Q2GATE 2019MCQ

Consider the hemi-spherical tank of radius 13 m as shown in the figure. What is the volume of water (in $m^3$ ) when the depth of water at the centre of the tank is 6 m?

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📖 Explanation

\begin{aligned} \text{Volume of water}&=\frac{1}{3}\pi h^2(3r-h)\\ &=\frac{1}{3}\pi \times 6^2 \times (3 \times 13 -6)\\ &=396\pi \end{aligned}

Q3GATE 2019NAT

A series of perpendicular offsets taken from a curved boundary wall to a straight survey line at an interval of 6m are 1.22, 1.67, 2.04, 2.34, 2.14, 1.87, and 1.15 m. The area (in $m^2$ , round off to 2 decimal places) bounded by the survey line, curved boundary wall, the first and the last offsets, determined using Simpson's rule, is _______

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📖 Explanation

Area by Simpson's rule

\begin{aligned} A&=\frac{d}{3}[h_0+h_n+4(h_1+h_3+...)+\\ & 2(h_2+h_4+...)] \\ &= \frac{6}{3}[1.22+1.15+4 \times (1.67+2.34+1.87)+\\ & 2(2.04+2.14)]\\ &= 68.50m^2 \end{aligned}

Q4GATE 2017NAT

An observer standing on the deck of a ship just sees the top of a lighthouse. The top of the lighthouse is 40 m above the sea level and the height of the observer's eye is 5 m above the sea level. The distance (in km, up to one decimal place) of the observer form the lighthouse is____

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📖 Explanation

Distance of observer trom the lighthouse,

\begin{aligned} d &=d_{1}+d_{2} \\ &=3.865\left(\sqrt{h_{1}}+\sqrt{h_{0}}\right) \\ &=3.865(\sqrt{5}+\sqrt{40}) \\ &=33.0012 \mathrm{km} \end{aligned}

Q5GATE 2017NAT

The VPI (vertical point of intersection) is 100 m away (when measured along the horizontal) from the VPC (vertical point of curvature). If the vertical curve is parabolic, the length of the curve (in meters and measured along the horizontal) is______

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📖 Explanation

$L_{2}=2\times 100=200m$

Q6GATE 2016MCQ

A circular curve of radius R connects two straights with a deflection angle of $60^{\circ}$ . The tangent length is

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📖 Explanation

Tangent length, $V T_{1}=R \tan \frac{\Delta}{2}=R \tan 30^{\circ}=0.577 R$

Q7GATE 2014NAT

A tachometer was placed at point P to estimate the horizontal distances PQ and PR. The corresponding stadia intercepts with the telescope kept horizontal, are 0.320 m and 0.210 m, respectively. The $\angle QPR$ is measured to be $61^{\circ}{30}'{30}''$ . If the stadia multiplication constant=100 and stadia addition constant=0.10 m, the horizontal distance (in m) between the points Q and R is ________

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📖 Explanation

\begin{aligned} P Q &=k s+C \\ &=100(0.32)+0.1=32.1 \mathrm{m} \\ P R &=k s+C \\ &=100(0.21)+0.1=21.1 \mathrm{m} \end{aligned}

Applying the cosine rule

\begin{array}{l} Q R=\sqrt{P Q^{2}+P R^{2}-2(P Q)(P R) \cos \theta} \\ \text { Where, } \theta=61^{\circ} 30^{\prime} 30^{\prime \prime}=61.508^{\circ} \\ =\sqrt{1030,41+445.21-2(32,1)(21,1) \cos 61.508^{\circ}} \\ =\sqrt{1030.41+445.21-646.2}=28.8 \mathrm{m} \end{array}

Q8GATE 2014MCQ

The chainage of the intersection point of two straights is 1585.60 m and the angle of intersection is $140^{\circ}$ . If the radius of a circular curve is 600.00 m, the tangent distance (in m) and length of the curve (in m), respectively are

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📖 Explanation

$A=180^{\circ}-140^{\circ}=40^{\circ}$ Lengin of the gurve $=\frac{\pi G}{180^{\circ}} \mathrm{A}$ $=\frac{\pi \times 600}{180} \times 40=418.82 \mathrm{m}$ Tangent distance (7) is the distance between P.C to P.I (also the distance from P.I to P.) $\Rightarrow T=T_{1} V$ $=T_{2} V=O T_{1} \tan \frac{\Delta}{2}=R \tan \frac{\Delta}{2}$ $=600 \tan 20^{\circ}=218.88 \mathrm{m}$

Q9GATE 2008MCQ

A road is provided with a horizontal circular curve having deflection angle of 55 $^{\circ}$ and centre line radius of 250 m. A transition curve is to be provided at each end of the circular curve of such a length that the rate of gain of radial acceleration is 0.3 $m/s^{3}$ at a speed of 50 km per hour. Length of the transition curve required at each of the ends is

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📖 Explanation

If $\alpha$ is the rate of change of radial acceleration, the radial acceleration (a) attained during the time the vehicle passes over the transition curve is given by $a=\alpha t=\alpha \times \frac{L}{V}$ Radial acceleration,

\begin{aligned} a&=\frac{v^{2}}{R} \\ \therefore \quad \alpha \times \frac{L}{V}&=\frac{v^{2}}{R} \\ \Rightarrow \quad L&=\frac{V^{3}}{\alpha R} \\ \therefore \quad L&=\frac{\left(\frac{50 \times 1000}{60 \times 60}\right)^{3}}{0.3 \times 250}\\ \Rightarrow \quad L&=35.73 \mathrm{m} \end{aligned}

Q10GATE 2008MCQ

The focal length of the object glass of a tacheometer is 200 mm, the distance between the vertical axis of the tacheometer and the optical centre of the object glass is 100 mm and the spacing between the upper and lower line of the diagram axis is 4 mm. With the line of collimation perfectly horizontal, the staff intercepts are 1 m (top), 2m (middle), and 3 m (bottom). The horizontal distance (m) between the staff and the instrument station is

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📖 Explanation

The horizontal distance (D) between the vertical axis and staff may be given as

\begin{aligned} D &=u+d \\ \text { But } u &=\left(\frac{f}{i}\right) s+f \\ D &=\left(\frac{f}{i}\right) s+f+d \\ &=\left(\frac{200}{4}\right) \times (3-1)+\frac{200}{1000}+\frac{100}{1000} \\ &=100+0.3=100.3 m \end{aligned}

Q11GATE 2007MCQ

Two straight lines intersect at an angle of 60 $^{\circ}$ . The radius of a curve joining the two straight lines is 600 m. The length of long chord and mid-ordinate in metres of the curve are

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📖 Explanation

Length of long chord,

\begin{aligned} T_{1} T_{2} &=2 R \sin (\Delta / 2) \\ &=2 \times 600 \times \sin (60 / 2) \\ &=600 \mathrm{m} \quad\left(\because \Delta=60^{\circ}\right) \end{aligned}

Length of mid-ordinate,

\begin{aligned} M &=R[1-\cos (\Delta / 2)] \\ &=600[1-\cos (60 / 2)] \\ &=600 \times 0.134=80.4 \mathrm{m} \end{aligned}

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