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1. REAL NUMBERS 1
MATHEMATICS COMPENDIUM
REAL NUMBERS
Points to Remember :
1. Euclid’s division lemma : Given positive integers a and b, there exists whole numbers q and r satisfying
a = bq + r, 0  r < b.
2. Euclid’sdivision algorithm: This is based on Euclid’s division lemma.According tothis, the HCFofany
two positive integers a and b, with a > b, is obtained as follows :
ApplyEuclid’s division lemma to find q and r where a = bq + r, 0  r < b.
If r = 0, the HCF is b. Ifr  0, applythe Euclid’s lemma tob and r.
Continue the process till the remainder is zero. The divisor at this stage will be HCF (a, b).
3. The fundamental theorem of arithmetic : Everycomposite number can be expressed (factorised) as a
product ofprimes, and this factorisation is unique, except for the order in which the prime factors occur.
4. For anytwo positive integers a and b, HCF (a, b) × LCM (a, b) = a × b.
 HCF (a, b)
LCM( , )
a b
a b

 and
LCM ( , )
HCF( , )
a b
a b
a b


5. For any three positive integes a, b and c, we have
HCF (a, b, c)
LCM ( , , )
LCM ( , ) LCM ( , ) LCM ( , )
a b c a b c
a b b c a c
  

 
and LCM (a, b, c) HCF ( , , )
HCF ( , ) HCF ( , ) HCF ( , )
a b c a b c
a b b c a c
  

 
6. Let a be a positive integer and p be a prime number such that p/a2
, then p/a.
7. If p is a positive prime, then p is an irrational number...
8. Let x be a rational number, whose decimal expansion terminates. Then we can express x in the form
p
q
,
where p and q are co-prime and the prime factorisation of q is of the form 2n
5m
, where n, m are non-
negative integers.
9. Let 
p
x
q
be a rational number, such that the prime factorisation ofq is ofthe form 2n
5m
, where n, m are
non-negative integers. Then x has a decimal expansion which terminates.
10. Let 
p
x
q
be a rational number, such that the prime factorisation of q is not ofthe form 2n
5m
, where n,
m are non-negative integers. Then x has a decimal expansion which is non terminating repeating (recur-
ring).

2. POLYNOMIALS MATHEMATICS–X
POLYNOMIALS
Points to Remember :
1. Let x be a variable, n be a positive integer and a0
, a1
, a2
, ......., an
be constants. Then
1
1 1 0( ) ,n n
n nf x a x a x a x a
     is called a polynomial in variable x.
2. The exponent of the highest degree term in a polynomial is known as its degree.
3. Degree Name of Polynomial Formof the Polynomial
0 Constant Polynomial f(x) = a, a is constant
1 Linear Polynomial f(x) = ax + b, a  0
2 Quadratic Polynomial f(x) = ax2
+ bx + c; a  0
3 Cubic Polynomial f(x) = ax3
+ bx2
+ cx + d; a  0
4. If f(x) is a polynomial and  is anyreal number, then the real number obtained byreplacing x by  in f(x)
at x =  and is denoted by f().
5. A real number  is a zeroofa polynomial f(x), if f() = 0.
6. A polynomial of degree n can have at most n real zeroes.
7. Geometrically, the zeroes of a polynomial f(x) are the x-coordinates of the points where the graph y = f(x)
intersects x-axis.
8. For any quadratic polynomial ax2
+ bx + c = 0, a  0, the graph of the corresponding equation
y = ax2
+ bx + c has one of the two shapes either open upwards like  or downwards like  , depending
on whether a > 0 or a < 0. These curves are called Parabolas.
9. If  and  are the zeroes of a quadratic polynomial f(x) = ax2
+ bx + c, a  0 then
2
coefficient of
coefficient of
b x
a x
     
2
constant term
coefficient of
c
a x
  
10. If  are the zeroes of a cubic polynomial f(x) = ax3
+ bx2
+ cx + d, a  0 then
2
3
coefficient of
coefficient of
b x
a x
 
      
3
coefficient of
coefficient of
c x
a x
     
3
constant term
coefficient of
d
a x
 
  
11. Division Algorithm : If f(x) is a polynomial and g(x) is a non-zero polynomial, then there exist two
polynomials q(x) and r(x) such that ( ) ( ) ( ) ( ),f x g x q x r x   where r(x) = 0 or degree of r(x) < degree
of g(x).

3. PAIR OF LINEAR EQUATIONS IN TWO VARIABLES 25
PAIR OF LINEAR EQUATIONS
IN TWO VARIABLES
Points to Remember :
1. A pair of linear equations in two variables x and y can be represented as follows :
a1
x + b1
y + c1
= 0; a2
x + b2
y + c2
= 0,
where a1
, a2
, b1
, b2
, c1
, c2
are real numbers such that 2 2 2 2
1 1 2 20, 0.a b a b   
2. Graphically, a pair oflinear equations a1
x + b1
y + c1
= 0, a2
x + b2
y + c2
= 0 in twovariablesrepresents a pair
of straight lines which are :
(i) Intersecting, if 1 1
2 2
a b
a b

here, the equations have a unique solution, and pair of equations is said to be consistent.
(ii) parallel, if 1 1 1
2 2 2
a b c
a b c
 
here, the equations have No solution, and pair of equations is said to be inconsistent.
(iii) Coincident, if 1 1 1
2 2 2
a b c
a b c
 
here, the equations have infinitely many solutions, and pair of equations is said to be consistent.
3. A pair of linear equations in two variables can be solved by the :
(i) Graphical method
(ii) Algebraic Methods; which are of three types :
(a) Substitution method
(b) Elimination method
(c) Cross-multiplication method
ILLUSTRATIVE EXAMPLES
Example1. Draw the graph of linear equation 2x + 3y = 7.
Solution. 2x + 3y = 7  3y = 7 – 2x 
7 2
3
x
y


Give atleast two suitable values to x to find the corresponding value of y.
If
7 2(2) 3
2, 1
3 3
x y

   
If
7 2(5) 3
5, 1
3 3
x y
 
    

4. QUADRATIC EQUATIONS MATHEMATICS–X
QUADRATIC EQUATIONS
Points to Remember :
1. A quadratic equation in the variable x is of the form ax2
+ bx + c = 0, where a, b, c are real numbers and
a  0.
2. A real number a is said to be a root of the quadratic equation ax2
+ bx + c = 0, if a2
+ b + c = 0. The
zeroes of the quadratic polynomial ax2
+ bx + c and the roots of the quadratic equation ax2
+ bx + c = 0
are the same.
3. If ax2
+ bx + c, a  0 is factorisable into a product of two linear factors, then the roots of the quadratic
equation ax2
+ bx + c = 0 can be found by equating each factor to zero.
4. The roots of a quadratic equation can also be found by using the method of completing the square.
5. Quadratic Formula(Shreedharacharya’s rule) :The roots ofa quadratic equation 2
0ax bx c   are
given by ,
2
b D
a
 
where D = b2
– 4ac is known as discriminant.
6. A quadratic equation ax2
+ bx + c = 0 has
(i) Two distinct real roots, if b2
– 4ac > 0 i.e. D > 0
(ii) Two equal roots (i.e., coincident roots), if b2
– 4ac = 0 i.e. D = 0
(iii) No real roots, if b2
– 4ac < 0 i.e. D < 0.
7. 2 2
orx a x a x a    
8. 2 2
x a a x a    

5. ARITHMETIC PROGRESSIONS 71
ARITHMETIC PROGRESSION
Points to Remember :
1. A sequence is an arrangement of numbers or objects in a definite order.
2. Asequence a1
, a2
, a3
, ......., an
, ...... is called an Arithmetic Progression (A.P) if there exists a constant d
such that a2
– a1
= d, a3
– a2
= d, a4
– a3
= d, ....., an
– an–1
= d and so on. The constant d is called the
common difference.
3. If ‘a’is the first term and ‘d’the common difference of an A.P., then theA.P. is a, a + d, a + 2d, a + 3d....
4. The nth
term of anA.P. with first term ‘a’and common difference ‘d’is given by ( 1) .na a n d  
5. The sumto n terms of anA.P. with first term ‘a’and common difference ‘d’is given by
[2 ( 1) ]
2
n
n
S a n d  
Also, [ ]
2
n
n
S a l  , where l = last term.
6. Sum of first n natural numbers = 1 + 2 + 3 + ..... n
( 1)
2
n n 
 .

6. 87
TRIANGLES
Points to Remember :
1. Two figures having the same shape but not necessarily the same size are called similar figures.
2. All the congruent figures are similar but the converse is not true.
3. Two polygons having the same number of sides are similar, if
(i) their corresponding angles are equal and
(ii) their corresponding sides are proportional (i.e. in the same ratio)
4. Basic proportionality theorem(Thales theorem) : Ifa line is drawn parallel to one side ofa triangle to
intersect the other two sides in distinct points, then the others two sides are divided in the same ratio.
5. Converse of Thales’ theorem : Ifa line divides anytwosides ofa triangle in the same ratio, then the line
is parallel to the third side of the triangle.
6. The line drawn from the mid-point of one side of a triangle is parallel to another side bisects the third
side.
7. The line joining the mid-points of two sides of a triangle is parallel to the third side.
8. AAA similarity criterion : If in two triangles, corresponding angles are equal, then the triangles are
similar.
9. AA similarity criterion : Ifin two triangles, twoangles of one triangle are respectivelyequal tothe two
angles of the other triangle, then the twotriangles are similar.
10. SSS similarity criterion : If in two triangles, corresponding sides are in the same ratio, then the two
trianglesare similar.
11. SAS similarity criterion : If one angle of a triangle is equal to one angle of another triangle and the
sides including these angles are in the same ratio, then the triangles are similar.
12. The ratio of the areas of two similar triangles is equal to square of the ratio of their corresponding sides.
13. If the areas of two similar triangles are equal, then the triangles are congruent.
14. If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, then
the triangles on both sides of the perpendicular are similar to the whole triangle and also to each other.
15. Pythagoras Theorem : In a right triangle, the square of the hypotenuse is equal to the sum of the
squares of the other two sides.
16. Converse of Pythagoras Theorem : Ifin a triangle, square of one side is equal tothe sum ofthe squares
of the other two sides, then the angle opposite to first side is a right angle.

7. CO-ORDINATE GEOMETRY 117
CO-ORDINATE GEOMETRY
Points to Remember :
1. The abscissa and ordinate of a given point are the distances of the point from y-axis and x-axis respec-
tively.
2. The coordinates of a point on the x-axis are of the form (x, 0) and ofa point on the y-axis are of the form
(0, y).
3. Distance Formula : The distance between two points P(x1
, y1
) and Q(x2
, y2
) is given by
2 2
2 1 2 1( ) ( )PQ x x y y   
4. Distance of a point P(x, y) from the origin O(0, 0) is given by 2 2
OP x y  .
5. Three points P, Q and R are collinear, then PQ + QR = PR or PQ + PR = QRor PR + RQ =PQ.
6. A ABC is an isosceles triangle, ifAB= BC or AB =AC or BC =AC.
7. A ABC is an equilateral triangle, ifAB= BC =AC.
8. A ABC is a right-angle triangle, ifAB2
+ BC2
= AC2
or AB2
+ AC2
= BC2
or BC2
+ AC2
=AB2
.
9. Section Formula : The co-ordinates of the point which divides the join of points P(x1
, y1
) and Q(x2
, y2
)
internallyin the ratio m : n are 2 1 2 1
,
mx nx my ny
m n m n
  
   
.
10. Mid-point Formula : The co-ordinates of the mid-point ofthe line segment joining the points P(x1
, y1
)
and Q(x2
, y2
) are 1 2 1 2
,
2 2
x x y y  
 
 
.
11. The coordinates of the centroid of a triangle formed by the points P(x1
, y1
), Q(x2
, y2
) and R(x3
, y3
) are
1 2 3 1 2 3
,
3 3
x x x y y y    
 
 
12. The area of the triangle formed bythe points P(x1
, y1
), Q(x2
, y2
) and R(x3
, y3
) is given by
1 2 3 2 3 1 3 1 2
1
.| ( ) ( ) ( )|
2
x y y x y y x y y    
1 2 2 3 3 1 2 1 3 2 1 3
1
| |
2
x y x y x y x y x y x y     
13. The points P(x1
, y1
), Q(x2
, y2
) and R(x3
, y3
) are collinear ifarea of PQR = 0 i.e. x1
(y2
– y3
) + x2
(y3
– y1
) +
x3
(y1
– y2
) = 0.
14. The area of a quadrilateral formed by the points P(x1
, y1
), Q(x2
, y2
), R(x3
, y3
) and S(x4
, y4
), taken in order is
given by :
1 2 2 3 3 4 4 1 2 1 3 2 4 3 1 4
1
| |
2
x y x y x y x y x y x y x y x y      

8. INTRODUCTION TO TRIGONOMETRY MATHEMATICS–X
INTRODUCTION TO TRIGONOMETRY
Points to Remember :
1. IfABC is a right triangle right angled at B and BAC = , 0°    90°, we have :
Base =AB, Perpendicular = BC and, Hypotenuse =AC
here,
BC Perpendicular
sin =
AC Hypotenuse
  ;
AB Base
cos =
AC Hypotenuse
 
BC Perpendicular
tan =
AB Base
  ;
AC Hypotenuse
cosec =
BC Perpendicular
 
AC Hypotenuse
sec
AB Base
   ;
AB Base
cot
BC Perpendicular
  
2. We have,
1 1 1
cosec , sec , cot
sin cos tan
     
  
Also, sin cos
tan , cot
cos sin
 
   
 
3. Values of various Trigonometric ratios :
T-ratio
 0° 30° 45° 60° 90°
sin  0
1
2
1
2
3
2
1
cos  1
3
2
1
2
1
2
0
tan  0
1
3
1 3 not defined
cosec  not defined 2 2
2
3
1
sec  1
2
3 2 2 not defined
cot  not defined 3 1
1
3
0
C
B A90°

9. 133
4. The value of sin  increases from 0 to1 as  increases from 0 ° to 90°. Also, the value of cos  decreases
from 1 to 0 as  increases from 0° to 90°.
5. If  is an acute angle, then
sin (90° – ) = cos , cos (90° – ) = sin 
tan (90° – ) = cot , cot (90° – ) = tan 
sec (90° – ) = cosec , cosec (90° – ) = sec 
6. Basic trigonometric identities :
(i) 2 2
sin cos 1   or 2 2
1 cos sin    or 2 2
1 sin cos   
(ii) 2 2
1 tan sec    or 2 2
sec tan 1  
(iii) 2 2
1 cot cosec    or cosec2
 – cot2
 = 1

10. SOMEAPPLICATIONS OFTRIGONOMETRY
Points to Remember :
1. The line of sight is the line drawn from the eye of an observer to the point in the object viewed by the
observer.
2. The angle of elevation of an object viewed, is the angle formed by the line of sight with the horizontal
when it is above the horizontal level, i.e. the case when we raise our head to look at the object.
angle of elevation
lineofsight
3. The angle of depression ofan object viewed, is the angle formed bythe line of sight with the horizontal
when it is below the horizontal level, i.e. the case when we lower our head to look at the object.
angle of depression
Line of sight
4. The height or length of an object or the distance between two distant objects can be determined with the
help of trigonometric ratios.
5. The observer is taken as a point unless the height of the observer is given.

11. AREAS RELATED TO CIRCLES MATHEMATICS–X
AREAS RELATED TO CIRCLES
Points to Remember :
1. A circle is a collection ofpoints which moves in a plane in such a waythat its distance from a fixed point
always remains the same. The fixed point is called the centre and the fixed distance is known as radius
ofthe circle.
2. Area and circumference of a circle : If r is the radius ofthe circle, then
(i) circumference = 2r = d where d = 2r (diameter)
(ii) Area =
2
2
4
d
r

 
(iii)Area of a semicircle
21
2
r 
(iv) Area of a quadrant
21
4
r 
3. Area of a circular ring : If R and r (R > r) are radii of two concentric circles, then area enclosed bythe
two circles.
=  (R2
– r2
)
4. Number of revolutions completed by a rotating wheel
circumference
Distance moved

5. If a sector of a circle of a radius r contains an angle of . Then,
(i) length of the arc of the sector 2
360
r

  

(ii) Perimeter of the sector 2 2
360
r r

  
 r
O
r
A B

(iii) Area of the sector
2
360
r

  

(iv) Area of the segment
= Area of the corresponding segment – Area of the corresponding triangle.
2 2 2 21
sin , or sin cos
360 2 360 2 2
r r r r
   
     
 
6. (i) In a clock, a minute hand rotates through an angle of 6° in one minute.
(ii) In a clock, an hour hand rotates through an angle of
1
2
 
 
 
in one minute.

12. SURFACE AREAS AND VOLUMES MATHEMATICS–X
SURFACE AREAS AND VOLUMES
Points to Remember :
1. Cuboid
(i)Volume = lbh
(ii) Curved surface area = 2h (l + b)
(iii) Total surface area = 2 (lb + bh + lh)
(iv) Diagonal 222
hbl 
2. Cube
(i)Volume = a3
(ii) Curved surface area = 4a2
(iii) Total surface area = 6a2
(iv) Diagonal a.3
3. Cylinder
(i)Volume = r2
h
(ii) Curved surface area = 2rh
(iii) Total surface area = 2r (r + h)
4. HollowCylinder
(i) Volume =  h (R2
– r2
)
(ii) Curved surface area = 2  h (R + r)
h
R
r
(iii) Total surface area = 2h (R + r) + 2 (R2
– r2
)
= 2 (R + r) (h + R – r)

13. SURFACE AREAS AND VOLUMES 215
5. Cone
(i)Volume hr 2
3
1

(ii) slant height, 22
rhl 
(iii) curved surface area lr
(iv) Total surface area )( rlr 
6. Sphere
(i)Volume 3
3
4
r
(ii) Total surface area = 4  r2 r
7. Spherical Shell
(i)Volume 3 34
( )
3
R r 
r
R(ii) Surface area (outer) = 4R2
8. Hemi-sphere
(i)Volume 3
3
2
r
r
(ii) Curved surface area = 2r2
(iii) Total surface area = 3r2
9. Frustumof a cone :
(i)Volume 2 21
( )
3
h r R rR   
(ii) Slant height 2 2
( ) ( )l h R r  
l
h
R
r
(iii) Curved surface area =  l (r + R)
(iv) Total surface area = l (r + R) +  (r2
+ R2
)

14. STATISTICS 239
STATISTICS
Points to Remember :
1. Mean of ungrouped data : 1 2
1
....... 1 n
n
i
i
x x x
x x
n n 
  
   where x1
, x2
, ....., xn
are n observations.
2. Meanofgroupeddata:
(i) Direct method : Ifthe values ofthe variables be x1
, x2
, ....., xn
and their corresponding frequencies
are f1
, f2
, ......, fn
then the mean of the data is given by







 n
i
i
n
i
ii
n
nn
f
xf
f........ff
xf......xfxf
x
1
1
21
2211
(ii) Short-cutmethod : 1
1
where is assumed mean
,
and,
n
i i
i
n
i i
i
i
f d
A
x A
d x A
f


 
 


(iii) Step-deviationmethod:
1
i
1
where, is assumed mean, = class width
. ,
and
n
i i
i
n i
i
i
f u A h
x A h x A
u
f h


 
  
   
 
  


3. Mode is the value ofthe variable which has the maximum frequency.
4. The mode of a continuousor grouped frequency distribution :
mode = 1 0
1 0 2
, where
2
f f
l h
f f f

 
 
l = lower limit of the modal class
f1
= frequency of the modal class
f0
= frequency of the class preceding the modal class.
f2
= frequency of the class following the modal class.
h = width of the modal class
5. The median is the middle value ofa distribution i.e. median of a distribution is the value of the variable
which divides it into two equal parts.
6. Median for individual series (ungrouped data) : Let n be the number ofobservations.
(i) Arrange the data in ascending or descending order.

15. (ii) (a) If n is odd, then median = value of
th
n





 
2
1
observation.
(b) If n is even, then median =
1
2
 value of
th th
+ 1
2 2
n n    
    
     
observations.
7. Medianfor continous groupeddata:
Median 2 , where
N
F
l h
f

  
l = lower limit of the median class
f = frequency of the median class
h = width of the median class
F = cumulative frequency of the class preceding the median class.
1
n
i
i
N f

 
8. Emperical formula : Mode = 3 Median – 2 Mean
9. Cumulative frequency distribution can be represented graphically using curve, known as ‘ogive’ ofthe
less than type and of the more than type.
10. The median of grouped data can be obtained graphicallyas the x-coordinate of the point of intersection
of the two ogives for the data.

16. PROBABILITY
Points to Remember :
1. In the experimental approach to probability, we find the probability of the occurence of an event by
actuallyperforming the experiment a number oftimes and adequate recording ofthe happening ofevent.
2. In a theoretical approach to probability, we trytopredict what will happen without actuallyperforming
the experiment.
3. An outcome of a random experiment is called an elementary event.
4. The theoretical (classical) probabilityof an event E, written as P(E), is defined as
Number of outcomes favourable to E
P(E)=
Numberof all possible outcomes of the experiment
5. The probability of an impossible event is 0, and that of sure event is 1.
6. The probability of an event E is a real number P(E) such that 0  P(E)  1.
7. An event having onlyone outcome is called an elementary event. The sum of the probabilities of all the
elementaryevents ofan experiment is 1.
8. For any event E, P(E) + P( E ) = 1, where E stands for ‘not E’.
9. Total possible outcomes, when a coin is tossed n times, is 2n
.
10. Total possible outcomes, when a die is thrown n times, is 6n
.
11.
Playing Cards (Total 52)
1313 1313
Spade ( )
(Black coloured)
Club ( )
(Black coloured)
 Heart ( )
(Red coloured)
 Diamond ( )
(Red coloured)


The cards in each suit are ace, king, queen, jack and number cards 2 to 10. Kings, queens and jacks are
called face cards.