Precalculus Review
- Page ID
- 10610
This page is a draft and is under active development.
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Exercise \(\PageIndex{1}\)
Write down an equation of a line passing through the points \((1,2)\) and \((3,5)\).
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Exercise \(\PageIndex{2}\)
For what value(s) of \(m\) is \(x^{3}+6x^{2}-m\) divisible by \((x+4)\).
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Exercise \(\PageIndex{3}\)
The expression \(\displaystyle\frac{\left(1/x+1/y\right) } {x+y}\) is equal to
- \(1\)
- \(1/(xy)\)
- \(xy/(x+y)^{2}\)
- \(x+y\)
- \(1/(x+y)^{2}\)
- Answer
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Exercise \(\PageIndex{4}\)
Solve the following quadratic equation for \(m\), where \[m^2-5m=-6\]
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Exercise \(\PageIndex{5}\)
Simplify \(\displaystyle {\left( {xy \over y^{-3}z^{-1}} \right)^{-1}}\).
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Exercise \(\PageIndex{6}\)
If \(\displaystyle{f(x)={1-x \over 1+x}}\), find \(\displaystyle{f \left( {-1 \over x} \right)}\)
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Exercise \(\PageIndex{7}\)
A hollow cylinder has height \(10{\rm cm}\) and radius \(5{\rm cm}\).
Another cylinder has height \(10{\rm cm}\) and radius \(4{\rm cm}\). The
smaller cylinder is inserted into the larger one. How much space is left
inside the larger cylinder but outside the smaller one (in \({\rm cm}^{3}\))
- Answer
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Exercise \(\PageIndex{8}\)
An equation for a line parallel to the line \(5x-y=2\) is
- \(5y-x=1\),
- \(5x+y=0\),
- \(x+5y=3\)
- \(y-5x=2\)
- Answer
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Exercise \(\PageIndex{9}\)
Find \(x\) if \(\displaystyle{ -4 = {2x+3 \over 1-x}}\).
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Exercise \(\PageIndex{10}\)
Write down an equation of a line passing through the point \((1,3)\) and with the slope \(m=-1/2\)'
- Answer
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Exercise \(\PageIndex{11}\)
Express the function in piecewise form without using absolute values \(F(x)=3|x+2|-2|x-1|\).
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Exercise \(\PageIndex{12}\)
In a general triangle \(ABC\), the angle at \(A\) is \(30^{\circ}\),
the length of \(AB\) is \(6\), and the length of \(AC\) is \(4\). Find the length
of \(CE\). (\(E\) is the point on \(AB\) closest to \(C\).)
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Exercise \(\PageIndex{13}\)
Suppose \(\log(a)=2\), \(\log(b)=3\) and \(\log(c)=5\). Find \(\displaystyle{ \log \left( {c \over ab} \right)}\).
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Exercise \(\PageIndex{14}\)
A rectangular field is bounded by a fence \(500{\rm m}\) long. The field is \(150{\rm m}\) longer than it is wide. Find the area of the field (in \({\rm m}^{2}\))
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