A, B, C......Z represent the numbers from 1, 2, 3....26. There is an encryption technique which uses a key. The key can be any number less than 26 and greater than 0. For a message ABCD, if the key is 1, the encrypted message becomes BCDE. Similarly, if the key is 5 and the message is XYZA, the encrypted message becomes CDEF. If you get an encrypted message from a friend as 'FCJN'. This was encrypted using a key of 24. The original message is _.

$\left|z-z_{1}\right|^{2} + \left|z-z_{2}\right|^{2} = a$ will represent a real circle on the Argand plane if

(A) $a \geq \left|z_{1} - z_{2}\right|^{2}$ (B) $2a \geq \left|z_{1} - z_{2}\right|^{2}$ (C) $a \geq \left|z_{1} - z_{2}\right|$ (D) $2a \geq \left|z_{1} - z_{2}\right|$

Let $A, B, C$ be three sets of complex numbers defined as $$ \begin{array}{l} A={z: \operatorname{Im}(z) \geq 1}, B={z:|z-2-i|=3}, \ C={z: \operatorname{Re}((1-i) z)=\sqrt{2}}. \end{array} $$

If $z$ be any point in $A \cap B \cap C$, then $|z+1-i|^{2}+|z-5-i|^{2}$ lies between A 25 and 29 B 30 and 34 C 35 and 39 D 40 and 44

The element Lawrencium (${103}^{256} \text{Lr}$) is named after Ernest Lawrence, whose invention, the cyclotron, helped discover many radioactive nuclei. Consider now the alpha decay of ${103}^{256} \text{Lr}$ as:

$$ _{103}^{256} \text{Lr} \rightarrow _{b}^{a} \text{Md} + \text{a}, $$

where \textit{a} represents the alpha particle given out. The values of \textit{a} and \textit{b} are:

A. $a=252$; $b=105$
B. $a=252$; $b=101$
C. $a=260$; $b=101$
D. $a=260$; $b=105$

Let $C_{1}$ and $C_{2}$ are concentric circles of radius 1 unit and $8 / 3$ unit, respectively, having centre at (3,0) on the Argand plane. If the complex number $z$ satisfies the inequality $\log {1 / 3}\left(\frac{|z-3|^{2}+2}{11|z-3|-2}\right)>1$ then: A $z$ lies outside $C{1}$ but inside $C_{2}$ B $z$ lies inside $C_{1}$ C $z$ lies outside $C_{2}$ D none of these

The value of $i^{1+3+5+\ldots . .+(2 n+1)}$ is:

A. $i$ if $n$ is even, $-i$ if $n$ is odd B. 1 if $n$ is even, -1 if $n$ is odd C. 1 if $n$ is odd, -1 if $n$ is even D. $i$ if $n$ is even, -1 if $n$ is odd