# Recordings & Questions

## Practice Questions

Consider the following sequence:

\begin{align}7,11,15,19…\end{align}

(a) Show that this sequence is arithmetic.

(b) What is the next term in this sequence?

(c) What is the general term for this sequence?

(d) What is the $$100$$th term in this sequence?

The general term of a sequence is given by:

\begin{align}T_n=4n+1\end{align}

(e) What are the first three terms of this sequence?

(f) What is $$30$$th term of this sequence?

(g) Which term of the sequence is $$61$$?

The general term $$T_n$$ of a particular sequence is given by

\begin{align}T_n = 2n+1\end{align}

(h) Show that that the sum of the first $$n$$ terms of this sequence is $$n^2+2$$.

(a) There is a common difference between successive terms of $$4$$.

(b) $$23$$

(c) $$T_n=4n+3$$

(d) $$403$$

(e) $$5,9,13$$

(f) $$121$$

(g) $$15$$th

## Practice Questions

Consider the points $$(1,2)$$, $$(3,-4)$$ and $$(5,-6)$$.

(a) Find the equation of the circle whose circumference passes through these three points.

Consider the circle $$x^2+y^2=10$$.

(b) Show that $$y=-3x-10$$ is a tangent to this circle.

A circle has a centre $$(3,-1)$$ and the point $$(2,5)$$ on its circumference.

(c) What is the equation of that circle?

Circle $$A$$ has a centre $$(-2,4)$$ and the origin on its circumference.

Circle $$B$$ has a centre at the origin and the point $$(-2,4)$$ on its circumference.

(d) What are the equations of both circles?

(a) $$x^2+y^2-22x-4y+25=0$$

(c) $$(x-3)^2+(y+1)^2=38$$

(d) A: $$(x+2)^2+(y-4)^2=20$$ B: $$x^2+y^2=20$$

## Practice Questions

Consider the following function:

\begin{align}y(x)=3x^2+6x+4\end{align}

(a) Calculate $$\dfrac{dy}{dx}$$

(b) What is the turning point of $$y(x)$$?

Consider the following function:

\begin{align}y(x)=x^2-10x\end{align}

(c) What is the turning point of $$y(x)$$?

(d) By using the second derivative test, show that this turning point is a local minimum.

Consider the following function:

\begin{align}y(x)=x^3-3x^2-9x+6\end{align}

(e) What are the turning points of $$y(x)$$?

(f) By using the second derivative test, state whether each turning point is a local maximum or a local minimum.

(a) $$\dfrac{dy}{dx}=6x+6$$

(b) $$(-1,1)$$

(c) $$(5,-25)$$

(d) Minimum as the second derivative is positive.

(e) $$(-1,11)$$ and $$(3,-21)$$

(f) $$(-1,11)$$ is a maximum as the second derivative is negative at that point. $$(3,-21)$$ is a minimum as the second derivative is positive at that point.