The length of a rectangular plot is thrice its breadth. If the area of the rectangular plot is 867 sq m, then what is the breadth of the rectangular plot?
Answer: B
With \( l = 3b \):
\[ 3b \times b = 3b^{2} = 867 \quad\Longrightarrow\quad b^{2} = 289 \]
\[ b = 17 \text{ m} \]
The length is then 51 m, which is offered as a distractor for anyone who solves for the wrong side.
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Find the surface area of a cuboid 16 m long, 14 m broad and 7 m high.
Answer: C
A cuboid has three pairs of identical faces.
\[ \text{TSA} = 2(lb + bh + hl) \]
\[ = 2(16 \times 14 + 14 \times 7 + 7 \times 16) \]
\[ = 2(224 + 98 + 112) = 2 \times 434 = 868 \text{ m}^{2} \]
The volume, a different quantity, would be \( 16 \times 14 \times 7 = 1568 \) m³.
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The length, breadth and height of a room is 5m, 4m and 3m, respectively. Find the length of the largest bamboo that can be kept inside the room
Answer: D
The longest bamboo lies along the room's space diagonal.
\[ d = \sqrt{l^{2} + b^{2} + h^{2}} = \sqrt{25 + 16 + 9} = \sqrt{50} \]
\[ = 5\sqrt{2} \approx 7.07 \text{ m} \]
The 7 m option is close but not exact; the floor diagonal alone would be \( \sqrt{41} = 6.4 \) m.
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By converting 3π⁄2 from radians to degrees, answer will be
Answer: C
Use \( \pi \text{ radians} = 180^{\circ} \).
\[ \frac{3\pi}{2} = \frac{3}{2} \times 180^{\circ} = 270^{\circ} \]
That is three quarters of a full turn, which is a useful check: \( 2\pi \) radians is \( 360^{\circ} \), so \( \tfrac{3\pi}{2} \) is three quarters of it.
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New unit to express magnitude of angle is called
Answer: B
The radian is the unit of angle defined by the circle itself: the angle subtended at the centre by an arc equal in length to the radius.
\[ 2\pi \text{ radians} = 360^{\circ}, \qquad 1 \text{ radian} \approx 57.3^{\circ} \]
Unlike the degree, it is a pure ratio of two lengths, which is why formulas such as \( s = r\theta \) hold only in radians.
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What will be the cost of building a fence around a square plot with area equal to 289 sq ft, if the price per foot of building the fence is Rs. 58?
Answer: A
Find the side, then the perimeter, then price it.
\[ \text{side} = \sqrt{289} = 17 \text{ ft} \]
\[ \text{perimeter} = 4 \times 17 = 68 \text{ ft} \]
\[ \text{cost} = 68 \times 58 = \text{Rs. } 3944 \]
The fence follows the boundary, so it is priced by perimeter, not area - the data are complete, so "cannot be determined" is a trap.
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Sector at center of circle having an angle of 90° is called
Answer: A
A sector of \( 90^{\circ} \) is one quarter of the circle, and is called a quadrant.
Its area is \( \tfrac{1}{4}\pi r^{2} \) and its arc \( \tfrac{1}{2}\pi r \). Two perpendicular diameters divide a circle into four such quadrants.
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Quarter of circle is
Answer: C
Two perpendicular radii cut a circle into four equal parts, and each is a quadrant - a sector of \( 90^{\circ} \).
A semicircle is half the circle, a chord is a line segment, and a minor segment is the smaller region cut off by a chord rather than by radii.
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The ratio of the length and the breadth of a rectangle is 4 : 3 and the area of the rectangle is 6912 sq cm. Find the ratio of the breadth and the area of the rectangle?
Answer: A
Let the length and the breadth of the rectangle be 4x cm and 3x respectively.
(4x)(3x) = 6912
12x2 = 6912
x2 = 576 = 4 * 144 = 22 * 122 (x > 0)
=> x = 2 * 12 = 24
Ratio of the breadth and the areas = 3x : 12x2 = 1 : 4x = 1: 96.
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If circumference of circle is 82π then value of 'r' is
Answer: A
\[ 2\pi r = 82\pi \quad\Longrightarrow\quad 2r = 82 \]
\[ r = 41 \text{ cm} \]
The 82 cm on offer is the diameter, one step short of the answer.
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