Use app×
Join Bloom Tuition
One on One Online Tuition
JEE MAIN 2025 Foundation Course
NEET 2025 Foundation Course
CLASS 12 FOUNDATION COURSE
CLASS 10 FOUNDATION COURSE
CLASS 9 FOUNDATION COURSE
CLASS 8 FOUNDATION COURSE
0 votes
4.3k views
in Binomial theorem by (46.6k points)
edited by

If the sum of the coefficients in the expansion of (αx2 − 2x + 1)35 is equal to the sum of the coefficients in the expansion of (x − αy)35, then find the value of α .

1 Answer

+1 vote
by (52.7k points)
selected by
 
Best answer

Sum of the coefficients in the expansion of (α x2 − 2x + 1)35

= Sum of the coefficients in the expansion of (x − αy)35 Putting x = y = 1. Therefore,

(α− 1)35 = (1 − α)35

⇒ (α − 1)35 = − (α − 1)35

⇒ 2(α − 1)35 = 0

⇒ α − 1 = 0 or α   = 1

The sum of the coefficients of the odd terms in the expansion of (1 + x)n is equal to the sum of the coefficients of the even terms and each is equal to 2n−1

Since, (1 + x) n = C0 + C1 + C2x2 + C3x3 + … + Cnxn Putting x = −1,

0 = C0 − C1 + C2 − C3 + … + (−1)n Cn

and 2n = C0 + C1 + C2 + C3 + … + Cn

{from Eq. (2)} 

Adding and subtracting these two equations, we get

2n = 2 (C0 + C2 + C4 + …) and 2n = 2(C1 + C3 + C5 + …)

Therefore, 

C0 + C2 + C4 + … = C1 + C3 + C5 + … = 2n−1

Sum of coefficients of odd terms = sum of coefficients of even terms = 2n−1

Welcome to Sarthaks eConnect: A unique platform where students can interact with teachers/experts/students to get solutions to their queries. Students (upto class 10+2) preparing for All Government Exams, CBSE Board Exam, ICSE Board Exam, State Board Exam, JEE (Mains+Advance) and NEET can ask questions from any subject and get quick answers by subject teachers/ experts/mentors/students.

...