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Akshay (Student) September 26 2016
This is the sum of a geometric series with an infinite number of terms.
First, find the common ratio:
(-2/9)/(2/3) = -1/3 , (2/27)/(-2/9) = -1/3
Therefore, the common ratio is -1/3
-1<(-1/3)<1 therefore the series will converge to a finite number.
The general formula for a sum of an infinite geometric series is a/(1-r) where a is the first number in the sequence and r is the common ratio.
So, substituting in the numbers, the sum of the series = (2/3)/(1-[-1/3]) = 1/2see more
This is a question testing the knowledge of the equations of constant acceleration (Suvat equations)
First convert the question into a standard form. For example by writing out the variables as follows
s (displacement) = unknown
u (initial velocity) = 10m/s
v (final velocity) = 0m/s
a (acceleration) = 2ms^-2
t (time) = unknown
You know v, u and a and you want to calculate t, therefore the equation you need to use is v = u + a*t
Re-arranging t = (v - u) / a
Finally substituting in v, u and a, t = (10 - 0) / 2 = 5s
So the time taken for the car to stop is 5 seconds.see more
The easiest way of approaching this question is to use De Moivre's formula:
e^(inx) = cos(nx) + isin(nx)
from which it is simple to show that cos(nx) = (e^(inx) + e^(-inx)) / 2 and sin(nx) = (e^(inx))- e^(-inx)) /2i
therefore, cos(4x)sin(x) = (e^(4ix) + e^(-4ix)) * ((e^(ix)) - (e^(-ix)) / 4i
= [e^(5ix) - e^(-5ix) - e^(3ix) + e^(-3ix)] / 4i
= sin(5x)/2 - sin(3x)/2
Finally, integrating, this gives cos(3x)/6 - cos(5x)/10 + integration constant
This can also be done by using various trigonometric identities, however this method is simpler and can continue to be applied to more complex questions.see more