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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Formula used to measure circumference of circle is
Answer: A
\[ C = 2\pi r \]
A circumference is a length, so it must be proportional to \( r \) rather than \( r^{2} \) - which rules out the area formula \( \pi r^{2} \) on dimensional grounds alone.
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A circle is inscribed in a square. If the difference between area of the square and circle is 262.5 cm², then find the area of the rectangle whose perimeter is same as that of circle while length of rectangle is 20% more than the breadth of rectangle (in cm²).
Answer: D
Let the side of square be 2a.
So, the diameter of circle = 2a
Radius of circle = a
WKT, Area of square = 4a
Area of circle = π r²
Perimeter of circle = 2π r
Area of rectangle = l x b
Perimeter of rectangle = 2(l + b)
Given that difference between area of the square and circle is 262.5 cm².
Area of square - Area of circle = 262.5
4a² - (22/7)a² = 262.5
28a² - 22a² = 1837.5
6a² = 1837.5
a² = 306.25
a = 17.5 cm
Perimeter of circle = 2πa = 2 x (22/7) x 17.5 = 110 cm
Also given length of rectangle is 20% more than the breadth of rectangle.
Let the breadth of rectangle be X.
So, length of rectangle = (120/100)X = 1.2X
As per the question,
Perimeter of rectangle = Perimeter of circle
2(1.2X + X) = 110
2.4X + 2X = 110
4.4X = 110
X = 25 cm
Breadth of rectangle = 25 cm
Length of rectangle = 1.2(25) = 30 cm
Area of rectangle = 25 x 30 = 750 cm².
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What is the curved surface area of a cone whose radius is 7 cm and slant height 10 cm ? (Take \(\pi = 22/7\))
Answer: A
Correct answer: 220 sq. cm
\[\text{Curved surface area} = \pi r l = \frac{22}{7} \times 7 \times 10 = 220 \text{ sq. cm}\]
Adding the base, \(\pi r^2 = 154\) sq. cm, would give a total surface area of 374 sq. cm, so read carefully which is asked.
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