The total income of $A$ and $B$ is greater than the total income of $C$ and $D$. The total income of $A$ and $C$ is equal to the total income of $B$ and $D$. Furthermore,the income of $A$ is equal to half the total income of $B$ and $D$. Which of the following statements is $NOT$ true?

$A$ training college has to conduct a refresher course for teachers of seven different subjects: Mechanics,Psychology,Philosophy,Sociology,Economics,Science,and Engineering from $22^{nd}$ July to $29^{th}$ July.
$1.$ The course should start with Psychology.
$2.$ $23^{rd}$ July,being Sunday,should be a holiday.
$3.$ Science subject should be on the previous day of the Engineering subject.
$4.$ The course should end with Mechanics.
$5.$ Philosophy should be immediately after the holiday.
$6.$ There should be a gap of one day between Economics and Engineering.
The refresher course will start with which one of the following subjects?

Hitesh is richer than Jaya,whereas Mohan is richer than Pritam. Lalit is as rich as Jaya. Amit is richer than Hitesh. What conclusion can be definitely drawn from the above statements?

In an examination,Raj got more marks than Mukesh but not as many as Priya. Priya got more marks than Gaurav and Kavita. Gaurav got less marks than Mukesh but his marks are not the lowest in the group. Who is second in the descending order of marks?

$(i)$ Seven books are placed one above the other in a particular way.
$(ii)$ History book is placed exactly above Civics book.
$(iii)$ Geography book is fourth from the bottom and English book is fifth from the top.
$(iv)$ There are two books in between Civics and Economics books.
How many books are there between Civics and Science books? To answer this question,which other extra information is required,if any,from the following?

Six persons $A, B, C, D, E$ and $F$ are sitting in two rows,three in each. $E$ is not at the end of any row.
$D$ is second to the left of $F$.
$C$,the neighbor of $E$,is sitting diagonally opposite to $D$.
$B$ is the neighbor of $F$.
Which of the following are sitting diagonally opposite to each other?