Find the first three terms in the expansion of:
\((4a - 3b)^5\)
\(=1024a^5 - 3840a^4b \\+5760a^3b^2 ...\)
If £200 is invested with an interest rate of 2% compounded quarterly, find the value of the investment after 6 years. £225.43
Here are the coordinates of 3 vertices of a square, what are the coordinates of the 4th?
\((4,1),(9,5),(0,6)\)
(5,10)
\( X \sim N(65, 9^2)\)
Find
\( P(36\lt X \lt48) \)
\(0.0288\)
Factorise:
\(x^2-x-2\)
\((x+1)(x-2)\)
Factorise:
\(5x^2+11x-12\)
\((x+3)(5x-4)\)
Draw a rough sketch of the graph of:
\(2y=x-2\)
Gradient 0.5
y intercept -1
What is the value of:
\(5^{-2}\)
\(= \frac{1}{25}\)
Find angle ABC if AC = 5.1m and AB = 6.3m. 39.0o
Find AC if angle ABC = 64o and BC = 3.8m. 3.42m
Describe the red region.
\(y = 5x^3 - 8x^2 + 2x\)
Find \( \dfrac{dy}{dx}\)
\(15x^2 - 16x + 2\)
\(y = \dfrac{8}{x^3} - 8\sqrt[9]{x}\)
Find \( \frac{dy}{dx}\)
\(-\frac{24}{x^4} - \frac{8}{9}x^{-\frac{8}{9}}\)
\(y=4\ln (7x^2+8)\)
Find \( \dfrac{dy}{dx}\)
\(56x(7x^2+8)^{-1}\)
\(y=x \tan x\)
Find \( \dfrac{dy}{dx}\)
\(tanx+\frac{x}{cos^2x}\)
\(y=\frac{x^2+5}{3x-7}\)
Find \( \dfrac{dy}{dx}\)
\(\frac{(3x^2-14x-15)}{(3x-7)^2}\)
Find the equation of the tangent to the curve:
\(y = 2x^2 - x + 3\)
where \(x = -1\)
\(y = 1 - 5x\)
Find the equation of the normal to the curve:
\(y = 2x^2 - x + 3\)
where \(x = -1\)
\(y = \frac{x}{5} + 6\frac{1}{5}\)
\(y =21x^2 - 8x + 6\)
Find \( \int y \quad dx\)
\(7x^3 - 4x^2 + 6x+c\)
A game is played 19 times and the probability of winning is 0.3. Calculate the probability of winning exactly 16 times. 0.00000143
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:
\(u_{5} = 27\)
\(u_{19} = 83\)
Find the sum of the first 21 terms.1071
Find the equations of the asymptotes of:
\(y=\dfrac{2x-7}{6-2x}-5\)
\(x=3,y=-6\)
In the triangle ABC,
BĈA = 39.5°.
BC = 5.7cm.
AB̂C = 103.97°.
Find CA to 1 dp.
9.3cm
Evaluate:
$$\sum_{n=0}^{6} 114 - n^2$$
707
\(f(x)=6x^2+6x-1\)
What is the value of the discriminant and what does it indicate?
60, Two distinct roots
\(f(x)=x^2+5x+8\)
By completing the square find the coordinates of the vertex.
(-2.5, 1.75)
Evaluate \(\log_2(32) \)
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:
(-1, 5) and (3, 1)
\(y=-x+4\)
Find the inverse of the function \(f\):
\(f(x)=\frac{\sqrt{x}-17}{16}\)
\((16x+17)²\)
\(f(x)=3x+2 \\ g(x)=2x^2 \\[1cm] \text{Find }gf(x)\)
\(18x^2+24x+8\)
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
\(y=x^3-4x\)
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
$$\sin{780°}$$\(\dfrac{\sqrt{3}}{2}\)
Solve:
\( 5a+2b+c=32 \\ 3a+4b+2c= 36 \\ a+5b+c=28\)
a = 4, b = 4, c = 4
Find the area of a sector with radius 7.2cm and angle \( \frac{\pi}{6}\)
🍕
13.6cm2
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:
$$y=\dfrac{x^2+x-6}{x-1}$$x=1,y=x+2
Evaluate:
$$ \sum_{k=1}^{8} \left( \dfrac12 \right)^{k-3} $$
7.97
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:
\(\int^{3}_{0} e^x dx\)
\(e^{3}- 1 \approx 19.1\)
Tin A contains 3 red balls and 5 green balls. Tin B contains 6 red balls and 8 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.
\(\frac{16}{23}\)
Find the parametric equation of the line:
\( \dfrac{x-2}{3} = \dfrac{2-y}{7} = \dfrac{z}{3} \)
\( x=2+3\lambda \quad y = 2 -7\lambda \quad z=3 \lambda \)
Simplify
$$ \sqrt{5-12i} $$
\(3-2i \; \text{ or } -3+2i\)
Evaluate:
\(\int x\sec^2x\; dx\)
\(xtanx+\ln|cosx|+c\)
Simplify:
$$\dfrac{\tan{x}}{\sec{x}}$$\(\sin{x}\)
$$ \DeclareMathOperator{cosec}{cosec} $$Find the volume of revolution when \(y=\sqrt[3]{x^2}\) is rotated about the y-axis for \(2 \le y \le 3\)
\(\frac{65\pi}{4}\) 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
\(f(x) = \cos(x)\)
\(1 - \frac{x^2}{2} + \frac{x^4}{24} - \frac{x^6}{720}\)
Solve for \(z\)
$$ z^4 = - 16 $$
\(\sqrt{2}+i\sqrt{2},-\sqrt{2}+i\sqrt{2} \\ \sqrt{2}-i\sqrt{2},-\sqrt{2}-i\sqrt{2}\)
6 girls sit at random on 6 seats in a row. Determine the probability that the two friends Karen and Julie sit at the ends of the row.
1/15 or 6.67%
Prove by mathematical induction that \( n! > 2^n \) for all integers \( n \) greater than 4
Show true for n=1, assume true for n=k, prove for n=k+1
Simplify
\((3 + 2\sqrt{5})(6 - 3\sqrt{5})\)
\(3\sqrt{5}-12\)
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