Tag: maths

Putting x=2, we get $2^2\left(2^2-1\right)= 12. So, x^2\left(x^2-1\right)$ is always divisible by 12

Suresh travelled 1200 km by air which form (2/5) of his trip. 
Then total distance of journey=$\frac{5}{2}\times 1200=3000 km$
Suresh travelled $\frac{1}{3}$of journey by car =$\frac{1}{3}\times 3000=1000 km$
Then suresh travelled distance by train =3000-1200-1000=800 km
If whole journey by train in 8 hours 
So speed of train=$\frac{3000}{8}$=375 km/Hr

Three lightships flash simultaneously at 6;00 a.m. The first lightship flashes every 12 seconds, the seconds lightship every 30 seconds and the third lightship every 66 seconds. 
Then we take common multiple of 12,30 or 66 
That is $2\times 6\times 5\times 11=660$
So  the three lightships next flash together=660 seconds =$\frac{660}{60}=11$mints
The time next flash is after 11 mints=6.11 A.M 

$\left(2^2 + 4^2 + 6^2 + ... + 20^2\right) = \left(1 \times 2\right)^2 + \left(2\times 2\right)^2 + \left(2 \times 3\right)^2 + ... + \left(2 \times 10\right)^2$

$= \left(2^2 \times 1^2\right) + \left(2^2 \times 2^2\right) + \left(2^2 \times 3^2\right) + ... + \left(2^2 \times 10^2\right)$

$= 2^2 \times \left [1^2 + 2^2 + 3^2 + ... + 10^2\right]$

$\left(1^2+2^2+3^2+....+n^2\right) = \frac{1}{6}n\left(n+1\right)\left(2n+1\right)$

$=\left(4\times\frac{1}{6}\times 10\times 11\times 21\right)$

$=\left(4\times 5\times 77\right)$

$= 1540$

In every 15 min there was a break of 5 min
If total 60 min break in all movie then number of commercial break =$\frac{60}{5}=12$
If 12 breaks are in movie then the duration of movie=$12\times 15+15=195    minutes$

Let, $\frac {2x}{3}=96$
$x=\frac {96\times 3}{2}=144$
$\therefore \frac {3}{4}\times 144=3\times 36=108$

$\displaystyle \left ( \left ( 3^2 \right )^{4/3} \div \left ( 3^3 \right )^{2/3} \right ) \times 3^{3/2}$
$= 3^{8/3} \div 3^2 \times 3^{3/2}$
$= \displaystyle 3^{8/3 - 2 + 3/2} $
$= \displaystyle  3^{ \displaystyle \frac{16 - 12 + 9}{6}}$
$= 3^{13/ 6}$

$1+9\times 10 = 91$

$\displaystyle\frac{0.24\times0.35}{0.02\times0.14\times0.15}= \frac{0.084}{0.00042}=200$

Let the smaller number is x