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1) I am fairly certain that you are referring to the midsegment theorem. Here is a diagram for you to reference as I prove:

We know that by the midsegment theorem that the segment bisects both of its intersecting sides and is parallel to the third side. \(\angle CDE\cong\angle CAB\) and \(\angle CED\cong\angle CBA\) by the Corresponding Angles Postulate. By the Angle-Angle Similarity Theorem, \(\triangle CDE \sim \triangle CAB\). We know that \(\frac{CD}{CA}\) equals some ratio. We know that \(\overline{CA}\) is broken up into two smaller segments, \(\overline{CD}\) and \(\overline{DA}\). The ratio is now \(\frac{CD}{CD+DA}\). By the given info, \(CD=DA\), so, by the substitution property of equality, the ratio can be written as\(\frac{CD}{CD+CD}=\frac{CD}{2CD}=\frac{1}{2}\). Since the ratio of corresponding sides of a similar triangle are equal, the ratio of \(\frac{DE}{AB}=\frac{1}{2}\). Rewriting this ratio turns into \(2DE=AB\Rightarrow DE=\frac{1}{2}AB\).

2) Here is a diagram again!
 

In this diagram here, \(AB=AC\), and \(D\) is the midpoint of \(\overline{AB}\) and \(E\) is the midpoint of \(\overline{AC}\), which makes \(\overline{BE}\) and \(\overline{CD}\) medians of this isosceles triangle.

By the given info, \(AB=AC\) and \(AD=AE\) because they are midpoints of equal-length segments. \(\angle A\cong\angle A\) by the reflexive property of congruence. Therefore, by Side-Angle-Side Congruency Theorem, \(\triangle ABE\cong\triangle ACD\). Utilizing the fact that corresponding parts of congruent triangles are congruent, \(CD=BE\). We are done!

Thank you, once again! I can't believe you spent an hour doing this for me. 

Of course! I will gladly delve deeper into this subject.

The explanation I provided is probably unsatisfactory and shoddy anyway. I believe that these rules are best demonstrated by example.

1. Divisibility by 7

Is \(205226\) divisible by 7? Well, let's use the process!

How about \(22604\)? Well, let's check it!

2. Divisibility by 11

Let's check if \(43923\) is divisible.

How about \(123567\)?

3. Divisibility by 13

Is \(19704\) divisible? Let's find out!

Is \(9321 \) able to be divised?
 

f(0) = c so c = 4,

f(1) = a + b + 4, implies that a + b = 11, so 2a + 2b = 22 --- (1)

f(2) = 4a + 2b + 4, implies that 4a + 2b = 32 --- (2)

(2) - (1) gives 2a = 10, a = 5.

5 + b = 11, b = 6.

Therefore abc = 4*5*6 = 120.

I understand the divisibility rules except 7, 11, and 13. What do you mean by the "rest?" Can you kindly explain further? 

For powers of x, we have:

\(\int{x^n}dx=\frac{x^{n+1}}{n+1}\text{ as long as }n\ne-1\)   

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Mary and John have both completed a trip up an escalator.

The speed and number of steps on the escalator is set.

So the time the escalator takes is also set - assuming no steps

Let there be x steps in 1 trip.

John took 15 steps on the way up and took 12 seconds to reach the top.

\(\frac{x-15}{12}\)   steps per sec

Mary took only 7 steps and took 18 seconds to reach the top. How many steps does the escalator have in total?

\(\frac{x-7}{18}\)      steps per sec

These are equal

\(\frac{x-15}{12}=\frac{x-7}{18}\\ 18(x-5)=12(x-7)\\ 3(x-5)=2(x-7)\\ 3x-45=2x-14\\ x=31 steps\)

The escalator has 31 steps.

Nice work guest :)

\(\angle ODC=\angle OBC=90^\circ\qquad \) A tangent is at right angles to the radius that it intersects with

\(\angle BOD=2*\angle BFD \)               

             When subtended from the same arc (BC), the angle at the  centre = 2* the angle at the circumference

\(\therefore \angle BOD=2*76=152^\circ\)

\(152+90+90+\angle BCD=360 \qquad \text{Angle sum of quadrilateral}\\ 332+\angle BCD=360 \\ \angle BCD=28^\circ\)

Let the time of the slower bike =t
The time of the faster bike =t + 4
Distance=Time x rate(speed)
2(90t) =120[t + 4], solve for t
t=8 hours
8 x 90 =720 km - the distance traveled by the slower bike.
120 x[8 + 4] =1,440 km - the distance traveled by the faster bike.

Integration Definitions:

1.

The action or process of integrating.

2. In mathamatics:

The finding of an integral or integrals.

If that did not answer your question then look at this page for a more mathamatical explanation:

http://mathsfirst.massey.ac.nz/Calculus/integration/basics/definition.html

Let the amount of money invested =M, then we have:

{M x [1.08]^2 - M} - {M x 0.08 x 2} = 768, solve for M

{ 1.1664M - M} - {0.16M} =768

0.0064M =768  Divide both sides by 0.0064

M = 768 / 0.0064

M =120,000 Rs - The principal amount invested. 

Compound interest =19,968

Simple interest =19,200

19,968 - 19,200 =768 Rs.

Correct!

So the answer is 5 - x^-3/2 - 3x^-2

The answer sheet has the results as indicated by the red boxes below (you still need to add them together):

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I understood how u did but i cant confirm ur answer with the answer of the answer sheet

I’m not surprised - it is the muddiest explanation I’ve ever come across!

Do you understand the derivatives in my previous reply, or do you need more explanation?

I am confused at what the answer sheet says

http://egyptigstudentroom.com/pastpapers/edexcel/alevel/gce/Math/Jan%202008/ms//8371_9374_GCE_Mathematics_msc_20080306_UA019869.pdf go to page number 9

This method utilizes Euler's theorem for computing modular inverses. This is especially useful when the moduli are prime numbers because calculating totients for prime numbers is a trivial process. The rest of the algorithm is an extension of cross multiplication similar to the Chinese remainder theorem, but much less cumbersome. 

GA

"Differentiate 5x - 7 + 2x^-1/2 + 3x^-1"

\(\frac{d}{dx}5x=5\\\frac{d}{dx}7=0\\\frac{d}{dx}2x^{-1/2}\rightarrow (-1/2)2x^{-1/2-1}\rightarrow -x^{-3/2}\\ \frac{d}{dx}3x^{-1}\rightarrow (-1)3x^{-1-1}\rightarrow -3x^{-2}\\\)

I'll leave you to add these together!

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Thank you.  I’ve studied and learned the techniques of two LaTex masters: Heureka and Lancelot Link.