AQA A Level Maths Paper 1 November 2021 question paper 7357/1

State the set of values of x which satisfies the inequality

Tick () one box.

Given that y ln (5x)

Circle your answer.

A geometric sequence has a sum to infinity of 3

A second sequence is formed by multiplying each term of the original sequence by 2

What is the sum to infinity of the new sequence?

Circle your answer.

The sum to infinity does not 6 3 6 exist

Millie is attempting to use proof by contradiction to show that the result of multiplying an irrational number by a non-zero rational number is always an irrational number.

Select the assumption she should make to start her proof.

Tick () one box.

Every irrational multiplied by a non-zero rational is irrational.

Every irrational multiplied by a non-zero rational is rational.

There exists a non-zero rational and an irrational whose product is irrational.

There exists a non-zero rational and an irrational whose product is rational.

The line L has equation

The point P has coordinates (15, 2)

Find the equation of the line perpendicular to L which passes through P.

Hence, find the shortest distance from P to L.

The sum of the first n terms of this series is T_n

Find the value of n such that T_n S_n

2 3 7 The equation x x x 3 has a single solution, x

By considering a suitable change of sign, show that lies between 1.5 and 1.6

Show that the equation x² x³ x 3 can be rearranged into the form

Hence, deduce an interval of width 0.001 in which lies.

Given that

show that

Hence, solve

in the interval 0 2

Give your answers to two decimal places.

Solve

Give your answers to one decimal place.

The table below shows the annual global production of plastics, P, measured in millions of tonnes per year, for six selected years.

Year	1980	1985	1990	1995	2000	2005
	75	94	120	156	206	260

It is thought that P can be modelled by

where t is the number of years after 1980 and A and k are constants.

Show algebraically that the graph of log₁₀ P against t should be linear.

Complete the table below.

	0	5	10	15	20	25
log10	1.88	1.97	2.08		2.31

Plot log₁₀ P against t, and draw a line of best fit for the data.

Hence, show that k is approximately 0.02

Find the value of A.

Using the model with k 002 predict the number of tonnes of annual global production of plastics in 2030.

Using the model with k 002 predict the year in which P first exceeds 8000

Give a reason why it may be inappropriate to use the model to make predictions about future annual global production of plastics.

Given that

use the quotient rule to show that

2 10 (b) The region enclosed by the curve y tan x and the horizontal line, which intersects the curve at x and x is shaded in the diagram below.

Show that the area of the shaded region is

Fully justify your answer.

A curve, C, passes through the point with coordinates (1, 6)

The gradient of C is given by

Show that C intersects the coordinate axes at exactly one point and state the coordinates of this point.

Fully justify your answer.

The equation of a curve is

The curve intersects the positive x-axis at the point P.

Show that the gradient of the curve at P is

Find the equation of the normal to the curve at P, giving your answer in the form

Given that

use the factor theorem to prove that (5x 1) is a factor of P(x).

Factorise P (x) completely.

Hence, prove that 250n³ 300n² 110n 12 is a multiple of 12 when n is a positive whole number.

The curve C is defined for t 0 by the parametric equations

2 2 3

C is shown in the diagram below.

Find the gradient of C at the point where it intersects the positive x-axis.

The area A enclosed between C and the x-axis is given by

Find the value of b.

Use the substitution y 4t² t³ to show that

2 3 4

Find the value of A.

Show that

for small values of x.

Hence, show that the area between the graph with equation

the positive x-axis and the line x 025 can be approximated by

where m and n are integers to be found.

Explain why

is not a suitable approximation for

Explain how

may be approximated by

for suitable values of a and b.