Question Solved1 Answer 9. Show that the vector \( \vec{a}=(5,9,14) \) can be written as a linear combination of the vectors \( \vec{b} \) and \( \vec{c} \), where \( \vec{b}=(-2,3,1) \) and \( \vec{c}=(3,1,4) \). (4 marks)

2. Entropy of functions. Let X be a random variable taking on a finite number of values. What is the (general) inequality relationship of H(X) and H(Y) if (a) Y = 2X2 (b) Y = cos X?

Parts E and B please 5. Compute the power Px and the energy Ex for each of the following signals: (a) x(t) = e-3tu(t) (b) x(t) = ej(2t+31/4) (c) x(t) = cos(t/2) (d) x[n] = (?)"u[n] (e) x[n] = ej(1/2+31/8) (f) x[n] = cos (31n/4)

1. Derive an expression for the horizontal distance of a vertical staff from a tacheometer if the line of sight of the telescope is horizontal 2. Derive expressions for the horizontal distance D and the vertical intercept (difference) V when the staff is (a) elevation and (b) depression. 3. The distance between a tachometer station and a staff station is measured to be 50.85 m. with the line of sight horizontal, the stadia hair readings are found to be 1.99 m and 1.395 m. Determine the instrument constant. 4. The following readings were taken with a tacheometer on to a vertical staff and zero vertical angle. Horizontal Distance Stadia Readings 46.2 m 0.78; 1.01; 1.24 51.2 m 1.86; 2.16; 2.47 Calculate the tacheometric constants. 5. The following readings were taken with tachometers on a vertical staff stand on three points, if the elevation the station point is 45.9 m and the height of instrument is 17.6 dm. Stadia readings Vertical angle 0.795, 1.025, 1.255 1.875, 2.180, 2.485 5 30 1.368, 2.359, 3.35 -6° 45' Find the distance from station to each staff point and their reduced levels? 6. In determining the elevation of point A and the distance between two points, A and B, a theodolite is set up at A and the following data are obtained: Z or a = 92° 45', stadia interval = 1.311m, hi = 1.28 m, and the line of sight at 2.62 m (central staff reading). The instrument constant K = 100 and C = 0.03 m. The elevation of B is 38.28 m. compute the distance AB and elevation of point A. 7. The elevation of station P is 345.67m. The height of theodolite over that station is 1.23m. When observing a rod held on station Q, the following data are recorded, Vertical angle - - 9° 45', lower staid hair = 0.83, center hair = 1.23, upper stadia hair = 1.63. Determine the distance between P and Q, and elevation of Q. 8. Stadia readings were taken with a theodolite on a vertical staff with the telescope inclined at a V.C.R of 93°30'. The staff readings were 299, 205 and 112 cm. The reduced level of the staff station is 100.75m, and the height of the instrument is 14 Dm. What is the reduced level of the ground at the instrument? Take constants as 100 and zero. 9. A tacheometer is setup at an intermediate point on a traverse course PQ and the following observations are made on a staff held vertical. Staff Station V.C.R + 279° 30' + 275900 Staff Intercept (S) Central Hair Readings 2.250 2.105 2.055 1.975 The constants are 100 and 0. Compute the length PQ and the reduced level of Q. if RL of P- 350.50 m. 10. From an instrument station Pof R.L 218.56 m two depressed lines of sight were taken to a staff held vertically over a station Q. The staff readings obtained is 3.160 m and 1.310m. The angels of depression of the lines of sight are 7 15 and 87. Find the horizontal distance between the two stations. Find also the R.L of the station Q. The height of instrument above the station P is 1.51m. 11. A theodolite was set over a station A of reduced level 44.620m, the instrument height being 1.425m. Observations were taken by tangent method to the 3m and 0.60m marks on each staff held vertically at three stations B, C and D with the following results. Instrument station Staff Station V.C.R 27910 885 95°15 273 30 9224 112 10 Find the distances from A to each station and their reduced levels? 12. Observations with a tachometer gave staff intercepts of 0.595m and 1.490m with telescope Horizontal and the staff held vertically at Horizontal distance of 61.062m and 152.40m respectively from the instrument, calculate the instrument constant and additive constant of the tachometer. The instrument was then set over a station with R.L 80.559m. the height of the instrument axis above the station point being 1.29 m. the readings of the three hairs on a vertical staff were 0.915m, 1895m and 2.875m, the telescope being inclined at 8 above the Horizontal. Determine the R.L of the staff station and its distance from the center the instrument.

Question 2 From the table below, calculate the actual distance for each point given the multiplying and additive constants equivalent to 98.4 and 0.5, respectively. Point Top stadia Middle stadia Below stadia Distance А 2.00 1.50 1.00 a) B 1.95 1.55 0.25 b) с 2.35 1.85 0.95 c) D 1.98 1.78 0.48 d) E 1.56 1.26 0.76 e)

em 6 Find the direction of the line normal to the surface defined by the function \[ f(x, y, z)=x^{2} y+y^{2} z+z^{2} x+1=0, \] at the point \( (1,2,-1) \). Find the equation of the tangent plane and the normal line at this point. Make a 3D plot to show the surface, the tangent plane, and the normal line.

A simple pendulum of mass m2 and length l is constrained to move in a single plane. The point of support is attached to a mass m1 which can move on a horizontal line in the same plane. Find the lagrangian of the system in terms of suitable generalized coordinates. Derive the equations of motion. Find the frequency of small oscillations of the pendulum.