Practice Paper 1 (Pure) (OCR A Level Maths: Pure)

Show that   can be written as .

Show that , where a is a rational number to be found.

Show that    can be written as , where and a and b are integers to be found.

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 .
 

Show that the perimeter, P, of the table top is given by the formula

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.

Prove by contradiction that there are an infinite number of even numbers.

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.

By setting up and solving suitable vector equations, prove that each of BE and CD divides the other in the ratio 1:2.

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.

Use your answers to part (a) to write down an equation for the line of best fit in the form ,  where and  are constants.

Show that  can be rearranged to give 

Hence find estimates for the constants and .

Find the time when the amount of the pain-relieving drug in the patient’s bloodstream is 1.5 mg/ml.

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

Show that at the end of year , the amount of money, in pounds, Stephen will have in his account is given by

Hence show that the total amount, in pounds, in Stephen’s account after  years is

Find the year in which the amount in Stephen’s account first exceeds £10 000.

State one assumption that has been made about this scenario.

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 .

Show that the coordinates of point A are (–2, 0).

Find the equation of the tangent to the curve at point A.

Solve the equation .

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.

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.

Find

Find

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 .

Express    as partial fractions.

Given that , find the general solution to the differential equation