7/29/2019 Exponentials and Logarithms
1/2
Exponentials and logarithms1Q
The number n, of insects in a colony, is given by
n = 3500e 0.08t
Where t is the number of days after observations commence.
(a) Sketch the graph of n against t. [2](b) Find the population
of the colony after 50 days. [2]
(c) How long does it take the population to double from when
observations
commenced? [2]
2Q
(a) Assuming that x = e lnx where x > 0, show that
lnx m = m lnx
[3]
(b) If 4 (x + 1) = 7 8 (x 2)
find x to 3 decimal places [6]
3Q
Using the substitution u = 7x in the equation
7x = 5 + 24 7x
find the value ofx correct to 4 decimal places [4]
4Q
(a) Factorise the expression 3x 3 4x 2 5x + 2 [4]
(b) Hence, using a suitable substitution, solve the equation
3e 3y 4e 2y 5e y + 2 = 0
giving your values ofy correct to 4 decimal places [3]
5Q
(a) Using the factor theorem, factorise the expression
2u 3 3u 2 8u + 12 [4]
(b) By using the substitution u = ex , solve the equation
2ex 8ex + 12e2x = 3
giving your answers correct to 3 decimal places. [3]
6Q
(a) Solve the equation
2 lnx = ln 4 + ln(2x + 5) [2]
(b) Solve the equation
e 2x 3ex = 54,
giving your answers correct to 3 decimal places. [3]
7/29/2019 Exponentials and Logarithms
2/2
7Q
After police arrive at the scene of a murder the pathologist has
determined that
the body is cooling according to the law
TT0 =A0e0.0503t
where Tis the temperature of the body, T0 is the temperature of
the surroundings
(the ambient temperature), t is the time in hours since the time
of death and A isa constant to be found.
If the temperature of the body at 9.30 p.m. is 35.3C, estimate
the time the
murder occurred, given that the surrounding temperature is 15C
and the normal
body temperature is 37C. [4]
8Q
Using the substitution u = 4x , solve the equation
16x = 21 4x + 1
giving your answer correct to three decimal places [4]
9Q
Given that
3 ln (x + 1) = ln (5 + 4xx 2) ln 2
show that
2x 3 + 7x 2 + 2x 3 = 0
Solve this equation using the factor theorem [6]
10Q
When Emily did a parachute jump for charity the parachute opened
shortly after
she left the aircraft. Her velocity at time t seconds from when
the parachute
opened is v ms1 where
v = 9 + 29e0.063t
(a) Sketch the graph of v against t. [2]
(b) What is Emilys speed at the instant the parachute opened?
[1]
(c) What is the lowest speed she can possibly attain if she fell
from a very
great height? [1]
(d) If she actually landed after 45 seconds what was her speed
on landing?
[2]
(e) How long did it take her to reach half the speed she had
when the
parachute opened? [3]
