Directions (61-63): The following questions are accompanied by two statements i.e. statement (I) and statement (II). You have to determine which statement (s) is/are sufficient/necessary to answer the questions.

Two equations are given below
(i) : b$\sqrt{A{\mathrm{^b}}^{\mathrm{}}}\times (C+1)=24$  (C is the largest even prime number)

(ii) : ${(A+X-Y)}^2=49$

Find the sum of A, X and Y

Statement (I) : Roots of the equation Xm³ + Ym² − 2 m − 3 = 0 are (m-1) and (m+1)

Statement (II) : A = Y + 5 , 3 X + A = 14

To find the sum of $A$, $X$, and $Y$, let us first analyze the information given in the main body of the question.

Step 1: Simplify Equation (i)
Equation (i) is given as:
$b\sqrt{A{b}^{}}\times (C+1)=24$
We are given that $C$ is the largest even prime number. The only even prime number is $2$, so:
$C=2$
Substituting this value back into the equation:
$b\sqrt{A{b}^{}}\times (2+1)=24$
$b\sqrt{A{b}^{}}\times 3=24$
$b\sqrt{A{b}^{}}=8$
Assuming $b$ is the index of the root (i.e., the $b^{\text{th}}$ root of $A$, represented as $\sqrt[b]{A}$):
$\sqrt[b]{A}=\frac{8}{C+1}=\frac{8}{3}$ (which is not an integer), or if it is written as $A^{1/b}\times 3=24\Rightarrow A^{1/b}=8\Rightarrow A=8^b$.
If $b=2$, then $A=8^2=64$. If $b=1$, then $A=8$. Thus, $A$ cannot be uniquely determined from Equation (i) alone without knowing $b$.

Step 2: Analyze Equation (ii)
Equation (ii) is given as:
${(A+X-Y)}^2=49$
Taking the square root on both sides:
$A+X-Y=\pm 7$

Step 3: Evaluate Statement (I)
Statement (I) states: Roots of the equation $X{m}^3+Y{m}^2-2m-3=0$ are $(m-1)$ and $(m+1)$ (which means the factors are $m=1$ and $m=-1$).
If $m=1$ is a root:
$X{\left(1\right)}^3+Y{\left(1\right)}^2-2\left(1\right)-3=0\Rightarrow X+Y=5$
If $m=-1$ is a root:
$X{(-1)}^3+Y{(-1)}^2-2(-1)-3=0\Rightarrow -X+Y-1=0\Rightarrow -X+Y=1$
Solving these two equations:
Adding them gives $2Y=6\Rightarrow Y=3$.
Substituting $Y=3$ gives $X=2$.
We get the values of $X$ and $Y$, but we do not know the value of $A$. Therefore, Statement (I) alone is not sufficient to find the sum of $A$, $X$, and $Y$.

Step 4: Evaluate Statement (II)
Statement (II) states:
$A=Y+5\Rightarrow A-Y=5$
And:
$3X+A=14$
Substitute $A-Y=5$ into Equation (ii):
$(A-Y)+X=\pm 7\Rightarrow 5+X=\pm 7$
This gives two cases:
1) $5+X=7\Rightarrow X=2$
2) $5+X=-7\Rightarrow X=-12$

Let us check both cases using the second equation in Statement (II), $3X+A=14$:
Case 1: $X=2$
$3\left(2\right)+A=14\Rightarrow A=8$
Using $A-Y=5$:
$8-Y=5\Rightarrow Y=3$
Here, the sum of $A+X+Y=8+2+3=13$.

Case 2: $X=-12$
$3(-12)+A=14\Rightarrow A=50$
Using $A-Y=5$:
$50-Y=5\Rightarrow Y=45$.

Now we can verify if the values of $A$ satisfy Equation (i) which is simplified to $A^{1/b}=8$ for integer variables:
For Case 1, $A=8$ is a solution (with $b=1$), whereas $A=50$ is not a power of $8$ for any integer $b$.
Thus, Case 1 ($A=8$, $X=2$, $Y=3$) is the unique solution. Consequently, Statement (II) alone is sufficient to answer the question, whereas Statement (I) alone is not sufficient.