Download App

Differentiation — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDifferentiation5 Marks
Question
If $\text{x}=\text{e}^{\cos2\text{t}}$ and $\text{y}=\text{e}^{\sin2\text{t}},$ prove that $\frac{\text{dy}}{\text{dx}}=-\frac{\text{y}\log\text{x}}{\text{x}\log\text{y}}$
✓

Answer

We have, $\text{x}=(\text{e}^{\cos2\text{t}})$ and $\text{y}=\text{e}^{\sin2\text{t}}$
$\Rightarrow\frac{\text{dx}}{\text{dt}}=\frac{\text{d}}{\text{dt}}\big(\text{e}^{\cos2\text{t}}\big)$ and $\frac{\text{dy}}{\text{dt}}=\frac{\text{d}}{\text{dt}}\big(\text{e}^{\sin2\text{t}}\big)$
$\Rightarrow\frac{\text{dx}}{\text{dt}}=\text{e}^{\cos2\text{t}}\frac{\text{d}}{\text{dt}}(\cos2\text{t})$ and $\frac{\text{dx}}{\text{dt}}=\text{e}^{\cos2\text{t}}\frac{\text{d}}{\text{dt}}(\cos2\text{t})$
$\Rightarrow\frac{\text{dx}}{\text{dt}}=\text{e}^{\cos2\text{t}}(-\sin2\text{t})\frac{\text{d}}{\text{dt}}(2\text{t})$ and $\frac{\text{dy}}{\text{dt}}=\text{e}^{\sin2\text{t}}(\cos2\text{t})\frac{\text{d}}{\text{dt}}(2\text{t})$
$\Rightarrow\frac{\text{dx}}{\text{dt}}=-2\sin 2\text{t}\text{e}^{\cos2\text{t}}$ and $\frac{\text{dy}}{\text{dt}}=2\cos 2\text{t}\text{e}^{\sin2\text{t}}$
$\because\frac{\frac{\text{dy}}{\text{dt}}}{\frac{\text{dx}}{\text{dt}}}=\frac{2\cos2\text{te}^{\sin2\text{t}}}{-2\sin2\text{te}^{\cos2\text{t}}}$
$\begin{bmatrix} \because\text{x}=\text{e}^{\cos2\text{t}}\Rightarrow\log\text{x}=\cos2\text{t} \\ \text{y}=\text{e}^{\sin2\text{t}}\Rightarrow\log\text{y}=\sin2\text{t} \end{bmatrix}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=-\frac{\text{y}\log\text{x}}{\text{x}\log\text{y}}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "Solve the Following Question.(5 Marks)" from Differentiation→Differentiation — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Three cards are cdrawn successively with replacement from a well shffled deck of 52 cards. A random variable X denotes the number of hearts in the three cards drawn. Determine the probability distribution of X.
→
Maximum Z = 15x + 10y
Subject to
$3\text{x}+2\text{y}\leq80$
$2\text{x}+3\text{y}\leq70$
$\text{x},\text{y}\geq0$
→
Solve $\cos^{-1}\sqrt3\text{x}+\cos^{-1}\text{x}=\frac{\pi}{2}$
→
Solve the following differential equation
$(1+\text{x}^2)\text{dy}=\text{xy dx}$
→
If $\begin{bmatrix}\text{x}&4&1\end{bmatrix}\begin{bmatrix}2&1&2\\1&0&2\\0&2&-4\end{bmatrix}\begin{bmatrix}\text{x}\\4\\-1\end{bmatrix}=0,$ find x.
→
A and B take turns in throwing two dice, the first to throw $10$ being awarded the prize, show that if A has the first throw, their chance of winning are in the ratio $12 : 11.$
→
If $\text{x}=\cot\text{t and y}=\sin\text{t},$ prove that $\frac{\text{dy}}{\text{dx}}=\frac{1}{\sqrt{3}}\text{ at t}=\frac{2\pi}{3}$
→
Find the second order derivatives of the following functions:$\log(\sin\text{x})$
→
A chemical company produces a chemical containing three basic elements A, B, C so that it has at least 16 liters of A, 24 liters of B and 18 liters of C. This chemical is made by mixing two compounds I and II. Each unit of compound I has 4 liters of A, 12 liters of B, 2 liters of C. Each unit of compound II has 2 liters of A, 2 liters of B and 6 liters of C. The cost per unit of compound I is ₹ 800/- and that of compound II is ₹ 640/-. Formulate the problem as L.P.P. and solve it to minimize the cost.
→
If $f(x) = x^3 + 4x^2 - x$, find f(A), where $\text{A}=\begin{bmatrix}0&1&2\\2&-3&0\\1&-1&0\end{bmatrix}$
→