# Center for Development Studies (CDS) - Entrance Solutions (2019)

### Answer Key

1. $\boxed{B} \ 4$
2. $\boxed{A} \ \frac{5}{14}$
3. $\boxed{A} \ \frac{27}{35}$
4. $\boxed{B} \ 63$
5. $\boxed{C} \ \frac{1}{8}$
6. $\boxed{A}$ The probability of correctly rejecting the null hypothesis.
7. $\boxed{B} \ 4\%$ probability that average marks are equal.
8. $\boxed{B} \ a^2\sigma_{x}+b^2\sigma_{y}$.
9. $\boxed{A}$ A $\chi^{2}$ distribution with mean $n$ and variance $2n$.
10. $\boxed{A} \ Z^2 \succ F[1,n]$.
11. $\boxed{C}$ Coefficient of Variation.
12. $\boxed{C}$ with mean $\mu$ and variance $\frac{\sigma^{2}}{n}$.
13. $\boxed{A}$ The mean of the sampling distribution is $\theta$.
14. $\boxed{A}$ Its sampling distribution has the least variance.
15. $\boxed{B}$ $\frac{1}{108}$
16. $\boxed{B}$ $\frac{29}{72}$
17. $\boxed{C}$ $\frac{25}{69}$
18. $\boxed{B}$  Maximum allowable probability of type $I$ error.
19. $\boxed{C}$ $\beta_{xy}=\beta_{yx}=0$
20. $\boxed{C}$ Biased and inconsistent.
21. $\boxed{C}$ $L_{B}<L_{A}$ and $K_{A}<K_{B}$ . Since, $Q_{B}$ have higher $A$ or the Total Factor Productivity compared to $Q_{A}$ ceteris paribus.
22. $\boxed{C}$ He prefers black tea over lemon tea. Since it is given that, $Coffee \succ Black \ tea$,  $Black \ tea \sim Green \ tea$ and $Lemon \ tea \succ Green \ tea$. This implies, the only false statement is $Black \ tea \succ Lemon \ tea$
23. $\boxed{A}$ $(tx_{1}+(1-t)x_{2},ty_{1}+(1-t)y_{2})\succcurlyeq (x_{1},y_{2})$
24. $\boxed{C}$ $f(B)-f(C)<0$
25. $\boxed{A}$ $U=5x_{1}+5x_{2}$ represents a straight line with slope 1 or it represents the indifference curve of perfect substitutes.
26. $\boxed{C}$ Figure C represents the Engel curve derived from homothetic preferences
27. $\boxed{A}$ AC continuously falls as output rises since $AC=\frac{TC}{q}=\frac{q(VC)+FC}{q}=VC+\frac{FC}{q}$
28. $\boxed{C}$ Greater than one, but less than 2.
29. $\boxed{A}$ Increase
30. $\boxed{C}$ Figure C represents the IC of strict complementarity.
31. $\boxed{C}$ Consumer prefernces have changed in favour of A.
32. $\boxed{B}$ Producing same output with more inputs.
33. $\boxed{B}$ Variable cost
34. $\boxed{B}$ Decrease
35. $\boxed{B}$ Figure B represents the IC of Ramu.
36. $\boxed{B}$ $MR=MC$
37. $\boxed{D}$ Total utlity
38. $\boxed{D}$ Limiting production
39. $\boxed{A}$ Increased barrier to entry
40. $\boxed{D}$ None of these
41. $\boxed{B}$ No
42. $\boxed{B}$ $8.5$
43. $\boxed{A}$ True
44. $\boxed{A}$ 25
45. $\boxed{A}$ True
46. $\boxed{D}$ Figure D represents the Saving-Income relationship.
47. $\boxed{B}$$3\% 48. \boxed{B} The stability criteria of the model. 49. \boxed{A} Fully on permanent income. 50. \boxed{A} To finance old age consumption 51. \boxed{B} The change in the GDP. 52. \boxed{D} Remains unchanged under 100-percent reserve banking but decrease under fractional -reserve banking 53. \boxed{B} Always 54. \boxed{C} Equilibrium output will increase but private consumption will not change. 55. \boxed{C} Two 56. \boxed{D} All of the above 57. \boxed{A} MPC will be lower and interst sensitivity of the investment will be lower. 58. \boxed{C} For a special case, we can get a vertical LM curve. 59. \boxed{B} Average Propensity to Consume (APC) falls as income rises. 60. \boxed{D} Greater than one. 61. \boxed{D} The rate of technological progress. 62. \boxed{D} The persistence of fiscal shock and elasticity of labour supply. 63. \boxed{A} Interest rate is low and savings rate is high. 64. \boxed{B} Only fiscal policy is effective. 65. \boxed{B} Accelerator principle 66. \boxed{C} Inflation is a monetary phenomenon. 67. \boxed{C} Real variables can be analysed without analyzing their nominal counterparts. 68. \boxed{A} Y=1000 ,i=5\% 69. \boxed{C} Though money is neutral in long run but not in short run, governments cannot exploit because agents are rational. 70. \boxed{C} basically arising from interaction between workers choice of labour and liesure and productivity shocks. 71. \boxed{C} e^{3x}(3x+1) 72. \boxed{A} \ d=\frac{1}{4} 73. \boxed{C} tx+(1-t)y \in X \ \ \ \forall x,y \in X and \forall t \in (0,1) 74. \boxed{A} +1 75. \boxed{C} f(\lambda x+(1- \lambda )y) \geq \lambda f(x)+(1- \lambda)f(y) \ \ \ \forall x,y \in S and \lambda \in (0,1) 76. \boxed{B} e 77. \boxed{A} (A \cap C) \cup (B \cap C) 78. \boxed{C} \frac{1-r^{n+1}}{1-r} 79. \boxed{A} Figure A represents the fucntion. 80. \boxed{D} \frac{4}{3} 81. \boxed{C} It's maximal rank is n. 82. \boxed{B}$$B^{-1}A^{-1}$
83. $\boxed{D}$ $4$
84. $\boxed{D}$ $a=1$ ,$b=0$
85. $\boxed{C}$ Limit does not exist
86. $\boxed{B}$ $(-3, \inf)$
87. $\boxed{C}$ $36^o$
88. $\boxed{B}$ A relation
89. $\boxed{B}$ $m \times n \times p$
90. $\boxed{A}$ $135$
91. $\boxed{D}$ $17 \%$
92. $\boxed{A}$ $4+/-2$
93. $\boxed{C}$ Fiscal deficit
94. $\boxed{D}$ Software and non-factor service
95. $\boxed{C}$ Private Corporate Sector
96. $\boxed{A}$ $0.70 \%$
97. $\boxed{D}$ China
98. $\boxed{D}$ India
99. $\boxed{B}$ Rice
100. $\boxed{A}$ Household sector

#### Detailed Solutions

1. Given,
$$Expected \ Value \ of \ x, \ E(x)=2 \ \\ Variance \ of \ x, \ V(x)=1$$
We know,
$$E(x)=NP=2 \\ \implies P=\frac{2}{N} \ ...(1)$$ $$V(x)=NP(1-P) \ ...(2)$$
Substitute $(1)$ in $(2)$,
$$N \times \frac{2}{N} (1-\frac{2}{N})=1 \\ \implies 2N-4=N \\ \boxed{\therefore N=4}$$

2. Given,
$$Number \ of \ black \ balls, \ \ n(B)=5 \\ Number \ of \ red \ balls, \ \ n(R)=3 \\ Total \ number \ of \ balls, \ \ n(T)=5+3=8 \\$$ $$\implies Pr(Both \ balls \ being \ black) \\ = \frac{n(B)}{n(T)} \times \frac{n(B-1)}{n(T-1)} \\ = \frac{5}{8} \times \frac{4}{7} \\ \boxed{\therefore Pr(BB)=\frac{5}{14}}$$