Question 1 |

Let M^{4}=I, (where I denotes the identity matrix) and M \neq I, M^{2} \neq I and M^{3} \neq I. Then, for any natural number k,M^{-1} equals

M^{4k+1} | |

M^{4k+2} | |

M^{4k+3} | |

M^{4k} |

Question 1 Explanation:

Given that M^{4}=I or M^{4 k}=I or M^{4(k+1)}=I

\begin{aligned} \therefore \quad M^{-1} \times I & =M^{4(k+1)} \times M^{-1} \\ \therefore \quad M^{-1} & =M^{4 k+3}\end{aligned}

\begin{aligned} \therefore \quad M^{-1} \times I & =M^{4(k+1)} \times M^{-1} \\ \therefore \quad M^{-1} & =M^{4 k+3}\end{aligned}

Question 2 |

The second moment of a Poisson-distributed random variable is 2. The mean of the random variable is _______

0.5 | |

1 | |

2 | |

3 |

Question 2 Explanation:

In Poisson distribution,

Mean = First moment =\lambda

secondmoment =\lambda^{2}+\lambda

Given, that second moment is 2

\begin{array}{r} \lambda^{2}+\lambda=2 \\ \lambda^{2}+\lambda-2=0 \\ (\lambda+2)(\lambda-1)=0 \\ \lambda=1 \end{array}

Mean = First moment =\lambda

secondmoment =\lambda^{2}+\lambda

Given, that second moment is 2

\begin{array}{r} \lambda^{2}+\lambda=2 \\ \lambda^{2}+\lambda-2=0 \\ (\lambda+2)(\lambda-1)=0 \\ \lambda=1 \end{array}

Question 3 |

Given the following statements about a function f:\mathbb{R}\rightarrow \mathbb{R}, select the right option:

P: If f(x) is continuous at x=x_{0}, then it is also differentiable at x =x_{0}

Q: If f(x) is continuous at x = x_{0}, then it may not be differentiable at x= x_{0}.

R: If f(x) is differentiable at x= x_{0}, then it is also continuous at x= x_{0}.

P: If f(x) is continuous at x=x_{0}, then it is also differentiable at x =x_{0}

Q: If f(x) is continuous at x = x_{0}, then it may not be differentiable at x= x_{0}.

R: If f(x) is differentiable at x= x_{0}, then it is also continuous at x= x_{0}.

P is true, Q is false, R is false | |

P is false, Q is true, R is true | |

P is false, Q is true, R is false | |

P is true, Q is false, R is true |

Question 3 Explanation:

P: If f(x) is continuous at x=x_{0}, then it is also
differentiable at x=x_{0}

Q: If f(x) is continuous at x=x_{0}, then it may or may not be derivable at x=x_{0}

R: If f(x) is differentiable at x=x_{0}, then it is also continuous at x=x_{0}

P is false

Q is true

R is true

Option (B) is correct

Q: If f(x) is continuous at x=x_{0}, then it may or may not be derivable at x=x_{0}

R: If f(x) is differentiable at x=x_{0}, then it is also continuous at x=x_{0}

P is false

Q is true

R is true

Option (B) is correct

Question 4 |

Which one of the following is a property of the solutions to the Laplace equation: \bigtriangledown ^{2} f= 0?

The solutions have neither maxima nor minima anywhere except at the boundaries. | |

The solutions are not separable in the coordinates. | |

The solutions are not continuous | |

The solutions are not dependent on the boundary conditions |

Question 5 |

Consider the plot of f(x) versus x as shown below.

Suppose F(x)=\int_{-5}^{x}f(y)dy . Which one of the following is a graph of F(x) ?

Suppose F(x)=\int_{-5}^{x}f(y)dy . Which one of the following is a graph of F(x) ?

A | |

B | |

C | |

D |

Question 5 Explanation:

F^{\prime}(x)=f(x) which is density function

F^{\prime}(x)=f(x) \lt 0 when x \lt 0

\therefore \quad F(x) is decreasing for x \lt 0

F^{\prime}(x)=f(x) \gt 0

when x\gt 0

\therefore \quad F(x) is increasing for x\gt 0.

F^{\prime}(x)=f(x) \lt 0 when x \lt 0

\therefore \quad F(x) is decreasing for x \lt 0

F^{\prime}(x)=f(x) \gt 0

when x\gt 0

\therefore \quad F(x) is increasing for x\gt 0.

There are 5 questions to complete.