Considering radian, value of cos 2.3π is
Answer: B
Convert the radian measure to degrees, using \( \pi \text{ radians} = 180^{\circ} \).
\[ 2.3\pi = 2.3 \times 180^{\circ} = 414^{\circ} \]
Subtracting a full turn:
\[ 414^{\circ} - 360^{\circ} = 54^{\circ} \]
\[ \cos 54^{\circ} = 0.587 \]
Cosine never exceeds 1, so the options above 1 can be discarded before any calculation.
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Formula used to measure area of circle is
Answer: B
\[ A = \pi r^{2} \]
An area must vary as the square of a length, which alone rules out \( \pi r^{2} + 2r \) as dimensionally inconsistent. The other two are real formulas from elsewhere: \( 4\pi r^{2} \) is a sphere's surface area and \( 2\pi r^{2} \) a hemisphere's curved surface.
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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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If arc of circle of radius 30 cm has a length of 19 cm then angle subtended at center of circle is
Answer: D
Arc length and central angle are related by \( s = r\theta \) with \( \theta \) in radians.
\[ \theta = \frac{s}{r} = \frac{19}{30} = 0.6333 \text{ rad} \]
Converting to degrees:
\[ 0.6333 \times \frac{180}{\pi} = 36.28^{\circ} \]
The formula \( s = r\theta \) works only in radians; in degrees it needs the extra factor \( \tfrac{\theta}{360} \times 2\pi r \).
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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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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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In formula 2πr, 'r' is considered as
Answer: D
In \( C = 2\pi r \), the letter \( r \) is the radius - the distance from the centre to any point on the circle.
The expression \( 2\pi r \) is itself the circumference, so \( r \) cannot be that; area and perimeter are likewise the results of formulas, not the input to this one.
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Considering radian, value of tan 1.1 is
Answer: A
The angle is in radians, so convert it to degrees first.
\[ 1.1 \text{ rad} = 1.1 \times \frac{180}{\pi} = 63.03^{\circ} \]
\[ \tan 63.03^{\circ} = 1.964 \]
Since \( 63^{\circ} \) is above \( 45^{\circ} \), the tangent must exceed 1 - which eliminates every other option at once.
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If radius is 18 cm and angle at center of circle is 120° then area of sector is
Answer: B
\[ A = \frac{\theta}{360} \times \pi r^{2} = \frac{120}{360} \times \pi \times 18^{2} \]
\[ = \frac{1}{3} \times 324\pi = 108\pi \]
\[ = 108 \times 3.1416 = 339.3 \text{ cm}^{2} \]
\( 120^{\circ} \) is a third of a full turn, so the sector is a third of the circle's 1017.9 cm².
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Largest part of circle divided by chord is called
Answer: D
A chord divides a circle into two regions called segments. The larger, on the side of the centre, is the major segment; the smaller is the minor segment.
A quadrant is bounded by two radii, not a chord, so it is not the answer here.
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