1. Evaluate a Limit
Evaluate the limit as \(x\) approaches infinity of:
\( \frac{x^3 - 3x^2 + 2x - 1}{x^3 + 2x^2 - x + 1} \)
2. Sum of an Infinite Series
Find the sum of the series:
\( \sum_{n=1}^{\infty} \frac{n^2}{n^4 + 4} \)
3. Rational Root Theorem
Prove that there is no rational root for the equation:
\( x^5 - x^4 + x^3 - x^2 + x - 1 = 0 \)
4. Differential Equation
Determine the general solution of the differential equation:
\( y' - 2y = e^{2x} \)
5. Area Between Curves
Determine the area enclosed by the curves:
\( y = x^2 \) and \( y = x^3 - x \)
6. Solve a Trigonometric Equation
Solve the equation:
\( \sin x + \cos x = 1 \), \( 0 \le x \le 2\pi \)
7. Sum of Cubes
Prove that:
\( \sum_{k=1}^{n} k^3 = \left(\sum_{k=1}^{n} k\right)^2 \)
for all natural numbers \(n\).
8. Tangent Line to a Curve
Find the equation of the tangent line to the curve \( y = e^{3x} \ln x \) at the point \((1,0)\).
9. Radius of Convergence
Determine the radius of convergence for the power series:
\( \sum_{n=1}^{\infty} \frac{(x-2)^n}{n} \)
10. Prove the Binomial Theorem
Prove that:
\( (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} b^k a^{n-k} \)
11. Volume of Revolution
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. Limit of a Sequence
Prove that \( \underset{n \to \infty}{\lim} \frac{n!}{n^n} = 0 \).
13. Diophantine Equation
Find the smallest positive integer solution to the Diophantine equation:
\( 7x + 11y = 2023 \).
14. Inequality in a Triangle
Prove that for any triangle with sides \(a\), \(b\), and \(c\), the following inequality holds:
\( \frac{a^3 + b^3 + c^3}{3} \geq abc \).
15. Evaluate a Definite Integral
Determine the value of the integral:
\( \int_0^\infty \frac{x^3}{{(x^2+1)^2}} \, dx \).
16. Extrema of a Function
Find the maximum and minimum values of the function \( f(x) = 3x^4 - 8x^3 + 5x^2 \) on the interval \( [0, 2] \).
17. Prove Irrationality
Show that \( \sqrt[3]{7} + \sqrt[3]{49} \) is irrational.
18. Recurrence Relation
Find the general solution of the recurrence relation \( a_n = 5a_{n-1} - 6a_{n-2} \), given \( a_0 = 1 \) and \( a_1 = 3 \).
19. Evaluate a Double Integral
Evaluate the double integral:
\( \int_{0}^{1} \int_{0}^{\sqrt{1-x^2}} \frac{1}{{(x^2+y^2)^2}} \, dy \, dx \).
20. Sum of Angles in a Polygon
Prove that the sum of the angles in an \(n\)-sided polygon is equal to \( 180^\circ (n-2) \).
21. Sum of an Infinite Geometric Series
Find the sum of the infinite geometric series:
\( \frac{1}{2} - \frac{1}{4} + \frac{1}{8} - \frac{1}{16} + \ldots \).
22. Properties of a Prime Number
Prove that for any prime number \(p\), the number \( \frac{p-1}{2} \) is odd if and only if \( p \equiv 3 \pmod{4} \).
23. Arc Length of a Curve
Determine the arc length of the curve \( y = \frac{1}{3} x^3 - x \) from \(x = 0\) to \(x = 2\).
24. Distinct Arrangements
Determine the number of distinct ways to arrange the letters of the word "MATHEMATICS" such that no two "M"s are adjacent.
25. Sum of Cubes Formula
Prove that for all positive integers \(n\), the following is true:
\( 1^3 + 2^3 + \ldots + n^3 = (1 + 2 + \ldots + n)^2 \).
26. Area in the Complex Plane
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. Roots of a Polynomial
Prove that the roots of the polynomial \( P(x) = x^n - a_1 x^{n-1} + a_2 x^{n-2} - \ldots + (-1)^n a_n \) are all real if and only if \( a_i \geq 0 \) for all \( 1 \leq i \leq n \).
28. Improper Integral
Evaluate the improper integral:
\( \int_{-\infty}^{\infty} \frac{1}{{1 + x^2}} \, dx \).
29. Complex Numbers
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. Infinitely Many Primes
Show that there are infinitely many prime numbers of the form \(4k + 3\), where \(k\) is an integer.