dx}\)
\(-\frac{4}{x^{3}} - \frac{4}{5}x^{-\frac{4}{5}}\)
\(y=\frac{1}{(4x+5)^6}\)
Find \( \dfrac{dy}{dx}\)
\(-\frac{24}{(4x+5)^7}\)
\(y=9x^2e^x\)
Find \( \dfrac{dy}{dx}\)
\(18xe^x+9x^2e^x\)
\(y=\frac{2x^2}{4x-1}\)
Find \( \dfrac{dy}{dx}\)
\(\frac{(8x^2-4x)}{(4x-1)^2}\)
Find the equation of the tangent to the curve:
\(y = 5x^2 + 7x + 3\)
where \(x = 1\)
\(y = 17x - 2\)
Find the equation of the normal to the curve:
\(y = 2x^2 - x + 3\)
where \(x = -1\)
\(y = \frac{x}{5} + 6\frac{1}{5}\)
\(y =24x^2 - 6x + 7\)
Find \( \int y \quad dx\)
\(8x^3 - 3x^2 + 7x+c\)
A game is played 16 times and the probability of winning is 0.4. Calculate the probability of winning exactly 13 times. 0.000812
Make up a maths question using this:
\(u_n=u_1+(n-1)d\)
The nth term of an arithmetic sequence
What letter is this?
Two terms of an arithmetic sequence:
\(u_{7} = 52\)
\(u_{16} = 124\)
Find the sum of the first 49 terms.9604
Find the equations of the asymptotes of:
\(y=3\left(\dfrac{2x+3}{7-x}\right)\)
\(x=7,y=-6\)
In the triangle ABC,
BĈA = 60.7°.
BC = 5.7cm.
AB̂C = 66.06°.
Find CA to 1 dp.
6.5cm
Evaluate:
$$\sum_{n=0}^{6} n^2 - 4n$$
7
\(f(x)=2x^2+9x+1\)
What is the value of the discriminant and what does it indicate?
73, Two distinct roots
\(f(x)=x^2+8x-9\)
By completing the square find the coordinates of the vertex.
(-4, -25)
Solve for x:
\( \log(x) + \log(29-x) = 2\)
\(x = 4 \text{ or } x = 25 \)
Find the integral:
\(\int \cos(x)e^{\sin(x)} \;dx\)
\(e^{\sin(x)}+c\)
Find the equation of the straight line that passes through:
(-8, -18) and (6, 10)
\(y=2x-2\)
Find the inverse of the function \(f\):
\(f(x)= \sqrt{x+15}\)
\(x²-15\)
\(\text{Find }f(x) \text{ if} \\ f(1-b)=3-b \\\)
\(f(x)=x+2\)
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
$$\cos{30°} \div \sin{\frac{\pi}{3}}$$\(1\)
Without a calculator find the exact value of
$$\cos{7\pi}$$\(-1\)
Solve:
\(2d+3e-4f = 12 \\ d-e-f= 3\\ 9d+2e-2f=63\)
d = 7, e = 2, f = 2
Find the area of a sector with radius 2.3cm and angle \( \frac{5\pi}{6}\)
🍕
6.92cm2
How many ways can twenty two people be divided into two equal groups?
352716
Find the equations of the asymptotes of:
$$y=\dfrac{-x^2+3x-2}{x}$$x=0,y=3-x
The first term of a geometric sequence is 33 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:
\(\dfrac{1}{2-x}\)
\(\frac{1}{2}+\frac{x}{4}+\frac{x^2}{8}+\frac{x^3}{16}\)
Evaluate:
\(\int^{160}_{80} \dfrac{1}{x} dx\)
\(\ln{2} \approx 0.693\)
Box A contains 7 red and 9 blue cubes, and box B contains 10 red and 13 blue cubes. Finn selects a box at random and takes a cube from that box. Given that the cube is red, what is the probability that it came from box B?
\(\dfrac{160}{321}\)
Find the point of intersection of \(L_1\) and \(L_2\) if:
\(L_1: \quad \dfrac{x+4}{3} = y-2 = \dfrac{z+1}{2} \)
\(L_2: \quad x = \dfrac{y-5}{2} = \dfrac{-z-1}{2} \)
\( (-1,3,1) \)
Simplify
$$ \sqrt{5-12i} $$
\(3-2i \; \text{ or } -3+2i\)
Evaluate:
\(\int 4x\sin{\left( \frac{x}{2} \right)}\; dx\)
\(-8xcos\frac{x}{2}+16sin\frac{x}{2}+c\)
Simplify:
$$\dfrac{\sin^2{x}-1}{\cos{x}}$$\(-\cos{x}\)
$$ \DeclareMathOperator{cosec}{cosec} $$Find the volume of revolution when \(y=x^2\) is rotated about the x-axis for \(-2 \le x \le 2\)
\(\frac{64\pi}{5}\) cubic units
Describe the behavior of a function at its inflection point.
The concavity of the function changes
Show how the first four terms of the Maclaurin series are obtained for
\(f(x) = (1 + x)^n\)
\(1 + nx + \frac{n(n-1)x^2}{2} + \frac{n(n-1)(n-2)x^3}{6}\)
Find the five 5th roots of 1
\(1, cis\frac{2\pi}{5}, cis\frac{4\pi}{5},\\ cis\frac{-2\pi}{5}, cis\frac{-4\pi}{5}\)
6 children, three boys and three girls, are randomly seated on a row of 6 chairs. Determine the likelihood that the three boys are seated together.
1/5 or 20%
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
Simplify:
$$\sqrt{32}$$
\(4\sqrt{2}\)
Simplify:
$$\dfrac{3}{4\sqrt{2}}$$\(\frac{3\sqrt{2}}{8}\)
Simplify
\((3 + \sqrt{5})(3 - \sqrt{5})\)
\(4\)
Simplify:
$$\dfrac{9}{2 + \sqrt{5}}$$\(\frac{18 - 9\sqrt{5}}{-1} = -18 + 9\sqrt{5}\)
Calculate the standard deviation of the following numbers:
11, 17, 20, 23, 29
6
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