Further Mathematics Questions and Answers

dy/dx = 2x - 6

y = ∫ 2x - 6

y = $\frac{2x^2}{2} - 6 + c$
y = x$^2$ - 6x + c
passes through (1,2)
2 = 1$^2$ - 6(1) + c
2 = 1 - 6 + c
c = 7
y = x$^2$ - 6x + c

y = x$^2$ -  6x + 7

2x$^2$ + 2y$^2$ - x - 3y - 41

standard equation of circle
(x-a)$^2$ + (x-b)$^2$ = r$^2$
General form of equation of a circle.
x$^2$ + y$^2$ + 2gx + 2fy + c = 0
a = -g, b = -f., r2 = g2 + f2 - c
the centre of the circle is (a,b)
comparing the equation with the general form of equation of circle.
2x$^2$ + 2y$^2$ - x - 3y - 41

= x$^2$ + y$^2$ + 2gx + 2fy + c
2x$^2$ + 2y$^2$ - x - 3y - 41 = 0
divide through by 2

g = $\frac{-1}{4}$ ; 2g = $\frac{-1}{2}$

f = $\frac{-3}{4}$ ; 2f = $\frac{-3}{2}$

a = -g  → - $\frac{-1}{4}$ ; = $\frac{1}{4}$

b = -f → - (\frac{-3}{4}\) = (\frac{3}{4}\)

therefore the centre is ($\frac{1}{4}$, $\frac{3}{4}$)

No explanation has been provided for this answer.

2x$^2$ + 7x - 15 ≥ 0

2x$^2$ -3x + 10x - 15 ≥ 0
x(2x - 3) + 5(2x - 3) ≥ 0
(x+5)(2x-3) ≥ 0
the points on x-axis where the graph ≥ 0

x ≤ -5 or x ≥ $\frac{3}{2}$

4sin$^2$θ + 1 = 2

4sin$^2$θ  = 2 - 1

4sin$^2$θ = 1

$\sqrt sin^2θ$ = $\sqrt \frac{1}{4}$

sinθ = $\frac{1}{2}$

θ = $sin^{-1} \frac{1}{2}$

θ = 30º 0r 150º