Evaluate \[\int{\frac{\cos \,x}{(1-sinx)(2-\sin x)}dx.}\] |
OR |
Prove that \[\int_{0}^{\pi }{\frac{x}{(1+sin\,x)}}\,dx=\pi .\] |
Find the coordinates of the point, where the line through \[(3,\,\,-\,4,\,\,-5)\] and \[(2,\,\,-\,3,\,\,1)\] crosses the plane \[2x+y+z=7.\] |
OR |
Find the foot of perpendicular from the point |
(2, 3, 4) to the line \[\frac{4-x}{2}=\frac{y}{6}=\frac{1-z}{3}.\] Also, find the length of the perpendicular segment. |
A class has 15 students whose ages are 14, 17, 15, 14, 21, 17, 19, 20, 16, 18, 20, 17, 16, 19 and 20 yr. One student is selected in such a manner that each has the same chance of being of chosen and the age X of the selected student is recorded. What is the probability distribution of the random variable X? |
Find mean, variance and standard deviation of X. |
OR |
3 bad eggs are mixed with 7 good eggs 3 eggs are taken at random from the 10 eggs. Find the probability distribution of number of bad eggs drawn. Also, find the mean and variance of the probability distribution. |
Show that the relation R in the S at \[A=\{x:x\in W,0\le x\le 12\}\] given by \[R=\{(a,\,\,b):|a-b|\] is multiple of 4} equivalence relation. Also, find the set of all elements related to 2. |
OR |
Show that the function |
\[f:R\to \{x\in R:-1<x<1\}\] defined by |
\[f(x)=\frac{x}{1+|x|},\] \[x\in R\] is one-one and onto function. |
Find the vector equation of the line passing through (1, 2, 3) and parallel to the planes \[\vec{r}\cdot (\hat{i}-\hat{j}+2\hat{k})=5\] and \[\vec{r}\cdot (3\hat{i}+\hat{j}+\hat{k})=6.\] |
OR |
Find the distance of the point \[(1,\,\,-\,2,\,\,3)\] from the plane \[x-y+z=5\] measured parallel to the line |
\[\frac{x}{2}=\frac{y}{3}=\frac{z}{-\,6}.\] |
The rate of increase of bacteria in c culture is proportional to the number of bacteria present. If the original number of bacteria doubles in two hours, in how many hours will it be five times? |
OR |
At any point (x, y) of a curve, the slope of the tangent is twice the slope of the line segment joining the point of contact to the point \[(-\,4,\,\,-\,3).\] Find the equation o: the curve given that it passes through \[(-\,2,\,\,1).\] |
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