Find the first three terms in the expansion of:
\((3a - 4b)^9\)
\(=19683a^9 - 236196a^8b \\+1259712a^7b^2 ...\)
If £220 is invested with an interest rate of 5% compounded monthly, find the value of the investment after 6 years. £296.78
Here are the coordinates of 3 vertices of a square, what are the coordinates of the 4th?
\((5,1),(10,6),(0,6)\)
(5,11)
\( X \sim N(50, 5^2)\)
Find
\( P(40\lt X \lt60) \)
\(0.955\)
Factorise:
\(x^2+x-6\)
\((x+3)(x-2)\)
Factorise:
\(x^2-16\)
\((x+4)(x-4)\)
Draw a rough sketch of the graph of:
\(y=x+2\)
Gradient 1
y intercept 2
What is the value of:
\(4^{0}\)
\(= 1\)
Find angle BCA if AB = 4.3m and AC = 5.4m. 38.5o
Find BC if angle BCA = 31o and AC = 4.4m. 5.13m
Describe the red region.
\(y = 3x^3 - 8x^2 + 7x\)
Find \( \dfrac{dy}{dx}\)
\(9x^2 - 16x + 7\)
\(y = \dfrac{2}{x^5} - 3\sqrt[4]{x}\)
Find \( \frac{dy}{dx}\)
\(-\frac{10}{x^6} - \frac{3}{4}x^{-\frac{3}{4}}\)
\(y=e^{2x+3}\)
Find \( \dfrac{dy}{dx}\)
\(2e^{2x+3}\)
\(y=7x^2e^x\)
Find \( \dfrac{dy}{dx}\)
\(14xe^x+7x^2e^x\)
\(y=\frac{ \ln x}{x^2}\)
Find \( \dfrac{dy}{dx}\)
\(\frac{(1-2lnx)}{x^3}\)
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:
\(y = x^2 - 2x + 1\)
where \(x = 0\)
\(y = \frac{x}{2} + 1\)
\(y =27x^2 - 10x + 5\)
Find \( \int y \quad dx\)
\(9x^3 - 5x^2 + 5x+c\)
A game is played 19 times and the probability of winning is 0.3. Calculate the probability of winning exactly 9 times. 0.0514
Make up a maths question using this:
\(S_n = \dfrac{u_1(r^n-1)}{r-1}\)
The sum of a geometric sequence
What letter is this?
Two terms of an arithmetic sequence:
\(u_{6} = 13\)
\(u_{12} = 19\)
Find the sum of the first 41 terms.1148
Find the equations of the asymptotes of:
\(y=\dfrac{2x-7}{6-2x}-5\)
\(x=3,y=-6\)
In the triangle ABC,
AB = 7.7cm.
BC = 8.2cm.
CA = 13.6cm.
Find angle CÂB.
32.3°
Evaluate:
$$\sum_{n=0}^{7} 91 - n^2$$
588
\(f(x)=5x^2-9x-8\)
What is the value of the discriminant and what does it indicate?
241, Two distinct roots
\(f(x)=x^2+7x-4\)
By completing the square find the coordinates of the vertex.
(-3.5, -16.25)
Express \(\log_2(32)\) in terms of a log to base 4.
\( 10\log_4(2) \text{ or } \log_4(1024) \)
Find the integral:
\(\int 3xe^{x^2} \;dx\)
\(\frac{3}{2}e^{x^2}+c\)
Find the equation of the straight line that passes through:
(-6, 5) and (2, -3)
\(y=-x-1\)
Find the inverse of the function \(f\):
\(f(x)=\frac{\sqrt{x-3}}{2}\)
\(4x²+3\)
\(f(x)=3x+2 \\ g(x)=2x^2 \\[1cm] \text{Find }gf(x)\)
\(18x^2+24x+8\)
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
\(y=x(5-x)\)
Sketch a height-time graph as this jar is filled.
Without a calculator find the exact value of
$$\sin{\frac{\pi}{4}} \times \cos{45°}$$\(\dfrac{1}{2}\)
Without a calculator find the exact value of
$$\sin{5\pi}$$\(0\)
Solve:
\(2x+y-3z= -7 \\ 3x+y+z= 35 \\ x-y+2z = 16\)
x = 6, y = 8, z = 9
Find the perimeter of a sector with radius 5.9cm and angle \( \frac{2\pi}{3}\)
🍕
24.2cm
A safe has a nine-digit code. How many possibilities are there if no digit can be repeated and the code must be odd?
1814400
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:
\((1+2x)^{\frac12}\)
\(1+x-\frac{x^2}{2}+\frac{x^3}{2}\)
Evaluate:
\(\int^{60}_{30} \dfrac{1}{x} dx\)
\(\ln{2} \approx 0.693\)
Box A contains 4 red and 5 blue cubes, and box B contains 6 red and 7 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?
\(\dfrac{27}{53}\)
Find the vector product:
\( \begin{pmatrix} 6 \\ 7 \\ 0 \end{pmatrix} \; \times \; \begin{pmatrix} 6 \\ -2 \\ 7 \end{pmatrix} \)
\( \begin{pmatrix} 49 \\ -42 \\ -54 \end{pmatrix} \)
Simplify
$$ \sqrt{5-12i} $$
\(3-2i \; \text{ or } -3+2i\)
Evaluate:
\(\int (2x+1)e^{-x}\; dx\)
\(-\frac{2x+3}{e^x}+c\)
Simplify:
$$\cosec{x}\tan{x}$$\(\sec{x}\)
$$ \DeclareMathOperator{cosec}{cosec} $$Find the volume of revolution when \(y=x\) is rotated about the x-axis for \(0 \le x \le 1\)
\(\frac{\pi}{3}\) cubic units
Describe the graph of an exponential function.
Clue: grow or decay rapidly, horizontal asymptote
Show how the first four terms of the Maclaurin series are obtained for
\(f(x) = \ln(1 + x)\)
\(x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4}\)
Solve for \(z\)
$$ z^3 = - 8i $$
\(\sqrt{3}-i,2i,-\sqrt{3}-i\)
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 \( 5^n - 1 \) is divisible by 4 for all natural numbers \( n \)
Show true for n=1, assume true for n=k, prove for n=k+1
Simplify:
$$\sqrt{12}$$
\(2\sqrt{3}\)
Simplify:
$$\dfrac{9}{\sqrt{6}}$$\(\frac{9\sqrt{6}}{6} = \frac{3\sqrt{6}}{2}\)
Simplify
\(7\sqrt{7} - 3\sqrt{7}\)
\(4\sqrt{7}\)
Simplify:
$$\dfrac{3}{7 - \sqrt{2}}$$\(\frac{21 + 3\sqrt{2}}{47}\)
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