Question 1 |

For what value of p the following set of equations will have no solution?

2x+3y=5

3x+py=10

2x+3y=5

3x+py=10

2.5 | |

5.25 | |

4.5 | |

None |

Question 1 Explanation:

Given system of equations has no solution if the lines are parallel i.e., their slopes are equal

\begin{aligned} \frac{2}{3}&=\frac{3}{p} \\ \Rightarrow \;\; p&=4.5 \end{aligned}

\begin{aligned} \frac{2}{3}&=\frac{3}{p} \\ \Rightarrow \;\; p&=4.5 \end{aligned}

Question 2 |

The integral \int_{x_1}^{x_2}x^{2}dx with {x_1} \gt {x_2} \gt 0 is evaluated analytically as well as numerically using a single application of the trapezoidal rule. If I is the exact value of the integral obtained analytically and J is the approximate value obtained using the trapezoidal rule, which of the following statements is correct about their relationship?

J \gt I | |

J \lt I | |

J=I | |

Insufficient data to determine the relationship |

Question 2 Explanation:

Exact value is computed by integration which follows the exact shape of graph while computing the area.

Whereas, in Trapezoidal rule, the lines joining each points are considered straight lines which is not the exact variation of graph all the time like as shown figure.

\therefore J\gt I

OR

\text{Error } =-\frac{h^{3}}{12}{f}''\left ( \Im \right )\times n_{i}

Here, f\left ( x \right )=x^{2}

or, {f}'\left ( x \right )=2x

or, {f}''\left ( x \right )=2\gt 0

Since {f}''\left ( x \right ) is positive, the error is negative.

Since error = exact-approximate.

=I-J

and since error is negative in this case J\gt I is true.

Question 3 |

Consider the following probability mass function (p.m.f) of a random variable X :

p(x,q)=\begin{cases} q & \text{ if } X=0\\ 1-q& \text{ if } X=1 \\ 0 & \text otherwise \end{cases}

If q = 0.4, the variance of X is _______.

p(x,q)=\begin{cases} q & \text{ if } X=0\\ 1-q& \text{ if } X=1 \\ 0 & \text otherwise \end{cases}

If q = 0.4, the variance of X is _______.

0.15 | |

0.24 | |

1.25 | |

2.52 |

Question 3 Explanation:

Given, \; \; \; \; q=0.4

Required value =V\left ( X \right )=E\left ( X^{2} \right )-\left [ E\left ( X \right ) \right ]^{2}

E\left ( X \right )=\sum_{i}^{}\, X_{i}p_{i}=0\times 0.4+1\times 0.6=0.6

E\left ( X^{2} \right )=\sum_{i}^{}\, X_{i}^{2}p_{i}=0^{2}\times 0.4+1^{2}\times 0.6=0.6

\therefore \; \; \; V\left ( X \right )=E\left ( X^{2} \right )-\left [ E\left ( X \right ) \right ]^{2}=0.6-0.36=0.24

Required value =V\left ( X \right )=E\left ( X^{2} \right )-\left [ E\left ( X \right ) \right ]^{2}

E\left ( X \right )=\sum_{i}^{}\, X_{i}p_{i}=0\times 0.4+1\times 0.6=0.6

E\left ( X^{2} \right )=\sum_{i}^{}\, X_{i}^{2}p_{i}=0^{2}\times 0.4+1^{2}\times 0.6=0.6

\therefore \; \; \; V\left ( X \right )=E\left ( X^{2} \right )-\left [ E\left ( X \right ) \right ]^{2}=0.6-0.36=0.24

Question 4 |

Workability of concrete can be measured using slump, compaction factor and Vebe time. Consider the following statements for workability of concrete:

(i) As the slump increases, the Vebe time increases

(ii) As the slump increases, the compaction factor increases

Which of the following is TRUE?

(i) As the slump increases, the Vebe time increases

(ii) As the slump increases, the compaction factor increases

Which of the following is TRUE?

Both (i) and (ii) are True | |

Both (i) and (ii) are False | |

(i) is True and (ii) is False | |

(i) is False and (ii) is True |

Question 4 Explanation:

As slump increase, water content increases means
workability increases.

Compaction factor

=\frac{\text { Weight of partially compacted concrete }}{\text { Weight of fully compacted concrete }}

\therefore Compaction factor will also increase.

The time required for complete remoulding in seconds is considered as a measure of workability and is expressed as the number of Vee-Bee seconds.

\therefore As workability increases, Vee-Bee time decreases.

Compaction factor

=\frac{\text { Weight of partially compacted concrete }}{\text { Weight of fully compacted concrete }}

\therefore Compaction factor will also increase.

The time required for complete remoulding in seconds is considered as a measure of workability and is expressed as the number of Vee-Bee seconds.

\therefore As workability increases, Vee-Bee time decreases.

Question 5 |

Consider the following statements for air-entrained concrete.

(i) Air-entrainment reduces the water demand for a given level of workability

(ii) Use of air-entrained concrete is required in environments where cyclic freezing and thawing is expected

Which of the following is TRUE?

(i) Air-entrainment reduces the water demand for a given level of workability

(ii) Use of air-entrained concrete is required in environments where cyclic freezing and thawing is expected

Which of the following is TRUE?

Both (i) and (ii) are True | |

Both (i) and (ii) are False | |

(i) is True and (ii) is False | |

(i) is False and (ii) is True |

Question 5 Explanation:

An air-entraining agent introduces air in the form
of bubbles that occupy upto 5% of the volume of
concrete distributed uniformly throughout cement
paste. Thus, for the same amount of water/cement
ratio, we get higher workability.

There are 5 questions to complete.