We are given the matrix equation: \((A - 2I)(A - 4I) = 0\).


Expand the equation: \(A^2 - 4A - 2IA + 8I^2 = 0\).


Simplify: \(A^2 - 6A + 8I = 0\).


Since \(A\) is non-singular, multiply the entire equation by \(A^{-1}\):

\(A^2 A^{-1} - 6A A^{-1} + 8I A^{-1} = 0\).

\(A - 6I + 8A^{-1} = 0\).


Rearrange the equation: \(A + 8A^{-1} = 6I\).


Divide both sides by 6: \(\frac{1}{6}A + \frac{8}{6}A^{-1} = \frac{6I}{6}\).


Simplify the coefficients: \(\frac{1}{6}A + \frac{4}{3}A^{-1} = I\).
Quick Tip: To find an expression involving a matrix \(A\) and its inverse \(A^{-1}\) from a polynomial equation in \(A\), multiply the polynomial by \(A^{-1}\). Remember that \(A A^{-1} = I\) and \(I A^{-1} = A^{-1}\).

The complex number is \(Z = -\frac{1}{2} + i\frac{\sqrt{3}}{2}\).


Let \(x = -\frac{1}{2}\) and \(y = \frac{\sqrt{3}}{2}\).


Since \(x < 0\) and \(y > 0\), \(Z\) lies in the second quadrant.


Calculate the reference angle \(\alpha = \arctan \left| \frac{y}{x} \right|\).

\(\alpha = \arctan \left| \frac{\sqrt{3}/2}{-1/2} \right| = \arctan(\sqrt{3})\).

\(\alpha = \frac{\pi}{3}\).


The amplitude \(\theta\) in the second quadrant is \(\theta = \pi - \alpha\).

\(\theta = \pi - \frac{\pi}{3} = \frac{2\pi}{3}\).
Quick Tip: When a complex number \(Z=x+iy\) lies in the second quadrant (\(x<0, y>0\)), its amplitude \(\theta\) is calculated as \(\pi\) minus the reference angle \(\alpha = \arctan(|y/x|)\).

Rewrite the equation by completing the square for \(x\): \(4x^2 - 16x + 9y^2 = 20\).

\(4(x^2 - 4x) + 9y^2 = 20\).

\(4(x^2 - 4x + 4) - 16 + 9y^2 = 20\).

\(4(x - 2)^2 + 9y^2 = 36\).


Divide by 36 to get the standard form: \(\frac{(x - 2)^2}{9} + \frac{y^2}{4} = 1\).


We have \(a^2 = 9\) and \(b^2 = 4\). Since \(a^2 > b^2\), the major axis is horizontal.


Eccentricity \(e\) is defined by \(b^2 = a^2 (1 - e^2)\).

\(4 = 9 (1 - e^2)\).

\(1 - e^2 = \frac{4}{9}\).

\(e^2 = 1 - \frac{4}{9} = \frac{5}{9}\).

\(e = \frac{\sqrt{5}}{3}\).
Quick Tip: The key to identifying properties of non-standard conic section equations is completing the square to isolate the variables and determine \(a^2\) and \(b^2\). The eccentricity formula \(e = \sqrt{1 - b^2/a^2}\) applies when \(a>b\).

Given \(|\vec{a}| = 1\) and \(|\vec{b}| = 1\).


We start with the magnitude squared of the difference vector:

\(|\vec{a} - \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 - 2(\vec{a} \cdot \vec{b})\).

\(|\vec{a} - \vec{b}|^2 = 1^2 + 1^2 - 2|\vec{a}||\vec{b}|\cos\theta\).

\(|\vec{a} - \vec{b}|^2 = 2 - 2\cos\theta = 2(1 - \cos\theta)\).


Using the half-angle identity \(1 - \cos\theta = 2\sin^2\frac{\theta}{2}\):

\(|\vec{a} - \vec{b}|^2 = 2 \left( 2\sin^2\frac{\theta}{2} \right) = 4\sin^2\frac{\theta}{2}\).


Taking the square root of both sides: \(|\vec{a} - \vec{b}| = 2\sin\frac{\theta}{2}\).


Solving for \(\sin\frac{\theta}{2}\): \(\sin\frac{\theta}{2} = \frac{1}{2}|\vec{a} - \vec{b}|\).
Quick Tip: Remember the cosine rule for triangles applied to vectors, which leads directly to the half-angle relation: the length of the vector difference \(|\vec{a}-\vec{b}|\) is proportional to \(\sin(\theta/2)\), where \(\theta\) is the angle between \(\vec{a}\) and \(\vec{b}\).

Let \(P(1, 2, 4)\) and the plane coefficients be \((A, B, C) = (2, -1, 1)\), \(D=2\).


The formula for the image \(P'(x', y', z')\) is:
\(\frac{x'-x_1}{A} = \frac{y'-y_1}{B} = \frac{z'-z_1}{C} = k\), where \(k = -2 \frac{Ax_1 + By_1 + Cz_1 + D}{A^2 + B^2 + C^2}\).


Calculate the numerator of the fraction:
\(Ax_1 + By_1 + Cz_1 + D = 2(1) - 1(2) + 1(4) + 2 = 2 - 2 + 4 + 2 = 6\).


Calculate the denominator of the fraction:
\(A^2 + B^2 + C^2 = 2^2 + (-1)^2 + 1^2 = 4 + 1 + 1 = 6\).


Substitute to find \(k\): \(k = -2 \frac{6}{6} = -2\).


Solve for \(x'\): \(\frac{x' - 1}{2} = -2 \implies x' = 1 + 2(-2) = -3\).


Solve for \(y'\): \(\frac{y' - 2}{-1} = -2 \implies y' = 2 + (-1)(-2) = 4\).


Solve for \(z'\): \(\frac{z' - 4}{1} = -2 \implies z' = 4 + 1(-2) = 2\).


The image point is \((-3, 4, 2)\).
Quick Tip: Be sure to substitute all components correctly, paying careful attention to signs (especially \(B=-1\) and the overall factor of \(-2\) in the image formula). This ensures accurate calculation of the ratio \(k\).

The expression is \(L = \lim_{x\to 0}[1 + x\sin(\pi - x)]^{\frac{1}{x}}\).


Use the identity \(\sin(\pi - x) = \sin x\).

\(L = \lim_{x\to 0}[1 + x\sin x]^{\frac{1}{x}}\).


This is of the indeterminate form \(1^\infty\). We use the limit rule \(L = e^{\lim_{x\to a} g(x)[f(x)-1]}\).


Here \(f(x) = 1 + x\sin x\) and \(g(x) = \frac{1}{x}\).


The exponent \(E\) is: \(E = \lim_{x\to 0} \frac{1}{x} [(1 + x\sin x) - 1]\).

\(E = \lim_{x\to 0} \frac{x\sin x}{x}\).

\(E = \lim_{x\to 0} \sin x\).

\(E = \sin(0) = 0\).


Therefore, \(L = e^E = e^0 = 1\).
Quick Tip: Always simplify trigonometric functions before applying limit identities. The appearance of \(\sin(\pi-x)\) should immediately signal simplification to \(\sin x\). For \(1^\infty\) limits, the conversion to exponential form \(e^E\) is standard.

Let \(I = \int_{0}^{\pi} \log (\sin^2 x) dx\).


Use the logarithm property: \(I = 2 \int_{0}^{\pi} \log (\sin x) dx\).


Use the property \(\int_{0}^{2a} f(x) dx = 2 \int_{0}^{a} f(x) dx\) when \(f(2a-x) = f(x)\).


Here \(2a = \pi\), so \(a = \pi/2\). Since \(\sin(\pi - x) = \sin x\), the property holds.

\(I = 2 \left[ 2 \int_{0}^{\pi/2} \log (\sin x) dx \right] = 4 \int_{0}^{\pi/2} \log (\sin x) dx\).


Recall the standard integral result: \(\int_{0}^{\pi/2} \log (\sin x) dx = -\frac{\pi}{2} \log 2\).

\(I = 4 \left( -\frac{\pi}{2} \log 2 \right) = -2\pi \log 2\).


Rewrite the result using logarithm properties to match the options:

\(I = 2\pi (-\log 2) = 2\pi (\log 2^{-1}) = 2\pi \log \left(\frac{1}{2}\right)\).
Quick Tip: Problems involving \(\int \log(\sin x)\) or \(\int \log(\cos x)\) often require repeated use of standard integral properties, specifically \(\int_{0}^{\pi/2} \log (\sin x) dx = -\frac{\pi}{2} \log 2\) and the \(f(2a-x)\) property.

Rearrange the equation into the linear first-order form \(\frac{dy}{dx} + P(x)y = Q(x)\):

\(\frac{dy}{dx} - y = 3 - 2x\).


Here \(P(x) = -1\) and \(Q(x) = 3 - 2x\).


Calculate the Integrating Factor (IF): \(IF = e^{\int P(x) dx} = e^{\int -1 dx} = e^{-x}\).


The general solution form is \(y \cdot IF = \int Q(x) \cdot IF dx + C_1\).

\(y e^{-x} = \int (3 - 2x) e^{-x} dx + C_1\).


Using integration by parts, we find \(\int (3 - 2x) e^{-x} dx = (2x - 1)e^{-x}\).


Substitute back: \(y e^{-x} = (2x - 1)e^{-x} + C_1\).


Multiply by \(e^x\): \(y = (2x - 1) + C_1 e^x\).

\(y = C_1 e^x + 2x - 1\).



\begin{quicktipbox
A linear first-order differential equation \(\frac{dy}{dx} + P(x)y = Q(x)\) is solved by multiplying by the Integrating Factor \(e^{\int P(x) dx}\) and then integrating the right side. The integration \(\int (3-2x)e^{-x} dx\) can be efficiently solved using tabular integration or careful application of parts.

\end{quicktipbox Quick Tip: A linear first-order differential equation \(\frac{dy}{dx} + P(x)y = Q(x)\) is solved by multiplying by the Integrating Factor \(e^{\int P(x) dx}\) and then integrating the right side. The integration \(\int (3-2x)e^{-x} dx\) can be efficiently solved using tabular integration or careful application of parts.

This follows a binomial distribution \(B(n, p)\).


Number of trials \(n = 100\).


Success (\(p\)) is getting an even number \(\{2, 4, 6\}\) on a standard die.


Probability of success \(p = \frac{3}{6} = \frac{1}{2}\).


Probability of failure \(q = 1 - p = \frac{1}{2}\).


The variance \(Var(X)\) for a binomial distribution is given by \(Var(X) = npq\).

\(Var(X) = 100 \times \frac{1}{2} \times \frac{1}{2}\).

\(Var(X) = 100 \times \frac{1}{4} = 25\).



\begin{quicktipbox
In statistical problems involving repeated independent trials with two outcomes (success/failure), identify the binomial distribution parameters \(n\) and \(p\). Variance is a quick calculation once \(n, p, q\) are known.

\end{quicktipbox Quick Tip: In statistical problems involving repeated independent trials with two outcomes (success/failure), identify the binomial distribution parameters \(n\) and \(p\). Variance is a quick calculation once \(n, p, q\) are known.

Let \(e\) be the identity element in the set of integers.


By the definition of the identity element, \(a \times e = a\) must hold for all \(a\).


Substitute the operation: \(a + e - ae = a\).


Subtract \(a\) from both sides: \(e - ae = 0\).


Factor out \(e\): \(e(1 - a) = 0\).


For this equation to be true for all integers \(a\), the identity element \(e\) must be 0.


(If \(e=0\), \(0(1-a) = 0\), which is true for all \(a\)).


The identity element is \(0\).



\begin{quicktipbox
The identity element must work for *all* elements in the set. If \(e(1-a)=0\), and \((1-a)\) is not universally zero, \(e\) must be zero. Always verify that \(e*a = a\) also holds (here \(0+a-0a = a\)).

\end{quicktipbox Quick Tip: The identity element must work for *all* elements in the set. If \(e(1-a)=0\), and \((1-a)\) is not universally zero, \(e\) must be zero. Always verify that \(e*a = a\) also holds (here \(0+a-0a = a\)).