Alan

Efficiency = 1 - Tcold/Thot  where Thot and Tcold are absolute temperatures. So:

0.34 = 1 - (273 + 35)/Thot   Rearrange to find Thot in Kelvin (then subtract 273 if you want the result in Celsius).

Efficiency = Work out/Energy_in     Substitute the given values and rearrange to find Energy_in

Almost right guest, but a+ b = -5 not +5

Using that your final formula becomes  -11 - 3*5 + 9 = -17

-(x⁹) + x⁹ = 0

and  

(-x)⁹ + x⁹ = 0

df/dx ≈ (f(9.01) - f(9))/(9.01 - 9)  = (3 - 3)/0.01  = 0

In general: cos(a+b) = cos(a)cos(b) - sin(a)sin(b)

So:  cos(θ + π) = cos(θ)cos(π) - sin(θ)sin(π)  =  -cos(θ) - 0    = -0.4   since cos(π) = -1  and  sin(π) = 0

As follows:

"Find the dervative of y with respect to the given independent variable y=log2 ( 8t^ln2 )"

Using the properties of logarithms:

y = log₂(8) + log₂(t^ln2)

y =  log₂(8) + ln(2)*log₂(t)

y = log₂(8) + ln(2)*ln(t)/ln(2)

y = log₂(8) + ln(t)

Now differentiate y wrt t:

dy/dt = 1/t

As follows:

Force (weight) is proportional to the mass of the planet and inversely proportional to the square of its radius, so

new weight = 800*2/2²  = 400N

How about  

1 and 400

2 and 400

4 and 400

5 and 400

...

I get 45 in total; assuming (1, 400) is different from (400, 1) etc.