131 Advanced Mathematics • Real Analysis Let f(x)=1/x^2-5 be defined on [1,4]. Which of the following is true? Let f(x)=1/x^2-5 be defined on [1,4]. Which of the following is true? सेव करें Save सेव्ड Saved A. There exists a point x in [1,4] such that f is not differentiable at x. There exists a point x in [1,4] such that f is not differentiable at x. B. The function has a tangent line between 1 and 4 with slope -1/64. The function has a tangent line between 1 and 4 with slope -1/64. C. The function has a tangent line between 1 and 4 with slope 3/2. The function has a tangent line between 1 and 4 with slope 3/2. D. The function has a tangent line between 1 and 4 with slope -5/16. The function has a tangent line between 1 and 4 with slope -5/16. उत्तर और व्याख्या देखें Show answer and explanation सही उत्तर: D Correct Answer: D The secant slope is [f(4)-f(1)]/(4-1)=[1/16-1]/3=-5/16. By the Mean Value Theorem, some tangent has this slope. The secant slope is [f(4)-f(1)]/(4-1)=[1/16-1]/3=-5/16. By the Mean Value Theorem, some tangent has this slope.
132 Advanced Mathematics • Dynamics The moment of inertia of an ellipse of mass M and semiaxes a and b about a tangent is, where p is the perpendicular from the centre to the tangent: The moment of inertia of an ellipse of mass M and semiaxes a and b about a tangent is, where p is the perpendicular from the centre to the tangent: सेव करें Save सेव्ड Saved A. (5M/4)p^2 (5M/4)p^2 B. (5M/6)p^2 (5M/6)p^2 C. (5M/2)p^2 (5M/2)p^2 D. (5M/3)p^2 (5M/3)p^2 उत्तर और व्याख्या देखें Show answer and explanation सही उत्तर: A Correct Answer: A Using the moment of inertia of the elliptic lamina about a parallel central axis together with the parallel-axis theorem gives (5M/4)p^2 for the stated tangent. Using the moment of inertia of the elliptic lamina about a parallel central axis together with the parallel-axis theorem gives (5M/4)p^2 for the stated tangent.
133 Advanced Mathematics • Statics Two equal uniform rods AB and AC, each of length 2b, are freely joined at A and rest on a smooth vertical circle of radius a. If 2theta is the angle between them, then: Two equal uniform rods AB and AC, each of length 2b, are freely joined at A and rest on a smooth vertical circle of radius a. If 2theta is the angle between them, then: सेव करें Save सेव्ड Saved A. a sin^3(theta)=b cos(theta) a sin^3(theta)=b cos(theta) B. b sin^2(theta)=a cos(theta) b sin^2(theta)=a cos(theta) C. a sin^2(theta)=b cos(theta) a sin^2(theta)=b cos(theta) D. b sin^3(theta)=a cos(theta) b sin^3(theta)=a cos(theta) उत्तर और व्याख्या देखें Show answer and explanation सही उत्तर: D Correct Answer: D Applying the equilibrium condition to the symmetric rod configuration gives b sin^3(theta)=a cos(theta), as marked in the source paper. Applying the equilibrium condition to the symmetric rod configuration gives b sin^3(theta)=a cos(theta), as marked in the source paper.
134 Advanced Mathematics • Dynamics A particle starts from the origin. The components of its velocity parallel to the coordinate axes at time t are 2t+3 and 4t. The path travelled by the particle is: A particle starts from the origin. The components of its velocity parallel to the coordinate axes at time t are 2t+3 and 4t. The path travelled by the particle is: सेव करें Save सेव्ड Saved A. 4x^2-y^2-4xy+18y=0 4x^2-y^2-4xy+18y=0 B. x^2+y^2+xy-12y=0 x^2+y^2+xy-12y=0 C. 3x^2+y^2+4xy-18y=0 3x^2+y^2+4xy-18y=0 D. 4x^2+y^2-4xy-18y=0 4x^2+y^2-4xy-18y=0 उत्तर और व्याख्या देखें Show answer and explanation सही उत्तर: D Correct Answer: D Integration gives x=t^2+3t and y=2t^2. Eliminating t from these relations yields 4x^2+y^2-4xy-18y=0. Integration gives x=t^2+3t and y=2t^2. Eliminating t from these relations yields 4x^2+y^2-4xy-18y=0.
135 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.
136 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.
137 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.
138 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.
139 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.
140 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.