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Andrew D.

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Pre-Calculus

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Question:

Find all solutions of the equation: cos(x)^2=1.

Andrew D.

Answer:

In order to solve this equation for its multiplicity of solutions, it is necessary to raise both sides of the equation by the power/exponent of (1/2), i.e. square root of both sides for equal reduction: (cos(x)^2)^(1/2)=(1)^(1/2) Note: the property/law of exponents dictates that any integer exponent, n, of a base is multiplied by m, the exponent of the exponential function/expression. Thus, it certainly follows that: *(b^n)^m=b^(nm)* (cos(x)^2)^(1/2)=cos(x)^(2/2)=cos(x) & 1^(1/2) = + or - 1 Then, cos(x)=+ or - 1 Observe that: cos(x)=1 for all x=2n(pi), where n is set of all integers, and that: cos(x)=(-1) for all x=(2n+1)(pi), where n is the set of all integers. This is due to cos(x)=1 initially at 0 radians, thus permitting the generalization to the sum of any multiple of 2(pi) with 0, whether it be positive or negative; the same reasoning applies to the latter equation of: cos(x)=(-1), where it exists at (pi) radians, again permitting the same treatment, except where (pi) is being added/aggregated with any multiple of 2(pi).

Computer Certification and Training

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Question:

If a computer cold boot is attempted, assuming the mains power cable is plugged in and is supplying the machine the according wattage of the geographical region fails to startup/excite the machine, betraying no indication of actuation aside from a single oscillation/flash of the front panel power supply light, what are the possible symptoms?

Andrew D.

Answer:

The symptoms elaborated above ultimately indicate a power supply (PSU) failure, which may be precipitated by any of the following conditions: - Atrophy due to age and frequency of use - Electric turbulence/inference due to lightning strikes, power surges/spikes, etc. - Debris, dirt, and any other obtrusive substances which may impede air flow/contribute to poor ventilation - Brown-outs, blackouts, etc.

Algebra

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Question:

(8-i)/(3-2i); If the preceding expression is rewritten in the form a+bi, where a and b are real numbers, what is the value of a? (Note: i = square root of (-1))

Andrew D.

Answer:

In order to revise/convert the phrasing of the expression (8-i)/(3-2i) into the standard form of a+bi, it is necessary to multiply both the numerator and denominator by the complex conjugate, or the complementary binomial to the negative b in (3-2i), of the denominator as such: ((8-i)/(3-2i))*((3+2i)/(3+2i)) Recognize first the multiplication of the sum and difference of the same two terms in the denominators, frequently identified as the decomposed/factored form of the difference of two squares, and reduce them accordingly: (3-2i)(3+2i)=((3^2)-((2^2)(i^2))=(9-4(-1))=(9+4)=13 Then, distribute the binomials (by means of the FOIL method or any alternative, algebraic method) in the numerator for the product thereof: (8-i)(3+2i)=24+16i-3i-(2i^2)=24+13i-2(-1)=24+13i+2=26+13i Reflect/apply the multiplication in the original expression: (26+13i)/13 Now, separate the fraction into two parts with the constant. 13. in the denominator and reduce: (26/13)+(13i/13)=2+1i=2+i Once in its standard form, it is certain that the value to which a is assigned in this expression, as it was requested in the prompt, corresponds to the numerical, constant value of 2. Therefore, the answer to the prompt's inquiry is a=2.

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