JAMIA 2024 — Mathematics PYQ

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Question

JAMIA 2024 — Mathematics PYQ

JAMIA | Mathematics | 2024

If A.M. of two numbers is 15/2 and their G.M. is 6, then find the two numbers

Choose the correct answer:

Explanation

Step 1: Let the two numbers be $a$ and $b$

We are given:

Arithmetic Mean (A.M.) = $\frac{15}{2}$
Geometric Mean (G.M.) = $6$

Step 2: Set up the equations

From the definition of A.M.:

\frac{a + b}{2} = \frac{15}{2}

a + b = 15 \quad \text{--- (Equation 1)}

From the definition of G.M.:

\sqrt{ab} = 6

ab = 36 \quad \text{--- (Equation 2)}

Step 3: Form a quadratic equation

The numbers $a$ and $b$ are the roots of the quadratic equation:

x^2 - (a+b)x + ab = 0

Substitute the values from Equation 1 and Equation 2:

x^2 - 15x + 36 = 0

Step 4: Solve the quadratic equation

We factor the equation by finding two numbers that multiply to $36$ and add to $-15$ :

x^2 - 12x - 3x + 36 = 0

x(x - 12) - 3(x - 12) = 0

(x - 12)(x - 3) = 0

So, the values of $x$ are:

x = 12 \quad \text{or} \quad x = 3