This set of Surveying Multiple Choice Questions & Answers (MCQs) focuses on “Survey Adjustments and Errors Theory – Normal Equations”.
1. Which of the following can be used for finding a normal equation?
a) Unknown values
b) Algebraic coefficients
c) Probability law
d) Probability curve
Explanation: A normal equation is formed by the multiplication of unknown coefficients by which, the obtained equation is added and leads to the formation of normal equation. If the number of equations formed is equal to the number of unknowns the n the most probable value values can be found by the equations.
2. Normal equation is used in case of________________
a) Probability equation
b) Arithmetic method
c) Indirect method
d) Direct method
Explanation: Since the usage of normal equation involves finding unknowns of an equation, it is more used in the direct method. The found unknowns are used for the determination of most probable values and in the method of correlates.
3. Which of the following processes is a tedious one?
a) Probability law
b) Normal equation
c) Probability equation
d) Most probable value
Explanation: The process involved in finding and using the unknowns in a normal equation is tedious because it involves deriving the unknowns and using them for finding the most probable value, solving their cases and in the method of correlates.
4. Determine the normal equation for x for the equations, 5x+2y+3z-6 = 0 and 2x+4y+6z-10 = 0, having equal weight.
a) 29x+18y+27z-50 = 0
b) 18x+29y+27z-50 = 0
c) 27x+18y+29z-50 = 0
d) 29x+81y+27z-50 = 0
Explanation: The normal equation for x can be found out by multiplying the equations with the constant of x i.e.,
5*(5x+2y+3z-6 = 0) = 25x+10y+15z-30 = 0 and similarly,
2*(2x+4y+6z-10 = 0) = 4x+8y+12z-20 = 0. On adding we get the normal equation for x i.e.,
29x+18y+27z-50 = 0.
5. If different weights of the equations are involved then they are to be subtracted with the coefficients.
Explanation: The presence of unequal weights may be a problem for having a normal equation. In that case, the weights are to be multiplied with the constants present with the variables for obtaining the required equation.
6. Formation of normal equation with unknown quantities must be multiplied with the algebraic coefficient.
Explanation: In order to form a normal equation with unknown quantities it is necessary to multiply every observation with an algebraic coefficient of an unknown quantity. By this, a relation can be established.
7. Find the normal equation for y of the equations 2x+3y+4z-7 = 0, x-4y+6z-9 = 0, having weights 3 and 2 respectively.
a) 48x-32y+8z-72 = 0
b) 32x-8y+48z-72 = 0
c) 8x-32y+48z-72 = 0
d) 8x-48y+32z-72 = 0
Explanation: In order to find normal equations for different weights we have to multiply with the weight along with the constant of the variable i.e.,
(3*3)*(2x+3y+4z-7=0) = 18x+27y+36z-54 = 0 and similarly,
(4*2)*(x-4y+6z-9=0) = 8x-32y+48z-72 = 0.
8. What will be the normal equation for z if the equations are given as 3x+9y+4z-43=0, 2x+6y+z-5=0. Assume these are having equal weights.
Explanation: The normal equation for the variable z can be found out by multiplying those equations with their respective variable constants i.e.,
4*(3x+9y+4z-43=0) = 12x+36y+16z-172=0 and
1*(2x+6y+z-5=0) = 2x+6y+z-5=0. On addition, we get the required normal equation,
9. A normal equation is formed by ____________
a) Subtracting algebraic coefficients
b) Adding algebraic coefficients
c) Dividing algebraic coefficients
d) Multiplying algebraic coefficients
Explanation: In general, normal equation can be found out by multiplying every given equation with the coefficient of unknown whose normal equation has to be found out. By adding all those equations, the complete normal equation can be found out.
10. Which of the following represents the correct set of constants and variables present in a normal equation?
a) 4 consonants, 3 variables
b) 3 consonants, 4 variables
c) 2 consonants, 2 variables
d) 1 consonant, 1 variable
Explanation: In general, any normal equation contains four constants and three variables. It can be found out by multiplying with the weight value provides with the constant of the variable required.
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