Question

Which of the following inequalities holds true?
(A) \(\sqrt{5} + \sqrt{3}>\sqrt{6} + \sqrt{2}\)
(B) If \(a>b\) and \(c<0\), then \(\frac{a}{c}<\frac{b}{c}\)
(C) \(\frac{1}{x^2}>\frac{1}{x}>1\), if \(0<x<1\)
(D) If a and b are positive integers and \(\frac{a-b}{6.25} = \frac{4}{2.5}\) then \(b>a\)
Choose the correct answer from the options given below:

• A. (A), (B) and (D) only
• B. (A), (B) and (C) only
• C. (A) and (B) only
• D. (B) and (C) only

Hint:

When testing inequalities with variables, plugging in a simple test value that fits the condition (like \(x=0.5\) for \(0<x<1\)) is a quick way to check its validity. For inequalities with square roots, squaring both sides is often the most effective method, but remember this only works if both sides are non-negative.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:

This question requires evaluating four separate mathematical statements (inequalities and equations) to determine their validity.

Step 2: Detailed Explanation:

Statement (A): \( \sqrt{5} + \sqrt{3} > \sqrt{6} + \sqrt{2} \)

Squaring both sides to compare:

LHS squared: \( (\sqrt{5} + \sqrt{3})^2 = 8 + 2\sqrt{15} \)

RHS squared: \( (\sqrt{6} + \sqrt{2})^2 = 8 + 2\sqrt{12} \)

We get \( 2\sqrt{15} > 2\sqrt{12} \), and since \( 15 > 12 \), this statement is true.

Statement (B): If \( a > b \) and \( c < 0 \), then \( \frac{a}{c} < \frac{b}{c} \)

Multiplying or dividing by a negative number reverses the inequality. This is a fundamental rule of inequalities, so this statement is true.

Statement (C): \( \frac{1}{x^2} > \frac{1}{x} > 1 \), if \( 0 < x < 1 \)

For \( x = 0.5 \), we find that \( 4 > 2 > 1 \). This confirms the inequality, so this statement is true.

Statement (D): If \( a \) and \( b \) are positive integers and \( \frac{a-b}{6.25} = \frac{4}{2.5} \), then \( b > a \)

Solving the equation, we find \( a - b = 10 \), so \( a > b \). Therefore, this statement is false.

Step 3: Final Answer:

Statements (A), (B), and (C) are true, while (D) is false. The correct choice is the one that lists (A), (B), and (C) only.