Principal Value of sin⁻¹(-√3/2) Question: Find the principal value of: \[ \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) \] Concept: The principal value range of \( \sin^{-1}x \) is: \[ -\frac{\pi}{2} \leq y \leq \frac{\pi}{2} \] — Solution: Step 1: Recall standard value \[ \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \] Step 2: Apply negative sign Since sine is negative, the angle lies in […]
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Number of Commutative Binary Operations on 2 Elements Question: The number of commutative binary operations that can be defined on a set of 2 elements is: (a) 8 (b) 6 (c) 4 (d) 2 Concept: For a set with \( n \) elements: Total binary operations = \( n^{n^2} \) Commutative operations depend only on
Number of Binary Operations on Set of 2 Elements Question: The number of binary operations that can be defined on a set with 2 elements is: (a) 8 (b) 4 (c) 16 (d) 64 Concept: Number of binary operations on a set with \( n \) elements is: \[ n^{n^2} \] Solution: Here \( n
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: