Edexcel A Level Maths Paper 1: Pure Mathematics 1 October 2020 question paper 9MA0/01

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

giving each term in simplest form.

Explain how you could use x = in the expansion to find an approximation for There is no need to carry out the calculation.

By taking logarithms of both sides, solve the equation

giving the value of p to one decimal place.

Relative to a fixed origin O ● point A has position vector 2i + 5j − 6k ● point B has position vector 3i − 3j − 4k ● point C has position vector 2i − 16j + 4k

Find AB

Show that quadrilateral OABC is a trapezium, giving reasons for your answer.

The function f is defined by

Find f⁻¹(7)

Show that ff (x) = where a and b are integers to be found.

A car has six forward gears. The fastest speed of the car ● in 1^st gear is 28 km h^–1 ● in 6^th gear is 115 km h^–1 Given that the fastest speed of the car in successive gears is modelled by an arithmetic sequence,

find the fastest speed of the car in 3^rd gear. Given that the fastest speed of the car in successive gears is modelled by a geometric sequence,

find the fastest speed of the car in 5^th gear.

Express sinx + 2 cosx in the form Rsin (x + α) where R and α are constants, R 0

Give the exact value of R and give the value of α in radians to 3 decimal places. The temperature, θ°C, inside a room on a given day is modelled by the equation

where t is the number of hours after midnight. Using the equation of the model and your answer to part (a),

deduce the maximum temperature of the room during this day,

find the time of day when the maximum temperature occurs, giving your answer to the nearest minute.

Figure 1 shows a sketch of a curve C with equation y = f (x) and a straight line l. The curve C meets l at the points (−2, 13) and (0, 25) as shown. The shaded region R is bounded by C and l as shown in Figure 1. Given that ● f (x) is a quadratic function in x ● (−2, 13) is the minimum turning point of y = f (x) use inequalities to define R.

A new smartphone was released by a company. The company monitored the total number of phones sold, n, at time t days after the phone was released. The company observed that, during this time, the rate of increase of n was proportional to n Use this information to write down a suitable equation for n in terms of t. (You do not need to evaluate any unknown constants in your equation.)

Figure 2 shows a sketch of the curve C with equation y = f (x) where

Show that f ʹ(x) = 8(2 + x − x²)e^−2x

Hence find, in simplest form, the exact coordinates of the stationary points of C. The function g and the function h are defined by g (x) = 2f (x) x h (x) = 2f (x) − 3 x 0

(c) Find (i) the range of g

the range of h

Use the substitution x = u² + 1 to show that

where p and q are positive constants to be found.

Hence, using algebraic integration, show that

where a is a rational constant to be found.

Circle C₁ has equation x² + y² = 100 Circle C₂ has equation (x − 15)² + y² = 40 The circles meet at points A and B as shown in Figure 3.

Show that angle AOB = 0.635 radians to 3 significant figures, where O is the origin. The region shown shaded in Figure 3 is bounded by C₁ and C₂

Find the perimeter of the shaded region, giving your answer to one decimal place.

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

Show that cosecθ − sinθ ≡ cosθcotθ θ ≠ (180n)° n

Hence, or otherwise, solve for 0 < x < 180° cosec x − sin x = cos x cot (3 x − 50°)

A sequence of numbers a₁, a₂, a₃, … is defined by

where k is a constant. Given that ● the sequence is a periodic sequence of order 3 ● a₁ = 2

show that

For this sequence explain why k 1

Find the value of

A large spherical balloon is deflating. At time t seconds the balloon has radius rcm and volume Vcm³ The volume of the balloon is modelled as decreasing at a constant rate.

Using this model, show that

where k is a positive constant. Given that ● the initial radius of the balloon is 40 cm ● after 5 seconds the radius of the balloon is 20 cm ● the volume of the balloon continues to decrease at a constant rate until the balloon is empty

solve the differential equation to find a complete equation linking r and t.

Find the limitation on the values of t for which the equation in part (b) is valid.

The curve C has equation

Show that

Prove by contradiction that there are no positive integers p and q such that