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

braintic

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

so I let t = tan ( x/2)

Domain 0< x/2 < 180 ( less than and equal to)

then use t formula

1 + t^2 / 1 - t^2 + 2t / 1 + t^2 = 1 - t^2/ 1 + t^2


1 + t^2 + 2t / 1 - t^4 = 1 - t^2 / 1 + t^2


Then I cross multiply and expand and simplify to get

t^6 - 2t^4 - 2t^3 - 3t^2 - 2t = 0

Not sure where to go from here.

Looks terribly wrong.
When you multiply through by (1-t²) (1+t²), you can't get a t^6 term.
 

rand_althor

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



If you solve this using your calculator, you get . However, since these are not standard values, you need to prove it. Look at this post to see a possible method.
 
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appleibeats

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

Find an approximation of Integral from 1 to 3 of dx/ x^2 + 1 using the trapezoidal rule with 3 subintervals

I am unsure about the numbers to use as it doesn't evenly split for 3 subintervals.
 

dan964

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

x values of 1, 5/3, 7/3, 3
and their function values.
(gaps of 2/3)
 

appleibeats

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

Use the substitution u = x - 2 to evaluate Integral from 2 to 3 of x (x - 2)^1/2 dx giving your answer correct to 2 decimal places

= approx 1.73

ii)

Suppose that the area of a particular function is given by the integral above, Use the simpson rule with 3 functional values and the trapezoidal rule with 5 functional values to estimate the area, and determine which gives the better aprrox.

So using the trapezoidal rule

Integral = approx 1.69358

Simpsons Rule:

Integral = approx 1.67851

Therefore Trapezoidal Rule give the better approx.

I was marked wrong for my trapezoidal rule and my final conclusion but right for the simpsons rule. Not sure where I have went wrong, I checked and I don't seem to see any mistake made.
 

appleibeats

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

Find the general solution to the equation cosx = cos3x
 

appleibeats

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

The lines y = sinx and y = 3/5pi intersect at the origin and at A .

i) Show that x = 5pi/ 6 is a root to the equation f(x) = sinx - 3x/5pi
 

braintic

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

Find the general solution to the equation cosx = cos3x
We don't need triple angle formulae for this:

3x = x + 2kπ
2x = 2kπ
x = kπ

OR

3x = (2π - x) + 2kπ
4x = 2(k+1)π
x = (k+1)π/2
which is identical to:
x = kπ/2

This answer includes all answers from the first solution.
So the simplest answer is:
x = kπ/2
 
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appleibeats

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

The lines y = sinx and y = 3/5pi intersect at the origin and at A .

i) Show that x = 5pi/ 6 is a root to the equation f(x) = sinx - 3x/5pi
 

braintic

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

The lines y = sinx and y = 3/5pi intersect at the origin and at A .

i) Show that x = 5pi/ 6 is a root to the equation f(x) = sinx - 3x/5pi
just substitute .....
 

appleibeats

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

Use a graphical approach to determine the number of positive solutions of sin x = x / 200.
 

kawaiipotato

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

sinx = x/200
200sinx = x

Sketch y = 200sinx
Sketch y = x
Both on the same cartesian plane.
The amount of solutions will be evident
 

appleibeats

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

sinx = x/200
200sinx = x

Sketch y = 200sinx
Sketch y = x
Both on the same cartesian plane.
The amount of solutions will be evident
So doing this I get the answer 3. But the answer in the textbook is 63. Still unsure how to do the question.
 

Zlatman

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

So doing this I get the answer 3. But the answer in the textbook is 63. Still unsure how to do the question.
Are you using radians for sin x?

EDIT: Once y > 200, x and sin x will have no more solutions. Therefore, we're looking at the domain of x < 200 (since y = x, and y must be less than 200). In this domain, sin x completes 31.8 cycles (200 divided by 2pi), and y = x passes through sin x twice in the first half of each cycle. This gives us 64 solutions. But x = 0 is included in this, and that is not positive; hence we have 63 solutions.

I hope that makes some sense. :)
 
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appleibeats

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

a) carefully sketch the curve y = sin^2 x for 0 < or equal to x < or equal to 2pi

b) Explain why y = sin^2x has range 0 </= y </= 1

c) Write down the period and amplitude of y = sin^2 x .
 

InteGrand

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appleibeats

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

I have sketched y = sinx and y = cosx, for 0 </= x </= 2pi

But am unsure how to sketch y = sinx + cosx for 0 </= x </= 2pi.


Then how do you calculate the period and amplitude of y = sinx + cosx
 

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