121 Advanced Mathematics • Digital Logic The minimized form of the logical expression A^cB^cC^c + A^cBC^c + A^cBC + ABC^c is: The minimized form of the logical expression A^cB^cC^c + A^cBC^c + A^cBC + ABC^c is: सेव करें Save सेव्ड Saved A. A^cC^c + BC^c + A^cB A^cC^c + BC^c + A^cB B. (AC)^c + (BC)^c + (AB)^c (AC)^c + (BC)^c + (AB)^c C. A^cC + B^cC + A^cB^c A^cC + B^cC + A^cB^c D. AC^c + B^cC + AB^c AC^c + B^cC + AB^c उत्तर और व्याख्या देखें Show answer and explanation सही उत्तर: A Correct Answer: A Grouping minterms 0 and 2 gives A^cC^c, 2 and 6 gives BC^c, and 2 and 3 gives A^cB. Hence the minimized sum is A^cC^c+BC^c+A^cB. Grouping minterms 0 and 2 gives A^cC^c, 2 and 6 gives BC^c, and 2 and 3 gives A^cB. Hence the minimized sum is A^cC^c+BC^c+A^cB.
122 Advanced Mathematics • Limits Evaluate lim as (x,y,z) approaches (3,0,1) of exp(-x)y sin(xy/z). Evaluate lim as (x,y,z) approaches (3,0,1) of exp(-x)y sin(xy/z). सेव करें Save सेव्ड Saved A. pi pi B. 1 1 C. 0 0 D. Does not exist Does not exist उत्तर और व्याख्या देखें Show answer and explanation सही उत्तर: C Correct Answer: C The function is continuous at (3,0,1), since z is nonzero there. Direct substitution gives e^(-3) x 0 x sin(0)=0. The function is continuous at (3,0,1), since z is nonzero there. Direct substitution gives e^(-3) x 0 x sin(0)=0.
123 Advanced Mathematics • Digital Logic The inputs of a NAND gate are connected together. The resulting circuit is: The inputs of a NAND gate are connected together. The resulting circuit is: सेव करें Save सेव्ड Saved A. OR OR B. XOR XOR C. NOT NOT D. AND AND उत्तर और व्याख्या देखें Show answer and explanation सही उत्तर: C Correct Answer: C If both NAND inputs equal A, the output is (A.A)^c=A^c. Therefore the tied-input NAND acts as a NOT gate. If both NAND inputs equal A, the output is (A.A)^c=A^c. Therefore the tied-input NAND acts as a NOT gate.
124 Advanced Mathematics • Differential Equations Which of the following is a Cauchy-Euler equation? Which of the following is a Cauchy-Euler equation? सेव करें Save सेव्ड Saved A. x^3 d^3y/dx^3 - 4x^2 d^2y/dx^2 + 8x dy/dx - 8y = 4ln(x) x^3 d^3y/dx^3 - 4x^2 d^2y/dx^2 + 8x dy/dx - 8y = 4ln(x) B. x^3 d^3y/dx^3 - 2x d^2y/dx^2 + 18y = 2x^3 x^3 d^3y/dx^3 - 2x d^2y/dx^2 + 18y = 2x^3 C. x^2 d^4y/dx^4 - x^3 d^3y/dx^3 + x d^2y/dx^2 = e^x x^2 d^4y/dx^4 - x^3 d^3y/dx^3 + x d^2y/dx^2 = e^x D. d^2y/dx^2 + 2xy = x^3 d^2y/dx^2 + 2xy = x^3 उत्तर और व्याख्या देखें Show answer and explanation सही उत्तर: A Correct Answer: A In a Cauchy-Euler equation, the coefficient of the kth derivative is proportional to x^k. Only option A follows this pattern throughout. In a Cauchy-Euler equation, the coefficient of the kth derivative is proportional to x^k. Only option A follows this pattern throughout.
125 Advanced Mathematics • Differential Equations If y=y1(x) is not a known solution, then by removal of the first derivative, the general solution of y''+P(x)y'+Q(x)y=R(x) is obtained, where y1 is given by: If y=y1(x) is not a known solution, then by removal of the first derivative, the general solution of y''+P(x)y'+Q(x)y=R(x) is obtained, where y1 is given by: सेव करें Save सेव्ड Saved A. e^(integral P dx) e^(integral P dx) B. e^((1/2)integral P dx) e^((1/2)integral P dx) C. e^((-1/2)integral P dx) e^((-1/2)integral P dx) D. e^(-integral P dx) e^(-integral P dx) उत्तर और व्याख्या देखें Show answer and explanation सही उत्तर: C Correct Answer: C The standard substitution that removes the first-derivative term is y=u exp[-(1/2)integral P(x)dx]. Thus the multiplying factor y1 is option C. The standard substitution that removes the first-derivative term is y=u exp[-(1/2)integral P(x)dx]. Thus the multiplying factor y1 is option C.
126 Advanced Mathematics • Probability Let X and Y be two independent N(0,1) random variables. Find Cov(X+3Y, 5X-2Y). Let X and Y be two independent N(0,1) random variables. Find Cov(X+3Y, 5X-2Y). सेव करें Save सेव्ड Saved A. 1 1 B. -1 -1 C. 0 0 D. 11 11 उत्तर और व्याख्या देखें Show answer and explanation सही उत्तर: B Correct Answer: B Using independence and unit variances, Cov(X+3Y,5X-2Y)=5Var(X)-6Var(Y)=5-6=-1. Using independence and unit variances, Cov(X+3Y,5X-2Y)=5Var(X)-6Var(Y)=5-6=-1.
127 Advanced 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: सेव करें Save सेव्ड Saved 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.
128 Advanced Mathematics • Probability Which one of the following statements is false? Which one of the following statements is false? सेव करें Save सेव्ड Saved 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.
129 Advanced 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: सेव करें Save सेव्ड Saved 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).
130 Advanced 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? सेव करें Save सेव्ड Saved 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.