AQA A Level Further Maths Paper 1 June 2023 question paper 7367/1

Find the number of solutions of the equation tanh x cosh x

Circle your answer.

0 1 2 3

The diagram below shows a locus on an Argand diagram.

Which of the equations below represents the locus shown above?

Circle your answer.

Which one of the points below is an invariant point under this transformation?

Circle your answer.

(1, 1)
(0, 2)
(3, 0)
(2, 1)

The solution of a second order differential equation is f (t)

The differential equation models heavy damping.

Which one of the statements below could be true?

Tick () one box.

The function f is defined by

Show that

Use the method of differences to show that

Find the value of ₁

Find the value of a

Find the integer c and the value of ₃

The function f is defined by

Find the set of values of x for which

Give your answer in set notation.

Find the x-coordinate of each of the stationary points of the graph of y g(x), giving your answers in exact form.

Explain how Simpson’s rule could be used to find a more accurate estimate of the integral in part (b).

The position vectors of the points A, B and C are

respectively.

Find the area of the triangle ABC

The points A, B and C all lie in the plane

Find an equation of the plane , in the form r n d

The point P has position vector p i 4j 2k

Find the exact distance of P from

The matrix M is defined as

where c is a real number.

The linear transformation T is represented by the matrix M

Show that, for one particular value of c, the image under T of every point lies in the plane

State the value of c for which this occurs.

It is given that M is a non-singular matrix.

State any restrictions on the value of c

Using your answer from part (b)(ii), solve

The function f is defined by

Fully factorise f (x)

Hence, solve the inequality f (x) 0

The new graph is then reflected in the x-axis, to give the graph of y g(x)

Starting from the identities for sinh 2 x and cosh 2 x, prove the identity

The function f is defined by

State the range of f

Use part (a) and part (b)(i) to prove that tanh 2x tanh x if x 0

Use l’Hoˆ pital’s rule to prove that

Show that r takes values in the range r k, where k is an integer.

Find the Cartesian equation of C in the form y² f (x)

The ellipse E has equation

Find the transformation that maps the graph of E onto C

Find the general solution of the differential equation

Show that

where a is a rational number to be found.

A curve C is defined parametrically for t 0 by

The arc formed by the graph of C from t 05 to t 4 is rotated through 2 radians about the x-axis to generate a surface with area S

Find the exact value of S, giving your answer in the form

where b, c and d are rational numbers to be found.