JEE 2023 — Mathematics PYQ

Binomial expansion $(a+b{)}^n$ ke liye, general term hota hai:

Yahan $n=680$, $a=3^{1/2}$, aur $b=5^{1/4}$ hai. Isliye:

2. Integrality ki shart (Condition):

Ek term "integral" tabhi hoga jab $3$ aur $5$ ki powers integers (purnank) hongi.

• $\frac{680-r}{2}$ ek integer hona chahiye $⟹(680-r)$ 2 se divisible hona chahiye. Iska matlab $r$ ek even number hona chahiye.

• $\frac{r}{4}$ ek integer hona chahiye $⟹r$ 4 se divisible hona chahiye.

Dono conditions ko milane par, $r$ ko 4 ka multiple hona chahiye (kyunki jo 4 se divide hoga, wo 2 se apne aap ho jayega).

3. $r$ ki values find karna:

Hume pata hai ki binomial expansion mein $0\le r\le n$, yani:

$r$ ki possible values jo 4 se divisible hain:

4. Number of terms count karna:

Yeh ek Arithmetic Progression (AP) hai jahan:

• First term ($a$) = $0$

• Last term ($l$) = $680$

• Common difference ($d$) = $4$

Formula: $l=a+(n-1)d$