1. Evaluate the limit as \(x\) approaches infinity of
\( \frac{x^3-3x^2+2x-1}{x^3+2x^2-x+1} \)
2. Find the sum of the series: \( \sum_{n=1}^{\infty} \frac{n^2}{n^4 + 4} \)
3. Prove that there is no rational root for the equation:
\( x^5 –x^4 +x^3 –x^2 +x–1 = 0 \)
4. Determine the general solution of the differential equation:
\( y'-2y = e^{2x} \)
5. Determine the area enclosed by the curves:
\( y=x^2 \) and \( y=x^3–x \)
6. Solve the equation:
\( sin x + cos x = 1 \), \( 0 \le x \le 2 \pi \)
7. Prove that \( \sum_{k=1}^{n} k^3 = \left(\sum_{k=1}^{n} k\right)^2 \) for all natural numbers \(n\).
8. Find the equation of the tangent line to the curve \( y = e^{3x} lnx \) at the point \((1,0)\).
9. Determine the radius of convergence for the power series:
\( \sum_{n=1}^{\infty} \frac{(x-2)^n}{n} \)
10. Prove the binomial theorem:
\( (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} b^k a^{n-k} \)
11. Determine the volume of the solid generated by revolving the region bounded by the curves \(y=x^2\) and \(y = x\) about the \(x\)-axis.
12. Prove that \( \underset{n \to \infty}{\lim} \frac{n!}{n^n} = 0 \)
13. Find the smallest positive integer solution to the Diophantine equation:
\( 7x+11y=2023 \)
14. Prove that for any triangle with sides of lengths a,b,c, the following inequality holds:
\( \frac{a^3 +b^3 +c^3}{3} \ge abc \)
15. Determine the value of the integral:
\( \int_0^\infty \frac{x^3}{{(x^2+1)^2}} \, dx \)
16. Find the maximum and minimum values of the function \( f(x)=3x^4–8x^3+5x^2 \) on the interval \( [0,2] \).
17. Show that \( \sqrt[3]{7} + \sqrt[3]{49} \) is irrational.
18. Find the general solution of the recurrence relation \( a_n=5a_{n-1}–6a_{n-2} \), given that \(a_0=1\) and \(a_1=3\).
19. Evaluate the double integral:
\( \int_{0}^{1} \int_{0}^{\sqrt{1-x^2}} \frac{1}{{(x^2+y^2)^2}} \, dy \, dx \)
20. Prove that the sum of the angles in an \(n\)-sided polygon is equal to \( 180^{\circ } (n-2) \).
21. Find the sum of the infinite geometric series:
\( \frac{1}{2} - \frac{1}{4} + \frac{1}{8} - \frac{1}{16} + \ldots \)
22. Prove that for any prime number \(p\), the number \( \frac{p-1}{2} \) is odd if and only if \( p \equiv 3 \) (mod 4).
23. Determine the arc length of the curve \( y = \frac{1}{3} x^3 –x \) from \(x=0\) to \(x=2\).
24. Determine the number of distinct ways to arrange the letters of the word "MATHEMATICS" such that no two "M"s are adjacent.
25. Prove that for all positive integers n, the following is true:
\( 1^3 +2^3 + \ldots +n^3 = (1+2+ \dots +n)^2 \)
26. Find the area of a triangle with vertices at the complex numbers \( z_1 = 1+2i \), \(z_2 = 2+i \) and \( z_3 = 1+i \) in the complex plane.
27. Prove that the roots of the polynomial \( P(x) = x^n –a_1 x^{n-1} +a_2 x^{n-2} - \ldots \) \( \ldots +(-1)^{n-1} a_{n-1} x–(-1)^n a_n \) are all real if and only if \( a_i \ge 0 \) for all \( 1 \le i \le n \) .
28. Evaluate the improper integral: \( \int_{-\infty}^{\infty} \frac{1}{{1 + x^2}} \, dx \)
29. If \(z\) is a complex number such that \(z^4=1\), prove that \(z^2–z+1=0\) if and only if \( z \neq 1 \).
30. Show that there are infinitely many prime numbers of the form \(4k + 3\), where \(k\) is an integer.