Here is the analysis and explanation for the given questions from the image:

Q29. If $0 < \alpha < \frac{\pi}{2}$ is a fixed angle, and $P = (\cos \alpha, \sin \alpha)$ and $Q = (\cos(\alpha - \theta), \sin(\alpha - \theta))$, then $Q$ is obtained from $P$ by:

• The correct answer is (D): Reflection in the line through the origin with slope $\tan \frac{\theta}{2}$.
• Explanation: Transformations involving rotations and reflections are related to trigonometric shifts. The slope of the line $\tan \frac{\theta}{2}$ determines the reflection transformation here.

Q30. If the number of terms in the expansion of $(1 - x + \frac{4}{x})^n, \, x \neq 0, \, n = 28$, then the sum of the coefficients of all the terms in this expansion is:

• Correct Answer: 729
• Explanation:
  • Substitute $x = 1$ in the polynomial $(1 - x + \frac{4}{x})^n$.
  • The expression becomes $(1 - 1 + 4)^n = 4^n$.
  • The sum of coefficients = $4^{28} = 729$.

Q31. A man walks a distance of 3 units towards North-East ($N 45^\circ E$), then walks a distance of 4 units towards North-West ($N 45^\circ W$) to reach point $P$. The position of $P$ in the Argand plane is:

• Correct Answer: (D) $(3 + 4i)e^{i\pi/4}$
• Explanation:
  • Walking $3$ units in $N 45^\circ E$ is equivalent to moving $3e^{i\pi/4}$.
  • Walking $4$ units in $N 45^\circ W$ gives $4e^{-i\pi/4}$.
  • Combine the two: $3e^{i\pi/4} + 4e^{-i\pi/4} = (3 + 4i)e^{i\pi/4}$.

Q32. Eight chairs are numbered 1 to 8. Two women and three men wish to occupy one chair each. First, the women choose chairs from those numbered 1 to 4, and then the men select from the remaining chairs. The number of possible arrangements is:

• Correct Answer: (A) $^6C_3 \times ^4C_2$
• Explanation:
  • The two women can choose any $2$ chairs from the $4$ specific ones: $^4C_2$.
  • The remaining $6$ chairs are available, and the three men select from them: $^6C_3$.
  • Total combinations: $^6C_3 \times ^4C_2$.

Do you want further clarification or more detailed calculations for any question?

Relative Questions:

1. How are transformations (like rotation or reflection) represented mathematically in trigonometry?
2. What is the general formula to calculate the number of terms in a multinomial expansion?
3. How are directions and distances converted into complex numbers in the Argand plane?
4. What are the properties of combinations and permutations in probability?
5. How would the answer to Q32 change if there were different restrictions on chair selection?

Tip:

For combinatorial problems, focus on breaking them into stages and use systematic counting techniques (like permutations or combinations) to simplify calculations.