Question

If \( a \perp b = a + 2b + 5 \), \( a \top b = 3a - b + 4 \) for all \( a,b \in \mathbb{R} \), then \( \{(3 \perp 4)\top 5\\perp(6\top 7) = \)}

• A. 72
• B. 78
• C. 82
• D. 86

Hint:

For custom operators, always evaluate inner brackets first and substitute values carefully according to the given definitions.

Answer 1

The Correct Option is C

Concept: Evaluate the custom operations one by one using the definitions provided.

Step 1: Calculating \(3 \perp 4\).
Using \[ a\perp b=a+2b+5 \] we get \[ 3\perp4=3+2(4)+5 \] \[ =3+8+5 \] \[ =16 \]

Step 2: Calculating \((3\perp4)\top5\).
\[ 16\top5=3(16)-5+4 \] \[ =48-5+4 \] \[ =47 \]

Step 3: Calculating \(6\top7\).
\[ 6\top7=3(6)-7+4 \] \[ =18-7+4 \] \[ =15 \]

Step 4: Calculating the final expression.
\[ 47\perp15 \] \[ =47+2(15)+5 \] \[ =47+30+5 \] \[ =82 \] {82}