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If A and B are whole numbers such that $9 A^{2} = 12 A + 96$ and $B^{2} = 2 B + 3$ , find the value of $5 A + 7 B$ .

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31

37

41

43

answer is C.

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Detailed Solution

It is given that A and B are whole numbers such that

9 A^{2} = 12 A + 96

and

B^{2} = 2 B + 3

.
Using factorisation method to solve both the given equations.
So,

9 A^{2} = 12 A + 96

\Rightarrow 9 A^{2} - 12 A - 96 = 0

Dividing 3 from all the coefficients,

\Rightarrow 3 A^{2} - 4 A - 32 = 0

\Rightarrow 3 A^{2} - 12 A + 8 A - 32 = 0

\Rightarrow (3 A + 8) (A - 4) = 0

The values of A are

\frac{- 8}{3}

and 4.
Solving the second equation,

B^{2} = 2 B + 3

{\Rightarrow B}^{2} - 2 B - 3 = 0

\Rightarrow B^{2} - 3 B + B - 3 = 0

\Rightarrow (B - 3) (B + 1) = 0

The value of B are 3 and -1.

∴ A = 4, B = 3 .

(

∵ A, B

are whole numbers

∵ (A \neq \frac{- 8}{3})

and

(B \neq - 1

)
To find the value of

5 A + 7 B

we substitute the value of A and B in the expression.

5 A + 7 B = 5 (4) + 7 (3)

= 41

Hence, the correct option is 3.

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