IPMAT Binomial Theorem PYQs

2 solved Binomial Theorem previous year questions (PYQs) from past year papers — attempt each and check the answer.

Q1:ipmat indore 2023QA › Binomial TheoremMediumSA · TITA

If three consecutive coefficients in the expansion of

(x+y)^n

are in the ratio

1:9:63

, then the value of

n

Enter your answer to attempt

The Setup: We are given three consecutive coefficients in the binomial expansion of

(x+y)^n

strictly in the ratio

1:9:63

. We must determine the power

n

. Step 1: Express the coefficients algebraically. Let the three consecutive coefficients be

\binom{n}{r-1}

\binom{n}{r}

, and

\binom{n}{r+1}

. We are given two ratio equations:

\frac{\binom{n}{r}}{\binom{n}{r-1}} = \frac{9}{1} = 9

\frac{\binom{n}{r+1}}{\binom{n}{r}} = \frac{63}{9} = 7

Step 2: Apply the standard binomial coefficient ratio formula. The ratio

\frac{\binom{n}{k}}{\binom{n}{k-1}}

simplifies universally to

\frac{n - k + 1}{k}

. Apply this to our first equation (

k = r

\frac{n - r + 1}{r} = 9 \implies n - r + 1 = 9r \implies n = 10r - 1

Apply this to our second equation (

k = r + 1

\frac{n - (r+1) + 1}{r+1} = 7 \implies \frac{n - r}{r + 1} = 7 \implies n - r = 7r + 7 \implies n = 8r + 7

Step 3: Solve the linear system for

r

and

n

. Equate the two expressions for

n

10r - 1 = 8r + 7

2r = 8 \implies r = 4

Substitute

r

back into either equation to find

n

n = 10(4) - 1 = 39

Final Answer: 39

Q2:ipmat indore 2022QA › Binomial TheoremEasySA · TITA

The sum of the coefficients of all the terms in the expansion of

(5 x-9)^{4}

is __________.

Enter your answer to attempt

The Setup: We are asked to find the arithmetic sum of the coefficients of all terms within the binomial expansion of

(5x-9)^4

. Step 1: Apply the polynomial coefficient property. For any polynomial

P(x)

, the sum of its fully expanded coefficients is mathematically obtained by evaluating the polynomial at

x = 1

. Step 2: Evaluate the expression at

x = 1

P(1) = (5(1) - 9)^4

P(1) = (5 - 9)^4

P(1) = (-4)^4

P(1) = 256

Final Answer: 256

Binomial Theorem — PYPs