Find the equation of the tangent line to the curve \(y=x^2-2x+7\) which is
(i) parallel to the line \(2x-y+9=0\)
(ii) perpendicular to the line \(5y-15x=13\)

6

(a)

Find the intervals in which f(x) = sin3x - cos3x,0 < x < π is strictly increasing or strictly decreasing.

6

(a)

A tank with rectangular base and rectangular sides open at the top is to be constructed so that its depth is 3 mand volume is 75 cm³. If building of tank costs Rs. 100 per square metre for the base and Rs. 50 per square metre for the sides, find the cost of least expensive tank.

6

(a)

A metal box with a square base and vertical sides is to contain 1024 cm³. The material for the top and bottom costs Rs. 5 per cm² and the material for the sides cost Rs. 2.50/cm². Find the least cost of the box.

6

(a)

Show that semi-vertical angle of a cone of maximum volume and given slant height is \(\cos^{-1}(\frac{1}{\sqrt3})\)

6

(a)

Prove that the semi-vertical angle of the right circular cone of given volume and least curved surface area is \(\cot^{-1}\sqrt2\)

6

(a)

Prove that the height of the cylinder of maximum volume, that can be inscribed in a sphere of radius R is \(\frac{2R}{\sqrt3}\) . Also find the maximum volume.

6

(a)

Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.

6

(a)

Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is \(\frac { 4r }{ 3 } \) Also show that the maximum volume of the cone is \(\frac { 8 }{ 27 } \) of the volume of the sphere.

6

(a)

If length of three sides of a trapezium other than base are equal to 10 cm, then find the area of the trapezium when it is maximum.

6

(a)