Question
Find the maximum and minimum values of the function $f(x)=\cos ^2 x+\sin x$.
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A medical company has factories at two places, A and B. From these places, supply is made to each of its three agencies situated at P, Q and R. The monthly requirements of the agencies are respectively 40, 40 and 50 packets of the medicines, while the production capacity of the factories, A and B, are 60 and 70 packets respectively. The transportation cost per packet from the factories to the agencies are given below:
How many packets from each factory be transported to each agency so that the cost of transportation is minimum? Also find the minimum cost?
How many packets from each factory be transported to each agency so that the cost of transportation is minimum? Also find the minimum cost?
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.
Find the shortest distance between the lines
$\vec{\text{r}}=6\hat{\text{i}}+2\hat{\text{j}}+2\hat{\text{k}}+\lambda\big(\hat{\text{i}}-2\hat{\text{j}}+2\hat{\text{k}}\big)$ and $\vec{\text{r}}=-4\hat{\text{i}}-\hat{\text{k}}+\mu\big(3\hat{\text{i}}-2\hat{\text{j}}+2\hat{\text{k}}\big)$
→$\vec{\text{r}}=6\hat{\text{i}}+2\hat{\text{j}}+2\hat{\text{k}}+\lambda\big(\hat{\text{i}}-2\hat{\text{j}}+2\hat{\text{k}}\big)$ and $\vec{\text{r}}=-4\hat{\text{i}}-\hat{\text{k}}+\mu\big(3\hat{\text{i}}-2\hat{\text{j}}+2\hat{\text{k}}\big)$
Find the equation of the plane passing through the points (-1, 2, 0), (2, 2, -1) and parallel to the line $\frac{\text{x}-1}{1}=\frac{2\text{y}+1}{2}=\frac{\text{z}+1}{-1}.$
→Evaluate the following definite integrals:$\int_{0}^\limits{\infty}\frac{1}{\text{a}^2+\text{b}^2\text{x}^2} \text{ dx}$
→Show that the following system of linear equation is inconsistent:
4x − 2y = 3
6x − 3y = 5
Find the particular solution of $\text{e}^{\frac{\text{dy}}{\text{dx}}}=\text{x}+1,$ that $\text{y}=3,$ when $\text{x}=0.$
→Find the condition for the following set of curves to intersect orthogonally
$\frac{\text{x}^2}{\text{a}^2}-\frac{\text{y}^2}{\text{b}^2}=1\text{ and }\frac{\text{x}^2}{\text{A}^2}-\frac{\text{y}^2}{\text{B}^2}=1$
→$\frac{\text{x}^2}{\text{a}^2}-\frac{\text{y}^2}{\text{b}^2}=1\text{ and }\frac{\text{x}^2}{\text{A}^2}-\frac{\text{y}^2}{\text{B}^2}=1$
If $\vec{\text{a}}=2\hat{\text{i}}+5\hat{\text{j}}-7\hat{\text{k}},\vec{\text{b}}=-3\hat{\text{i}}+4\hat{\text{j}}+\hat{\text{k}}$ and $\vec{\text{c}}=\hat{\text{i}}-2\hat{\text{j}}-3\hat{\text{k}},$ compute $\big(\vec{\text{a}}\times\vec{\text{b}}\big)\times\vec{\text{c}}$ and $\vec{\text{a}}\times\big(\vec{\text{b}}\times\vec{\text{c}}\big)$ and verify that these are not equal.
→Show that $\begin{vmatrix}\text{x}-3&\text{x}-4&\text{x}-\alpha\\\text{x}-2&\text{x}-3&\text{x}-\beta\\\text{x}-1&\text{x}-2&\text{x}-\gamma\end{vmatrix}=0,$ where $\alpha,\beta,\gamma$ are in A.P.
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