# Thread: Cambridge Prelim MX1 Textbook Marathon/Q&A

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

Thanks Ambility for answer that question.
Also wondering if you could help me with Q 5 from 6G:

The question is:

1/ (k^1/2 + (k + 1 ) ^1/2 (k^1/4 +( k +1 )^ 1/4 = ( k + 1) ^ 1/4 - k^1/4

I know its hard to read, best to look at the textbook. Also wondering how do you write your maths on BOS, so I don't have to write it like I do above.

Thanks.  Reply With Quote

2. ## Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread Originally Posted by appleibeats Thanks Ambility for answer that question.
Also wondering if you could help me with Q 5 from 6G:

The question is:

1/ (k^1/2 + (k + 1 ) ^1/2 (k^1/4 +( k +1 )^ 1/4 = ( k + 1) ^ 1/4 - k^1/4

I know its hard to read, best to look at the textbook. Also wondering how do you write your maths on BOS, so I don't have to write it like I do above.

Thanks.
Honestly, I've spent 40 minutes on this problem and I can't get at it. Either it's something simple that I've been overlooking, or it requires some method I haven't yet learnt.

To answer you question on writing maths, I use a thing called LaTeX. It allows you to type in a code between ["tex"] tags (without the quotes), and it will format it in mathematics. If you want to learn how to use it, I recommend looking at this guide and looking at how some people set their LaTeX out. If you click the "Reply with Quote" button on the bottom left of people's comments, you should be able to see their LaTeX.

Anyway, if anyone wants to work out his question, this is it:  Reply With Quote

3. ## Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread  Reply With Quote

4. ## Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread Originally Posted by SpiralFlex How silly of me to overlook this.  Reply With Quote

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

^Not silly at all, we all overlook things.  Reply With Quote

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

How would you rationalise the denominator of:

1/(cuberoot(2)-1)  Reply With Quote

7. ## Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread Originally Posted by yog How would you rationalise the denominator of:

1/(cuberoot(2)-1)  Reply With Quote

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

Question 18 a and c from 6A:

Simplify:

a)

the denominator is meant to be 2^n+1 +2, can't get it to work for some reason.

c)  Reply With Quote

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

Also Question 20F from 6A:

if a = 2^ 1/2 + 2^-1/2 and b = 2^1/2 - 2 ^-1/2

find:

a^3 + b^3

answer is 7 root 2

Also 21a)

If x = 2^1/3 + 4^ 1/3, show that x^3 = 6(1 + x )  Reply With Quote

10. ## Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread Originally Posted by appleibeats Question 18 a and c from 6A:

Simplify:

a)

the denominator is meant to be 2^n+1 +2, can't get it to work for some reason.

c)

To write , the LaTeX code needs to be written like this: 2^{n+1} (put curly braces around the thing you want in the exponent).

Part a)

Based on what you say the answer is, I think you made a typo in the numerator, it should be instead (OR, the denominator should be . I'll assume the numerator is .

Then the numerator can be written as:

(as )

(using the index law )

(factorising).

The denominator can be written as:

(as , using the index law )

(factorising).

So the original fraction is (cancelling ).

Part c)

The idea is to rewrite the numerator in terms of and using index laws, and then hopefully we will be able to cancel something with the denominator (notice that 12 and 18 can both be written in terms of powers of 2 and 3).

The numerator can be written as:

(as and )

(using the index law )

(using the index law )

(factorising by taking out a common factor of )

(as ).

So the original fraction becomes .  Reply With Quote

11. ## Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread Originally Posted by appleibeats Also Question 20F from 6A:

if a = 2^ 1/2 + 2^-1/2 and b = 2^1/2 - 2 ^-1/2

find:

a^3 + b^3

answer is 7 root 2

Also 21a)

If x = 2^1/3 + 4^ 1/3, show that x^3 = 6(1 + x )
First question:

Find the values of a, b, a squared, b squared and ab from previous parts of the question. Take the positive square roots of a squared and b squared to find a and b (because the question gives us positive values for a and b).

Factor to make calculating easier (sum of two cubes):

Substitute the values:

which becomes

Multiply:

Second question:  Reply With Quote

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

Question 18b and c from 6B:

Simplify SD, S + D, S - D, S^2 - D^2

b) Rewrite the formulae for S and D as quadratic equations in . Hence express x in terms of S, and in terms of D, in the case where x > 1

c)  Reply With Quote

13. ## Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread  Reply With Quote

14. ## Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread Originally Posted by Feynman I guess no one has any more questions...  Reply With Quote

15. ## Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread Originally Posted by appleibeats Hi, not sure how to go about question 16 from chapter 4J. The question is

the side of a triangle are n2 + n + 1 , 2n + 1 and n2 - 1, where n>1. Find the largest angle of the triangle.

( n2 being --> n squared )

Thanks.
use cosine rule  Reply With Quote

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

Q 10 from 7H:

Sand being poured from a conveyor belt forms a cone with heigh h and semivertical angle 60 degrees. Show that the volume of the pile is V = pih^3 ( can do this), and differentiate with respect to t. (can do this)

a) Suppose that the sand is being poured at a constant rate of 0.3 m^3 /min , and let A be the area of the base. Find the rate at which the height is increasing:

i) when the height is 4 metres, ( can do this), ii) when the radius is 4 metres ( CANT DO THIS)

b) (CANT DO THIS) Show that dA/dt = 6pih dh/dt, and find the rate of increase o the base area at these times.

c) (CANT DO THIS) At what rate mist the sand be poured if it is required that the height increase at 8 cm/min, when the height is 4m?

ALSO

Q11)

An upturned cone of semivertical angle 45 degrees is being filled with water at a constnat rate of 20cm^3 /s. Find the rate at which the height, the area of the water surface, and the area of the cone wetted by the water, are increasing when the height is 50 cm.

Thanks.  Reply With Quote

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

Hi, worked out the previous question.

Have another one though.

Question 14 from 7I.

Find zeroes and discontinuities of:

a) y = cosx + sinx / cosx - sinx

b) y = cosx - sinx / cosx + sinx

Thanks  Reply With Quote

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

hey, does anyone know how to do question 17 and 20 from 2G?  Reply With Quote

19. ## Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread Originally Posted by appleibeats Hi, worked out the previous question.

Have another one though.

Question 14 from 7I.

Find zeroes and discontinuities of:

a) y = cosx + sinx / cosx - sinx

b) y = cosx - sinx / cosx + sinx

Thanks
Seeing as the problem doesn't give a domain, and the functions go on forever, we need to represent the answer as a general solution because there is going to be infinitely many possible solutions. These fractions will have zeros when the numerator is equal to zero ("zero divided by anything is zero") and will have discontinuities when the denominator is equal to zero ("dividing by zero is undefined"). With that in mind, let's work on 14a:

You can use the answers from 14a. to work out 14b. 14b's denominator is the same as 14a's numerator, and 14b's numerator is the same as 14a's denominator.  Reply With Quote

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

Thanks for that help.

Also I don't quite understand how to get the answers to the questions in 4 in 7J.

I know how to sketch them and their continuity, but am unsure about how to find where it is not differentiable.

eg, question 4a)

y = |x + 2 |

Also, question 7d from chapter 8A.

write down the general form of a monic quadratic for which one of the zeroes is x = 1. (I know how to do this.) Then find the equation of such a quadratic in which :

d) the curve passes through (3,9).

Thank you.  Reply With Quote

21. ## Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread Originally Posted by appleibeats Thanks for that help.

Also I don't quite understand how to get the answers to the questions in 4 in 7J.

I know how to sketch them and their continuity, but am unsure about how to find where it is not differentiable.

eg, question 4a)

y = |x + 2 |

Also, question 7d from chapter 8A.

write down the general form of a monic quadratic for which one of the zeroes is x = 1. (I know how to do this.) Then find the equation of such a quadratic in which :

d) the curve passes through (3,9).

Thank you.
I have a method to solve 8A 7d, but it's hardly ideal. I haven't studied that chapter so I recommend asking a teacher.

As for 7J 4a, the graph is not differentiable at a point where there is a sharp turn. This is because there are multiple tangent lines which can be drawn to the graph, multiple slopes of those tangent lines, multiples derivatives, no defined derivative. If you look at the graph of , there is a point which takes a sudden turn at x=-2. So it's not differentiable at x=-2.  Reply With Quote

22. ## Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread Originally Posted by appleibeats Thanks for that help.

Also I don't quite understand how to get the answers to the questions in 4 in 7J.

I know how to sketch them and their continuity, but am unsure about how to find where it is not differentiable.

eg, question 4a)

y = |x + 2 |

Also, question 7d from chapter 8A.

write down the general form of a monic quadratic for which one of the zeroes is x = 1. (I know how to do this.) Then find the equation of such a quadratic in which :

d) the curve passes through (3,9).

Thank you.
Q 7d) from Chapter 8A:  Reply With Quote

23. ## Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread Originally Posted by InteGrand Q 7d) from Chapter 8A:

There does exist other suitable quadratics though, right? I came up with .  Reply With Quote

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

The answer in the text book for that question is y = 1/2 (2x + 3)(x - 1)

Still unsure about how to get this answer.

Also does anyone know how to do Question 23 from 8 F.

A piece of string of length l is bent to form the sector of a circle of radius r. Show that the area of the sector is maximised when r = 1/4l .

Also Question 27 from the same chapter.

A rectangle is inscribed in an isosceles triangle with one of the sides of the rectangle on the base of the triangle. Prove that the rectangle of greatest area occupies half the area of the triangle.

Thanks.  Reply With Quote

25. ## Re: Year 11 Mathematics 3 Unit Cambridge Question & Answer Thread Originally Posted by Ambility There does exist other suitable quadratics though, right? I came up with .
Yep there are infinitely many possible quadratics we could use.

The general form was . In order for the point (3, 9) to lie on the curve, we just need to choose a and A (with ) satisfying , and there are clearly an infinite number of pairs satisfying this condition.  Reply With Quote