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APPLICATION OF DERIVATIVES — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAPPLICATION OF DERIVATIVES4 Marks
Question
A spherical ball of salt is dissolving in water in such a manner that the rate of decrease of the volume at any instant is propotional to the surface. Prove that the radius is decreasing at a constant rate.
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Answer

We have, rate of decrease of the volume of spherical ball of salt at any instant is surface. Let the radius of the spherical ball of the salt be r.
$\therefore$ Volume of the ball $(\text{V})=\frac{4}{3}\pi\text{r}^3$
and surface area $(\text{S})=4\pi\text{r}^2$
$\because\ \frac{\text{dV}}{\text{dT}}\propto\text{S}$
$\Rightarrow\ \frac{\text{d}}{\text{dt}}\Big(\frac{4}{3}\pi\text{r}^3\Big)\propto4\pi\text{r}^2$
$\Rightarrow\ \frac{4}{3}\pi3\text{r}^2\frac{\text{dr}}{\text{dt}}\propto4\pi\text{r}^2$
$\Rightarrow\ \frac{\text{dr}}{\text{dt}}\propto\frac{4\pi\text{r}^2}{4\pi\text{r}^2}$
$\Rightarrow\ \frac{\text{dr}}{\text{dt}}=\text{k.1}$ [where, k is the proportionality constant]
$\Rightarrow\ \frac{\text{dr}}{\text{dt}}=\text{k}$
Hence, the radius of ball is decreasing at a constant rate.

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Based on the above information, answer the following questions.
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  1. $2\hat{\text{i}}+3\hat{\text{j}}+6\hat{\text{k}}$
  2. $2\hat{\text{i}}-3\hat{\text{j}}-6\hat{\text{k}}$
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What will be the height of the plant after 2 days?

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Based on the above information, answer the following questions.
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Now, suppose the given equation is $(1+\sin\text{x})\frac{\text{dy}}{\text{dx}}+\text{y}\cos\text{x}+\text{x}=0.$
Based on the above information, answer the following questions.
  1. The value of P and Q respectively are:
  1. $\frac{\sin\text{x}}{1+\cos\text{x}},\ \frac{\text{x}}{1+\sin\text{x}}$
  2. $\frac{\cos\text{x}}{1+\sin\text{x}},\ \frac{\text{-x}}{1+\sin\text{x}}$
  3. $\frac{-\cos\text{x}}{1+\sin\text{x}},\ \frac{\text{x}}{1+\sin\text{x}}$
  4. $\frac{\cos\text{x}}{1+\sin\text{x}},\ \frac{\text{x}}{1+\sin\text{x}}$
  1. The value of I.F is:
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  2. $\cos\text{x}$
  3. $1+\sin\text{x}$
  4. $1-\cos\text{x}$
  1. Solution of given equation is:
  1. $\text{y}(1-\sin\text{x})=\text{x+c}$
  2. $\text{y}(1+\sin\text{x})=-\text{x}^2+\text{c}$
  3. $\text{y}(1-\sin\text{x})=\frac{\text{-x}^2}{2}\text{+c}$
  4. $\text{y}(1+\sin\text{x})=\frac{\text{-x}^2}{2}\text{+c}$
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  2. $\frac{2+\text{x}^2}{2(1+\sin\text{x})}$
  3. $\frac{2-\text{x}^2}{2(1-\sin\text{x})}$
  4. $\frac{2+\text{x}^2}{2(1-\sin\text{x})}$
  1. Value of is $\text{y}\Big(\frac{\pi}{2}\Big)$ is:
  1. $\frac{4-\pi^2}{2}$
  2. $\frac{8-\pi^2}{16}$
  3. $\frac{8-\pi^2}{4}$
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  3. 2
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$\text{p}(\text{x})=15-\frac{\text{x}}{3000}$ where x is the number of tickets sold.

Based on the above information, answer the following questions.
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  1. $15\text{x}-\frac{\text{x}^2}{3000}$
  2. $15-\frac{\text{x}^2}{3000}$
  3. $15\text{x}-\frac{1}{3000}$
  4. $15\text{x}-\frac{\text{x}}{3000}$
  1. The range of x is.
  1. [24000, 36000]
  2. [0, 24000]
  3. [0, 36000]
  4. None of these
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  1. 20000
  2. 21000
  3. 22500
  4. 25000
  1. When the revenue is maximum, the price of the ticket is.
  1. ₹ 5
  2. ₹ 5.5
  3. ₹ 7
  4. ₹ 7.5
  1. How many spectators should be present to maximize the revenue?
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  2. 21000
  3. 22000
  4. 22500
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Order: The order of a differential equation is the order of the highest order derivative appearing in the differential equation.
Degree: The degree of differential equation is the power of the highest order derivative, when differential coefficients are made free from radicals and fractions. Also, differential equation must be a polynomial equation in derivatives for the degree to be defined.
Based on the above information, answer the following questions.
  1. Find the degree of the differential equation $2\frac{\text{d}^2\text{y}}{\text{dx}^2}+3\sqrt{1-\Big(\frac{\text{dy}}{\text{dx}}\Big)^2-\text{y}=0}.$
  1. $3$
  2. $4$
  3. $3$
  4. $1$
  1. Order and degree of the differential equation $\text{y}\frac{\text{dy}}{\text{dx}}=\frac{\text{x}}{\frac{\text{dy}}{\text{dx}}+\Big(\frac{\text{dy}}{\text{dx}}\Big)^3}$ are respectively.
  1. $1, 1$
  2. $1, 2$
  3. $1, 3$
  4. $1, 4$
  1. Find order and degree of the equation $y'" + y^2 + e^{y'} = 0.$
  1. Order $= 3,$ degree $=$ undefined.
  2. Order $= 1,$ degree $= 3.$
  3. Order $= 2,$ degree $=$ undefined.
  4. Order $= 1,$ degree $= 2.$
  1. Determine degree of the differential equation $(\sqrt{\text{a+x}})\times\Big(\frac{\text{dy}}{\text{dx}}\Big)+\text{x}=0.$
  1. $3$
  2. Not defined
  3. $1$
  4. $2$
  1. Order and degree of the differential equation $\Bigg(1+\Big(\frac{\text{dy}}{\text{dx}}\Big)^3\Bigg)^\frac{7}{3}=7\frac{\text{d}^2\text{y}}{\text{dx}^2}$ are respectively.
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  2. $2, 3$
  3. $1, 3$
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Three car dealers, say A, Band C, deals in three types of cars, namely Hatchback cars, Sedan cars, SUV cars. The sales figure of 2019 and 2020 showed that dealer A sold 120 Hatchback, 50 Sedan, 10 SUV cars in 2019 and 300 Hatchback, 150 Sedan, 20 SUV cars in 2020; dealer B sold 100 Hatchback, 30 Sedan, 5 SUV cars in 2019 and 200 Hatchback, 50 Sedan, 6 SUV cars in 2020; dealer C sold 90 Hatchback, 40 Sedan, 2 SUV cars in 2019 and 100 Hatchback, 60 Sedan, 5 SUV cars in 2020.

Based on the above information, answer the following questions.
  1. The matrix summarizing sales data of 2019 is:
  1. $\begin{matrix}&\text{Hatchback}&\text{Sedan}&\text{SUV}\end{matrix}\\\begin{matrix}\text{A}\\\text{B}\\\text{C}\end{matrix}\begin{bmatrix}\ \ \ \ \ \ 300\ \ \ &\ \ 150&\ \ \ \ \ 20\\\ \ \ 200&\ \ 50&\ \ \ \ \ 6\\\ \ \ 100&\ \ 30&\ \ \ \ \ 5\end{bmatrix}$
  2. $\begin{matrix}&\text{Hatchback}&\text{Sedan}&\text{SUV}\end{matrix}\\\begin{matrix}\text{A}\\\text{B}\\\text{C}\end{matrix}\begin{bmatrix}\ \ \ \ \ \ 120\ \ \ &\ \ 100&\ \ \ \ \ 20\\\ \ \ 100&\ \ 30&\ \ \ \ \ 5\\\ \ \ 90&\ \ 40&\ \ \ \ \ 2\end{bmatrix}$
  3. $\begin{matrix}&\text{Hatchback}&\text{Sedan}&\text{SUV}\end{matrix}\\\begin{matrix}\text{A}\\\text{B}\\\text{C}\end{matrix}\begin{bmatrix}\ \ \ \ \ \ 100\ \ \ &\ \ 30&\ \ \ \ \ 5\\\ \ \ 120&\ \ 50&\ \ \ \ \ 10\\\ \ \ 90&\ \ 40&\ \ \ \ \ 2\end{bmatrix}$
  4. $\begin{matrix}&\text{Hatchback}&\text{Sedan}&\text{SUV}\end{matrix}\\\begin{matrix}\text{A}\\\text{B}\\\text{C}\end{matrix}\begin{bmatrix}\ \ \ \ \ \ 200\ \ \ &\ \ 50&\ \ \ \ \ 6\\\ \ \ 100&\ \ 30&\ \ \ \ \ 5\\\ \ \ 300&\ \ 150&\ \ \ \ \ 20\end{bmatrix}$
  1. The matrix summarizing sales data of 2020 is:
  1. $\begin{matrix}&\text{Hatchback}&\text{Sedan}&\text{SUV}\end{matrix}\\\begin{matrix}\text{A}\\\text{B}\\\text{C}\end{matrix}\begin{bmatrix}\ \ \ \ \ \ 300\ \ \ &\ \ 150&\ \ \ \ \ 20\\\ \ \ 200&\ \ 50&\ \ \ \ \ 6\\\ \ \ 100&\ \ 60&\ \ \ \ \ 5\end{bmatrix}$
  2. $\begin{matrix}&\text{Hatchback}&\text{Sedan}&\text{SUV}\end{matrix}\\\begin{matrix}\text{A}\\\text{B}\\\text{C}\end{matrix}\begin{bmatrix}\ \ \ \ \ \ 120\ \ \ &\ \ 50&\ \ \ \ \ 10\\\ \ \ 100&\ \ 60&\ \ \ \ \ 5\\\ \ \ 90&\ \ 40&\ \ \ \ \ 2\end{bmatrix}$
  3. $\begin{matrix}&\text{Hatchback}&\text{Sedan}&\text{SUV}\end{matrix}\\\begin{matrix}\text{A}\\\text{B}\\\text{C}\end{matrix}\begin{bmatrix}\ \ \ \ \ \ 100\ \ \ &\ \ 60&\ \ \ \ \ 5\\\ \ \ 120&\ \ 50&\ \ \ \ \ 10\\\ \ \ 90&\ \ 40&\ \ \ \ \ 2\end{bmatrix}$
  4. $\begin{matrix}&\text{Hatchback}&\text{Sedan}&\text{SUV}\end{matrix}\\\begin{matrix}\text{A}\\\text{B}\\\text{C}\end{matrix}\begin{bmatrix}\ \ \ \ \ \ 200\ \ \ &\ \ 50&\ \ \ \ \ 6\\\ \ \ 100&\ \ 60&\ \ \ \ \ 5\\\ \ \ 300&\ \ 150&\ \ \ \ \ 20\end{bmatrix}$
  1. The cost incurred by the organisation on village Z is:
  1. $\begin{matrix}&\text{Hatchback}&\text{Sedan}&\text{SUV}\end{matrix}\\\begin{matrix}\text{A}\\\text{B}\\\text{C}\end{matrix}\begin{bmatrix}\ \ \ \ \ \ 190\ \ \ &\ \ 100&\ \ \ \ \ 7\\\ \ \ 300&\ \ 80&\ \ \ \ \ 11\\\ \ \ 420&\ \ 200&\ \ \ \ \ 30\end{bmatrix}$
  2. $\begin{matrix}&\text{Hatchback}&\text{Sedan}&\text{SUV}\end{matrix}\\\begin{matrix}\text{A}\\\text{B}\\\text{C}\end{matrix}\begin{bmatrix}\ \ \ \ \ \ 300\ \ \ &\ \ 80&\ \ \ \ \ 11\\\ \ \ 190&\ \ 100&\ \ \ \ \ 7\\\ \ \ 420&\ \ 200&\ \ \ \ \ 30\end{bmatrix}$
  3. $\begin{matrix}&\text{Hatchback}&\text{Sedan}&\text{SUV}\end{matrix}\\\begin{matrix}\text{A}\\\text{B}\\\text{C}\end{matrix}\begin{bmatrix}\ \ \ \ \ \ 420\ \ \ &\ \ 200&\ \ \ \ \ 30\\\ \ \ 300&\ \ 80&\ \ \ \ \ 11\\\ \ \ 190&\ \ 100&\ \ \ \ \ 7\end{bmatrix}$
  4. None of these
  1. The increase in sales from 2019 to 2020 is given by the matrix.
  1. $\begin{matrix}&\text{Hatchback}&\text{Sedan}&\text{SUV}\end{matrix}\\\begin{matrix}\text{A}\\\text{B}\\\text{C}\end{matrix}\begin{bmatrix}\ \ \ \ \ \ 180\ \ \ &\ \ 100&\ \ \ \ \ 10\\\ \ \ 10&\ \ 20&\ \ \ \ \ 1\\\ \ \ 100&\ \ 20&\ \ \ \ \ 3\end{bmatrix}$
  2. $\begin{matrix}&\text{Hatchback}&\text{Sedan}&\text{SUV}\end{matrix}\\\begin{matrix}\text{A}\\\text{B}\\\text{C}\end{matrix}\begin{bmatrix}\ \ \ \ \ \ 10\ \ \ &\ \ 20&\ \ \ \ \ 3\\\ \ \ 100&\ \ 20&\ \ \ \ \ 1\\\ \ \ 180&\ \ 100&\ \ \ \ \ 10\end{bmatrix}$
  3. $\begin{matrix}&\text{Hatchback}&\text{Sedan}&\text{SUV}\end{matrix}\\\begin{matrix}\text{A}\\\text{B}\\\text{C}\end{matrix}\begin{bmatrix}\ \ \ \ \ \ 180\ \ \ &\ \ 100&\ \ \ \ \ 10\\\ \ \ 100&\ \ 20&\ \ \ \ \ 1\\\ \ \ 10&\ \ 20&\ \ \ \ \ 3\end{bmatrix}$
  4. $\begin{matrix}&\text{Hatchback}&\text{Sedan}&\text{SUV}\end{matrix}\\\begin{matrix}\text{A}\\\text{B}\\\text{C}\end{matrix}\begin{bmatrix}\ \ \ \ \ \ 100\ \ \ &\ \ 20&\ \ \ \ \ 3\\\ \ \ 180&\ \ 100&\ \ \ \ \ 10\\\ \ \ 10&\ \ 20&\ \ \ \ \ 3\end{bmatrix}$
  1. If each dealer receive profit of ₹ 50000 on sale of a Hatchback. ₹ 100000 on sale of a Sedan and ₹ 200000 on sale of a SUV, then amount of profit received in the year 2020 by each dealer is given by the matrix.
  1. $\begin{matrix}\text{A}\\\text{B}\\\text{C}\end{matrix}\begin{bmatrix}30000000\\15000000\\12000000\end{bmatrix}$
  2. $\begin{matrix}\text{A}\\\text{B}\\\text{C}\end{matrix}\begin{bmatrix}12000000\\16200000\\34000000\end{bmatrix}$
  3. $\begin{matrix}\text{A}\\\text{B}\\\text{C}\end{matrix}\begin{bmatrix}34000000\\16200000\\12000000\end{bmatrix}$
  4. $\begin{matrix}\text{A}\\\text{B}\\\text{C}\end{matrix}\begin{bmatrix}15000000\\30000000\\12000000\end{bmatrix}$
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