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Cambridge Prelim MX1 Textbook Marathon/Q&A (1 Viewer)

appleibeats

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

Water is leaking out through a hole in the bottom of a bucket at a rate dv/dt =-8(25 - t), where v cm^3 is the volume of liquid in the bucket at time T seconds
Initially there was 2 1/2 litres of water in the bucket. Find v in terms of t, and hence find the time taken for the bucket to be emptied.
 

InteGrand

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

Water is leaking out through a hole in the bottom of a bucket at a rate dv/dt =-8(25 - t), where v cm^3 is the volume of liquid in the bucket at time T seconds
Initially there was 2 1/2 litres of water in the bucket. Find v in terms of t, and hence find the time taken for the bucket to be emptied.






















 

appleibeats

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

y = ax(x - a) is the general form of a quadratic whose graph passes through the origin. Hence find the equation of such a quadratic:

d) with no other x -intercepts, and having gradient 1 at x = -2

e) with another x intercept at x = 5, and having gradient 2 at the origin


for d) i get the answer y = 1/4 x^2

however the answer in the book has it as negative my answer. Not sure why.

For e) i know that to have a x intercept at 5

it becomes y = x(x -5)

but not sure how to work out the gradient bit.
 

rand_althor

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread



 
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appleibeats

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

If y = 1 - 2x is a tangent to x^2 = 4ay , find a and the point of contact

I though you first find the gradient of the parabola; y' = x / 2a

Then that equals the gradients of the line; so

x / 2a = - 2

x = -4a

Now not sure where to go from here.
 

dan964

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

Sub the x coordinate into the two equations
one you will get y=8a+1 and the other y=4a
Solve simultaenously to get a and hence y and x.
 

rand_althor

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread



 
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appleibeats

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

Consider the parabola x^2 = 12y. Line ax + by = 1 is tangent to this parabola.

Show that 3a^2 + b = 0

found the gradient for each function. Thus the x coordinate. Sub it into one of the functions and get the y coordinate. Then subbed it into the line and simplified. However still didn't get the right answer.

Is this the correct method of answer this question?
 

rand_althor

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

Your method is correct.

 

appleibeats

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

For the same question,

Hence, find the tangents to the parabola with y intercept -27

Do I sub in (0, -27) into the equation of the line: ax + by = 1

Then find b = - 1/27

Then sub this into 3a^2 + b = 0

So, 3a^2 = 1/27

a^2 = 1/81

a = + 1/9 and - 1/9

So, the equations of the tangents are:

1/9 x - 1/27 y = 1 ; 3x - y - 27 = 0

and

-1/9 x - 1/27 y = 1 ; 3x + y + 27 = 0

Is this correct? Not really sure if my method is right.
 
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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

How do you graph the regions of: (a) |y| > |x|
(b) 1/x > 1/y
 

appleibeats

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

AB is the chord contact of the parabola x^2 = 4ay from P (x0, y0). The line AB meets the directrix of the parabola at D. Prove that PD subtends a right angle at the focus.
 

appleibeats

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

Indefinite Integral of ( (ax)^1/2 + 1/(ax)^1/2 ) dx

Question is to find each of the indefinite integrals:

I get the answer 2(ax)^3/2 / 3 + 2(ax)^1/2 + C

The answer in the book gives instead of my 2(ax)^ 1/2 as 2(ax)^1/2 all over a

Not sure where the a comes from?? Have i done the integral wrong???
 

appleibeats

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

Indefinite Integral of ( (ax)^1/2 + 1/(ax)^1/2 ) dx

Question is to find each of the indefinite integrals:

I get the answer 2(ax)^3/2 / 3 + 2(ax)^1/2 + C

The answer in the book gives instead of my 2(ax)^ 1/2 as 2(ax)^1/2 all over a

Not sure where the a comes from?? Have i done the integral wrong???
 

InteGrand

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

Indefinite Integral of ( (ax)^1/2 + 1/(ax)^1/2 ) dx

Question is to find each of the indefinite integrals:

I get the answer 2(ax)^3/2 / 3 + 2(ax)^1/2 + C

The answer in the book gives instead of my 2(ax)^ 1/2 as 2(ax)^1/2 all over a

Not sure where the a comes from?? Have i done the integral wrong???










 

InteGrand

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Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread

The answer in the book says 25 seconds. I understand your working, but why does neither of your two answer equal 25 seconds??
I realised I made a silly mistake, I should have converted the initial volume V(0) to cubic cm, since the dV/dt is given in cubic cm per second.
 

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