Line segment which joins distinct points on circumference is called
Answer: B
A chord is a line segment whose two endpoints both lie on the circle.
A radius joins the centre to a point on the circle, so only one of its ends is on the circumference. The longest chord of a circle is its diameter, which passes through the centre.
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A conical cavity is drilled in a circular cylinder of 15 cm height and 16 cm base diameter. The height and the base diameter of the cone are the same as those of the cylinder. Determine the total surface area of the remaining solid.
Answer: A
The cone is drilled out of the cylinder, so three surfaces remain exposed: the cylinder's curved side, its base, and the cone's curved inner surface. The cylinder's top face is gone, replaced by the cavity's opening.
With \( r = 8 \) cm and \( h = 15 \) cm, the cone's slant height is
\[ l = \sqrt{r^{2} + h^{2}} = \sqrt{64 + 225} = \sqrt{289} = 17 \text{ cm} \]
\[ \text{TSA} = 2\pi r h + \pi r^{2} + \pi r l \]
\[ = 2\pi(8)(15) + \pi(64) + \pi(8)(17) \]
\[ = 240\pi + 64\pi + 136\pi = 440\pi \text{ cm}^{2} \]
The cavity's wall is exposed and must be counted, while the flat top is not - the drilling removes it entirely.
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The volume of a wall, 5 times as high as it is broad and 8 times as long as it is high, is 12.8 cu. meters. Find the breadth of the wall.
Answer: D
Let the breadth be \( b \). Then the height is \( 5b \) and the length \( 8 \times 5b = 40b \).
\[ V = b \times 5b \times 40b = 200b^{3} = 12.8 \]
\[ b^{3} = 0.064 \quad\Longrightarrow\quad b = 0.4 \text{ m} = 40 \text{ cm} \]
"Eight times as long as it is high" refers to the height \( 5b \), not the breadth - reading it as \( 8b \) would give a different wall.
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A conical vessel, whose internal radius is 12 cm and height 50 cm, is full of some liquid. The contents of this vessel are emptied into a cylindrical vessel with an internal radius of 10 cm. Find the height to which the liquid rises in the cylindrical vessel.
Answer: C
Pouring conserves volume, so equate the cone's with the cylinder's.
\[ V_{\text{cone}} = \frac{1}{3}\pi (12)^{2}(50) = \frac{1}{3}\pi (7200) = 2400\pi \text{ cm}^{3} \]
\[ V_{\text{cyl}} = \pi (10)^{2} h = 100\pi h \]
\[ 100\pi h = 2400\pi \quad\Longrightarrow\quad h = 24 \text{ cm} \]
The \( \pi \) cancels, so no approximation is needed. The one-third factor is what keeps the liquid from filling the wider cylinder to 50 cm.
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The theatre in the town is built in the form of a kite. It's perimeter is 100 m. If one of the sides measure 20 m, then what are the measurements of others sides?
Answer: D
W.K.T, the shape of Kite resembles Quadrilateral.
The adjacent sides of a kite are of equal length.
Given, perimeter of Quadrilateral = 100 m
Length of one of the side = x = 20 m
Let the Length of other side = y
W.K.T, perimeter of Quadrilateral = sum of the length of all the four sides
---> 100 = 20 + 20 + y + y
---> 100 = 40 + 2y
---> 60 = 2y
---> y = 30 m
Hence, the length of sides of the kite (ie., Quadrilateral) = 20 m, 20 m, 30 m, 30 m
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By converting 3.5rad into degrees, answer will be
Answer: A
\[ \pi \text{ radians} = 180^{\circ} \]
\[ 3.5 \text{ rad} = 3.5 \times \frac{180}{\pi} = 3.5 \times 57.2958 \]
\[ = 200.54^{\circ} \]
Since \( \pi \approx 3.14 \) radians is \( 180^{\circ} \), 3.5 radians must be somewhat more than a straight angle - which rules out every option below \( 180^{\circ} \).
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How many spherical bullets can be made out of a lead cylinder 28 cm high and with base radius 6 cm, each bullet being 1.5 cm in diameter?
Answer: D
Melting conserves volume, so divide the cylinder's by one bullet's. The bullet's radius is \( 1.5 \div 2 = 0.75 \) cm.
\[ V_{\text{cyl}} = \pi (6)^{2}(28) = 1008\pi \text{ cm}^{3} \]
\[ V_{\text{bullet}} = \frac{4}{3}\pi (0.75)^{3} = \frac{4}{3}\pi (0.421875) = 0.5625\pi \]
\[ n = \frac{1008}{0.5625} = 1792 \]
The diameter must be halved before cubing - using 1.5 as the radius would cut the count eightfold.
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The area of a field in the shape of a trapezium measures 1440 m². The perpendicular distance between the parallel side is 24m. If the ratio of the parallel sides is 5:3, what is the length of the longer parallel side?
Answer: A
Given, ratio of the parallel sides of a trapezium = 5 : 3
---> Let the parallel sides (ie., 'a' and 'b') be 5x and 3x respectively.
Given, perpendicular distance = height = 24 m
and Area = 1440 m2
Formula: Area of Trapezium = (1/2) * h * (a + b)
---> 1440 = (1/2) * 24 * (5x + 3x)
---> 1440 = 12 * (8x)
---> 120 = 8x
---> 15 = x
Thus, parallel sides, a = 5x = 5 * 15 = 75 m
and b = 3x = 3 * 15 = 45 m
Hence, Longer parallel side is 'a' = 75 m
Smaller parallel side is 'b' = 45 m.
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Find the area of trapezium whose parallel sides are 20 cm and 18 cm long, and the distance between them is 15 cm.
Answer: C
\[ A = \frac{1}{2}(a + b)h = \frac{1}{2}(20 + 18) \times 15 \]
\[ = \frac{1}{2} \times 38 \times 15 = 285 \text{ cm}^{2} \]
The two parallel sides are averaged, then multiplied by the perpendicular distance between them.
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If sector of circle of radius 16 cm has an area of 140 cm² then angle subtended at center of circle is
Answer: A
\[ A = \frac{\theta}{360} \times \pi r^{2} \]
\[ 140 = \frac{\theta}{360} \times \pi \times 256 \]
\[ \theta = \frac{140 \times 360}{256\pi} = \frac{50400}{804.25} \]
\[ \theta = 62.66^{\circ} \]
The whole circle covers 804.2 cm², and 140 is a little over a sixth of it - consistent with an angle a little over \( 60^{\circ} \).
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