Question

There are three-three sets are given in column-A and column-B \begin{tabular}{l|l} \(\frac{\text { Column-A }}{(1)\\{\mathrm{L}, \mathrm{A}, \mathrm{T}\\}}\) & (A) \(\left\\{\mathrm{x} / \mathrm{x} \in \mathrm{z}, \mathrm{x}^{2}<5\right\\}\) \\\ (2) \(\left\\{\mathrm{x} \in \mathrm{z} / \mathrm{x}^{3}-\mathrm{x}=0\right\\}\) & (B) \(\\{\mathrm{x} / \mathrm{x}\) is a letter of the word LATA \(\\}\) \\ (3) \(\\{-2,-1,0,1,2\\}\) & (C) \(\\{\sin 0, \sin (3 \pi / 2), \tan (5 \pi / 4)\\}\) \end{tabular} Which one of the following matches is correct? (a) \(1-\mathrm{A}, 2-\mathrm{B}, 3-\mathrm{C}\) (b) \(1-\mathrm{B}, 2-\mathrm{A}, 3-\mathrm{C}\) (c) \(1-\mathrm{B}, 2-\mathrm{C}, 3-\mathrm{A}\) (d) \(1-\mathrm{A}, 2-\mathrm{C}, 3-\mathrm{B}\)

Short answer

(a) 1-B, 2-A, 3-C

Step-by-step solution

Analyze set 1 in Column-A

Set 1 in Column-A, denoted as (1), contains the elements: L, A, and T. Now, let's look at the sets in Column-B and try to find a match. - Set A: Contains integer values of x for which \(x^2 < 5\). Since this set consists of numbers, it cannot be a match with set (1) which consists of letters. - Set B: Contains values of x which are letters of the word "LATA". This set has the elements L, A, and T, which match exactly with set (1) in Column-A. So, we have 1-B as the first match. - Set C: Contains the values of sin(0), sin(3π/2), and tan(5π/4). This set consists of trigonometric values and cannot be a match with set (1) which consists of letters.

Analyze set 2 in Column-A

Set 2 in Column-A, denoted as (2), contains integer values of x for which \(x^3 - x = 0\). Now, let's look at the remaining sets in Column-B and try to find a match. - Set A: Contains integer values of x for which \(x^2 < 5\). The elements in this set are -2, -1, 0, 1, and 2, which are the integer values of x that satisfy the given condition \(x^3 - x = 0\). So, we have 2-A as the second match. - Set C: Since Set A already matched with set 2 in Column-A, there is no need to look for any further matches.

Analyze set 3 in Column-A

Set 3 in Column-A, denoted as (3), contains the integer elements: -2, -1, 0, 1, and 2. Now, as we have already found the matches for set 1 and set 2 in Column-A, the only remaining set in Column-B to match with is Set C, which contains the trigonometric values sin(0), sin(3π/2), and tan(5π/4). This means 3-C is the last match.

Determine the correct match

Based on our analysis in the previous steps, we have the following matches between Column-A and Column-B sets: - 1-B - 2-A - 3-C Looking at the given options, the correct answer is: (a) 1-B, 2-A, 3-C

Key concepts

Trigonometric Functions

Trigonometric functions are vital in mathematics and are used to relate angles to side lengths in right-angled triangles. The key trigonometric functions include sine, cosine, and tangent, commonly denoted as \( \sin \), \( \cos \), and \( \tan \) respectively. These functions also extend their utility to circular functions, handling angles beyond the limitations of a triangle.

Here are a few common trigonometric values often seen in problems:

• For \( \sin 0 \), the value is 0 because the angle is at the start of the circle.
• For \( \sin \left(\frac{3\pi}{2}\right) \), the value is -1, positioned at the bottom of the circle.
• For \( \tan \left(\frac{5\pi}{4}\right) \), the value is 1, as this angle has its tangent line crossing the unit circle in both x and y directions equally.

Trigonometric functions form the backbone of an array of mathematical concepts, especially in fields like calculus, physics, and engineering, making their understanding crucial for students.

Integer Solutions

Integer solutions refer to solutions of equations where the values sought are whole numbers. Discrete mathematics frequently deals with integer solutions because many real-world applications demand whole number constraints.

In the given exercise, one task is to solve for integer values of \( x \) such that \( x^3 - x = 0 \). To find these solutions, factor the expression:

• Factor as \( x(x^2-1) = 0 \), so \( x = 0 \) or \( x^2 = 1 \).
• Simplifying \( x^2=1 \) gives solutions \( x = 1 \) and \( x = -1 \).

Thus, the integer solutions to the equation are \( -1, 0, \) and \( 1 \). These solutions guide the identification of the correct set match in the original problem, underlying the importance of solving algebraic equations efficiently.

Problem Solving Analysis

Effective problem solving in math involves systematically analyzing potential solutions and narrowing down possibilities until the correct answer is clearly identified.

In the exercise, we must match sets from two columns based on their properties. Start by identifying what each set represents and what makes the elements within them unique:

• Analyze each property of column sets separately—here, one set relates to letters, one to integers from an equation, and another to trigonometric values.
• Compare the unique properties of each set in one column to those in the other column to find corresponding matches.

The key lies in understanding each set individually and identifying overlaps or matches. By following through the logical steps and permutations, we can isolate the correct set correspondences, reflecting a structured approach to tackling these sometimes deceptively intricate problems.