Evaluate \[\int_{3}^{4}{\frac{\sqrt{x}}{\sqrt{x}+\sqrt{7-x}}dx.}\] |
OR |
Evaluate \[\int_{0}^{1}{\frac{\log (1+x)}{1+{{x}^{2}}}\,dx.}\] |
A company has two plants for manufacturing scooters. Plant I manufactures 70% of the scooters and plant II manufactures 30% of the scooters. At plant I, 30% of the scooters are maintaining pollution norms and plant II, 90% of the scooters are maintaining pollution norms. A scooter chosen at random and is found to be fit oh pollution norms. Find the probability that it has come from plant II. |
OR |
Events A and B are such that \[P(A)=\frac{1}{2},\] \[P(B)=\frac{7}{12}\] and P (not A or not B) \[=\frac{1}{4}.\] State whether A and B are independent. |
Find the coordinates of point on line \[\frac{x-1}{2}=\frac{y+2}{3}=\frac{z-3}{6},\] which are at a distance of 3 units from the point \[(1,\,\,-2,\,\,3).\] |
OR |
Show that the lines \[\vec{r}=(\hat{i}+\hat{j}-\hat{k})+\lambda (3\hat{i}-\hat{j})\] and \[\vec{r}-=(4\hat{i}-\hat{k})+\mu (2\hat{i}+3\hat{k})\] are coplanar. Also, find the equation of the plane containing them. |
Consider \[f:{{R}^{+}}\to [-\,9,\,\,\infty ]\] given by \[f(x)=5{{x}^{2}}+6x-9.\] Prove that f is invertible with \[{{f}^{-1}}(y)=\left( \frac{\sqrt{54+5y}-3}{5} \right)\] |
(Where, \[{{R}^{+}}\] is the set of all positive real numbers). |
OR |
Let '*' be a binary operation on the set |
{0, 1, 2, 3, 4, 5} defined as |
\[a*b=\left\{ \begin{matrix} a+b, & \text{if}\,\,a+b<6 \\ a+b-6, & \text{if}\,\,a+b\ge 6 \\ \end{matrix} \right.\] |
Show that zero is the identity for this operation and each element a of the set is invertible with \[b-a,\] being the inverse of a. |
Find the equation of the plane passing through the line of intersection of planes |
\[2x+y-z=3,\] \[5x-3y+4z+9=0\] and parallel to the line \[\frac{x-1}{2}=\frac{y-3}{4}=\frac{z-5}{5}.\] |
OR |
Find the distance of the point \[(-\,2,\,\,3,\,\,-4)\] from the line \[\frac{x+2}{3}=\frac{2y+3}{4}=\frac{3z+4}{5}\] measured parallel to the plane \[4x+12y-3z+1=0.\] |
Find the volume of the largest cylinder that can be inscribed in sphere of radius r. |
OR |
A given quantity of metal is to be cast into a half cylinder with a rectangular base and semicircular ends. Show that in order that total surface area is minimum, the ratio of length of cylinder to the diameter of its semicircular ends is \[\pi :(\pi +2).\] |
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