NIMCET 2015 — Mathematics PYQ

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Question

NIMCET 2015 — Mathematics PYQ

NIMCET | Mathematics | 2015

A matrix $M_r$ is defined as $M_r = \begin{bmatrix} r & r-1 \\ r-1 & r \end{bmatrix}, r \in N$ , then the value of $\det(M_1) + \det(M_2) + \dots + \det(M_{2015})$ is:

Choose the correct answer:

Explanation

1. Find the Determinant of the General Matrix $M_r$ :

Given the matrix:

M_r = \begin{bmatrix} r & r-1 \\ r-1 & r \end{bmatrix}

The determinant $\det(M_r)$ is calculated as:

\det(M_r) = (r)(r) - (r-1)(r-1)

\det(M_r) = r^2 - (r-1)^2

2. Simplify the Determinant Expression:

Expanding the square:

\det(M_r) = r^2 - (r^2 - 2r + 1)

\det(M_r) = r^2 - r^2 + 2r - 1

\det(M_r) = 2r - 1

3. Evaluate the Summation:

We need to find the sum $S = \sum_{r=1}^{2015} \det(M_r)$ .

Substituting the expression for the determinant:

S = \sum_{r=1}^{2015} (2r - 1)

This is the sum of the first $n$ odd numbers, where $n = 2015$ .

The sum of the first $n$ odd numbers is given by the formula:

\sum_{r=1}^{n} (2r - 1) = n^2

4. Final Calculation:

For $n = 2015$ :

S = 2015^2

Correct Option:

(d) $2015^2$