Inverse of 4*6 for a*b = ab/3 Question: Let \( * \) be defined on \( \mathbb{Q}^+ \) by: \[ a * b = \frac{ab}{3} \] Find the inverse of \( 4 * 6 \). Options: (a) \( \frac{9}{8} \) (b) \( \frac{2}{3} \) (c) \( \frac{3}{2} \) (d) None of these Solution: Step 1: […]
Inverse of 8 for a*b = ab/2 Question: Let \( * \) be defined on \( \mathbb{Q}^+ \) by: \[ a * b = \frac{ab}{2} \] Find the inverse of \( 8 \). Options: (a) \( \frac{1}{8} \) (b) \( \frac{1}{2} \) (c) 2 (d) 4 Solution: Step 1: Find identity element Let identity be
Inverse of Matrix [2 -3; 3 2] Question: Find the inverse of the matrix: \[ \begin{bmatrix} 2 & -3 \\ 3 & 2 \end{bmatrix} \] Concept: For a matrix of the form: \[ \begin{bmatrix} a & -b \\ b & a \end{bmatrix} \] Its inverse is: \[ \frac{1}{a^2 + b^2} \begin{bmatrix} a & b \\
Inverse of a for a*b = a + b + ab Question: Let \( * \) be defined on \( \mathbb{R} – \{-1\} \) by: \[ a * b = a + b + ab \] Find the inverse of \( a \). Options: (a) \( -a \) (b) \( -\frac{a}{a+1} \) (c) \( \frac{1}{a}
Identity Element for a*b = a + b – ab Question: Let \( * \) be defined on \( \mathbb{Q} – \{1\} \) by: \[ a * b = a + b – ab \] Find the identity element. Options: (a) 0 (b) 1 (c) \( \frac{1}{2} \) (d) -1 Solution: Step 1: Let identity
Identity Element for a*b = a + b + 10 Question: Let \( * \) be defined on \( \mathbb{N} \) by: \[ a * b = a + b + 10 \] Find the identity element. Options: (a) -10 (b) 0 (c) 10 (d) Non-existent Solution: Step 1: Let identity be \( e \),
Inverse of 0.1 for a*b = ab/100 Question: Let \( * \) be defined on \( \mathbb{Q}^+ \) by: \[ a * b = \frac{ab}{100} \] Find the inverse of \(0.1\). Options: (a) \(10^5\) (b) \(10^4\) (c) \(10^6\) (d) None of these Solution: Step 1: Find identity element Let identity be \( e \), then:
Check Properties of a*b = 3a + b Question: A binary operation \( * \) on \( \mathbb{Z} \) is defined by: \[ a * b = 3a + b \] Determine its properties. Options: (a) Commutative (b) Associative (c) Not commutative (d) Commutative and associative Solution: Step 1: Check Commutativity \[ a * b
Check Properties of a*b = a² + b² Question: On \( \mathbb{Z} \), define: \[ a * b = a^2 + b^2 \] Determine its properties. Options: (a) Commutative and associative (b) Associative but not commutative (c) Not associative (d) Not a binary operation Solution: Step 1: Check Closure Since \(a^2 + b^2 \in \mathbb{Z}\),
Properties of a*b = a/b on Integers Question: Let \( * \) be defined on non-zero integers by: \[ a * b = \frac{a}{b} \] Which property is satisfied? (a) Closure (b) Commutative (c) Associative (d) None of these Solution: 1. Closure: For closure, result must be an integer. Example: \[ 1 * 2 =