Alan

Think about

\(\mathbf{A^7}\mathbf{x}= \mathbf{A^6}(\mathbf{A}\mathbf{x})\)

As follows

You can crunch the numbers.

Divide 10389 by 12.  n is the remainder.

The first graph has the x values within the range 0 to 2pi.

The following graphs should help.  You will notice that x = 0, pi/4 and 5pi/4 are not roots!

The line 2y=3x intersects the ellipse (x²/8)+(y²/18)=1 at the points Q and R. What is the length of segment QR?

This should help

Be careful at x =1, as it is a strict inequality!

Mean of the set of four numbers:

  (x1 + x2 + x3 + x4)/4 = 18   

Multiply both sides by 4, so

x1 + x2 + x3 + x4 = 4*18   ...(1)

Mean of the five numbers

(x1 + x2 + x3 + x4 + x5)/5 = 16

Multiply both sides by 5, so

x1 + x2 + x3 + x4 + x5 = 5*16   ...(2)

Replace x1 + x2 + x3 + x4 in (2) by using (1)

4*18 + x5 = 5*16

Subtract 4*18 from both sides

x5 = 5*16 - 4*18 

x5 = 80 - 72

x5 = 8

1.  Subtract (p.c)²  from both sides. to get  E² - (p.c)² = (m.c²)² 

2.  Take the square root of both sides to get \(\sqrt{E^2-(p.c)^2}=m.c^2\)

3. Divide both sides by c² to get \(\frac{\sqrt{E^2-(p.c)^2)}}{c^2}=m\)

4. Substitute for E and p using the expressions given.

Like so