Pure Mathematics and Statistics.Mathematics B(MEI) (H630, Questions and answers.
Arithmetic series
Sn 1 n(a l) 1 n{2a (n
... [Show More] 1)d}
Geometric series
a(1 rn )
Sn
S
1 r
a
1 r
for
r 1
Binomial series
(a b)n an nC1 an1b nC2 an2b2 K
n Cr anrbr K
bn
(n ¥ ) ,
where n C
C n n!
r n r
r!(n r)!
(1 x)n 1 nx n(n 1) x2 K
2!
n(n 1)K (n r 1) xr K
r!
x 1,
n ¡
Differentiation
f (x) f (x)
tan kx k sec2 kx
sec x sec x tan x
cot x cosec2 x
cosec x cosec x cot x
v du u dv
Quotient Rule
y u ,
dy dx dx
v dx v2
Differentiation from first principles
f (x) lim f (x h) f (x)
h 0 h
Integration
f (x) dx ln f (x) c
f (x)
f (x)f (x)n dx 1 f (x)n1 c
n 1
Integration by parts u dv dx uv v du dx
Small angle approximations
sin , cos 1 1 2, tan where θ is measured in radians
Trigonometric identities
sin( A B) sin Acos B cos Asin B
cos( A B) cos Acos B msin Asin B
tan( A B)
tan A tan B
1 mtan A tan B
( A B (k 1 ) )
Numerical methods
Trapezium rule:
b y dx 1 h{( y y ) 2( y y
… y
) }, where h b a
a 2 0 n 1 2
n1 n
The Newton-Raphson iteration for solving f(x) 0 : x
x f(xn )
n1
f (xn )
Probability
P( A B) P( A) P(B) P( A B)
P( A B) P( A) P(B | A) P(B) P( A | B ) or
P( A | B) P( A B)
P(B )
Sample variance
1
x 2
s2 S where S
(x
x )2 x2
i x2 nx 2
n 1
xx xx i i n i
Standard deviation, s
The binomial distribution
If X ~ B(n, p) then P(X r) nCr prqnr where q 1 p
Mean of X is np
Hypothesis testing for the mean of a Normal distribution
2 2 X
If X ~ N,
then X ~ N , n and /
~ N(0, 1)
n
Percentage points of the Normal distribution
p 10 5 2 1
z 1.645 1.960 2.326 2.576
Kinematics
Motion in a straight line Motion in two dimensions
v u at
s ut 1 at2 s 1 u vt v2 u2 2as s vt 1 at2
v u at
s ut 1 at2
s 1 u vt
s vt 1 at2
Answer all the questions
1 The masses, x grams, of a sample of 800 apples are summarised in the histogram, fig. 1.
Fig. 1 Find an estimate of the median mass of the apples.
[4]
2 Solve the equation 24x1 352x , giving your answer in the form
x log10 a .
log10 b
[6]
3 The value £V of a car t years after it is new is modelled by the equation V Aekt , where A and k
are positive constants which depend on the make and model of the car.
(a) Brian buys a new sports car. Its value is modelled by the equation V 20000e0.2t . Calculate how much value, to the nearest £100, this car has lost after 1 year.
[2]
At the same time as Brian buys his car, Kate buys a new hatchback for £15 000. Her car loses
£2000 of its value in the first year.
(b) (i) Show that, for Kate’s car, k 0.143 correct to 3 significant figures.
[3]
In this question you must show detailed reasoning
(ii) Find how long it is before Brian’s and Kate’s cars have the same value.
[3]
4 In this question you must show detailed reasoning
(a) Show that the equation sin x cos x 6cos x
tan x
can be expressed in the form tan2 x tan x 6 0 .
[2]
(b) Hence solve the equation sin x cos x 6cos x
tan x
for 0o x 360o
[4]
5 Fig. 5 shows sketch of the curve with equation
y x 4
x2
(a) Show that
Fig. 5
d2 y 24
dx2 x4
[3]
(b) (i) Hence determine the coordinates of the stationary point on the curve.
[3]
(ii) Verify that the stationary point is a maximum.
[2]
6 A politics student wishes to investigate average house prices across England.
The student decides to compare employment rates with median house prices, as shown in Fig 6.1 [Show Less]