CUET Mathematics 2024 - An objective function Z=a x+b y is maximum at points (8,2) and (4,6). If a ≥ 0 and b ≥ 0 and a b=25, then the maximum value of the function is equal to : | PYQs + Solutions

CUET Mathematics 2024

Algebra

Linear Programming

Easy

An objective function $Z=a x+b y$ is maximum at points $(8,2)$ and $(4,6)$ . If $a \geq 0$ and $b \geq 0$ and $a b=25$ , then the maximum value of the function is equal to :

60

50

40

80

✅ Correct Option: 2

Since both points

(8,2)

and

(4,6)

give the same maximum value of

Z

, we can write:

Z_{max} = a(8) + b(2) = a(4) + b(6)

\newline

8a + 2b = 4a + 6b

\newline

4a = 4b

\newline

a = b

Using the constraint

ab = 25

and knowing that

a = b

a^2 = 25

\newline

a = 5

(since

a \geq 0

)

\newline

Thus,

b = 5

as well.

Calculating the maximum value:

Z_{max} = a(8) + b(2) = 5(8) + 5(2) = 40 + 10 = 50

We can verify using the second point:

Z_{max} = 5(4) + 5(6) = 20 + 30 = 50

Therefore, the maximum value of the function is

50