AQA A Level Maths Paper 1 June 2022 question paper 7357/1

A curve is defined by the parametric equations

Which of the options shown below is a Cartesian equation for this curve?

Circle your answer.

A periodic sequence is defined by

State the period of the sequence.

Circle your answer.

The curve

is transformed by a stretch, scale factor 2, parallel to the y-axis.

State the equation of the curve after it has been transformed.

Circle your answer.

The graph of

where

is shown in Figure 1.

Figure 1

Tick () one box.

Find an equation of the tangent to the curve

Find the first two terms, in ascending powers of x, of the binomial expansion of

Hence, for small values of x, show that

where A, B and C are constants to be found.

Sketch the graph of

The lines L₁ and L₂ are parallel.

L₁ intersects the y-axis at the point P.

The point Q is the point on L₂ closest to P, as shown in the diagram.

Find the coordinates of Q.

Hence show that PQ k34, where k is an integer to be found.

A circle, C, has centre (a, 17).

L₁ and L₂ are both tangents to C.

Find a.

Find the equation of C.

The first three terms of an arithmetic sequence are given by

Show that x 5 is the only value which gives an arithmetic sequence.

Write down the value of the first term of the sequence.

Find the value of the common difference of the sequence.

The sum of the first N terms of the arithmetic sequence is S_N where

Find the value of N.

The diagram shows a sector of a circle OAB.

The point C lies on OB such that AC is perpendicular to OB.

Angle AOB is radians.

Given the area of the triangle OAC is half the area of the sector OAB, show that

Use a suitable change of sign to show that a solution to the equation

lies in the interval given by

The Newton-Raphson method is used to find an approximate solution to the equation

Using ₁ as a first approximation for apply the Newton-Raphson method twice

Give your answer to three decimal places.

Explain how a more accurate approximation for can be found using the Newton-Raphson method.

Explain why using ₁ as a first approximation in the Newton-Raphson method does not lead to a solution for .

The polynomial p (x) is given by

where b is a constant.

Use the factor theorem to prove that (x 2) is a factor of p(x) for all values of b.

The graph of y p(x) meets the x-axis at exactly two points.

Sketch a possible graph of y p(x)

Given p (x) can be written as

find the value of b.

Fully justify your answer.

Find the sum to infinity of the sequence.

Hence, or otherwise, evaluate

Find the smallest positive exact value of , in radians, which satisfies the equation

Figure 2 shows the approximate shape of the vertical cross section of the entrance to a cave. The cave has a horizontal floor.

The entrance to the cave joins the floor at the points O and P.

Garry models the shape of the cross section of the entrance to the cave using the

where a is a constant, and x and y are the horizontal and vertical distances respectively, in metres, measured from O.

The distance OP is 16 metres.

Find the value of a that Garry should use in the model.

Show that the maximum height of the cave above OP is approximately 10.5 metres.

Suggest one limitation of the model Garry has used.

The region bounded by the curve

and the x-axis is shaded in the diagram below.

Use the trapezium rule with 5 ordinates to find an estimate for the area of the shaded region.

Give your answer correct to three significant figures.

Show that the exact area is given by

Fully justify your answer.

Given that

Express y in terms of sin .

Hence, prove that

Show that

Use the substitution

to show that

can be written as

where k is a constant to be found.

Hence, show

where c is a constant.