Question 1 |
Four red balls, four green balls and four blue balls are put in a box. Three balls are pulled out of the box at random one after another without replacement. The probability that all the three balls are red is
\frac{1}{72} | |
\frac{1}{55} | |
\frac{1}{36} | |
\frac{1}{27} |
Question 1 Explanation:
Probability that all the three balls are red is
\begin{array}{l} =\mathrm{R} \cdot \mathrm{R} \cdot \mathrm{R} \\ =\frac{4}{12} \times \frac{3}{11} \times \frac{2}{10}=\frac{24}{1320}=\frac{1}{55} \end{array}
Question 2 |
The rank of the matrix \begin{bmatrix} -4 & 1 & -1 \\ -1 & -1 & -1 \\ 7 & -3 & 1 \end{bmatrix} is
1 | |
2 | |
3 | |
4 |
Question 2 Explanation:
\begin{aligned} &\left[\begin{array}{ccc}-4 & 1 & -1 \\-1 & -1 & -1 \\7 & -3 & 1\end{array}\right]\\ R_{1} \longleftrightarrow R_{2} \qquad&{\left[\begin{array}{ccc}-1 & -1 & -1 \\-4 & 1 & -1 \\7 & -3 & 1\end{array}\right]} \\ R_{2}-4 R_{1}, R_{3}+7 R_{1} \qquad&{\left[\begin{array}{ccc}-1 & -1 & -1 \\0 & 5 & 3 \\0 & -10 & -6\end{array}\right]} \\ R_{3}+2 R_{2} \qquad &{\left[\begin{array}{ccc}-1 & -1 & -1 \\0 & 5 & 3 \\0 & 0 & 0\end{array}\right]} \end{aligned}
No. of non zero rows =2
rank=2
No. of non zero rows =2
rank=2
Question 3 |
According to the Mean Value Theorem, for a continuous function f(x) in the interval [a, b], there exists a value \xi in this interval such that \int_{a}^{b}f(x)dx =
f(\xi)(b-a) | |
f(b)(\xi-a) | |
f(a)(b-\xi) | |
0 |
Question 3 Explanation:
\int_{a}^{b} f(\xi) d x=f(\xi)(b-a)
Question 4 |
F(z) is a function of the complex variable z = x + iy given by
F(z)=iz+kRe(z)+ i \; lm(z)
For what value of k will F(z) satisfy the Cauchy-Riemann equations?
F(z)=iz+kRe(z)+ i \; lm(z)
For what value of k will F(z) satisfy the Cauchy-Riemann equations?
0 | |
1 | |
-1 | |
y |
Question 4 Explanation:
\begin{aligned} F(z) &=i z+k \operatorname{Re}(z)+i \operatorname{Im}(z) \\ u+i v &=i(x+i y)+k x+i y \\ u+i v &=k x-y+i(x+y) \\ u &=k x-y, v=x+y \\ u_{x} &=k, u_{y}=-1 \\ V &=x+y \\ v_{x} &=1 \\ v_{y} &=1 \\ u_{x} &=v_{y} \\ k &=1 \end{aligned}
Question 5 |
A bar of uniform cross section and weighing 100 N is held horizontally using two massless and inextensible strings S1 and S2 as shown in the figure.
The tension of the strings are
The tension of the strings are
T_{1} = 100 N \; and \; T_{2} = 0 N | |
T_{1} = 0 N \; and \; T_{2} = 100 N | |
T_{1} = 75 N \; and \; T_{2} = 25 N | |
T_{1} = 25 N \; and \; T_{2} = 75 N |
Question 5 Explanation:
\begin{aligned} T_{1}+T_{2} &=100 \mathrm{N} \qquad \ldots(i)\\ \Sigma M_{A} &=0 \\ T_{2} \cdot \frac{L}{2} &=100 \times \frac{L}{2} \\ \therefore \qquad T_{2} &=100 \mathrm{N} \\ T_{1} &=0 \mathrm{N} \end{aligned}
There are 5 questions to complete.