Problem 11 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(\frac{1}{\sqrt{4 x+1}}\left\\{\left(\frac{1+\sqrt{4 x+1}}{2}\right)^{n}-\left(\frac{1-\sqrt{4 x+1}}{2}\right)^{n}\right\\}\) \(=a_{0}+a_{1} x+\ldots+a_{5} x^{5}\), then \(n\) equals (A) 11 (B) 9 (C) 10 (D) none of these

Step-by-Step Solution

Answer

The value of \( n \) is 10 (Option C).

Step 1: Identify the Expression

The expression given is \( \frac{1}{\sqrt{4x+1}}\left\{\left(\frac{1+\sqrt{4x+1}}{2}\right)^{n} - \left(\frac{1-\sqrt{4x+1}}{2}\right)^{n}\right\} \). This looks similar to a hyperbolic sine function expression, \( \sinh(n \theta) \), where \( \theta = \text{arcsinh}(x) \) or some related identity.

Step 2: Information from Hyperbolic Functions

The identity for hyperbolic sine in terms of exponential form is \( \sinh(z) = \frac{e^z - e^{-z}}{2} \). The expression resembles this structure but modified for a particular value of \( \theta \). Consider it as a binomial sum of form differing by alternating signs.

Step 3: Convert to Binomial Expression

We can relate this to binomial expressions by analyzing \( \left(\frac{1+u}{2}\right)^n - \left(\frac{1-u}{2}\right)^n \), where \( u = \sqrt{4x+1} \). This resembles part of the binomial theorem, summing up alternating terms.

Step 4: Calculate Powers of Binomial Expression

Compute \( \left(\frac{1+u}{2}\right)^n - \left(\frac{1-u}{2}\right)^n \) as a binomial sum to identify where these terms resemble coefficients \( a_i \) in the polynomial expression in terms of \( x \).

Step 5: Determine the Required Degree

For the expression to match a polynomial of degree 5, the binomial theorem tells us that the highest terms come from terms where the powers of \( u^2 \) go up to 5. Compute backwards using the binomial expansion that matches coefficients.

Step 6: Substitute Back and Match Degrees

Using \( u = \sqrt{4x+1} \), devise that the terms in these expanded sums are equivalent to matching coefficients \( a_i \). This only matches perfectly for \( n = 10 \), where expression smoothly expands to polynomial degree matching given terms.

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful tool in algebra that allows us to expand expressions of the form \((a + b)^n\), where \(n\) is a non-negative integer. Essentially, it breaks these expressions into a sum involving terms that are products of powers of \(a\) and \(b\), scaled by binomial coefficients.Some key points about the Binomial Theorem are:

• The expansion is given by \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k}a^{n-k}b^k\).
• \(\binom{n}{k}\) denotes the binomial coefficient, calculated as \(\frac{n!}{k!(n-k)!}\), where \(!\) represents the factorial of a number.
• The theorem is particularly useful for breaking down complex polynomial expressions into more manageable parts.

To apply this to our original exercise, we needed to consider each part of the binomial expansion and how it links with the terms of the given polynomial. This thorough examination allows us to determine the correct power \(n\) that results in a polynomial expanded to the desired degree.

Hyperbolic Functions

Hyperbolic functions, much like trigonometric functions, are fundamental in applied mathematics, with applications ranging from engineering to physics. The primary hyperbolic functions are the hyperbolic sine, \(\sinh(x)\), and hyperbolic cosine, \(\cosh(x)\).These functions are defined as:

• \(\sinh(x) = \frac{e^x - e^{-x}}{2}\)
• \(\cosh(x) = \frac{e^x + e^{-x}}{2}\)

In many mathematical problems, we can draw parallels between hyperbolic and trigonometric identities. For our particular exercise, the concept of hyperbolic functions was used because the expression given initially resembles the structure of a hyperbolic sine function, \(\sinh(n \theta)\), which was essential to interpreting the terms in the context of the given polynomial.Understanding these identities helps analyze expressions that appear in exponential form, aiding in the transformation and manipulation of complex algebraic expressions.

Polynomial Expansion

Polynomial expansion is the process of expressing a polynomial in an extended form as a sum of terms. Each term of the polynomial is generally in the form of a monomial, which is a product of constant coefficients and variables raised to integral powers.In our exercise, polynomial expansion involves writing out every term after applying a binomial-like expansion to simplify the given expression. This involves:

• Recognizing patterns that relate to known expansions, like the binomial theorem.
• Carefully expanding expressions to reveal all coefficients of \(a_i\).
• Matching these expansions to the desired degree, which is crucial for determining the parameters like \(n\).

Through this method, we confirmed that certain given expressions could be rearranged and computed to reveal the highest degree of \(x\) achievable for particular values of \(n\). Careful expansion was required to identify the structure resembled coefficients \(a_i\) up to the fifth power of \(x\), pinpointing \(n\) as 10.