Find the radius of convergence of the series \[ \sum_{n=0}^{\infty} \frac{(n!)^2}{(2n)!}\, x^n. \]

Determine whether the sequence \[ a_n = 1 - (-1)^n + \frac{1}{n} \]
is convergent or divergent.

Let \(G = P(N)\), where the operation is
\[ A \Delta B = A \cup B - A \cap B \]
Which of the following is true?

• (A) \(G\) is abelian but not cyclic
• (B) \(G\) has elements of order 4
• (C) \(G\) has elements of order 8
• (D) \(\emptyset\) is the identity element of \(G\)

Solve the system: \[ x + 2y + 2z = 1 \] \[ 2x + 3y + 2z = 2 \] \[ ax + 5y + bz = b \]
Find \(a + b\) for infinite solutions.

If \[ f(x) = \big( f(x) - \pi x \big) + \pi, \]
then the possible value(s) of \( f(3) - f(2) \) is/are:

• (A) \( \pi + \dfrac{1}{6} \)
• (B) \( \pi - \dfrac{1}{6} \)
• (C) \( \dfrac{\pi}{2} + 1 \)
• (D) \( \dfrac{\pi}{6} \)

Evaluate: \[ {}^{5}C_{0} + {}^{6}C_{1} + {}^{7}C_{2} + {}^{8}C_{3} + {}^{9}C_{4} + {}^{10}C_{5} + {}^{11}C_{6}. \]

Which of the following statements are false?

• (A) \(S_3\) is a subgroup of \(S_4\)
• (B) \(\mathbb{Z}_3\) is a subgroup of \(S_4\)
• (C) \(S_3\) is a quotient group of \(S_4\)
• (D) \(\mathbb{Z}_6\) is a quotient group of \(S_4\)

Find the number of automorphisms of the cyclic group \(\mathbb{Z}_n\) for \(n = 30\).

Let \(P\) be a \(5 \times 5\) matrix such that \(\det(P) = 2\).
If \(Q\) is the cofactor matrix of \(P\), then find \(\det(Q)\).

Given that the solution of \[ \frac{d^2y}{dx^2} + \alpha \frac{dy}{dx} + \beta y = -e^{-x} \]
is \[ y(x) = C_1 e^{-x} + C_2 e^{2x} + x e^{-x}, \]
find the values of \(\alpha\) and \(\beta\).

There are four different types of bananas. In how many ways can 12 children select bananas so that at least one child selects different types of bananas?

If the function \( f(x) \) satisfies \[ f'(x) = f(x) - \pi x + \pi, \]
then the possible value of \( f(1) \) is

• (A) \( \pi + \frac{1}{6} \)
• (B) \( \pi - \frac{1}{6} \)
• (C) \( \frac{\pi}{2} + 1 \)
• (D) \( 1 - \frac{1}{2} \)

There are four different types of bananas. In how many ways can 12 children select bananas so that at least one banana is selected from each type?

Find the radius of convergence of the series

\[ \sum_{n=0}^{\infty} \frac{\binom{n}{6}^2}{(2n)!} x^n \]

Given
\[ y = -3x - 3 + m e^{2x} \]

Find the Orthogonal Trajectories (O.T.).

Let \[ A = \begin{pmatrix} 0 & 1 & 0 & 0 & 0
1 & 0 & 0 & 0 & 0
0 & 0 & 0 & 1 & 0
0 & 0 & 0 & 0 & 1
0 & 0 & 1 & 0 & 0 \end{pmatrix}. \]
Which of the following statements is correct?

• (A) \( A \) has four distinct eigenvalues in \( \mathbb{C} \).
• (B) \( A \) has three distinct eigenvalues in \( \mathbb{R} \).
• (C) \( (A - I) \) has nullity 3.
• (D) \( A \) has two real and three complex eigenvalues.

Let \( P \) be a \(6 \times 4\) matrix and \( Q \) be a \(4 \times 6\) matrix such that \( PQ = 0 \). Which of the following statements is correct?

• (A) Row space\((P)\subseteq\) Null space\((Q)\)
• (B) Column space\((P)\subseteq\) Null space\((Q)\)
• (C) \( r(P) + r(Q) \ge 4 \)
• (D) \( r(P) + r(Q) = 4 \)