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1. Preliminaries
0
1) What is 0 ?
2)
Here 16 = (5 × 3) + 1
i.e. dividend = (divisor × quotient) + remainder
a
3) For the ratio a: b i.e. b , a is called as an
antecedent and b is called as a consequent.
4) If there are n elements in a set then there
are 2n subsets of that set.
e.g. If A = 1, 2, 3 then subsets are
, 1 , 2 , 3 , 1, 2 , 2, 3 , 1, 3 and 1, 2, 3

2. 5) Absolute value i.e. modulus value
x means the absolute value or modulus
value of x.
It x is a positive real number then x x
It x is a negative real number then x x
If x is zero then x 0
In short
x x for x 0
x for x 0
Note:
-x does not necessarily mean negative.
If x a then x=±a
Graph of y x

3. x: -2 -1 0 1 2
y: 2 1 0 1 2
Graph is a right angle with vertex at origin.
Poll Question
1 1
Which is greater 5 or 3
3 2
?

4. Ans: LCM of 2 and 3 is 6
1 1 1 1
2
3 6 2 6 6
5 5 (5 ) 25
1 3 1 1
2 6 3 6 6
and 3 3 (3 ) 27
As 27 25
1 1
6 6
27 25
1 1
2 3
3 5
6) x co-ordinate is also known as abscissa
y co-ordinate is also known as ordinate
Where does the point (k, 0) lie ?

5. Poll Question
a, b, c, d………x, y, z are distinct non-zero
real numbers. Then the product
(x-a).)(x-b).(x-c)………(x-z) is zero or non-
zero ?
7) Zeroes of the polynomial.
If P(x) is a polynomial such that P(α)=0 for
any real number α, then α is called the zero
or root of the polynomial P(x).
Consider P(x)=x2+10x+16
Now P(x)=x2+10x+16=(x+2) (x+8)
Clearly, P(x) = 0 if x = -2 or x = -8
∴-2 and -8 are zeroes of the polynomial
x2+10x+16.

6. 1
8) 2
0.5 is terminating decimal
1
3
0.333....... i.e. 0.3 or 0.3 is recurring decimal

9) For a triangle point of intersection of three
i) Medians is called as centroid.
ii) Altitudes is called as orthocentre.
iii) Perpendicular bisectors is called as
circumcentre.
iv) Angle bisectors is called as incentre.

7. 10) Pythagoras theorem, Apollonius theorem,
30°-60°-90° triangle theorem, 45°-45°-90°
triangle theorem.
Poll Question
1) How many maximum number of common
tangents can be drawn to two distinct
circles 2 or 4 ?

8. Ans: 4

9. 11) Trigonometry
Trigonon = Triangle
Metry = Measures
Poll Question
What is unit of sin θ, degree or radian?

10. Ans: It’s a ratio. It has no unit.
Express sin θ entirely in terms of tan θ
Ans:
sin
tan
cos
1
sin cos tan tan
sec
1 1
tan tan
2 2
sec 1 tan