Question 1

Consider the curve $2x^2=sin^3 y$ for$0≤y≤π$.  Using implicit differentiation, find an expression for $\frac{dy}{dx}$.

Accepted Answer

$\frac{dy}{dx}=\frac{4x}{3 sin^2 y cos y}$

Answer

$\frac{dy}{dx}=\frac{4x}{sin^2 y cos y}$

Answer

$\frac{dy}{dx}=\frac{4x}{ sin^3 y cos y}$

Answer

$\frac{dy}{dx}=\frac{4x}{3 sin^2 y cos^3 y}$

Question 2

Consider the curve $2x^2=sin^3 y$ for$0≤y≤π$.  Find the equation of the tangent to the curve at the point $(\frac{1}{4},  \frac{5π}{6})$

Accepted Answer

$y=-1.54x+3. $

Answer

$y=1.54x+3. $

Answer

$y=1.54x-3. $

Answer

$y=-1.54x-3. $

Question 3

The curve $C$ is given by the equation $y=x tan(\frac{πxy}{4})$.  At the point $(1, 1)$ ,  $\frac{dy}{dx}=$

Accepted Answer

$\frac{2+π}{2-π}$

Answer

$\frac{π-2}{2+π}$

Answer

$\frac{2-π}{2+π}$

Answer

$\frac{2+π}{π-2}$

Question 4

The function f is defined by $f(x)=\frac{2 ln x+1}{x-3}$, $0 < x < 3$.
Find $f^{'} (x)$.

Accepted Answer

$f^{'} (x)=\frac{2(x-3)-x(2 ln x+1)}{x(x-3)^2}$

Answer

$f^{'} (x)=\frac{(x-3)-x(2 ln x+1)}{x(x-3)^2}$

Answer

$f^{'} (x)=\frac{2(x-3)-(2 ln x+1)}{x(x-3)^2}$

Answer

$f^{'} (x)=\frac{2(x-3)-x(2 ln x+1)}{(x-3)^2}$

Question 5

The function f is defined by $f(x)=\frac{2 ln x+1}{x-3}$, $0 < x < 3$.
Find the coordinates of the point of inflexion on the graph of $y=f(x)$.

Accepted Answer

$(0.899, −0.375)$

Answer

$(0.009, 2.82)$

Answer

$(1.22, −0.785)$

Answer

$(1.56, −1.31)$

Question 6

Let $y=arccos(\frac{x}{2})$
Find $\frac{dy}{dx}$.

Accepted Answer

$\frac{dy}{dx} =-\frac{1}{\sqrt{4-x^2}}$

Answer

$\frac{dy}{dx} =\frac{2}{\sqrt{4-x^2}}$

Answer

$\frac{dy}{dx} =\frac{1}{\sqrt{4-x^2}}$

Answer

$\frac{dy}{dx} =-\frac{2}{\sqrt{4-x^2}}$

Question 7

Consider the curve $y=\frac{1}{1-x}+\frac{4}{x-4}$.
Find the $x$-coordinates of the points on the curve where the gradient is zero.

Accepted Answer

$x=2$,  $x=-2$

Answer

$x=2$,  $x=-1$

Answer

$x=-1$,  $x=-2$

Answer

$x=3$,  $x=-1$

Question 8

A curve C is given by the implicit equation $x+y-cos(xy)=0$.
Find $\frac{dy}{dx}$

Accepted Answer

$\frac{dy}{dx}=-\frac{1+y sin(xy)}{1+x sin(xy)}$

Answer

$\frac{dy}{dx}=-\frac{1-y sin(xy)}{1-x sin(xy)}$

Answer

$\frac{dy}{dx}=\frac{1-y sin(xy)}{1-x sin(xy)}$

Answer

$\frac{dy}{dx}=\frac{1+y sin(xy)}{1+x sin(xy)}$

Question 9

A water trough which is $10$ metres long has a uniform cross-section in the shape of a semicircle with radius $0.5$ metres. It is partly filled with water as shown in the following diagram of the cross-section. The centre of the circle is $O$ and the angle $KOL$ is $θ$ radians.  Find an expression for the volume of water $V (m^3)$ in the trough in terms of $θ$.

Accepted Answer

$V=\frac{5}{4}(θ-sinθ)$

Answer

$V=\frac{5}{2}(sinθ-θ)$

Answer

$V=\frac{5}{2}(θ-sinθ)$

Answer

$V=\frac{5}{4}(sinθ-θ)$

Question 10

A water trough which is $10$ metres long has a uniform cross-section in the shape of a semicircle with radius $0.5$ metres. It is partly filled with water as shown in the following diagram of the cross-section. The centre of the circle is $O$ and the angle $KOL$ is $θ$ radians.  Find an expression for the volume of water $V (m^3)$ in the trough in terms of $θ$. The volume of water is increasing at a constant rate of $0.0008 m^3 s^{-1}$.  Calculate $\frac{dθ}{dt}$ when $θ=\frac{π}{3}$.

Accepted Answer

$\frac{dθ}{dt}=0.00128$  $rad s^{-1}$

Answer

$\frac{dθ}{dt}=1.28$  $rad s^{-1}$

Answer

$\frac{dθ}{dt}=0.0128$  $rad s^{-1}$

Answer

$\frac{dθ}{dt}=0.128$  $rad s^{-1}$

Question 11

Consider the curve defined by the equation $4x^2+y^2=7$.
Find the equation of the NORMAL to the curve at the point $(1, \sqrt{3})$.

Accepted Answer

$y=0.433x+1.30$

Answer

$y=0.433x-1.30$

Answer

$y=-0.433x-1.30$

Answer

$y=-0.433x+1.30$

Question 12

An earth satellite moves in a path that can be described by the curve $72.5x^2+71.5y^2=1$ where $x=x(t)$ and $y=y(t)$ are in thousands of kilometres and $t$ is time in seconds.  Given that $\frac{dx}{dt}=7.75×10^{-5}$ when $x=3.2×10^{-3}$, find the possible values of $\frac{dy}{dt}$.

Accepted Answer

$\frac{dy}{dt}=±2.13×10^{-6}$

Answer

$\frac{dy}{dt}=±2.33×10^{-6}$

Answer

$\frac{dy}{dt}=±2.33×10^{-5}$

Answer

$\frac{dy}{dt}=±2.13×10^{-5}$

Question 13

Consider two functions $f$ and $g$ and their derivatives $f^{'}$ and $g^{'}$. The following table shows the values for the two functions and their derivatives at $x=1, 2$ and $3$.
Given that $p(x)=f(x)g(x)$, find $p^{'} (3)$

Accepted Answer

$11$

Answer

$6$

Answer

$7$

Answer

$12$

Question 14

Consider two functions $f$ and $g$ and their derivatives $f^{'}$ and $g^{'}$. The following table shows the values for the two functions and their derivatives at $x=1, 2$ and $3$.
Given that $h(x)=g∘f(x)$, find $h^{'} (2)$

Accepted Answer

$16$

Answer

$8$

Answer

$12$

Answer

$14$

Question 15

Consider the function $f(x)=\frac{lnx}{x}$, ${x>0}.$
The sketch below shows the graph of $y= f(x)$ and its tangent at a point $A$.
Find $f^{'} (x)$.

Accepted Answer

$f^{'} (x)=\frac{1-lnx}{x^2}$

Answer

$f^{'} (x)=\frac{x-lnx}{x^3}$

Answer

$f^{'} (x)=\frac{x-lnx}{x^2}$

Answer

$f^{'} (x)=\frac{lnx-1}{x^2}$

Question 16

Consider the function $f(x)=\frac{lnx}{x}$, ${x>0}.$
The sketch below shows the graph of $y= f(x)$ and its tangent at a point $A$.
Find the coordinates of $B$, at which the curve reaches its maximum value.

Accepted Answer

$(e,  \frac1e)$

Answer

$(e^2,  \frac{2}{e^2})$

Answer

$(1,0)$

Answer

$(\sqrt{e},  \frac{1}{2\sqrt{e}})$

Question 17

Consider the function $f(x)=\frac{lnx}{x}$, ${x>0}.$
The sketch below shows the graph of $y= f(x)$ and its tangent at a point $A$.
Find the coordinates of $C$, the point of inflexion on the curve..

Accepted Answer

$(e^{\frac32}, \frac32 e^{-\frac{3}2})$

Answer

$(e^{\frac12}, \frac12 e^{-\frac{1}2})$

Answer

$(e^{\frac32}, \frac32 e^{\frac{3}2})$

Answer

$(e^{\frac12}, \frac12 e^{\frac{1}2})$

Question 18

Consider the function $f(x)=\frac{lnx}{x}$, ${x>0}.$
The sketch below shows the graph of $y= f(x)$ and its tangent at a point $A$.
The graph of $y= f(x)$ crosses the $x$-axis at the point $A$.
Find the equation of the tangent to the graph of f at the point $A$.

Accepted Answer

$y=x-1$

Answer

$y=1.25x-1$

Answer

$y=x-1.25$

Answer

$y=0.75x-1$

Question 19

A window is made in the shape of a rectangle with a semicircle of radius $r$ metres on top, as shown in the diagram. The perimeter of the window is a constant $P$ metres.
Find the area of the window in terms of $P$ and $r$.

Accepted Answer

$A=Pr-2r^2-\frac{πr^2}{2}$

Answer

$A=Pr+2r^2-\frac{πr^2}{2}$

Answer

$A=Pr-2r^2+\frac{πr^2}{2}$

Answer

$A=Pr+2r^2+\frac{πr^2}{2}$

Question 20

A window is made in the shape of a rectangle with a semicircle of radius $r$ metres on top, as shown in the diagram. The perimeter of the window is a constant $P$ metres.
Find the area of the window in terms of $P$ and $r$.  Find the width of the window in terms of $P$ when the area is a maximum.

Accepted Answer

$\frac{2P}{4+π}$

Answer

$\frac{P}{2+π}$

Answer

$\frac{P}{4+π}$

Answer

$\frac{2P}{2+π}$

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