Board Previous Year Questions (Board PYQs)

Board Previous Year Questions

PYQ 1 (1 mark): The total surface area of a hemisphere of radius $r$ is: (A) $2\pi r^2$ (B) $3\pi r^2$ (C) $4\pi r^2$ (D) $\tfrac23\pi r^3$. [CBSE] Solution: $3\pi r^2$. Answer: (B).

PYQ 2 (1 mark): A cone and a cylinder have the same base and height. The ratio of their volumes is: (A) 1:1 (B) 1:2 (C) 1:3 (D) 3:1. [CBSE] Solution: Cone is one-third. Answer: (C).

PYQ 3 (2 marks): Two cubes each of volume 64 cm$^3$ are joined end to end. Find the surface area of the resulting cuboid. [CBSE] Solution: edge 4; cuboid $4\times4\times8$; TSA $=160$ cm$^2$. Answer: 160 cm$^2$.

PYQ 4 (2 marks): A medicine capsule is a cylinder with two hemispheres at its ends. Its length is 14 mm and diameter 5 mm. Find its surface area. $(\pi=\tfrac{22}{7})$ [CBSE] Solution: $r=2.5$, $h=9$; $2\pi r(h+2r)=2\times\tfrac{22}{7}\times2.5\times14=220$ mm$^2$. Answer: 220 mm$^2$.

PYQ 5 (3 marks): A toy is a cone of radius 3.5 cm mounted on a hemisphere of the same radius; total height 15.5 cm. Find the total surface area. $(\pi=\tfrac{22}{7})$ [CBSE] Solution: cone $h=12$, $l=12.5$; TSA $=\pi r(l+2r)=11\times19.5=214.5$ cm$^2$. Answer: 214.5 cm$^2$.

PYQ 6 (3 marks): From a solid cylinder ($h=2.4$ cm, diameter 1.4 cm) a cone of the same height and diameter is hollowed out. Find the TSA of the remaining solid. $(\pi=\tfrac{22}{7})$ [CBSE] Solution: $r=0.7$, $l=2.5$; TSA $=\pi r(2h+l+r)=2.2\times8=17.6$ cm$^2$. Answer: ~17.6 cm$^2$.

PYQ 7 (3 marks): A solid is a cone standing on a hemisphere, both of radius 1 cm and the cone's height equal to its radius. Find the volume of the solid in terms of $\pi$. [CBSE] Solution: $\tfrac13\pi(1)(1)+\tfrac23\pi(1)=\pi$ cm$^3$. Answer: $\pi$ cm$^3$.

PYQ 8 (3 marks): A metallic sphere of radius 4.2 cm is melted and recast into a cylinder of radius 6 cm. Find the height of the cylinder. $(\pi\text{ cancels})$ [CBSE] Solution: $\tfrac43\pi(4.2)^3=\pi(36)h\Rightarrow h=\dfrac{4\times74.088}{3\times36}=2.744$ cm. Answer: 2.744 cm.

PYQ 9 (3 marks): A sphere of diameter 6 cm is dropped into a cylindrical vessel of diameter 12 cm containing water. Find the rise in water level. [CBSE] Solution: sphere $=36\pi$; $\pi(36)h=36\pi\Rightarrow h=1$ cm. Answer: 1 cm.

PYQ 10 (3 marks): How many silver coins of diameter 1.75 cm and thickness 2 mm must be melted to form a cuboid $5.5\times10\times3.5$ cm? $(\pi=\tfrac{22}{7})$ [CBSE] Solution: cuboid $=192.5$ cm$^3$; coin $=\pi r^2 t=\tfrac{22}{7}\times(0.875)^2\times0.2=0.4813$ cm$^3$; number $=192.5/0.4813\approx400$. Answer: 400.

PYQ 11 (3 marks): A bucket is a frustum with radii 20 cm and 12 cm and height 15 cm. Find its capacity. $(\pi=3.14)$ [CBSE] Solution: $V=\tfrac13\times3.14\times15\times(400+144+240)=\tfrac13\times3.14\times15\times784=12308.8$ cm$^3\approx12.31$ L. Answer: ~12.31 L.

PYQ 12 (3 marks): The slant height of a frustum is 4 cm and the perimeters of its circular ends are 18 cm and 6 cm. Find its curved surface area. [CBSE] Solution: $2\pi R=18$, $2\pi r=6$; CSA $=\pi l(R+r)=\tfrac{l}{2}(2\pi R+2\pi r)=\tfrac{4}{2}(18+6)=48$ cm$^2$. Answer: 48 cm$^2$.

PYQ 13 (2 marks): A cubical block of side 7 cm is surmounted by the largest hemisphere. Find its greatest diameter and the surface area. $(\pi=\tfrac{22}{7})$ [CBSE] Solution: diameter $=7$ cm; SA $=6a^2+\pi r^2=294+38.5=332.5$ cm$^2$. Answer: 7 cm; 332.5 cm$^2$.

PYQ 14 (3 marks): Water flows through a cylindrical pipe of internal diameter 2 cm at 6 m/s into a cylindrical tank of radius 60 cm. Find the rise in water level in 30 minutes. $(\pi\text{ cancels})$ [CBSE] Solution: in 30 min water length $=6\times60\times30=10800$ m $=1080000$ cm; pipe volume $=\pi(1)^2\times1080000$; tank $\pi(60)^2 h=\pi\times1080000\Rightarrow h=\dfrac{1080000}{3600}=300$ cm $=3$ m. Answer: 3 m.

PYQ 15 (3 marks): A wooden article is a cylinder ($h=10$ cm, radius 3.5 cm) with a hemisphere scooped from each end. Find the TSA. $(\pi=\tfrac{22}{7})$ [CBSE] Solution: $2\pi r(h+2r)=2\times\tfrac{22}{7}\times3.5\times17=374$ cm$^2$. Answer: 374 cm$^2$.

PYQ 16 (1 mark): The volume of the largest sphere that can be carved from a cube of edge $a$ is: (A) $\tfrac16\pi a^3$ (B) $\tfrac43\pi a^3$ (C) $\pi a^3$ (D) $\tfrac13\pi a^3$. [CBSE] Solution: $r=\tfrac a2$; $\tfrac43\pi(\tfrac a2)^3=\tfrac16\pi a^3$. Answer: (A).

PYQ 17 (2 marks): The radii of the ends of a frustum 45 cm high are 28 cm and 7 cm. Find its slant height. [CBSE] Solution: $l=\sqrt{45^2+(28-7)^2}=\sqrt{2025+441}=\sqrt{2466}\approx49.66$ cm. Answer: ~49.66 cm.

PYQ 18 (3 marks): 504 cones of radius 3.5 cm and height 3 cm are melted to form a sphere. Find its radius. $(\pi\text{ cancels})$ [CBSE] Solution: $504\times\tfrac13\pi(12.25)(3)=\tfrac43\pi R^3\Rightarrow6174\pi=\tfrac43\pi R^3\Rightarrow R^3=4630.5\Rightarrow R\approx16.67$ cm. Answer: ~16.67 cm.

PYQ 19 (3 marks): A hemispherical bowl of internal radius 9 cm is full of liquid, poured into cylindrical bottles of radius 1.5 cm and height 4 cm. How many bottles are needed? [CBSE] Solution: bowl $=\tfrac23\pi(729)=486\pi$; bottle $=\pi(2.25)(4)=9\pi$; number $=54$. Answer: 54.

PYQ 20 (4 marks): A solid is a right circular cone (height 120 cm, radius 60 cm) on a hemisphere (radius 60 cm), placed in a cylinder (radius 60 cm, height 180 cm) full of water so it touches the bottom. Find the volume of water left. $(\pi=3.14)$ [CBSE] Solution: cylinder $=\pi(3600)(180)=648000\pi$; solid $=\tfrac13\pi(3600)(120)+\tfrac23\pi(216000)=144000\pi+144000\pi=288000\pi$; water left $=(648000-288000)\pi=360000\pi=1130400$ cm$^3$. Answer: 1130400 cm$^3$ (~1130.4 L).