PUC Science 2nd PUC Class 12 - Karnataka Board PUC Question Bank Solutions for Mathematics

If y = cot x show that \[\frac{d^2 y}{d x^2} + 2y\frac{dy}{dx} = 0\] ?

Find \[\frac{d^2 y}{d x^2}\] where \[y = \log \left( \frac{x^2}{e^2} \right)\] ?

If y = ae^2x + be^−x, show that, \[\frac{d^2 y}{d x^2} - \frac{dy}{dx} - 2y = 0\] ?

If y = e^x (sin x + cos x) prove that \[\frac{d^2 y}{d x^2} - 2\frac{dy}{dx} + 2y = 0\] ?

If y = cos⁻¹ x, find \[\frac{d^2 y}{d x^2}\] in terms of y alone ?

If  \[y = e^{a \cos^{- 1}} x\] ,prove that \[\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} - a^2 y = 0\] ?

If y = 500 e^7x + 600 e^−7x, show that \[\frac{d^2 y}{d x^2} = 49y\] ?

If x = 2 cos t − cos 2t, y = 2 sin t − sin 2t, find \[\frac{d^2 y}{d x^2}\text{ at } t = \frac{\pi}{2}\] ?

If x = 4z² + 5, y = 6z² + 7z + 3, find \[\frac{d^2 y}{d x^2}\] ?

If y log (1 + cos x), prove that \[\frac{d^3 y}{d x^3} + \frac{d^2 y}{d x^2} \cdot \frac{dy}{dx} = 0\] ?

If y = sin (log x), prove that \[x^2 \frac{d^2 y}{d x^2} + x\frac{dy}{dx} + y = 0\] ?

If y = 3 e^2x + 2 e^3x, prove that  \[\frac{d^2 y}{d x^2} - 5\frac{dy}{dx} + 6y = 0\] ?

If y = (cot⁻¹ x)², prove that y₂(x² + 1)² + 2x (x² + 1) y₁ = 2 ?

If y = cosec⁻¹ x, x >1, then show that \[x\left( x^2 - 1 \right)\frac{d^2 y}{d x^2} + \left( 2 x^2 - 1 \right)\frac{dy}{dx} = 0\] ?

\[\text { If x } = \cos t + \log \tan\frac{t}{2}, y = \sin t, \text { then find the value of } \frac{d^2 y}{d t^2} \text { and } \frac{d^2 y}{d x^2} \text { at } t = \frac{\pi}{4} \] ?

\[ \text { If x } = a \sin t \text { and y } = a\left( \cos t + \log \tan\frac{t}{2} \right), \text { find } \frac{d^2 y}{d x^2} \] ?

\[\text { If x } = a\left( \cos t + t \sin t \right) \text { and y} = a\left( \sin t - t \cos t \right),\text { then find the value of } \frac{d^2 y}{d x^2} \text { at } t = \frac{\pi}{4} \] ?

\[\text{ If x } = a\left( \cos t + \log \tan\frac{t}{2} \right) \text { and y } = a\left( \sin t \right), \text { evaluate } \frac{d^2 y}{d x^2} \text { at t } = \frac{\pi}{3} \] ?

\[\text { If x } = a\left( \cos2t + 2t \sin2t \right)\text {  and y } = a\left( \sin2t - 2t \cos2t \right), \text { then find } \frac{d^2 y}{d x^2} \] ?

If \[x = 3 \cos t - 2 \cos^3 t, y = 3\sin t - 2 \sin^3 t,\] find \[\frac{d^2 y}{d x^2} \] ?