Let A be a 3 \(\times\) 3 matrix such that \(A^2 = I\). If det(A) = \(-\)1 and the sum of eigenvalues is 1, find the set of eigenvalues of A.

• (A) 1, 1, 1
• (B) 1, 1, \(-\)1
• (C) \(-\)1, \(-\)1, 1
• (D) \(-\)1, \(-\)1, \(-\)1

Analyze the function \(f(x) = |x - 1| + |x - 2| + |x - 3|\). Determine the points where the derivative \(f'(x)\) is undefined.

• (A) \(x = 1, 3\)
• (B) \(x = 1, 2, 3\)
• (C) \(x = 1.5, 2.5\)
• (D) \(x = 0\)

Identify the order and degree of the differential equation:

\(\left(\frac{d^3y}{dx^3}\right)^2 + 4\left(\frac{dy}{dx}\right)^4 + y = \sin(x)\)

• (A) Order 3, Degree 4
• (B) Order 3, Degree 2
• (C) Order 4, Degree 3
• (D) Order 1, Degree 4

For the linear differential equation \[ \frac{dy}{dx} + \frac{2}{x}y = x^2 \]
calculate the integrating factor (I.F.).

• (A) \( x \)
• (B) \( x^2 \)
• (C) \( x^3 \)
• (D) \( \ln(x) \)

Maximize \( Z = 5x + 3y \) subject to \( x + y \le 6 \) and \( x, y \ge 0 \). Where does the maximum value occur?

• (A) \( (0,0) \)
• (B) \( (0,6) \)
• (C) \( (6,0) \)
• (D) \( (3,3) \)

A bag has 4 red and 6 black balls. Two balls are drawn without replacement. What is the probability that the second is red given the first was black?

• (A) \( 4/10 \)
• (B) \( 4/9 \)
• (C) \( 6/9 \)
• (D) \( 2/9 \)

Find the domain of the function \(f(x) = \sin^{-1}(3x - 1)\).

• (A) [-1, 1]
• (B) [0, 2/3]
• (C) [-1/3, 1/3]
• (D) [1/3, 1]

For vectors \(\vec{a} = 3\hat{i} - \hat{j} + 2\hat{k}\) and \(\vec{b} = \hat{i} + 2\hat{j} - \hat{k}\), find the scalar projection of \(\vec{a}\) onto \(\vec{b}\).

• (A) -1/\(\sqrt{6}\)
• (B) 1/\(\sqrt{6}\)
• (C) -1/6
• (D) 1/6

Let R be a relation on {1,2,3} defined by R = {(1,1), (2,2), (3,3), (1,2)}. Identify the properties satisfied by R.

• (A) Symmetric only
• (B) Reflexive and Transitive
• (C) Equivalence relation
• (D) None of these

Find the local maximum point of the function f(x) = -x³ + 3x + 1

• (A) x = 1
• (B) x = -1
• (C) x = 0
• (D) x = \(\sqrt{3}\)