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can someone help me solve numbers 25 and 27? Thanks you :)

without mistake:

25.
\(\begin{array}{rcll} \text{Isosceles triangle: } & \angle{OAD} &=& \angle{ODA} \\\\ \text{Exterior Angle Theorem: } & \angle{AOP} &=& \angle{OAD} + \angle{ODA} \\ & \angle{AOP} &=& \angle{OAD} + \angle{OAD} \\ & \angle{AOP} &=& 2\cdot \angle{OAD} \qquad | \qquad : 2\\ & \frac{\angle{AOP} } {2} &=& \angle{OAD} \\ & \frac{ 41^{\circ} } {2} &=& \angle{OAD} \\ & 20.5^{\circ} &=& \angle{OAD} \\ \end{array}\)

Well done heureka!  I knew there had to be a simple way - I just didn't know the secant-tangent rule.  Must remember that in future.

I did it as follows:

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can someone help me solve numbers 25 and 27? Thanks you :)

25.

\(\begin{array}{rcll} \text{Isosceles triangle: } & \angle{OAD} &=& \angle{ODA} \\\\ \text{Exterior Angle Theorem: } & \angle{OAP} &=& \angle{OAD} + \angle{ODA} \\ & \angle{OAP} &=& \angle{OAD} + \angle{OAD} \\ & \angle{OAP} &=& 2\cdot \angle{OAD} \qquad | \qquad : 2\\ & \frac{\angle{OAP} } {2} &=& \angle{OAD} \\ & \frac{ 41^{\circ} } {2} &=& \angle{OAD} \\ & 20.5^{\circ} &=& \angle{OAD} \\ \end{array} \)

I should have said that acceleration is change in velocity / change in time

If the train is travelling at a constant 90mph then the acceleration is 0.

The instantaneous velocity has nothing to do with the acceleration at that point.

The acceleration is a measure of the change in velocity.

\(\frac{(12a^4b^5)^3}{(12a^6b^8)^8}\)

\(=\frac{1728a^{12}b^{15}}{1728a^{48}b^{64}}\)

\(=\frac{1}{a^{36}b^{49}}\)

\(\frac{14}{20}\times100\)

\(=70\%\)

You need another equation in order to solve with matrices, unless you just want to express x in terms of y or vice versa.

33)

\((x-4)^2+(y-1)^2=(6\sqrt3)^2\\ (x-4)^2+(y-1)^2=108\)

The centre looks like  (-2,3)   and the radius looks like 4

so

The equation would be

\((x+2)^2+(y-3)^2=16\)

Mmm Interesting

27.   Mmm

Well it has to be more than 3 because the two tangents to a circle subtended from a point will be equal.

(I not sure how I should word that) 

I can't be 9 either.  For the same reason.

So it has to be between 3 and 9.

What about 8 ......

No, it'll have to be less than that.

OK I don't think it can be any of those but I would like someone to give a more mathematical precise......

25.

<POD is 180 degrees,  

<POA=41

so <AOD=139

triangle AOD is isoceles so 

<OAD= 0.5*(180-139)=0.5*41=   20 degrees and 30 minutes

This is not a diophantine equation because ther is no suggestion in the question that the answers have to be integers. :)

It is however a linear equation :)

By 4th term, I assume you mean that n=4

What is the fourth term in this multiplication pattern? multiplication pattern using the formula 2 • 6n – 1

2*6*4-1=  47

sin^2 (x) = sin(x)*sin(x)  and cos^2(x) =  cos(x) * cos(x)   .......but..........

sin^-1(x)    and  cos^-1(x)     indicate the "sine inverse" and "cosine inverse".........given a ratio, these return an angle,   either in radians or in degrees

I assume that 6 is the radius.

Let the centre be O

The longer interval from O to the line AB is 6+5=11cm long

6,8,11 is not a pythagorean triad so angle OAB is not a right angled triangle so

AB is not a tangent to the circle.

The tangent to a circle is at right angles to the radius that it touches :)

We have  a triangle whose 3 three sides are 6 , 8 and 11...........to be tangent to the circle, AB would have to meet the radius of the circle at a right angle at the point of tangency.

Thus, we would have have to have the relationship that

6^2  + 8^2  = 11^2     .......but

36 + 64   ≠  121

So.....AB is not tangent to the circle

what is negative 8 divided by negative 3

\(\frac{-8}{-3}=\frac{8}{3}=2\frac{2}{3}\)

It already is fully factorised :)

ax + by = 1        This is a linear Diophantine equation. x , y are the unknowns, a, b are given constants.

https://en.wikipedia.org/wiki/Diophantine_equation

Look under "Examples"

A Diophantine equation is an equation in which only integer solutions are allowed.

http://mathworld.wolfram.com/DiophantineEquation.html

It is not a diophantine equation.  Or at least there is no reason to think that it is.

It is just the equation of a line, passing through(0,19/4) and (-19/5,0),  and yes there are infinite solutions.

To find 75 degrees in radians:

If you let  R  represent radians and  D  represent degrees, you can use this formula:

R / D  =  pi / 180

R / 75  =  pi / 180

R  =  (pi / 180) x 75

R  =  5 pi / 12