If x/(b -c)(b+c—2a)= y/(c -a)(c+a-2b)= z/(a -b)(a+b-2c) then the value of (x + y * z) is?

If x/(b -c)(b+c—2a)= y/(c -a)(c+a-2b)= z/(a -b)(a+b-2c)

then the value of (x + y * z) is?

Solution :Let all three ratios are equal to a constant value of K
Then it goes like 

1. X=K{(b -c)(b+c—2a)}
2. Y=K{(c -a)(c+a-2b)}
3. C=K{(a -b)(a+b-2c)}

Then adding X,Y and Z would be =K{(b -c)(b+c—2a)+(c -a)(c+a-2b)+(a -b)(a+b-2c)}=KX0=0(Answer).

If α+β+γ=1, α^2+β^2+γ^2=6, α^3+β^3+γ^3=8 then α^4+β^4+γ^4=?

If α+β+γ=1, α²+β²+γ²=6, α³+β³+γ³=8 then α⁴+β⁴+γ⁴=?

Solution: If x, y and z replaces α β and γ for easy writing, then

(X²+y²+z²)=x⁴+y⁴+z⁴+2{(xy)²+(yz)²+(zx)²}   -----(1)

Therefore, solving with given values it goes like x⁴+y⁴+z⁴=36-2{(xy)²+(yz)²+(zx)²}   -(2)

(X+Y+Z)²=X²+y²+z²+2XY+2YZ+2ZX-------(3)

Therefore, solving with given values it goes like (xy)²+(yz)²+(zx)²=1-4xyz-----(4)

X³+y³+z³-3xyz=(x+y+z) (X²+y²+z²-XY-YZ-ZX) -----(5)

Therefore, solving with given values it goes like xyz=-2---------(7)

Then using (7) in (4 ) and followed by solving 2 by using (4) , it goes like  x⁴+y⁴+z⁴=18 (Answer).

The number of numbers from 1 to 200 which are divisible by neither 3 nor 7 is?

Answer: You can do this by A.P.

First, let’s see how many numbers are divisible by 3.

It goes like this: 3,6,9,12.......198

By using the formula, we get, a+(n-1) d = an; 3+(n-1)3 = 198

n=66

So the total no. of numbers that are divisible by 3 is 66.

Now the numbers that are divisible by 7.

It goes like, 7,14,21....196

Use the formula again a+(n-1) d = an; 7+(n-1)7 = 196

n=28

So the total no. of numbers that are divisible by 7 is 28.

Now the numbers that are divisible with both 3 and 7.

It goes like: 21, 42, 63....189

Use the formula a+(n-1) d = an; 21+(n-1)21 = 189

n=9

So the total no. of numbers that are divisible by both 3 and 7 are 9.

Finally, from 1 to 200, there are 200 numbers. We have to remove those numbers that are either divisible by 3 or 7. Here it goes, 200 - (66 + 28) = 106.

Now certain numbers are repeated in both 3 and 7, like 21,42,63 etc. These numbers have been removed twice. So we can add these numbers once. So the final answer is 106 + 9 = 115. Hope it helps!