Let p and p + 2 be prime numbers and let
\(Δ=\begin{vmatrix} p! & (p+1)! & (p+2)! \\ (p+1)! & (p+2)! & (p+3)! \\ (p+2)! & (p+3)! & (p+4)! \\ \end{vmatrix}\)
Then the sum of the maximum values of α and β, such that pα and (p + 2)β divide Δ, is _______.
Let p and p + 2 be prime numbers and let
\(Δ=\begin{vmatrix} p! & (p+1)! & (p+2)! \\ (p+1)! & (p+2)! & (p+3)! \\ (p+2)! & (p+3)! & (p+4)! \\ \end{vmatrix}\)
Then the sum of the maximum values of α and β, such that pα and (p + 2)β divide Δ, is _______.
Correct Answer: 4
Solution and Explanation
The correct answer is 4
\(Δ=\begin{vmatrix} p! & (p+1)! & (p+2)! \\ (p+1)! & (p+2)! & (p+3)! \\ (p+2)! & (p+3)! & (p+4)! \\ \end{vmatrix}\)
\(=p!(p+1)!⋅(p+2)!\)\(\begin{vmatrix} 1 & p+1 & (p+1)(p+2) \\ 1 & (p+2) & (p+2)(p+3) \\ 1 & (p+3) & (p+3)(p+4) \\ \end{vmatrix}\)
\(=p!(p+1)!⋅(p+2)!\)\(\begin{vmatrix} 1 & p+1 & p^2+3p+2\\ 0 & 1 & 2p+4 \\ 0 & 1 & 2p+6 \\ \end{vmatrix}\)
\(=2(p!)⋅((p+1)!)⋅((p+2)!)\)
\(=2(p+1)⋅(p!)2⋅((p+2)!)\)
\(=2(p+1)2⋅(p!)3⋅((p+2)!)\)
∴ Maximum value of α is 3 and β is 1.
∴ α + β = 4
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- Four friends Abhay, Bina, Chhaya, and Devesh were asked to simplify \( 4 AB + 3(AB + BA) - 4 BA \), where \( A \) and \( B \) are both matrices of order \( 2 \times 2 \). It is known that \( A \neq B \) and \( A^{-1} \neq B \). Their answers are given as:
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- Given \[ A = \begin{bmatrix} -4 & 4 & 4 \\ -7 & 1 & 3 \\ 5 & -3 & -1 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & -1 & 1 \\ 1 & -2 & -2 \\ 2 & 1 & 3 \end{bmatrix} \] find \( AB \). Hence, solve the system of linear equations: \[ x - y + z = 4, \] \[ x - 2y - 2z = 9, \] \[ 2x + y + 3z = 1. \]
- If \[ A = \begin{bmatrix} 1 & 2 & 0 \\ -2 & -1 & -2 \\ 0 & -1 & 1 \end{bmatrix} \] then find \( A^{-1} \). Hence, solve the system of linear equations: \[ x - 2y = 10, \] \[ 2x - y - z = 8, \] \[ -2y + z = 7. \]
- Assertion (A): \( A = \text{diag} [3, 5, 2] \) is a scalar matrix of order \( 3 \times 3 \).
Reason (R): If a diagonal matrix has all non-zero elements equal, it is known as a scalar matrix.
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Concepts Used:
Matrices
Matrix:
A matrix is a rectangular array of numbers, variables, symbols, or expressions that are defined for the operations like subtraction, addition, and multiplications. The size of a matrix is determined by the number of rows and columns in the matrix.
The basic operations that can be performed on matrices are:
- Addition of Matrices - The addition of matrices addition can only be possible if the number of rows and columns of both the matrices are the same.
- Subtraction of Matrices - Matrices subtraction is also possible only if the number of rows and columns of both the matrices are the same.
- Scalar Multiplication - The product of a matrix A with any number 'c' is obtained by multiplying every entry of the matrix A by c, is called scalar multiplication.
- Multiplication of Matrices - Matrices multiplication is defined only if the number of columns in the first matrix and rows in the second matrix are equal.
- Transpose of Matrices - Interchanging of rows and columns is known as the transpose of matrices.