Exam Questions & Step-by-Step Markschemes (5 Problems)
Question 1 • 2024
5 Marks
Show that the volume of the largest cone that can be inscribed in a sphere of radius R is \frac{8}{27} of the volume of the sphere.
Answer: V_{\text{cone}} = \frac{8}{27} V_{\text{sphere}}
Question 2 • 2023
4 Marks
Water is dripping out from a conical funnel at a uniform rate of 4\text{ cm}^3/\text{s} through a tiny hole at the vertex in the bottom. When the slant height of water is 3\text{ cm} , find the rate of decrease of the slant height of water, given that the semi-vertical angle of the funnel is 30^\circ .
Answer: \text{Rate of decrease} = \frac{32\sqrt{3}}{27\pi}\text{ cm/s}
Question 3 • 2024
5 Marks
Show that the height of the right circular cylinder of maximum volume that can be inscribed in a given sphere of radius R is \frac{2R}{\sqrt{3}} . Also find the maximum volume.
Answer: h = \frac{2R}{\sqrt{3}}, \quad V_{\text{max}} = \frac{4\pi R^3}{3\sqrt{3}}
Question 4 • 2024
5 Marks
Show that the semi-vertical angle of a right circular cone of given surface area and maximum volume is \sin^{-1}\left(\frac{1}{3}\right) .
Answer: \alpha = \sin^{-1}(1/3)
Question 5 • 2023
3 Marks
Sand is pouring from a pipe at the rate of 12\text{ cm}^3/\text{s} . The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4\text{ cm} ?
Answer: \frac{dh}{dt} = \frac{1}{48\pi} \text{ cm/s}