# GATE CE 2015 SET-1

 Question 1
For what value of p the following set of equations will have no solution?
2x+3y=5
3x+py=10
 A 2.5 B 5.25 C 4.5 D None
Engineering Mathematics   Linear Algebra
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}
 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?
 A $J \gt I$ B $J \lt I$ C J=I D Insufficient data to determine the relationship
Engineering Mathematics   Numerical Methods
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 _______.
 A 0.15 B 0.24 C 1.25 D 2.52
Engineering Mathematics   Probability and Statistics
Question 3 Explanation:
Given, $\; \; \; \; q=0.4$ Required value $=V\left ( X \right )=E\left ( X^{2} \right )-\left [ E\left ( X \right ) \right ]^{2}$
$E\left ( X \right )=\sum_{i}^{}\, X_{i}p_{i}=0\times 0.4+1\times 0.6=0.6$
$E\left ( X^{2} \right )=\sum_{i}^{}\, X_{i}^{2}p_{i}=0^{2}\times 0.4+1^{2}\times 0.6=0.6$
$\therefore \; \; \; V\left ( X \right )=E\left ( X^{2} \right )-\left [ E\left ( X \right ) \right ]^{2}=0.6-0.36=0.24$
 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?
 A Both (i) and (ii) are True B Both (i) and (ii) are False C (i) is True and (ii) is False D (i) is False and (ii) is True
RCC Structures   Working Stress and Limit State Method
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.
 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?
 A Both (i) and (ii) are True B Both (i) and (ii) are False C (i) is True and (ii) is False D (i) is False and (ii) is True
RCC Structures   Working Stress and Limit State Method
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.