Is Binomial Theorem part of the EAPCET Mathematics syllabus?

Yes. Binomial Theorem is taught in Intermediate second year · Paper IIA Mathematics, and the EAPCET Engineering syllabus is based on the Intermediate first- and second-year syllabus. Check the official notification each year for any changes.

How many Mathematics questions are there in EAPCET?

80 of the 160 questions in the Engineering paper are Mathematics, spread across 37 chapters.

What mistakes do students make in Binomial Theorem?

Treating Tᵣ as the general term ⁿCᵣ aⁿ⁻ʳ bʳ; that expression is Tᵣ₊₁, the (r + 1)th term. Forgetting to raise the constant in the second term to the power r, such as dropping the 2ʳ or the (−1)ʳ in the coefficient. Counting the middle term as the (n/2)th; for even n there are n + 1 terms, so the middle one is the (n/2 + 1)th. Using the rational-index expansion of (1 + x)ⁿ when |x| ≥ 1, where the series does not converge.

EAPCET Mathematics · Intermediate second year · Paper IIA

Binomial Theorem

Q: How should I prepare Binomial Theorem for EAPCET?

Write the general term once, collect the power of x, and solve for r — most coefficient questions are just that. For coefficient-sum questions, substitute x = 1 or x = −1 in the expansion.

Binomial Theorem expands (a + b)ⁿ for positive integers and (1 + x)ⁿ for rational n. Questions centre on the general term, middle terms, coefficients and approximations.

Subject: Mathematics
Taught in: Intermediate second year · Paper IIA
Mathematics in the paper: 80 of 160 questions

Key formulas and results

General term: $T_{r+1} = {}^{n}C_{r}\, a^{n-r} b^{r}$
$C_{0} + C_{1} + \dots + C_{n} = 2^{n}$
$C_{0} + C_{2} + C_{4} + \dots = C_{1} + C_{3} + C_{5} + \dots = 2^{n-1}$
Middle term: $T_{n/2 + 1}$ when $n$ is even; $T_{(n+1)/2}$ and $T_{(n+3)/2}$ when $n$ is odd
$(1 + x)^{-1} = 1 - x + x^{2} - x^{3} + \dots$ for $|x| < 1$
For small $x$ , $(1 + x)^{n} \approx 1 + nx$
$\sum r \cdot {}^{n}C_{r} = n \cdot 2^{n-1}$ and $\sum r^{2} \cdot {}^{n}C_{r} = n(n + 1) \cdot 2^{n-2}$
$\displaystyle\sum_{r=0}^{n} \dfrac{{}^{n}C_{r}}{r + 1} = \dfrac{2^{n+1} - 1}{n + 1}$
$C_{0}^{2} + C_{1}^{2} + \dots + C_{n}^{2} = {}^{2n}C_{n}$
$C_{0}C_{r} + C_{1}C_{r+1} + \dots + C_{n-r}C_{n} = {}^{2n}C_{n-r}$ ; in particular $C_{0}C_{1} + C_{1}C_{2} + \dots + C_{n-1}C_{n} = {}^{2n}C_{n-1}$
Vandermonde: ${}^{n}C_{0} \cdot {}^{m}C_{r} + {}^{n}C_{1} \cdot {}^{m}C_{r-1} + \dots + {}^{n}C_{r} \cdot {}^{m}C_{0} = {}^{m+n}C_{r}$
General term of $(1 - x)^{-m}$ : ${}^{m+r-1}C_{r}\, x^{r}$ ; of $(1 + x)^{-m}$ : $(-1)^{r}\,{}^{m+r-1}C_{r}\, x^{r}$ $(|x| < 1)$
Multinomial: the coefficient of $x_{1}^{a_{1}} x_{2}^{a_{2}} \cdots x_{k}^{a_{k}}$ in $(x_{1} + x_{2} + \dots + x_{k})^{n}$ is $\dfrac{n!}{a_{1}!\,a_{2}! \cdots a_{k}!}$ , with $a_{1} + \dots + a_{k} = n$
$\dfrac{{}^{n}C_{r}}{{}^{n}C_{r-1}} = \dfrac{n - r + 1}{r}$ and ${}^{n}C_{r} = \dfrac{n}{r} \cdot {}^{n-1}C_{r-1}$

What this chapter covers

Binomial theorem for a positive integral index
General term and middle terms
Properties of binomial coefficients
Numerically greatest term
Binomial theorem for a rational index
Approximations

How it is usually asked

Find the coefficient of xᵏ in an expansion
Find the term independent of x
Evaluate sums of binomial coefficients
Approximate values such as √(1.02)

Understanding Binomial Theorem

For a positive integer n, (a + b)ⁿ expands into n + 1 terms, and the (r + 1)th term is Tᵣ₊₁ = ⁿCᵣ aⁿ⁻ʳ bʳ. The coefficients ⁿC₀, ⁿC₁, …, ⁿCₙ read the same from both ends, rise to the middle and fall again, which is why the middle term or terms carry the largest binomial coefficient. When n is even there is one middle term; when n is odd there are two.

Coefficient questions all follow one routine: write the general term, collect the power of x it contains, set that power equal to the one required and solve for r. A term independent of x has power zero. Sums of coefficients come from substituting values: x = 1 gives C₀ + C₁ + … + Cₙ = 2ⁿ, and x = −1 shows the even-place and odd-place sums are equal, each 2ⁿ⁻¹.

For a negative or fractional index, (1 + x)ⁿ becomes an infinite series 1 + nx + n(n − 1)x²/2! + …, valid only when |x| < 1. Keeping the first two or three terms gives quick approximations such as √(1.02) ≈ 1.01. EAPCET questions ask for specific coefficients, independent terms, middle terms, coefficient sums and such approximations.

Worked example

EAPCET-style question

The term independent of $x$ in the expansion of $\left(x^{2} + \dfrac{2}{x}\right)^{6}$ is

(1)60
(2)160
(3)240
(4)480

Solution

General term: $T_{r+1} = {}^{6}C_{r}\,(x^{2})^{6-r}\left(\dfrac{2}{x}\right)^{r} = {}^{6}C_{r}\, 2^{r} x^{12-3r}$ .
For the term independent of $x$ , $12 - 3r = 0$ , so $r = 4$ .
$T_{5} = {}^{6}C_{4} \times 2^{4} = 15 \times 16 = 240$ .

Answer: 240

Shortcuts and results to remember

The $k$ th term from the end of $(a + b)^{n}$ is the $(n - k + 2)$ th term from the beginning.
Remainder of $a^{b}$ on division by $c$ : write a power of $a$ as (multiple of $c \pm 1$ ) and expand; every term except the last is a multiple of $c$ .
$C_{0}^{2} - C_{1}^{2} + C_{2}^{2} - \dots + (-1)^{n}C_{n}^{2} = 0$ if $n$ is odd and $(-1)^{n/2}\,{}^{n}C_{n/2}$ if $n$ is even.
For a negative or fractional index, $(a + b)^{n} = a^{n}\left(1 + \dfrac{b}{a}\right)^{n}$ is valid only when $\left|\dfrac{b}{a}\right| < 1$ .

Common mistakes to avoid

Treating Tᵣ as the general term ⁿCᵣ aⁿ⁻ʳ bʳ; that expression is Tᵣ₊₁, the (r + 1)th term.
Forgetting to raise the constant in the second term to the power r, such as dropping the 2ʳ or the (−1)ʳ in the coefficient.
Counting the middle term as the (n/2)th; for even n there are n + 1 terms, so the middle one is the (n/2 + 1)th.
Using the rational-index expansion of (1 + x)ⁿ when |x| ≥ 1, where the series does not converge.

How to prepare it

Write the general term once, collect the power of x, and solve for r — most coefficient questions are just that.

For coefficient-sum questions, substitute x = 1 or x = −1 in the expansion.

Once the formulas feel familiar, test them the way the exam does: practise Binomial Theorem questions chapter-wise, then attempt a full previous year paper to see whether it holds up under a 3-hour timer.

Related Mathematics chapters

See every chapter in the EAPCET Mathematics syllabus.

Practise Binomial Theorem chapter-wise

Chapter-wise quizzes show exactly which topics in this chapter you still get wrong — before a full paper does.

Frequently asked questions

Is Binomial Theorem part of the EAPCET Mathematics syllabus?: Yes. Binomial Theorem is taught in Intermediate second year · Paper IIA Mathematics, and the EAPCET Engineering syllabus is based on the Intermediate first- and second-year syllabus. Check the official notification each year for any changes.
How should I prepare Binomial Theorem for EAPCET?: Write the general term once, collect the power of x, and solve for r — most coefficient questions are just that. For coefficient-sum questions, substitute x = 1 or x = −1 in the expansion.
How many Mathematics questions are there in EAPCET?: 80 of the 160 questions in the Engineering paper are Mathematics, spread across 37 chapters.
What mistakes do students make in Binomial Theorem?: Treating Tᵣ as the general term ⁿCᵣ aⁿ⁻ʳ bʳ; that expression is Tᵣ₊₁, the (r + 1)th term. Forgetting to raise the constant in the second term to the power r, such as dropping the 2ʳ or the (−1)ʳ in the coefficient. Counting the middle term as the (n/2)th; for even n there are n + 1 terms, so the middle one is the (n/2 + 1)th. Using the rational-index expansion of (1 + x)ⁿ when |x| ≥ 1, where the series does not converge.