Formulae

AS Level Mathematics A (H230)

Binomial series

a b a a C C b a b a C b b n n n n n n n

r

n r r n

1

1

2 + = + + 2 2 + + f f+ + - - - ^ h ^n ! Nh,

where C

! !

n !

r r n r

n C

n

r n r = = = - c ^ m h

Differentiation from first principles

x lim h

x h x

f

f f

h 0 = + -

" l^ ^ ^ h h h

Standard deviation

n

x x

n

x

x

2 2

2 - = - /^ h /

 or f

f x x

f

fx

x

2 2

2 - = - ^ h

/

/

/

/

The binomial distribution

If X n + B^ , ph then X x

n

x

P p p 1 x n x = = - - ^ h c m ^ h , mean of X is np, variance of X is np^1-ph

Kinematics

v u = +at

s ut at 2

1 2 = +

s u v t 2

1 = + ^ h

v u 2as

2 2 = +

s vt at 2

1 2 = -

3

© OCR 2022 H230/01 Jun22 Turn over

 Section A: Pure Mathematics

Answer all the questions.

1 Find the term in x3

 in the binomial expansion of ( ) 3 2x 5 - . [2]

2 In this question you must show detailed reasoning.

The cubic polynomial f(x) is defined by f x x5 4x ax 2 3 2 ^ h = - + - , where a is a constant.

You are given that ( ) x-2 is a factor of f(x).

(a) Find the value of a. [2]

(b) Find all the factors of f(x). [3]

3 The diagram in the Printed Answer Booklet shows part of the graph of y x 4 3 x

2 = - + .

(a) It is required to solve the equation x x3 1 0 2 - + = graphically by drawing a straight line with

equation y m = +x c on the diagram, where m and c are constants.

 Find the values of m and c. [2]

(b) Use the graph to find approximate values of the roots of the equation x x3 1 0 2 - + = . [2]

(c) By shading, or otherwise, indicate clearly the regions where all of the following inequalities

are satisfied. You should use the values of m and c found in part (a).

 x H 0 x G 4 y x 4 3 x

2 G - + y m H x c + [3]

4 In this question you must show detailed reasoning.

Solve the following equations, for 0° ° G Gx 360 .

(a) 2 1 tan x+ = 4 [3]

(b) 5 1 sin c x x 2 os2 - = [5]

5 The gradient of a curve is given by d

d

x x

x

y 3 2 = - . The curve passes through the point (6, 20).

(a) Determine the equation of the curve. [4]

(b) Hence determine y x d p

1

y in terms of the constant p.