JEE Main 2024 (30 Jan Shift 2)

LEVELJEE Main

Animated Solution for Mathematics - Binomial Theorem: Suppose $2 - p, p, 2 - α, α$ are the coefficient of four consecutive terms in the expansion of $(1 + x)^{n}$ . Then the value of $p^{2} - α^{2} + 6 α + 2 p$ equals

Select Answer:

Visualized Solution

Define Consecutive Coefficients

Let the four consecutive coefficients in $(1 + x)^{n}$ be:
$^{n} C_{r} = 2 - p$
$^{n} C_{r + 1} = p$
$^{n} C_{r + 2} = 2 - α$
$^{n} C_{r + 3} = α$

The Pascal's Addition Rule

Instead of complex ratios, we use Pascal's Identity:
$^{n} C_{k} +^{n} C_{k + 1} =^{n + 1} C_{k + 1}$

Adding the First Pair

Add the first two coefficients:
$^{n} C_{r} +^{n} C_{r + 1} = (2 - p) + p$
$^{n + 1} C_{r + 1} = 2$

Adding the Second Pair

Add the next two coefficients:
$^{n} C_{r + 2} +^{n} C_{r + 3} = (2 - α) + α$
$^{n + 1} C_{r + 3} = 2$

Analyzing the Binomial Value

We have a binomial coefficient equal to $2$ :
$^{m} C_{k} = 2$
From Pascal's Triangle, the value $2$ only appears when $m = 2$ and $k = 1$ .

Solving for n

Therefore, the upper index must be $2$ :
$n + 1 = 2$
$n = 1$

Checking Constraints (The Trap)

If $n = 1$ , the expansion is $(1 + x)^{1}$ .
Total number of terms = $n + 1 = 2$ .
But the problem gives four consecutive terms.

Final Conclusion

It is impossible to have $4$ terms when $n = 1$ .
The given coefficients cannot exist simultaneously.
Final Answer: Data Inconsistent

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The Sigma Insight: Properties of Binomial Coefficients

Solution Diagram

Analyzing the Setup

We are tasked with evaluating the expression

p^{2} - α^{2} + 6 α + 2 p

given four consecutive coefficients in the expansion of

(1 + x)^{n}

2 - p

p

2 - α

, and

α

These correspond to the binomial coefficients:

^{n} C_{r} = 2 - p

^{n} C_{r + 1} = p

^{n} C_{r + 2} = 2 - α

^{n} C_{r + 3} = α

The Elegance of Pascal's Identity

Rather than diving into complex ratios, we utilize Pascal's Identity:

^{n} C_{k} +^{n} C_{k + 1} =^{n + 1} C_{k + 1}

Applying this to our pairs, we observe a significant simplification. For the first pair:

^{n} C_{r} +^{n} C_{r + 1} = (2 - p) + p = 2

By Pascal's Identity, this implies:

^{n + 1} C_{r + 1} = 2

Repeating this logic for the second pair:

^{n} C_{r + 2} +^{n} C_{r + 3} = (2 - α) + α = 2

This yields:

^{n + 1} C_{r + 3} = 2

The Moment of Revelation

We now have two binomial coefficients,

^{n + 1} C_{r + 1}

and

^{n + 1} C_{r + 3}

, both equal to

2

In the context of Pascal's Triangle, the value

2

is unique. It occurs only at the position

^{2} C_{1}

This forces the upper index to be:

n + 1 = 2 ⟹ n = 1

The Final Twist

We must now verify this result against the constraints of the binomial theorem. If

n = 1

, the expansion is

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

This expansion contains only two terms. It is mathematically impossible for an expansion with only two terms to possess four consecutive coefficients.

The conditions provided in the problem statement are mutually exclusive. Because the premise itself is logically inconsistent, the requested expression cannot be evaluated under the given constraints.

Common Traps & FAQs

Wait, why do we reject the ratio method? Isn't that the standard way to solve for

n

and

r

in binomial problems?

The ratio method is indeed a powerful tool, but it is not the only one! In competitive exams like JEE, time is your most precious resource. Using ratios here would create a system of equations involving

n

and

r

that is algebraically tedious. By using Pascal's Identity, we exploit the symmetry of the coefficients to make the variables

p

and

α

vanish instantly. It is about choosing the most elegant path, not just the standard one.

Why does the value

2

^{m} C_{k} = 2

force

m = 2

and

k = 1

Great observation! If you look at Pascal's Triangle, the value

2

is quite rare. It only appears in the third row (where

m = 2

), specifically as

^{2} C_{1}

. Since binomial coefficients grow rapidly as

m

increases, and for a fixed

m

, they are symmetric,

2

cannot appear for any

m > 2

except in very specific cases that don't apply here. This uniqueness is the key to unlocking the problem.

Is it possible for a JEE question to have 'Data Inconsistent' as the answer? I feel like I did something wrong.

Absolutely! In fact, identifying an inconsistent or impossible scenario is a higher-order skill. It tests whether you are blindly calculating or truly understanding the constraints of the problem. If your derivation leads to

n = 1

, but the problem demands four consecutive terms, you have correctly identified that the premise is flawed. Trust your logic!

Animated Solution for Mathematics - Binomial Theorem: Suppose $2 - p, p, 2 - α, α$ are the coefficient of four consecutive terms in the expansion of $(1 + x)^{n}$ . Then the value of $p^{2} - α^{2} + 6 α + 2 p$ equals

Select Answer:

Visualized Solution

Define Consecutive Coefficients

The Pascal's Addition Rule

Adding the First Pair

Adding the Second Pair

Analyzing the Binomial Value

Solving for n

Checking Constraints (The Trap)

Final Conclusion

The Sigma Insight: Properties of Binomial Coefficients

Analyzing the Setup

The Elegance of Pascal's Identity

The Moment of Revelation

The Final Twist

Similar Questions

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Select Answer:

Visualized Solution

Define Consecutive Coefficients

The Pascal's Addition Rule

Adding the First Pair

Adding the Second Pair

Analyzing the Binomial Value

Solving for n

Checking Constraints (The Trap)

Final Conclusion

The Sigma Insight: Properties of Binomial Coefficients

Analyzing the Setup

The Elegance of Pascal's Identity

The Moment of Revelation

The Final Twist

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