Question 1 |

Which of the following is/are the eigenvector(s) for the matrix given below?

\begin{pmatrix} -9 &-6 &-2 &-4 \\ -8& -6 & -3 & -1 \\ 20 & 15 & 8 & 5 \\ 32& 21& 7&12 \end{pmatrix}

\begin{pmatrix} -9 &-6 &-2 &-4 \\ -8& -6 & -3 & -1 \\ 20 & 15 & 8 & 5 \\ 32& 21& 7&12 \end{pmatrix}

**MSQ**\begin{pmatrix}
-1\\
1\\
0\\
1
\end{pmatrix} | |

\begin{pmatrix}
1\\
0\\
-1\\
0
\end{pmatrix} | |

\begin{pmatrix}
-1\\
0\\
2\\
2
\end{pmatrix} | |

\begin{pmatrix}
0\\
1\\
-3\\
0
\end{pmatrix} |

Question 1 Explanation:

Question 2 |

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 as

L= \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?

\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 as

L= \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?

L_{32}=2,U_{33}=-\frac{1}{2},x_1=-1 | |

L_{32}=2,U_{33}=2,x_1=-1 | |

L_{32}=-\frac{1}{2},U_{33}=2,x_1=0 | |

L_{32}=-\frac{1}{2},U_{33}=-\frac{1}{2},x_1=0 |

Question 2 Explanation:

Question 3 |

The value of the following limit is _____

\lim_{x \to 0+} \frac{ \sqrt{x}}{1-e^{2 \sqrt{x}}}

\lim_{x \to 0+} \frac{ \sqrt{x}}{1-e^{2 \sqrt{x}}}

-0.5 | |

0.5 | |

0 | |

1 |

Question 3 Explanation:

Question 4 |

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?

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?

Statement 1 is correct and Statement 2 is wrong. | |

Statement 1 is wrong and Statement 2 is correct. | |

Both Statement 1 and Statement 2 are correct. | |

Both Statement 1 and Statement 2 are wrong. |

Question 4 Explanation:

Question 5 |

For two n-dimensional real vectors P and Q, the operation s(P,Q) is defined as follows:

s(P,Q) = \displaystyle \sum_{i=1}^n (P[i] \cdot Q[i])

Let \mathcal{L} be a set of 10-dimensional non-zero real vectors such that for every pair of distinct vectors P,Q \in \mathcal{L}, s(P,Q)=0. What is the maximum cardinality possible for the set \mathcal{L}?

s(P,Q) = \displaystyle \sum_{i=1}^n (P[i] \cdot Q[i])

Let \mathcal{L} be a set of 10-dimensional non-zero real vectors such that for every pair of distinct vectors P,Q \in \mathcal{L}, s(P,Q)=0. What is the maximum cardinality possible for the set \mathcal{L}?

9 | |

10 | |

11 | |

100 |

Question 5 Explanation:

Question 6 |

Suppose that f: \mathbb{R} \rightarrow \mathbb{R} is a continuous function on the interval [-3,3] and a differentiable function in the interval (-3,3) such that for every x in the interval, f'(x) \leq 2. If f(-3) =7, then f(3) is at most __________

19 | |

32 | |

11 | |

54 |

Question 6 Explanation:

Question 7 |

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 _______

1 | |

2 | |

3 | |

4 |

Question 7 Explanation:

Question 8 |

Consider the following matrix.

\begin{pmatrix} 0 & 1 & 1 & 1\\ 1& 0& 1 & 1\\ 1& 1 & 0 & 1 \\1 & 1 & 1 & 0 \end{pmatrix}

The largest eigenvalue of the above matrix is __________.

\begin{pmatrix} 0 & 1 & 1 & 1\\ 1& 0& 1 & 1\\ 1& 1 & 0 & 1 \\1 & 1 & 1 & 0 \end{pmatrix}

The largest eigenvalue of the above matrix is __________.

1 | |

3 | |

4 | |

6 |

Question 8 Explanation:

Question 9 |

Consider the following expression.

\lim_{x\rightarrow-3}\frac{\sqrt{2x+22}-4}{x+3}

The value of the above expression (rounded to 2 decimal places) is ___________.

\lim_{x\rightarrow-3}\frac{\sqrt{2x+22}-4}{x+3}

The value of the above expression (rounded to 2 decimal places) is ___________.

0.25 | |

0.45 | |

0.75 | |

0.85 |

Question 9 Explanation:

Question 10 |

If x+2 y=30,then \left(\frac{2 y}{5}+\frac{x}{3}\right)+\left(\frac{x}{5}+\frac{2 y}{3}\right) will be equal to

8 | |

16 | |

18 | |

20 |

Question 10 Explanation:

There are 10 questions to complete.

Update Q184 as A instead of C

The answer is D

I think Q160 is broken (all options are becoming wrong)