Class: 12 Subject: Math Type: Long

Question 1: Find the maximum area of an isosceles triangle inscribed in the ellipse \( \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 \) with its vertex at one end of the major axis....

Class: 12 Subject: Math Type: Long

Question 2: A tank with a rectangular base and rectangular sides, open at the top is to be constructed so that its depth is \( 2\,\text{m} \) and volume is \( 8\,\text{m}^3 \). If the building of tank costs Rs 70 per square meter for the base and Rs 45 per squar...

Class: 12 Subject: Math Type: Long

Question 3: The sum of the perimeter of a circle and a square is \( k \), where \( k \) is some constant. Prove that the sum of their areas is least when the side of the square is double the radius of the circle....

Class: 12 Subject: Math Type: Long

Question 4: A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is \( 10\,\text{m} \). Find the dimensions of the window to admit maximum light through the whole opening....

Class: 12 Subject: Math Type: Long

Question 5: Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius \( r \) is \( \dfrac{4r}{3} \)....

Class: 12 Subject: Math Type: Long

Question 6: Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius \( R \) is \( \dfrac{2R}{\sqrt{3}} \). Also, find the maximum volume....

Class: 12 Subject: Math Type: Long

Question 7: Show that the height of the cylinder of greatest volume which can be inscribed in a right circular cone of height \( h \) and semi-vertical angle \( a \) is one-third that of the cone and the greatest volume of the cylinder is \( \dfrac{4}{27} \pi h^...