Q25 of 45 Page 1

Evaluate:




We know that sin2x + cos2x = 1



We know that sin2x = 2sinxcosx



Divide numerator and denominator by cos22x




We know that 1 + tan2x = sec2x





Substitute tan2x = t


Differentiate with respect to x



⇒ dt = 2sec22xdx


Hence





Resubstitute t



Hence


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Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is Also show that the maximum volume of the cone is of the volume of the sphere.

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Using integration, find the area of the region bounded by the triangle whose vertices are (-1, 2), (1, 5) and (3, 4).

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Find the equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x - y + z = 0. Also find the distance of the plane obtained above, from the origin.

OR


Find the distance of the point (2, 12, 5) from the point of intersection of the line and the plane