The file you uploaded contains a mathematics exam paper for Class XI, Pure Mathematics. Below are the solutions for each section and selected problems:

Section A (Multiple Choice)

1. For $|A|$:

   • If $|A| = 1$, the correct alternative is C.

2. For real and rational solutions:

   • A (Real and rational).

3. For trigonometric identity $\cos \theta = \frac{13}{12}$, and $\sin \theta$:

   • $\sin \theta$ should be calculated using the identity $\sin^2 \theta + \cos^2 \theta = 1$, resulting in C.

4. For $\lim_{x \to 2} \frac{x^2 + 5x + 6}{2x^2 - 3x}$, simplify and calculate the limit:

   • The result is B (11).

5. The slope of the curve $y = 2x^2$ at $x = 1$:

   • First derivative is $dy/dx = 4x$, so slope is $4 \times 1 = 4$. The correct alternative is A (4).

6. For the integral $\int (2x + 4)\sqrt{x^2 + 4x + 3} dx$:

   • The correct answer is A.

7. For the quadratic equation roots $2$ and $3$:

   • The quadratic is $(x - 2)(x - 3) = x^2 - 5x + 6$, so the answer is A.

Section B

1. Cramer's Rule for the system of equations: $x + 2y - z = 3, \quad 3x - y + 2z = 1, \quad 2x - 2y + 3z = 3$ Steps:

   • Form the coefficient matrix and compute the determinant. Use the determinant to solve for $x$, $y$, and $z$.

2. Collinear points:

   • The given points are $(k, 1)$, $(2, -3)$, $(3, 4)$. To find $k$, use the condition of collinearity by equating the slopes between pairs of points.

3. Distance between parallel lines:

   • For lines $3x + 4y + 3 = 0$ and $3x + 4y + 15 = 0$, the distance is: $\text{Distance} = \frac{|3 - 15|}{\sqrt{3^2 + 4^2}} = 3$

4. Logarithmic form of $53 = 125$:

   • The logarithmic form is $\log_5{125} = 3$.

These are some of the main solutions for the questions in the exam. Let me know if you'd like details or specific steps for any particular question.

5 Further Questions

1. Can you calculate other limits and slopes given different equations?
2. How do you solve for complex numbers like $(3 - 4i)(x + yi) = 1 + 0i$?
3. What is the best approach to solving Cramer's Rule for systems of equations?
4. How can the properties of determinants be applied to problems like the one in Question 8?
5. What is the method to calculate probabilities involving combinatorics?

Tip

For trigonometric problems, always try to use basic identities such as $\sin^2 \theta + \cos^2 \theta = 1$ for faster solutions.