The uploaded document contains a set of calculus and linear algebra problems. Here’s an overview of the questions and the steps required to solve them:

1. Taylor’s Theorem for Polynomial Expansion
   Expand $2x^3 + 7x^2 + x - 6$ in powers of $(x-2)$.

2. Taylor Expansion for $\sin x$
   Expand $\sin x$ around $x = \frac{\pi}{2}$ and find $\sin 91^\circ$ to 4 decimal places.

3. Verification Using Partial Derivatives
   Prove that $x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = 3$ where $u = \log \frac{x^4 + y^4}{x+y}$.

4. Higher Order Partial Derivative
   For $u = \log(x^3 + y^3 + z^3 - 3xyz)$, show that:
   $\left( \frac{\partial}{\partial x} + \frac{\partial}{\partial y} + \frac{\partial}{\partial z} \right)^2 u = -\frac{9}{(x+y+z)^2}.$

5. Angle Between Surfaces
   Find the angle between $x^2 + y^2 + z^2 = 9$ and $x^2 + y^2 - z = 3$ at $(2, -1, 2)$.

6. Vector Solenoidality and Curl
   For $\mathbf{f} = (ax^2y + yz)\mathbf{i} + (xy^2 - xz^2)\mathbf{j} + (2xyz - 2x^2y^2)\mathbf{k}$, determine $a$ such that the vector is solenoidal, and find its curl.

7. Double Integration with Change of Order
   Evaluate $\int_0^1 \int_0^{\sqrt{1-x^2}} y^2 \, dy \, dx$ by changing the order of integration.

8. Double Integration Over a Region
   Evaluate $\iint_R y \, dx \, dy$, where $R$ is bounded by $y^2 = 4x$ and $x^2 = 4y$.

9. Series Convergence
   Various series are tested for convergence using methods like $p$-series, root test, ratio test, and Raabe’s test.

10. Matrix Operations
    Problems include:

    • Verifying Cayley-Hamilton theorem and finding $A^{-1}$.
    • Reducing matrices to normal forms using nonsingular matrices $P$ and $Q$.
    • Solving systems of linear equations based on specific conditions.

11. Geometric Problems
    Radius of curvature for given curves and angle between surfaces.

12. Vector Calculus
    Find divergence and curl of given vectors at specified points.

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Here are some relative questions for exploration:

1. How does Taylor's theorem generalize for functions?
2. What are practical applications of double integration in physics?
3. How does the Cayley-Hamilton theorem simplify matrix computations?
4. What is the significance of solenoidality in vector fields?
5. How do radius of curvature computations apply to real-world geometry?

Tip: Understanding how to use convergence tests for series is crucial for solving advanced mathematical problems.