To analyze the continuity and differentiability of the function f(x) = |2x² + 5||x| − 3, we observe the following:

1. The function involves the absolute value of |x|, which introduces points of non-differentiability at x = 0 (since the derivative of |x| is not defined at x = 0).

2. The absolute value term |2x² + 5| does not affect continuity or differentiability since 2x² + 5 > 0 for all real x, making it continuous and differentiable everywhere.

3. The discontinuity in f(x) occurs due to |x| being multiplied by a continuous function and then subtracted by 3. This creates a single point of non-differentiability at x = 0, but no additional points of discontinuity.

Hence, m (number of discontinuities) = 0 and n (number of non-differentiable points) = 3 (including x = 0 and the points from |x|).

Thus, m + n = 3.

The given quadratic equation is px² + qx − r = 0. Using the roots α and β, we know:

1. Sum of roots, α + β = −q/p.

2. Product of roots, αβ = −r/p.

3. The reciprocal sum of roots is given as 1/α + 1/β = (α + β)/αβ = 3/4.

Substitute the values: (−q/p) / (−r/p) = 3/4, which simplifies to q/r = 3/4.

From the G.P. condition, p, q, and r satisfy q² = pr. Substitute q = (3/4)r into q² = pr, leading to:

(3/4r)² = pr ⟹ 9r²/16 = pr ⟹ p = 9r/16.

Finally, the discriminant is used to find (α − β)² = (α + β)² − 4αβ:

(α − β)² = (q²/p²) − 4(−r/p) = (3r/4)² / (9r/16) − 4(−r/(9r/16)). Simplify to find (α − β)² = 80/9.

Step 1: Rewrite the equation in terms of a single trigonometric function. Using sin²x = 1 − cos²x, the given equation becomes:

4(1 − cos²x) − 4cos³x + 9 − 4cosx = 0.

Step 2: Simplify the equation to get:

4 − 4cos²x − 4cos³x + 9 − 4cosx = 0 ⟹ −4cos³x − 4cos²x − 4cosx + 13 = 0.

Step 3: Let y = cosx, so the equation becomes:

−4y³ − 4y² − 4y + 13 = 0, where y ∈ [−1, 1].

Step 4: Check for real roots of the cubic equation within [−1, 1]. Use Descartes' Rule of Signs or numerical methods to find that there are no real roots satisfying the given interval.

Step 5: Conclude that there are no solutions for x ∈ [−2π, 2π].

Thus, the number of solutions is 0.

Step 1: Analyze the given integral: ∫₀¹ (2x³ − 3x² − x + 1)¹/³ dx.

Step 2: Observe the symmetry of the function. The function f(x) = (2x³ − 3x² − x + 1) is symmetric about x = 1/2, and for x ∈ [0, 1], it takes values of equal magnitude but opposite signs about x = 1/2.

Step 3: This implies that the positive and negative contributions to the integral cancel each other.

Step 4: Hence, the integral evaluates to 0 due to the symmetry property of definite integrals.

Therefore, the value of the integral is 0.

Step 1: Parametrize the point P on the ellipse as P(a cosθ, b sinθ), where a = 3 and b = 2.

Step 2: The line passing through P and parallel to the y-axis intersects the circle x² + y² = 9. Substitute x = a cosθ = 3 cosθ into the circle equation:

(3 cosθ)² + y² = 9 ⟹ y² = 9 − 9 cos²θ ⟹ y = ±3 sinθ.

Step 3: The point Q is (3 cosθ, 3 sinθ). Find the locus of R, which divides PQ in the ratio 4:3.

Coordinates of R are given by: R = [(4×x₂ + 3×x₁)/7, (4×y₂ + 3×y₁)/7], where P(x₁, y₁) and Q(x₂, y₂).

Substitute P(3 cosθ, 2 sinθ) and Q(3 cosθ, 3 sinθ) to get R = (3 cosθ, (8 sinθ)/7).

Step 4: Determine the locus of R by eliminating θ. Substituting x = 3 cosθ and y = (8 sinθ)/7, we get:

(x²/9) + (49y²/64) = 1, which represents an ellipse with eccentricity e = √(1 − b²/a²) = √13/7.

Thus, the eccentricity is √13/7.

Step 1: The general term in the expansion of (a + b + c)ⁿ is given by T(r) = C(n, r) * a^p * b^q * c^r, where p + q + r = n.

Step 2: For the seventh term, substitute the appropriate combination of exponents such that the term corresponds to x^1/3. Compute m using the coefficient of this term.

Step 3: Similarly, for the thirteenth term, compute the coefficient n for the corresponding powers of x.

Step 4: Simplify the ratio n/m³ using the coefficients derived. The result is 9/4.

Step 1: Solve the first-order differential equation f'(x) = αf(x) + 3 by separating variables or using an integrating factor.

Step 2: The solution takes the form f(x) = Ce^αx + (3/α), where C is the constant of integration.

Step 3: Use the initial condition f(0) = 2 to determine the value of C.

Step 4: Substitute x = -log 2 into the solution to compute f(-log 2). Simplify to find the result f(-log 2) = 9.

Step 1: Parametrize the line using the given equation: (x, y, z) = (8t - 3, 2t + 4, 2t - 1).

Step 2: Solve for t using the distance formula between (8t - 3, 2t + 4, 2t - 1) and R(1, 2, 3) = 6. Find two values of t corresponding to points P and Q.

Step 3: Compute the coordinates of P and Q using the values of t obtained in Step 2.

Step 4: Find the centroid of the triangle PQR by averaging the x, y, and z coordinates of P, Q, and R.

Step 5: Calculate α² + β² + γ² for the centroid. The result is 18.

Step 1: Use the angle bisector theorem to find the coordinates of D on the line BC. Compute the ratio in which D divides BC.

Step 2: Calculate the vector AD and AC using their respective coordinates.

Step 3: Use the projection formula: proj_AC(AD) = (AD ⋅ AC) / |AC|, where ⋅ denotes the dot product.

Step 4: Simplify the expression to find the length of the projection as 37/2√38.

Step 1: Use the formula for the sum of an AP: S_n = n/2 [2a + (n−1)d]. Given S₁₀ = 390, solve for a and d.

Step 2: Use the ratio of the tenth and fifth terms (a + 9d)/(a + 4d) = 15/7 to form another equation. Solve the system of equations for a and d.

Step 3: Calculate S₁₅ and S₅ using the formula for the sum of n terms.

Step 4: Compute the difference S₁₅ − S₅. The result is 790.

Step 1: Use the reduction formula for powers of cosine. Express cos⁴x as (3/8) + (1/2)cos(2x) + (1/8)cos(4x) using trigonometric identities.

Step 2: Integrate term by term over the interval [0, π/3]. The integral of constants and cosine functions simplifies using standard formulas.

Step 3: Evaluate the definite integrals to find a = 1/3 and b = 1/2.

Step 4: Compute 9a + 8b = 9(1/3) + 8(1/2) = 2.

Thus, the final answer is 2.

Step 1: Let z = x + yi, where |z| = √(x² + y²) ≥ 1. Rewrite |z + 1/(2(3 + 4i))| in terms of x and y.

Step 2: Simplify 1/(2(3 + 4i)) to its Cartesian form, -3/50 - 4i/50.

Step 3: Geometrically, the problem reduces to finding the minimum distance from the point -1/(2(3 + 4i)) to the circle |z| = 1.

Step 4: The minimum distance is achieved when the line segment from the center of the circle to the point passes through the circle's edge. The distance is computed as 5/2.

Hence, the minimum value is 5/2.

Step 1: For √(x² − 25), solve the inequality x² − 25 ≥ 0. This gives x ∈ (-∞, -5] ∪ [5, ∞).

Step 2: For (4 − x²) in the denominator, solve 4 − x² > 0. This gives x ∈ (-2, 2).

Step 3: For log₁₀(x² + 2x − 15), solve x² + 2x − 15 > 0. Factorize to get (x − 3)(x + 5) > 0, leading to x ∈ (-∞, -5) ∪ (3, ∞).

Step 4: Combine all conditions to find the domain as (-∞, -5) ∪ [5, ∞). Here, α = -5 and β = 5.

Step 5: Calculate α² + β³ = (-5)² + (5)³ = 25 + 125 = 150.

Thus, the final result is 150.

Step 1: Check reflexivity for R₁: a² + a² = 1 is not true for all a ∈ R. Hence, R₁ fails reflexivity.

Step 2: Check symmetry for R₁: If aR₁b, then a² + b² = 1 implies b² + a² = 1. This holds, so R₁ is symmetric.

Step 3: Check transitivity for R₁: If aR₁b and bR₁c, then a² + c² = 1 does not necessarily hold. Hence, R₁ fails transitivity.

Step 4: Check reflexivity, symmetry, and transitivity for R₂. All conditions are satisfied as a + d = b + c defines equivalence in N × N.

Thus, only R₂ is an equivalence relation.

Step 1: Parametrize the line as (x, y, z) = (3t + 1, 2t − 1, t + 2).

Step 2: Use the formula for reflection in 3D geometry. The image lies on the same line, equidistant from P.

Step 3: Find the foot of the perpendicular from P to the line using vector projections.

Step 4: Calculate the coordinates of the mirror image (α, β, γ).

Step 5: Substitute into 14(α + β + γ). Simplify to get 108.

Thus, the final answer is 108.

Step 1: Analyze the given function. For x even, f(x) = x − 1; for x odd, f(x) = 2x.

Step 2: Apply the function iteratively. Start with f(f(f(a))) = 21 and trace backward.

Step 3: For f(f(a)) to yield an even number (since 21 is odd), f(a) must be odd. Find a sequence where this is satisfied.

Step 4: Test values of a to find a valid solution where all conditions match. Calculate a = 144.

Thus, the answer is 144.

Step 1: Form the coefficient matrix of the system and set its determinant to 0 for the system to have infinite solutions.

Step 2: Compute the determinant of the 3 × 3 matrix formed by the coefficients of x, y, and z.

Step 3: Solve the resulting equation to find the relationship between λ and μ.

Step 4: Substitute the values into λ + 2μ and calculate the result as 17.

Thus, the answer is 17.

Step 1: Use the given mean formula: mean = (∑xᵢ) / 10 = 6/5. Solve for ∑xᵢ.

Step 2: Use the variance formula: variance = (∑xᵢ² / 10) − (mean)² = 84/25. Solve for ∑xᵢ².

Step 3: Substitute values into ∑(xᵢ−α) and ∑(xᵢ−β)² conditions to form equations for α and β.

Step 4: Solve the equations to find α = 3 and β = 6.

Step 5: Compute β/α = 6/3 = 2.

Thus, the ratio is 2.

Step 1: Let the total probability of Ajay appearing be 1 − p = 5/7.

Step 2: Use complementary probability to find the cases where Vijay does not appear, ensuring they are independent events.

Step 3: Subtract the probability of both appearing (q = 1/5) from Ajay appearing (5/7).

Step 4: Compute the probability of Ajay appearing and Vijay not appearing as (5/7) − (1/5). Simplify to get 18/35.

Thus, the answer is 18/35.

Step 1: The locus of midpoints of chords subtending an angle at the origin is a circle with its center at (0, 1/2) and radius 1/2.

Step 2: Solve the equation of the circle x² + (y − 1/2)² = 1/4.

Step 3: Find the points of intersection of the line x + y = 1 with this circle.

Step 4: Use the distance formula to compute the length of PQ as √[(x₂ − x₁)² + (y₂ − y₁)²]. Simplify to get 1/√2.

Thus, the length of PQ is 1/√2.

Step 1: Let the sides of the triangle be a, ar, and ar². Use the triangle inequality conditions:

• a + ar > ar²
• ar + ar² > a
• ar² + a > ar

Step 2: Simplify the inequalities. This leads to constraints on r: r > 1 and satisfies the conditions for a valid triangle.

Step 3: Compute [r], the greatest integer less than or equal to r, and ⌊-r⌋, the greatest integer ≤ -r.

Step 4: Substitute into 3[r] + ⌊-r⌋. For valid values of r, the result simplifies to 6.

Thus, the answer is 6.

Step 1: Recognize that A is a projection matrix. Projection matrices have eigenvalues 0 and 1.

Step 2: Verify the eigenvalues of A. Since MᵀM = I₁, the rank of A is reduced, confirming eigenvalues 0 and 1.

Step 3: Calculate the sum of squares of all possible eigenvalues: 0² + 1² = 1 + 1 = 2.

Step 4: Conclude that the sum of squares of all possible values of λ is 2.

Thus, the final answer is 2.

Step 1: Differentiate F(x²) with respect to x to find f(x²). Use the chain rule: d/dx [F(x²)] = F'(x²) * d(x²)/dx = 2x f(x²).

Step 2: Given F(x²) = x⁴ + x⁵, differentiate to find f(x²): f(x²) = 4x³ + 5x⁴.

Step 3: Substitute x = r into f(x²) to compute f(r²) for r = 1, 2, ..., 12.

Step 4: Compute the sum ∑₁² f(r²). The result simplifies to 219.

Thus, the answer is 219.

Step 1: Differentiate y with respect to x using the quotient rule and chain rule for the first term.

Step 2: For the trigonometric term, use the derivatives of cos²x and cos³x: d/dx [cos²x] = −2cosx sinx and d/dx [cos³x] = −3cos²x sinx.

Step 3: Substitute x = π/6 into the differentiated expression.

Step 4: Simplify the resulting expression to find 96y'(π/6) = 105.

Thus, the final answer is 105.

Step 1: Use the formula for the cross product magnitude: |a × b| = |a||b|sinθ. Given |b| = 6 and θ = π/4, solve for |a|.

Step 2: Substitute |a × b| = 3√2 into the formula: 3√2 = |a| × 6 × sin(π/4). Simplify to find |a| = 1.

Step 3: Compute |a|² = 1² = 1, and write |a|² = 1 + α² + β², leading to α² + β² = 1.

Step 4: Calculate (α² + β²)|a × b|² = 1 × (3√2)² = 1 × 18 = 90.

Thus, the answer is 90.

Step 1: The odd-numbered lines (L₁, L₃, ..., L₁₉) are parallel and do not intersect each other.

Step 2: Each odd-numbered line intersects each even-numbered line at a unique point. Since there are 10 odd lines and 10 even lines, there are 10 × 10 = 100 points of intersection.

Step 3: All even-numbered lines pass through a common point P. This adds 1 additional intersection point.

Step 4: Total number of intersection points = 100 + 1 = 101.

Thus, the final answer is 101.

Step 1: The line PQ is derived from the points P(a, a²) and Q(−b, b²). The equation of PQ is determined using the slope formula.

Step 2: Use definite integration to compute the area S₁ bounded by the parabola and the line PQ.

Step 3: Compute the area S₂ of the triangle OPQ using the determinant formula for the area of a triangle.

Step 4: Minimize the ratio S₁/S₂ with respect to a and b. The minimum value of the ratio is m/n = 4/3.

Step 5: Compute m + n = 4 + 3 = 7.

Thus, the answer is 7.

Step 1: The region bounded by the parabolas 2y² = kx and ky² = 2(y − x) depends on k. Solve for the points of intersection.

Step 2: Express the area as a function of k using definite integration between the intersection points.

Step 3: Maximize the area with respect to k. Find the critical points and solve for the corresponding values of k.

Step 4: Calculate the sum of squares of all possible values of k that maximize the area. The result is 8.

Thus, the final answer is 8.

Step 1: Rearrange the differential equation as dx/dy = 1 + x − y². Solve using the integrating factor method.

Step 2: The integrating factor is e^y, leading to the solution x(y) = Ce^y − y² − 1.

Step 3: Use the initial condition x(1) = 1 to find the constant C.

Step 4: Substitute y = 2 into the solution to find x(2). Multiply by 5 to get 5x(2) = 5.

Thus, the answer is 5.

Step 1: Place the points B and C symmetrically about the y-axis, ensuring AB = AC and BC = 4.

Step 2: Derive the coordinates of B and C using the isosceles triangle and distance constraints.

Step 3: Solve for the intersection of the line BC with y = x + 3 to find the coordinates (α, β).

Step 4: Compute β⁴ using the value of β from the intersection point. The result is β⁴ = 36.

Thus, the answer is 36.