Question

Which of the following statements are true?
A. If  \(x : y = 3 : 1\) then \(x ^3 − y ^3 = \frac{10}{11}\)
B. If \(x=y+12,x:y=3:2\) and \(z:y=1:3\),then \(z+x=44\)
C. If \(3x=8y\) and \(5y=9z\),then \(\frac{x}{z}=\frac{72}{15}\)
Choose the most appropriate answer from the options given below:

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

Answer 1

The Correct Option is C

 To determine which statements are true, let's analyze each statement individually: 

1. Statement A: If \(x : y = 3 : 1\) then \(x^3 − y^3 = \frac{10}{11}\)
   • Given: \(x : y = 3 : 1\)
   • Let \(x = 3k\) and \(y = k\)
   • Then, \(x^3 - y^3 = (3k)^3 - (k)^3 = 27k^3 - k^3 = 26k^3\)
   • We need \(26k^3 = \frac{10}{11}\), which is not possible for any real value of \(k\). Therefore, Statement A is false.
2. Statement B: If \(x=y+12\), \(x:y=3:2\), and \(z:y=1:3\), then \(z+x=44\)
   • Given: \(x = y + 12\) and \(x : y = 3 : 2\)
   • Let \(x = 3p\) and \(y = 2p\)
   • From \(x = y + 12\), we have \(3p = 2p + 12\). Solving gives \(p = 12\)
   • Thus, \(x = 3 \times 12 = 36\) and \(y = 2 \times 12 = 24\)
   • For \(z : y = 1 : 3\), \(z = \frac{1}{3}y = \frac{1}{3} \times 24 = 8\)
   • Therefore, \(z + x = 8 + 36 = 44\), So, Statement B is true.
3. Statement C: If \(3x = 8y\) and \(5y = 9z\), then \(\frac{x}{z} = \frac{72}{15}\)
   • Given: \(3x = 8y\) implies \(x = \frac{8}{3}y\)
   • Also, \(5y = 9z\) implies \(y = \frac{9}{5}z\)
   • Substitute the value of \(y\) in \(x = \frac{8}{3}y\), we get \(x = \frac{8}{3} \times \frac{9}{5}z = \frac{72}{15}z\)
   • This implies \(\frac{x}{z} = \frac{72}{15}\), which means Statement C is true.

Therefore, the correct answer is: B and C only.