Question 1

Consider the curve $y=\frac{1}{(1-x)}$,  $x∈R$, $x≠1$. Find $\frac{dy}{dx}$

Accepted Answer

$\frac{1}{(1-x)^2}$

Answer

$-\frac{1}{x^2}$

Answer

$-\frac{1}{(1-x)^2}$

Answer

$\frac{1}{x^2}$

Question 2

Consider the curve $y=\frac{1}{(1-x)}$,  $x∈R$, $x≠1$. Determine the equation of the normal to the curve at the point $x=3$ in the form $ax+by+c=0$ where $a$, $b$, $c∈Z$.

Accepted Answer

$8x+2y-23=0$

Answer

$x+4y-5=0$

Answer

$8x-2y-23=0$

Answer

$x-4y-5=0$

Question 3

Consider the function defined by $f(x)=x^3-3x^2+4$.
Determine the values of $x$ for which $f(x)$ is a decreasing function.

Accepted Answer

$0<x<2$

Answer

$x>2$

Answer

$x<0$ and $x>2$

Answer

$x<0$

Question 4

Consider the function defined by $f(x)=x^3-3x^2+4$.
There is a point of inflection, $P$, on the curve $y=f(x)$.
Find the coordinates of $P$.

Accepted Answer

$(1, 2)$

Answer

$( 2,0)$

Answer

$(0, 4)$

Answer

$(1.5, 0.625)$

Question 5

The curve $C$ has equation $y=\frac{1}{8}(9+8x^2-x^4)$ .
Find the x-coordinates of the points on $C$ at which $\frac{dy}{dx}=0$ .

Accepted Answer

$0$, $-2$, and $2$

Answer

$0$, and $2$

Answer

$-2$, and $2$

Answer

$-3$, $-2$, and $0$

Question 6

The curve $C$ has equation $y=\frac{1}{8}(9+8x^2-x^4)$ .
The tangent to $C$ at the point $P(1, 2)$ cuts the x-axis at the point $T$. Determine the coordinates of $T$.

Accepted Answer

$(-\frac{1}{3},0)$

Answer

$(3,0)$

Answer

$(-\frac{1}{2},0)$

Answer

$(-1,0)$

Question 7

The curve $C$ has equation $y=\frac{1}{8}(9+8x^2-x^4)$ .
The tangent to $C$ at the point $P(1, 2)$ cuts the x-axis at the point $T$. The normal to $C$ at the point $P$ cuts the y-axis at the point $N$. Find the area of triangle $PTN$.

Accepted Answer

$\frac{13}{9}$

Answer

$\frac{13}{3}$

Answer

$\frac{13}{18}$

Answer

$\frac{26}{9}$

Question 8

The line $y=m(x-m)$ is a tangent to the curve $(1-x)y=1$.   
 Determine the value of $m$.

Accepted Answer

$2.31$

Answer

$-2.31$

Answer

$1.52$

Answer

$-1.52$

Question 9

The line $y=m(x-m)$ is a tangent to the curve $(1-x)y=1$.   
Determine the coordinates of the point where the tangent meets the curve.

Accepted Answer

$(1.66, –1.52)$

Answer

$(2.31, –0.763)$

Answer

$(2.5, –0.667)$

Answer

$(1.56, –1.79)$

Question 10

A family of cubic functions is defined as $f_k (x)=k^2 x^3-kx^2+x$, $k∈Z^+$ .
Express in terms of $k$ the coordinates of the points of inflection $P_k$ on the graphs of $f_k$ .

Accepted Answer

$(\frac{1}{3k},\frac{7}{27k})$

Answer

$(\frac{1}{k},\frac{1}{k})$

Answer

$(k,k^5-k^3)$

Answer

$(\frac{1}{2k},\frac{5}{24k})$

Question 11

To differentiate from first principles the function $f(x)=3x^3-x$ you must use the following equation:

Accepted Answer

$\frac{(3(x+h)^3-(x+h))-(3x^3-x)}{h}$

Answer

$\frac{(3(x+h)^3-(x+h))-(3x^3-x+h)}{h}$

Answer

$\frac{(3(x+h)^3(x+h))-(3x^3-x)}{h}$

Answer

$\frac{(3(x+h)^3+(x+h))-(3x^3-x)}{h}$

Question 12

Let $f(x)=\sqrt{\frac{x}{1-x}}$, 0<x<1.  $y=f(x)$ has one point of inflexion.  Find its coordinates.

Accepted Answer

$(\frac{1}{4},\frac{1}{\sqrt3})$

Answer

$(\frac{1}{3},\frac{1}{\sqrt2})$

Answer

$(\frac{1}{2},\frac{1}{\sqrt3})$

Answer

$(\frac{1}{4},\frac{1}{\sqrt2})$

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