JAMIA 2022 — Mathematics PYQ

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Question

JAMIA 2022 — Mathematics PYQ

JAMIA | Mathematics | 2022

If x, 2x + 2, 3x + 3 are in G.P., then the 4th term is

Choose the correct answer:

Explanation

Step 1: Find the value of $x$

If three terms $a, b, c$ are in G.P., then the middle term squared is equal to the product of the other two:

b^2 = ac

Given the terms $x, (2x + 2), (3x + 3)$ , we set up the equation:

(2x + 2)^2 = x(3x + 3)

Expand both sides:

(2(x + 1))^2 = 3x(x + 1)

4(x + 1)^2 = 3x(x + 1)

Step 2: Solve the Equation

We can rearrange the equation to one side:

4(x + 1)^2 - 3x(x + 1) = 0

Factor out $(x + 1)$ :

(x + 1) [4(x + 1) - 3x] = 0

(x + 1) [4x + 4 - 3x] = 0

(x + 1) (x + 4) = 0

This gives two possible values for $x$ :

$x = -1$
$x = -4$

Note: If $x = -1$ , the terms become $-1, 0, 0$ . In a standard G.P., terms are usually non-zero, so we proceed with $x = -4$ .

Step 3: Determine the Terms and Common Ratio

Substitute $x = -4$ into the expressions:

First term ( $a_1$ ) = $x = -4$
Second term ( $a_2$ ) = $2(-4) + 2 = -8 + 2 = -6$
Third term ( $a_3$ ) = $3(-4) + 3 = -12 + 3 = -9$

Now, find the common ratio ( $r$ ):

r = \frac{a_2}{a_1} = \frac{-6}{-4} = \frac{3}{2} = 1.5

Verify with the third term:

-6 \times 1.5 = -9

(Correct)

Step 4: Find the 4th Term

The fourth term ( $a_4$ ) is the third term multiplied by the common ratio:

a_4 = a_3 \times r

a_4 = -9 \times \frac{3}{2}

a_4 = -\frac{27}{2} = -13.5