Given that is small, show that sin sin cos .
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Given that is small, show that sin sin cos .
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Show that can be written as .
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Show that , where a is a rational number to be found.
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Show that can be written as , where and a and b are integers to be found.
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The top of a patio table is to be made in the shape of a sector of a circle with radius r and central angle , where .
Although r and may be varied, it is necessary that the table have a fixed area of A m2.
Explain why .
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Show that the perimeter, P, of the table top is given by the formula
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Show that the minimum possible value for P is equal to the perimeter of a square with area A. Be sure to prove that your value is a minimum.
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Prove by contradiction that there are an infinite number of even numbers.
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In triangle ABC, D is the midpoint of AB and E is the midpoint of AC. BE and CD intersect at point F.
Given that and , write the vectors and in terms of
a and b.
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By setting up and solving suitable vector equations, prove that each of BE and CD divides the other in the ratio 1:2.
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An exponential model of the form is used to model the amount of a pain-relieving drug (D mg/ml) there is in a patient’s bloodstream, hours after the drug was administered by injection. and are constants.
The graph below shows values of plotted against with a line of best fit drawn.
(i) Use the graph and line of best fit to estimate at time .
(ii) Work out the gradient of the line of best fit.
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Use your answers to part (a) to write down an equation for the line of best fit in the form , where and are constants.
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Show that can be rearranged to give
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Hence find estimates for the constants and .
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Find the time when the amount of the pain-relieving drug in the patient’s bloodstream is 1.5 mg/ml.
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Stephen opens a savings account with £600.
Compound interest is paid annually at a rate of 1.2%.
At the start of each new year Stephen pays another £600 into his account.
Show that at the end of two years Stephen has £1221.69 in the account
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Show that at the end of year , the amount of money, in pounds, Stephen will have in his account is given by
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Hence show that the total amount, in pounds, in Stephen’s account after years is
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Find the year in which the amount in Stephen’s account first exceeds £10 000.
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State one assumption that has been made about this scenario.
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The diagram below shows part of the graph of , where is the function defined by
Points and are the three places where the graph intercepts the -axis.
Find .
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Show that the coordinates of point A are (–2, 0).
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Find the equation of the tangent to the curve at point A.
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Solve the equation .
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On the same diagram, sketch the graphs of and .
Label the coordinates of the points where the two graphs intersect each other and the coordinate axes.
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A student is estimating the area bounded by the curve , the x-axis and the lines and .
The student intends to estimate the area by using trapezia of equal width.
Add to the diagram above to show how the student can use 4 trapezia to estimate the area.
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Find
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Find
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A circle has equation
The lines and are both tangents to the circle, and they intersect at the point (5, 0).
Find the equations of and , giving your answers in the form .
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Express