Evaluate \[\int{\sqrt{3-4x-4{{x}^{2}}}\,\,dx.}\] |
OR |
Evaluate \[\int{\frac{\sin (x-\alpha )}{\sin (x+\alpha )}\,dx.}\] |
If \[{{x}^{y}}+{{y}^{x}}={{a}^{b}},\] then find \[\frac{dy}{dx}.\] |
OR |
If \[y=\log [x+\sqrt{{{x}^{2}}+{{a}^{2}}}],\] show that \[({{x}^{2}}+{{a}^{2}})\frac{{{d}^{2}}y}{d{{x}^{2}}}+x\frac{dy}{dx}=0.\] |
Find the mean and variance of number of tails when a coin is tossed thrice. |
OR |
12 cards numbered 1 to 12 are placed in a box, mixed up thoroughly and then a card is drawn at random from the box. |
If it is known that the number on the drawn card is more than 3, then find the probability that it is an even number. |
For the curve \[y=4{{x}^{3}}-2{{x}^{5}},\] find all the point on the curve at which the tangent passes through the origin, |
OR |
Show that of all the rectangles with a given perimeter, the square has the largest area. |
Find the image of point (1, 0, 0) on the line \[\frac{x}{1}=\frac{y}{2}=\frac{z}{3}.\] |
OR |
Find the equation of the plane that contains the point \[(1,\,\,-\,1,\,\,2)\] and is perpendicular to both the planes \[2x+3y-2z=5\]and \[x+2y-3z=8.\] |
Hence, find the distance of point \[P(-\,2,\,\,5,\,\,5)\] from the plane obtained above. |
Show that \[\Delta ABC\] is an isosceles triangle, if the determinant |
OR |
If \[A+B+C=\pi ,\] show that |
\[=-\sin (A-B)\sin (B-C)\sin (C-A)\] |
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