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

Practice Paper Questions

1
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4 marks

Given that theta is small, show that sin 4 thetaalmost equal to 2 sin 2 theta cos 2 theta.

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2a
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2 marks

Show that fraction numerator open parentheses 5 plus 2 square root of x close parentheses squared over denominator square root of x end fraction  can be written as 25 x to the power of negative 1 half end exponent plus 20 plus 4 x to the power of 1 half end exponent.

2b
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3 marks

Show that fraction numerator 729 over denominator 4 square root of 3 minus square root of 27 end fraction equals 3 to the power of a, where a is a rational number to be found.

2c
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2 marks

Show that  fraction numerator 2 square root of x over denominator square root of 2 end fraction open parentheses square root of 8 x plus 6 x close parentheses  can be written as a y squared plus b y cubed, where space y equals square root of 2 x end root and a and b are integers to be found.

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3a
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2 marks

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 0 degree less than theta less than 360 degree.

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Although r and theta may be varied, it is necessary that the table have a fixed area of  A m2.

Explain why r greater than space square root of A over pi end root .
 

3b
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2 marks

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

P equals 2 r plus fraction numerator 2 A over denominator r end fraction

3c
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5 marks

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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4
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4 marks

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

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5a
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3 marks

In triangle ABC, D is the midpoint of AB and E is the midpoint of AC BE and CD intersect at point F.

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Given that  stack A B with rightwards arrow on top equals 2 bold a space and  stack A C with rightwards arrow on top equals 2 bold b, write the vectors space stack B C with rightwards arrow on top comma space space stack B E space with rightwards arrow on topand stack C D with rightwards arrow on top in terms of
a and b.

5b
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6 marks

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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6a
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2 marks

An exponential model of the form  D equals A e to the power of negative k t end exponent  is used to model the amount of a pain-relieving drug (D mg/ml) there is in a patient’s bloodstream, t hours after the drug was administered by injection. A  and k are constants.

The graph below shows values of In D plotted against t with a line of best fit drawn.

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(i)        Use the graph and line of best fit to estimate ln space D at time t equals 0.

(ii)       Work out the gradient of the line of best fit.

6b
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1 mark

Use your answers to part (a) to write down an equation for the line of best fit in the form ln space D equals m t plus ln space c,  where m and c are constants.

6c
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1 mark

Show that D equals A e to the power of negative k t end exponent can be rearranged to give ln space D equals negative k t plus ln space A

6d
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2 marks

Hence find estimates for the constants A spaceand k.

6e
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2 marks

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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7a
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2 marks

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

7b
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2 marks

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

         600 left parenthesis 1.012 plus 1.012 squared plus 1.012 cubed plus midline horizontal ellipsis plus 1.012 to the power of n right parenthesis

7c
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2 marks

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

         50 space 600 left parenthesis 1.012 to the power of n minus 1 right parenthesis.

7d
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2 marks

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

7e
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1 mark

State one assumption that has been made about this scenario.

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8a
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4 marks

The diagram below shows part of the graph of y space equals space straight f left parenthesis x right parenthesis, where straight f left parenthesis x right parenthesis is the function defined by

straight f left parenthesis x right parenthesis space equals space left parenthesis x squared space minus 1 right parenthesis space In left parenthesis x space plus space 3 right parenthesis comma space space space space space space x space greater than space minus 3

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Points A comma space B and C are the three places where the graph intercepts the x-axis.

Find straight f apostrophe left parenthesis x right parenthesis.

8b
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2 marks

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

8c
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3 marks

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

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9a
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3 marks

Solve the equation vertical line 6 minus 2 x vertical line space equals space 4.

9b
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4 marks

On the same diagram, sketch the graphs of vertical line 6 minus 2 x vertical line space equals space 4 and y space equals space 4.
Label the coordinates of the points where the two graphs intersect each other and the coordinate axes.

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10
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2 marks

A student is estimating the area bounded by the curve space y equals straight f left parenthesis x right parenthesis, the x-axis and the lines x equals a and x equals b.

The student intends to estimate the area by using trapezia of equal width.

q1-8-2-further-integration-easy-a-level-maths-pure-screenshots

Add to the diagram above to show how the student can use 4 trapezia to estimate the area.

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11a
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6 marks

Find

integral x squared sin space 3 x space space straight d x

11b
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4 marks

Find

integral fraction numerator ln space x over denominator space x cubed end fraction space straight d x

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12
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8 marks

A circle has equation x squared plus y squared plus 4 x plus 12 y equals negative 23.

The lines l subscript 1 and l subscript 2 are both tangents to the circle, and they intersect at the point (5, 0).

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Find the equations of l subscript 1 and l subscript 2, giving your answers in the form y equals m x plus c.

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13
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10 marks

Express  fraction numerator 2 x minus 5 over denominator x cubed minus 3 x squared minus 13 x plus 15 end fraction  as partial fractions.

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14
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4 marks

Given that y space greater than space 2, find the general solution to the differential equation

space fraction numerator straight d y over denominator straight d x end fraction space equals space x squared left parenthesis y space minus space 2 right parenthesis

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