Q: Given the matrix $$D = \begin{pmatrix}6 & 2 \\ 4 & 5\end{pmatrix}$$, what is the inverse $$D^{-1}$$?

$$\frac{1}{22}\begin{pmatrix}5 & -2 \\ -4 & 6\end{pmatrix}$$

Question 1

Given the formula for power dissipated by a resistor $$P = \frac{V^2$}{R}$$, use dimensional analysis to determine the dimensions of resistance $$R in $$terms of mass $$(M)$$, length $$(L)$$, and time $$(T). $$Which of the following is correct?

Accepted Answer

Question 2

A guitar string has mass $$M$$, length $$L$$, and tension $$F. A $$colleague proposes the period of vibration is $$t = 2\pi \frac{M^2$L}{F}. $$Using dimensional analysis, which of the following best explains why this formula is incorrect?

Accepted Answer

The dimensions of the right-hand side do not match those of time.

Question 3

The sum of the first $$n$$ terms of an arithmetic progression is given by $$S_n = \frac{n}{2}[2a + (n-1)d]. $$For the sequence $$3, 6, 9, ..., 45$$, what is the sum of the first 15 terms?

Accepted Answer

$$360$$

Question 4

A digital IC chip counts in the sequence $$512, 1024, 2048, .... $$What is the 11th count in this geometric progression?

Accepted Answer

$$524288$$

Question 5

A capacitor is charged by a DC supply through a resistor. The voltage across the capacitor is given by $$V_C = V_S(1 - e$$^{-t/RC}$$). If$$ $$V_S = 21V$$, $$R = 23M\Omega$$, $$V_C = 1V$$ after $$t = 4s$$, what is the approximate value of the capacitance $$C$$?

Accepted Answer

$$3.56 	imes 10$$^{-6}$$ F$$

Question 6

Given $$V_S = 8\sin(36\pi ft - \frac{\pi}{4})$$ with $$f = 2	ext{MHz}$$, at what time $$t$$ does the instantaneous voltage first reach $$+4V$$?

Accepted Answer

$$5.79 	ext{ ns}$$

Question 7

The curve of a heavy power cable is given by $$y = 80\cosh\left(\frac{x}{80}\right). $$What is the value of $$y$$ when $$x = 120$$?

Accepted Answer

$$188.19$$

Question 8

A simple pendulum has mass $$m$$, length $$l$$, and is influenced by gravity $$g. $$Using dimensional analysis, which formula best represents the period $$t$$?

Accepted Answer

$$t = k\sqrt{\frac{l}{g}}$$

Question 9

Convert the voltage $$V = 150 \angle 60^\circ to $$its complex form.

Accepted Answer

$$75 + 129.9j$$

Question 10

Given the matrix $$D = \begin{pmatrix}6 & 2 \ 4 & 5\end{pmatrix}$$, what is the inverse $$D^{-1}$$?

Accepted Answer

$$\frac{1}{22}\begin{pmatrix}5 & -2 \ -4 & 6\end{pmatrix}$$

Question 11

A sample of 10 wireless dongles has measured transmit powers (in dBm): $$22.3, 24.1, 22.8, 22.3, 24.9, 22.3, 21.5, 23.9, 22.3, 21.5. $$What is the mean transmit power?

Accepted Answer

$$22.79$$

Question 12

For the same sample of dongle powers, what is the standard deviation (rounded to two decimal places)?

Accepted Answer

$$1.14$$

Question 13

A factory produces bolts with 96% within tolerance. If 7 bolts are selected at random, what is the probability that exactly 2 exceed the tolerance?

Accepted Answer

$$0.0274$$

Question 14

If the mean capacitance of 500 capacitors is $$200\mu F$$ with a standard deviation of $$12\mu F$$, how many are likely to have values between $$185\mu F$$ and $$215\mu F ($$assuming normal distribution)?

Accepted Answer

$$394$$

Question 15

A fuel additive is tested on 120 cars. The mean mpg with additive is 51, without is 48, and the sample standard deviation is 15. What is the calculated $$z-$$score for the test?

Accepted Answer

$$2.19$$

Question 16

In the context of hypothesis testing for the fuel additive, what is the correct conclusion if the calculated $$z-$$score is $$2.19$$ and the critical value at 5% significance is $$1.96$$?

Accepted Answer

Reject the null hypothesis; the additive has a significant effect.

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