Question 1: If \(y=\sin^{-1}x,\) then \((1-x^{2})\frac{d^{2}y}{dx^{2}}\) is equal to:

• A. \(x\frac{dy}{dx}\)
• B. \(-x\frac{dy}{dx}\)
• C. \(x^{2}\frac{dy}{dx}\)
• D. \(-x^{2}\frac{dy}{dx}\)

Question 2: If \(y=a\cos(\log x)+b\sin(\log x)\), then \(x^{2}y_{2}+xy_{1}\) is:

• A. \(\cot(\log x)\)
• B. y
• C. -y
• D. \(\tan(\log x)\)

Question 3: If \(y=\log(\sqrt{x}+\frac{1}{\sqrt{x}})^{2}\), then show that \(x(x+1)^{2}y_{2}+(x+1)^{2}y_{1}=2.\)

Question 4: If \(x=a~sin^{3}\theta\), \(y=b~cos^{3}\theta\) then find \(\frac{d^{2}y}{dx^{2}}\) at \(\theta=\frac{\pi}{4}\)

Question 5: If \(x = a \left( \cos \theta + \log \tan \frac{\theta}{2} \right)\) and \(y = \sin \theta\), then find \(\frac{d^2y}{dx^2}\) at \(\theta = \frac{\pi}{4}\).