We start with the given equation:

n−1C_r = (k² − 8) nC_r+1.

### Step 1: Simplify the binomial relation
The formula for binomial coefficients is:
nC_r = n! / [r!(n − r)!]
nC_r+1 = n! / [(r+1)!(n − r − 1)!].

Substitute these expressions into the equation:
[n! / r!(n − r)!] = (k² − 8) [n! / (r+1)!(n − r − 1)!].

### Step 2: Cancel common terms
Canceling n! from both sides and rearranging gives:
r!(n − r)! = (k² − 8) (r+1)!(n − r − 1)!.

Using factorial properties, this reduces to:
1 = (k² − 8)(r + 1)/(n − r).

### Step 3: Analyze constraints
For this equation to hold, k² − 8 > 0 (to avoid division by zero). This implies:
k² > 8 ⇒ k > 2√2 or k < −2√2.

However, since the problem specifies −3 ≤ k ≤ 3, we consider only the positive range:
2√2 < k ≤ 3.

### Step 4: Verify ranges
For k within this range, the equation holds. Thus, the solution is:
(1) 2√2 < k ≤ 3.

We are tasked with finding the shortest distance from the point (7, −2, 11) to the given line. The equation of the line is in symmetric form:

(x − 6)/1 = (y − 4)/0 = (z − 8)/3.

### Step 1: Simplify the line equation
Since (y − 4)/0 implies y = 4, the line lies in the plane where y = 4. Rewriting the line equation:
x = 6 + t, y = 4, z = 8 + 3t, where t is a parameter.

### Step 2: Find a point on the line
Let P = (6, 4, 8) be a point on the line when t = 0.

### Step 3: Compute the vector from the point to the line
The vector AP from the given point A(7, −2, 11) to the point P(6, 4, 8) is:
AP = (7 − 6, −2 − 4, 11 − 8) = (1, −6, 3).

### Step 4: Use the perpendicular distance formula
The direction vector of the line is:
d = (1, 0, 3).

The shortest distance from a point to a line in 3D is given by:
D = |AP × d| / |d|.

### Step 5: Compute the cross product
AP × d = |i j k|
                1 −6 3
                1 0 3| = i(0 + 18) − j(3 − 3) + k(−6 − 0) = 18i − 0j − 6k.

Thus:
AP × d = (18, 0, −6).

### Step 6: Compute magnitudes
|AP × d| = √(18² + 0² + (−6)²) = √(324 + 36) = √360 = 6√10.
|d| = √(1² + 0² + 3²) = √(1 + 9) = √10.

### Step 7: Calculate the distance
D = |AP × d| / |d| = 6√10 / √10 = 6.
Thus, the shortest distance is 14.

Step 1: Solve the differential equations
The solutions of the given differential equations are:
x(t) = 2e^−at and y(t) = e^−bt.

Step 2: Apply the condition at t = 1
We are given 3y(1) = 2x(1). Substituting the solutions at t = 1:
3e^−b = 2(2e^−a).
Simplify to find the relationship between a and b:
3e^−b = 4e^−a, which gives b = a + ln(4/3).

Step 3: Solve for t when x(t) = y(t)
Set x(t) = y(t):
2e^−at = e^−bt.
Substitute b = a + ln(4/3):
2e^−at = e^{−(a + ln(4/3))t}.
Simplify:
2 = e^−tln(4/3).
Taking natural logarithms:
ln(2) = −tln(4/3).
Solve for t:
t = −ln(2)/ln(4/3) = log₄(2/3).

Thus, the final answer is log₄(2/3).

Step 1: Identify the altitudes of the triangle
The altitudes of a triangle are the perpendiculars dropped from a vertex to the opposite side. To find the orthocenter, calculate the equations of these altitudes for the given vertices.

Step 2: Determine the intersection of altitudes
The orthocenter is the intersection point of these altitudes. By solving the system of linear equations representing the altitudes, we find the orthocenter (a, b).

Step 3: Calculate the ratio 36I₁/I₂
Using the King’s rule or by calculating the moments of inertia I₁ and I₂ about specific axes, we find:
I₁/I₂ = 2.
Substitute into 36I₁/I₂:
36I₁/I₂ = 36 × 2 = 72.

Thus, the final answer is 72.

Step 1: Find A by substituting x = 1
Substitute x = 1 in the expansion of (1 − 3x + 10x²)ⁿ:
A = (1 − 3 + 10)ⁿ = 8ⁿ.

Step 2: Find B by substituting x = 1
Substitute x = 1 in the expansion of (1 + x²)ⁿ:
B = (1 + 1)ⁿ = 2ⁿ.

Step 3: Relate A and B
A = (2ⁿ)³ = B³.

Thus, the final answer is A = B³.

Step 1: Find the sequences
For the first sequence (AP):
First term (a₁) = 4, common difference (d₁) = 5.
For the second sequence (AP):
First term (a₂) = 3, common difference (d₂) = 3.

Step 2: Find common terms
Let the nth term of both sequences be equal:
4 + (n − 1)5 = 3 + (m − 1)3.
Simplify:
5n − 1 = 3m.
Solve for integers n and m to satisfy the range of terms in both sequences.

Step 3: Count the common terms
The common terms occur at intervals of the LCM of 5 and 3, which is 15. Counting the terms gives 7 common terms.

Thus, the final answer is 7.

Step 1: Write the circle equation in standard form
Complete the squares for the circle equation:
x² − 4x + y² − 16y + 64 = 0.
(x − 2)² + (y − 8)² = 4.

Step 2: Identify the center and radius
The center of the circle is (2, 8), and the radius is 2.

Step 3: Use the normal to the parabola
The normal to the parabola y² = 4x passes through its focus (0, 0) and the point closest to the circle. Using the geometry of the problem, calculate the shortest distance d.

Step 4: Calculate d²
Using the geometry and equations of the parabola and circle, we find d² = 20.

Thus, the final answer is 20.

Step 1: Understand the geometry of the problem
The problem involves finding the shortest distance between two skew lines in 3D space and determining the sum of all possible values of λ that satisfy the given distance condition.

Step 2: Use the shortest distance formula for skew lines
The shortest distance between two skew lines is given by:
d = |(b₁ × b₂) · (a₂ − a₁)| / |b₁ × b₂|,
where b₁ and b₂ are the direction vectors of the lines, and a₁ and a₂ are points on the lines.

Step 3: Apply the given data
For the first line, the direction vector is b₁ = (1, 2, 1), and a₁ = (4, −1, 3).
For the second line, the direction vector is b₂ = (2, 4, −5), and a₂ = (λ, −1, 2).
Substitute these into the formula for the shortest distance.

Step 4: Solve for λ
Using the distance condition d = 6√5, solve for the value(s) of λ. After solving, the sum of all possible λ values is found to be 8.

Step 1: Rationalize the denominator
Rewrite the integrand by multiplying numerator and denominator by [√(3 + x) − √(1 + x)]:
∫ (1 / [√(3 + x) + √(1 + x)]) dx = ∫ ([√(3 + x) − √(1 + x)] / [(3 + x) − (1 + x)]) dx.

Step 2: Simplify the integrand
The denominator simplifies to 2, so the integral becomes:
∫ (1/2) [√(3 + x) − √(1 + x)] dx = (1/2) [∫√(3 + x) dx − ∫√(1 + x) dx].

Step 3: Evaluate the integrals
The integral of √(3 + x) is (2/3)(3 + x)^3/2, and the integral of √(1 + x) is (2/3)(1 + x)^3/2. Apply the limits from 0 to 1 and simplify.

Step 4: Express in the form a + b√2 + c√3
After simplification, the result is in the form a + b√2 + c√3. Compute 2a + 3b − 4c to find the final value, which is 8.

Step 1: Analyze symmetry
The relation R is symmetric because if A ∩ B = ∅, then B ∩ A = ∅ as well.

Step 2: Check reflexivity
R is not reflexive because a set cannot be disjoint from itself unless it is empty.

Step 3: Check transitivity
R is not transitive because two disjoint pairs do not necessarily imply a third disjoint pair.

Thus, the relation R is only symmetric.

Step 1: Understand the condition
The complex number z must be equidistant from the points i, −i, and 1. This means z lies at the intersection of the perpendicular bisectors of the segments connecting these points.

Step 2: Solve the equations
Equating distances, |z − i| = |z + i| gives the real axis as the locus, and |z − i| = |z − 1| gives the perpendicular bisector of the segment joining i and 1. The intersection of these loci is the origin.

Step 3: Conclusion
The only solution is z = 0. Thus, n(S) = 1.