Question:
If \(x^3+y^3 = 6\) and \(\frac{d^2y}{dx^2}\cdot \frac{d^2x}{dy^2} = \frac{m}{(xy)^n}\), where \(m,n\in R\) then \(\frac{m}{n} =\)
If \(x^3+y^3 = 6\) and \(\frac{d^2y}{dx^2}\cdot \frac{d^2x}{dy^2} = \frac{m}{(xy)^n}\), where \(m,n\in R\) then \(\frac{m}{n} =\)
Show Hint
Find y double prime and x double prime separately, then multiply.
Updated On: Oct 1, 2026
- \(36\)
- \(9\)
- \(6\)
- \(4\)
Show Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Step 1: First derivative:
From \(x^3+y^3=6\): \(3x^2+3y^2y'=0\), so \(y'=-\dfrac{x^2}{y^2}\).
Step 2: Second derivative d2y/dx2:
\(y''=-\dfrac{2xy^2-x^2\cdot2yy'}{y^4}=-\dfrac{2xy^2+2x^4/y}{y^4}=-\dfrac{2x(y^3+x^3)}{y^5}=-\dfrac{12x}{y^5}\), using \(x^3+y^3=6\).
Step 3: Second derivative d2x/dy2:
By symmetry of the equation in \(x\) and \(y\): \(\dfrac{d^2x}{dy^2}=-\dfrac{12y}{x^5}\).
Step 4: Multiply:
\[ \dfrac{d^2y}{dx^2}\cdot\dfrac{d^2x}{dy^2}=\dfrac{144xy}{x^5y^5}=\dfrac{144}{(xy)^4} \]
So \(m=144\) and \(n=4\), giving \(\dfrac mn=36\). Option A.
Final Answer:
m/n = 36. \[ \boxed{\text{(A) }36} \]
From \(x^3+y^3=6\): \(3x^2+3y^2y'=0\), so \(y'=-\dfrac{x^2}{y^2}\).
Step 2: Second derivative d2y/dx2:
\(y''=-\dfrac{2xy^2-x^2\cdot2yy'}{y^4}=-\dfrac{2xy^2+2x^4/y}{y^4}=-\dfrac{2x(y^3+x^3)}{y^5}=-\dfrac{12x}{y^5}\), using \(x^3+y^3=6\).
Step 3: Second derivative d2x/dy2:
By symmetry of the equation in \(x\) and \(y\): \(\dfrac{d^2x}{dy^2}=-\dfrac{12y}{x^5}\).
Step 4: Multiply:
\[ \dfrac{d^2y}{dx^2}\cdot\dfrac{d^2x}{dy^2}=\dfrac{144xy}{x^5y^5}=\dfrac{144}{(xy)^4} \]
So \(m=144\) and \(n=4\), giving \(\dfrac mn=36\). Option A.
Final Answer:
m/n = 36. \[ \boxed{\text{(A) }36} \]
Was this answer helpful?
0
0
Top MHT CET Mathematics Questions
- The line $5x + y - 1 = 0$ coincides with one of the lines given by $5x^2 + xy - kx - 2y + 2 = 0 $ then the value of k is
- If $\int\frac{f\left(x\right)}{log \left(sin\,x\right)}dx = log\left[log\,sin\,x\right]+c$ then $f\left(x\right)=$
- If $\int\limits^{K}_0 \frac{dx}{2 + 18 x^2} = \frac{\pi}{24}$, then the value of K is
- If $\int^{\pi/2}_{0} \log\cos x dx =\frac{\pi}{2} \log\left(\frac{1}{2}\right)$ then $ \int^{\pi/2}_{0} \log\sec x dx = $
- The point on the curve $y = \sqrt{x - 1}$ where the tangent is perpendicular to the line $2x + y - 5 = 0 $ is
View More Questions
Top MHT CET Second Order Derivative Questions
- If \( y = 5\cos x - 3\sin x \), then \( \frac{d^2y}{dx^2} + y \) equals:
- A wire of length 20 units is divided into two parts such that the product of one part and the cube of the other part is maximum. The product of these parts is:
- If \[ x = \sqrt{e^{\sin^{-1} t}} \quad \text{and} \quad y = \sqrt{e^{\cos^{-1} t}}, \] then find \[ \frac{d^2 y}{dx^2}. \]
- If $x = a \sin t - b \cos t$ and $y = a \cos t + b \sin t$, and it is given that $\frac{d^2y}{dx^2} = 0$, then:
- If \( y = \cos^{-1} x \), find \( \frac{d^2y}{dx^2} \) in terms of \( y \).
View More Questions
Top MHT CET Questions
- During r- DNA technology, which one of the following enzymes is used for cleaving DNA molecule ?
- In non uniform circular motion, the ratio of tangential to radial acceleration is (r = radius of circle, $v =$ speed of the particle, $\alpha =$ angular acceleration)
- The temperature of $32^{\circ}C$ is equivalent to
- The line $5x + y - 1 = 0$ coincides with one of the lines given by $5x^2 + xy - kx - 2y + 2 = 0 $ then the value of k is
- The heat of formation of water is $ 260\, kJ $ . How much $ H_2O $ is decomposed by $ 130\, kJ $ of heat ?
View More Questions