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Question 1
ప్రశ్న 1
In the horizontal alignment of a proposed access road, which statement correctly defines a SIMPLE CIRCULAR CURVE?
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Explanation:
A simple circular curve consists of one circular arc having a constant radius. It connects two straight tangent portions of a horizontal route alignment. The curve has a definite centre of curvature throughout its length. The transition between the straight portions and the circular arc occurs at the tangent points. A compound curve contains circular arcs of different radii bending in the same direction. A reverse curve consists of arcs bending in opposite directions, while a transition curve has varying curvature.
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Question 2
ప్రశ్న 2
For a simple circular curve inserted between two straight road tangents, which identification of the principal points is correct?
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Explanation:
The point of intersection, PI, is the theoretical intersection of the two straight tangents. The point of curvature, PC, is where the circular curve begins from the incoming tangent. The point of tangency, PT, is where the circular arc joins the outgoing tangent. The curve itself generally does not pass through PI. The distances from PI to PC and from PI to PT along their respective tangents are equal for a simple circular curve. These points provide the basic geometric control for curve setting out.
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Question 3
ప్రశ్న 3
For a simple circular curve having a direction-change angle Delta between the forward prolongation of the incoming tangent and the outgoing tangent, which relationship is correct?
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Explanation:
The direction-change angle describes the change in route direction between the two tangents. For a simple circular curve, the central angle subtended by the arc between PC and PT has the same magnitude. This central angle is commonly denoted by Delta. It is the angle used in the standard tangent-length, curve-length and long-chord formulas. The smaller angle formed at PI by the two rays pointing from PI toward PC and PT is supplementary to Delta. Distinguishing these angle conventions prevents incorrect substitution in circular-curve calculations.
వివరణ:
Question 4
ప్రశ్న 4
If R is the radius and Delta is the central angle of a simple circular curve, which formula gives the TANGENT LENGTH T from PI to either tangent point?
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Explanation:
The tangent length is measured along either straight tangent from PI to the corresponding tangent point. Joining PI and the circle centre forms a right triangle with either radius drawn to a tangent point. The tangent is perpendicular to the radius at the point of contact. From this geometry, tan(Delta/2) equals T/R. Rearranging gives T = R tan(Delta/2). The formula uses half the central angle rather than the full angle.
వివరణ:
Question 5
ప్రశ్న 5
A simple circular curve has a radius of 300 m and a central angle of 60 degrees. What is its tangent length from PI to PC, correct to three decimal places?
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Explanation:
The tangent length of a simple circular curve is T = R tan(Delta/2). The given radius is 300 m and the central angle is 60 degrees. The half angle is therefore 30 degrees. Substitution gives T = 300 tan(30 degrees). Since tan(30 degrees) is approximately 0.5773503, the tangent length is 173.205 m. The same tangent length applies from PI to PT along the outgoing tangent.
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Question 6
ప్రశ్న 6
The tangent length of a simple circular curve is 200 m, and its central angle is 90 degrees. What is the radius of the curve?
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Explanation:
The tangent-length relation is T = R tan(Delta/2). The tangent length is given as 200 m and the central angle is 90 degrees. Therefore the half angle is 45 degrees. Rearranging the equation gives R = T/tan(45 degrees). Since tan(45 degrees) equals one, R = 200 m. The result also shows that tangent length and radius are numerically equal when the central angle is 90 degrees.
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Question 7
ప్రశ్న 7
A simple circular curve has a radius of 240 m and a central angle of 75 degrees. What is the LENGTH OF THE CIRCULAR ARC between its tangent points, correct to three decimal places?
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Explanation:
The length of a circular curve is the arc distance between PC and PT. When the central angle is expressed in degrees, L = pi R Delta/180. Substituting R = 240 m and Delta = 75 degrees gives L = pi multiplied by 240 multiplied by 75 divided by 180. This simplifies to L = 100 pi m. The resulting curve length is approximately 314.159 m. The arc length must not be confused with the straight long chord joining the two tangent points.
వివరణ:
Question 8
ప్రశ్న 8
A simple circular curve has radius 300 m and measured arc length 157.080 m between its tangent points. Using L = R Delta, with Delta expressed in radians, what is the central angle to the NEAREST WHOLE DEGREE?
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Explanation:
The arc-length formula in radian measure is L = R Delta. Rearranging gives Delta = L/R. Substitution gives Delta = 157.080/300 = 0.523600 radians. Converting to degrees requires multiplication by 180/pi. The resulting central angle is approximately 30.000 degrees. Therefore the central angle to the nearest whole degree is 30 degrees.
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Question 9
ప్రశ్న 9
A simple circular curve has radius 250 m and central angle 60 degrees. What is the length of the LONG CHORD connecting PC and PT?
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Explanation:
The long chord is the straight-line distance between the two tangent points. For a simple circular curve, its length is C = 2R sin(Delta/2). The given central angle is 60 degrees, so the half angle is 30 degrees. Substitution gives C = 2 multiplied by 250 multiplied by sin(30 degrees). Since sin(30 degrees) equals 0.5, the long chord is 250.000 m. The corresponding circular arc is longer than the chord because it follows a curved path.
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Question 10
ప్రశ్న 10
The long chord of a simple circular curve is 300 m, and its central angle is 60 degrees. What is the radius of the curve?
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Explanation:
The long-chord formula is C = 2R sin(Delta/2). Rearranging gives R = C/[2 sin(Delta/2)]. The long chord is 300 m and the central angle is 60 degrees. Therefore R = 300/[2 sin(30 degrees)]. Since sin(30 degrees) equals 0.5, the denominator becomes one. The required radius is consequently 300 m.
వివరణ:
Question 11
ప్రశ్న 11
For a simple circular curve of radius 300 m and central angle 60 degrees, what is the EXTERNAL DISTANCE from PI to the midpoint of the circular arc, correct to three decimal places?
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Explanation:
External distance is measured from the point of intersection toward the midpoint of the circular arc along the curve's symmetry line. Its formula is E = R[sec(Delta/2) - 1]. The radius is 300 m and the half central angle is 30 degrees. Thus E = 300[sec(30 degrees) - 1]. Using sec(30 degrees) approximately equal to 1.1547005 gives E approximately 46.410 m. This is distinct from the tangent length of 173.205 m and the middle ordinate of approximately 40.192 m.
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Question 12
ప్రశ్న 12
A simple circular curve has radius 400 m and central angle 60 degrees. Calculate the MID-ORDINATE measured between the midpoint of the long chord and the midpoint of the arc.
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Explanation:
The middle ordinate is the maximum perpendicular separation between the long chord and the circular arc. For a simple circular curve, M = R[1 - cos(Delta/2)]. The given radius is 400 m and the half central angle is 30 degrees. Substitution gives M = 400[1 - cos(30 degrees)]. Since cos(30 degrees) is approximately 0.8660254, M is approximately 53.590 m. The middle ordinate is measured from the chord midpoint to the curve, not from PI to the curve.
వివరణ:
Question 13
ప్రశ్న 13
For a simple circular curve having a nonzero central angle less than 180 degrees, which statement correctly compares the CIRCULAR ARC LENGTH with the LONG CHORD length between the same tangent points?
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Explanation:
The long chord directly joins PC and PT along a straight line. The circular arc connects the same points along the curved circumference. For a nonzero central angle, the straight chord is shorter than the corresponding arc. The arc length is L = R Delta when Delta is expressed in radians. The long chord is C = 2R sin(Delta/2). The inequality follows because sin(x) is less than x for positive x in the applicable angular range.
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Question 14
ప్రశ్న 14
A simple circular curve has radius 200 m and central angle 90 degrees. Which pair correctly gives its EXTERNAL DISTANCE and MID-ORDINATE, respectively, to three decimal places?
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Explanation:
The external distance is E = R[sec(Delta/2) - 1]. With R = 200 m and Delta/2 = 45 degrees, E = 200[sec(45 degrees) - 1]. This gives E approximately 82.843 m. The mid-ordinate is M = R[1 - cos(Delta/2)]. Therefore M = 200[1 - cos(45 degrees)] = 58.579 m approximately. The external distance exceeds the middle ordinate because these quantities are measured from different reference points along the curve's symmetry line.
వివరణ:
Question 15
ప్రశ్న 15
A simple circular curve has radius 300 m and central angle 60 degrees. What is the difference between its ARC LENGTH and its LONG CHORD length, to three decimal places?
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Explanation:
The circular arc length is L = pi R Delta/180 when Delta is in degrees. Substituting R = 300 m and Delta = 60 degrees gives L = 314.159 m approximately. The long chord is C = 2R sin(Delta/2). Thus C = 2 multiplied by 300 multiplied by sin(30 degrees) = 300.000 m. The required difference is 314.159 - 300.000 = 14.159 m. This difference represents the additional distance travelled along the circular arc compared with the straight chord.
వివరణ:
Question 16
ప్రశ్న 16
For a simple circular curve of central angle Delta satisfying 0 < Delta < 180 degrees, E denotes the external distance and M denotes the middle ordinate. Which relationship is correct?
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Explanation:
The external distance is E = R[sec(Delta/2) - 1]. The middle ordinate is M = R[1 - cos(Delta/2)]. Dividing E by M and simplifying gives E/M = 1/cos(Delta/2). Therefore E = M sec(Delta/2). For the specified angular range, sec(Delta/2) is greater than one. Consequently the external distance is greater than the middle ordinate for any nonzero simple circular curve in this range.
వివరణ:
Question 17
ప్రశ్న 17
The chainage of PI for a proposed simple circular road curve is 12+450.000 m. The radius is 300 m and the central angle is 60 degrees. Assuming the PI chainage is assigned along the incoming tangent alignment, what is the chainage of PC to three decimal places?
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Explanation:
The point of curvature lies on the incoming tangent before PI. Its chainage is obtained by subtracting the tangent length from the assigned PI chainage. The tangent length is T = 300 tan(30 degrees) = 173.205 m approximately. The assigned PI chainage 12+450.000 represents 12450.000 m. Therefore the PC chainage is 12450.000 - 173.205 = 12276.795 m. In chainage notation, this is 12+276.795 m.
వివరణ:
Question 18
ప్రశ్న 18
A simple circular curve begins at PC chainage 8+650.000 m. Its radius is 300 m and central angle is 60 degrees. If chainage increases along the circular arc from PC to PT, what is the chainage of PT?
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Explanation:
The chainage of PT is obtained by adding the circular arc length to the PC chainage. The arc length formula is L = pi R Delta/180. For R = 300 m and Delta = 60 degrees, L is approximately 314.159 m. The starting chainage 8+650.000 represents 8650.000 m. Adding the arc length gives 8650.000 + 314.159 = 8964.159 m. The PT chainage is therefore 8+964.159 m, not a value obtained by adding the long chord.
వివరణ:
Question 19
ప్రశ్న 19
The assigned PI chainage of a simple circular curve is 5+000.000 m, its radius is 200 m, and its central angle is 90 degrees. Chainage increases along the incoming tangent to PC and then along the arc to PT. Which pair correctly gives the PC and PT chainages?
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Explanation:
The tangent length is T = 200 tan(45 degrees) = 200.000 m. Subtracting this from the assigned PI chainage gives PC = 5000.000 - 200.000 = 4800.000 m. The circular arc length is L = pi multiplied by 200 multiplied by 90 divided by 180. This gives L approximately equal to 314.159 m. Adding the arc length to PC gives PT = 4800.000 + 314.159 = 5114.159 m. The PT chainage is not generally equal to PI chainage plus tangent length because chainage follows the curved route.
వివరణ:
Question 20
ప్రశ్న 20
A simple circular curve begins at chainage 10+180.000 m. Its radius is 360 m and central angle is 50 degrees. What is the chainage of the MIDPOINT OF THE CIRCULAR ARC, measured along the curve, correct to three decimal places?
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Explanation:
The midpoint of a circular arc lies at half the total arc length measured from PC. The complete curve length is L = pi R Delta/180. Substitution gives L = pi multiplied by 360 multiplied by 50 divided by 180 = 100 pi m. Thus the curve length is approximately 314.159 m and half its length is 157.080 m. Adding this half-length to PC chainage 10180.000 m gives 10337.080 m. The midpoint chainage is therefore 10+337.080 m.
వివరణ:
Question 21
ప్రశ్న 21
In a simple circular curve, the DEGREE OF CURVE is defined by the 30 m ARC definition: it is the central angle subtended by 30 m of circular arc. If the degree of curve is 6 degrees, what is the radius to three decimal places?
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Explanation:
Under the 30 m arc definition, the degree of curve is the central angle subtended by an arc of exactly 30 m. The arc-length relation is 30 = pi R D/180 when D is expressed in degrees. Rearranging gives R = 30 multiplied by 180 divided by pi D. For D = 6 degrees, R = 5400/(6 pi). This evaluates to approximately 286.479 m. The 30 m chord definition uses a different expression and must not be substituted into this arc-based calculation.
వివరణ:
Question 22
ప్రశ్న 22
For a simple circular curve, the DEGREE OF CURVE is defined by the 30 m CHORD definition: it is the central angle subtended by a straight chord of length 30 m. If the degree of curve is 6 degrees, what is the radius to three decimal places?
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Explanation:
Under the chord definition, the specified 30 m distance is a straight chord rather than an arc. The chord formula is C = 2R sin(D/2). Substituting C = 30 m and D = 6 degrees gives 30 = 2R sin(3 degrees). Therefore R = 15/sin(3 degrees). Evaluating this expression gives R approximately 286.610 m. This is slightly different from the radius obtained using the 30 m arc definition, so the convention must always be stated.
వివరణ:
Question 23
ప్రశ్న 23
For circular curves defined by the SAME fixed 30 m arc convention or the SAME fixed 30 m chord convention, which statement correctly relates radius to degree of curve?
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Explanation:
The degree of curve expresses the central angular change corresponding to a specified standard arc or chord length. For a fixed standard arc length, degree of curve is inversely proportional to radius. Under a fixed chord definition, the inverse relationship is expressed through the chord-angle sine formula. Reducing the radius increases the angular change associated with the standard length. Therefore a smaller-radius curve is sharper and has a greater degree of curve. The chosen arc or chord convention must be kept unchanged when comparing numerical degrees of curvature.
వివరణ:
Question 24
ప్రశ్న 24
Two simple circular curves have the SAME central angle Delta, with 0 < Delta < 180 degrees. Curve B has TWICE the radius of Curve A. How do their tangent lengths, arc lengths, long chords, external distances and mid-ordinates compare?
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Explanation:
For a fixed central angle, tangent length is R tan(Delta/2). Arc length is R Delta when Delta is expressed in radians. Long chord is 2R sin(Delta/2), external distance is R[sec(Delta/2) - 1], and mid-ordinate is R[1 - cos(Delta/2)]. Each expression contains radius as a simple multiplying factor when the angle remains fixed. Therefore doubling the radius doubles each of these five linear geometric elements. The shapes of the curves remain geometrically similar, with Curve B being a two-times enlargement of Curve A.
వివరణ:
Question 25
ప్రశ్న 25
A simple circular curve has radius 350 m, central angle 40 degrees and assigned PI chainage 6+800.000 m along the incoming tangent. Chainage follows the incoming tangent to PC and then the circular arc to PT. Which pair gives the tangent-point chainages, rounded independently from full-precision calculations to three decimal places?
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Explanation:
First calculate the tangent length using T = R tan(Delta/2). With R = 350 m and Delta = 40 degrees, T = 350 tan(20 degrees) = 127.389582 m approximately. The PC chainage is 6800.000 - 127.389582 = 6672.610418 m, or 6+672.610 m. The curve arc length is L = pi multiplied by 350 multiplied by 40 divided by 180 = 244.346095 m approximately. Adding the unrounded arc length to the unrounded PC chainage gives PT = 6916.956513 m. Rounding the final results to three decimals gives PC = 6+672.610 m and PT = 6+916.957 m.
వివరణ:
Answer Key సమాధానాల పట్టిక
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Question 1 ప్రశ్న 1Answer: B. A single circular arc of constant radius connecting two straight tangents. సమాధానం: B.
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Question 2 ప్రశ్న 2Answer: C. PI is the intersection of the extended tangents, PC is the beginning of the circular curve, and PT is its ending tangent point. సమాధానం: C.
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Question 3 ప్రశ్న 3Answer: A. The central angle subtended by the full circular arc equals Delta. సమాధానం: A.
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Question 4 ప్రశ్న 4Answer: D. T = R tan(Delta/2). సమాధానం: D.
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Question 5 ప్రశ్న 5Answer: B. 173.205 m. సమాధానం: B.
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Question 6 ప్రశ్న 6Answer: A. 200 m. సమాధానం: A.
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Question 7 ప్రశ్న 7Answer: C. 314.159 m. సమాధానం: C.
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Question 8 ప్రశ్న 8Answer: D. 30 degrees. సమాధానం: D.
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Question 9 ప్రశ్న 9Answer: A. 250.000 m. సమాధానం: A.
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Question 10 ప్రశ్న 10Answer: C. 300 m. సమాధానం: C.
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Question 11 ప్రశ్న 11Answer: B. 46.410 m. సమాధానం: B.
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Question 12 ప్రశ్న 12Answer: D. 53.590 m. సమాధానం: D.
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Question 13 ప్రశ్న 13Answer: C. The circular arc is longer than the long chord because it follows the curved path between the endpoints. సమాధానం: C.
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Question 14 ప్రశ్న 14Answer: B. External distance 82.843 m; mid-ordinate 58.579 m. సమాధానం: B.
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Question 15 ప్రశ్న 15Answer: A. 14.159 m. సమాధానం: A.
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Question 16 ప్రశ్న 16Answer: D. E = M sec(Delta/2). సమాధానం: D.
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Question 17 ప్రశ్న 17Answer: C. 12+276.795 m. సమాధానం: C.
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Question 18 ప్రశ్న 18Answer: A. 8+964.159 m. సమాధానం: A.
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Question 19 ప్రశ్న 19Answer: B. PC = 4+800.000 m; PT = 5+114.159 m. సమాధానం: B.
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Question 20 ప్రశ్న 20Answer: D. 10+337.080 m. సమాధానం: D.
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Question 21 ప్రశ్న 21Answer: A. 286.479 m. సమాధానం: A.
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Question 22 ప్రశ్న 22Answer: B. 286.610 m. సమాధానం: B.
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Question 23 ప్రశ్న 23Answer: C. A smaller radius represents a sharper curve and corresponds to a larger degree of curve. సమాధానం: C.
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Question 24 ప్రశ్న 24Answer: D. Each of the five linear geometric elements becomes twice its corresponding value for Curve A. సమాధానం: D.
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Question 25 ప్రశ్న 25Answer: C. PC = 6+672.610 m; PT = 6+916.957 m. సమాధానం: C.