The series {n=1}^∞ (1/n) is : - Solved MCQ | Agrigyan

In complex analysis, considering complex numbers $z_1$ and $z_2$ as 2D vectors in the Argand plane, the dot product (scalar product) is defined as:
$z_1 \cdot z_2 = |z_1||z_2|\cos\theta = \text{Re}(z_1 \bar{z}_2)$,
where $\theta$ is the angle between them.

Detailed analysis of options:

• $|z_1||z_2|$ is the product of magnitudes, which only equals the dot product when $\theta = 0$.
• $|z_1||z_2|\sin\theta$ is the magnitude of the cross product (or $\text{Im}(z_1 \bar{z}_2)$).
• $|z_1||z_2|\cos\theta$ is the correct standard formula for the dot product.
• $|z_1||z_2|\tan\theta$ is incorrect.

Simplifying each term by multiplying numerator and denominator by the complex conjugate:
First term: $\frac{5+5i}{3-4i} = \frac{(5+5i)(3+4i)}{3^2 + (-4)^2} = \frac{15 + 20i + 15i + 20i^2}{25} = \frac{15 - 20 + 35i}{25} = \frac{-5 + 35i}{25} = -\frac{1}{5} + \frac{7}{5}i$.
Second term: $\frac{20}{4+3i} = \frac{20(4-3i)}{4^2 + 3^2} = \frac{80 - 60i}{25} = \frac{16}{5} - \frac{12}{5}i$.
Adding the two terms:
$\left(-\frac{1}{5} + \frac{7}{5}i\right) + \left(\frac{16}{5} - \frac{12}{5}i\right) = \frac{-1+16}{5} + \frac{7-12}{5}i = \frac{15}{5} - \frac{5}{5}i = 3 - i$.

Detailed analysis of options:

• $3 - i$ is correct.
• $1 + i$ is incorrect.
• $\frac{5}{2} - \frac{i}{2}$ is incorrect.
• $-2 + 9i$ is incorrect.

A subset $W$ of a vector space $V$ is a subspace if and only if:

• The zero vector $\mathbf{0} \in W$.
• It is closed under vector addition: $\mathbf{u}, \mathbf{v} \in W \implies \mathbf{u} + \mathbf{v} \in W$.
• It is closed under scalar multiplication: $c \in \mathbb{R}, \mathbf{u} \in W \implies c\mathbf{u} \in W$.
  For $W = \{(a_1, \dots, a_n) \mid \sum a_i = 0\}$:
  Condition: $\sum 0 = 0 \implies \mathbf{0} \in W$.
  Condition: If $\sum u_i = 0$ and $\sum v_i = 0$, then $\sum (u_i + v_i) = \sum u_i + \sum v_i = 0$.
  Condition: $\sum c u_i = c \sum u_i = 0$.