Find the minimum value of n for which |
\[{{\tan }^{-1}}\frac{n}{\pi }>\frac{\pi }{4},\] \[n\in N.\] |
OR |
Show that \[\tan \left( \frac{1}{2}{{\sin }^{-1}}\frac{3}{4} \right)=\frac{4-\sqrt{7}}{3}.\] |
Evaluate \[\int{\frac{1+{{x}^{2}}}{1+{{x}^{4}}}\,dx.}\] |
OR |
Evaluate \[\int{x\cdot {{(\log \,\,x)}^{2}}\,dx.}\] |
Find the image of the point (1, 6, 3) on the line \[\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}.\] |
Also, write the equation of the line joining the given point and its image and find the length of segment joining the given point and its image. |
OR |
Find the foot of the perpendicular from the point (0, 2, 3) on the line \[\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}.\] |
Also, find the length of the perpendicular. |
The sum of three numbers is 6. If we multiply third number by 3 and add second number to it, we get 11. By addinq first and third numbers, we get double of the second number. Represent it algebraically and find the numbers using matrix method. |
OR |
Solve the following system of equations by matrix method, where \[x\ne 0,\] \[y\ne 0\] and \[z\ne 0.\] |
\[\frac{2}{x}-\frac{3}{y}+\frac{3}{z}=10,\] \[\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=10\] |
and \[\frac{3}{x}-\frac{1}{y}+\frac{2}{z}=13\] |
Let \[A=\{x\in R:0\le x\le 1\}.\] If \[f:A\to A\] is defined by |
\[f(x)=\left\{ \begin{matrix} x, & i\text{f}\,\,x\in Q \\ 1-x & \text{if}\,\,x\notin Q \\ \end{matrix} \right.\] |
Then prove that \[fof(x)=x\] for all \[x\in A.\] |
OR |
Let \[A=N\times N\] and * be the binary operation on A defined by (a, b) * (c, d) = \[(a+c,\,\,b+d)\] |
Show that * is commutative and associative. |
Find the identity element for * on A, if any. |
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