Q1. Consider
=z^3+9z^2+28z+20, z\in C)
. Given that
=0)
, factorise
)
over
Q2. A body of mass 2kg is initially at rest and is acted on by a resultant force of v-4 newtons where v is the velocity in m/s. The body moves in a straight line as a result of the force.
a. Show that the acceleration is given be
b. Solve the differential equation in part a. to find v as a function of t.
Q3. Relative to an origin
, point A has cartesian coordinates (1,2,2) and point B has cartesian coordinates (-1,3,4)
a. Find an expressing for the vector
in the form
b. Show that the cosine of the angle between the vectors
and
is
c.
Hence find the exact area of triangle
Q4. Given that
, plot and label points for each of the following on the argand diagram below.
i.
ii.
iii.
Q5. Given that
=\tan^{-1}(2x))
, find
)
Q6. Evaluate
\sin(2x)\, dx)
Q7. Consider the differential equation:
^2})
,
, for which
when
and
when
Given that
=\frac{4x}{(1-x^2)^2})
, find the solution of this differential equation.
Q8. The path of a particle is given by
=t\sin(t)i-t\cos(t)j, t \geq 0)
. The particle leaves the origin at t=0 and then spirals outwards.
a. Show that the second time the particles crosses the x-axis after leaving the origin occurs when
b. Find the speed of the particle when
c. Let
be the acute angle at which the path of the particle crosses the x-axis. Find
)
when
Q9.
a. On the axes below, sketch the graph with equation
^2}{4}=1)
. State all intercepts with the coordinate axes and give the equations of any asymptotes.
b. Find the gradient of the curve with equation
at the point where
and
Q10. Part of the graph with equation
\sqrt{x+1})
is shown below. Find the area that is bounded by the curve and the x-axis. Give your answer in the form
where
,
and
are integers.
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