Binomial Theorem

01

PYQ 2023

medium

mathematics ID: ap-eamce

Find the coefficient of in ?

02

PYQ 2024

medium

mathematics ID: ap-eamce

The square root of the independent term in the expansion of is:

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3

4

03

PYQ 2024

easy

mathematics ID: ap-eamce

For the coefficient of in the expansion of is

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3

4

04

PYQ 2024

easy

mathematics ID: ap-eamce

If the coefficients of the,, and terms in the expansion of are in the ratio, then is:}

1

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3

4

Official Solution

Correct Option: (3)

$Step 1: Understanding binomial coefficients The general term in the binomial expansion of is given by: The coefficients of the,, and terms are given by: We are given the ratio: Step 2: Expressing binomial coefficients as ratios Using the property of binomial coefficients: Thus, we obtain the equations: Step 3: Solving for From the first equation: From the second equation: Solving these equations simultaneously gives: Step 4: Conclusion Thus, the final answer is:$

05

PYQ 2024

hard

mathematics ID: ap-eamce

$If the coefficients of the and terms in the expansion of are equal, then the value of is:}$

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3

4

Official Solution

Correct Option: (3)

$Step 1: Understanding binomial coefficients The general term in the expansion of is given by: The coefficients of the and terms are: Since these coefficients are equal, we set up the equation: Step 2: Using binomial coefficient property Using the symmetric property of binomial coefficients: we get: Since, we equate: Step 3: Solving for Rearrange the equation: Step 4: Conclusion Thus, the final answer is:$

06

PYQ 2024

easy

mathematics ID: ap-eamce

$The absolute value of the difference of the coefficients of and in the expansion of is:$

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4

Official Solution

Correct Option: (1)

$Step 1: Partial Fraction Expansion Expanding the function: Solving for and, we find: Step 2: Expanding Each Term Using binomial series, we expand: Multiplying by, we extract coefficients of and : Step 3: Finding Absolute Difference Thus, the correct answer is .$

07

PYQ 2024

easy

mathematics ID: ap-eamce

$The sum of the series is:$

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4

Official Solution

Correct Option: (1)

$The given series is a type of infinite geometric series. The general form of the series is: Step 1: Express this as a geometric series with first term and common ratio . The sum of an infinite geometric series is given by: Where is the first term and is the common ratio. Here, and .$

08

PYQ 2024

easy

mathematics ID: ap-eamce

$Numerically greatest term in the expansion of, when, is:$

Official Solution

Correct Option: (1)

09

PYQ 2024

medium

mathematics ID: ap-eamce

$If the,, and terms in the expansion of are 96, 216, and 216 respectively, and is a positive integer, then is:$

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4

Official Solution

Correct Option: (1)

$Given the expansion: The general term in the expansion is given by: Step 1: Write Equations for the Given Terms From the problem, we know: - 2nd term (when): - 3rd term (when): - 4th term (when): Step 2: Form Ratios for Simplification Dividing the second equation by the first: From binomial coefficients: Thus, Cross-multiplying, Step 3: Form a Second Ratio Dividing the third equation by the second: From binomial coefficients: Thus, From the earlier step, Now substitute this into the second equation: Cross-multiplying: Expanding: Dividing by 2: Using the quadratic formula: Since must be an integer, . Step 4: Compute Since, substituting : Now, Using the given conditions, this simplifies to . Step 5: Final Answer$

10

PYQ 2024

medium

mathematics ID: ap-eamce

$If, then :$

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Official Solution

Correct Option: (1)

$We are given: Step 1: Common Denominator The denominator on the right side is: Thus, Equating the numerators, Step 2: Expanding Both Terms Expanding the first term: Expanding the second term: Now combine like terms: Step 3: Equating Coefficients By comparing coefficients: - - - - Step 4: Solving for Since, substitute this into the second equation: Now from, let and . Step 5: Calculate Final Answer: (1)$

11

PYQ 2024

medium

mathematics ID: ap-eamce

$The coefficient of in the expansion of is:$

1

$18$

2

$540$

3

$1620$

4

$2178$

Official Solution

Correct Option: (4)

$We are asked to find the coefficient of in the expansion of . Step 1: Apply the Multinomial Theorem The expansion of is given by the multinomial expansion: In our case, the expression is , so we have , , and . The general term in the expansion will be: This simplifies to: Step 2: Find the Values of and for We need the exponent of to be 5, so: We consider the possible values of and that satisfy this equation: - If , then . - If , then . - If , then . Step 3: Calculate the Corresponding Coefficients For each valid pair , we substitute into the general term and calculate the coefficient: \begin{itemize} \item For , , the term is: \item For , , the term is: \item For , , the term is: \end{itemize} Step 4: Total Coefficient of The total coefficient of is the sum of the coefficients from each valid term: Thus, the coefficient of is .$

High-Yield Trend

Chapter Questions
11 MCQs

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Binomial Theorem

High-Yield Trend

Chapter Questions 11 MCQs

Official Solution

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Chapter Questions
11 MCQs