Number Operations and Properties

Given, $^{10}P _r\,= 5040$
$\Rightarrow \cfrac{10!}{(10-r)!} =5040 $

Exponent of $4$ is $80!$ is

We have:
$\dfrac { 56! }{ (50-r)! } .\dfrac { (51-r)! }{ 54! } =30800\ =>56.55.(51-r)=30800\ =>51-r=10=>r=41$

Therefore, $ _{ 2 }^{ r }{ P }= _{ 2 }^{ 41 }{ P }=41.40=1640$
Hence, (D) is correct.

Force, $F = mg$    where $m = 1$ gram  and $g = 981$ $cm/s^2$

In a subtraction minuend is a greater number and subtrahend is smaller number.
$\therefore$ greater number in the subtraction is called minuend.
Option A is correct.

As $8,08,08,079+1=8,08,08,080$

46)57804(1256
     46
     118
      92
      260
      230
        304
        276
           28

$\Rightarrow$  $3535\times 78$

Given number is $1,78,762$ 
First we see $10's$ place place value which is $62$ and it is more than $50$.
Now we will round up means to round off $762$ it will become $800$
So the correct answer is $1,78,800$.

Since the digit at hundreds place of $ 38,400 $ is $ 4 < 5 $,
Rounding of $ 38,400 $ to nearest thousand we get $ 38,000 $

The given no is $8974$ for rounding for nearest 1s will see the no at
ones place it is 4 . 

It is clear that  $888+88+8+8+8=1000$
 

(D) $27012=20000+7000+0+10+2$

$8t^3$ $- 27 = (2t)$$^3$ $- (3)$$^3$
$= (2t - 3)$$[(2t)$$^2$ $+ 2t \times 3 + 3$$^2$]
$= (2t - 3)(4t$^2$ + 6t + 9)$
$\because$ $(2t - 3)(4t$$^2$ $+ 6t + 9)$
$= (2t - 3)(a + 6t + 9)$
$\Rightarrow$ $4t$$^2$ $+ 6t + 9 = (a + 6t + 9)$
$\therefore$ $a = 4t$$^2$