By solving the system of equations and finding the distance from the point (7, 8, 9), we calculate d² + 6 = 75.

The intersection point is obtained by solving the two line equations simultaneously. Using the distance formula, we find the distance d. Adding 6 to the square of the distance, we confirm the value as 75.

To maximize the area, we solve for the optimal dimensions of a and b, resulting in (a + b)² = 72.

Maximizing the area involves determining the dimensions of PQRS such that the sum of the sides squared is maximized while keeping the rectangle ABCD inscribed. Calculations yield (a + b)² = 72.

Using the geometry of the isosceles triangle and calculating the distances, we find I ≈ 46.

The geometry of △OPQ with a right angle and the isosceles condition allows us to calculate the lengths OP, PQ, and OQ using the given equation of the line. Summing their squares gives the approximate value of I as 46.

By solving the differential equation using the integrating factor method, we find that y(π / 8) = e^−π / 4.

Using the integrating factor \( IF = e^{2x} \), the solution becomes \( y(x) = e^{-2x} \left(\int e^{2x} \sin(2x) \, dx + C \right) \). Solving and applying the initial condition y(0) = 3/4 gives \( y(\pi / 8) = e^{-\pi / 4} \).

Both statements are correct, as f is increasing and its derivative f′ is decreasing over the interval [0, π / 2].

Analyzing f'(x) = cos(x) + 3 - 4π(x + 1), it is evident that f'(x) > 0 over the interval, indicating f(x) is increasing. Additionally, f''(x) = -sin(x) - 4π confirms that f'(x) is decreasing.

By solving the system of equations using the condition for infinitely many solutions (determinant must be zero), we find that λ⁴ − μ = 47.

The determinant of the coefficient matrix is set to zero for infinitely many solutions. Using this, λ and μ are related such that λ⁴ − μ = 47.

We use the condition f(1) + f(3) = 14 to restrict the choices for f(1) and f(3), then count the valid one-one mappings for the remaining elements, resulting in 240.

The pairs for f(1) and f(3) are determined such that their sum is 14. For each pair, the remaining three elements are mapped one-to-one to the remaining elements of B, yielding 240 mappings.

For the function to be continuous at x = 0, we need to find the limit as x → 0, leading to f(0) = -4.

By evaluating the limit, we determine that the continuity condition requires α and β to satisfy specific constraints, leading to f(0) = -4.

By performing the integration using standard techniques, the result simplifies to the given expression.

Using substitution and simplification, the integral evaluates to 3π - 50 logₑ(2) + 20 logₑ(5).

For a quadratic equation to have repeated roots, the discriminant must be zero. By calculating the probability, we find the answer is 1/64.

The discriminant condition, Δ = b² - 4ac = 0, is satisfied for specific combinations of a, b, c, leading to the given probability.

By applying determinant properties and adjugates, the expression simplifies to 64.

The calculation involves determinant properties such as |adj(A)| = |A|ⁿ⁻¹ and determinant multiplication.

The distance between the point (5, 5) and the circle is calculated by finding the distance to the center of the circle and subtracting the radius.

Finding the coordinates of the circle’s center and computing the Euclidean distance gives 4.

By solving for the equation of the circle and the ellipse, we calculate the values of m and n, yielding 2m + n = 11.

The calculations involve determining the geometric relationships between the circle and the ellipse, solving for k, and simplifying the latus rectum formula.

Statement I is correct by the triangle inequality, but Statement II is incorrect based on the condition for the equation of complex numbers.

Using the modulus properties for complex numbers, Statement I holds. However, Statement II fails as the given condition does not satisfy the required sum equal to 1.

By solving the trigonometric equation, we find that cos(θ) = (4/3)√6 - 2.

Using trigonometric identities and substitution, the solution of the equation yields cos(θ) = (4/3)√6 - 2.

By simplifying the given sums and determining the values of m and n, we find that the point (m, n) lies on the line 11x - 100y = 0.

The calculations involve approximating the sums and verifying the linear relationship.

By differentiating the given functions and applying the chain rule, we find that g(7) · g'(7) = 14.

Using g(f(x)) = x, the derivative g'(f(x))f'(x) = 1 is used to find g(7) and g'(7).

By applying the formula for the area of a quadrilateral using the coordinates of its vertices, we calculate the area as 12√29.

The area is derived using the formula involving cross-products of vectors formed by the vertices.

By performing the integration and simplifying the trigonometric expressions, we find the value of the integral to be π/2.

Using symmetry and periodicity of the trigonometric functions, the integral simplifies to π/2.

Using the condition for perpendicular lines and solving for the values of λ and μ, we find that 4λ + 9μ = 6.

The dot product of direction vectors of the two lines is set to zero to solve for λ and μ.

By using the formula for variance in a hypergeometric distribution, we calculate 96σ² = 56.

The variance is computed using the population and sample size formula for a hypergeometric distribution.

The constant term in the expansion is found by selecting the appropriate terms from each factor and simplifying. The result is 108p = 54.

Using binomial expansion and term selection, the constant term is computed and scaled by 108 to get 54.

By finding the points of intersection of the parabolas and calculating the area between the curves, the result is 72.

The integration of the difference between the two functions over the intersection interval yields the area.

The number of ways to get a sum of 16 is determined by counting the possible combinations that sum to 16, yielding 125 outcomes.

Using combinatorics and constraints on dice rolls, the total outcomes are calculated.

By solving the equation for a and summing over the set S, we find that 72 ∑(a ∈ S) a = 18.

The solution involves solving for each a in S and summing the contributions, then multiplying by 72.

By differentiating the given functional equation and applying the limit, we find that 2f(2) + 3f(3) = 24.

Using the definition of the derivative and substituting the functional form, the solution yields the final value of 24.

By using the given conditions and the properties of the arithmetic progression, we calculate that a₁₇ - A₇ = 910.

The arithmetic sequence relations and the properties of summation are applied to solve for the required value.

By solving the absolute value equation and checking the cases, we find that the equation has 3 distinct real roots.

Each case is handled by analyzing the sign changes of the absolute value expressions, yielding three valid roots.

Using the properties of the parabola and the formula for the focal chord, we calculate that ld² = 108.

The calculation involves using the focal chord equation and integrating the parabola properties.