Question 1 |

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 1 Explanation:

Question 2 |

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 2 Explanation:

Question 3 |

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 3 Explanation:

Question 4 |

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 4 Explanation:

Question 5 |

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 5 Explanation:

Question 6 |

Let A and B be two nxn matrices over real numbers. Let rank(M) and det(M) denote the rank and determinant of a matrix M, respectively. Consider the following statements.

I. rank(AB)=rank (A)rank (B)

II. det(AB)=det(A)det(B)

III. rank(A+B) \leq rank (A) + rank (B)

IV. det(A+B) \leq det(A) + det(B)

Which of the above statements are TRUE?

I. rank(AB)=rank (A)rank (B)

II. det(AB)=det(A)det(B)

III. rank(A+B) \leq rank (A) + rank (B)

IV. det(A+B) \leq det(A) + det(B)

Which of the above statements are TRUE?

I and II only | |

I and IV only | |

II and III only | |

III and IV only |

Question 6 Explanation:

Question 7 |

Consider the functions

I. e^{-x}

II. x^2-\sin x

III. \sqrt{x^3+1}

Which of the above functions is/are increasing everywhere in [0,1] ?

I. e^{-x}

II. x^2-\sin x

III. \sqrt{x^3+1}

Which of the above functions is/are increasing everywhere in [0,1] ?

III only | |

II only | |

II and III only | |

I and III only |

Question 7 Explanation:

Question 8 |

Consider the following matrix:

\begin{bmatrix} 1 & 2 & 4 & 8\\ 1& 3 & 9 &27 \\ 1 & 4 & 16 &64 \\ 1 & 5 & 25 &125 \end{bmatrix}

The absolute value of the product of Eigenvalues of R is _________ .

\begin{bmatrix} 1 & 2 & 4 & 8\\ 1& 3 & 9 &27 \\ 1 & 4 & 16 &64 \\ 1 & 5 & 25 &125 \end{bmatrix}

The absolute value of the product of Eigenvalues of R is _________ .

10 | |

12 | |

25 | |

125 |

Question 8 Explanation:

Question 10 |

Compute \lim_{x \to 3}\frac{x^4-81}{2x^2-5x-3}

1 | |

53/12 | |

108/7 | |

Limit does not exist |

Question 10 Explanation:

There are 10 questions to complete.

Update Q184 as A instead of C