Problem 37 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If \(\Delta_{\mathrm{r}}=\left|\begin{array}{ccc}\mathrm{r} & 2 \mathrm{r}-1 & 3 \mathrm{r}-2 \\ \frac{\mathrm{n}}{2} & \mathrm{n}-1 & \mathrm{a} \\\ \frac{1}{2} \mathrm{n}(\mathrm{n}-1) & (\mathrm{n}-1)^{2} & \frac{1}{2}(\mathrm{n}-1)(3 \mathrm{n}-4)\end{array}\right|\) then the value of \(\sum_{\mathrm{r}=1}^{\mathrm{n}-1} \Delta_{\mathrm{r}}\) (a) depends only on a (b) depends only on \(\mathrm{n}\) (c) depends both on a and \(\mathrm{n}\) (d) is independent of both a and \(\mathrm{n}\)

Step-by-Step Solution

Answer

The solution is independent of both \( \mathrm{a} \) and \( \mathrm{n} \).

Step 1: Understand the Problem

The given problem involves calculating the value of a determinant \( \Delta_{\mathrm{r}} \) and then evaluating the summation of \( \Delta_{\mathrm{r}} \) over a range. We need to determine how this summation depends on the variables \( \mathrm{a} \) and \( \mathrm{n} \).

Step 2: Analyze the Determinant

Examine the determinant \( \Delta_{\mathrm{r}} \) of the 3x3 matrix with elements based on the variable \( \mathrm{r} \), \( \mathrm{n} \), and a parameter \( \mathrm{a} \). The matrix is defined as: \[ \Delta_{\mathrm{r}} = \begin{vmatrix} \mathrm{r} & 2\mathrm{r}-1 & 3\mathrm{r}-2 \ \frac{\mathrm{n}}{2} & \mathrm{n}-1 & \mathrm{a} \ \frac{1}{2} \mathrm{n}(\mathrm{n}-1) & (\mathrm{n}-1)^{2} & \frac{1}{2}(\mathrm{n}-1)(3\mathrm{n}-4) \end{vmatrix} \]

Step 3: Simplify the Determinant

To simplify the determinant, look for a way to express the determinant's value in terms of \( \mathrm{a} \) and \( \mathrm{n} \). Use matrix operations like row or column operations to reduce complexity. Observe that the presence of \( \mathrm{r} \) in the determinant suggests the determinant might simplify to a different form that depends purely on \( \mathrm{n} \).

Step 4: Calculate the Summation

The problem involves summing \( \Delta_{\mathrm{r}} \) over the range of \( \mathrm{r}=1 \) to \( \mathrm{r}=\mathrm{n}-1 \). The goal is to determine whether this summation shows dependence on \( \mathrm{a} \) or \( \mathrm{n} \). Given the way the determinant simplifies, establish whether there is any resultant term that includes \( \mathrm{a} \).

Step 5: Conclusion

After determining whether \( \Delta_{\mathrm{r}} \) simplifies to eliminate \( \mathrm{a} \) from dependence, check if any non-zero term remains dependent on \( \mathrm{n} \). Due to the structure of the matrix elements depending on \( \mathrm{n} \) but not \( \mathrm{a} \), deduce that the value does not depend on \( \mathrm{a} \). Often, such determinants further simplify to constants over their range of summation.

Key Concepts

Matrix operations

Matrix operations are fundamental tools in linear algebra and are vital for solving systems of linear equations, transforming geometric data, and processing multidimensional datasets. These operations include addition, subtraction, multiplication, finding determinants, and inverses, among others.

When dealing with a matrix like the one in our exercise, understanding how to manipulate it is crucial. One powerful operation is calculating the determinant, which provides information about the matrix's properties. For example, it can indicate whether a set of equations has a unique solution or if a system of vectors spans a space.

Matrix operations can also simplify complex expressions. By using row or column operations, we can reduce matrices, making them easier to work with without changing their fundamental properties. This simplification is often integral to finding solutions for equations or calculating sums as seen in this problem.

Summation of sequences

The summation of sequences is a process that involves adding up a series of numbers defined by a particular rule or pattern. In mathematical terms, this is often represented as \(\sum_{r=1}^{n-1} \Delta_r\), where each \( \Delta_r \) is a term in the sequence that needs to be summed.

Summing sequences can reveal patterns or allow us to calculate accumulated values over specific ranges. It plays a key role in various mathematical applications, such as calculus or series analysis. Understanding the rule that generates each term of the sequence is vital.

In the context of our original problem, understanding how to compute the sum of determinants \( \Delta_r \) involves inspecting how each element of the sequence behaves in relation to changing variables like \( \mathrm{n} \) and possibly eliminating dependence on certain variables, such as \( \mathrm{a} \), as the solution process refined observed.

Variable dependence in mathematics

Variable dependence is a concept that describes how the outcome of a mathematical expression relies on one or more variables. It's crucial to determine which parts of an expression genuinely affect the outcome, as this guides understanding and simplifies calculations.

In our exercise, we need to assess how the summation of determinants \( \Delta_r \) depends on the variables \( \mathrm{a} \) and \( \mathrm{n} \). This involves recognizing which operations or simplifications can obscure the presence of certain variables, leading to outcomes that may seem independent of certain parameters.

By examining the expression carefully, you can find which variables have a direct impact and which do not. Through simplification tricks, for instance, using matrix row or column operations, some dependencies can potentially be neutralized, indicating independence as concluded in this exercise. Understanding these relations helps better plan approaches and confirm whether an outcome depends on certain parameters or if it's constant across their changes.