Question

If $*$ is defined by a$*$ b $=$ $a - {b^2}$ and $\oplus$ is defined by $a \oplus b = {a^2} + b$, where $a$ and $b$ are integers, then $\left( {3 \oplus 4} \right)*$ $5$ is equal to
$A.164$
$B.38$
$C.12$
$D.28$
$E.144$

Answer 1

Hint – In this question, we have to apply the values of the given operations $' \oplus '$ and $'*'$, and then by considering the numbers as $'a'$ and $'b'$. It is a relation or expression involving one or more variables.
Complete step-by-step answer:
It is given that, symbol $*$ is defined by $a*b = a - {b^2}$ and the symbol $'\oplus'$ is defined by $a \oplus b$ $= {a^2}+b$ , where $'a'$ and $'b'$ are integers.
We have to find the value of $\left( {3 \oplus 5} \right)$ $*5$
Firstly, we will calculate $3 \oplus 4$
Let us consider, $a = 3$ and $b = 4$
Since, $a \oplus b = {a^2} + b$
$\Rightarrow 3 \oplus 4 = {3^2} + 4 \\\ \Rightarrow 3 \oplus 4 = 9 + 4 \\\ \Rightarrow 3 \oplus 4 = 13 \\\$
Now, we will find $\left( {3 \oplus 4} \right)*5$
Again, consider as $a = 3 \oplus 4$ that is 13 and $b = 5$
$a*b = a - {b^2} \\\ \Rightarrow 13*5 = 13 - {5^2} \\\ \Rightarrow 13*5 = 13 - 25 \\\ \Rightarrow 13*5 = - 12 \\\$
Therefore, the required values of $\left( {3 \oplus 4} \right)*5$ is -12
Note – In this type of question, one must know that the approach we have used in this particular question is binary operations that is defined as an operation which is performed on a set $A$. The function is given by : AA $\to$ A. So the operation * performed on operands a and b is denoted by
a*b.