Binomial Theorem Practice Questions JEE: 8 Worked Solutions

Should I start with chapter questions, PYQs or mocks?

Use worked chapter drills to repair concepts, authentic previous-year questions to test transfer to exam wording, and full-length mocks to practise whole-paper decisions. The eight free original Binomial Theorem practice questions for JEE below include complete worked solutions on this page, not an unverified download or external question bank.

Start with these questions, not a platform ranking. A full-paper mock is not a substitute for Binomial Theorem practice: it tests paper management, not every method within this chapter.

How do chapter drills, PYQs and full-length mocks compare?

Chapter drills let you target a method, PYQs test it in authentic exam wording, and mocks add whole-paper timing. Choose by the problem you need to fix, not by the size of a question bank.

Primary purpose: Original worked drills on this page repair specific methods. Authentic JEE PYQs test transfer to actual exam questions. Full-length mocks practise whole-paper execution.
Subtopic control: These drills deliberately cover coefficients, constant terms, middle terms, sums, products and term comparison. PYQ selection depends on the source’s organisation. No dedicated Binomial Theorem filter or chapterwise analytics has been verified for the mocks discussed here.
Exam wording: These drills are original, not PYQs. For authentic PYQs, check the paper and question reference. Mock questions are not necessarily past exam questions.
Solutions and feedback: Every drill below has a worked solution. PYQ solution quality depends on the source. Verified mock feedback is a scored per-subject breakdown, not a promise of worked chapter solutions.
Timing context: These drills are untimed learning exercises. Isolated PYQs lose the surrounding paper’s time pressure. For whole-paper practice, JEEnius full-length mocks offer full 75-question, 300-mark, 180-minute papers, with free tests included.
Main limitation: Eight drills cannot cover the chapter. Familiar PYQs can reward memory rather than understanding. No guaranteed Binomial Theorem representation in a mock has been verified.

That mock format is for JEE Main preparation. JEE Advanced has two compulsory three-hour papers, with question types and marking schemes that vary. Do not use a Main-format mock as an Advanced-format replica.

Which format should I choose for my current mistake?

Use drills when you cannot select the term, PYQs when you need to recognise the method, and mocks when whole-paper timing is the problem. Your class gives context, but the first incorrect step should decide the next task.

Class 11, expanding everything manually: Start with chapter drills on the general term and exponent matching. Write the required exponent condition before doing coefficient arithmetic.
Class 12, comfortable with direct coefficients but weak in products or sums: Complete the mixed questions below, especially Q4, Q5 and Q7. Then use authentic PYQs to check whether you recognise the method without a chapter-drill cue.
Dropper, recognising old solutions: Solve unfamiliar original variants without hints. Remembering a PYQ answer is not evidence that you can rebuild its reasoning.
Accurate in chapter practice, slow in mixed papers: Prioritise full-length mocks. Practise question selection, leaving a stalled question and switching subjects.
Advanced aspirant: Use identity-based and multi-step drills as a bridge to authentic Advanced PYQs. This short set builds foundations; it does not reproduce the range of Advanced difficulty.

For parents, another book is not the repair for a repeated term-indexing error. Ask the student to show the first incorrect step. The next resource should address that weakness, not add another unfinished question bank.

Can I try eight free Binomial Theorem practice questions for JEE now?

Attempt all eight questions before reading the solutions. Every question is original practice, not an actual JEE question or a predicted exam question. These are short-answer learning exercises, not a replica of either exam’s question-format distribution.

For each attempt, record unaided, hinted or incorrect, even if a hint led to the correct answer. Keep your working, including abandoned steps, to identify where the method broke down.

Q1, original practice: Find the coefficient of the stated power in the expansion.

x^{3} in (1 + 2 x)^{5}

Q2, original practice: Find the term independent of the variable.

{(x^{2} + \frac{1}{x})}^{9}, x \neq 0

Q3, original practice: Find the middle term or terms. $(1 + x)^{7}$

Q4, original practice: Evaluate the sum.

\sum_{k = 0}^{8} k (\binom{8}{k})

Q5, original practice: Find the coefficient of the stated power in the product.

x^{4} in (1 + x)^{6} (1 - x)^{4}

Q6, original practice: Find the integer exponent if the specified coefficient is 45.

Coefficient of x^{2} in (1 + x)^{n} = 45, n \in Z, n \geq 2

Q7, original practice: Evaluate the sum.

\sum_{k = 0}^{6} (^{\binom{6}{k}) 2}

Q8, original practice: Find the greatest numerical term, stating both its position and value.

{(1 + \frac{1}{2})}^{10}

What are the answers and worked solutions?

Q 1 : & 80 Q 2 : & 84 Q 3 : & 35 x^{3}, 35 x^{4} Q 4 : & 1024 Q 5 : & 2 Q 6 : & 10 Q 7 : & 924 Q 8 : & fourth term, value 15

Check your reasoning as well as your result: a correct answer reached through a guessed index is not a dependable method. All solutions use the finite binomial theorem for nonnegative integer powers.

The general term is the starting tool. The selection index is not the term number: the index starts at zero, while term positions start at one.

T_{r + 1} = (\binom{n}{r}) a^{n - r} b^{r}, r \in {0, 1, \dots, n}

Selectionindex = r, term number = r + 1

Q1: Match the power, then keep the numerical factor.

Choose the variable-containing part three times. Square-bracket notation means “coefficient of the stated power”.

[x^{3}] (1 + 2 x)^{5} = (^{\binom{53}{2}) 3} = 10 \cdot 8 = 80

If you obtained 10, you likely omitted the internal coefficient’s contribution. That factor is: $2^{3}$

Q2: Set the total exponent to zero.

Both parts contribute to the exponent. Combine their powers before setting the exponent to zero.

T_{r + 1} = (\binom{9}{r}) (x^{2})^{9 - r} (x^{- 1})^{r} = (\binom{9}{r}) x^{18 - 3 r}

18 - 3 r = 0 ⟹ r = 6

This is an allowed integer between zero and nine. The constant term is therefore:

(\binom{96}{=}) (\binom{93}{=}) 84

Q3: Count terms before selecting the middle.

The exponent is seven, but the expansion has eight terms. The middle positions are fourth and fifth, corresponding to selection indices three and four.

T_{4} = (^{\binom{73}{x}) 3} = 35 x^{3}

T_{5} = (^{\binom{74}{x}) 4} = 35 x^{4}

Do not confuse the exponent with the number of terms. For a nonnegative integer power, these are:

Exponent = n, number of terms = n + 1

Q4: Differentiate to create the index multiplier.

Differentiation brings the exponent down as the required multiplier. Start with the binomial identity:

(1 + x)^{8} = \sum_{k = 0}^{8} (\binom{8}{k}) x^{k}

8 (1 + x)^{7} = \sum_{k = 1}^{8} k (\binom{8}{k}) x^{k - 1}

Now substitute: $x = 1$

The zero-index term contributes nothing to the requested sum. Hence:

\sum_{k = 0}^{8} k (\binom{8}{k}) = 8 \cdot 2^{7} = 1024

Substituting one before differentiating would miss the index multiplier.

Q5: Restructure before multiplying coefficients.

Pair equal powers of the two conjugate factors. This requires fewer operations than expanding both factors separately.

(1 + x)^{6} (1 - x)^{4} = (1 - x^{2})^{4} (1 + x)^{2}

(1 - x^{2})^{4} = 1 - 4 x^{2} + 6 x^{4} + \dots

(1 + x)^{2} = 1 + 2 x + x^{2}

The linear term contributes nothing because the other factor contains only even powers. The requested coefficient is:

[x^{4}] = 6 \cdot 1 + (- 4) \cdot 1 = 6 - 4 = 2

Q6: Convert the coefficient condition into an equation.

Use the binomial coefficient for the second power. Solve the resulting quadratic, then check the stated domain.

(\binom{n}{2}) = \frac{n (n - 1)}{2} = 45

n^{2} - n - 90 = (n - 10) (n + 9) = 0

n = 10 or n = - 9

The negative root fails the domain condition. The required exponent is:

n = 10

Q7: Read the sum as a coefficient of a product.

Compare the coefficient of the sixth power on both sides. In the product, the paired exponents must add to six.

(1 + x)^{6} (1 + x)^{6} = (1 + x)^{12}

[x^{6}] = \sum_{k = 0}^{6} (\binom{6}{k}) (\binom{6}{6 - k})

Binomial symmetry turns each product into a square. Then read the coefficient from the right-hand side.

(\binom{6}{6 - k}) = (\binom{6}{k})

\sum_{k = 0}^{6} (^{\binom{6}{k}) 2} = (\binom{12}{6}) = 924

Q8: Compare consecutive terms, not their distance from the middle.

All terms are positive, so a ratio above one means the next term is larger. The consecutive-term ratio is:

\frac{T_{r + 1}}{T_{r}} = \frac{11 - r}{2 r}, r = 1, 2, \dots, 10

It exceeds one through the third comparison, then falls below one from the fourth comparison onward. At the change:

\frac{T_{4}}{T_{3}} = \frac{8}{6} > 1, \frac{T_{5}}{T_{4}} = \frac{7}{8} < 1

The terms therefore increase through the fourth term and then decrease. Its value is:

T_{4} = (\binom{10}{3}) {(\frac{1}{2})}^{3} = \frac{120}{8} = 15

The greatest numerical term is not automatically a middle term. Its position depends on both the binomial coefficient and the numerical factors.

What should I practise next, and is this enough for Advanced?

Repair the method behind your error, then test it on authentic PYQs. Eight questions can reveal a gap; they cannot certify chapter mastery or sufficient Advanced preparation. Use this error map to choose the next exercise.

Q1 to Q3: Repair coefficient extraction, exponent matching and term indexing.
Q4 and Q7: Practise binomial-sum identities through differentiation and coefficient comparison.
Q5: Practise algebraic restructuring before expansion.
Q6: Practise recovering a parameter and checking its domain.
Q8: Practise consecutive-term ratios, including cases where adjacent terms tie.

Use a changed version to check whether you know the method rather than remember the answer. Then retry the original without viewing its solution.

A left-to-right revision flow with five labelled boxes, “Attempt without hints”, “Find the first incorrect step”, “Name the error”, “Solve a changed version” and “Retry the original unaided”, connected by arrows in that order.

Attempt without hints.
Check the first incorrect step, not just the answer.
Name the error precisely, such as “forgot the internal coefficient”.
Solve a changed version using another power or target exponent.
Retry the original without viewing its solution.

When should I move from chapter drills to PYQs?

Move when you can reproduce the relevant method unaided and explain why it applies. Choose authentic questions with a traceable paper and question reference, and check that the solution justifies its steps. You need not finish every possible drill before attempting a PYQ.

Are these questions enough for JEE Advanced?

No. They provide foundation practice in identities and restructuring, but Advanced preparation also needs authentic Advanced PYQs and unfamiliar multi-step problems. Getting all eight right shows success with these methods, not complete preparation.

Should I take a mock or solve more chapter questions now?

Take a mock when whole-paper decisions are the weakness; repair the chapter first when the method still needs hints. For whole-paper execution, JEEnius full-length mocks include free tests and a scored per-subject breakdown. After your next attempt, separate a wrong method from a poor time decision before choosing what to practise again.

Next step: full-length mock tests on JEEnius and sit a full 75-question, 300-mark, 180-minute paper and get a scored per-subject breakdown (free tests included).

If that step was the hard part, work through Differential Equations JEE 2026: Option D and the Mismatch.

Frequently asked questions

Are these Binomial Theorem questions actual JEE PYQs?

No. All eight questions are original practice exercises with complete worked solutions, not actual JEE questions or predictions. For authentic PYQs, use questions with a traceable paper and question reference.

How do I find the constant term in a binomial expansion?

Write the general term and combine all powers of the variable. Set the total exponent to zero, then check that the resulting selection index is an allowed integer. Substitute that index into the general term to obtain the constant term.

Are these eight questions enough for JEE Advanced?

No. They build foundations in coefficient extraction, identities, restructuring and term comparison. Advanced preparation also requires authentic Advanced PYQs and unfamiliar multi-step problems.

Should I solve chapter questions, PYQs or mocks for Binomial Theorem?

Use chapter drills when you still need hints to select or apply a method. Move to authentic PYQs when you can reproduce the method unaided and explain why it applies. Choose full-length mocks when question selection and whole-paper timing are the main weaknesses.