Step 1: Understanding the Concept:

The sum and product of the roots of a quadratic equation \(ax^2 + bx + c = 0\) are given by \(-b/a\) and \(c/a\), respectively.

We apply these relations to the given quadratic equations to find the values of \(\alpha, \beta\), and \(r\).


Step 2: Key Formula or Approach:

For the first equation \(x^2 - 3x + r = 0\), the sum of roots is \(\alpha + \beta = 3\) and the product is \(\alpha\beta = r\).

For the second equation \(x^2 + 3x + r = 0\), the sum of roots is \(\frac{\alpha}{2} + 2\beta = -3\) and the product is \(\left(\frac{\alpha}{2}\right)(2\beta) = r\).


Step 3: Detailed Explanation:

The product of roots condition for the second equation yields \(\alpha\beta = r\), which is consistent with the first equation.

Now we solve the system of linear equations for the sum of roots.
\[ \alpha + \beta = 3 \] \[ \frac{\alpha}{2} + 2\beta = -3 \implies \alpha + 4\beta = -6 \]
Subtracting the first equation from the second yields a new relation.
\[ 3\beta = -9 \implies \beta = -3 \]
Substituting \(\beta = -3\) into the first equation gives the value of \(\alpha\).
\[ \alpha - 3 = 3 \implies \alpha = 6 \]
Now we find \(r\) by multiplying the roots.
\[ r = \alpha\beta = (6)(-3) = -18 \]
Next, we calculate the roots \(R_1\) and \(R_2\) for the final equation \(x^2 + 6x - m = 0\).
\[ R_1 = 2\alpha + \beta + 2r = 2(6) + (-3) + 2(-18) = 12 - 3 - 36 = -27 \] \[ R_2 = \alpha - 2\beta - \frac{r}{2} = 6 - 2(-3) - \left(\frac{-18}{2}\right) = 6 + 6 + 9 = 21 \]
The sum of these roots is \(R_1 + R_2 = -27 + 21 = -6\), which matches the coefficient of \(x\) in \(x^2 + 6x - m = 0\).

The product of the roots is given by the constant term \(-m\).
\[ R_1 R_2 = -m \] \[ (-27)(21) = -m \] \[ -567 = -m \implies m = 567 \]

Step 4: Final Answer:

The value of \(m\) is \(567\).
Quick Tip: When given multiple equations with shared parameters, use the sum and product of roots simultaneously to quickly isolate the variables.
Always rewrite equations like \(x^2 + 6x = m\) into standard form \(x^2 + 6x - m = 0\) to correctly identify the constant term.

Step 1: Understanding the Concept:

The distance between two complex numbers \(z_1\) and \(z_2\) is geometrically represented by \(|z_1 - z_2|\).

For two circles where one lies completely inside the other, the minimum and maximum distances between points on the two circles occur along the line connecting their centers.


Step 2: Key Formula or Approach:

Let \(C_1\) have center \(O_1(0,0)\) and radius \(r\).

Let \(C_2\) have center \(O_2(3,4)\) and radius \(R = 5\).

The distance between their centers is \(d = |3 + 4i| = \sqrt{3^2 + 4^2} = 5\).

Since \(C_2\) lies completely inside \(C_1\), the minimum distance is \(r - (d + R)\) and the maximum distance is \(r + d + R\).


Step 3: Detailed Explanation:

Given that the minimum distance is 2, we can set up the equation for \(r\).
\[ \min |z_1 - z_2| = r - (d + R) = 2 \]
Substitute the known values \(d = 5\) and \(R = 5\) into the equation.
\[ r - (5 + 5) = 2 \] \[ r - 10 = 2 \implies r = 12 \]
Now we need to find the maximum possible distance between a point on \(C_1\) and a point on \(C_2\).
\[ \max |z_1 - z_2| = r + (d + R) \] \[ \max |z_1 - z_2| = 12 + (5 + 5) = 12 + 10 = 22 \]

Step 4: Final Answer:

The maximum distance \(\max |z_1 - z_2|\) is \(22\).
Quick Tip: For nested circles, the minimum distance is \(R_{outer} - (d + R_{inner})\) and the maximum distance is \(R_{outer} + d + R_{inner}\), where \(d\) is the distance between their centers.

Step 1: Understanding the Concept:

A system of three linear equations has infinitely many solutions if the determinant of the coefficient matrix is zero and the equations are linearly dependent.

Alternatively, we can eliminate one variable and force the resulting two equations in two variables to be identical.


Step 2: Key Formula or Approach:

We will eliminate \(x\) from the given system to form two identical equations in \(y\) and \(z\).

Equation 1 is \(x + 5y + 6z = 4\).

Equation 2 is \(2x + 3y + 4z = 7\).

Equation 3 is \(x + 6y + az = b\).


Step 3: Detailed Explanation:

Perform the operation (Equation 2) - \(2 \times\) (Equation 1) to eliminate \(x\).
\[ (2x + 3y + 4z) - 2(x + 5y + 6z) = 7 - 2(4) \] \[ -7y - 8z = -1 \implies 7y + 8z = 1 \]
This can be written as \(y + \frac{8}{7}z = \frac{1}{7}\).

Now, perform (Equation 3) - (Equation 1) to eliminate \(x\) again.
\[ (x + 6y + az) - (x + 5y + 6z) = b - 4 \] \[ y + (a - 6)z = b - 4 \]
For the system to have infinitely many solutions, these two newly formed equations must represent the same line.

By comparing the coefficients of \(z\) and the constant terms, we get the values of \(a\) and \(b\).
\[ a - 6 = \frac{8}{7} \implies a = 6 + \frac{8}{7} = \frac{50}{7} \] \[ b - 4 = \frac{1}{7} \implies b = 4 + \frac{1}{7} = \frac{29}{7} \]
Now we have the coordinates of the point \((a, b) = \left(\frac{50}{7}, \frac{29}{7}\right)\).

We check which option satisfies this point by finding \(a - b\).
\[ a - b = \frac{50}{7} - \frac{29}{7} = \frac{21}{7} = 3 \]
Thus, the point satisfies the line equation \(x - y = 3\).


Step 4: Final Answer:

The point \((a, b)\) lies on the line \(x - y = 3\).
Quick Tip: Instead of using Cramer's rule which can be lengthy, eliminating one variable and comparing the remaining equations is often much faster for "infinitely many solutions" problems.

Step 1: Understanding the Concept:

An Arithmetic Progression (A.P.) has a common difference \(d\), so \(a_n = a + (n-1)d\).

A Geometric Progression (G.P.) has a common ratio \(r\), so \(g_n = a \cdot r^{n-1}\).

We can use the given equations to find the first term \(a\), the difference \(d\), and the ratio \(r\).


Step 2: Key Formula or Approach:

Let the first term be \(a_1 = g_1 = a\).

From the problem, \(a_1 = 1\), so \(a = 1\).

The first given condition is \(a_2 + g_2 = 1\).

The second given condition is \(a_3 + g_3 = 4\).


Step 3: Detailed Explanation:

Substitute the general forms into the first condition.
\[ (a + d) + ar = 1 \]
Since \(a = 1\), this simplifies nicely.
\[ (1 + d) + r = 1 \implies d + r = 0 \implies d = -r \]
Now, substitute the general forms into the second condition.
\[ (a + 2d) + ar^2 = 4 \]
Substitute \(a = 1\) and \(d = -r\).
\[ 1 + 2(-r) + r^2 = 4 \] \[ r^2 - 2r - 3 = 0 \]
Factoring the quadratic equation yields the possible values for \(r\).
\[ (r - 3)(r + 1) = 0 \]
This gives \(r = 3\) or \(r = -1\).

Since the G.P. is increasing and \(g_1 = 1 > 0\), the common ratio must be greater than 1.

Therefore, we select \(r = 3\).

This means \(d = -r = -3\).

We need to find the value of \(a_{10} + g_5\).

Calculate the 10th term of the A.P.
\[ a_{10} = a + 9d = 1 + 9(-3) = 1 - 27 = -26 \]
Calculate the 5th term of the G.P.
\[ g_5 = a \cdot r^4 = 1 \cdot (3)^4 = 81 \]
Finally, compute the required sum.
\[ a_{10} + g_5 = -26 + 81 = 55 \]

Step 4: Final Answer:

The sum \(a_{10} + g_5\) is \(55\).
Quick Tip: Pay attention to keywords like "increasing G.P.".
Since the first term is positive, an increasing G.P. requires a common ratio \(r > 1\).
This instantly helps in discarding the negative root.

Step 1: Understanding the Concept:

We first need to find the general \(n\)-th term \(T_n\) of the given series.

The numerator of the \(n\)-th term is the sum of the cubes of the first \(n\) natural numbers.

The denominator is the sum of the first \(n\) odd numbers.


Step 2: Key Formula or Approach:

The formula for the numerator sum is \(1^3 + 2^3 + \dots + n^3 = \left( \frac{n(n+1)}{2} \right)^2 = \frac{n^2(n+1)^2}{4}\).

The formula for the denominator sum is \(1 + 3 + 5 + \dots + (2n-1) = n^2\).

Thus, the general term is given by \(T_n = \frac{n^2(n+1)^2 / 4}{n^2} = \frac{(n+1)^2}{4}\).


Step 3: Detailed Explanation:

We need the sum up to 8 terms, which is \(S_8 = \sum_{n=1}^8 T_n\).
\[ S_8 = \sum_{n=1}^8 \frac{(n+1)^2}{4} = \frac{1}{4} \sum_{n=1}^8 (n+1)^2 \]
To evaluate this efficiently, let \(k = n + 1\).

As \(n\) goes from \(1\) to \(8\), \(k\) goes from \(2\) to \(9\).
\[ S_8 = \frac{1}{4} \sum_{k=2}^9 k^2 \]
We know the sum of squares formula is \(\sum_{k=1}^m k^2 = \frac{m(m+1)(2m+1)}{6}\).

For \(m = 9\), we calculate the total sum.
\[ \sum_{k=1}^9 k^2 = \frac{9 \times 10 \times 19}{6} = 15 \times 19 = 285 \]
Since our sum starts from \(k = 2\), we subtract the \(k = 1\) term which is \(1^2 = 1\).
\[ \sum_{k=2}^9 k^2 = 285 - 1 = 284 \]
Now, calculate the final sum \(S_8\).
\[ S_8 = \frac{1}{4} \times 284 = 71 \]

Step 4: Final Answer:

The sum up to 8 terms is \(71\).
Quick Tip: Memorizing standard series sums is crucial for competitive exams.
The sum of the first \(n\) odd numbers is always \(n^2\), and the sum of cubes is the square of the sum of the first \(n\) integers.

Step 1: Understanding the Concept:

We use the symmetric property of binomial coefficients which states \(\binom{n}{k} = \binom{n}{n-k}\).

This property allows us to simplify the given expression into a much more recognizable form.


Step 2: Key Formula or Approach:

Apply the symmetry property to each term individually.
\(\binom{30}{30-r} = \binom{30}{r}\)
\(\binom{30}{31-r} = \binom{30}{r-1}\)
\(\binom{30}{32-r} = \binom{30}{r-2}\)
\(\binom{30}{33-r} = \binom{30}{r-3}\)

The given expression simplifies to the following sum.
\[ S = \binom{30}{r} + 3\binom{30}{r-1} + 3\binom{30}{r-2} + \binom{30}{r-3} \]

Step 3: Detailed Explanation:

Notice that the coefficients \(1, 3, 3, 1\) are exactly the binomial coefficients of \((1+x)^3\).

This suggests finding the coefficient of \(x^r\) in a product of two polynomials.

Consider the algebraic identity:
\[ (1+x)^{30} \cdot (1+x)^3 = (1+x)^{33} \]
Let's find the coefficient of \(x^r\) on both sides of this identity.

On the left side, we expand both binomials.
\[ \left( \dots + \binom{30}{r}x^r + \binom{30}{r-1}x^{r-1} + \dots \right) \times \left( \binom{3}{0} + \binom{3}{1}x + \binom{3}{2}x^2 + \binom{3}{3}x^3 \right) \]
The term with \(x^r\) is obtained by multiplying pairs that give a combined degree of \(r\).
\[ \binom{30}{r}\binom{3}{0} + \binom{30}{r-1}\binom{3}{1} + \binom{30}{r-2}\binom{3}{2} + \binom{30}{r-3}\binom{3}{3} \]
Substitute the known values of \(\binom{3}{k}\).
\[ = \binom{30}{r}(1) + \binom{30}{r-1}(3) + \binom{30}{r-2}(3) + \binom{30}{r-3}(1) \]
This matches our simplified expression \(S\) perfectly.

On the right side of the identity \((1+x)^{33}\), the coefficient of \(x^r\) is simply \(\binom{33}{r}\).

Therefore, we can equate the two representations.
\[ \binom{33}{r} = \binom{m}{r} \implies m = 33 \]

Step 4: Final Answer:

The value of \(m\) is \(33\).
Quick Tip: Vandermonde's Identity, \(\sum \binom{m}{k}\binom{n}{r-k} = \binom{m+n}{r}\), is highly useful for solving sums of products of binomial coefficients.
Recognizing the sequence \(1, 3, 3, 1\) as \(\binom{3}{k}\) is the key here.

Step 1: Understanding the Concept:

To form a triangle, we need to choose 3 distinct vertices from the \(n\) vertices of the polygon.

The number of ways to do this is given by the combination formula \(\binom{n}{3}\).


Step 2: Key Formula or Approach:

The total number of triangles formed is \(p_n = \binom{n}{3}\).

We are given the mathematical relation:
\[ p_{n+1} - p_n = 66 \]
Substituting the combination formula into this relation yields:
\[ \binom{n+1}{3} - \binom{n}{3} = 66 \]

Step 3: Detailed Explanation:

Using Pascal's Identity \(\binom{n}{k} + \binom{n}{k-1} = \binom{n+1}{k}\), we can rewrite the left side.
\[ \binom{n+1}{3} - \binom{n}{3} = \binom{n}{2} \]
So, we have a simplified equation.
\[ \binom{n}{2} = 66 \]
Expand the combination formula for further solving.
\[ \frac{n(n-1)}{2} = 66 \implies n(n-1) = 132 \]
We need to find two consecutive positive integers whose product is 132.

Since \(11 \times 12 = 132\), we can determine that \(n = 12\).

The prime factorization of \(12\) is \(2^2 \times 3\).

The distinct prime divisors of \(12\) are \(2\) and \(3\).

The sum of these distinct prime divisors is \(2 + 3 = 5\).


Step 4: Final Answer:

The sum of all distinct prime divisors of \(n\) is \(5\).
Quick Tip: Pascal's identity \(\binom{n+1}{r} - \binom{n}{r} = \binom{n}{r-1}\) simplifies consecutive binomial terms instantly.
Always look for this pattern to avoid expanding factorials manually.

Step 1: Understanding the Concept:

Let's simplify the points system by dividing everything by 5.

So, Heads (H) gives 2 points, and Tails (T) gives 1 point.

We now need the probability of obtaining exactly 6 points in this scaled system.


Step 2: Key Formula or Approach:

The man can reach exactly 6 points in multiple sequences of coin tosses.

Let's list all possible combinations of Heads and Tails that sum to exactly 6 points.

- \(0\) Heads, \(6\) Tails: \(0(2) + 6(1) = 6\)

- \(1\) Head, \(4\) Tails: \(1(2) + 4(1) = 6\)

- \(2\) Heads, \(2\) Tails: \(2(2) + 2(1) = 6\)

- \(3\) Heads, \(0\) Tails: \(3(2) + 0(1) = 6\)


Step 3: Detailed Explanation:

Now, we calculate the probability for each valid combination using permutations.

Case 1: 6 Tails in a total of 6 tosses.

The number of ways is \(\frac{6!}{6!} = 1\).

The probability is \(1 \times \left(\frac{1}{2}\right)^6 = \frac{1}{64}\).


Case 2: 1 Head, 4 Tails in a total of 5 tosses.

The number of ways is \(\frac{5!}{1! 4!} = 5\).

The probability is \(5 \times \left(\frac{1}{2}\right)^5 = \frac{5}{32} = \frac{10}{64}\).


Case 3: 2 Heads, 2 Tails in a total of 4 tosses.

The number of ways is \(\frac{4!}{2! 2!} = 6\).

The probability is \(6 \times \left(\frac{1}{2}\right)^4 = \frac{6}{16} = \frac{24}{64}\).


Case 4: 3 Heads, 0 Tails in a total of 3 tosses.

The number of ways is \(\frac{3!}{3!} = 1\).

The probability is \(1 \times \left(\frac{1}{2}\right)^3 = \frac{1}{8} = \frac{8}{64}\).


The total probability of scoring exactly 30 points (or scaled 6 points) is the sum of these probabilities.
\[ P = \frac{1}{64} + \frac{10}{64} + \frac{24}{64} + \frac{8}{64} = \frac{43}{64} \]
We are given that this probability is \(\frac{m}{n} = \frac{43}{64}\).

Since 43 is a prime number, \(\gcd(43, 64) = 1\) is satisfied.

Therefore, we deduce that \(m = 43\) and \(n = 64\).

Finally, we calculate \(m + n = 43 + 64 = 107\).


Step 4: Final Answer:

The value of \(m + n\) is \(107\).
Quick Tip: To simplify probability state-space problems like this, always scale down the values by dividing by their GCD.
It makes listing the valid combinations much easier and significantly less prone to arithmetic errors.

Step 1: Understanding the Concept:

For \(n\) observations \(x_1, x_2, \dots, x_n\), the mean is \(\bar{x} = \frac{\sum x_i}{n}\) and the variance is \(\sigma^2 = \frac{\sum x_i^2}{n} - \bar{x}^2\).

We are given partial sums for the first \((n-1)\) terms, which allows us to find expressions for the \(n\)-th term \(x_n\) and its square \(x_n^2\).


Step 2: Key Formula or Approach:

Using the given mean \(\bar{x} = 8\), we get \(\sum_{i=1}^n x_i = 8n\).

Using the given variance \(\sigma^2 = 16\), we get \(\frac{\sum_{i=1}^n x_i^2}{n} - 8^2 = 16\).

This simplifies to \(\sum_{i=1}^n x_i^2 = 80n\).


Step 3: Detailed Explanation:

We are given the sum of the first \((n-1)\) observations.
\[ \sum_{i=1}^{n-1} x_i = 48 \]
This allows us to write the \(n\)-th observation in terms of \(n\).
\[ x_n = \sum_{i=1}^n x_i - \sum_{i=1}^{n-1} x_i = 8n - 48 \]
We are also given the sum of squares of the first \((n-1)\) observations.
\[ \sum_{i=1}^{n-1} x_i^2 = 496 \]
This allows us to write the square of the \(n\)-th observation.
\[ x_n^2 = \sum_{i=1}^n x_i^2 - \sum_{i=1}^{n-1} x_i^2 = 80n - 496 \]
Equating the square of \(x_n\) from the first relation to the second relation gives a quadratic equation.
\[ (8n - 48)^2 = 80n - 496 \]
Factor out constants to heavily simplify the calculation.
\[ [8(n - 6)]^2 = 16(5n - 31) \] \[ 64(n^2 - 12n + 36) = 16(5n - 31) \]
Divide both sides by 16.
\[ 4(n^2 - 12n + 36) = 5n - 31 \] \[ 4n^2 - 48n + 144 - 5n + 31 = 0 \] \[ 4n^2 - 53n + 175 = 0 \]
Now, solve this quadratic equation for \(n\).
\[ n = \frac{53 \pm \sqrt{53^2 - 4(4)(175)}}{2(4)} \] \[ n = \frac{53 \pm \sqrt{2809 - 2800}}{8} = \frac{53 \pm \sqrt{9}}{8} = \frac{53 \pm 3}{8} \]
This yields two mathematical possibilities.
\(n = \frac{56}{8} = 7\) or \(n = \frac{50}{8} = 6.25\).

Since the number of observations \(n\) must be a positive integer, we conclude that \(n = 7\).


Step 4: Final Answer:

The value of \(n\) is \(7\).
Quick Tip: Always simplify large algebraic equations by factoring out common terms before expanding squares.
It drastically reduces calculation errors when applying the quadratic formula.

Step 1: Understanding the Concept:

Two lines are geometrically perpendicular if the product of their slopes is \(-1\).

After finding \(k\), we can find the intersection of the two lines to determine the center of the circle \(C\).

The distance from the center \(C\) to the origin \((0,0)\) directly gives the radius \(r\).

Using the perpendicular distance \(d\) from \(C\) to the given secant line, we can find the length of the chord \(AB\).


Step 2: Key Formula or Approach:

First, identify the slopes of the given lines.

For \(L_1: x + (k - 1)y + 3 = 0\), the slope is \(m_1 = \frac{-1}{k-1}\).

For \(L_2: 2x + k^2y - 4 = 0\), the slope is \(m_2 = \frac{-2}{k^2}\).

The condition for perpendicular lines is \(m_1 m_2 = -1\).

The length of a chord is given by Pythagoras theorem: \(AB = 2\sqrt{r^2 - d^2}\).


Step 3: Detailed Explanation:

Apply the perpendicularity condition.
\[ \left(\frac{-1}{k-1}\right) \left(\frac{-2}{k^2}\right) = -1 \] \[ \frac{2}{k^2(k-1)} = -1 \implies k^3 - k^2 + 2 = 0 \]
By inspection of small integers, \(k = -1\) satisfies the equation since \((-1)^3 - (-1)^2 + 2 = -1 - 1 + 2 = 0\).

Substitute \(k = -1\) back into the equations of the lines to get their concrete forms.
\(L_1: x - 2y + 3 = 0\)
\(L_2: 2x + y - 4 = 0 \implies y = 4 - 2x\)

Substitute \(y\) from \(L_2\) into \(L_1\) to find the intersection.
\[ x - 2(4 - 2x) + 3 = 0 \implies x - 8 + 4x + 3 = 0 \implies 5x = 5 \implies x = 1 \] \[ y = 4 - 2(1) = 2 \]
So, the center of the circle is \(C(1, 2)\).

Since the circle passes through the origin \((0,0)\), the radius squared is the squared distance to the origin.
\[ r^2 = (1 - 0)^2 + (2 - 0)^2 = 1 + 4 = 5 \]
Now, calculate the perpendicular distance \(d\) from the center \(C(1, 2)\) to the line \(x - y + 2 = 0\).
\[ d = \frac{|1(1) - 1(2) + 2|}{\sqrt{1^2 + (-1)^2}} = \frac{|1|}{\sqrt{2}} = \frac{1}{\sqrt{2}} \]
The length of the chord squared can now be computed.
\[ (AB)^2 = (2\sqrt{r^2 - d^2})^2 = 4(r^2 - d^2) \]
Substitute the values \(r^2 = 5\) and \(d^2 = \frac{1}{2}\) into the expression.
\[ (AB)^2 = 4\left(5 - \frac{1}{2}\right) = 4\left(\frac{9}{2}\right) = 18 \]

Step 4: Final Answer:

The value of \((AB)^2\) is \(18\).
Quick Tip: The length of a chord intersected by a line on a circle is always \(2\sqrt{r^2 - d^2}\).
You do not need to find the actual intersection points \(A\) and \(B\); just use the center's perpendicular distance \(d\) and the circle's radius \(r\).

Step 1: Understanding the Concept:

For a conic section, the equation of a chord with a given midpoint \((x_1, y_1)\) is given by \(T = S_1\).

After finding the equation of the line \(PQ\), we can solve it simultaneously with the curve \(xy = 12\) to find the exact coordinates of points \(P\) and \(Q\).

Finally, we use the determinant method to find the area of triangle \(OPQ\).


Step 2: Key Formula or Approach:

The equation of the hyperbola is \(S \equiv xy - 12 = 0\).

The midpoint is \((x_1, y_1) = \left(\frac{1}{2}, -\frac{1}{2}\right)\).

The chord formula is \(T = S_1\), which translates to \(\frac{xy_1 + yx_1}{2} - 12 = x_1y_1 - 12\).


Step 3: Detailed Explanation:

Substitute the coordinates of the midpoint into the chord formula.
\[ \frac{x(-1/2) + y(1/2)}{2} = (1/2)(-1/2) \] \[ \frac{-x + y}{4} = \frac{-1}{4} \]
Multiply by 4 to get the equation of the line \(PQ\).
\[ -x + y = -1 \implies y = x - 1 \]
Now, substitute \(y = x - 1\) into the hyperbola equation \(xy = 12\) to find points \(P\) and \(Q\).
\[ x(x - 1) = 12 \] \[ x^2 - x - 12 = 0 \]
Factor the quadratic equation.
\[ (x - 4)(x + 3) = 0 \]
This gives the \(x\)-coordinates of \(P\) and \(Q\) as \(x = 4\) and \(x = -3\).

Using \(y = x - 1\), the corresponding \(y\)-coordinates are \(y = 3\) and \(y = -4\).

So, the points are \(P(4, 3)\) and \(Q(-3, -4)\).

The area of triangle \(OPQ\) with the origin \(O(0,0)\) is given by the formula \(\frac{1}{2}|x_1y_2 - x_2y_1|\).
\[ Area = \frac{1}{2} |(4)(-4) - (-3)(3)| \] \[ Area = \frac{1}{2} |-16 + 9| \] \[ Area = \frac{1}{2} |-7| = \frac{7}{2} \]

Step 4: Final Answer:

The area of the triangle \(OPQ\) equals \(\frac{7}{2}\).
Quick Tip: The chord formula \(T = S_1\) is a massive time-saver for any conic section when the midpoint of a chord is known.
For \(xy = c^2\), \(T\) is simply \(\frac{x y_1 + y x_1}{2}\).

Step 1: Understanding the Concept:

The vertex of a parabola given by \(y = ax^2 + bx + c\) is located at \(x = \frac{-b}{2a}\).

The distance from the vertex to the \(x\)-axis is simply the absolute value of the \(y\)-coordinate of the vertex.

We are given that the parabola passes through a specific point, which gives us a relation between \(p\) and \(q\).


Step 2: Key Formula or Approach:

For the parabola \(y = x^2 + px + q\), the \(x\)-coordinate of the vertex is \(x_v = \frac{-p}{2}\).

The \(y\)-coordinate of the vertex is \(y_v = \left(\frac{-p}{2}\right)^2 + p\left(\frac{-p}{2}\right) + q = \frac{-p^2}{4} + q\).

The condition that the parabola passes through \((1, -1)\) means \(-1 = 1^2 + p(1) + q\).


Step 3: Detailed Explanation:

From the passing point condition, we can express \(q\) in terms of \(p\).
\[ -1 = 1 + p + q \implies p + q = -2 \implies q = -2 - p \]
Substitute this expression for \(q\) into the \(y\)-coordinate of the vertex.
\[ y_v = \frac{-p^2}{4} + (-2 - p) = -\left(\frac{p^2}{4} + p + 2\right) \]
The distance from the vertex to the \(x\)-axis is \(D = |y_v|\).
\[ D = \left| \frac{p^2}{4} + p + 2 \right| \]
To minimize this distance, we complete the square for the quadratic expression inside the absolute value.
\[ D = \frac{1}{4} \left| p^2 + 4p + 8 \right| \] \[ D = \frac{1}{4} \left| (p^2 + 4p + 4) + 4 \right| \] \[ D = \frac{1}{4} \left| (p + 2)^2 + 4 \right| \]
Since \((p + 2)^2 \ge 0\), the minimum value of the expression inside the absolute value occurs when \((p + 2)^2 = 0\).

This happens when \(p = -2\).

With \(p = -2\), we find the corresponding value of \(q\).
\[ q = -2 - (-2) = 0 \]
We need to find the value of \(p^2 + q^2\).
\[ p^2 + q^2 = (-2)^2 + 0^2 = 4 + 0 = 4 \]

Step 4: Final Answer:

The value of \(p^2 + q^2\) is \(4\).
Quick Tip: To find the minimum or maximum of a quadratic function quickly, completing the square is often faster and less prone to sign errors than using calculus (derivatives).

Step 1: Understanding the Concept:

We are dealing with a trigonometric identity that needs simplification.

After simplifying the expression \(2(\cos^8\theta - \sin^8\theta)\sec 2\theta\), we will find its range of values over the allowed domain of \(\theta\).

Since \(a\) must be an integer, \(a^2\) must be a perfect square integer that falls within this range.


Step 2: Key Formula or Approach:

We use the difference of squares factorization: \(x^4 - y^4 = (x^2 - y^2)(x^2 + y^2)\).

Apply this to \(\cos^8\theta - \sin^8\theta = (\cos^4\theta - \sin^4\theta)(\cos^4\theta + \sin^4\theta)\).

Further, \(\cos^4\theta - \sin^4\theta = (\cos^2\theta - \sin^2\theta)(\cos^2\theta + \sin^2\theta)\).

Recall that \(\cos^2\theta + \sin^2\theta = 1\) and \(\cos^2\theta - \sin^2\theta = \cos 2\theta\).


Step 3: Detailed Explanation:

Let's simplify the given expression for \(a^2\).
\[ a^2 = 2(\cos^8\theta - \sin^8\theta)\sec 2\theta \] \[ a^2 = 2(\cos^4\theta - \sin^4\theta)(\cos^4\theta + \sin^4\theta) \left(\frac{1}{\cos 2\theta}\right) \] \[ a^2 = 2(\cos^2\theta - \sin^2\theta)(\cos^2\theta + \sin^2\theta)(\cos^4\theta + \sin^4\theta) \left(\frac{1}{\cos 2\theta}\right) \]
Substitute the basic identities.
\[ a^2 = 2(\cos 2\theta)(1)(\cos^4\theta + \sin^4\theta) \left(\frac{1}{\cos 2\theta}\right) \]
Since \(\theta \in P \implies \tan^2\theta \neq 1 \implies \cos 2\theta \neq 0\), we can safely cancel \(\cos 2\theta\).
\[ a^2 = 2(\cos^4\theta + \sin^4\theta) \]
We can rewrite \(\cos^4\theta + \sin^4\theta\) by completing the square.
\[ \cos^4\theta + \sin^4\theta = (\cos^2\theta + \sin^2\theta)^2 - 2\sin^2\theta \cos^2\theta = 1 - \frac{1}{2}(4\sin^2\theta \cos^2\theta) = 1 - \frac{1}{2}\sin^2 2\theta \]
Substitute this back into the equation for \(a^2\).
\[ a^2 = 2\left(1 - \frac{1}{2}\sin^2 2\theta\right) = 2 - \sin^2 2\theta \]
Now we determine the range of this expression.

We know that \(0 \le \sin^2 2\theta \le 1\).

However, the domain condition \(\tan^2\theta \neq 1\) means \(\theta \neq \frac{\pi}{4}, \frac{3\pi}{4}, \dots\), which implies \(2\theta \neq \frac{\pi}{2}, \frac{3\pi}{2}, \dots\).

Therefore, \(\sin^2 2\theta\) can never be exactly equal to \(1\).

So the range of \(\sin^2 2\theta\) is \([0, 1)\).

This means the range of \(a^2\) is \(2 - [0, 1) = (1, 2]\).

We need \(a \in \mathbb{Z}\), which means \(a^2\) must be a perfect square integer.

The only integer in the interval \((1, 2]\) is \(2\).

If \(a^2 = 2\), then \(a = \pm \sqrt{2}\), which are not integers.

Since there is no integer \(a\) that satisfies the equation, the set \(S\) is empty.


Step 4: Final Answer:

The number of elements \(n(S)\) is \(0\).
Quick Tip: Always strictly evaluate domain constraints.
Here, \(\tan^2\theta \neq 1\) subtly removes the possibility of \(a^2 = 1\), restricting the interval to \((1, 2]\) and leaving no perfect squares available.

Step 1: Understanding the Concept:

We are given \(\vec{c}\) as a linear combination of two known vectors \(\vec{a}\) and \(\vec{b}\).

By applying the distributive property of the dot product, we can generate a system of two linear equations in terms of the unknown scalars \(\lambda\) and \(\mu\).

Once \(\lambda\) and \(\mu\) are found, we can construct \(\vec{c}\) and compute its squared magnitude.


Step 2: Key Formula or Approach:

The dot product distributes over vector addition: \((\lambda \vec{a} + \mu \vec{b}) \cdot \vec{v} = \lambda(\vec{a} \cdot \vec{v}) + \mu(\vec{b} \cdot \vec{v})\).

Let \(\vec{v}_1 = 3\hat{i} - 6\hat{j} + 2\hat{k}\) and \(\vec{v}_2 = \hat{i} + \hat{j} + \hat{k}\).

The given conditions are \(\vec{c} \cdot \vec{v}_1 = 10\) and \(\vec{c} \cdot \vec{v}_2 = -2\).


Step 3: Detailed Explanation:

First, calculate the dot products of \(\vec{a}\) and \(\vec{b}\) with \(\vec{v}_1\).
\[ \vec{a} \cdot \vec{v}_1 = (-1)(3) + (1)(-6) + (3)(2) = -3 - 6 + 6 = -3 \] \[ \vec{b} \cdot \vec{v}_1 = (1)(3) + (3)(-6) + (1)(2) = 3 - 18 + 2 = -13 \]
Substitute these into the first condition.
\[ \lambda(-3) + \mu(-13) = 10 \implies -3\lambda - 13\mu = 10 \quad --- (Eq 1) \]
Next, calculate the dot products of \(\vec{a}\) and \(\vec{b}\) with \(\vec{v}_2\).
\[ \vec{a} \cdot \vec{v}_2 = (-1)(1) + (1)(1) + (3)(1) = -1 + 1 + 3 = 3 \] \[ \vec{b} \cdot \vec{v}_2 = (1)(1) + (3)(1) + (1)(1) = 1 + 3 + 1 = 5 \]
Substitute these into the second condition.
\[ \lambda(3) + \mu(5) = -2 \implies 3\lambda + 5\mu = -2 \quad --- (Eq 2) \]
Now, solve the linear system by adding (Eq 1) and (Eq 2).
\[ (-3\lambda - 13\mu) + (3\lambda + 5\mu) = 10 - 2 \] \[ -8\mu = 8 \implies \mu = -1 \]
Substitute \(\mu = -1\) into (Eq 2) to find \(\lambda\).
\[ 3\lambda + 5(-1) = -2 \implies 3\lambda = 3 \implies \lambda = 1 \]
Now that we have \(\lambda\) and \(\mu\), we construct the vector \(\vec{c}\).
\[ \vec{c} = (1)\vec{a} + (-1)\vec{b} = \vec{a} - \vec{b} \] \[ \vec{c} = (-\hat{i} + \hat{j} + 3\hat{k}) - (\hat{i} + 3\hat{j} + \hat{k}) = -2\hat{i} - 2\hat{j} + 2\hat{k} \]
Finally, calculate the squared magnitude \(|\vec{c}|^2\).
\[ |\vec{c}|^2 = (-2)^2 + (-2)^2 + 2^2 = 4 + 4 + 4 = 12 \]

Step 4: Final Answer:

The value of \(|\vec{c}|^2\) is \(12\).
Quick Tip: Pre-calculating the dot products of the basis vectors (\(\vec{a}\) and \(\vec{b}\)) with the target vectors prevents messy algebra and reduces the problem to a simple \(2 \times 2\) linear system.

Step 1: Understanding the Concept:

We are given a point \(P\) and the midpoint of the segment connecting \(P\) to its foot of perpendicular \(A\) on a line.

Using the midpoint formula, we can express the coordinates of \(A\) in terms of the unknowns \(a\) and \(b\).

Since \(A\) is the foot of the perpendicular, it must lie on the given line, and the vector \(\vec{PA}\) must be orthogonal to the line's direction vector.


Step 2: Key Formula or Approach:

The midpoint \(M\) of points \(P(x_1, y_1, z_1)\) and \(A(x_2, y_2, z_2)\) is \(\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2}\right)\).

The direction vector of the line is \(\vec{d} = 2\hat{i} + \hat{j} + 3\hat{k}\).

The orthogonality condition is \(\vec{PA} \cdot \vec{d} = 0\).


Step 3: Detailed Explanation:

Let the coordinates of point \(A\) be \((x_2, y_2, z_2)\).

Using the given midpoint \(\left(0, \frac{3}{4}, -\frac{1}{4}\right)\), we can set up equations.
\[ \frac{a + x_2}{2} = 0 \implies x_2 = -a \] \[ \frac{b + y_2}{2} = \frac{3}{4} \implies y_2 = \frac{3}{2} - b \] \[ \frac{0 + z_2}{2} = -\frac{1}{4} \implies z_2 = -\frac{1}{2} \]
So, point \(A\) is \(\left(-a, \frac{3}{2} - b, -\frac{1}{2}\right)\).

Since \(A\) lies on the line \(\frac{x - 1}{2} = \frac{y - 2}{1} = \frac{z - \alpha}{3}\), we substitute its coordinates.
\[ \frac{-a - 1}{2} = \frac{\frac{3}{2} - b - 2}{1} = \frac{-\frac{1}{2} - \alpha}{3} = k \]
Simplify the middle term.
\[ \frac{-a - 1}{2} = -\frac{1}{2} - b \implies -a - 1 = -1 - 2b \implies a = 2b \]
Now, form the vector \(\vec{PA}\).
\[ \vec{PA} = (-a - a)\hat{i} + \left(\frac{3}{2} - b - b\right)\hat{j} + \left(-\frac{1}{2} - 0\right)\hat{k} = -2a\hat{i} + \left(\frac{3}{2} - 2b\right)\hat{j} - \frac{1}{2}\hat{k} \]
Apply the orthogonality condition \(\vec{PA} \cdot \vec{d} = 0\).
\[ 2(-2a) + 1\left(\frac{3}{2} - 2b\right) + 3\left(-\frac{1}{2}\right) = 0 \] \[ -4a + \frac{3}{2} - 2b - \frac{3}{2} = 0 \] \[ -4a - 2b = 0 \implies 2a + b = 0 \implies b = -2a \]
We now have a system: \(a = 2b\) and \(b = -2a\).

Substitute \(b\) into the first equation: \(a = 2(-2a) \implies a = -4a \implies 5a = 0 \implies a = 0\).

Since \(a = 0\), it follows that \(b = 0\).

Now substitute \(a\) and \(b\) back into the line equation to find \(\alpha\).
\[ \frac{0 - 1}{2} = \frac{-\frac{1}{2} - \alpha}{3} \] \[ -\frac{1}{2} = \frac{-1 - 2\alpha}{6} \] \[ -3 = -1 - 2\alpha \implies 2\alpha = 2 \implies \alpha = 1 \]
Finally, compute the required sum.
\[ a^2 + b^2 + \alpha^2 = 0^2 + 0^2 + 1^2 = 1 \]

Step 4: Final Answer:

The value of \(a^2 + b^2 + \alpha^2\) is \(1\).
Quick Tip: When a point lies on a line, equating the parametric fractions immediately gives you relations between variables.
Always use the orthogonality condition (\(\vec{PA} \cdot \vec{d} = 0\)) to generate the final required equation.

Step 1: Understanding the Concept:

We first need to find the initial angle between the adjacent sides \(\vec{PQ}\) and \(\vec{PS}\) using the dot product.

The problem states that side PS is rotated by an acute angle \(\alpha\) until it becomes completely perpendicular to PQ.

This implies the new angle between the vectors is \(90^\circ\).


Step 2: Key Formula or Approach:

The angle \(\theta\) between two vectors \(\vec{u}\) and \(\vec{v}\) is given by \(\cos\theta = \frac{\vec{u} \cdot \vec{v}}{|\vec{u}| |\vec{v}|}\).

The trigonometric identity \(\sin^2 A - \sin^2 B = \sin(A - B)\sin(A + B)\) will be used to simplify the final expression.


Step 3: Detailed Explanation:

Let's calculate the initial angle \(\theta\) between \(\vec{PQ}\) and \(\vec{PS}\).
\[ \vec{PQ} = 0\hat{i} + 1\hat{j} + 1\hat{k} \implies |\vec{PQ}| = \sqrt{0^2 + 1^2 + 1^2} = \sqrt{2} \] \[ \vec{PS} = 1\hat{i} - 1\hat{j} + 0\hat{k} \implies |\vec{PS}| = \sqrt{1^2 + (-1)^2 + 0^2} = \sqrt{2} \] \[ \cos\theta = \frac{(0)(1) + (1)(-1) + (1)(0)}{\sqrt{2}\sqrt{2}} = \frac{-1}{2} \]
Since \(\cos\theta = -1/2\), the initial angle is \(\theta = 120^\circ\).

The vector is rotated by an acute angle \(\alpha\) to make the new angle \(90^\circ\).

The new angle is \(\theta \pm \alpha = 90^\circ\).

So, \(120^\circ \pm \alpha = 90^\circ\).

Since \(\alpha\) must be an acute angle (\(\alpha < 90^\circ\)), we take \(\alpha = 120^\circ - 90^\circ = 30^\circ\).

Now we evaluate the required trigonometric expression.
\[ E = \sin^2\left(\frac{5\alpha}{2}\right) - \sin^2\left(\frac{\alpha}{2}\right) \]
Substitute \(\alpha = 30^\circ\).
\[ E = \sin^2(75^\circ) - \sin^2(15^\circ) \]
Apply the identity \(\sin^2 A - \sin^2 B = \sin(A - B)\sin(A + B)\) with \(A = 75^\circ\) and \(B = 15^\circ\).
\[ E = \sin(75^\circ - 15^\circ)\sin(75^\circ + 15^\circ) \] \[ E = \sin(60^\circ)\sin(90^\circ) \]
We know that \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\) and \(\sin(90^\circ) = 1\).
\[ E = \left(\frac{\sqrt{3}}{2}\right)(1) = \frac{\sqrt{3}}{2} \]

Step 4: Final Answer:

The value of the expression is \(\frac{\sqrt{3}}{2}\).
Quick Tip: Whenever you see a difference of squares of sine functions, immediately use the identity \(\sin^2 A - \sin^2 B = \sin(A-B)\sin(A+B)\).
It turns complex fractional angles into standard, easy-to-evaluate angles.

Step 1: Understanding the Concept:

The integrand \(f(x) = \sin^4 x + \cos^4 x\) is a periodic function.

We need to determine its fundamental period and use the property of definite integrals for periodic functions: \(\int_0^{nT} f(x) dx = n \int_0^T f(x) dx\).


Step 2: Key Formula or Approach:

The period of both \(\sin^4 x\) and \(\cos^4 x\) is \(\pi\), but their sum actually has a shorter period.

Observe that \(f(x + \pi/2) = \sin^4(x + \pi/2) + \cos^4(x + \pi/2) = \cos^4 x + \sin^4 x = f(x)\).

Thus, the fundamental period \(T\) is \(\frac{\pi}{2}\).

Wallis' formula for integrals of the form \(\int_0^{\pi/2} \sin^n x dx = \int_0^{\pi/2} \cos^n x dx = \frac{(n-1)(n-3)\dots}{n(n-2)\dots} \cdot \frac{\pi}{2}\) (for even \(n\)).


Step 3: Detailed Explanation:

Since the period is \(T = \frac{\pi}{2}\), we can rewrite the upper limit \(20\pi\) as \(40 \times \frac{\pi}{2}\).
\[ I = \int_0^{40 \cdot \frac{\pi}{2}} (\sin^4 x + \cos^4 x) dx = 40 \int_0^{\pi/2} (\sin^4 x + \cos^4 x) dx \]
Using Wallis' formula for \(n=4\), we evaluate the integrals of \(\sin^4 x\) and \(\cos^4 x\) separately.
\[ \int_0^{\pi/2} \sin^4 x dx = \frac{3 \cdot 1}{4 \cdot 2} \cdot \frac{\pi}{2} = \frac{3\pi}{16} \]
Similarly, by symmetry:
\[ \int_0^{\pi/2} \cos^4 x dx = \frac{3\pi}{16} \]
Now, substitute these back into the total integral.
\[ I = 40 \left( \frac{3\pi}{16} + \frac{3\pi}{16} \right) = 40 \left( \frac{6\pi}{16} \right) \]
Simplify the fraction.
\[ I = 40 \left( \frac{3\pi}{8} \right) = 5 \cdot 3\pi = 15\pi \]

Step 4: Final Answer:

The value of the integral is \(15\pi\).
Quick Tip: Always check for the smallest period of trigonometric sums before integrating.
Using Wallis' integrals \(\int_0^{\pi/2} \sin^n x dx\) is the fastest way to evaluate powers of sine and cosine over \([0, \pi/2]\).

Step 1: Understanding the Concept:

The limit condition tells us about the lowest degree terms of the polynomial.

For the limit \(\lim_{x \to 0} \frac{f(x)}{x^3}\) to exist and equal a non-zero finite value, the polynomial \(f(x)\) must not have any \(x^0\), \(x^1\), or \(x^2\) terms.

The extrema conditions tell us that the derivative \(f'(x)\) evaluates to zero at specific points.


Step 2: Key Formula or Approach:

Let the 5th degree polynomial be \(f(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + f\).

Due to the limit condition \(\lim_{x \to 0} \frac{f(x)}{x^3} = -5\), we must have \(f = 0\), \(e = 0\), \(d = 0\), and \(c = -5\).

So, \(f(x) = ax^5 + bx^4 - 5x^3\).

For extrema at \(x = \pm 1\), we use \(f'(1) = 0\) and \(f'(-1) = 0\).


Step 3: Detailed Explanation:

Find the first derivative of the simplified polynomial.
\[ f'(x) = 5ax^4 + 4bx^3 - 15x^2 \]
Apply the condition \(f'(1) = 0\).
\[ 5a(1)^4 + 4b(1)^3 - 15(1)^2 = 0 \implies 5a + 4b - 15 = 0 \quad --- (Eq 1) \]
Apply the condition \(f'(-1) = 0\).
\[ 5a(-1)^4 + 4b(-1)^3 - 15(-1)^2 = 0 \implies 5a - 4b - 15 = 0 \quad --- (Eq 2) \]
Add (Eq 1) and (Eq 2) to eliminate \(b\).
\[ 10a - 30 = 0 \implies a = 3 \]
Subtract (Eq 2) from (Eq 1) to find \(b\).
\[ 8b = 0 \implies b = 0 \]
Now we have the full equation for the polynomial.
\[ f(x) = 3x^5 - 5x^3 \]
We need to calculate \(f(2) - f(-2)\).

First, calculate \(f(2)\).
\[ f(2) = 3(2)^5 - 5(2)^3 = 3(32) - 5(8) = 96 - 40 = 56 \]
Since \(f(x)\) only has odd powers, it is an odd function, meaning \(f(-x) = -f(x)\).

So, \(f(-2) = -f(2) = -56\).

Finally, compute the difference.
\[ f(2) - f(-2) = 56 - (-56) = 112 \]

Step 4: Final Answer:

The value of \(f(2) - f(-2)\) is \(112\).
Quick Tip: A limit of the form \(\lim_{x\to 0} \frac{f(x)}{x^n} = c\) acts as a sieve.
It immediately tells you that all terms in \(f(x)\) with degree less than \(n\) are zero, and the coefficient of \(x^n\) is \(c\).
Also, recognizing odd functions (\(f(-x) = -f(x)\)) saves calculation time.

Step 1: Understanding the Concept:

We are given an indefinite integral of a rational function.

The denominator can be factored, meaning we can use the method of partial fractions to integrate it.

After finding the general antiderivative \(f(x)\), we will use the initial condition \(f(4)\) to find the constant of integration \(C\).

Finally, we evaluate \(f(7)\) and match it to the given logarithmic form to find \(\alpha\) and \(\beta\).


Step 2: Key Formula or Approach:

Factor the denominator: \(x^2 + 2x - 15 = (x + 5)(x - 3)\).

Set up the partial fraction decomposition.
\[ \frac{16x + 24}{(x + 5)(x - 3)} = \frac{A}{x + 5} + \frac{B}{x - 3} \]
Use the cover-up method to find constants \(A\) and \(B\).


Step 3: Detailed Explanation:

To find \(A\), cover \((x+5)\) and substitute \(x = -5\).
\[ A = \frac{16(-5) + 24}{-5 - 3} = \frac{-80 + 24}{-8} = \frac{-56}{-8} = 7 \]
To find \(B\), cover \((x-3)\) and substitute \(x = 3\).
\[ B = \frac{16(3) + 24}{3 + 5} = \frac{48 + 24}{8} = \frac{72}{8} = 9 \]
Now integrate the separated fractions.
\[ f(x) = \int \left( \frac{7}{x + 5} + \frac{9}{x - 3} \right) dx = 7\ln|x + 5| + 9\ln|x - 3| + C \]
Use the condition \(f(4) = 14\ln(3)\) to find \(C\).
\[ f(4) = 7\ln|4 + 5| + 9\ln|4 - 3| + C \] \[ f(4) = 7\ln(9) + 9\ln(1) + C \]
Since \(\ln(9) = \ln(3^2) = 2\ln(3)\) and \(\ln(1) = 0\):
\[ f(4) = 7(2\ln 3) + 0 + C = 14\ln 3 + C \]
Given \(f(4) = 14\ln 3\), we conclude that \(C = 0\).

So the function is \(f(x) = 7\ln|x + 5| + 9\ln|x - 3|\).

Now, evaluate \(f(7)\).
\[ f(7) = 7\ln(12) + 9\ln(4) \]
Express everything in terms of prime bases 2 and 3.
\[ \ln(12) = \ln(2^2 \cdot 3) = 2\ln 2 + \ln 3 \] \[ \ln(4) = \ln(2^2) = 2\ln 2 \]
Substitute these back.
\[ f(7) = 7(2\ln 2 + \ln 3) + 9(2\ln 2) \] \[ f(7) = 14\ln 2 + 7\ln 3 + 18\ln 2 = 32\ln 2 + 7\ln 3 \]
Combine into a single logarithm using logarithm power rules.
\[ f(7) = \ln(2^{32}) + \ln(3^7) = \ln(2^{32} \cdot 3^7) \]
Comparing this to \(\ln(2^\alpha \cdot 3^\beta)\), we get \(\alpha = 32\) and \(\beta = 7\).

The requested sum is \(\alpha + \beta = 32 + 7 = 39\).


Step 4: Final Answer:

The value of \(\alpha + \beta\) is \(39\).
Quick Tip: The "cover-up" method is the fastest way to decompose proper rational functions into partial fractions with distinct linear factors.
It bypasses the need to set up and solve a system of equations.

Step 1: Understanding the Concept:

We are given a first-order non-linear differential equation.

By rearranging the terms and dividing by \(y^2\), we can check if it is a homogeneous differential equation.

Once verified, the standard substitution \(x = vy\) will transform it into a separable differential equation.


Step 2: Key Formula or Approach:

Rewrite the equation by dividing entirely by \(y^2\).
\[ 2 \frac{dx}{dy} - 2\frac{x}{y} + \left(\frac{x}{y}\right)^2 = 0 \]
This is clearly a homogeneous equation of degree zero.

Let \(x = vy\), which implies \(\frac{dx}{dy} = v + y\frac{dv}{dy}\).


Step 3: Detailed Explanation:

Substitute \(x/y = v\) and \(\frac{dx}{dy}\) into the modified equation.
\[ 2\left(v + y\frac{dv}{dy}\right) - 2v + v^2 = 0 \]
Distribute the 2.
\[ 2v + 2y\frac{dv}{dy} - 2v + v^2 = 0 \]
The \(2v\) terms cancel out perfectly.
\[ 2y\frac{dv}{dy} + v^2 = 0 \implies 2y\frac{dv}{dy} = -v^2 \]
Now separate the variables \(v\) and \(y\).
\[ \frac{-2}{v^2} dv = \frac{dy}{y} \]
Integrate both sides.
\[ \int -2v^{-2} dv = \int \frac{1}{y} dy \] \[ \frac{2}{v} = \ln|y| + C \]
Substitute back \(v = \frac{x}{y}\).
\[ \frac{2y}{x} = \ln y + C \quad (since y > 1) \]
Use the initial condition \(x(e) = e\) to find the constant \(C\).
\[ \frac{2e}{e} = \ln(e) + C \implies 2 = 1 + C \implies C = 1 \]
So, the particular solution curve is:
\[ \frac{2y}{x} = \ln y + 1 \]
Rearrange to express \(x\) as a function of \(y\).
\[ x = \frac{2y}{\ln y + 1} \]
Finally, evaluate \(x\) at \(y = e^2\).
\[ x(e^2) = \frac{2e^2}{\ln(e^2) + 1} \] \[ x(e^2) = \frac{2e^2}{2\ln(e) + 1} = \frac{2e^2}{2(1) + 1} = \frac{2e^2}{3} \]

Step 4: Final Answer:

The value of \(x(e^2)\) is \(\frac{2}{3} e^2\).
Quick Tip: Always look for homogeneous structures like \(x/y\) or \(y/x\).
Dividing by the highest power of the independent variable (like \(y^2\) here) makes the substitution incredibly obvious and straightforward.

Step 1: Understanding the Concept:

The relation \(R\) is defined on the Cartesian product \(A \times A\).

An element in \(R\) is a pair of ordered pairs \(((x, y), (z, w))\) such that the specified conditions are met.

The conditions are independent: \(x\) must divide \(z\), and \(y\) must be less than or equal to \(w\).

Since they are independent, we can find the number of valid pairs \((x, z)\) and the number of valid pairs \((y, w)\) separately, and then multiply them.


Step 2: Key Formula or Approach:

Total elements in \(R\) = (Number of pairs \((x, z)\) such that \(x | z\)) \(\times\) (Number of pairs \((y, w)\) such that \(y \le w\)).

We evaluate each condition for the set \(A = \{2, 3, 4, 5, 6\}\).


Step 3: Detailed Explanation:

First, find the number of pairs \((x, z)\) where \(x\) divides \(z\):

If \(x = 2\), \(z \in \{2, 4, 6\}\) (3 pairs).

If \(x = 3\), \(z \in \{3, 6\}\) (2 pairs).

If \(x = 4\), \(z \in \{4\}\) (1 pair).

If \(x = 5\), \(z \in \{5\}\) (1 pair).

If \(x = 6\), \(z \in \{6\}\) (1 pair).

Total valid \((x, z)\) pairs = \(3 + 2 + 1 + 1 + 1 = 8\).

Next, find the number of pairs \((y, w)\) where \(y \le w\):

If \(y = 2\), \(w \in \{2, 3, 4, 5, 6\}\) (5 pairs).

If \(y = 3\), \(w \in \{3, 4, 5, 6\}\) (4 pairs).

If \(y = 4\), \(w \in \{4, 5, 6\}\) (3 pairs).

If \(y = 5\), \(w \in \{5, 6\}\) (2 pairs).

If \(y = 6\), \(w \in \{6\}\) (1 pair).

Total valid \((y, w)\) pairs = \(5 + 4 + 3 + 2 + 1 = 15\).

Finally, the total number of elements in \(R\) is the product of these independent choices.
\[ Total elements = 8 \times 15 = 120 \]

Step 4: Final Answer:

The number of elements in \(R\) is \(120\).
Quick Tip: When a relation on a Cartesian product has independent conditions for each coordinate, you can always count the satisfying pairs for each coordinate separately and multiply the results.

Step 1: Understanding the Concept:

We are given two matrix equations involving an invertible matrix \(A\).

The "sum of the diagonal elements" of a matrix is its Trace, denoted as \(Tr(M)\).

A key property of the trace operator is that it is invariant under cyclic permutations, meaning \(Tr(XY) = Tr(YX)\).


Step 2: Key Formula or Approach:

From \(PA = B\), we get \(P = BA^{-1}\).

From \(AQ = B\), we get \(Q = A^{-1}B\).

We need the trace of \(2(P+Q)\), which is \(2(Tr(P) + Tr(Q))\).

Using the trace property: \(Tr(P) = Tr(BA^{-1}) = Tr(A^{-1}B) = Tr(Q)\).


Step 3: Detailed Explanation:

First, calculate the determinant of \(A\) to find \(A^{-1}\).
\[ |A| = (2)(-2) - (-2)(4) = -4 + 8 = 4 \] \[ A^{-1} = \frac{1}{4} \begin{bmatrix} -2 & 2
-4 & 2 \end{bmatrix} = \begin{bmatrix} -1/2 & 1/2
-1 & 1/2 \end{bmatrix} \]
Now, calculate the matrix \(Q = A^{-1}B\).
\[ Q = \begin{bmatrix} -1/2 & 1/2
-1 & 1/2 \end{bmatrix} \begin{bmatrix} 3 & 9
1 & 3 \end{bmatrix} \]
Compute the diagonal elements of \(Q\) only, since we only need its trace.
\[ Q_{11} = (-1/2)(3) + (1/2)(1) = -3/2 + 1/2 = -1 \] \[ Q_{22} = (-1)(9) + (1/2)(3) = -9 + 3/2 = -7.5 \]
The trace of \(Q\) is:
\[ Tr(Q) = Q_{11} + Q_{22} = -1 - 7.5 = -8.5 \]
By the cyclic property of the trace, \(Tr(P) = Tr(Q) = -8.5\).

Now, find the trace of \(2(P+Q)\).
\[ Tr(2(P+Q)) = 2(Tr(P) + Tr(Q)) \] \[ Tr(2(P+Q)) = 2(-8.5 - 8.5) = 2(-17) = -34 \]
The question asks for the absolute value of this sum.
\[ |-34| = 34 \]

Step 4: Final Answer:

The absolute value of the sum is \(34\).
Quick Tip: Always use the cyclic property of traces (\(Tr(AB) = Tr(BA)\)) to save time.
You don't need to compute matrix \(P\) at all, and you only need to compute the diagonal elements of \(Q\).

Step 1: Understanding the Concept:

Let the given circle be \(S_1: x^2 + y^2 = 36\). Its center is \(O(0,0)\) and radius is \(R = 6\).

Let the variable circle be \(S_2\). Its diameter is \(AB\), so its center \(M\) is the midpoint of \(AB\) and its radius is \(r = MA\).

Since \(S_2\) touches \(S_1\) internally, the distance between their centers must be equal to the difference of their radii.


Step 2: Key Formula or Approach:

Condition for internal touching: \(OM = R - r\).

Here, \(R = 6\) and \(r = MA\), so \(OM = 6 - MA \implies OM + MA = 6\).

This means the sum of distances from \(M\) to two fixed points \(O\) and \(A\) is constant, which is the definition of an ellipse.


Step 3: Detailed Explanation:

The locus of \(M\) is an ellipse with foci at \(O(0,0)\) and \(A(3,0)\).

The length of the major axis of this ellipse is \(2a_M = 6 \implies a_M = 3\).

The distance between the foci is \(2c_M = OA = 3\).

The eccentricity of the locus of \(M\) is:
\[ e_M = \frac{2c_M}{2a_M} = \frac{3}{6} = \frac{1}{2} \]
We are asked to find the eccentricity of the locus of \(B\).

Since \(M\) is the midpoint of \(AB\), we can write \(\vec{M} = \frac{\vec{A} + \vec{B}}{2} \implies \vec{B} = 2\vec{M} - \vec{A}\).

This equation represents a homothety (scaling and translation).

Transformations like scaling and translation do not change the shape or eccentricity of a conic section.

Therefore, the locus of \(B\) is an ellipse with the exact same eccentricity as the locus of \(M\).

So, \(e = \frac{1}{2}\).

We need to calculate the value of \(72e^2\).
\[ 72e^2 = 72 \left(\frac{1}{2}\right)^2 = 72 \left(\frac{1}{4}\right) = 18 \]

Step 4: Final Answer:

The value of \(72e^2\) is \(18\).
Quick Tip: Recognizing the geometric definition of an ellipse (\(PS + PS' = 2a\)) saves you from doing messy algebraic substitutions.
Additionally, remember that any linear transformation (like \(B = 2M - A\)) preserves the eccentricity of conic sections.

Step 1: Understanding the Concept:

We are dealing with the area between a hyperbola and a straight line.

First, we find the points of intersection by solving the two equations simultaneously.

Then, we set up a definite integral of the upper curve minus the lower curve between the intersection points.


Step 2: Key Formula or Approach:

The hyperbola is \(\frac{x^2}{9} - \frac{y^2}{16} = 1\).

The line is \(8x - 3y = 24 \implies y = \frac{8}{3}x - 8\).

Substitute \(y\) from the line into the hyperbola equation to find intersection limits \([x_1, x_2]\).

Area \(A = \int_{x_1}^{x_2} (y_{upper} - y_{lower}) dx\).

Standard integral: \(\int \sqrt{x^2 - a^2} dx = \frac{x}{2}\sqrt{x^2 - a^2} - \frac{a^2}{2}\ln|x + \sqrt{x^2 - a^2}|\).


Step 3: Detailed Explanation:

Find the intersection points:
\[ 16x^2 - 9\left(\frac{8}{3}x - 8\right)^2 = 144 \] \[ 16x^2 - 9\left(\frac{64}{9}x^2 - \frac{128}{3}x + 64\right) = 144 \] \[ 16x^2 - 64x^2 + 384x - 576 = 144 \] \[ -48x^2 + 384x - 720 = 0 \]
Divide by \(-48\):
\[ x^2 - 8x + 15 = 0 \implies (x-3)(x-5) = 0 \]
The intersection points are at \(x = 3\) and \(x = 5\).

In the interval \([3, 5]\), let's check which curve is higher. At \(x=4\):

Line: \(y = \frac{8}{3}(4) - 8 = \frac{32}{3} - \frac{24}{3} = \frac{8}{3} \approx 2.67\)

Hyperbola: \(y = 4\sqrt{\frac{4^2}{9} - 1} = \frac{4\sqrt{7}}{3} \approx \frac{4(2.64)}{3} \approx 3.52\)

Thus, the hyperbola is above the line.
\[ y_{hyp} = 4\sqrt{\frac{x^2}{9} - 1} = \frac{4}{3}\sqrt{x^2 - 9} \] \[ A = \int_{3}^{5} \left( \frac{4}{3}\sqrt{x^2 - 9} - \left(\frac{8}{3}x - 8\right) \right) dx \]
Integrate the linear part first:
\[ \int_{3}^{5} \left(\frac{8}{3}x - 8\right) dx = \left[ \frac{4}{3}x^2 - 8x \right]_{3}^{5} \] \[ = \left( \frac{100}{3} - 40 \right) - \left( \frac{36}{3} - 24 \right) = \left( -\frac{20}{3} \right) - (-12) = 12 - \frac{20}{3} = \frac{16}{3} \]
Now integrate the hyperbola part:
\[ \int_{3}^{5} \frac{4}{3}\sqrt{x^2 - 9} dx = \frac{4}{3} \left[ \frac{x}{2}\sqrt{x^2 - 9} - \frac{9}{2}\ln|x + \sqrt{x^2 - 9}| \right]_{3}^{5} \] \[ = \frac{4}{3} \left[ \left( \frac{5}{2}(4) - \frac{9}{2}\ln(5+4) \right) - \left( 0 - \frac{9}{2}\ln(3) \right) \right] \] \[ = \frac{4}{3} \left[ 10 - \frac{9}{2}\ln 9 + \frac{9}{2}\ln 3 \right] = \frac{4}{3} \left[ 10 - 9\ln 3 + \frac{9}{2}\ln 3 \right] = \frac{4}{3} \left[ 10 - \frac{9}{2}\ln 3 \right] = \frac{40}{3} - 6\ln 3 \]
Subtract the linear part from the hyperbola part to get the area:
\[ A = \left( \frac{40}{3} - 6\ln 3 \right) - \frac{16}{3} = \frac{24}{3} - 6\ln 3 = 8 - 6\ln 3 \]
We need to evaluate \(3(A + 6\ln 3)\).
\[ 3(8 - 6\ln 3 + 6\ln 3) = 3(8) = 24 \]

Step 4: Final Answer:

The value is \(24\).
Quick Tip: Always pick a test point between the limits of integration to confirm which curve is the upper boundary to ensure the calculated area is positive.

Step 1: Understanding the Concept:

A function of the form \(f(x) = [g(x)]\), where \([\cdot]\) is the greatest integer function, is generally discontinuous at all points where \(g(x)\) takes an integer value.

We must analyze the behavior of the inner function \(g(x)\) over the given interval.


Step 2: Key Formula or Approach:

Let \(g(x) = x^2 - x - 0.5\).

Find the derivative \(g'(x)\) to check for monotonicity.

Determine the range of \(g(x)\) on the interval \([2, 4]\) by evaluating the endpoints.

Count the number of distinct integers within this range.


Step 3: Detailed Explanation:

The inner function is \(g(x) = x^2 - x - 0.5\).

Differentiating it, we get \(g'(x) = 2x - 1\).

For \(x \in [2, 4]\), \(g'(x) \ge 2(2) - 1 = 3 > 0\).

Since \(g'(x) > 0\), the function \(g(x)\) is strictly increasing over the interval \([2, 4]\).

Evaluate \(g(x)\) at the endpoints of the interval:
\[ g(2) = 2^2 - 2 - 0.5 = 4 - 2.5 = 1.5 \] \[ g(4) = 4^2 - 4 - 0.5 = 16 - 4.5 = 11.5 \]
Since \(g(x)\) is continuous and strictly increasing from \(1.5\) to \(11.5\), it will cross every integer value between \(1.5\) and \(11.5\) exactly once.

The integer values within this range are: \(2, 3, 4, 5, 6, 7, 8, 9, 10, 11\).

Each time \(g(x)\) hits one of these integers, the greatest integer function \([g(x)]\) will have a jump discontinuity.

There are exactly 10 such integer values.


Step 4: Final Answer:

The number of points of discontinuity is \(10\).
Quick Tip: For \(f(x) = [g(x)]\) on \([a, b]\), if \(g(x)\) is strictly monotonic, the number of discontinuities is simply the number of integers strictly between \(g(a)\) and \(g(b)\).