Evaluate \[\int_{0}^{\pi /2}{\log (\sin \,x)\,dx.}\] |
OR |
Evaluate \[\int{\frac{{{\sin }^{6}}x+{{\cos }^{6}}x}{{{\sin }^{2}}{{\cos }^{2}}x}}\,dx.\] |
For 6 trials of an experiment, let X be a binomial variate which satisfies the relation 9P(X = 4) = P(X = 2). Find the probability of success. |
OR |
A bag A contains 4 black and 6 red balls and bag B contains 7 black and 3 balls. A die is thrown. If 1 or 2 appears on it, then bag A is chosen, otherwise bag B. If two balls are drawn at random (without replacement) from the selected bag, find the probability of one of them being red and another black. |
Show that the differential equation \[\left[ x\,{{\sin }^{2}}\left( \frac{y}{x} \right)-y \right]\,dx+x\,dy=0\] |
Is homogeneous. Find the particular solution of this differential equation, given that \[y=\frac{\pi }{4},\] when x = 1. |
OR |
Find the solution of differential equation |
\[{{x}^{2}}dy+y(x+y)dx=0,\] if x = 1 and y = 1. |
Consider the function \[f:{{R}^{+}}\to [4,\,\,\infty )\] defined by \[f(x)={{x}^{2}}+4,\] where \[{{R}^{+}}\] is the set of all non-negative real numbers. Show that f is invertible. Also, find the inverse of f. |
OR |
Show that the relation S in the set |
\[A=\{x\in Z:0\,\,\le x\le 12\}\] given by |
\[S=\{(a,\,\,b):a,\,\,b\in Z,\,\,|a-b|\]is divisible by 4} is an equivalence relation. Find the set of all elements related to 4. |
Find the equation of plane determined |
by points \[A(3,\,\,-\,1,\,\,2),\] B(5, 2, 4), \[C(-\,1,\,\,-\,1,\,\,6)\] and hence find the distance between plane and point P(6, 5, 9). |
OR |
Show that the lines |
\[\vec{r}=3\hat{i}+2\hat{j}-4\hat{k}+\lambda (\hat{i}+2\hat{j}+2\hat{k})\] and |
\[\vec{r}=5\hat{i}-2\hat{j}+\mu (3\hat{i}+2\hat{j}+6\hat{k})\] are intersecting. |
Hence, find their point of intersection. |
If then show that A satisfies the following equation. |
\[{{A}^{3}}-4{{A}^{2}}+11I-3A=O\] |
OR |
If \[A+B+C=\pi ,\]show that |
\[=-\sin (A\,-B)sin(B\,-C)sin(C\,-A).\] |
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