Find the first three terms in the expansion of
Given that is small such that and higher powers of can be ignored show that
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Find the first three terms in the expansion of
Given that is small such that and higher powers of can be ignored show that
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The trapezium rule is to be used to find an estimate for the integral
The table below shows values for x and f(x), rounded to three significant figures where appropriate.
x |
4 |
4.5 |
5 |
5.5 |
6 |
6.5 |
7 |
7.5 |
8 |
f(x) |
3.16 |
3.39 |
3.61 |
3.81 |
4 |
4.18 |
4.36 |
4.53 |
4.69 |
Using the values in the table find
(i) an estimate for the integral using 2 strips,
(ii) an estimate for the integral using 4 strips,
(iii) an estimate for the integral using 8 strips.
Justify which of the estimates from part (a) will be the most accurate estimate for the integral.
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The function has as a factor, and when is divided by the remainder is 7.
Show that and must satisfy the simultaneous equations:
Hence find and
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A curve has the equation .
Find expressions for and .
Determine the coordinates of the local minimum of the curve.
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A circle has centre (-2, 3) and passes through the point (6, -3).
Find an equation for the circle .
Find an equation for the tangent to the circle at P.
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A training track for cyclists is in the shape of a circle made up of several lane
One lap of the inner lane is m, with each lane working outwards having a lap distance of m more than the lane immediately inside it.
During a training session, a cyclist is expected to complete one lap of each lane, starting with the inner lane, before moving onto the next one.
A cyclist trains until they have completed the first five lanes.
Find the total distance travelled by the cyclist.
There are 8 lanes in total on the training track.
Find the lap distance of the outside lane.
During a particular training session, a cyclist completes a lap of each lane but in addition, also completes a further 10 laps of the outside lane.
Find the total distance the cyclist travels during this training session.
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The diagram below shows the graphs of
Find the area of the shaded region, .
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Show that
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Solve the equation tan for .
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A convergent geometric series has first term 64, and the sum to infinity of the series is 384.
Show that the common ratio, , of the series is .
Find the difference between the ninth and tenth terms of the series, giving your answer correct to 3 significant figures.
Calculate the sum of the first eight terms in the series, giving your answer correct to 3 significant figures.
Given that the sum of the first terms of the series is greater than 380, find the smallest possible value of .
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Solve the equation
giving your answers correct to 3 significant figures.
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Solve the equation
giving your answers in the form , where and are rational numbers to be found.
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