Middle term and term independent of x: Class 11 practice
In (a + b)ⁿ the general term is Tᵣ₊₁ = ⁿCᵣ aⁿ⁻ʳ bʳ. Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Laws of exponents
- ⁿCᵣ
The idea behind middle term and term independent of x
In (a + b)ⁿ the general term is Tᵣ₊₁ = ⁿCᵣ aⁿ⁻ʳ bʳ. For the term independent of x, set the total power of x to zero and solve for r. If n is even, the middle term is T(n/2 + 1).
Step-by-step method
- Write Tᵣ₊₁.
- Collect the power of x.
- Set it equal to the required power and solve for r.
- Substitute r back.
Worked example 1 (Basic)
Find the middle term of (x + 1)^6.
- n = 6 is even → middle term T4
- T4 = 6C3 x^3 (1)^3 = 20x^3
- Answer: 20x^3
Worked example 2 (Standard)
Find the term independent of x in (x + 2/x)^8.
- Tr+1 = 8Cr x^(8 − 2r) (2)^r
- 8 − 2r = 0 → r = 4
- Term = 1120
- Answer: 1120
Worked example 3 (Challenge)
Find the term independent of x in (x − 1/x)^8.
- Tr+1 = 8Cr x^(8 − 2r) (−1)^r
- 8 − 2r = 0 → r = 4
- Term = 70
- Answer: 70
Common mistakes and misconceptions
- Using r instead of r + 1 for the term number
- Losing the coefficient's power
Practice questions
- (Basic) Find the middle term of (x + 1)^10.
- (Basic) Find the middle term of (x + 3)^4.
- (Standard) Find the term independent of x in (x + 3/x)^6.
- (Standard) Find the term independent of x in (x + 3/x)^4.
- (Challenge) Find the term independent of x in (x − 3/x)^10.
- (Challenge) Find the term independent of x in (x − 1/x)^6.
Answer key with full working
- Q1: n = 10 is even → middle term T6 → T6 = 10C5 x^5 (1)^5 = 252x^5 → Answer: 252x^5
- Q2: n = 4 is even → middle term T3 → T3 = 4C2 x^2 (3)^2 = 54x^2 → Answer: 54x^2
- Q3: Tr+1 = 6Cr x^(6 − 2r) (3)^r → 6 − 2r = 0 → r = 3 → Term = 540 → Answer: 540
- Q4: Tr+1 = 4Cr x^(4 − 2r) (3)^r → 4 − 2r = 0 → r = 2 → Term = 54 → Answer: 54
- Q5: Tr+1 = 10Cr x^(10 − 2r) (−3)^r → 10 − 2r = 0 → r = 5 → Term = -61236 → Answer: -61236
- Q6: Tr+1 = 6Cr x^(6 − 2r) (−1)^r → 6 − 2r = 0 → r = 3 → Term = -20 → Answer: -20
Syllabus note
This skill is listed under "Binomial Theorem" for Class 11 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.
Related practice and notes
Frequently asked questions
Which chapter is middle term and term independent of x in?
"Binomial Theorem", Class 11 Maths (NCERT/CBSE). Other boards may place it differently.
How were the answers checked?
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.