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DSAT Math For Scoring 750-800 (Part 1)

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 1. What is the length of major axis of  \frac{{{x^2}}}{5} + \frac{{{y^2}}}{6} = 24

\begin{array}{l} A.{\rm{ }}\sqrt {30} \\ B.{\rm{ 2}}\sqrt {30} \\ C.{\rm{ 4}}\sqrt {30} \\ D.{\rm{ 24}}\\ E.{\rm{ 30}} \end{array}

Solution 

\frac{{{x^2}}}{5} + \frac{{{y^2}}}{6} = 24 \Rightarrow \frac{{{x^2}}}{{120}} + \frac{{{y^2}}}{{144}} = 1

\Rightarrow \left\{ \begin{array}{l} a = \sqrt {120} \\ b = \sqrt {144} \end{array} \right. \Rightarrow 2a = 2\sqrt {120} = 4\sqrt {30}

2. Bob averaged 40 mph going to work and 60 mph going home. What was his average speed for the trip ?      

\begin{array}{l} A.{\rm{ }}44{\rm{ }}mph\\ B.{\rm{ }}46{\rm{ }}mph\\ C.{\rm{ }}48{\rm{ }}mph\\ D.{\rm{ }}50{\rm{ }}mph\\ E.{\rm{ 52 }}mph \end{array}                       

Solution 

\frac{{40 + 60}}{2} = 50{\rm{ }}mph

3. In ∆ABC, AB=2, BC=3 and AC=4. What is the measure of the largest angle to the nearest degree ? 

\begin{array}{l} A.{\rm{ 10}}{{\rm{4}}^ \circ }\\ B.{\rm{ 10}}{{\rm{8}}^ \circ }\\ C.{\rm{ 11}}{{\rm{2}}^ \circ }\\ D.{\rm{ 11}}{{\rm{6}}^ \circ }\\ E.{\rm{ 12}}{{\rm{0}}^ \circ } \end{array}

Solution 

\begin{array}{l} {b^2} = {a^2} + {c^2} - 2ac\cos B \Rightarrow \cos B = \frac{{{a^2} + {c^2} - {b^2}}}{{2ac}} = - \frac{1}{4}\\ \Rightarrow B = {104^ \circ }29' \end{array}

4. If sinx=a, 0 \le x \le {30^ \circ }, what is sin2x in terms of a ?

\begin{array}{l} A.{\rm{ }}a\sqrt {1 + {a^2}} \\ B.{\rm{ }}a\sqrt {1 - {a^2}} \\ C.{\rm{ 2}}a\sqrt {1 + {a^2}} \\ D.{\rm{ 2}}a\sqrt {1 - {a^2}} \\ E.{\rm{ 2}}a\sqrt {4 - {a^2}} \end{array}

Solution 
\begin{array}{l} \cos x = \sqrt {1 - {{\sin }^2}x} = \sqrt {1 - {a^2}} \\ \Rightarrow \sin 2x = 2\sin x\cos x = 2a\sqrt {1 - {a^2}} \end{array}

 6. What are the y intercepts of  3{x^4}{y^2} + {y^2} + 2y = 4xy + 15

A.{\rm{ (0,5) and (0,}} - {\rm{3)}}
B.{\rm{ (0,}} - {\rm{5) and (0,3)}}
C.{\rm{ (0,5) and (0,3)}}
D.{\rm{ (0,}} - {\rm{3) and (0,}} - {\rm{5)}}
E.{\rm{ (0,3) and (0,}} - {\rm{5)}}

Solution 

x = 0 \Rightarrow {y^2} + 2y - 15 = 0 \Rightarrow y = - 5{\rm{ }}and{\rm{ }}3

7. The half-life of titanium-44 is 63 years. After how many years will 1 gram be left of 70 grams titanimum-44 to the nearest year ?

\begin{array}{l} A.{\rm{ 368}}\\ B.{\rm{ 386}}\\ C.{\rm{ 638}}\\ D.{\rm{ 683}}\\ E.{\rm{ 836}} \end{array}
Solution 

8. Bob invested $100,000 and 20 years later it was $500,000. What was the average percent increase per year ?

\begin{array}{l} A.{\rm{ 7}}{\rm{.5\% }}\\ B.{\rm{ 7}}{\rm{.7\% }}\\ C.{\rm{ 7}}{\rm{.9\% }}\\ D.{\rm{ 8}}{\rm{.1\% }}\\ E.{\rm{ 8}}{\rm{.3\% }} \end{array}
Solution 

9. {x^4} - 5{x^2} - 36 = 0. What are the solutions over complex number ?

\begin{array}{l} A.{\rm{ 3, }} - {\rm{2, 2i, }} - 3i\\ B.{\rm{ 3, }} - {\rm{2, 3i, }} - 2i\\ C.{\rm{ 2, }} - 3{\rm{, 2i, }} - 3i \end{array}

\begin{array}{l} D.{\rm{ 3, }} - 3{\rm{, 2i, }} - 2i\\ E.{\rm{ 2, }} - {\rm{2, 3i, }} - 3i \end{array}

Solution 

10. tan θ=3a, 0 \le \theta \le \frac{\pi }{2}, what is cscθ ? 

\begin{array}{l} A.{\rm{ }}\frac{{\sqrt {9{a^2} + 1} }}{{3a}}\\ B.{\rm{ }}\frac{{\sqrt {9{a^2} + 2} }}{{2a}}\\ C.{\rm{ }}\frac{{\sqrt {9{a^2} + 3} }}{a} \end{array}
\begin{array}{l} D.{\rm{ }}\frac{{\sqrt {9{a^2} - 2} }}{{2a}}\\ E.{\rm{ }}\frac{{\sqrt {9{a^2} - 1} }}{{3a}} \end{array}

Solution 




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