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Class
CONCEPTS IN MATHS CLASS IX UNIT I ALGEBRA
IX

Number Systems :-

 1.                  Rational number – A number of form p/a where  pฮ Z, q ฮ Z ; Z = integers.  Note that for q = 1, an integer is also a rational number.

2.                  Non Rational Number – A number which cannnot be represented in the form p/q for any integers p and q. Eg  ึ2 = 1.414.... cannot be written as p/q.

3.                  Non terminating Rational Number – eg 1/3 = 0.3333..... where the sequence never ends is called non terminating.

ฝ = .50 is a terminating rational.

Note that there can have a set of digits repeating like 15/7 = 2.142857 1428571.....

4.                  Decimal representation of Intelrnal Numbers :-

eg ึ2 =1.4142135...... is non repeating and non terminating decimal hence irrational.

5.                  Absolute value – Gives the positive integer.|a| = a if aณo      
       eg |2| = 2        -a if a<0           eg |-2| = 2

6.                  Surds :- If a number is cannot be represented as nth power of some rational number, then the irrational number n√a = (a)1/n is called surds.  N – ordder of surd.   A – radical.

7.                  Laws of Radicals.

(a)                            (nึa)n = a
(b)                            (nึa nึb)n = (nึa)n (nึb)n = ab = (nึab)n
 (c)                            nึa/nึb =  nึ(a/b)
(d)           mึ(nึa) = mn/a = nึ(mึa)   where m and n are positive integers.

8.                  Comparison of surds – Two surds of same order can be compared else if the orders are different then they can be compared by working their orders elqual.

9.                  Addition and subtraction of surds :-

Eg. 5ึ3 + 6ึ3 = (5+6)ึ3 = 11ึ3 i.e. Surds having same radical can be added or substracted.

10.              nึa x nึb = nึab – for multiplication or division order of surds should be made same.

Eg. ึ24/3ึ2000       = 6ึ13824/6ึ40000  =          6/13824/40000   =      6ึ216/62

11.       Rationalization -   1/ึ3+ึ2       = 1/(ึ3+ึ2)   x (ึ3-ึ2)/( ึ3-ึ2)       =ึ3-ึ2

also ึ32 = 4ึ2 can be rationalised by multiplying by ึ2.

So ึ2 is called simples Rationalising factor.

11.              Quadratic Surds – Surds of second order.

Eg.       6(3ึ5-5ึ3) = 6x(3ึ5+5ึ3)/(3ึ5)2-(5ึ3)2 = 6/(45-75)x(3ึ5+5ึ3)  = -1/5(3ึ5+5ึ3).

Polnomicals.

1.                  Remainder Theorem :

Let p(x) be any polynomial of degreee≥ >, 1 and a any real number, if p(x) is divided by (x-a) then remainder is p(a).

2.                  Factorisation :-ax2+bx+c = 0  - (x+b+√(b2-4ac)/2a) (+b-√b2-4ac/2a) = 0

3.                  Method of spliting middle term –

(a)                x2+bx+c = (x+p) (x+q) such that p+q = b and pq = c.
(b)               ax2+bx+c = (px+q)(rx+s) – a = pr, b = ps+qr, c = qs.

Linear Equation in one Variable :-
1.                  Linear equation – Equations of the form – ax+b = c., where a, b and c are real numbers.
2.                  To solve linear equations of type :-
  ax+b=cx+d
  x(a-c)=d-b-x=d-b/a-c

3.                  Applications –

Eg. Find a number which tripled and added 6 yields 48.
Eg 3x+6 = 48 = x = 14.
Eg Age Problems – son”s age now is half his fathers age and 10 years ago he was 1/3 rd his father age.  So find sons age.
Eg If x is son’s age then, (x *2) – 10 = (x-10)*3 = x=20
So one needs to formulate linear problems.

Unit 2- Logarithms.

  1. an is called nth power of the base a.
  2. Laws of integral exponent.
    (a)am an = am+n  
    (b)am/an = am-n.
    (c) ((a)m)n - amn
    (d)  (ab)n – an bn.
  1. a 1/n is called nth root of base a.
  2. Logarithm Definition – given a five real number a= 1, if am = b, then we say 
    log ab = m, ie m is the logarithm of b to the base a.
  1. Logarithm to base 10.   Log10 10 = 1, log 10 0.01 = -2.
  2. Log a1 = 0 for any base 1 as (a)0 = 1 holds for any a.
  3. Laws of Logarithms :-
    (a)                log a(mn) = log am + log an.
    (b)                   Log a(m/n) = log am – log an
    (c)                    Log a(m)n = n Log am.
    Note that Log a(m+n)≠ Log am + Log an, ie log a is not a linear operator.
  1. Characterstic and Mantissa.
    N = mx10p can be written as log n = log m + log 10 (10) p
    Log n = p+log mn.
    Where p is the characterstic and m is mantissa of log n.
  1. Antilogarithm –
    If Log n = f then n = antilog f that mean n is called the antilogrithm of t.
  1. To solve numerical problems :-
    If x= 1.23 x 11.2.
     Then log x =Log 1.23 + log 11.2 = 1.1391., x = Antilog 1.1391.Unit

3_Trignometry.                                                                                               

C1.
(a)  Sin ุ = BC/AC
(b)  Cos ุ = AB/AC
(c)  Tan ุ = BC/AB.

2.
(i) Tan ุ = Sin Q/Cos Q 
(ii) Consec ุ = 1/Sin ุ
 (iii) Sec ุ = 1/Cos ุ
 (iv) Cot ุ = 1/Tan ุ

4.                  Trigonometric rations of certain Angles.

For 30 deg.
(i)  Sin 30=1/2
(ii) Cos 30=3/2
(iii) Tan 30=1/3
(iv) Cosec 30=2
(v)  Sec 30=2/3
(vi) Cot 30=ึ3.

For 60 deg
(i) Sin 30=3/2
(ii) Cos 30=ฝ
(iii) Tan 30=1/3

For 45 deg.
(i) Sin 45=(1/2)ฝ
(ii)Cos 45 =  ฝ
(iii)Tan 45 =1.

For 0 deg and 90 deg.
(i)  Sin 0บ=0
(ii) Cos 0บ=1
(iii) Tan 0บ=0
(iv) Sin 90°=1 
(v) Cos 90°=0
(vi) Tan 90บ=๘ (not defined)

5.                  Heights and Distance


Unit 4 -   Geometry.

LINES AND ANGLES.

1.                  Types of Angles.
(i)   Right Angle – Angle equal to 90 บ.
(ii) Obtuse Angle - Angle greater than 90บ.
(iii) Acute Angle - Angle lower than to 90บ

2.                  Supplimentalry angles – Two angles whose sum is 180 deg.

3.                  Complimentary angles – Two angles whose sum is 90 deg.

4.                  Angle bisector – A line Bx is called the angle bisector of XBC if XBC = XBC.

5.                  Interior of an angle – The interor of an angle BAC of the set of all points P in its  plane, which lie on same side of line AB as C and also on the same side of line AC as B.

6.                  Exterior of an angle – The exterior of an angle BAC is the set of all points P in its plane, which do not lie on the angle on in its interior.

7.                  Congeant Angles – Two angles are congreant if a trace of one copy can be superimposed on the BAC is congreant to FEG then we write BAC = PEG.

8.                  Adjacent Angles – Two angles are called adjacent angles, if
(a)   They have the same vertex.
(b)   They have a common arm.
(c)   Uncommon arm are on either side of common arm.

9.                  Vertically opposite angles – Two angles are called vertifically opposite if their arms form two pairs of opposite rays.

10.              Theorem – If two lines intersect, the vertically opposite angles are equal.

11.              Traversal of given lines – A line which intersects two or more given lines at distinct points is called traversal of the given lines.

12.              Corresponding angles – following pairs are called pair of corresponding angles

(i)                  (a) 1 and 5    (a) 2 and 6   (b) 3 and 7

(ii)                Alternate Interior Angles.(a) 3 and 5.   (b) 2 and 8.

(iii)               Consecutibve Interior Angles.(a) 4 and 5.   (b) 3 and 6.

13.              Theorm – If a traversal intersects two parallel lines each pair of alternate angles are equal.  
Converse – If a teraversal intersects two lines in such a way that a pair of altelrnate angles are equal then the lines are paralle.

14.              If a traversal intersects two parallel lines then each pair of consecutive linterior angles are supplimentary.

Note _ The proofs of above results should be done.

15.              Types of Triangles.

A – on basis of sides :-

(i)                  Equilateral – All three sides are elqual.
(ii)                Isosceles    -           Any two sides are equal.
(iii)               Scaline       -           None of the sides are equal.

B-    On basis of Angles.

(i)       Acute Angle -   Each of the anges are acute.
(ii)                Obtuse Angle – One angle is obtuse.
(iii)               Right angle – One angle is right angle.
16.              The sum of three angles of a triange is 180 deg.
(a)                ACD – Exterior angle.
(b)               ACD = ABD + BAD – Theorem.                                   
(c)                ACD > ABD and ACD > < BAD.

17.              Converse Polygon – A polygon P1 P2 .... Pn is called converse if for each side of the polygon, the line containing that side has all the other vertices on the same side of it.

18.              Regular Polygon – If all sides and all angels of a polygon are equal, its called regular polygon.