Arjun and Bhavesh launch a joint business venture. Arjun contributes 25% more initial capital than Bhavesh, but keeps his funds invested for 4 months less than Bhavesh. At the end of the term, Bhavesh receives Rs.4000 as his profit share, which exceeds Arjun's share by Rs.1000. For how many months did Arjun invest his capital?

The correct option is 6 months.

Step 1: Understand the concept of profit sharing
In a joint business venture, profit is divided among partners in proportion to the product of their capital investment and the time period for which the capital was invested.
$\text{Profit Share Ratio}=\frac{{\text{Capital}}_1\times {\text{Time}}_1}{{\text{Capital}}_2\times {\text{Time}}_2}$

Step 2: Express capital and investment duration for both partners
Let Bhavesh's initial capital be $C_B=x$.
Arjun contributes 25% more capital than Bhavesh:
$C_A=x+0.25x=1.25x=\frac{5}{4}x$
Let the time duration for Bhavesh's investment be $T_B=t$ months.
Since Arjun keeps his funds invested for 4 months less than Bhavesh, Arjun's investment duration is:
$T_A=t-4$ months.

Step 3: Calculate the profit share ratio
Bhavesh's profit share = Rs. 4000.
Bhavesh's share exceeds Arjun's share by Rs. 1000, so Arjun's profit share is:
$\text{Arjun's Profit}=4000-1000=\text{Rs.}3000$
The ratio of Arjun's profit share to Bhavesh's profit share is:
$\frac{\text{Arjun's Profit}}{\text{Bhavesh's Profit}}=\frac{3000}{4000}=\frac{3}{4}$

Step 4: Set up the equation and solve for Bhavesh's investment time (t)
Equating the ratio of profit shares to the ratio of (Capital × Time):
$\frac{C_A\times T_A}{C_B\times T_B}=\frac{3}{4}$
Substitute the values into the equation:
$\frac{\frac{5}{4}x\times (t-4)}{x\times t}=\frac{3}{4}$
Cancel out $x$ from both numerator and denominator:
$\frac{5(t-4)}{4t}=\frac{3}{4}$
Multiply both sides by 4:
$\frac{5(t-4)}{t}=3$
Cross-multiply to solve for $t$:
$5(t-4)=3t$
$5t-20=3t$
$5t-3t=20$
$2t=20$
$t=10$
So, Bhavesh invested his capital for 10 months.

Step 5: Determine Arjun's investment duration
Arjun invested for 4 months less than Bhavesh:
$T_A=10-4=6\text{months}$
Therefore, Arjun invested his capital for 6 months.