Number Operations and Properties

We know the identity $a^{2}+b^{2}-2ab = (a-b)^{2}$

Given, $2l-3m= -1$

Squaring on both sides, we get 

$(2l-3m)^{2}= (-1)^{2}$

$\Rightarrow 4l^{2}+9m^{2}-12lm = 1$     .....Also given that $lm =20 $

$\Rightarrow 4l^{2}+9m^{2}-12 \times 20 = 1$

$\Rightarrow 4l^{2}+9m^{2}-240 = 1$

$\Rightarrow 4l^{2}+9m^{2}= 241$

Hence, option C is correct.

$(8x)^2+(6x)^2=d^2$
or $64x^2+36x^2=d^2$
or $100x^2=d^2$
or $d=\sqrt {100x^2}=10x$
$10x=200$
$\therefore x=\frac {200}{10}=20$
Hence, $8x\times 6x=8\times 20\times 6\times 20=19,200$

$2\pi r h=2\pi r^2$
$2\pi \times r\times 5=2\pi r^2$ or $r=5$

$R=gS-4, S=8, R=16$
or $16=g\times 8-4$ or $16+4=g\times 8$
or $g=\frac {20}{8}=\frac {5}{2}$
Therefore, when $S=10$
$R=\frac {5}{2}\times 10-4=21$

$3.14\times { 10 }^{ 6 }=3.14\times 1000000=3140000$

There are $10$ letters in the word 'LOGARITHMS'.
So, the number of $4$-letter word$=$Number of arrangements of $10$ letters, taken $4$ at a time
$=$ $^{10}P _4=5040$.

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.

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 $1,78,800$.