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.