Practice Paper 1 (Pure 1) (CIE A Level Maths: Probability & Statistics 2)

Practice Paper Questions

1
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3 marks

The line segment A B is the diameter of a circle.

A has coordinates (—7, —9) and B has coordinates (9, 3).

Find the coordinates of the centre of the circle and the length of the diameter.

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2
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4 marks

Find the equation of the curve passing through the point (-2, 3) and given by

         y space equals space integral open parentheses 3 minus 2 x plus 4 over x squared close parentheses space straight d x

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3
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4 marks

In a computer animation, the radius of a circle increases at a constant rate of 1 millimetre per second. Find the rate, per second, at which the area of the circle is increasing at the time when the radius is 8 millimetres. Give your answer as a multiple of straight pi.

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4a
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2 marks

The diagram shows the graph of y equals straight f left parenthesis t right parenthesis, where straight f left parenthesis t right parenthesis equals cos space t,  0 degree less or equal than x less or equal than 360 degree.

q7a-2-10-combinations-of-transformations-a-level-only-edexcel-a-level-pure-maths-hard

(i)

Write down the maximum value of y when y equals negative 2 straight f left parenthesis 3 t right parenthesis.

(ii)
Write down the value of t for which this maximum occurs.
4b
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2 marks

Find, in terms of straight f left parenthesis t right parenthesis, the combination of transformations that would map the graph of y equals straight f left parenthesis t right parenthesis onto the graph of y equals 2 minus 4 sin space t,0 degree less or equal than x less or equal than 180 degree  .

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5
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5 marks

In the expansion of left parenthesis a plus b x right parenthesis to the power of 4 comma the coefficient of the x cubed term is 216.

In the expansion of left parenthesis a plus b x right parenthesis to the power of 6 comma the coefficient of the x to the power of 4 term is 4860.

Find the possible values of a and b.

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6a
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2 marks

The curve C has equation y equals 3 x squared minus 6 plus 4 over x.  The point Popen parentheses 1 comma space 1 close parentheses lies on C.

Find an expression for fraction numerator d y over denominator d x end fraction.

6b
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3 marks

Show that an equation of the normal to C at point P is x plus 2 y equals 3.

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7a
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2 marks

Show that open parentheses x plus 1 close parentheses open parentheses x minus 2 close parentheses open parentheses x minus 3 close parentheses identical to x cubed minus 4 x squared plus x plus 6.

7b
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5 marks

Hence, or otherwise, solve the equation tan3 xnegative 4 tan2 xplustan xplus 6 equals 0  for  0 degree less or equal than space x less or equal than 360 degree, giving your answers to 1 decimal place where appropriate.

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8a
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3 marks

The kth term of a geometric progression is given by u subscript k space equals space 2401 space open parentheses 2 over 7 close parentheses to the power of k.

Calculate, giving your answers as exact values

The sum to infinity of the progression starting with the seventh term.

8b
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2 marks

The sum to infinity of the progression whose kth term is given by v subscript k space equals space u subscript k plus 4 end subscript, where u subscript k is defined as above.

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9
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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 l1 and l2 are both tangents to the circle, and they intersect at the point (5, 0).

q7-3-2-circles-vhard-a-level-maths-pure-screenshot

Find the equations of l1 and l2, giving your answers in the form y equals m x plus c.

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10a
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4 marks

The circle sector O A B is shown in the diagram below.

The angle at the centre is theta radians , and the radii O A and O B are each equal to r cm

Additionally,C D is parallel to A B, so that  A D equals B C and O D equals O C.

q7a-5-4-radian-measure-a-level-only-edexcel-a-level-pure-maths-hard

In the case when A D equals B C equals 1 cm, show that the area of the shaded shape A B C D is given by 1 half straight theta r squared minus 1 half open parentheses straight r minus 1 close parentheses squared sin theta.

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10b
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3 marks

Show that for small values of theta, the area of A B C D is approximately 1 half theta open parentheses 2 straight r minus 1 close parentheses.

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11a
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3 marks

The functions f(x) and g(x) are defined as follows

 straight f left parenthesis x right parenthesis space equals space left parenthesis x space minus space 1 right parenthesis squared space minus space 4 space space space space space space space space space space space x element of straight real numbers comma x greater or equal than 1

g left parenthesis x right parenthesis equals space 1 space plus space square root of left parenthesis x plus 4 right parenthesis space end root space space space space space space space space space space x element of straight real numbers comma x greater or equal than negative 4

Find 

(i)  fg(x)

(ii)  gf(x)

11b
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2 marks

Write down straight f to the power of negative 1 end exponent left parenthesis x right parenthesis and state its domain and range.

11c
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2 marks

The graphs of f(x) and straight f to the power of negative 1 end exponent left parenthesis x right parenthesis are drawn on the same axes.
Describe the transformation that would map one graph onto the other.

11d
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2 marks

Find the coordinates of the point where the graphs of  y = f(x) and  y space equals space straight f to the power of negative 1 end exponent left parenthesis x right parenthesis meet. 

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12a
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3 marks

A curve has the equation y equals x cubed minus 12 x plus 7.

Find expressions forfraction numerator d y over denominator d x end fractionand fraction numerator d squared y over denominator d x squared end fraction.

12b
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3 marks

Determine the coordinates of the local minimum of the curve.

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13
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8 marks

The diagram below shows the graphs of  y equals 12 x minus x squared minus 27 space and space y equals x squared minus 14 x plus 45.

screenshot-2023-08-09-at-12-10-41-pm

Find the area of the shaded region, R.

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