JEE Advanced 1992

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Animated Solution for Mathematics - Binomial Theorem: The expression $[x + (x^{3} - 1)^{1/2}]^{5} + [x - (x^{3} - 1)^{1/2}]^{5}$ is a polynomial of degree

Q: How do I know which terms cancel out?

Think about the binomial expansion of $(x+y)^n$ and $(x-y)^n$. The terms with odd powers of $y$ (like $y^1, y^3, y^5$) have a negative sign in the second expansion. When you add them, they vanish! Only the even powers of $y$ survive.

Select Answer:

Visualized Solution

Identifying the Structure

Given expression: $[x + (x^{3} - 1)^{\frac{1}{2}}]^{5} + [x - (x^{3} - 1)^{\frac{1}{2}}]^{5}$
Let $y = (x^{3} - 1)^{\frac{1}{2}}$
The expression simplifies to: $(x + y)^{5} + (x - y)^{5}$

Applying the Binomial Identity

Recall the identity: $(x + y)^{n} + (x - y)^{n} = 2 [^{n} C_{0} x^{n} +^{n} C_{2} x^{n - 2} y^{2} +^{n} C_{4} x^{n - 4} y^{4} + \dots]$
Notice that all odd powers of $y$ cancel out.
Only even powers of $y$ ( $y^{0}, y^{2}, y^{4}, \dots$ ) remain in the final expression.

Expanding for $n = 5$

For $n = 5$ , the expansion is: $2 [^{5} C_{0} x^{5} +^{5} C_{2} x^{3} y^{2} +^{5} C_{4} x y^{4}]$
Calculating coefficients: $^{5} C_{0} = 1$ , $^{5} C_{2} = 10$ , $^{5} C_{4} = 5$
Substitute coefficients: $2 [1 \cdot x^{5} + 10 x^{3} y^{2} + 5 x y^{4}]$

Substituting $y^{2} = x^{3} - 1$

Substitute $y^{2} = (x^{3} - 1)$ and $y^{4} = (x^{3} - 1)^{2}$
Expression becomes: $2 [x^{5} + 10 x^{3} (x^{3} - 1) + 5 x (x^{3} - 1)^{2}]$

Expanding the Polynomial Terms

Expand the terms inside the bracket:
$10 x^{3} (x^{3} - 1) = 10 x^{6} - 10 x^{3}$
$5 x (x^{3} - 1)^{2} = 5 x (x^{6} - 2 x^{3} + 1) = 5 x^{7} - 10 x^{4} + 5 x$

Finding the Highest Power

Combine all terms: $2 [x^{5} + 10 x^{6} - 10 x^{3} + 5 x^{7} - 10 x^{4} + 5 x]$
Final simplified polynomial: $10 x^{7} + 20 x^{6} + 2 x^{5} - 20 x^{4} - 20 x^{3} + 10 x$
The highest power of $x$ is $7$ .

Final Degree of the Polynomial

Key Takeaway: The degree of $(x + y)^{n} + (x - y)^{n}$ depends on the highest power resulting from the even powers of $y$ .
Final Answer: The degree of the polynomial is 7.
Next Challenge: What would be the degree if the expression was $[x + (x^{3} - 1)^{\frac{1}{2}}]^{5} - [x - (x^{3} - 1)^{\frac{1}{2}}]^{5}$ ?

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The Sigma Insight: Binomial Expansion for Positive Integral Index

Analyzing the Setup

The expression

[x + (x^{3} - 1)^{\frac{1}{2}}]^{5} + [x - (x^{3} - 1)^{\frac{1}{2}}]^{5}

may appear intimidating due to the nested radicals. However, JEE Advanced problems often rely on spotting hidden structures rather than brute force.

Let us define

y = (x^{3} - 1)^{\frac{1}{2}}

. The expression simplifies significantly to the form:

(x + y)^{5} + (x - y)^{5}

The Binomial Magic

We invoke the Binomial Theorem to expand the expression. When calculating

(x + y)^{n} + (x - y)^{n}

, terms with odd powers of

y

cancel out, leaving twice the sum of terms with even powers of

y

For

n = 5

, the expansion becomes:

2 [^{5} C_{0} x^{5} +^{5} C_{2} x^{3} y^{2} +^{5} C_{4} x y^{4}]

Calculating the binomial coefficients, we have

^{5} C_{0} = 1

^{5} C_{2} = 10

, and

^{5} C_{4} = 5

. Substituting these values, the expression becomes:

2 [1 \cdot x^{5} + 10 x^{3} y^{2} + 5 x y^{4}]

The Final Reveal

Now, we substitute the original variable back into the expression. Given

y^{2} = (x^{3} - 1)

and

y^{4} = (x^{3} - 1)^{2}

, the expression is:

2 [x^{5} + 10 x^{3} (x^{3} - 1) + 5 x (x^{3} - 1)^{2}]

To determine the degree of the resulting polynomial, we identify the highest power of

x

in each term:

1. The first term

x^{5}

has degree

5

. 2. The second term

10 x^{3} (x^{3} - 1) = 10 x^{6} - 10 x^{3}

has degree

6

. 3. The third term

5 x (x^{3} - 1)^{2} = 5 x (x^{6} - 2 x^{3} + 1) = 5 x^{7} - 10 x^{4} + 5 x

has degree

7

Comparing these, the highest power is

7

. Therefore, the degree of the polynomial is 7.

Common Traps & FAQs

Why can't I just expand

[x + (x^{3} - 1)^{1/2}]^{5}

directly?

Expanding that directly would lead to a nightmare of nested radicals and a high probability of algebraic errors. In JEE Advanced, the goal is to find the most elegant path. By using the symmetry of the expression, we eliminate the radicals immediately.

How do I know which terms cancel out?

Think about the binomial expansion of

(x + y)^{n}

and

(x - y)^{n}

. The terms with odd powers of

y

(like

y^{1}, y^{3}, y^{5}

) have a negative sign in the second expansion. When you add them, they vanish! Only the even powers of

y

survive.

What if the question had a minus sign instead of a plus?

That's a fantastic question! If it were a minus sign, the even powers of

y

would cancel out, and you would be left with twice the sum of the odd powers of

y

. The degree would then be determined by the highest power of

x

in those odd-powered terms.

Animated Solution for Mathematics - Binomial Theorem: The expression $[x + (x^{3} - 1)^{1/2}]^{5} + [x - (x^{3} - 1)^{1/2}]^{5}$ is a polynomial of degree

Select Answer:

Visualized Solution

Identifying the Structure

Applying the Binomial Identity

Expanding for $n = 5$

Substituting $y^{2} = x^{3} - 1$

Expanding the Polynomial Terms

Finding the Highest Power

Final Degree of the Polynomial

The Sigma Insight: Binomial Expansion for Positive Integral Index

Analyzing the Setup

The Binomial Magic

The Final Reveal

Similar Questions

The sum of the co-efficients of all odd degree terms in the expansion of $(x + x^{3} - 1)^{5} + (x - x^{3} - 1)^{5}, (x > 1)$ is :

If n is the degree of the polynomial, $[\frac{2}{5 x ^{3} + 1 - 5 x ^{3} - 1}]^{8} + [\frac{2}{5 x ^{3} + 1 + 5 x ^{3} - 1}]^{8}$ and m is the coefficient of $x^{n}$ in it, then the ordered pair (n,m) is equal to:

The sum of the co-efficients of all even degree terms in $x$ in the expansion of $(x + x^{3} - 1)^{6} + (x - x^{3} - 1)^{6}, (x > 1)$ is equal to :

The coefficient of $x^{101}$ in the expression $(5 + x)^{500} + x (5 + x)^{499} + x^{2} (5 + x)^{498} + \dots + x^{500}$ , $x > 0$ , is

If in the expansion of $(1 + x)^{m} (1 - x)^{n}$ , the coefficients of $x$ and $x^{2}$ are 3 and $- 6$ respectively, then $m$ is

The coefficient of $x^{7}$ in the expansion of $(1 - x - x^{2} + x^{3})^{6}$ is

If the coefficients of $x$ and $x^{2}$ in $(1 + x)^{p} (1 - x)^{q}$ are 4 and $- 5$ respectively, then $2 p + 3 q$ is equal to

If $α$ and $β$ be the coefficients of $x^{4}$ and $x^{2}$ respectively in the expansion of $(x + x^{2} - 1)^{6} + (x - x^{2} - 1)^{6}$ , then:

Let $α, β, γ$ and $δ$ be the coefficients of $x^{7}$ , $x^{5}$ , $x^{3}$ and $x$ respectively in the expansion of $(x + x^{3} - 1)^{5} + (x - x^{3} - 1)^{5}$ , $x > 1$ . If u and v satisfy the equations $α u + β v = 18$ and $γ u + δ v = 20$ , then $u + v$ equals:

The coefficient of $x^{7}$ in the expression $(1 + x)^{10} + x (1 + x)^{9} + x^{2} (1 + x)^{8} + \dots + x^{10}$ is :

Animated Solution for Mathematics - Binomial Theorem: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The expression [x+(x3−1)1/2]5+[x−(x3−1)1/2]5 is a polynomial of degree

Select Answer:

Visualized Solution

Identifying the Structure

Applying the Binomial Identity

Expanding for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } n=5

Substituting .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } y2=x3−1

Expanding the Polynomial Terms

Finding the Highest Power

Final Degree of the Polynomial

The Sigma Insight: Binomial Expansion for Positive Integral Index

Analyzing the Setup

The Binomial Magic

The Final Reveal

Similar Questions

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum of the co-efficients of all odd degree terms in the expansion of (x+x3−1​)5+(x−x3−1​)5,(x>1) is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If n is the degree of the polynomial, [5x3+1​−5x3−1​2​]8+[5x3+1​+5x3−1​2​]8 and m is the coefficient of xn in it, then the ordered pair (n,m) is equal to:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum of the co-efficients of all even degree terms in x in the expansion of (x+x3−1​)6+(x−x3−1​)6,(x>1) is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The coefficient of x101 in the expression (5+x)500+x(5+x)499+x2(5+x)498+⋯+x500, x>0, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If in the expansion of (1+x)m(1−x)n, the coefficients of x and x2 are 3 and −6 respectively, then m is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The coefficient of x7 in the expansion of (1−x−x2+x3)6 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the coefficients of x and x2 in (1+x)p(1−x)q are 4 and −5 respectively, then 2p+3q is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If α and β be the coefficients of x4 and x2 respectively in the expansion of (x+x2−1​)6+(x−x2−1​)6, then:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The coefficient of x7 in the expression (1+x)10+x(1+x)9+x2(1+x)8+⋯+x10 is :

Animated Solution for Mathematics - Binomial Theorem: The expression $[x + (x^{3} - 1)^{1/2}]^{5} + [x - (x^{3} - 1)^{1/2}]^{5}$ is a polynomial of degree

Expanding for $n = 5$

Substituting $y^{2} = x^{3} - 1$

The sum of the co-efficients of all odd degree terms in the expansion of $(x + x^{3} - 1)^{5} + (x - x^{3} - 1)^{5}, (x > 1)$ is :

If n is the degree of the polynomial, $[\frac{2}{5 x ^{3} + 1 - 5 x ^{3} - 1}]^{8} + [\frac{2}{5 x ^{3} + 1 + 5 x ^{3} - 1}]^{8}$ and m is the coefficient of $x^{n}$ in it, then the ordered pair (n,m) is equal to:

The sum of the co-efficients of all even degree terms in $x$ in the expansion of $(x + x^{3} - 1)^{6} + (x - x^{3} - 1)^{6}, (x > 1)$ is equal to :

The coefficient of $x^{101}$ in the expression $(5 + x)^{500} + x (5 + x)^{499} + x^{2} (5 + x)^{498} + \dots + x^{500}$ , $x > 0$ , is

If in the expansion of $(1 + x)^{m} (1 - x)^{n}$ , the coefficients of $x$ and $x^{2}$ are 3 and $- 6$ respectively, then $m$ is

The coefficient of $x^{7}$ in the expansion of $(1 - x - x^{2} + x^{3})^{6}$ is

If the coefficients of $x$ and $x^{2}$ in $(1 + x)^{p} (1 - x)^{q}$ are 4 and $- 5$ respectively, then $2 p + 3 q$ is equal to

If $α$ and $β$ be the coefficients of $x^{4}$ and $x^{2}$ respectively in the expansion of $(x + x^{2} - 1)^{6} + (x - x^{2} - 1)^{6}$ , then:

The coefficient of $x^{7}$ in the expression $(1 + x)^{10} + x (1 + x)^{9} + x^{2} (1 + x)^{8} + \dots + x^{10}$ is :