Solve each equation and check your solutions. 0.75a = 14.25

Use a Maclaurin expansion to find each integral. $\int \cos x^{2} d x$

Solve the inequality. Graph and check the solution.

$$55>-7 v+6$$

Solve the system by using Gaussian elimination or Gauss-Jordan elimination.

$$2(x-6 z)=3 y+x+17$$

$$2 x-19=3 y+18 z$$

$$-3 x+7 y+36 z=-41$$

Refer to a die that is loaded so that the numbers 1 and 3 are equally likely to appear and 2, 4, 5, and 6 are equally likely to appear and 2, 4, 5, and 6 are equally likely to occur. However, 1 is three times as likely to appear as 2. One die is rolled. Assign probability to the outcomes that accurately model the likelihood of the various numbers to appear.

Find a polynomial $p_3$ such that $\left\{p_{0}, p_{1}, p_{2}, p_{3}\right\}$ is an orthogonal basis for the subspace $\mathbb{P}_3$ of $\mathbb{P}_4$. Scale the polynomial $p_3$ so that its vector of values is (1, 2, 0, -2, 1).

Write a brief definition for the following term, using the language of the cardiovascular system.

ventricle: $\rule{10cm}{0.15mm}$

Investigate how the locations of the centroid, the orthocentre, and the circumcentre of a triangle are related. Describe your method and your findings.

Discuss similarities and differences in the way terrestrial planets and giant planets formed in our Solar System.

How did Romans improve life in the empire by using science and mathematics?

Show that $\frac { 10 ! } { 6 ! 4 ! } + \frac { 10 ! } { 5 ! 5 ! } = \frac { 11 ! } { 6 ! 5 ! }$

The following steps occur in the reaction of ethyl alcohol $\left(\mathrm{CH}_3 \mathrm{CH}_2 \mathrm{OH}\right)$ with oxygen to yield acetic acid $\left(\mathrm{CH}_3 \mathrm{CO}_2 \mathrm{H}\right)$. Show that equations 1 and 2 sum to give the net equation and calculate $\Delta H^{\circ}$ for the net equation.

$$\begin{aligned} & \mathrm{CH}_3 \mathrm{CH}_2 \mathrm{OH}(l)+1 / 2 \mathrm{O}_2(g) \longrightarrow \\ & \mathrm{CH}_3 \mathrm{CHO}(g)+\mathrm{H}_2 \mathrm{O}(l) \quad \Delta \mathrm{H}^{\circ}=-174.2 \mathrm{~kJ}\end{aligned}$$

$$\mathrm{CH}_3 \mathrm{CHO}(g)+1 / 2 \mathrm{O}_2(g) \longrightarrow$$

(Net) $\mathrm{CH}_3 \mathrm{CH}_2 \mathrm{OH}(l)+\stackrel{\mathrm{CH}_3 \mathrm{CO}_2 \mathrm{H}(l)}{\mathrm{O}_2(g) \longrightarrow} \longrightarrow \mathrm{H}^{\circ}=-318.4 \mathrm{~kJ}$ $\mathrm{CH}_3 \mathrm{CO}_2 \mathrm{H}(l)+\mathrm{H}_2 \mathrm{O}(l) \quad \Delta \mathrm{H}^{\circ}=$ ?

Show that each conjecture is false by finding a counterexample.

Assume that i = 1, j = 2, k = 3 and m = 2.What does each of the following statements print?

a) System.out.println( i == 1 );

b) System.out.println( j == 3 );

c) System.out.println( ( i >= 1 ) && ( j < 4 ) );

d) System.out.println( ( m <= 99 ) & ( k < m ) );

e) System.out.println( ( j >= i ) || ( k == m ) );

f) System.out.println( ( k + m < j ) | ( 3 - j >= k ) );

g) System.out.println( !( k > m ) );