(e) (a? + b2 + c?) x (a – b + c)
Answers
Answer:
How can you say einstein was not a bad people?
Answer:
Algebra
1. (a + b)2 = a2 + 2ab + b2; a2 + b2 = (a + b)2 − 2ab
2. (a − b)2 = a2 − 2ab + b2; a2 + b2 = (a − b)2 + 2ab
3. (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
4. (a + b)3 = a3 + b3 + 3ab(a + b); a3 + b3 = (a + b)3 − 3ab(a + b)
5. (a − b)3 = a3 − b3 − 3ab(a − b); a3 − b3 = (a − b)3 + 3ab(a − b)
6. a2 − b2 = (a + b)(a − b)
7. a3 − b3 = (a − b)(a2 + ab + b2)
8. a3 + b3 = (a + b)(a2 − ab + b2)
9. an − bn = (a − b)(an−1 + an−2b + an−3b2 + ··· + bn−1)
10. an = a.a.a . . . n times
11. am.an = am+n
12. am
an = am−n if m>n
= 1 if m = n
= 1
an−m if m<n; a ∈ R, a 6= 0
13. (am)n = amn = (an)m
14. (ab)n = an.bn
15. a
b
n
= an
bn
16. a0 = 1 where a ∈ R, a 6= 0
17. a−n = 1
an , an = 1
a−n
18. ap/q = √q ap
19. If am = an and a 6= ±1, a 6= 0 then m = n
20. If an = bn where n 6= 0, then a = ±b
21. If √x, √y are quadratic surds and if a + √x = √y, then a = 0 and x = y
22. If √x, √y are quadratic surds and if a+ √x = b+ √y then a = b and x = y
23. If a, m, n are positive real numbers and a 6= 1, then loga mn = loga m+loga n
24. If a, m, n are positive real numbers, a 6= 1, then loga
m
n
= loga m−loga n
25. If a and m are positive real numbers, a 6= 1 then loga mn = n loga m
26. If a, b and k are positive real numbers, b 6= 1, k 6= 1, then logb a = logk a
logk b
27. logb a = 1
loga b where a, b are positive real numbers, a 6= 1, b 6= 1
28. if a, m, n are positive real numbers, a 6= 1 and if loga m = loga n, then
m = n
29. if a + ib = 0 where i = √−1, then a = b = 0
30. if a + ib = x + iy, where i = √−1, then a = x and b = y
31. The roots of the quadratic equation ax2+bx+c = 0; a 6= 0 are −b ± √
b2 − 4ac
2a
The solution set of the equation is (
−b + √
∆
2a ,
−b − √
∆
2a
)
where ∆ = discriminant = b2 − 4ac
32. The roots are real and distinct if ∆ > 0.
33. The roots are real and coincident if ∆ = 0.
34. The roots are non-real if ∆ < 0.
35. If α and β are the roots of the equation ax2 + bx + c = 0, a 6= 0 then
i) α + β = −b
a = − coeff. of x
coeff. of x2
ii) α · β = c
a = constant term
coeff. of x2
36. The quadratic equation whose roots are α and β is (x − α)(x − β)=0
i.e. x2 − (α + β)x + αβ = 0
i.e. x2 − Sx + P = 0 where S =Sum of the roots and P =Product of the
roots.
37. For an arithmetic progression (A.P.) whose first term is (a) and the common
difference is (d).
i) nth term= tn = a + (n − 1)d
ii) The sum of the first (n) terms = Sn = n
2 (a + l) = n
2 {2a + (n − 1)d}
where l =last term= a + (n − 1)d.
38. For a geometric progression (G.P.) whose first term is (a) and common ratio
is (γ),
i) nth term= tn = aγn−1.
ii) The sum of the first (n) terms:
Sn = a(1 − γn)
1 − γ
ifγ < 1
= a(γn − 1)
γ − 1
if γ > 1
= na if γ = 1
.