Question
Solving an integer programming problem by rounding off answers obtained by solving it as a linear programming problem (using simplex), we find that.

Answer

  1. The value of the objective function for a maximization problem will likely be less than that for the simplex solution.
Solution:
Solving an integer programming problem by rounding off answers obtained by solving it as a linear programming problem, we find that the value of the objective function for a maximization problem will likely be less than that for the simplex solution.

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Let $f : R \rightarrow R$ be defined as $\text{f(x)}=\begin{cases}2\text{x},&\text{if x}>3\\\text{x}^2,&\text{if }1<\text{x}\leq3\\3\text{x},&\text{if x}\leq1\end{cases}.$ Then, find $f(-1) + f(2) + f(4):$
$\int\frac{\cos2\text{x dx}}{(\sin\text{x}+\cos\text{x})^2}=$
  1. $-\frac{1}{\sin\text{x}+\cos\text{x}}+\text{c}$
  2. $\log|\sin\text{x}+\cos\text{x }|+\text{c}$
  3. $\log|\sin\text{x}-\cos\text{x }|+\text{c}$
  4. $\frac{1}{(\sin\text{x}+\cos\text{x})^2}$
Choose the correct answer from the given four options.
$\text{P}\Big(\frac{\text{B}}{\text{A}}\Big)$ is equal to:
  1. $\frac{1}{5}$
  2. $\frac{3}{10}$
  3. $\frac{1}{2}$
  4. $\frac{3}{5}$
Consider the following statements on a set $A=\{1,2,3\}$ :
(i) $\quad R=\{(1,1),(2,2)\}$ is a reflexive relation on $A$.
(ii) $R=\{(3,3)\}$ is symmetric and transitive but not a reflexive relation on $A$.
Which of the statements given above is/are correct?
$\int\frac{\text{x}^3}{\text{x}+1}\text{ dx}$ is equal to:
  1. $\text{x}+\frac{\text{x}^2}{2}+\frac{\text{x}^3}{3}-\log|1-\text{x}|+\text{C}$
  2. $\text{x}+\frac{\text{x}^2}{2}-\frac{\text{x}^3}{3}-\log|1-\text{x}|+\text{C}$
  3. $\text{x}-\frac{\text{x}^2}{2}-\frac{\text{x}^3}{3}-\log|1-\text{x}|+\text{C}$
  4. $\text{x}-\frac{\text{x}^2}{2}+\frac{\text{x}^3}{3}-\log|1-\text{x}|+\text{C}$
Let $A=\{a, b, c\}$ and let $R=\{(a, a),(a, b)$, $(b, a)\}$. Then, $R$ is
The cartesian equation of the line which passes through the point $(-2,4,-5)$ and parallel to the line given by $\frac{x+3}{3}=\frac{y-4}{5}=\frac{z+8}{6}$ is
If $\begin{vmatrix}\text{a}&\text{p}&\text{x}\\\text{b}&\text{q}&\text{y}\\\text{c}&\text{r}&\text{z}\end{vmatrix}=16,$ then the value of $\begin{vmatrix}\text{p}+\text{x}&\text{a}+\text{x}&\text{a}+\text{p}\\\text{q}+\text{y}&\text{b}+\text{y}&\text{b}+\text{q}\\\text{r}+\text{z}&\text{c}+\text{z}&\text{c}+\text{r}\end{vmatrix}$ is:
  1. 4
  2. 8
  3. 16
  4. 32
The differential coefficient of $\text{f}(\log\text{x})$ w.r.t. x, where $\text{f(x)}=\log\text{x}$ is:
  1. $\frac{\text{x}}{\log\text{x}}$
  2. $\frac{\log\text{x}}{\text{x}}$
  3. $(\text{x}\log\text{x})^{-1}$
  4. $\text{None of these.}$
In each of the following, choose the correct answer:
The probability that a student is not a swimmer is $\frac{1}{5}.$ Then the probability that out of five students, four are swimmers is
$\ ^5\text{C}_\text{4}\Big(\frac{4}{5}\Big)^4\frac{1}{5}$
$\Big(\frac{4}{5}\Big)^4\frac{1}{5}$
$\ ^5\text{C}_1\frac{1}{5}\Big(\frac{4}{5}\Big)^4$
None of these