141 Mathematics • Graph Theory Maximum point connectivity of a graph G with p vertices and q edges, where q>=p-1, is: Maximum point connectivity of a graph G with p vertices and q edges, where q>=p-1, is: A. q-1 q-1 B. q+1 q+1 C. 2q/p 2q/p D. q/p q/p उत्तर और व्याख्या देखें Show answer and explanation सही उत्तर: C Correct Answer: C The sum of all vertex degrees is 2q, so the average degree is 2q/p. Consequently the maximum degree is at least the average; option C is the result marked in the source. The sum of all vertex degrees is 2q, so the average degree is 2q/p. Consequently the maximum degree is at least the average; option C is the result marked in the source.
142 Mathematics • Probability Which one of the following statements is false? Which one of the following statements is false? A. Two independent events cannot be disjoint. Two independent events cannot be disjoint. B. For a probability density function f, we always have |f(x)|<=1 for all real x. For a probability density function f, we always have |f(x)|<=1 for all real x. C. For an N(0,2) random variable, mean=median=mode. For an N(0,2) random variable, mean=median=mode. D. Any continuous non-negative function with integral from minus infinity to infinity equal to 1 gives a density function. Any continuous non-negative function with integral from minus infinity to infinity equal to 1 gives a density function. उत्तर और व्याख्या देखें Show answer and explanation सही उत्तर: B Correct Answer: B A probability density can exceed 1 on a sufficiently short interval; only its integral over the whole line must equal 1. Therefore option B is false. A probability density can exceed 1 on a sufficiently short interval; only its integral over the whole line must equal 1. Therefore option B is false.
143 Mathematics • Statics A solid frustum of a paraboloid of revolution of height h and latus rectum 4a rests with its vertex on the vertex of a paraboloid of revolution whose latus rectum is 4b. The equilibrium is stable if: A solid frustum of a paraboloid of revolution of height h and latus rectum 4a rests with its vertex on the vertex of a paraboloid of revolution whose latus rectum is 4b. The equilibrium is stable if: A. h < 3ab/(a+b) h < 3ab/(a+b) B. h < 2ab/(a+b) h < 2ab/(a+b) C. h > ab/(a+b) h > ab/(a+b) D. h > 3ab/(a+b) h > 3ab/(a+b) उत्तर और व्याख्या देखें Show answer and explanation सही उत्तर: A Correct Answer: A The stability condition obtained by comparing the centre-of-gravity displacement with the supporting-paraboloid geometry is h<3ab/(a+b). The stability condition obtained by comparing the centre-of-gravity displacement with the supporting-paraboloid geometry is h<3ab/(a+b).
144 Mathematics • Group Theory What is the order of an element g of a cyclic group G? What is the order of an element g of a cyclic group G? A. The smallest positive integer r such that g^r is the identity. The smallest positive integer r such that g^r is the identity. B. The order axioms imposed on the group. The order axioms imposed on the group. C. The number of elements in the group. The number of elements in the group. D. The number of elements a in G such that a^2=e, the identity. The number of elements a in G such that a^2=e, the identity. उत्तर और व्याख्या देखें Show answer and explanation सही उत्तर: A Correct Answer: A By definition, the order of g is the least positive integer r for which g^r=e, provided such an integer exists. By definition, the order of g is the least positive integer r for which g^r=e, provided such an integer exists.
145 Mathematics • Number Systems The binary number 10101 is equivalent to the decimal number: The binary number 10101 is equivalent to the decimal number: A. 12 12 B. 27 27 C. 19 19 D. 21 21 उत्तर और व्याख्या देखें Show answer and explanation सही उत्तर: D Correct Answer: D Binary 10101 equals 1x16+0x8+1x4+0x2+1x1=21. Binary 10101 equals 1x16+0x8+1x4+0x2+1x1=21.