The method for all three questions is the same: For (1):
Rearranging 2x - y + 1 = 0:
y = 2x + 1
Gradient = 2
diff. y = x^2:
dy/dx = 2x
therefore:
2x = 2
x =1
ie at x=1 the tangent is parralel to 2x- y +1 = 0
let the parabola be y =ax^2 + bx + c
Sub (0,-2):
-2 = C
Sub (1,0):
0 = a + b -2
2 = a + b ____________(1)
Sub (3,-8):
-8 = 9a + 3b -2
-6 = 9a + 3b
-2 = 3a + b____________(2)
(2)-(1):
-4 = 2a
-2 = a
b = 4
therefore parabola is y = -2x^2 +4x -2
HOBr is Hypobromous acid. I agree with your tutors equation.
Also the first equation does have water:
C6H10 + Br2 <b>(aq)</b> --> C6 H10 Br2
Ill take a stab in saying that the top layer is 1,2-dibromocyclohexane since it would have a density lower than water.
ie Prove: a + ar + ar^2 .......+ar^(n-1) = a(1-r^n)/(1-r)
Test n =1:
LHS = a
RHS = a(1-r)/(1-r) = a
Assume n=k:
a + ar + ar^2 .......+ar^(k-1) = a(1-r^k)/(1-r)
Prove n=k+1:
a + ar + ar^2 .......+ar^(k-1) + ar^k
= a(1-r^k)/(1-r) +ar^k
=a(1-r^k)/(1-r) +ar^n(1-r)/(1-r)
=a(1-r^k + r^k -...
Chem:
Teacher: "And the electron comes over here and goes. oooooh thats a bit alright!"
---------------------------
Me: Can we make rainbow fizz sir?
Teacher: No
Me: Can we make rainbow fizz sir?
Teacher: No
Me: Can we make rainbow fizz sir?
Teacher: Oh alright but ony...
alternatively you could consider the:
area = are of rectangle - the area between the parabola and the x-axis
ie: Area = 4a*a - Int(between -2a and 2a)[(x^2)/4a]
= 4a^2 - (4a^2)/3
= (8a^2)/3
first you need the subsitution
u=x - π/2
du = dx
and also to recognise that:
cosx = cos(u + π/2)
= cos(u)cos(π/2) - sin(u)sin(π/2)
= - sin(u)
then get you to first part, then since its odd and the intergral is between a and -a (in this case pi/2) its equal zero.