20230807-AM1

Find the mean of for the data $$14,\ 19,\ 12,\ 24,\ x,\ y,\ 14,\ 11,\ 19,\ 27$$
where the mode is $$14$$ and the median is $$15$$ .

 If $$f\left(x\right)=x^3-2x^2-x+5$$ ,
find the remainder when the polynomial $$g\left(x\right)=f\left(f\left(x\right)\right)$$  is divided by $$\left(x-2\right)$$ .

Given that $$a\log_{520}2+b\log_{520}5+c\log_{520}13=3$$ ,
find the value of $$a+b+c$$ .

If $$\left(1+x\right)\left(1+x^2\right)\left(1+x^4\right)\cdots\left(1+x^{1024}\right)=1+x+x^2+x^3+\cdots+x^n$$ ,
find the value of $$n$$ .

Given that the coefficient of $$x^2$$ in the expansion of $$\left(\frac{2}{x^2}+3x\right)^n$$  is $$81648$$ ,
find the value of $$n$$ .

How many solutions to the equation $$\cos2023x=-0.5$$ are there in the interval $$\left(0\degree,180\degree\right)$$ ?

Solve $$\sin5x=\sin x+\sin2x$$ , $$0<x<\pi$$ .

Find the acute angle between the lines $$2x-6y+13=0$$  and $$3y-2x+7=0$$ .

The mid-point of the three sides of $$\Delta ABC$$ are $$\left(2,0\right)$$ , $$\left(2,3\right)$$ and $$\left(8,7\right)$$ .
Find the area of $$\Delta ABC$$ .

Find the equation of the circle passing through the points $$\left(2,-7\right)$$ , $$\left(2,1\right)$$  and $$\left(-2,-3\right)$$ .