What is the surface area of a sphere of radius 7 cm ? (Take \(\pi = 22/7\))
Answer: C
Correct answer: 616 sq. cm
\[\text{Surface area} = 4\pi r^2 = 4 \times \frac{22}{7} \times 49 = 4 \times 154 = 616 \text{ sq. cm}\]
Notice that this is exactly four times the area of a great circle of the sphere, which is a useful check.
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A special type of cylindrical vessel with radius and height of 24.5 cm and 5 cm respectively is used to hold Cognac. The vessel is filled upto 80% of its capacity and then total Cognac from cylindrical vessel transferred to 9 cuboidal vessels whose length and breadth is 7 cm & 8 cm respectively. Find the height of each cuboidal vessel?
Answer: D
Cylinder: Radius - 24.5 cm; Height - 5 cm
Cuboid: Length - 7 cm; Breadth - 8 cm
WKT, Volume of cylinder = πr²h
Volume of cubiod = l x b x h
Volume of 80% of Cognac in cylindrical vessel = Volume of 9 cuboidal vessels
(22/7) x 24.5 x 24.5 x 5 x 0.8 = 9 x 7 x 8 x h
h = [(22/7) x 24.5 x 24.5 x 5 x 0.8 ]/[9 x 7 x 8]
h = 7546/504
h = 14.9 cm.
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If circumference of circle is 64π then area of circle (in terms of π) is
Answer: B
Find the radius from the circumference, then use it for the area.
\[ 2\pi r = 64\pi \quad\Longrightarrow\quad r = 32 \text{ cm} \]
\[ A = \pi r^{2} = \pi (32)^{2} = 1024\pi \text{ cm}^{2} \]
The \( \pi \) cancels in the first step and returns in the second, so no numerical value for it is ever needed.
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Sector at center of circle having an angle of 180° is called
Answer: A
A sector whose central angle is \( 180^{\circ} \) covers half the circle, bounded by a diameter and the arc - a semicircle.
Its area is \( \tfrac{1}{2}\pi r^{2} \) and its perimeter \( \pi r + 2r \), the arc plus the diameter.
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The area of a square is equal to five times the area of a rectangle of dimensions 125 cm * 64 cm. What is the perimeter of the square?
Answer: B
\[ \text{rectangle} = 125 \times 64 = 8000 \text{ cm}^{2} \]
\[ \text{square} = 5 \times 8000 = 40{,}000 \text{ cm}^{2} \]
\[ \text{side} = \sqrt{40000} = 200 \text{ cm} \]
\[ \text{perimeter} = 4 \times 200 = 800 \text{ cm} \]
The question asks for the perimeter, so the square root must be multiplied by four - stopping at the side gives 200.
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By converting angle from degrees to radians, 40° is equal to
Answer: B
\[ \pi \text{ radians} = 180^{\circ} \]
\[ 40^{\circ} = 40 \times \frac{\pi}{180} = \frac{2\pi}{9} \]
\[ = 0.698 \text{ rad} \]
A radian is about \( 57.3^{\circ} \), so \( 40^{\circ} \) must come to a little under 1 radian - a quick check on the size of the answer.
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Two identical cubes each of total surface area of 6 cm² are joined end to end. Which of the following is the total surface area of the cuboid so formed?
Answer: C
Each cube has total surface area 6 cm², so a single face is 1 cm² and the edge is 1 cm.
Joined end to end they form a cuboid of \( 2 \times 1 \times 1 \):
\[ \text{TSA} = 2(lb + bh + hl) = 2(2 + 1 + 2) = 10 \text{ cm}^{2} \]
The two faces that meet are sealed inside and no longer count. Adding the cubes' areas would give 12 cm², which is exactly 2 cm² too many - the pair of hidden faces.
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The perimeter of a triangle is 28 cm and the inradius of the triangle is 2.5 cm. What is the area of the triangle?
Answer: D
The inradius relates the area to the semi-perimeter:
\[ A = r \times s, \qquad s = \frac{28}{2} = 14 \]
\[ A = 2.5 \times 14 = 35 \text{ cm}^{2} \]
35 cm² is not among the choices, so the answer is none of these. The 70 cm² on offer comes from using the full perimeter instead of the semi-perimeter.
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Two circular cylinders of equal volume have their heights in the ratio 1: 2. Ratio of their radii is
Answer: B
Equal volumes with heights in the ratio \( 1 : 2 \):
\[ \pi r_{1}^{2} h = \pi r_{2}^{2} (2h) \]
\[ r_{1}^{2} = 2 r_{2}^{2} \quad\Longrightarrow\quad \frac{r_{1}}{r_{2}} = \sqrt{2} \]
So the ratio is \( \sqrt{2} : 1 \). The shorter cylinder must be wider, and because volume depends on \( r^{2} \), halving the height needs only a \( \sqrt{2} \) increase in radius rather than a doubling.
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A rectangular field measures 60 m by 40 m. What is the cost of fencing it at Rs. 25 per metre ?
Answer: B
Correct answer: Rs. 5000
Fencing follows the perimeter, not the area.
\[\text{Perimeter} = 2(60 + 40) = 200 \text{ m}\]
\[\text{Cost} = 200 \times 25 = \text{Rs. } 5000\]
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