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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.