Engineering Mathematics
Q43.
A point on a curve is said to be an extremum if it is a local minimum or a local maximum. The number of distinct extrema for the curve 3x^{4}-16x^{3}+24x^{2}+37 isQ44.
Let x be a random variable following normal distribution with mean +1 and variance 4. Let Y be another normal variable with mean -1 and variance unknown. If P (x \leq -1) = P (Y \geq2) , the standard deviation of Y isQ45.
Which one of the following is a closed form expression for the generating function of the sequence \left \{ a_{n} \right \}, where a_{n}=2n+3 for all n = 0, 1, 2,...?Q46.
Let A=\begin{bmatrix} 1 & 2 & 3 &4 \\ 4& 1& 2 &3 \\ 3& 4 & 1 &2 \\ 2 &3 &4 &1 \end{bmatrix} and B=\begin{bmatrix} 3& 4 & 1 &2 \\ 4& 1& 2 &3 \\ 1 & 2 & 3 &4 \\ 2 &3 &4 &1 \end{bmatrix}Let det(A) and det(B) denote the determinants of the matrices A and B, respectively. Which one of the options given below is TRUE?Q47.
Let A be the adjacency matrix of the graph with vertices {1, 2, 3, 4, 5}.Let \lambda _1,\lambda _2,\lambda _3,\lambda _4,\; and \; \lambda _5 be the five eigenvalues of A. Note that these eigenvalues need not be distinct.The value of \lambda _1+\lambda _2+\lambda _3+ \lambda _4+ \lambda _5 = _____Q48.
Consider the following two statements with respect to the matrices A_{m \times n},B_{n \times m},C_{n \times n} \text{ and }D_{n \times n}, Statement 1: tr(AB) = tr(BA) Statement 2: tr(CD) = tr(DC) wheretr() represents the trace of a matrix. Which one of the following holds?Q49.
Suppose that P is a 4x5 matrix such that every solution of the equation Px=0 is a scalar multiple of \begin{bmatrix} 2 & 5 & 4 &3 & 1 \end{bmatrix}^T. The rank of P is _______Q50.
Consider solving the following system of simultaneous equations using LU decomposition. \begin{aligned} x_1+x_2-2x_3&=4 \\ x_1+3x_2-x_3&=7 \\ 2x_1+x_2-5x_3&=7 \end{aligned}where L and U are denoted asL= \begin{bmatrix} L_{11} & 0 & 0 \\ L_{21}& L_{22} & 0 \\ L_{31} & L_{32} & L_{33} \end{bmatrix}, U= \begin{bmatrix} U_{11} & U_{12} & U_{13} \\ 0& U_{22} & U_{23} \\ 0 & 0 & U_{33} \end{bmatrix}Which one of the following is the correct combination of values for L32, U33, and x_1?