1. Which equation represents the eigenvalues of the matrix \( \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix} \)?

1. \(\lambda^2 – 4\lambda + 3 = 0\)
2. \(\lambda^2 – 4\lambda + 4 = 0\)
3. \(\lambda^2 – 3\lambda + 2 = 0\)
4. \(\lambda – 3 = 0\)

2. The area of the region bounded by the curves \( y = x^2 \) and \( y = x \) is:

1. 0.1667 square units
2. 0.3333 square units
3. 0.5 square units
4. 1 square unit

3. Solve the linear system using matrix inversion method: \( x + 2y = 5 \) and \( 3x + 4y = 11 \).

1. \(x = 1, y = 2\)
2. \(x = 2, y = 1.5\)
3. \(x = 3, y = 1\)
4. \(x = 7, y = -1\)

4. What is the Fourier transform of \( f(t) = e^{-|t|} \)?

1. \(\frac{2}{1 + \omega^2}\)
2. \(\frac{1}{1 + \omega^2}\)
3. \(\frac{2}{\omega^2 – 1}\)
4. \(\frac{2\omega}{1 + \omega^2}\)

5. A function \( f(x) \) is defined as \( f(x) = \left\{ \begin{array}{ll} x^2 & \quad x < 1 \\ 2x + 1 & \quad x \geq 1 \end{array} \right. \). Find the limit \( \lim_{x \to 1} f(x) \).

1. 1
2. 2
3. 3
4. Does not exist

6. Calculate the derivative of the function \( g(x) = \ln(x^2 + 1) \).

1. \(\frac{2x}{x^2 + 1}\)
2. \(\frac{2}{x^2 + 1}\)
3. \(\frac{x}{x^2 + 1}\)
4. \(\frac{1}{2x(x^2 + 1)}\)

7. The integral \( \int \frac{x^2}{x^2 + 1} dx \) can be solved by:

1. Partial fractions
2. Substitution
3. Integration by parts
4. Direct integration

8. A probability function \( P(x) \) is defined for \( x = 0, 1, 2, \ldots \). If \( E(X) = 1 \) and \( \operatorname{Var}(X) = 2 \), what is the distribution of \( X \)?

1. Poisson distribution
2. Binomial distribution
3. Normal distribution
4. Exponential distribution

9. The solution to the differential equation \( \frac{d^2y}{dx^2} – 4\frac{dy}{dx} + 4y = 0 \) is:

1. \(y = (C_1 + C_2x)e^{2x}\)
2. \(y = C_1 e^{2x} + C_2 e^{-2x}\)
3. \(y = C_1 e^{4x} + C_2 e^{-4x}\)
4. \(y = C_1 \cos(2x) + C_2 \sin(2x)\)

10. The length of the curve \( y = \ln(\cos x) \) from \( x = 0 \) to \( x = \frac{\pi}{4} \) is:

1. \(\frac{\pi}{4}\)
2. \(\frac{1}{4}\)
3. \(\ln(2)\)
4. Cannot be determined