Hg
Name: Class 10 STICK TO YOUR WALL IN STUDY AREA
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The Hg Classes (8
th
to 12
th
) By: Er Hershit Goyal (B.Tech. IIT BHU), 134-SF, Woodstock Floors, Nirvana Country, Sector 50, GURUGRAM +91 9599697178.
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ALGEBRA
POLYNOMIALS:
Graphs of different types of polynomials:
Polynomial
Max # of zeros = Max # of
points graph cuts x-axis
Graph Type
Possible Graphs
Linear
Ax + B
1
Straight Line
Quadratic
Ax
2
+ Bx + C
2
Parabola
• If a > 0, graph is
open at top.
• If a < 0, graph is
open from bottom
Cubic
Ax
3
+ Bx
2
+ Cx + D
3
Curve
Relation between roots & coefficients:
If α, β are roots of a quadratic equation ax
2
+ bx + c = 0, then:
Sum of the roots
α + β
Product of the roots
αβ
If α, β, γ are roots of the cubic equation ax
3
+ bx
2
+ cx + d = 0, then:
Sum of the roots
α + β + γ
Product of roots taken 2 at a time
αβ + βγ + γα
Product of the roots
αβγ
The eq
n
is → x
2
- (sum)x + (product) = 0
The eq
n
is → x
3
– (sum)x
2
+ (product of 2 at time)x – (product) = 0
LINEAR EQUATIONS:
Conditions for consistency & number of solutions: For pair of linear equations, a
1
x + b
1
y + c
1
= 0 & a
2
x + b
2
y + c
2
= 0:
If a
1
/a
2
≠ b
1
/b
2
Unique solution
Lines are intersecting.
Consistent
If a
1
/a
2
= b
1
/b
2
≠ c
1
/c
2
No Solution
Lines are parallel.
Inconsistent
If a
1
/a
2
= b
1
/b
2
= c
1
/c
2
Infinite Solutions
Lines are coinciding.
Consistent
Cross-Product Formula:
x y 1
b
1
c
1
a
1
b
1
b
2
c
2
a
2
b
2
____x____ = ____y____ = ____1____
b
1
c
2
− b
2
c
1
c
1
a
2
− c
2
a
1
a
1
b
2
− a
2
b
1
QUADRATIC EQUATIONS: For a quadratic equation, ax
2
+ bx + c = 0:
If b
2
– 4ac > 0,
If b
2
– 4ac = 0
If b
2
– 4ac < 0
• If one root is negative of the
other, then: α + β = 0 and
• If one root is reciprocal of the
other, then: αβ = 1
Roots are real and distinct, given by:
Roots are real and equal, given by:
No real roots exist.
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ARITHMETIC PROGRESSION: a, a + d, a + 2d, a + 3d … is an AP, where a is ‘first term’ and d is ‘common difference’.
S
n
= Sum of n terms = n[2a + (n-1)d]/2 or n(a + l)/2 where l is the last term
T
n
= n
th
term = a + (n-1)d
If a, b and c are in AP, then b – a = c – b or b =
.
b is known as the arithmetic mean of a and c.
3 terms in A.P. can be assumed as: a + d, a, a – d
4 terms in A.P. can be assumed as: a – 3d, a – d, a + d, a + 3d
5 terms in A.P. can be assumed as: a – 2d, a – d, a, a + d, a + 2d
Sum of first n natural numbers = n(n+1)/2
