Step 1: Understanding the Concept:

The given equation involves a sum of terms of the form \((x+k-1)(x+k+1)\).

Since \((a-1)(a+1) = a^2 - 1\), each term can be simplified to a square minus one.

The resulting sum will form a quadratic equation in \(x\).

The roots are given as \(\alpha\) and \(\alpha+2\), allowing us to use the properties of quadratic equations like the sum and product of roots or the discriminant.


Step 2: Key Formula or Approach:

1. General term: \(T_k = (x+k-1)(x+k+1) = (x+k)^2 - 1\).

2. Sum of series: \(\sum_{k=1}^{n} ((x+k)^2 - 1) = 4n\).

3. Roots of \(ax^2 + bx + c = 0\) are \(\alpha_1, \alpha_2\), then \(|\alpha_1 - \alpha_2| = \frac{\sqrt{D}}{|a|}\).


Step 3: Detailed Explanation:

Expand the summation:
\[ \sum_{k=1}^{n} (x^2 + 2kx + k^2 - 1) = 4n \]
\[ nx^2 + 2x \left(\sum_{k=1}^{n} k\right) + \sum_{k=1}^{n} k^2 - n = 4n \]

Substitute the standard summation formulas:
\[ nx^2 + 2x \frac{n(n+1)}{2} + \frac{n(n+1)(2n+1)}{6} - n = 4n \]

Divide by \(n\) (since \(n \in \mathbb{N}\)):
\[ x^2 + (n+1)x + \frac{(n+1)(2n+1)}{6} - 1 = 4 \]
\[ x^2 + (n+1)x + \left( \frac{2n^2 + 3n + 1}{6} - 5 \right) = 0 \]

Comparing with \(ax^2 + bx + c = 0\), we have \(a=1, b=n+1\).

The roots are \(\alpha\) and \(\alpha+2\). The difference of roots is \(|(\alpha+2) - \alpha| = 2\).

Using \(D = b^2 - 4ac = a^2(difference of roots)^2\):
\[ (n+1)^2 - 4(1) \left( \frac{2n^2 + 3n + 1}{6} - 5 \right) = 2^2 \]
\[ (n+1)^2 - \frac{2(2n^2 + 3n + 1)}{3} + 20 = 4 \]

Multiply by 3 to clear the fraction:
\[ 3(n^2 + 2n + 1) - 4n^2 - 6n - 2 + 60 = 12 \]
\[ 3n^2 + 6n + 3 - 4n^2 - 6n - 2 + 60 - 12 = 0 \]
\[ -n^2 + 49 = 0 \Rightarrow n^2 = 49 \Rightarrow n = 7 \]

Now, use the sum of roots formula:
\[ \alpha + (\alpha + 2) = -(n+1) \]
\[ 2\alpha + 2 = -(7+1) = -8 \]
\[ 2\alpha = -10 \Rightarrow \alpha = -5 \]

The value of \(n + \alpha = 7 + (-5) = 2\).


Step 4: Final Answer:

The required value of \(n + \alpha\) is 2.
Quick Tip: When a quadratic equation is formed by a summation, simplify the sum first by dividing out common variables like \(n\). Using the difference of roots formula \(|r_1 - r_2| = \frac{\sqrt{D}}{|a|}\) is often faster than calculating roots explicitly.

Step 1: Understanding the Concept:

To solve an equation involving complex numbers, we simplify each term to the form \(a + bi\).

Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal.


Step 2: Key Formula or Approach:

1. Rationalize denominators: Multiply the numerator and denominator by the conjugate of the denominator.

2. Compare Real Parts and Imaginary Parts.


Step 3: Detailed Explanation:

Rationalize the fractions:
\[ \frac{2x}{1+3i} = \frac{2x(1-3i)}{(1+3i)(1-3i)} = \frac{2x(1-3i)}{1^2 + 3^2} = \frac{2x(1-3i)}{10} = \frac{x(1-3i)}{5} \]
\[ \frac{y}{1-2i} = \frac{y(1+2i)}{(1-2i)(1+2i)} = \frac{y(1+2i)}{1^2 + 2^2} = \frac{y(1+2i)}{5} \]

Substitute back into the equation:
\[ 50 \left( \frac{x(1-3i)}{5} - \frac{y(1+2i)}{5} \right) = 31 + 17i \]
\[ 10 [x(1-3i) - y(1+2i)] = 31 + 17i \]
\[ 10x - 30xi - 10y - 20yi = 31 + 17i \]

Group real and imaginary parts:
\[ (10x - 10y) + i(-30x - 20y) = 31 + 17i \]

Equate parts:

1) \(10x - 10y = 31\)

2) \(-30x - 20y = 17\)

Solve for \(x\) and \(y\):

From (1), \(30x - 30y = 93\) (multiplied by 3).

Add (2) to this result:
\[ (30x - 30y) + (-30x - 20y) = 93 + 17 \]
\[ -50y = 110 \Rightarrow y = -\frac{11}{5} = -2.2 \]

Substitute \(y\) in (1):
\[ 10x - 10(-2.2) = 31 \Rightarrow 10x + 22 = 31 \Rightarrow 10x = 9 \Rightarrow x = 0.9 \]

Calculate the required value:
\[ 10(x - 3y) = 10(0.9 - 3(-2.2)) = 10(0.9 + 6.6) = 10(7.5) = 75 \]


Step 4: Final Answer:

The value of \(10(x-3y)\) is 75.
Quick Tip: Always simplify denominators of complex fractions immediately. If there is a constant factor outside a bracket (like 50), it usually cancels with denominators inside, making calculations much cleaner.

Step 1: Understanding the Concept:

A system of linear equations \(AX = B\) has no solution if the determinant of the coefficient matrix (\(\Delta\)) is 0, but at least one of the Cramer's determinants (\(\Delta_x, \Delta_y, \Delta_z\)) is non-zero. Alternatively, we can use row reduction to find an inconsistent row (like \(0 = non-zero\)).


Step 2: Key Formula or Approach:

Row reduction is often more direct for finding specific parameter relationships.


Step 3: Detailed Explanation:

Write the augmented matrix:
\[ \left[ \begin{array}{ccc|c} 1 & 2 & 1 & 5
2 & 1 & \alpha & 5
8 & 4 & \beta & 18 \end{array} \right] \]

Perform row operations to eliminate entries in the first column:
\(R_2 \to R_2 - 2R_1\)
\(R_3 \to R_3 - 8R_1\) (Wait, looking at the coefficients, \(R_3\) is related to \(R_2\). Let's use \(R_3 \to R_3 - 4R_2\)):

First, let's keep it simple:
\(R_3 \to R_3 - 4R_2\)

Row 3 becomes:
\[ (8 - 4 \times 2)x + (4 - 4 \times 1)y + (\beta - 4 \alpha)z = 18 - 4 \times 5 \]
\[ 0x + 0y + (\beta - 4\alpha)z = -2 \]

For the system to have no solution, the equation for Row 3 must be inconsistent.

This happens when the coefficient of \(z\) is zero, while the constant is non-zero.
\[ \beta - 4\alpha = 0 \Rightarrow \beta = 4\alpha \]

Since we need \(\beta/\alpha\):
\[ \frac{\beta}{\alpha} = 4 \]

We also verify the determinant \(\Delta = 0\):
\[ \Delta = \begin{vmatrix} 1 & 2 & 1
2 & 1 & \alpha
8 & 4 & \beta \end{vmatrix} = 1(\beta - 4\alpha) - 2(2\beta - 8\alpha) + 1(8-8) = (\beta - 4\alpha) - 4(\beta - 4\alpha) = -3(\beta - 4\alpha) \]

Setting \(\Delta = 0\) also gives \(\beta = 4\alpha\).


Step 4: Final Answer:

The ratio \(\beta/\alpha\) is 4.
Quick Tip: In competitive exams, look for linear dependencies between equations. Here, \(R_3\) was almost a multiple of \(R_2\). If you see a relationship like \(Row_3 - k \cdot Row_2\), the "No Solution" condition typically forces the left side to zero and the right side to non-zero.

Step 1: Understanding the Concept:

Every square matrix satisfies its own characteristic equation \(|M - \lambda I| = 0\). We can find \(\alpha\) and \(\beta\) by comparing the given matrix equations with the characteristic equations of \(A\) and \(B\). Then we perform matrix arithmetic and determinant calculations.


Step 2: Key Formula or Approach:

1. Characteristic Equation: \(\lambda^2 - tr(M)\lambda + \det(M) = 0\).

2. Properties of Adjoints: \(\det(adj(M)) = (\det M)^{n-1}\). For \(2 \times 2\), \(\det(adj(M)) = \det M\).


Step 3: Detailed Explanation:

For matrix \(A\): \(tr(A) = 1+\alpha\), \(\det(A) = \alpha - 2\).

Characteristic eq: \(A^2 - (1+\alpha)A + (\alpha-2)I = O\).

Comparing with \(A^2 - 4A + I = O\):
\(1+\alpha = 4 \Rightarrow \alpha = 3\).

Check: \(\alpha-2 = 3-2 = 1\) (Matches). So, \(A = \begin{bmatrix} 1 & 2
1 & 3 \end{bmatrix}\).

For matrix \(B\): \(tr(B) = 3+2 = 5\), \(\det(B) = 6 - 3\beta\).

Characteristic eq: \(B^2 - 5B + (6-3\beta)I = O\).

Comparing with \(B^2 - 5B - 6I = O\):
\(6 - 3\beta = -6 \Rightarrow 3\beta = 12 \Rightarrow \beta = 4\).

So, \(B = \begin{bmatrix} 3 & 3
4 & 2 \end{bmatrix}\).


Now evaluate statements:

Testing (S1):
\(B-A = \begin{bmatrix} 3-1 & 3-2
4-1 & 2-3 \end{bmatrix} = \begin{bmatrix} 2 & 1
3 & -1 \end{bmatrix}\)
\(B+A = \begin{bmatrix} 3+1 & 3+2
4+1 & 2+3 \end{bmatrix} = \begin{bmatrix} 4 & 5
5 & 5 \end{bmatrix}\)
\((B-A)(B+A) = \begin{bmatrix} 2 & 1
3 & -1 \end{bmatrix} \begin{bmatrix} 4 & 5
5 & 5 \end{bmatrix} = \begin{bmatrix} 8+5 & 10+5
12-5 & 15-5 \end{bmatrix} = \begin{bmatrix} 13 & 15
7 & 10 \end{bmatrix}\)

Taking Transpose: \([(B-A)(B+A)]^T = \begin{bmatrix} 13 & 7
15 & 10 \end{bmatrix}\).

(S1) claims it is \(\begin{bmatrix} 13 & 15
7 & 10 \end{bmatrix}\), so (S1) is incorrect.


Testing (S2):
\(A+B = \begin{bmatrix} 4 & 5
5 & 5 \end{bmatrix}\)
\(\det(A+B) = (4 \times 5) - (5 \times 5) = 20 - 25 = -5\).

For a \(2 \times 2\) matrix, \(\det(adj(M)) = \det(M)^1 = -5\).

(S2) claims it is \(-5\), so (S2) is correct.


Step 4: Final Answer:

Only (S2) is correct.
Quick Tip: Cayley-Hamilton Theorem is your best friend here. If a matrix equation \(M^2 + pM + qI = O\) is given, \(p = -tr(M)\) and \(q = \det(M)\). This quickly finds unknown parameters without solving matrix squares.

Step 1: Understanding the Concept:

The intersection of two arithmetic progressions (A.P.s) is also an A.P. whose common difference is the least common multiple (LCM) of the common differences of the two original series.

We need to find common terms that satisfy the range constraints of both sets and then count those that are divisible by 3.


Step 2: Key Formula or Approach:

1. General term of A.P.: \(T_n = a + (n-1)d\).

2. Common difference of intersection (\(d_c\)): \(LCM(d_1, d_2)\).

3. Range constraint: \(T_n \leq \min(max value of A, max value of B)\).


Step 3: Detailed Explanation:

For set A: First term \(a_1 = 1\), difference \(d_1 = 5\), number of terms \(n_1 = 101\).

Max term of A = \(1 + (101-1)5 = 501\).

For set B: First term \(a_2 = 9\), difference \(d_2 = 7\), number of terms \(n_2 = 71\).

Max term of B = \(9 + (71-1)7 = 499\).

Common terms (\(A \cap B\)) must be \(\leq 499\).

Let's find the first common term:

Terms of A: 1, 6, 11, 16, 21, ...

Terms of B: 9, 16, 23, ...

First common term \(a = 16\).

Common difference \(d_c = LCM(5, 7) = 35\).

Common terms are of form: \(16, 51, 86, 121, 156, 191, 226, 261, 296, 331, 366, 401, 436, 471, 506 \dots\)

Within range \(\leq 499\), terms are: 16, 51, 86, 121, 156, 191, 226, 261, 296, 331, 366, 401, 436, 471.

Now filter for divisibility by 3:

- 51 (sum of digits \(5+1=6\)): Yes

- 156 (sum of digits \(1+5+6=12\)): Yes

- 261 (sum of digits \(2+6+1=9\)): Yes

- 366 (sum of digits \(3+6+6=15\)): Yes

- 471 (sum of digits \(4+7+1=12\)): Yes

Total count = 5.


Step 4: Final Answer:

The number of elements in \(A \cap B\) divisible by 3 is 5.
Quick Tip: Once you find the first common term \(a=16\) and \(d_c=35\), observe that \(16 \equiv 1 \pmod{3}\) and \(35 \equiv 2 \pmod{3}\).
The general term is \(16 + 35k\).
For divisibility by 3: \(1 + 2k \equiv 0 \pmod{3} \Rightarrow 2k \equiv -1 \equiv 2 \pmod{3} \Rightarrow k \equiv 1 \pmod{3}\).
Since \(k\) ranges from 0 to 13 (because \(16+35(13) = 471\)), values of \(k\) are 1, 4, 7, 10, 13. Total 5 terms.

Step 1: Understanding the Concept:

We need to form 7-digit numbers using 5 distinct digits where every digit is used at least once.

Since we have 7 slots and 5 digits, two digits must be extra. This leads to two distinct cases for the distribution of frequencies of the digits.


Step 2: Key Formula or Approach:

Identify possible frequency partitions of 7 into 5 parts where each part \(\geq 1\):

Case 1: One digit is used 3 times, others used 1 time each. (Partition: 3, 1, 1, 1, 1)

Case 2: Two digits are used 2 times each, others used 1 time each. (Partition: 2, 2, 1, 1, 1)


Step 3: Detailed Explanation:

Case 1: One digit appears thrice, others once.

1. Select which digit is repeated 3 times: \({}^{5}C_{1} = 5\) ways.

2. Arrange the 7 digits (with one digit repeated thrice): \(\frac{7!}{3! \times 1! \times 1! \times 1! \times 1!} = \frac{5040}{6} = 840\) ways.

Total for Case 1 = \(5 \times 840 = 4200\).


Case 2: Two digits appear twice, others once.

1. Select which two digits are repeated 2 times each: \({}^{5}C_{2} = 10\) ways.

2. Arrange the 7 digits (with two digits repeated twice): \(\frac{7!}{2! \times 2! \times 1! \times 1! \times 1!} = \frac{5040}{4} = 1260\) ways.

Total for Case 2 = \(10 \times 1260 = 12600\).


Total Number of 7-digit numbers = \(4200 + 12600 = 16800\).


Step 4: Final Answer:

The number of such seven-digit numbers is 16800.
Quick Tip: Always handle such problems by creating partitions first. If the number of digits was smaller or the number of slots larger, Principle of Inclusion-Exclusion (PIE) could be used: \(5^7 - {}^{5}C_{1} \cdot 4^7 + {}^{5}C_{2} \cdot 3^7 - {}^{5}C_{3} \cdot 2^7 + {}^{5}C_{4} \cdot 1^7\). However, partitioning is much faster for small differences.

Step 1: Understanding the Concept:

We use the binomial coefficient identity \({}^{n}C_{r} = \frac{n}{r} \cdot {}^{n-1}C_{r-1}\) to simplify the equation.

Then, we analyze the resulting integer constraints on \(k\) and the range constraints on \(r\).


Step 2: Key Formula or Approach:

1. \({}^{36}C_{r+1} = \frac{36}{r+1} \cdot {}^{35}C_r\).

2. Range of \(r\): \(0 \leq r \leq 35\) for \({}^{35}C_r\) to be defined, and \(0 \leq r+1 \leq 36\) for \({}^{36}C_{r+1}\).


Step 3: Detailed Explanation:

Substitute the identity into the given equation:
\[ \frac{36}{r+1} \cdot {}^{35}C_r = \frac{6}{k^2-3} \cdot {}^{35}C_r \]

Since binomial coefficients are non-zero in the defined range, we can divide both sides:
\[ \frac{36}{r+1} = \frac{6}{k^2-3} \]
\[ \frac{6}{r+1} = \frac{1}{k^2-3} \Rightarrow k^2 - 3 = \frac{r+1}{6} \]
\[ k^2 = 3 + \frac{r+1}{6} \]

Since \(k \in \mathbb{Z}\), \(k^2\) must be a perfect square.

Also, \(0 \leq r \leq 35 \Rightarrow 1 \leq r+1 \leq 36\). Thus \(0 < \frac{r+1}{6} \leq 6\).

Possible values for \(\frac{r+1}{6}\):

- If \(\frac{r+1}{6} = 1 \Rightarrow r=5\). Then \(k^2 = 3+1 = 4 \Rightarrow k = \pm 2\).

Solutions: \((5, 2), (5, -2)\).

- If \(\frac{r+1}{6} = 2, 3, 4, 5\), then \(k^2 = 5, 6, 7, 8\). No integer \(k\) exists.

- If \(\frac{r+1}{6} = 6 \Rightarrow r=35\). Then \(k^2 = 3+6 = 9 \Rightarrow k = \pm 3\).

Solutions: \((35, 3), (35, -3)\).

Total solutions in set S = 4.


Step 4: Final Answer:

The number of elements in set S is 4.
Quick Tip: Remember to consider both positive and negative values of \(k\) as the condition is \(k \in \mathbb{Z}\). Often students forget the \(\pm\) and find only half the solutions.

Step 1: Understanding the Concept:

For grouped data, the mean is given by \(\bar{x} = \frac{\sum f_i x_i}{\sum f_i}\), where \(x_i\) is the mid-point of the class intervals. We first find \(k\) and then check which quadratic equation it satisfies.


Step 2: Key Formula or Approach:

1. Midpoints (\(x_i\)): 7.5, 12.5, 17.5, 22.5, 27.5, 32.5.

2. Frequencies (\(f_i\)): 2, \(k\), 28, 54, \(k+1\), 5.


Step 3: Detailed Explanation:

Calculate \(\sum f_i\):
\(\sum f_i = 2 + k + 28 + 54 + (k+1) + 5 = 2k + 90\).

Calculate \(\sum f_i x_i\):
\(\sum f_i x_i = (7.5 \times 2) + (12.5 \times k) + (17.5 \times 28) + (22.5 \times 54) + 27.5(k+1) + (32.5 \times 5)\)
\(\sum f_i x_i = 15 + 12.5k + 490 + 1215 + 27.5k + 27.5 + 162.5\)
\(\sum f_i x_i = 40k + 1910\).

Set mean to 21:
\[ \frac{40k + 1910}{2k + 90} = 21 \]
\[ 40k + 1910 = 21(2k + 90) \]
\[ 40k + 1910 = 42k + 1890 \]
\[ 2k = 20 \Rightarrow k = 10 \].

Now, test \(k=10\) in the options:

(C) \(2(10)^2 - 19(10) - 10 = 200 - 190 - 10 = 0\). (Correct)


Step 4: Final Answer:

The value of \(k\) is 10, which is a root of the equation \(2x^2 - 19x - 10 = 0\).
Quick Tip: To simplify arithmetic with mid-points, you can use step-deviation: \(u_i = \frac{x_i - A}{h}\). Choosing \(A=22.5, h=5\), the values of \(u_i\) would be -3, -2, -1, 0, 1, 2. This makes the multiplication much easier.

Step 1: Understanding the Concept:

Given mid-points of sides, we can find the vertices of the triangle. Then, using side lengths and vertex coordinates, we apply the Incentre formula: \(I = \left( \frac{ax_1+bx_2+cx_3}{a+b+c}, \frac{ay_1+by_2+cy_3}{a+b+c} \right)\).


Step 2: Key Formula or Approach:

1. Vertices: \(x_1+x_2=2x_{m1}\), etc. Or \(A = M_2+M_3-M_1\).

2. Midpoints: \(M_1(2.5, 7), M_2(2.5, 3), M_3(4, 5)\).


Step 3: Detailed Explanation:

Let the vertices be \(A, B, C\).
\(A = M_2 + M_3 - M_1 = (2.5+4-2.5, 3+5-7) = (4, 1)\).
\(B = M_1 + M_3 - M_2 = (2.5+4-2.5, 7+5-3) = (4, 9)\).
\(C = M_1 + M_2 - M_3 = (2.5+2.5-4, 7+3-5) = (1, 5)\).

Vertices are \(A(4, 1), B(4, 9), C(1, 5)\).

Side lengths:
\(c = AB = \sqrt{(4-4)^2 + (9-1)^2} = 8\).
\(a = BC = \sqrt{(4-1)^2 + (9-5)^2} = \sqrt{3^2 + 4^2} = 5\).
\(b = AC = \sqrt{(4-1)^2 + (1-5)^2} = \sqrt{3^2 + 4^2} = 5\).

Incentre \((h, k)\):
\(h = \frac{5(4) + 5(4) + 8(1)}{5+5+8} = \frac{20+20+8}{18} = \frac{48}{18} = \frac{8}{3}\).
\(k = \frac{5(1) + 5(9) + 8(5)}{18} = \frac{5+45+40}{18} = \frac{90}{18} = 5\).

Calculate \(3h + k = 3(8/3) + 5 = 8 + 5 = 13\).


Step 4: Final Answer:

The value of \(3h + k\) is 13.
Quick Tip: To find vertices from midpoints quickly, use the vector property: \(A = M_{AB} + M_{AC} - M_{BC}\). Also, if two sides are equal (here \(a=b=5\)), the triangle is isosceles, which often simplifies coordinate calculations.

Step 1: Understanding the Concept:

For an ellipse with \(a < b\), the major axis is along the y-axis.

The eccentricity relation is \(a^2 = b^2(1 - e^2)\).

Length of Latus Rectum = \(\frac{2a^2}{b}\).


Step 2: Key Formula or Approach:

1. Eccentricity: \(e^2 = 1 - a^2/b^2 \Rightarrow \frac{a^2}{b^2} = 1 - e^2\).

2. Point substitution: \(\frac{16}{a^2} + \frac{9}{b^2} = 1\).


Step 3: Detailed Explanation:

Given \(e = \frac{\sqrt{5}}{3} \Rightarrow e^2 = \frac{5}{9}\).

Using \(a < b\):
\(\frac{a^2}{b^2} = 1 - \frac{5}{9} = \frac{4}{9} \Rightarrow a^2 = \frac{4}{9}b^2\).

Substitute point \((4, 3)\) into ellipse equation:
\(\frac{16}{a^2} + \frac{9}{b^2} = 1\)
\(\frac{16}{(4b^2/9)} + \frac{9}{b^2} = 1\)
\(\frac{16 \times 9}{4b^2} + \frac{9}{b^2} = 1 \Rightarrow \frac{36}{b^2} + \frac{9}{b^2} = 1\)
\(\frac{45}{b^2} = 1 \Rightarrow b^2 = 45 \Rightarrow b = \sqrt{45} = 3\sqrt{5}\).

Now find \(a^2 = \frac{4}{9} \times 45 = 20\).

Length of Latus Rectum = \(\frac{2a^2}{b} = \frac{2 \times 20}{3\sqrt{5}} = \frac{40}{3\sqrt{5}}\).

Rationalize the denominator: \(\frac{40\sqrt{5}}{3 \times 5} = \frac{8\sqrt{5}}{3}\).


Step 4: Final Answer:

The length of the latus rectum is \(\frac{8\sqrt{5}}{3}\).
Quick Tip: Always check whether \(a < b\) or \(a > b\) before calculating the Latus Rectum. For vertical ellipses (\(a < b\)), the formula is \(2a^2/b\), whereas for horizontal ellipses (\(a > b\)), it is \(2b^2/a\).

Step 1: Understanding the Concept:

The given expression is a product of sine functions with specific angles. We can convert these radian measures to degrees for easier manipulation and use trigonometric product-to-sum identities or specialized sine product formulas.


Step 2: Key Formula or Approach:

1. Radian to Degree: \(\frac{\pi}{18} = 10^\circ, \frac{5\pi}{18} = 50^\circ, \frac{7\pi}{18} = 70^\circ\).

2. Identity: \(\sin \theta \sin(60^\circ - \theta) \sin(60^\circ + \theta) = \frac{1}{4} \sin 3\theta\).


Step 3: Detailed Explanation:

Let the product be \(K\):
\[ K = \sin 10^\circ \sin 50^\circ \sin 70^\circ \]

Using the identity with \(\theta = 10^\circ\):
\[ K = \sin 10^\circ \sin(60^\circ - 10^\circ) \sin(60^\circ + 10^\circ) = \frac{1}{4} \sin(3 \times 10^\circ) \]
\[ K = \frac{1}{4} \sin 30^\circ = \frac{1}{4} \times \frac{1}{2} = \frac{1}{8} \]

Now, we need to find the value of \(\sin \left(\frac{10 K \pi}{3}\right)\):
\[ \sin \left(\frac{10 \times (1/8) \times \pi}{3}\right) = \sin \left(\frac{10\pi}{24}\right) = \sin \left(\frac{5\pi}{12}\right) \]

Converting back to degrees: \(\frac{5\pi}{12} = \frac{5 \times 180^\circ}{12} = 75^\circ\).

Using the value of \(\sin 75^\circ\):
\[ \sin 75^\circ = \sin(45^\circ + 30^\circ) = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ \]
\[ = \left(\frac{1}{\sqrt{2}}\right)\left(\frac{\sqrt{3}}{2}\right) + \left(\frac{1}{\sqrt{2}}\right)\left(\frac{1}{2}\right) = \frac{\sqrt{3}+1}{2\sqrt{2}} \]


Step 4: Final Answer:

The value is \(\frac{\sqrt{3}+1}{2 \sqrt{2}}\).
Quick Tip: Standard product results like \(\sin 20^\circ \sin 40^\circ \sin 80^\circ = \frac{\sqrt{3}}{8}\) and \(\sin 10^\circ \sin 50^\circ \sin 70^\circ = \frac{1}{8}\) are extremely common in competitive exams and should be memorized to save time.

Step 1: Understanding the Concept:

We need to find the number of solutions for the equation \(\sin^2 x + \sin x \cos x = a\) where \(a\) is an integer and \(x \in [-\pi, \pi]\). We first determine the range of the left-hand side function to find which integers \(a\) are possible.


Step 2: Key Formula or Approach:

1. Simplify function: \(f(x) = \sin^2 x + \sin x \cos x = \frac{1 - \cos 2x}{2} + \frac{\sin 2x}{2}\).

2. Transformation: \(A \sin \theta + B \cos \theta = \sqrt{A^2+B^2} \sin(\theta + \phi)\).


Step 3: Detailed Explanation:

Let \(f(x) = \frac{1}{2} + \frac{1}{2}(\sin 2x - \cos 2x)\).

Using the transformation, \(\sin 2x - \cos 2x = \sqrt{2} \sin(2x - \pi/4)\).

So, \(f(x) = \frac{1}{2} + \frac{1}{\sqrt{2}} \sin(2x - \frac{\pi}{4})\).

Range of \(\sin(2x - \frac{\pi}{4})\) is \([-1, 1]\).

Range of \(f(x)\) is \([\frac{1}{2} - \frac{1}{\sqrt{2}}, \frac{1}{2} + \frac{1}{\sqrt{2}}]\).

Approximate values: \(\frac{1}{\sqrt{2}} \approx 0.707\).

Range \(\approx [0.5 - 0.707, 0.5 + 0.707] \approx [-0.207, 1.207]\).

The only integers in this range are \(a = 0\) and \(a = 1\).

Case 1: \(a = 0\)
\(\sin x (\sin x + \cos x) = 0 \implies \sin x = 0\) or \(\tan x = -1\).

For \(x \in [-\pi, \pi]\): \(\sin x = 0 \implies x = \{-\pi, 0, \pi\}\) (3 solutions).
\(\tan x = -1 \implies x = \{-\pi/4, 3\pi/4\}\) (2 solutions).

Total solutions for \(a=0\) is \(3 + 2 = 5\).

Case 2: \(a = 1\)
\(\frac{1}{2} + \frac{1}{\sqrt{2}} \sin(2x - \frac{\pi}{4}) = 1 \implies \sin(2x - \frac{\pi}{4}) = \frac{1}{\sqrt{2}}\).

Let \(\theta = 2x - \frac{\pi}{4}\). Since \(x \in [-\pi, \pi]\), \(\theta \in [-2\pi - \frac{\pi}{4}, 2\pi - \frac{\pi}{4}] = [-\frac{9\pi}{4}, \frac{7\pi}{4}]\).

Solutions for \(\sin \theta = \frac{1}{\sqrt{2}}\) in this range:
\(\theta = \{ \dots, -2\pi+\frac{\pi}{4}, -2\pi+\frac{3\pi}{4}, \frac{\pi}{4}, \frac{3\pi}{4} \} = \{-\frac{7\pi}{4}, -\frac{5\pi}{4}, \frac{\pi}{4}, \frac{3\pi}{4}\}\).

All these values are within the range. This gives 4 solutions.

Total elements in set \(S = 5 + 4 = 9\).


Step 4: Final Answer:

The number of elements \(n(S)\) is 9.
Quick Tip: To count solutions accurately, plot a quick rough graph or use the unit circle. For equations involving \(2x\), the number of solutions usually doubles compared to equations in \(x\).

Step 1: Understanding the Concept:

A point lying on the \(xy\)-plane has its \(z\)-coordinate equal to zero. If the intersection point lies on this plane, we can set \(z=0\) in the parametric forms of both lines and solve for the unknown constants \(a\) and \(b\).


Step 2: Key Formula or Approach:

1. Intersection condition: Set both lines equal to parameters \(\lambda\) and \(\mu\).

2. \(xy\)-plane condition: \(z = 0\).


Step 3: Detailed Explanation:

Let the intersection point be \(P(x, y, z)\). Since it is on the \(xy\)-plane, \(z = 0\).

For Line 1: \(\frac{x+1}{3} = \frac{y+a}{5} = \frac{0+b+1}{7}\).

From this, \(\frac{x+1}{3} = \frac{b+1}{7} \implies x = \frac{3b+3}{7} - 1 = \frac{3b-4}{7}\).

And \(\frac{y+a}{5} = \frac{b+1}{7} \implies y = \frac{5b+5}{7} - a = \frac{5b+5-7a}{7}\).

For Line 2: \(\frac{x-2}{1} = \frac{y-b}{4} = \frac{0-2a}{7}\).

From this, \(x-2 = -\frac{2a}{7} \implies x = 2 - \frac{2a}{7} = \frac{14-2a}{7}\).

And \(\frac{y-b}{4} = -\frac{2a}{7} \implies y = b - \frac{8a}{7} = \frac{7b-8a}{7}\).

Equating the \(x\)-coordinates:
\(\frac{3b-4}{7} = \frac{14-2a}{7} \implies 2a + 3b = 18 \quad \dots (Eq. 1)\).

Equating the \(y\)-coordinates:
\(\frac{5b+5-7a}{7} = \frac{7b-8a}{7} \implies 5b + 5 - 7a = 7b - 8a \implies a - 2b = -5 \implies 2a - 4b = -10 \quad \dots (Eq. 2)\).

Subtracting Eq. 2 from Eq. 1:
\((2a + 3b) - (2a - 4b) = 18 - (-10) \implies 7b = 28 \implies b = 4\).

Substitute \(b=4\) into \(a - 2b = -5\):
\(a - 8 = -5 \implies a = 3\).

Finally, \(a + b = 3 + 4 = 7\).


Step 4: Final Answer:

The value of \(a+b\) is 7.
Quick Tip: When a question mentions a point of intersection lying on a specific plane (\(xy, yz, or zx\)), use the condition for that plane immediately. It reduces the 3D problem to a 2D coordinate system, making the system of equations much easier to solve.

Step 1: Understanding the Concept:

We need to find the maximum value of an expression involving magnitudes of linear combinations of vectors. We let \(\theta\) be the angle between \(\vec{a}\) and \(\vec{b}\) and express the magnitudes as functions of \(\theta\).


Step 2: Key Formula or Approach:

1. Magnitude: \(|\vec{u} + \vec{v}| = \sqrt{|\vec{u}|^2 + |\vec{v}|^2 + 2|\vec{u}||\vec{v}|\cos\theta}\).

2. Trigonometric Max: Max of \(A \sin \phi + B \cos \phi = \sqrt{A^2 + B^2}\).


Step 3: Detailed Explanation:

Let \(|\vec{a}| = 2\) and \(|\vec{b}| = 3\). Let the angle between \(\vec{a}\) and \(\vec{b}\) be \(\theta\).

Consider \(|3\vec{a} + 2\vec{b}|\):
\[ |3\vec{a} + 2\vec{b}| = \sqrt{9|\vec{a}|^2 + 4|\vec{b}|^2 + 12|\vec{a}||\vec{b}|\cos\theta} = \sqrt{9(4) + 4(9) + 12(2)(3)\cos\theta} \]
\[ = \sqrt{36 + 36 + 72\cos\theta} = \sqrt{72(1 + \cos\theta)} = \sqrt{144 \cos^2(\theta/2)} = 12|\cos(\theta/2)| \]

Similarly for \(|3\vec{a} - 2\vec{b}|\):
\[ |3\vec{a} - 2\vec{b}| = \sqrt{36 + 36 - 72\cos\theta} = \sqrt{72(1 - \cos\theta)} = \sqrt{144 \sin^2(\theta/2)} = 12|\sin(\theta/2)| \]

The expression is \(E = 3(12|\cos(\theta/2)|) + 4(12|\sin(\theta/2)|)\):
\[ E = 36 |\cos(\theta/2)| + 48 |\sin(\theta/2)| \]

For maximum value, we use the property that \(\max(a \cos \phi + b \sin \phi) = \sqrt{a^2 + b^2}\):
\[ Max E = \sqrt{36^2 + 48^2} = \sqrt{(12 \times 3)^2 + (12 \times 4)^2} \]
\[ = 12 \sqrt{3^2 + 4^2} = 12 \times 5 = 60 \]


Step 4: Final Answer:

The maximum value is 60.
Quick Tip: When you see terms like \(|X+Y|\) and \(|X-Y|\), look for trigonometric half-angle simplifications. In this problem, the coefficients were deliberately chosen (\(3|\vec{a}| = 6 = 2|\vec{b}|\)) to make the identity \(1 \pm \cos \theta\) work perfectly.

Step 1: Understanding the Concept:

The direction of a line perpendicular to two given vectors is found using their cross product. Once we have the direction and a point on the line, we find the foot of the perpendicular from the origin by ensuring the vector from the origin to the point is orthogonal to the line's direction.


Step 2: Key Formula or Approach:

1. Direction vector \(\vec{v} = \vec{n_1} \times \vec{n_2}\).

2. Condition for foot of perpendicular from origin: \(\vec{OP} \cdot \vec{v} = 0\).


Step 3: Detailed Explanation:

Let \(\vec{n_1} = (2, 2, 1)\) and \(\vec{n_2} = (1, 2, 2)\).

Direction of line \(L\) is \(\vec{v} = \vec{n_1} \times \vec{n_2}\):
\[ \vec{v} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k}
2 & 2 & 1
1 & 2 & 2 \end{vmatrix} = \hat{i}(4-2) - \hat{j}(4-1) + \hat{k}(4-2) = (2, -3, 2) \]

Equation of Line \(L\) passing through \((1, 1, 1)\): \(\vec{r} = (1, 1, 1) + \lambda(2, -3, 2)\).

Any point \(P\) on line \(L\) is \((1+2\lambda, 1-3\lambda, 1+2\lambda)\).

If \(P\) is the foot of perpendicular from origin \(O(0, 0, 0)\), then \(\vec{OP} \cdot \vec{v} = 0\):
\[ (1+2\lambda)(2) + (1-3\lambda)(-3) + (1+2\lambda)(2) = 0 \]
\[ 2 + 4\lambda - 3 + 9\lambda + 2 + 4\lambda = 0 \implies 17\lambda + 1 = 0 \implies \lambda = -\frac{1}{17} \]

Now find the sum of coordinates \(a+b+c\):
\[ a+b+c = (1+2\lambda) + (1-3\lambda) + (1+2\lambda) = 3 + \lambda \]
\[ a+b+c = 3 - \frac{1}{17} = \frac{51-1}{17} = \frac{50}{17} \]

Finally, \(34(a+b+c) = 34 \times \frac{50}{17} = 2 \times 50 = 100\).


Step 4: Final Answer:

The value of \(34(a+b+c)\) is 100.
Quick Tip: To find the sum of coordinates for a parametric point \(P(x(\lambda), y(\lambda), z(\lambda))\), calculate the sum as a function of \(\lambda\) first before substituting the specific \(\lambda\) value. This often simplifies the fractions and reduces arithmetic errors.

Step 1: Understanding the Concept:

For the limit to result in a finite non-zero value \(m\), given that the denominator tends to 0 as \(x \to 2\), the numerator must also tend to 0.

Furthermore, the order of the zero in the denominator must match the order of the zero in the numerator for the limit to be finite.


Step 2: Key Formula or Approach:

1. Use approximations for \(x \to 2\): \(\sin(f(x)) \approx f(x)\), \(\log_e(1+t) \approx t\), and \(\sqrt{1+t}-1 \approx t/2\).

2. If \(x \to a\) is a zero of order \(n\), then \((x-a)^n\) is a factor of the polynomial.


Step 3: Detailed Explanation:

Let \(x - 2 = t\). As \(x \to 2, t \to 0\).

The denominator becomes \((\sqrt{1+t}-1) \log_e(1+t)\).

Applying limits: \(\lim_{t \to 0} \frac{\sqrt{1+t}-1}{t} \cdot t \cdot \frac{\log_e(1+t)}{t} \cdot t = \frac{1}{2} \cdot t^2\).

The denominator is of the order \((x-2)^2\).

Thus, the polynomial \(P(x) = x^3 - 5x^2 + ax + b\) must have \((x-2)^2\) as a factor.

Let \(x^3 - 5x^2 + ax + b = (x-2)^2(x-k)\).

Expanding: \((x^2 - 4x + 4)(x-k) = x^3 - (k+4)x^2 + (4k+4)x - 4k\).

Comparing the coefficients of \(x^2\): \(k+4 = 5 \implies k = 1\).

Now find \(a\) and \(b\):
\(a = 4k + 4 = 4(1) + 4 = 8\).
\(b = -4k = -4(1) = -4\).

Now find \(m\):
\[ m = \lim_{x \to 2} \frac{(x-2)^2(x-1)}{\frac{1}{2}(x-2)^2} = 2 \lim_{x \to 2} (x-1) = 2(2-1) = 2 \]

Calculation of \(a + b + m = 8 + (-4) + 2 = 6\).


Step 4: Final Answer:

The value of \(a+b+m\) is 6.
Quick Tip: In limits involving polynomials inside trigonometric or logarithmic functions, always check the power of the infinitesimal in the denominator. This power tells you the multiplicity of the root in the numerator's polynomial.

Step 1: Understanding the Concept:

The given equation is a first-order variable separable differential equation. We can separate the variables \(y\) and \(x\) to integrate both sides.


Step 2: Key Formula or Approach:

1. \(\int \frac{1}{y} dy = \int (2 + \log_e x) dx\).

2. Integration by parts: \(\int \log_e x dx = x \log_e x - x\).


Step 3: Detailed Explanation:

Separate the variables:
\[ \frac{dy}{y} = (2 + \log_e x) dx \]

Integrate both sides:
\[ \int \frac{dy}{y} = \int 2 dx + \int \log_e x dx \]
\[ \log_e y = 2x + (x \log_e x - x) + C \]
\[ \log_e y = x + x \log_e x + C \]

The curve passes through \((1, e)\). Substitute \(x=1, y=e\):
\[ \log_e e = 1 + 1 \log_e 1 + C \]
\[ 1 = 1 + 0 + C \implies C = 0 \].

The curve is \(\log_e y = x(1 + \log_e x)\).

To find \(f(e)\), substitute \(x=e\):
\[ \log_e y = e(1 + \log_e e) \]
\[ \log_e y = e(1 + 1) = 2e \]
\[ y = e^{2e} \].


Step 4: Final Answer:

The value of \(f(e)\) is \(e^{2e}\).
Quick Tip: For integration of \(\ln x\), always remember the result \(x \ln x - x\). In differential equations, always solve for the constant \(C\) immediately after integration using the initial point to avoid errors in later steps.

Step 1: Understanding the Concept:

Critical points are points in the domain where the function's derivative is zero or the derivative does not exist. For absolute value functions, we check where the expression inside the absolute value is zero (potential non-differentiability) and where its derivative is zero.


Step 3: Detailed Explanation:

Let \(g(x) = \frac{\sin x}{x}\). The function is \(f(x) = |g(x)|\).

1. Points where \(f'(x) = 0\):

This occurs where \(g'(x) = 0\).
\[ g'(x) = \frac{x \cos x - \sin x}{x^2} \]
\(g'(x) = 0 \implies x \cos x = \sin x \implies x = \tan x\).

In the interval \((-2\pi, 2\pi)\), the graphs of \(y=x\) and \(y=\tan x\) intersect at 3 points:

- \(x = 0\) (The limit is 1, and the derivative of \((\sin x)/x\) at 0 is 0 by expansion).

- One point in \((\pi, 3\pi/2)\).

- One point in \((-3\pi/2, -\pi)\).

Total 3 points where the derivative is zero.

2. Points where \(f'(x)\) does not exist:

This occurs where \(g(x) = 0\) (where the function crosses the x-axis).
\(\frac{\sin x}{x} = 0 \implies \sin x = 0\) for \(x \neq 0\).

In \((-2\pi, 2\pi)\), this occurs at \(x = \pi\) and \(x = -\pi\).

At these 2 points, the absolute value creates a sharp corner (cusp), so the derivative does not exist.

Total critical points = \(3 + 2 = 5\).


Step 4: Final Answer:

The number of critical points is 5.
Quick Tip: For \(f(x) = |g(x)|\), critical points consist of the roots of \(g'(x)=0\) and the roots of \(g(x)=0\). Visualizing the graph of \((\sin x)/x\) and "reflecting" the negative portions across the x-axis helps identify local extrema and corners easily.

Step 1: Understanding the Concept:

The Greatest Integer Function \([x]\) remains constant between integer intervals. We must break the integral into sub-intervals \([0, 1), [1, 2), [2, 3)\) to simplify the denominator.


Step 2: Key Formula or Approach:

1. \(\int (e^x + e^{-x}) dx = e^x - e^{-x}\).

2. Break the integral: \(\int_0^3 f(x) dx = \int_0^1 f(x) dx + \int_1^2 f(x) dx + \int_2^3 f(x) dx\).


Step 3: Detailed Explanation:

Break the integral:

- For \(x \in [0, 1), [x] = 0 \implies [x]! = 0! = 1\).

- For \(x \in [1, 2), [x] = 1 \implies [x]! = 1! = 1\).

- For \(x \in [2, 3), [x] = 2 \implies [x]! = 2! = 2\).

Integral \(I = \int_0^1 (e^x + e^{-x}) dx + \int_1^2 (e^x + e^{-x}) dx + \int_2^3 \frac{e^x + e^{-x}}{2} dx\).

Combine the first two terms:
\[ I = \int_0^2 (e^x + e^{-x}) dx + \frac{1}{2} \int_2^3 (e^x + e^{-x}) dx \]
\[ I = [e^x - e^{-x}]_0^2 + \frac{1}{2} [e^x - e^{-x}]_2^3 \]
\[ I = (e^2 - e^{-2}) - (e^0 - e^0) + \frac{1}{2} [(e^3 - e^{-3}) - (e^2 - e^{-2})] \]
\[ I = e^2 - \frac{1}{e^2} + \frac{1}{2} e^3 - \frac{1}{2 e^3} - \frac{1}{2} e^2 + \frac{1}{2 e^2} \]
\[ I = \frac{1}{2} e^2 + \frac{1}{2} e^3 - \frac{1}{2 e^2} - \frac{1}{2 e^3} = \frac{1}{2} (e^2 + e^3 - \frac{1}{e^2} - \frac{1}{e^3}) \]


Step 4: Final Answer:

The integral evaluates to \(\frac{1}{2}(e^2 + e^3 - \frac{1}{e^2} - \frac{1}{e^3})\).
Quick Tip: When an integrand has the same value across consecutive intervals (like here from 0 to 2), combine them into one integral to reduce the number of limit substitutions and potential calculation errors.

Step 1: Understanding the Concept:

This is a variable separable differential equation. We rearrange terms to separate \(y\) and \(x\) and then integrate to find the general solution.


Step 2: Key Formula or Approach:

1. Separate variables: \(\frac{1}{y+1} dy = -\frac{\cos x}{1 + \sin x} dx\).

2. Integrate using \(\int \frac{f'(x)}{f(x)} dx = \ln|f(x)|\).


Step 3: Detailed Explanation:

The DE is: \((1 + \sin x) \frac{dy}{dx} = -(y+1) \cos x\).

Separating variables:
\[ \frac{dy}{y+1} = -\frac{\cos x}{1 + \sin x} dx \]

Integrating:
\[ \ln |y+1| = -\ln |1 + \sin x| + \ln C \]
\[ \ln |y+1| + \ln |1 + \sin x| = \ln C \]
\[ (y+1)(1 + \sin x) = C \]

Given \(y(0) = 0\):
\[ (0+1)(1 + \sin 0) = C \implies 1(1) = C \implies C = 1 \].

Equation of curve: \((y+1)(1 + \sin x) = 1\).

Substituting the point \((\alpha, -1/2)\):
\[ (-1/2 + 1)(1 + \sin \alpha) = 1 \]
\[ \frac{1}{2}(1 + \sin \alpha) = 1 \implies 1 + \sin \alpha = 2 \implies \sin \alpha = 1 \].

So, \(\alpha = \pi/2\).


Step 4: Final Answer:

A value of \(\alpha\) is \(\pi/2\).
Quick Tip: Sometimes you can recognize the equation as a derivative of a product: \(\frac{d}{dx}[(y+1)(1+\sin x)] = 0\). This immediately leads to \((y+1)(1+\sin x) = C\), saving several steps of integration.

Step 1: Understanding the Concept:

For the function \(f(x) = \sqrt{\log_{0.6} (g(x))}\) to be defined, two conditions must be satisfied:

1. The argument of the logarithm must be positive: \(g(x) > 0\).

2. The expression inside the square root must be non-negative: \(\log_{0.6} (g(x)) \geq 0\).

Since the base of the logarithm is \(0.6\), which is less than 1, the logarithmic inequality reverses: \(\log_{0.6} (g(x)) \geq 0 \implies 0 < g(x) \leq 1\).


Step 2: Key Formula or Approach:

We need to find the intersection of the two inequalities:
\( \frac{2x-5}{x^2-4} > 0 \) and \( \frac{2x-5}{x^2-4} \leq 1 \).


Step 3: Detailed Explanation:

First, solve \(\frac{2x-5}{(x-2)(x+2)} > 0\):

The critical points are \(-2, 2\), and \(2.5\).

Using the wavy curve method, the expression is positive in the intervals \(x \in (-2, 2) \cup (2.5, \infty)\).

Second, solve \(\frac{2x-5}{x^2-4} \leq 1\):
\[ \frac{2x-5}{x^2-4} - 1 \leq 0 \implies \frac{2x-5 - x^2 + 4}{x^2-4} \leq 0 \implies \frac{-x^2+2x-1}{x^2-4} \leq 0 \]
\[ \frac{x^2-2x+1}{x^2-4} \geq 0 \implies \frac{(x-1)^2}{(x-2)(x+2)} \geq 0 \]

The term \((x-1)^2\) is always non-negative.

The fraction is \(\geq 0\) if the denominator is positive: \(x^2 - 4 > 0 \implies x \in (-\infty, -2) \cup (2, \infty)\).

Also, the fraction is zero at \(x = 1\).

So the solution set for the second inequality is \((-\infty, -2) \cup \{1\} \cup (2, \infty)\).

Third, find the intersection of the two sets:
\[ [(-2, 2) \cup (2.5, \infty)] \cap [(-\infty, -2) \cup \{1\} \cup (2, \infty)] = \{1\} \cup (2.5, \infty) \]

Comparing this result with the form \((-\infty, a] \cup \{b\} \cup [c, d) \cup (e, \infty)\), we identify \(b = 1\) and \(e = 2.5\).

Other variables \(a, c, d\) are not present in this specific domain structure.

Assuming the question requires the numerical result based on these values: \(e - b = 2.5 - 1 = 1.5\).


Step 4: Final Answer:

The value is 1.5.
Quick Tip: Remember the direction of logarithmic inequalities: if base \(a < 1\), \(\log_a x \geq 0 \implies x \leq a^0 = 1\). Always find the domain of the internal log argument (\(x > 0\)) first before solving the square root condition.

Step 1: Understanding the Concept:

We are given the sum of the first \(n\) terms \(S_n\) of a sequence. The general term \(a_n\) is found using \(a_n = S_n - S_{n-1}\). After finding \(a_n\), we compute the difference of consecutive terms and evaluate the given sum.


Step 2: Key Formula or Approach:

1. \(a_n = S_n - S_{n-1}\).

2. Standard sum: \(\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}\).


Step 3: Detailed Explanation:

Given \(S_n = 6n^3\).

The general term \(a_n\) is:
\[ a_n = 6n^3 - 6(n-1)^3 = 6[n^3 - (n^3 - 3n^2 + 3n - 1)] \]
\[ a_n = 6(3n^2 - 3n + 1) = 18n^2 - 18n + 6 \].

Now, calculate the difference \(a_{k+1} - a_k\):
\[ a_{k+1} - a_k = [18(k+1)^2 - 18(k+1) + 6] - [18k^2 - 18k + 6] \]
\[ = 18((k+1)^2 - k^2) - 18((k+1) - k) \]
\[ = 18(2k+1) - 18 = 36k + 18 - 18 = 36k \].

Substitute this into the required summation:
\[ \sum_{k=1}^6 \left( \frac{36k}{36} \right)^2 = \sum_{k=1}^6 k^2 \]

Using the sum of squares formula for \(n=6\):
\[ \sum_{k=1}^6 k^2 = \frac{6(6+1)(2 \times 6 + 1)}{6} = 7 \times 13 = 91 \].


Step 4: Final Answer:

The final sum is 91.
Quick Tip: When \(S_n\) is a cubic polynomial in \(n\), the general term \(a_n\) will always be a quadratic in \(n\), and the first difference \(a_{k+1} - a_k\) will be a linear function. Recognizing these polynomial degrees helps verify your algebra quickly.

Step 1: Understanding the Concept:

For a quadratic expression \(f(x) = Ax^2 + Bx + C\) to be strictly positive for all \(x\), the conditions are \(A > 0\) and discriminant \(D < 0\). Since \(a \in \{1, 2, 3, 4\}\), \(a > 0\) is always true. We just need to check \(D < 0\).


Step 2: Key Formula or Approach:

1. Discriminant \(D = (2\sqrt{2}b)^2 - 4ac < 0\).

2. Total outcomes in the sample space: \(4 \times 4 \times 4 = 64\).


Step 3: Detailed Explanation:

The condition \(D < 0\) implies:
\[ (2\sqrt{2}b)^2 - 4ac < 0 \implies 8b^2 < 4ac \implies 2b^2 < ac \]

We check valid triplets \((a, b, c)\) by testing cases for \(b\):

Case 1: \(b = 1\)
\(2(1)^2 < ac \implies ac > 2\).

Total pairs \((a, c)\) are \(4 \times 4 = 16\).

Pairs where \(ac \leq 2\) are \((1, 1), (1, 2), (2, 1)\) (3 pairs).

Favorable pairs for \(b=1\) is \(16 - 3 = 13\).

Case 2: \(b = 2\)
\(2(2)^2 < ac \implies ac > 8\).

Valid \((a, c)\) pairs: \((3, 3), (3, 4), (4, 3), (4, 4)\) (4 pairs).

Case 3: \(b = 3\)
\(2(3)^2 < ac \implies ac > 18\).

Since the maximum value of \(ac\) is \(4 \times 4 = 16\), no pairs exist.

Case 4: \(b = 4\)
\(2(4)^2 < ac \implies ac > 32\). No pairs exist.

Total favorable outcomes = \(13 + 4 = 17\).

Probability \(P = \frac{17}{64} = \frac{m}{n}\).

Since \(gcd(17, 64) = 1\), \(m = 17\) and \(n = 64\).

Calculating \(m + n = 17 + 64 = 81\).


Step 4: Final Answer:

The value of \(m+n\) is 81.
Quick Tip: Systematically listing values for the variable with the smallest set or the one that has the largest effect (like \(b^2\)) is the most reliable way to solve counting-based probability problems. Always subtract small counts from the total (complement method) when it's faster.

Step 1: Understanding the Concept:

A circle cutting equal intercepts on both axes must have its center on the line \(y = x\) or \(y = -x\). Since the center is in the first quadrant, it lies on \(y = x\). For a circle to intersect the coordinate axes at exactly three points, it must pass through the origin \((0, 0)\) and intersect the \(x\) and \(y\) axes at one other point each.


Step 2: Key Formula or Approach:

1. Center \((h, h)\). Passes through \((0, 0) \implies r^2 = h^2 + h^2 = 2h^2\).

2. Chord length \(L = 2\sqrt{r^2 - d^2}\), where \(d\) is the distance from the center to the line.


Step 3: Detailed Explanation:

Let the center be \(O(h, h)\) and radius be \(r\). Since it passes through origin, the equation is:
\[ (x-h)^2 + (y-h)^2 = 2h^2 \implies r^2 = 2h^2 \]

Calculate the distance \(d\) from \((h, h)\) to the line \(x+y-1=0\):
\[ d = \frac{|h + h - 1|}{\sqrt{1^2 + 1^2}} = \frac{|2h - 1|}{\sqrt{2}} \]

Given chord length is \(\sqrt{14}\), so half-length is \(\frac{\sqrt{14}}{2}\).

Using Pythagoras theorem in the circle-chord triangle:
\[ \left( \frac{\sqrt{14}}{2} \right)^2 = r^2 - d^2 \]
\[ \frac{14}{4} = 2h^2 - \frac{(2h-1)^2}{2} \]
\[ 3.5 = 2h^2 - \frac{4h^2 - 4h + 1}{2} \]
\[ 7 = 4h^2 - (4h^2 - 4h + 1) \]
\[ 7 = 4h - 1 \implies 4h = 8 \implies h = 2 \].

Square of the radius \(r^2 = 2h^2 = 2(2^2) = 8\).


Step 4: Final Answer:

The square of the radius of the circle is 8.
Quick Tip: The condition "exactly 3 points" on coordinate axes for a circle centered at \((h, h)\) always implies that the circle passes through the origin. This allows you to write the radius immediately in terms of the center coordinates.

Step 1: Understanding the Concept:

This problem relies on the property of the sum of integrals of a function and its inverse:
\[ \int_a^b f(x) dx + \int_{f(a)}^{f(b)} f^{-1}(y) dy = b f(b) - a f(a) \]


Step 2: Key Formula or Approach:

1. Identify \(f(x) = \log_2(x^2+4)\).

2. Find its inverse \(f^{-1}(x)\).


Step 3: Detailed Explanation:

Let \(y = \log_2(x^2+4)\).

Then \(2^y = x^2 + 4 \implies x^2 = 2^y - 4 \implies x = \sqrt{2^y - 4}\).

Thus, \(f^{-1}(x) = \sqrt{2^x - 4}\).

Now check the limits:

For \(f(x)\): Lower limit \(a = 0 \implies f(0) = \log_2(4) = 2\).

Upper limit \(b = 2\sqrt{3} \implies f(2\sqrt{3}) = \log_2((2\sqrt{3})^2 + 4) = \log_2(12 + 4) = 4\).

Notice the second integral is \(\int_2^4 f^{-1}(x) dx\), which matches the limits \([f(0), f(2\sqrt{3})]\).

Applying the formula:
\[ \alpha = [x \cdot f(x)]_0^{2\sqrt{3}} = (2\sqrt{3} \times f(2\sqrt{3})) - (0 \times f(0)) \]
\[ \alpha = 2\sqrt{3} \times 4 = 8\sqrt{3} \].

The required value is \(\alpha^2\):
\[ \alpha^2 = (8\sqrt{3})^2 = 64 \times 3 = 192 \].


Step 4: Final Answer:

The value of \(\alpha^2\) is 192.
Quick Tip: Always check for the inverse relation when you see two definite integrals of dissimilar functions added together, especially if the limits of one are the functional outputs of the other's limits. It avoids the need for complex integration by parts.