Q.(a)

(a)(i) Maths methods give economics precision, brevity and logical/empirical testing; (a)(ii) rural housing suffers from poverty and lack of finance/materials. (b) By Cramer's rule the solution is x1 = 1, x2 = 3, x3 = 4.

(a)(i) Three uses of Mathematical Methods in Economics (in the Tamil Nadu HSC Class-11 Economics syllabus):

1. Precision and brevity — mathematical symbols and equations state economic relationships exactly and in a short, clear form, avoiding the vagueness of ordinary language.
2. Logical derivation of results — using functions, slopes and derivatives, marginal concepts (marginal utility, marginal cost, marginal revenue) can be derived and economic conclusions reached logically.
3. Testing and verification of theories — with statistics and econometrics, economic theories and relationships can be tested against real data, and equilibrium values (e.g. market equilibrium) can be computed.

(a)(ii) Two causes for housing problems in rural areas:

1. Poverty and low income — most rural households have low and irregular incomes, so they cannot afford to build or maintain proper, durable houses.
2. Lack of finance and building materials — inadequate institutional housing credit, along with a shortage and high cost of building materials and the absence of proper planning, leaves many rural homes small, kutcha and overcrowded.

(b) Solving the linear equations by Cramer's rule:

The system is:

7x1 − x2 − x3 = 0

10x1 − 2x2 + x3 = 8

6x1 + 3x2 − 2x3 = 7

Coefficient determinant D:

D = | 7 −1 −1 ; 10 −2 1 ; 6 3 −2 |

D = 7[(−2)(−2) − (1)(3)] − (−1)[(10)(−2) − (1)(6)] + (−1)[(10)(3) − (−2)(6)]

D = 7[4 − 3] + 1[−20 − 6] − 1[30 + 12]

D = 7(1) + 1(−26) − 1(42) = 7 − 26 − 42 = −61

D1 (replace column 1 by constants 0, 8, 7):

D1 = | 0 −1 −1 ; 8 −2 1 ; 7 3 −2 |

D1 = 0[(−2)(−2) − (1)(3)] − (−1)[(8)(−2) − (1)(7)] + (−1)[(8)(3) − (−2)(7)]

D1 = 0 + 1(−16 − 7) − 1(24 + 14) = 0 − 23 − 38 = −61

D2 (replace column 2 by constants 0, 8, 7):

D2 = | 7 0 −1 ; 10 8 1 ; 6 7 −2 | …