Question 1 |
Match the items in columns I and II.
P-1 Q-4 R-3 S-2 | |
P-1 Q-4 R-2 S-3 | |
P-1 Q-3 R-2 S-4 | |
P-4 Q-1 R-2 S-3 |
Question 1 Explanation:
(P) Gauss- Seidal method \rightarrow Linear algebraic equation
(Q) Forward Newton \rightarrow Gauss method Interpolation
(R) Runge-Kutta method \rightarrow Non-linear differential equations
(S) Trapezoidal Aule \rightarrow Numerical integration
(Q) Forward Newton \rightarrow Gauss method Interpolation
(R) Runge-Kutta method \rightarrow Non-linear differential equations
(S) Trapezoidal Aule \rightarrow Numerical integration
Question 2 |
The solution of the differential equation \frac{\mathrm{d} y}{\mathrm{d} x}+ 2xy = e^{-x^{2}}
with y(0)=1
is:
(1+x) e^{+x^{2}} | |
(1+x) e^{-x^{2}} | |
(1-x) e^{+x^{2}} | |
(1-x) e^{-x^{2}} |
Question 2 Explanation:
Given equation
\frac{d y}{d x}+2 x y=\sigma^{x^{2}}
Integrating factor
\begin{aligned} \text { I.F. } &=\mathrm{e}^{\int 2 x \;dx}=e^{x^{2}} \\ \text { Solution is } y x^{e^{x^{2}}}&=\int e^{-x^{2}} \cdot e^{x^{2}} d x \\ \therefore \quad y x^{ e^{x^{2}}}&=x+c\\ d t x &=0 \\ y &=1\\ \therefore \; \; c&=1 \\ y&=(1+x) e^{-x^{2}} \end{aligned}
\frac{d y}{d x}+2 x y=\sigma^{x^{2}}
Integrating factor
\begin{aligned} \text { I.F. } &=\mathrm{e}^{\int 2 x \;dx}=e^{x^{2}} \\ \text { Solution is } y x^{e^{x^{2}}}&=\int e^{-x^{2}} \cdot e^{x^{2}} d x \\ \therefore \quad y x^{ e^{x^{2}}}&=x+c\\ d t x &=0 \\ y &=1\\ \therefore \; \; c&=1 \\ y&=(1+x) e^{-x^{2}} \end{aligned}
Question 3 |
Let x denote a real number. Find out the INCORRECT statement.
S = \left \{ x : x \gt 3 \right \}
represents the set of all real numbers greater than 3 | |
S=\left \{ x : x^{2} \lt 0 \right \}
represents the empty set | |
S=\left \{ x : x \in A \; and \; x \in B \right \}
represents the union of set A and set B | |
S = \left \{ x : a \lt x \lt b \right \}
represents the set of all real numbers between a and b, where a and b are real numbers. |
Question 3 Explanation:
The incorrect statement is, S=\left \{ x : x \in A \; and \; x \in B \right \} represents the union of set A and set B
As and represents the intersection not union.
As and represents the intersection not union.
Question 4 |
A box contains 20 defective items and 80 non-defective items. If two items are selected at random without replacement, what will be the probability that both items are defective.?
\frac{1}{5}
| |
\frac{1}{25} | |
\frac{20}{99} | |
\frac{19}{495} |
Question 4 Explanation:
Probability of first item being defective is
P_{1}=\frac{20}{100}
Probability that second item being defective is
P_{2}=\frac{19}{99}
Probability that both are defective
P=P_{1} P_{2}=\frac{20}{100} \times \frac{19}{99}=\frac{19}{495}
P_{1}=\frac{20}{100}
Probability that second item being defective is
P_{2}=\frac{19}{99}
Probability that both are defective
P=P_{1} P_{2}=\frac{20}{100} \times \frac{19}{99}=\frac{19}{495}
Question 5 |
For a circular shaft of diameter d subjected to torque T, the maximum value of the shear stress is:
\frac{64T}{\pi d^{3}} | |
\frac{32T}{\pi d^{3}} | |
\frac{16T}{\pi d^{3}} | |
\frac{8T}{\pi d^{3}} |
Question 5 Explanation:
\begin{aligned} \frac{T}{J} &=\frac{\tau}{d / 2} \\ \Rightarrow \qquad \frac{T}{\frac{\pi}{32} d^{4}} &=\frac{\tau}{d / 2} \\ \therefore \qquad & \tau=\frac{16 T}{\pi d^{3}} \end{aligned}
There are 5 questions to complete.