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Question 1
ప్రశ్న 1
Which group consists entirely of recognised LINEAR methods used for setting out a simple circular curve in surveying?
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Explanation:
Linear methods establish points on a circular curve primarily through measured distances and offsets. Offsets from tangents use the incoming or outgoing tangent as the reference line. Offsets from the long chord use the straight line joining the two tangent points. Successive bisection of chords locates additional curve points by dividing existing chords and setting out corresponding ordinates. The chord-produced method advances from previously established curve points by extending chords and measuring offsets. Rankine's method and the two-theodolite method are angular methods rather than purely linear setting-out methods.
వివరణ:
Question 2
ప్రశ్న 2
A surveyor intends to set out a simple circular road curve using perpendicular offsets from its incoming tangent. Which field procedure is correct?
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Explanation:
The tangent-offset method requires a correctly established tangent line. Its point of curvature serves as the origin for measuring distances along the tangent. Suitable points are marked on the tangent at the calculated longitudinal distances. Perpendicular offsets are then erected toward the required circular arc. The calculated offset lengths locate the corresponding curve points, which are marked with pegs. The positions should be checked against the curve geometry and the available tangent-point control.
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Question 3
ప్రశ్న 3
For a simple circular curve of radius R, let x be the distance measured along the tangent from PC and y the PERPENDICULAR offset from that tangent to the circular curve. Which is the EXACT geometric relationship for a point on the relevant arc?
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Explanation:
The centre of a simple circular curve is at a perpendicular distance R from the tangent at PC. A point on the curve and its perpendicular projection onto the tangent form a right-triangle relationship with the centre. The circle equation gives x^2 + (R - y)^2 = R^2. Solving for the offset toward the relevant arc gives y = R - sqrt(R^2 - x^2). This is the exact perpendicular-offset formula within the applicable tangent projection range. The expression sqrt(R^2 + x^2) - R instead represents the exact radial offset measured from a point on the tangent toward the circle centre.
వివరణ:
Question 4
ప్రశ్న 4
The exact perpendicular offset from a tangent is y = R - sqrt(R^2 - x^2). Under which condition is the approximate formula y = x^2/(2R) most appropriate?
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Explanation:
The approximate tangent-offset formula is obtained by expanding the exact square-root expression. Its first nonzero term is x^2/(2R). The omitted higher-order terms become relatively small when x/R is small. Therefore the approximation is suitable for short offsets on comparatively flat circular curves. For larger ratios of x to R, the approximation can produce a significant positional error. The required setting-out accuracy should determine whether the exact or approximate formula is used.
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Question 5
ప్రశ్న 5
A simple circular curve has radius R = 200 m and central angle 60 degrees. A point is to be located by a perpendicular offset from the incoming tangent at x = 40 m from PC. Using the EXACT formula y = R - sqrt(R^2 - x^2), what is the offset to three decimal places?
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Explanation:
The exact perpendicular-offset formula is y = R - sqrt(R^2 - x^2). Substituting R = 200 m and x = 40 m gives y = 200 - sqrt(40000 - 1600). The square-root term is sqrt(38400), approximately 195.959179 m. Therefore y = 200 - 195.959179 = 4.040821 m. Rounded to three decimal places, the offset is 4.041 m. The approximate formula would give 4.000 m, which is close but not identical to the exact result.
వివరణ:
Question 6
ప్రశ్న 6
A simple circular curve of radius 250 m is to be set out by perpendicular offsets from the tangent. Using the APPROXIMATE formula y = x^2/(2R), what offset is required at x = 50 m from PC?
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Explanation:
The question explicitly specifies the approximate perpendicular-offset formula. The tangent distance is x = 50 m and the radius is R = 250 m. Substitution gives y = 50^2/(2 multiplied by 250). This simplifies to y = 2500/500 = 5.000 m. The exact circular-curve offset is approximately 5.051 m. The difference illustrates why the formula convention and required precision must be considered during field setting out.
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Question 7
ప్రశ్న 7
For a circular curve of radius 100 m, the perpendicular offset from the tangent is required at x = 60 m from PC. The curve has a central angle of 90 degrees, so the specified point is within its tangent projection range. Compare the EXACT formula with the approximation y = x^2/(2R). Which statement is correct?
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Explanation:
The exact perpendicular offset is 100 - sqrt(10000 - 3600). Since sqrt(6400) = 80, the exact offset equals 20.000 m. The approximate formula gives 60^2/(2 multiplied by 100) = 18.000 m. Therefore the approximation underestimates the exact offset by 2.000 m. This is a relatively large discrepancy because x/R = 0.60 is not small. The example demonstrates that a short-offset approximation should not be used indiscriminately for sharp curves or large tangent distances.
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Question 8
ప్రశ్న 8
During the setting out of a simple circular curve, the surveyor measures a perpendicular offset of 5.000 m at a tangent distance of 50.000 m from PC. Assuming the observations represent the EXACT circular geometry, what is the radius of the curve?
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Explanation:
The exact relationship is x^2 + (R - y)^2 = R^2. Expanding and rearranging gives 2Ry = x^2 + y^2. Therefore R = (x^2 + y^2)/(2y). Substituting x = 50.000 m and y = 5.000 m gives R = (2500 + 25)/10. The resulting radius is 252.500 m. Using the approximate relation R = x^2/(2y) would instead give 250 m and would omit the y^2 term.
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Question 9
ప్రశ్న 9
A simple circular curve has radius 200 m and central angle 60 degrees. A curve point is required at an EXACT perpendicular offset of 10 m from the incoming tangent. What is its distance x from PC, measured along that tangent, to three decimal places?
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Explanation:
The exact relationship for perpendicular tangent offsets is x^2 + (R - y)^2 = R^2. Rearranging gives x = sqrt(2Ry - y^2). With R = 200 m and y = 10 m, x = sqrt(4000 - 100). Therefore x = sqrt(3900) = 62.449980 m approximately. Rounded to three decimal places, the required tangent distance is 62.450 m. The approximate inverse formula would give sqrt(4000), which is slightly larger than the exact result.
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Question 10
ప్రశ్న 10
For setting out a simple circular curve from an incoming tangent, which statement correctly distinguishes RADIAL offsets from PERPENDICULAR offsets?
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Explanation:
A perpendicular offset is set out at right angles to the tangent line. Its exact value at tangent distance x is R - sqrt(R^2 - x^2). For a radial offset, the marked point on the tangent is joined toward the circle centre. The corresponding exact distance to the nearer circular intersection is sqrt(R^2 + x^2) - R. The two constructions therefore follow different directions and generally require different offset lengths. For small x/R, both expressions have the same leading approximation x^2/(2R).
వివరణ:
Question 11
ప్రశ్న 11
A circular curve has radius 100 m and central angle 90 degrees. A point Q is marked 60 m along the incoming tangent from PC. A RADIAL offset is measured from Q directly toward the circle centre to the nearer intersection with the curve. Using y = sqrt(R^2 + x^2) - R, what is the offset to three decimal places?
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Explanation:
The stated construction is a radial offset from a marked tangent point toward the circle centre. Its exact value is y = sqrt(R^2 + x^2) - R. Substitution gives y = sqrt(10000 + 3600) - 100. The square root of 13600 is approximately 116.619038 m. Thus the radial offset is approximately 16.619 m. The perpendicular offset at the same tangent distance would be 20.000 m, demonstrating that the two exact constructions are different.
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Question 12
ప్రశ్న 12
Which procedure correctly describes setting out a simple circular curve by OFFSETS FROM THE LONG CHORD?
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Explanation:
The long chord is the straight line directly connecting the two tangent points of a simple circular curve. In this method, PC and PT are first established and the long-chord line is located on the ground. Suitable points are marked at measured distances along this baseline. Perpendicular offsets are then calculated and set out toward the circular arc. The largest offset occurs at the midpoint of the long chord for the ordinary minor circular arc. This method is convenient when the long chord is accessible and can be measured reliably.
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Question 13
ప్రశ్న 13
For a simple circular curve of radius R and long-chord length L, let x denote the SIGNED horizontal distance along the long chord from its midpoint. Which expression gives the exact perpendicular offset O(x) toward the MINOR circular arc?
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Explanation:
The long chord is perpendicular to the radius passing through its midpoint. Its perpendicular distance from the centre is sqrt(R^2 - (L/2)^2). At a position x from the chord midpoint, the corresponding radial projection toward the minor arc is sqrt(R^2 - x^2). The required ordinate is the difference between these two distances. Therefore O(x) = sqrt(R^2 - x^2) - sqrt(R^2 - (L/2)^2). The expression gives zero at the two tangent points where x equals plus or minus L/2. It gives the maximum ordinate at x = 0.
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Question 14
ప్రశ్న 14
A simple circular curve has radius 100 m and long-chord length 120 m. What is the maximum perpendicular offset from the LONG CHORD at its midpoint?
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Explanation:
The maximum long-chord offset is the middle ordinate of the circular arc. Its exact value is M = R - sqrt(R^2 - (L/2)^2). Here R = 100 m and half the long chord is 60 m. Substitution gives M = 100 - sqrt(10000 - 3600). Since sqrt(6400) = 80, M equals 20.000 m. The offset is measured perpendicular to the long chord at its midpoint toward the minor circular arc.
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Question 15
ప్రశ్న 15
A simple circular curve has radius 100 m and long-chord length 120 m. Using exact long-chord geometry, what is the perpendicular offset to the curve at a point 28 m from the MIDPOINT of the long chord?
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Explanation:
The long-chord half-length is 120/2 = 60 m. The perpendicular distance from the circle centre to the long chord is sqrt(10000 - 3600) = 80 m. At x = 28 m from the midpoint, the corresponding radial projection is sqrt(10000 - 28^2). This becomes sqrt(9216) = 96 m. The required offset is therefore 96 - 80 = 16.000 m. The result is smaller than the maximum mid-ordinate of 20 m because the point is away from the midpoint.
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Question 16
ప్రశ్న 16
A curve of radius 125 m has a long chord of length 150 m. A surveyor marks a point on the long chord at a distance of 119 m from PC, measured toward PT. What is the EXACT perpendicular offset from that point to the minor circular arc?
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Explanation:
The midpoint of the long chord is 150/2 = 75 m from PC. A point 119 m from PC is therefore 119 - 75 = 44 m from the midpoint. The distance from the circle centre to the long chord is sqrt(125^2 - 75^2) = sqrt(10000) = 100 m. At x = 44 m, the radial projection toward the arc is sqrt(125^2 - 44^2) = sqrt(13689) = 117 m. The exact perpendicular offset is 117 - 100 = 17.000 m. This calculation illustrates why the distance from PC must first be converted to the midpoint-based coordinate used in the offset formula.
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Question 17
ప్రశ్న 17
A simple circular curve has radius 300 m and central angle 60 degrees. An offset from the long chord is required at a point 75 m from its midpoint. Using the exact geometry, what is the offset to three decimal places?
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Explanation:
The long chord is L = 2R sin(Delta/2). For R = 300 m and Delta = 60 degrees, L = 600 sin(30 degrees) = 300 m. The perpendicular distance from the centre to the long chord is 300 cos(30 degrees), approximately 259.807621 m. At x = 75 m from the midpoint, the corresponding radial projection is sqrt(300^2 - 75^2), approximately 290.473751 m. Subtracting gives O = 290.473751 - 259.807621 = 30.666130 m. The required offset is 30.666 m to three decimal places.
వివరణ:
Question 18
ప్రశ్న 18
A simple circular curve has a measured LONG CHORD of 120 m and a middle ordinate of 20 m. Assuming exact circular geometry, what is the radius of the curve?
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Explanation:
The relationship between radius, long chord and middle ordinate is M = R - sqrt(R^2 - (L/2)^2). Rearranging this expression gives R = L^2/(8M) + M/2. Here L = 120 m and M = 20 m. Substitution gives R = 14400/160 + 10. Therefore R = 90 + 10 = 100.000 m. The result can be verified by calculating the middle ordinate again as 100 - sqrt(10000 - 3600) = 20 m.
వివరణ:
Question 19
ప్రశ్న 19
A circular road curve begins at PC chainage 2+107.000 m and ends at PT chainage 2+192.000 m, with chainage measured along the arc. Pegs are proposed at every whole 20 m chainage, including 2+120, 2+140, 2+160 and 2+180 m. What are the FIRST and LAST sub-arc CHAINAGE INTERVALS, respectively, before converting them into straight subchord lengths?
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Explanation:
The first regular peg is at chainage 2+120.000 m, or 2120 m. The PC chainage is 2107 m, so the initial arc-chainage interval is 2120 - 2107 = 13 m. The last regular peg is at chainage 2+180.000 m, or 2180 m. The PT chainage is 2192 m, giving a final arc-chainage interval of 2192 - 2180 = 12 m. These are chainage differences measured along the circular arc, not exact straight subchord lengths. If straight subchords are required for precise setting out, they must be calculated from the curve radius and the corresponding subtended arc angles.
వివరణ:
Question 20
ప్రశ్న 20
A simple circular curve has radius R = 200 m and central angle Delta = 60 degrees. The midpoint of its long chord has already been used to locate the midpoint of the circular arc. In SUCCESSIVE BISECTION OF CHORDS, what perpendicular offset is required at the midpoint of either new half-chord joining a tangent point to the arc midpoint, to three decimal places?
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Explanation:
The midpoint of the full circular arc divides its central angle into two equal parts. Each half-arc therefore subtends 30 degrees at the centre. The chord joining a tangent point to the arc midpoint subtends this 30-degree angle. Its midpoint ordinate is R[1 - cos(30 degrees/2)] = R[1 - cos(15 degrees)]. Substituting R = 200 m gives 200[1 - cos(15 degrees)] = 6.814835 m approximately. The required offset is therefore 6.815 m, measured perpendicular to either half-chord toward its associated circular arc.
వివరణ:
Question 21
ప్రశ్న 21
A circular curve has radius 200 m. Two consecutive curve points subtend a CENTRAL angle of 10 degrees at the circle centre. What is the STRAIGHT CHORD LENGTH between these points, to three decimal places?
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Explanation:
A straight chord subtending central angle theta in a circle of radius R has length C = 2R sin(theta/2). The given radius is 200 m and the central angle is 10 degrees. Therefore the half-angle is 5 degrees. Substitution gives C = 400 sin(5 degrees). This evaluates to approximately 34.862297 m. Rounded to three decimal places, the chord length is 34.862 m, slightly less than the corresponding circular arc length.
వివరణ:
Question 22
ప్రశ్న 22
For a circular curve of radius 300 m, two consecutive pegs are specified to be separated by exactly 30.000 m of ARC LENGTH. What is their corresponding STRAIGHT CHORD LENGTH to three decimal places?
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Explanation:
The central angle subtended by an arc of length s is theta = s/R when theta is in radians. Here theta = 30/300 = 0.100000 radian. The straight chord length is C = 2R sin(theta/2). Therefore C = 600 sin(0.050000 radian). Evaluating this expression gives approximately 29.987502 m. The corresponding chord length to three decimals is 29.988 m, demonstrating that chord length is slightly smaller than arc length.
వివరణ:
Question 23
ప్రశ్న 23
In the approximate OFFSET-FROM-TANGENT construction used for the FIRST peg of a chord-produced circular-curve setting-out procedure, the distance measured along the initial tangent is 20 m and the curve radius is 250 m. Using O1 = C1^2/(2R), what is the first offset?
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Explanation:
The first offset in the approximate chord-produced procedure can be obtained using the short-offset relation O1 = C1^2/(2R). The initial tangent measurement is C1 = 20 m. The curve radius is R = 250 m. Substitution gives O1 = 20^2/(2 multiplied by 250). Therefore O1 = 400/500 = 0.800 m. The approximation assumes that the initial tangent measurement is small relative to the radius and that the required offset is set toward the curve.
వివరణ:
Question 24
ప్రశ్న 24
In setting out a circular curve by the APPROXIMATE OFFSETS-FROM-CHORD-PRODUCED method, the first chord length is C1 = 15 m and the succeeding chord length is C2 = 20 m. The curve radius is R = 250 m. Using O2 = C2(C1 + C2)/(2R), what offset is required from the first chord produced to locate the next curve point?
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Explanation:
The approximate offset from the first chord produced depends on both the preceding and succeeding chord lengths. The prescribed formula is O2 = C2(C1 + C2)/(2R). Substitution gives O2 = 20(15 + 20)/(2 multiplied by 250). This simplifies to 20 multiplied by 35 divided by 500. The required offset is therefore 1.400 m. The method assumes small chord lengths relative to the radius and requires careful control of the produced chord direction to limit accumulated setting-out errors.
వివరణ:
Question 25
ప్రశ్న 25
A long simple circular road curve must be set out using linear measurements because a theodolite is unavailable. The full long chord is obstructed by existing structures, and continuous access along the tangents is also restricted, but successive curve-peg positions remain accessible. Which method is the most appropriate practical choice?
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Explanation:
The offsets-from-chords-produced method advances the curve using previously established curve points. It avoids the need to measure offsets from the full long chord at every stage. It can be useful for comparatively long, large-radius curves when sequential access along the intended alignment is available. Each new chord is produced from the preceding curve points, and the calculated offset is used to locate the next peg. Errors in previous pegs or produced directions can accumulate, so independent checks against available control are important. The method is therefore a suitable linear alternative when the principal tangent and long-chord baselines are obstructed but consecutive curve points remain accessible.
వివరణ:
Answer Key సమాధానాల పట్టిక
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Question 1 ప్రశ్న 1Answer: B. Offsets from tangents, offsets from the long chord, successive bisection of chords and offsets from chords produced. సమాధానం: B.
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Question 2 ప్రశ్న 2Answer: D. Locate PC and establish the tangent direction, mark calculated distances along that tangent, erect perpendiculars toward the curve and measure the corresponding offsets. సమాధానం: D.
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Question 3 ప్రశ్న 3Answer: A. y = R - sqrt(R^2 - x^2), where 0 <= x < R. సమాధానం: A.
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Question 4 ప్రశ్న 4Answer: C. When x/R is sufficiently small that higher-order terms in the expansion can be neglected. సమాధానం: C.
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Question 5 ప్రశ్న 5Answer: B. 4.041 m. సమాధానం: B.
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Question 6 ప్రశ్న 6Answer: A. 5.000 m. సమాధానం: A.
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Question 7 ప్రశ్న 7Answer: D. The exact offset is 20.000 m, while the approximation gives 18.000 m and underestimates by 2.000 m. సమాధానం: D.
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Question 8 ప్రశ్న 8Answer: C. 252.500 m. సమాధానం: C.
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Question 9 ప్రశ్న 9Answer: B. 62.450 m. సమాధానం: B.
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Question 10 ప్రశ్న 10Answer: A. A radial offset from a marked tangent point is measured along the line toward the circle centre and has exact length sqrt(R^2 + x^2) - R, whereas the perpendicular offset is measured normal to the tangent and equals R - sqrt(R^2 - x^2). సమాధానం: A.
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Question 11 ప్రశ్న 11Answer: D. 16.619 m. సమాధానం: D.
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Question 12 ప్రశ్న 12Answer: C. Establish the straight long chord joining PC and PT, mark selected distances along it, and erect calculated perpendicular offsets toward the circular arc. సమాధానం: C.
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Question 13 ప్రశ్న 13Answer: B. O(x) = sqrt(R^2 - x^2) - sqrt(R^2 - (L/2)^2). సమాధానం: B.
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Question 14 ప్రశ్న 14Answer: A. 20.000 m. సమాధానం: A.
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Question 15 ప్రశ్న 15Answer: C. 16.000 m. సమాధానం: C.
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Question 16 ప్రశ్న 16Answer: D. 17.000 m. సమాధానం: D.
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Question 17 ప్రశ్న 17Answer: B. 30.666 m. సమాధానం: B.
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Question 18 ప్రశ్న 18Answer: A. 100.000 m. సమాధానం: A.
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Question 19 ప్రశ్న 19Answer: C. 13 m and 12 m. సమాధానం: C.
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Question 20 ప్రశ్న 20Answer: D. 6.815 m. సమాధానం: D.
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Question 21 ప్రశ్న 21Answer: A. 34.862 m. సమాధానం: A.
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Question 22 ప్రశ్న 22Answer: B. 29.988 m. సమాధానం: B.
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Question 23 ప్రశ్న 23Answer: C. 0.800 m. సమాధానం: C.
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Question 24 ప్రశ్న 24Answer: D. 1.400 m. సమాధానం: D.
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Question 25 ప్రశ్న 25Answer: A. Use offsets from successive CHORDS PRODUCED, establishing and checking each curve peg before extending the next chord. సమాధానం: A.