JEE 2023 Mathematics PYQ — Let be the constant term in the binomial expansion of . If the su… | Mathem Solvex | Mathem Solvex
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JEE 2023 — Mathematics PYQ
JEE | Mathematics | 2023
Let α be the constant term in the binomial expansion of(x−x236)n,n≤15.If the sum of the coefficients of the remaining terms in theexpansion is 649 and the coefficient of x−n is λα,then λ is equal to ________.
Choose the correct answer:
A.
36
(Correct Answer)
B.
35
C.
34
D.
33
Correct Answer:
36
Explanation
Step 1: n ki value nikaalna
Kisi bhi binomial expansion (axp+bxq)n mein coefficients ka sum nikaalne ke liye hum x=1 rakhte hain.
Yahan expression hai: (x−x3/26)n
x=1 rakhne par total sum of coefficients:
S=(1−6)n=(−5)n
Question ke mutabik, "remaining terms" ka sum 649 hai. Iska matlab:
Total Sum−α=649
(−5)n−α=649⟹(−5)n=649+α…(1)
Kyunki n≤15 hai aur humein ek integer value chahiye, (−5)n ki powers check karte hain:
(−5)2=25
(−5)3=−125
(−5)4=625 (Ye 649 ke kaafi kareeb hai)
Agar hum n=4 lein, toh α (constant term) nikaal kar check karte hain.
Step 2: Constant term α nikaalna
General term Tr+1 ka formula:
Tr+1=(rn)(x)n−r(−x3/26)r
Tr+1=(rn)(x1/2)n−r(−6)r(x−3/2)r
Tr+1=(rn)(−6)rx2n−r−23r=(rn)(−6)rx2n−4r
Constant term ke liye x ki power 0 honi chahiye:
2n−4r=0⟹n=4r
Agar r=1 ho, toh n=4. Ye hamari pichli observation se match kar raha hai.
Toh, n=4 aur r=1:
α=(14)(−6)1=4×(−6)=−24
Ab equation (1) mein verify karte hain:
(−5)4=649+(−24)
625=625
Ye perfectly match kar raha hai! Iska matlab n=4 aur α=−24.
Step 3: Coefficient of x−n (yani x−4) nikaalna
Humein x ki power −4 chahiye:
24−4r=−4⟹4−4r=−8⟹4r=12⟹r=3
Coefficient of x−4:
Coeff=(34)(−6)3=4×(−216)=−864
Step 4: λ ki value find karna
Question ke anusar:
Coefficient of x−n=λα
−864=λ×(−24)
λ=−24−864
λ=36
Final Answer:
λ=36
Explanation
Step 1: n ki value nikaalna
Kisi bhi binomial expansion (axp+bxq)n mein coefficients ka sum nikaalne ke liye hum x=1 rakhte hain.
Yahan expression hai: (x−x3/26)n
x=1 rakhne par total sum of coefficients:
S=(1−6)n=(−5)n
Question ke mutabik, "remaining terms" ka sum 649 hai. Iska matlab:
Total Sum−α=649
(−5)n−α=649⟹(−5)n=649+α…(1)
Kyunki n≤15 hai aur humein ek integer value chahiye, (−5)n ki powers check karte hain:
(−5)2=25
(−5)3=−125
(−5)4=625 (Ye 649 ke kaafi kareeb hai)
Agar hum n=4 lein, toh α (constant term) nikaal kar check karte hain.
Step 2: Constant term α nikaalna
General term Tr+1 ka formula:
Tr+1=(rn)(x)n−r(−x3/26)r
Tr+1=(rn)(x1/2)n−r(−6)r(x−3/2)r
Tr+1=(rn)(−6)rx2n−r−23r=(rn)(−6)rx2n−4r
Constant term ke liye x ki power 0 honi chahiye:
2n−4r=0⟹n=4r
Agar r=1 ho, toh n=4. Ye hamari pichli observation se match kar raha hai.
Toh, n=4 aur r=1:
α=(14)(−6)1=4×(−6)=−24
Ab equation (1) mein verify karte hain:
(−5)4=649+(−24)
625=625
Ye perfectly match kar raha hai! Iska matlab n=4 aur α=−24.
