Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download 8th class Maths Bridge Program
Document related concepts
no text concepts found
Transcript
CLASS-VIII
MPC BRIDGE COURSE
CLASS - VIII
MPC BRIDGE COURSE
CONTENTS
MATHEMATICS : 2 TO 43
PHYSICS
: 44 TO 69
CHEMISTRY
: 70 TO 101
KEY
: 102 TO 108
NARAYANA GROUP OF SCHOOLS
1
CLASS-VIII
MPC BRIDGE COURSE
MATHEMATICS
DAY-1 : SYNOPSIS
Variable: A letter symbol which can take
various numerical values is called a
variable or literal. Examples: x, y, z etc.
Constant : Quantities which have only
one fixed value are called constants.
Term: Numericals or literals or their
combinations
by
operation
of
multiplication are called terms.
Constant Term : A term of an expression
having no literal is called a constant
term.
TYPES OF ALGEBRAIC EXPRESSIONS:
* An expression containing only one term
is called a monomial.
* An expression containing two terms is
called a binomial.
* An expression containing three terms is
called a trinomial.
* An expression containing two or more
terms is called a multinomial.
* An expression containing one or more
terms with positive integral indices
(powers) is called a polynomial.
Every non-zero number is
Note:
considered a monomial with degree zero.
Degree of polynomial : The highest power
of terms in a polynomial is called the
degree of a polynomial.
Zero polynomial : If all the coefficients
in a polynomial are zeroes, then it is
called a zero polynomial.
Zero of the polynomial : The number for
which the value of a polynomial is zero,
is called zero of the polynomial.
Degree of zero polynomial is not
Note:
defined.
Substitutions : The method of replacing
numerical values in the place of literal
numbers is called substitution.
Like and Unlike terms :
* Terms which contain the same literal
factors are called like terms or similar
terms. In like terms, the numerical
coefficient may be different.
NARAYANA GROUP OF SCHOOLS
2
2
Ex: 8ab , 5ab ,
2 2
8 2
ab ; 2m2n, 4nm2 ,
m n.
3
3
* Terms which do not have the same literal
factors are called Unlike terms.
Example : 5a,5a2,5ab; 2xy2 ,2x2 y
Addition of algebraic expressions :
Addition of algebraic expressions means
adding the like terms of the expressions.
* Combining the coefficients of like terms
of an expression through addition or
subtraction is called simplification of an
algebraic expression.
There are two methods of adding
algebraic expressions. They are
i) Horizontal method ii) Vertical method.
Horizontal method :
In this method, like terms should be
added and unlike terms should be
written separately
Subtraction of Algebraic expressions:
Additive Inverse of expression :
* The additive inverse or the negative of
an expression is obtained by replacing
each term of the expression by its additive
inverse.
Example: Additive inverse of -9x is 9x
* To subtract 1 st expression from the 2 nd
expression, additive inverse of the 1 st
expression should be added to the 2 nd
expression.
If P and Q are two algebraic expressions
then P – Q = P + (–Q).
Example: Subtract 11a – 6b from 7a + 4b
Solution: (7a + 4b) – (11a – 6b) = 7a +
4b – 11a + 6b = –4a +10b
Subtraction can also be done in two ways.
* Horizontal method
* Vertical method
Multiplication of Polynomials: Multiply
each term of the first Polynomial with
each term of the second and add the
like terms in the product.
Suppose (a+b) and (c + d) are two
Polynomials. By using the distributive law
of multiplication over addition, we can
find their product as given below.
2
CLASS-VIII
MPC BRIDGE COURSE
8. Divide 4x – 10x2 + 5x by 2x, then the
resultent value is
3
a b c d a c d b c d
a c a d b c b d
ac ad bc bd
Column Method of Multiplication: In this
method we write multiplicand and the
multiplier in descending powers of,
arrange one under another, and multiply
the multiplicand by every term of the
multiplier and add.
DAY-1 : WORKSHEET
Conceptual Understanding Questions :
1. The degree of the polynomial
2 4 1
7
x x + x 7 + 3 is
3
5
8
1) 1
2) 4
3) 7
4) 12
2. The zeroes of the polynomial 2x – 3 is
1)
3.
4.
5.
6.
2
2)
3
2
3) 0
4) 3
3
The simplified form of 1.5 x 3 – 1.7 x 3
+ 0.2x 3 + 2x 3 is
1) 3.7 x3
2) 3x3
3) 2x3
4) x3
2
The addition of 7x – 4x + 5 and
–3x2 + 2x – 1 is
1) 4x2 – 2x + 4
2) 4x2 + 2x + 4
3) 4x2 – 2x – 4
4) 4x2 + 2x – 4
3
2
Subtract 2x – 4x + 3x + 5 from
4x3 + x2 + x + 6, then the resultent
value is
1) 6x3 + 5x2 – 2x + 1 2) 2x3 + 5x2 – 2x+1
3) 2x3 – 5x2 – 2x + 1 4) 2x3 – 5x2 + 2x – 1
Additive inverse of ax2 + bx + c is
1) – ax2 – bx – c
2) –ax2 + bx – c
3) ax2 + bx + c
4) – ax2 + bx + c
7.The product of
4 2 3 2
a bc
1)
3
3)
5 2 3 2
a bc
3
6
5
12
ab, bc and
abc is
5
6
9
4 2 3 2
2) a b c
3
4)
2 2 3 2
a bc
3
NARAYANA GROUP OF SCHOOLS
2
1) 2x 5x
5
2
2
2) 4x 5x
5
2
2
3) 4x 5x
5
2
2
4) 4x 5x
5
2
Single Correct Choice Type :
9. Two adjacent sides of a rectangle are
3a – b and 6b – a then its perimeter is
1) 2a + 5b
2) 4a + 10b
3) 2a – 5b
4) 4a – 10b
10.The perimeter of a triangle whose sides
are 2y + 3z, z – y, 4y – 2z is
1) 5y + 2z
2) 10y + 4z
3) 5y – 2z
4) –5y + 2z
11.The perimeter of a rectangle,
16x3 – 6x2 + 12x + 4. If one of its sides is
8x2 + 3x, then the other side is
1) 16x3 – 14x2 + ax + 4
2) 8x3 – 11x2 + 3x + 2
3) 16x3 + 14x2 + ax – 4
4) 8x3 + 11x2 + 3x – 2
12. Subtract x3 – xy2 + 5x2y – y3 from
–y3 – 6x2y – xy2 + x3
1) 2y3 – 8x2y + 3xy2 – 2x3
2) 2x3 – 8xy2 + x2y – 2y3
3) –11x2y
4) –12x2y
13. What must be added to 2x2 – 3xy + 5y2
to get x2 – xy – y2 is
1) x2 + 2xy – 6y2
2) –x2 + 2xy – 6y2
2
2
3) x – 2xy + 6y
4) x2 – 2xy – 6y2
14. The value of
a
3
2a 2 4a 5 a 3 2a 2 8a 5 =
1) 2a3 4a2 12a 10
2) 2a3 4a2 12a 10
3) 2a 3 4a2 12a 10
4) 2a3 4a2 12a 10
15. What must be added to x3 + 3x – 8 to get
3x3 + x2 + 6?
1) 2x3 + x2 – 3x + 14
2) 2x2 + x2 + 14
3) 2x3 + x2 – 6x – 14
4) 2x3 + x2 – 14
3
CLASS-VIII
MPC BRIDGE COURSE
16.If A 16x 8 4 and B 15x 10 5 ,
then A – B =
1) 8x
2) 9x
3) 7x
4) 6x
17.The product of 4x 9y 3x 11y is
1)12x2 17xy 99y 2
2) 12x2 17xy 99y 2
3) 12x2 17xy 99y2
4) 12x2 17xy 99y2
18.The product of 1.5x(10x2y – 100xy2) is
1) 15x3y – 150x2y2 2) 15x3y + 150x2y2
3) 150x2y2 – 15x3y 4) 15x2y2 + 15x3y
19. If A
B
32x 8 y 4 16x 4 y 3 4x 2 y 2
,
4xy
AB
16x 5 y 4
is
2 2 , then
4x 3 y 3
4x y
1) 8x7y2 + 4x3y + x
3) 8x7 + 4x3y – xy2
2) 8x7y2 – 4x3y – xy2
4) 8x7 – 4x3 + xy2
DAY-2 : SYNOPSIS
MULTIPLICATION BY USING FORMULAE:
2
* x a x b x a b x ab
2
* x a x b x a b x ab
2
* x a x b x a b x ab
2
* x a x b x a b x ab
2
2
2
* a b a 2ab b
2
2
* a b a 2ab b
Conceptual Understanding Questions :
1. The product of (x + 5) and (x + 4) is
1) x2 + 9x + 20
2) x2 – 9x + 20
2
3) x – 9x – 20
4) x2 + 9x – 20
2. (x2 – ay)2 =
1) x4 – a2y2 – 2xay 2) x4 + a2y2 – 2axy
3) x4 – a2y2 + 2ax2y 4) x4 + a2y2 – 2ax2y
3. (2x + 3y) (2x – 3y) =
1) 4x2 + 9y2
2) 2x2 – 3y2
3) 2x2 + 3y2
4) 4x2 – 9y2
3
3
4. 2a 2a =
b
b
2
1) 4a
9
b2
2
2) 4a
9
b2
3
3
2
4) 2a 2
2
b
b
3
5. (2x + 3y) =
1) 8x3 + 27y3 + 18xy (2x +3y)
2) 8x3 + 27y3 + 36x2y + 54xy2
3) 8x2 + 27y3 + 18xy (x + y)
4) Both 1 & 2
6. (x + 2y) (x2 – 2xy + 4y2) =
1) x2 + 2y2
2) (x + 2y)2
3) x3 + 8y3
4) (x + 2y)3
7. If x = 2, y = 3 and z = – 5,
then x3 + y3 + z3 =
1) 90
2) –90
3) 0
4) –90xyz
2
3) 2a
Single Correct Choice Type :
2
8. Using the identity the value of 497
2
2
* a b a b a b .
*
*
*
*
*
DAY-2 : WORKSHEET
2
is
a3+b3=(a+b)(a2-ab+b2) or (a+b)3-3ab(a+b)
1) 247006
2) 247009
3) 257006
4) 2578009
9. The value of 0.768 × 0.768 – 2 × 0.768 ×
0.568 + 0.568 × 0.568 is
1) 0.04
2) 0.4
3) 0.004 4) 0.0004
a3-b3=(a-b)(a2+ab+b2) or (a-b)3+3ab(a-b)
10.If P 1162 162 , Q 1242 242 and R = 120,
(a+b)3=a3+3a2b+3ab2+b3 or a3+b3+3ab(a+b)
(a-b)3=a3-3a2b+3ab2-b3 or a3-b3-3ab(a-b)
a3 + b3 + c3 – 3abc
2
2
2
= (a + b + c)(a + b + c – ab – bc – ca)
* (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
NARAYANA GROUP OF SCHOOLS
then the value of
1) 20
2) 30
P
Q
R
100 100
3) 40
4) 10
4
CLASS-VIII
MPC BRIDGE COURSE
11. If p a 2 2ab b2 and q 4ab , then p + q
is
1) 2ab
2) a b
2
2
12.The expansion 4a
2
2
3) a b 4) a b
2
20.
1) 3xyz 2) xyz
21.
9 12a
as a perfect
25
5
3
1) 2a
5
3
3) 2a
5
3
2) 2a
5
2
2 3
4) 2a
5
1
12
2)
1
12
3)
1
18
159991
16
1
1
3
2) 1
1
3) abc 3
4) 27abc
DAY-3 : SYNOPSIS
4)
1
18
Literal factor (Divisor):
If two or more algebraic expressions are
multiplied, their products are obtained.
The algebraic expressions which
multiplied to form the product, are called
the factors of the product.
Example: 12xy = 3x × 4y
3x and 4y are factors of 12xy
2)
Greatest/Highest common factor (G.C.F./
H.C.F):
160001
16
3
15909
4)
4
16
2
2
15. 52 × 48 = (a) – (b) , then the values of a
and b are
1) 50, 2
2) 52, 48
3) 4, 50
4) 4, 52
3) 1991
2
2
4
4
16.The value of a b a b a b a b
when a = 0, b = 1 is___
1) 0
2) 1
3) –1
4) 2
1
1
5 , then x 3 3 =
x
x
1) 100
2) 125
3) 140
4
4) 145
4
18. If x + y = 8; xy = 12 then x y –––––
––––
1) 1012 2) 1112
3) 1212 4) 1312
1
1
4 then z 2 2 ––––––––––
z
z
1) 18 2) 16
3) 14
4) 8
If z
NARAYANA GROUP OF SCHOOLS
G.C.F./H.C.F of two or more monomials
is the highest monomial which divides
each of the given monomials completely.
L.C.M of Monomials:
The L.C.M of two or more monomials is a
monomial having the least powers of
constants and variables such that each
of the given monomials is a factor of it.
The sign of the coefficient of the L.C.M of
the monomials is the same as the sign
of the coefficient of the product of the
monomials.
Factorization: The process of resolving the
given expression into factors is called
factorization.
17.If x
19.
1
4) 3xyz
2
1
3
14. If 99 100 =
4
4
1)
3
If a 3 b 3 c 3 0 then a b c
1) 0
1
1
7
7
13. The product of 3 6 and 3 6 is
1)
3
4) 1
y z z x
3) 1
22.
2
3
x y
3) 0
1) 3(x–y) (y–z) (z–x) 2) 0
square is
2
If x + y + z = 0 then x 3 y 3 z 3
Different types of Factorization:
1. Taking out common factor: Steps :
* Find the H.C.F. of all the terms of the
given expression.
* Divide each term by this H.C.F. and
enclose the quotient within the brackets,
keeping the common factor outside the
brackets.
5
CLASS-VIII
Finding factors of multinomials :
To factorize a multinomial, in general we
have to express the multinomial as a
product of two or more expressions. These
two or more expressions whose product
is equal to the given multinomial, are
called the factors of the multinomial. This
is the reverse process of multiplication.
Prime multinomial: A multinomial is said
to be prime if it is divisible by one and
itself only.
Common factor: A number or a number
letter combination which divides all the
terms of a multinomial is called a
common factor of the terms of the
multinomial.
Common Binomial factor: The greatest
common factor of the terms of a
multinomial need not be a monomial
always. It can also be a binomial.
Factorisation and rearrangement of
terms: If we observe the terms of an
algebraic expression in the order they are
given, they may not have a common
factor. But by rearranging the terms in
such a way that each group of terms has
a common factor, some algebraic
expressions can be factorized.
While rearranging the sign and value of
each term should not be altered.
Example: ax by ay bx
= x(a b) y(a b) = (a b)(x y)
The factors are (a + b) and (x – y)
To find the factors of difference of two
squares: The difference of two squares
is always equal to the product of the sum
and difference of the square roots of the
square terms in the expression.
Example: a 2 b2 (a b)(a b)
Factorization of (a + b)3, (a – b)3 forms :
3
3
2
2
3
* a b a 3a b 3ab b or
a b
3
a 3 b3 3ab a b
3
3
3
3
2
2
3
* a b a 3a b 3ab b or
a b
3
a b 3ab a b
NARAYANA GROUP OF SCHOOLS
MPC BRIDGE COURSE
Factorization of a3 + b3, a3 – b3 forms
3
3
2
2
* a b a b a ab b (or)
3
a 3 b3 a b 3ab a b
3
3
2
2
* a b a b a ab b (or)
3
a 3 b3 a b 3ab a b
DAY-3 : WORKSHEET
Conceptual Understanding Questions :
1. The HCF of 2x2 and 12x2 is
1) 2x2 2) 12x2
3) 2x
4) 12x
3
2
2. The HCF of x and –yx is
1) x3
2) –yx2
3) x2
4) –yx5
3. The HCF of numerical coefficient of the
given monomials 6x3a2b2c, 8x2ab3c3 and
12a3b2c2 is
1) 12 2) 8
3) 6
4) 2
2
2
4. The factorization of 4x – 9y is
1) (2x – 3y) (2x + 3y)2) (2x – 3y) (3x – 2y)
3) (2x – 3y) (x – y) 4) (x – y) (3x – 2y)
5. The factorization of x4 – y4 is
1) (x + y)2 (x – y)2
2) (x + y) (x – y) (x2 + y2)
3) (x + y) (x – y) (x2 – y2)
4) (x2 + y2) (x + y) (y – x)
6. The factorization of x3 + 8y3 is
1) (x + 2y) (x2 + 2xy + 4y2)
2) (x + 2y) (x2 – 2xy + 4y2)
3) (x – 2y) (x2 + 2xy + 4y2)
4) (x – 2y) (x2 – 2xy + 4y2)
7. The factorization of 8a3 – 27 is
1) (2a + 3) (4a2 + 6a + 9)
2) (2a + 3) (4a2 – 6a + 9)
3) (2a – 3) (4a2 + 6a + 9)
4) (2a – 3) (4a2 – 6a + 9)
Single Correct Choice Type :
8. If A = 384 x4y5z3 and B = 256 x2y3z3, then
their G.C.D is
1) 27 x 2 y 3z 5
2) 27 x 2 y 5 z 3
3) 27 x 2 y 3z 3
4) 27 x 4 y 5 z 5
9. The factors of 19x – 57y is
1) – 19 (x + 3y)
2) 19 (x + 3y)
3) – 19 (x – 3y)
4) 19 (x – 3y)
6
CLASS-VIII
MPC BRIDGE COURSE
10.The H.C.F of 15pq, 20qr, 25rp is
1) 5pqr 2) 25pqr
3) 5
4) 25
11.The factors of – 10ab3 + 30ba3 – 50a2b3 is
1) – 10ab (b2 – 3a2 + 5ab2)
2) 10ab (b2 – 3a2 + 5ab2)
3) – 10ab (b2 + 3a2 + 5ab2)
4) 10ab (b2 + 3a2 + 5ab2)
12.The factors of (x2 + 3x)2 – 5(x2 +3x) – y(x2
+ 3x) + 5(x2 + 3x) are
1) (x2 + 3x)(x2 + 3x – y)
2) (x2 + 3x – 5)(x2 + 3x + y)
3) (x2 + 3x + 5)(x2 + 3x + y)
4) (x2 + 3x – 5)(x2 + 3x – y)
2
13.The factors of 2a 6b 3 a 3b are
1) a 3b 2 3a 9b
2) a 3b 2 3a 9b
18.
125 p 3 8q 3 =
3
1) 5 p 2q 30 pq 5 p 2q
3
2) 5 p 2q 30 pq 2q 5 p
2
2
3) 5 p 2 q 25 p 10 pq 4 q
4) both (1) and (3)
19.
1000a 3 8 =
2) 10 a 2 100 a
14.If you factorise 4p2 – (q + r)2 the factors
are
1) 2p q r
2) 2p q r
3) 2p q r
4) Both 2 and 3
15.The factors of x6 – y6 is/are
3
3
3
3
1) x y , x y
2
2
4
2 2
4
2) x y , x x y y
2
2
4
2 2
4
3) x y , x x y y
4) Both 1 and 2
16.The factor of 5x2 – 80y2 are
1) 5(x + 4y)(x – 4y) 2) 5(x – 4y)(x – 4y)
3) 5(x + 4y)(x + 4y) 4) 5(4y – 5)(x + y)
17.
20 a 4
2
1) 10 a 2 100 a 20 a 4
4) 10 a 2 100 a
2
20 a 4
2
3) 10a 2 100a 20 a 4
2
DAY-4 : SYNOPSIS
3) a 3b 2 3a 9b
4) a 3b 2 3a 9b
Intriduction: A second-degree equation is
a polynomial equation in which the
highest degree of the variable is 2. In
particular, a second-degree equation in
one unknown is called a quadratic
equation. We define the standard form of
a quadratic equation as
Ax2 + Bx + C = 0
A 0
The zero-product rule :
If a . b = 0, then a = 0 or b = 0
Solving a Quadratic Equation by using the
Quadratic Formula:
Consider the quadratic equation
ax 2 bx c 0 a 0 .
By solving this equation with completion
of square method we get
b b2 4ac
2a
The factors of 2x 3 + 5x 2 y -12xy 2 is
x
1) x x 4y 2x 3y
Let the roots are denoted by say
2) x x 4y 2x 3y
b b2 4ac
b b2 4ac
,
2a
2a
We use theseformulas to find the roots
of any other quadratic equation.
3) x x 4y 2x 3y
4) x x 4y 2x 3y
NARAYANA GROUP OF SCHOOLS
7
CLASS-VIII
MPC BRIDGE COURSE
NATURE OF ROOTS:
Discriminant :b2 – 4ac denoted by or D
is called the discriminant of a quadratic
equation ax 2 bx c 0 where a, b, c
R and a 0 . Thus b2 4ac
Nature of the Roots of a Quadratic Equation
2
ax bx c 0
where a, b, c R and a 0 :
DAY-4 : WORKSHEET
Conceptual Understanding Questions :
1. The solution set of
is
b d
a c
2) a,c
b d
,
a c
4) b, d
1) ,
3)
1) If 0, and a,b,c R , then the roots are
complex and conjugates. In this case the
graph of the curve y = f(x) does not
intersect x - axis.
2) If 0, and a,b,c R, then the roots are
real and each of the root is called a
double or repeated root and is equal to
b
. In this case the curve y = f(x) touch
2a
b
,0 . Also the
x - axis in one point
2a
quadratic expression will be a perfect
square expression when 0
3) If 0, and a,b,c R, then the roots are
real and distinct. In this case the curve
y = f(x) intersect x - axis in two distinct
points
4) If 0 and a,b,c Q and is a perfect
square, then the roots are rational and
distinct.
5) If 0 and a,b,c Q and is not a
perfect square, then the roots are
irrational and conjugates.
Nature of the Roots of a Quadratic
2
Equation ax bx c 0 where a, b, c C
and real part of a not equal to 0 :
In this case roots are complex and may
or may not be conjugates.
2. The roots of x 2 x 6 0 are
1) –2,3
2) 1,–2
3) –3,–2 4) –3,2
3. The roots of x 2 4x 12 0 are
1) 2,–6
2) –2,6
3) 2,6
4) –2,–6
4. The nature of the roots of the equation
x 2 4x 4 0 are
1) Real and rational 2) Real and irrational
3) Real and equal 4) all of these
Single Correct Choice Type :
5. x a x b 0 then the value of x
1) a,b
2)
1 1
,
a b
2
3) a, b
4)
a b
,
b a
2
6. x a x a 0 , then x is
1) 0
2)
1
4a
3)
1
4a
4)
1
2a
7. If a b the roots of the equation
x a x b b2 are
1) real and distinct 2) real and equal
3) real
4) imaginary
8. The number of real roots of the equation
x 1
2
2
2
x 2 x 3 0 is
1) 2
2) 1
3) 0
4) 3
9. The value of m for which the equation
1 m x 2 2 1 3m x 1 8m 0
equal roots, is
1) 0
2) 1
NARAYANA GROUP OF SCHOOLS
ax b cx d 0
3) 2
h a s
4) 5
8
CLASS-VIII
10.
11.
12.
The roots of 5 x 2 3x 2 0 are
1) Rational and equal
2) Rational and not equal
3) Irrational
4) Imaginary
If the roots of the equation
x2 - 15 - m(2x-8)=0 are equal then m =
1) 3, -5 2) 3, 5
3) -3, 5
4) -3, -5
Only one of the roots of ax 2 bx c 0,
a 0 , is zero if
1) c 0
2) c 0, b 0
3) b 0, c 0
4) b 0, c 0
MPC BRIDGE COURSE
For example : ( i )
16
1
16 i 16 i 4 (i 4) 2 16
( ii )
36
1
36 i 66 i 6 (i 6) 2 36
Imaginary number: Square root of a
negative number is called an imaginary
number.
For examle :
1,
5,
6, 25 etc.
are all imaginary numbers.
Integral powers of iota ( i ) :
( i ) Positve Integral Power of i :
we know that, i =
1 we can write
higher powers of i as follows :
13.
14.
6 6 6 ...... 2 2 2 ......
is
1) 1
2) 2
3) 3
The roots of the equation
a c b x 2 2cx b c a 0
1) 1,
3) 1,
15.
16.
a c b
bca
2) 1,
bca
a c b
4) 1,
4) 4
i 0 1, i1 i, i 2 1,
i 3 i 2 i 1 i i,
i 4 (i 2 ) 2 (1)2 1,
are
bca
2c
i 5 i 4 i 1 i i,
i 6 (i )4 i 2 1 (1) 1,
i 7 i 4 i 3 1 (i ) i,
2c
a c b
i 8 (i 4 )2 (1)2 1, i 9 i 8 i 1 i i
and
so on.
If a b c x b c a x c a b is a
perfect square, then a, b, c are in
1) A.P.
2) G.P.
3) H.P.
4) A.G.P.
If the roots of
2
p
2
q 2 x 2 2q p r x q 2 r 2 0 be
real and equal, then p, q, r will be in
1) A.P 2) G.P
3) H.P
4) None
DAY-5 : SYNOPSIS
Complex Numbers: Euler was he first
Mathematician. Who intoduced the
symbol i ( read as iota )
for
1 with
property i 2 1 0i.e. i 2 1 . He also
called this symbol as imaginary unit.
1 1
a
a 1. a i a a R
NARAYANA GROUP OF SCHOOLS
I order to compute i p for p> 4, we divide
p by 4 and obtain the remainder r. Let
q be the quotient, when p is divided by
4. Then, p = 4q + r. were 0 r 4
i p i 4 q r (i 4 )q i r 1. r r i r
Thus, the value of i p for p 4 is i r ,
where r is the remainder when p s
divided by 4.
( ii ) Negative integral Powers of i :
Complex Numbers: A number of the form
a+ib is called a comples number, wher
a and b are real
numbers and
i 1 . Complex number is generally
denoted by z i.e.
z a ib
For example : 3+2i, 5+i,2-3i etc. are
complex numbers.
9
CLASS-VIII
Real and imaginary parts of complex
number :
Let a+ib be a complex number then, a
is called real and b the imaginary part
of z and may be denoted by Re(z) and
Im(z) repectively.
For example : If z=3+2i, then Re(z) =3,
im(z) =2.
Purely real and purney imaginary
complex numbers: A complex
z a ib is called purely real, if b = 0 i.e
Im(z)=0 and is called purely imaginary,
if a =0. i.e., Re(z) =0.
For example : z=3 is purely real and
z=2i is purely imaginary
Set of complex numbers:
The Product set R R consisting of the
ordered pairs of real numbers is called
the set of complex numbers. The set of
complex number is denoted by C i.e.
C a ib : a, b R
Note: We observe that nthe system of
complex numbers includes the system
of real numbers i.e. R C.
Equality of complex numbers:
MPC BRIDGE COURSE
Note: Order relations “ greater than” and
“less than” are not defined for complex
numbers. The inequalities like
i>0, 3+i < 2 etc. are meaningless.
Addition of two complex numbers:
If z1 ( x , y ) C and z2 (a, b) C then
z1 z2 ( x, y ) (a, b) ( xa yb, xb ya) .
If z1 , z 2 C , then z1 z2 C and z1 z2 is
called the sum of two complex numbers
z1 and z2 .
Multiplication of two complex numbers:
The postulate (iii) defines the binary
operation of multiplication of two complex numbers.
If z1 ( x, y ) C and z2 (a, b) C , then
z1 z2 ( x, y ) (a, b) ( xa yb, xb ya )
e.g. (2,5) (3, 9) (2.3 5.9, 2.9 5.3) ( 39, 33)
If z1 , z2 C , then z1 z2 C and z1 z 2 is
called the product of two complex numbers z1 and z2 .
Division of complex numbers:
Two complex numbers z1 a ib and
The division of a complex number z1 by a
z2 c id are said to be equal, if a = c
non zero complex number z2 is defined
and b=d.
Proof: a ib c id
a c i (d b)
as the multiplicative inverse of z1 by
the multiplicative inverse of
( a c ) 2 ( d b ) 2 ( a c ) 2 ( d b ) 2 0
Here, sum of two positive numbers is
zero. This is only possible, if each number is zero.
(a c) 2 0 a c and
(d b)2 0 b d .
For example : If a + ib =3+2i, then
a=3,b=2
Zero complex number: A complex number z is said to be zero, if its both real
and imaginary parts are zero. In other
words, z = a+ib = 0, if and only if a = 0
and b = 0.
denoted by
Therefore,
z2 and is
z1
z2 .
1
z1
1
= z1 , z2 z1 . .
z2
z2
Let z1 a ib1 and z2 a2 ib2
Then
=
z1 a1 ib1 (a1 ib1 ) (a2 ib2 )
z2 a2 ib2 (a2 ib2 ) (a2 ib2 )
a1a2 b1 b2 i ( a1b2 b1a2
a22 b22
a1a2 b1 b2 a2b1 a1b2
= a 2 b 2 a 2 b2
2
2
2 2
NARAYANA GROUP OF SCHOOLS
10
CLASS-VIII
MPC BRIDGE COURSE
For example : If
z1 2 3i and z 2 5 4i, then
z1
2 3i (2 3i ) (5 4i ) 10 8i 15i 12
z 2 5 4i (54i ) (5 4i )
25 16
1
22 7
i.
22 7i
41
41 41
12.When simplified the value of i 57 (1/ i 25 )
is
1) 0
2) 2i
3) -2i
4) 2
13.If i 2 1 , then i 2 i 2 i 4 i 6 ........ to
( 2n+1 ) terms is equal to
1) 0
2) -i
3) 1
4) -1
14.The value of (1 i ) 2 (1 i) 4 is equal to
DAY-5 : WORKSHEET
1) 8
Conceptual Understanding Questions :
1. i
123
__________ .
1) i
2) -i
3) 1
4) -1
2. i120 i 2 ____________.
1) 0
2) -1
3) 1
4) 2
3. If z =1-2i then Real part of z is
___________.
1) i
2) -2
3) 1
4) 0
4. If z = 3+2i then imaginary part of z is
_________.
1) 2
2) 1
3) i
4) 3
5. If p+iq=0 then p+q is ___________.
1) 1
2) -1
3) -2
4) 0
6. If z1 a 2i, z2 a 2i then z1 z2 is
_________.
1) 2
2) 0
3) 2a
4) 2i
7. If z1 2 i, z2 2 i then z1 z2 is
__________.
1) 2 i
2) 0
3) 4
4) -4
3) -2
9. If z1 1 i, z 2 1 i, then
______________.
1) -i
2)1
1) i
1) (3i ) 6 i18
3)
4)
ab
2)
2) 1
3) ab
4)
3) -i
4) -1
NARAYANA GROUP OF SCHOOLS
2)
7
i7
i3
2
2) -i
3) 4
4) i
2
Express (4 3i ) 3 in the form of (a+ib) is
1) (-44-11 77i)
3) -44+11 7i)
ab i
11. i103
1) i
4) -1
21
48 5 27 i 6 3
4
17 1 34
19. i
i
23.
ab
3) -i
16 25 i9
19 1 25
18. i
i
22.
4) i
a . b
1)
2) 1
16.The value of 2 i 2 i 4 i 6 ___________
1) 0
2) -1
3) 2
4) 1
17.Which of the following is true?
Single Correct Choice Type :
10.
4) -4
1) -i
2) -2
3) 2i
4) 2
20. (1-i) (1+2i) (1-3i) =
1) 1-6i
2) 8-6i
3) 6-8i
4) 6+8i
21. Express ( 2+3i) (-4i) in the form of a+ib
1) 16+i
2) 17+i
3) 18+i
4) None
4) 1
z1
z2 is
3) 0
3) -8
i 4 n 1 i 4 n 1
15.If n N , the value of
is
2
1) -4
8. If z1 1 i, z 2 1 i, then z1 z2 is
____________.
1) 0
2) 2
2) 5
2) 44-117i)
4) Non e
If (3 y 2)i16 (5 2 x )i 0, then
1) x = 5/2
3) Both (1) & (2)
2) y = 2/3
4) None
11
CLASS-VIII
24. If
MPC BRIDGE COURSE
3
5 i 2 5 y 2 , then y =
5
1) 0
2) 1
3) 2
4) 3
25. Matrix match type:
Column-I
Column-II
a) If (3y-2) +i (7-2x) =0, then x = p) 33/4
b) If (3y-2) +i (7-2x) =0, then y = q) 3/4
c) If 4x+i (3x-y) =3-6i, then x =
r) 2/3
d) If 4x+i (3x-y) =3-6i, then y =
s) 7/2
26. If 2x+i4y=2i, then x is ___________
27. If 2x+4iy = i 3 x-y+3, then 3y = _______
28. Which of the following is true ?
1) (3 5) (3 5) 14 ( 44 117i )
2) ( 2 3) ( 3 2 3) 7 3i
3) (2 3i ) 2 ( 5 12i)
4) ( 5 7i ) 2 ( 44 14 14 5) i
DAY-6 : SYNOPSIS
Ordered Pair: A pair of numbers a and b
listed in a specific order with a at the
first place and b at the second place is
called an ordered pair (a, b).
Y
5
4
3
2
1
O
X' -5 -4 -3 -2 -1
-1
1
2 3
NARAYANA GROUP OF SCHOOLS
X
-3
-4
-5
Y'
The point 0 is called the origin.
The configuration so formed is called the
coordinate system or coordinate plane.
Coordinates of a point in a plane: Let P be
a point in a plane. Let the distance of P
from the y-axis = a units. And, the
distance of P from the x-axis = b units.
Then, we say that the coordinates of P
are (a, b).
a is called the x-coordinates or abscissa
of P.
b is called the y-coordinates or ordinate
of P.
Y
Note that a, b b, a .
Thus, (2, 5) is one ordered pair and (5, 2)
is another ordered pair.
CO-ORDINATE SYSTEM
We represent each point in a plane by
means of an ordered pair of real
numbers, called the coordinates of that
point.
The position of a point in a plane is
determined with reference to two fixed
mutually perpendicular lines, called the
coordinate axes.
On a graph paper, let us draw two
mutually perpendicular straight lines
X 'OX and YOY ' , intersecting each other
at the point O. These lines are known as
the coordinate axes or axes of reference.
The horizontal line X 'OX is called the
x-axis. The vertical line YOY ' is called
the y-axis.
4 5
-2
P(a, b)
b
X'
O
a
X
Y'
Consider the point P shown on the
adjoining graph paper. Draw PM OX .
Quadrants: Let X 'OX and YOY ' be the
coordinates axes. These axes divide the
plane of the graph paper into four
regions, called quadrants.
The region XOY is called the First
Quadrant.
The region YOX ' is called the Second
Quadrant.
The region X 'OY ' is called the Third
Quadrant.
The region Y 'OX is called the Fourth
Quadrant.
12
CLASS-VIII
MPC BRIDGE COURSE
Using the convention of signs, we have
the signs of the coordinates in various
quadrants as given below:
Region
Quadrant
Nature of x and y
Signs of coordinates
XOY
I
x > 0, y > 0
(+, +)
YOX'
II
x < 0, y > 0
(–, +)
X'OY '
III
x < 0, y < 0
(–, –)
Y 'OX
IV
x > 0, y < 0
(+, –)
Any point on x-axis: If we consider any
point on x-axis, then its distance from
x-axis is 0. So, its ordinate is zero. Thus,
the coordinates of any point on x-axis is
(x, 0).
Any point on y-axis: If we consider any
point on y-axis, then its distance from
y-axis is 0. So, its abscissa is zero. Thus,
the coordinates of any point on y-axis is
(0, y).
Slope of x-axis is 0; slope of y-axis not
defined.
Distance between two points
Let A
x1 , y1 , B x2 , y2 be
any two points
on a line not parallel to the axes. From
the adjacent figure we have the right
angle triangle ABC.
Y
B
x1, y 1
A
X
O
C y2
x 2 x1
y1
x1
AB2 AC2 BC2
But AC x2 x1,BC y2 y1
2
AB
2
2
x2 x1 y2 y1
2
NOTE : The distance to the point A x1 , y1
from origin is
x12 y12
NARAYANA GROUP OF SCHOOLS
Conceptual Understanding Questions :
1. If the x co-ordinate of a point is 2 and its
y co-ordinate is 3, then it is represented
as
1) 2, 3
2) 3, 2
3) (2, 3) 4) (3, 2)
2. If the abscissa & ordinate of a point are
3 and 2 respectively then the point is
represented as
1) 2, 3
2) 3, 2
3) (2, 3) 4) (3, 2)
3. If a point is at a distance of 2 units
from Y – axis and 3 units from X – axis
then the point is represented as
1) 2, 3
2) 3, 2
3) (2, 3) 4) (3, 2)
4. A Point (4, 0) lies on
1) X – axis
2) Y – axis
3) Origin
4) X and Y axes
5. A Point (0, 5) lies on
1) X – axis
2) Y – axis
3) Origin
4) X and Y axes
6. The distance between two points (0,0)
and ( 2,5) is
1) 10 units
2)
29 units
3)
4)
27 units
7 units
7. The distance between two points (2,2)
and ( 5,4) is
1) 13 units
2)
5 units
3)
4)
27 units
7 units
8. The distance between two points (–2,3)
and ( 4,0) is
x2
AB2 x2 x1 y2 y1
DAY-6 : WORKSHEET
1) 15 units
2) 2 5 units
3) 3 5 units
4)
35 units
Single Correct Choice Type :
9. In which of the following quadrant does
the given point (3, –8) lie?
1) I quadrant
2) II quadrant
3) III quadrant
4) IV quadrant
10.In which of the following quadrant does
the given point (–5, 1) lie?
1) I quadrant
2) II quadrant
3) III quadrant
4) IV quadrant
13
CLASS-VIII
MPC BRIDGE COURSE
11.In which of the following quadrant does
the given point (–6, –8) lie?
1) I quadrant
2) II quadrant
3) III quadrant
4) IV quadrant
12.The Horizantal axis is called.
1) X axis
2) Y axix
3) Origin
4) I Quardrant
13.The nearest point from the origin is
1) (2, –3) 2) (5, 0) 3) (0, –5) 4) (1, 3)
14.If Q(x,y) lies in the Fourth Quadrant then
x is
1) Positive
2) Negetive
3) Both 1 & 2
4) None
15.The triangle formed by (0, 1), (1, 0) and
(1, 1) is(through graph)
1) Right angle isoceles triangle
2) Scalene triangle
3) Equilateral triangle
4) Cannot form a triangle
16.The X co ordinate on Y axis is
1) 0
2) 1
3) Undifine
4) None
17.The distance between (4,–3) (–4,3) is
1) 10units
2) 12units
3)14units
4)11units
18. If the distance between the points
(5,–2) (1,a) is 5, then the value of a is
1) 5units
2) 2units
3)4units
4) 1unit
19. The point on x-axis which is equidistant
from the points (5, 4), (–2, 3) is
1) (2,0) 2) (4,0)
3)(12,0) 4) (5,0)
20. The distance between A(7,3) and B on
the x-axis whose abscissa is 11 is
1) 12units
2) 10 units
3) 5units
4) 15units
MATRIX MATCH TYPE
21.Column-I
Column-II
a) Distance between (5, 3), (8, 7) 1) -3
b) Distance between (0, 0), (-4, 3) 2) 5
1 3 3 1
c) Distance between , , , 3) 3
2 2 2 2
4) If (1, x) is at
10 units from(0, 0)
then x =
NARAYANA GROUP OF SCHOOLS
4) 5
5) 4
DAY-7 : SYNOPSIS
Dividing a line segment in a given ratio
(section formulae) :
P
A
(P divides AB in the ratio m:n internally.)
n
m
P
A
B
(P divides AB in the ratio m:n externally.)
Section formulae : The point ‘P’ which
divides the line segment joining the
points A x1 , y1 , B x2 , y2 in the ratio m:n
mx2 nx1 my2 ny1
,
i) internally is
; m n 0
mn
mn
mx2 nx1 my2 ny1
,
ii) externally is
; m n
mn
mn
Mid point of a line segment : The mid point
of
line
segment
joining
of x1 , y1
x1 x2 y1 y2
,
2
2
and x2 , y2 is
NOTE :
1. The point P(x,y) divides the line segment
joining A x1 , y1 and B x2 , y2 in
x1 x : x x 2 (or) y1 y : y y 2
i.e AP =
PB = x1 x x x 2
2. x-axis divides the line segment joining
x1 , y1
and
x2 , y2
in the ratio y1 : y2
3. y-axis divides the line segment joining
x1 , y1
and
x2 , y2
in the ratio x1 : x2
Second - order determinant :
The expression
a b
is called the secondc d
order determinant.
14
CLASS-VIII
MPC BRIDGE COURSE
a b
It is defined as
= ad-bc
c d
4 3
4 1 3 2 4 6 2.
2 1
Example :
Area of a triangle :
1. The area of the triangle formed by the
points A x1 , y1 , B x2 , y2 and C x3 , y3
=
1
2
(or)
x y
1
2
y3 (or)
1 x1 x2
2 y1 y2
x1 x3
y1 y3
1 x1 x2 x3 x1
2 y1 y2 y3 y1 sq.units
2. The area of the triangle formed by the
points O 0, 0 , A x1 , y1 , B x2 , y2
1
x1 y2 x2 y1 sq.units.
=
2
4. x - axis divides the line segment joining
(2,–3), (5,7) in the ratio is
1) 1 : 2
2) 3 : 7
3) 4 : 5
4) 3 : 4
5. The area of the triangle formed by the
points (0,0), (2,0), (0,2) is
1)4 sq. units
2) 2 sq.units
3) 3 sq.units
4) 0
Single Correct Choice Type :
6. If the points
3, 8 , 4, 11
and
5, k
are collinear then, the value of k is
1) 14
2) -8
3) 4
4) 5
7. The triangle formed by (0,1), (1,0) and
(1,1) is
1) Right angle isosceles triangle
2) Scalene triangle
3) Equilateral triangle
4) Cannot form a triangle
8. The mid point of the line joining the points
1, 4
1) 5
and x, y is
2) 5 2
2,3
then x y is
3) 7
4) -5
NOTE :
1. Three points A,B,C are collinear if the
area of ABC is zero.
9. If the point p 2,3 divides the line joining
2. If D,E,F are the mid points of the sides of
the ABC
then
the
area
of
is
1) 1: 2 internally 2) 1: 2 externally
3) 2 :1 internally 4) 2 :1 externally
10. The coordinates of the point which
divides the line segment joining points A
(0,0) and B(9, 12) in the ratio 1 : 2 are
1) (–3, 4)
2) (3, 4)
3) (3, –4)
4) None of these
11. The point which divides the line joining
ABC = 4 (area of DEF ).
3. If G is the centroid of the ABC then area
of ABC = 3(area of GAB )
DAY-7 : WORKSHEET
Conceptual Understanding Questions :
1. The vertices of a triangle are A(0,–4) ,
B(4,0) and C(0,0), so ABC is
1) Right angled triangle
2) Isosceles triangle
3) Right angled, Isoceles triangle
4) Equilateral triangle
2. The mid point of (1,2) and (3,4) is
1) (2,3)
2) (3,2)
3) (2,4)
4) (1,3)
3. The ratio in which (2,3) divides the line
segment joining (4,8), (–2,–7) is
1) 2 : 1 externally 2) 2 : 3 internally
3) 4 : 3 externally 4) 1 : 2 internally.
NARAYANA GROUP OF SCHOOLS
the points 5,6 and 8,9 ,then the ratio
the points
a b,a b
and
a b,a b
in the ratio a : b is
a 2 b2 a b 2
a 2 b2 b2 ab
2)
,
,
1)
ab
ab
ab
ab
a 2 b2 a 2 b2 2ab
,
3)
4) None of these
a b
ab
12.The ratio in which the line segment
joining the points (3, –4) and (–5, 6)
is divided by the x – axis is
1) 2 : 3
2) 3 : 2
3) –2 : 3 4) None
15
CLASS-VIII
MPC BRIDGE COURSE
13.Let P and Q be the points on the line
segment joining A(–2, 5) and B(3, 1) such
that AP = PQ = QB. Then the midpoint of
PQ is
1
1
1) ,3 2) ,4 3) 2,3
2
2
4) 1,4
14. The coordinates of points A, B, C are
x1 , y1 , x 2 , y 2 and x 3 , y 3 and
Exponential form: The product of a
number ‘x’ with itself n times (n is natural
number)
is
given
by
x x x x......... x (n times) and is
written as x n which is called the
exponential form. Here x is called base
and n is called the exponent (or) index of
‘x’. xn can be read as nth power of x (or) x
raised to the power ‘n’.
point D divides AB in the ratio l : k. If P
divides
line
DC
in
the
ratio
Exponential form is also called
as power notation.
m : k , then the coordinates of P are
Example: 2 2 2 2 2 25
kx1 lx 2 mx 3 ky1 ly 2 my 3
,
1)
k l m
k l m
Here 2 2 2 2 2 is called the product
form (or) expanded form and 25 is called
the exponential form.
lx1 mx 2 kx 3 ly1 my 2 ky 3
,
2)
l mk
l mk
mx1 kx 2 lx 3 my1 ky 2 ly 3
,
3)
m k l
m k l
4) None of these
15. P = (– 5,4) and Q = (–2,–3). If PQ is
produced to R such that P divides QR
externally in the ratio 1 : 2, then R is
1) 1,10
2) 1, 10 3) 10,1 4) 2, 10
DAY-8 : SYNOPSIS
Variable: A symbol which can take
various numerical values is called a
variable.
Example : x, y, z, a, b, c etc.
Note:
Variables are also known as
Note:
Note:
* The first power of a
number is the number itself. i.e., a1 = a.
* The second power is called square.
Example: Square of ‘3’ is 32
* The third power is called cube.
Example: Cube of x is x3
* 1 raised to any integral power gives ‘1’
Example: 1100 = 1
* (– 1)odd natural number = –1
Example: (– 1)375 = –1
Laws of Exponents (or) Indices:
1. The product of the two powers of the same
base is a power of the same base with
the index equal to the sum of the indices.
i.e., If a 0 be any rational number and
m, n be positive integers, then
a m a n a m n
m n
literals.
Constant: A symbol having a fixed value
is called a constant.
2. x
3
Example: 3, 2010, –5,
etc.
7
Term: Numericals or literals or their
combination formed by operation of
multiplication are called terms.
3. Power of Product ab a n bn where
Example: 5n,
x
, 10x 2 ,7l 5 m , y, 5, x 2 ,
6
x mn ,Where m,n are positive
integers, x 0
n
a 0, b 0 , and n is a positive integer
4. Quotient of powers of the same base.
a m n
am
1
a n nm
a
if
mn
if
nm
etc.
NARAYANA GROUP OF SCHOOLS
16
CLASS-VIII
MPC BRIDGE COURSE
5. Power of a quotient
m
m
a
a
i.e., m where a 0, b 0 , and m
b
b
is a positive integer.
6. Powers with exponent zero: If we apply
the above laws of indices of evaluate
am
where m = n, and a 0 ; then
an
10.
4
If p = 32
10
A) 32 5
C)
11.
32
10
3
5
5
If
16
8
B)
2
1. If x is the fourth power of 2 and y is the third
power of 4, then (x + y)2 + (y – x) + (y + x) is
A) 6528 B) 6526 C) 6524 D) 6520
2. If 212a = 224, 33x = 312, then ax × xa is
A) 256
B) 64
C) 512
D) 1024
3. If a = 4, b = 16 and c = 32, then a3b4c5 is
A) 247
B) 248
C) 245
D) 249
4. If x4 = 16 and y5 = 243, then x × y is
A) 6
B) 8
C) 4
D) 9
12
4
3
2
5. If P = 4 × 8 × 16 × 32 × 643, where
p = 2x, then (p)10 is
A) 2764
B) 2768
C) 2760
D) 2756
6. If xx + 3 = 32, yy + 3 = 729, then x x+y × y x+y is
32
10
5
10
D) 32 5
5
4
5
x
5
5
÷
=
and
16
16
6 6
×
19 19
DAY-8: WORKSHEET
4
m
where p5 = 32 , then
p × m is
17
am
a m n a 0 1
an
5
4
y
6
=
, then (x + y)3
19
A) 243
B) 126
C) 123
D) 244
12. If xa – b × xb – c × xc – a = p, then the value of
p is
A) 1
B) 0
C) 2
D) – 1
12
9
8
6
13. If a = 5 ÷ 5 , b = 4 ÷ 4 and c = 34 ÷ 32,
then (a2b3c4)2 is
A) (5 × 4 × 3)12 × 33 B) (5 × 4 × 3)12 × 32
C) (5 × 4 × 3)12 × 34 D) (5 × 4 × 3)12 × 35
14. If a 8 × 4 b = (2)8 × (4) 3, p q = 5 2, then
ap × qb
is
aq × qa
A) 16
B) 8
C) 32
D) 64
DAY-9 : SYNOPSIS
A) (6)5
B) (12)5
C) (3)10
3
D)
2
5
Natural Numbers: Counting numbers 1, 2,
3, 4,..... are called Natural numbers,
denoted by N. N = { 1, 2, 3, 4, ........}.
2
7. If (4
–1
2
1
× 3 ) = , then x is
x
–1 2
A) 24
B) 6
C) 18
D) 12
-3
23
2
8. If (3 – 2 ) × is multiplied by
,
5 × 33
3
then the resultant is
2
2
2
A) 1
B) 0
C) – 1
D)
3
3 2 2
2 3 3
9. If (2x y ) × (4a b ) = p where x = 2y, y =
3a, a = 4b and b = 2, then the value of p
is
A) 256 (48)6(24)4(8)6(2)9
B) 256(48)5(24)4(2)9
C) 256(48)6(24)4(8)5(2)9
D) 256(48)6(74)3(8)5(2)8
NARAYANA GROUP OF SCHOOLS
Note:
* The smallest Natural numbers
is 1 and the largest number can’t be
determined.
* The number of natural numbers between
a and b where a < b is b – a –1.
* The number of natural numbers from a
to b where a < b is b –a + 1.
Whole Numbers : The natural numbers
along with zero are called whole
numbers, denoted by W.
W = { 0, 1, 2,
3, 4, .......}.
Note:
* The smallest whole number is
‘0’ and the largest whole number cannot
be determined
* All natural numbers are whole numbers.
17
CLASS-VIII
* The difference of any two consecutive
whole numbers is ‘1’.
Fractions:
* Tej has taken a card board which is in
the shape of a square.
* He has cut the card board as shown.
He has observed that he got two pieces
of card board and each is ‘a part of the
whole’. Hence, we can say that each part
is a fraction.
* Therefore from the above illustrations we
say that “A part of the whole is called
Fraction.”
Compa rison of fractio ns: By c ross
multiplication: If two fractions
a
and
b
c
are to be compared ,we cross multiply
d
a c
(i) If a d b c, then
b d
a c
(ii) If a d b c, then
b d
a c
(iii) If a d b c, then
b d
2
5
Example: Compare the and
3
6
Solution: On cross multiplication we get
2 6 and 3 5 12 and 15 12 < 15
2
MPC BRIDGE COURSE
Solution: The L.C.M of 5,4,2,10 = 20
2 2 4
8 1 1 5
5
,
Now
,
5 5 4 20 4 4 5 20
3 3 10 30
9
9 2 18
,
2 2 10 20 10 10 2 20
Now compare the numerators of like
8 5 30 18
,
,
,
fractions
20 20 20 20
Arranging them in ascending order, we
5
8
18 30
1 2 9
3
get
So,
20 20 20 20
4 5 10 2
Fundamental operations on fractions:
Addition: While adding like terms, add
the numerators and retain the common
denominator.
In, general
a c a c
b b
b
Multiplication of fractions:
In, general
If
a
c
and
are two fractions, then the
b
d
product of these fractions =
product of their numerators
a c
=
b d product of their denomi n ators
Example: Multiply
Solution:
By taking the L.C.M: Taking the L.C.M of
the denominator of the given fraction.
Convert each of the fraction into an
equivalent fraction with denominator
equal to the L.C.M. Compare their
numerators . The higher the value of the
numerator, the greater is the fraction
Example: Arrange
2 1 3 5
, , ,
in ascending
5 4 2 10
order.
NARAYANA GROUP OF SCHOOLS
Example: Add
3 4 34 7
5 5
5
5
Subtraction: While subtracting like terms,
subtracting the numerators and retain
the common denominator.
5
36
a c a c
b b
b
3
2
and .
7
5
3 2 3 2
6
7 5 7 5 35
Division of fractions: If
fractions, then
a c
a d
=
.
b d
b c
Example: Divide
Solution:
a
c
and
are two
b
d
5 25
9 3
5 25 5 3
1
9 3
9 25 15
18
CLASS-VIII
MPC BRIDGE COURSE
DAY-9: WORKSHEET
Conceptual Understanding Questions :
2
3
1. Compare and
is
3
4
2 3
2 3
2 3
1)
2) 3) 4) Both 1 & 3
3 4
3 4
3 4
2
2. A Fraction greater than
is
5
1
1
1
1
1)
2)
3)
4)
3
2
5
4
3. 10 paise as fraction of Rs. 1 is
1
2
3
1
1)
2)
3)
4)
5
5
10
10
2 1
1
4. Ascending order of , ,and
is
3 2
6
1 1 2
2 1 1
1) , ,
2) , ,
6 2 3
3 6 2
1 2 1
2 1 1
3) , ,
4) , ,
6 3 2
3 2 6
2 1
3
5. Descending order of , and
is
5 3
7
2 1 3
1 2 3
1) , ,
2) , ,
5 3 7
3 5 7
3 2 1
3 1 2
3) , ,
4) , ,
7 5 3
7 3 5
2 3
6. =
5 5
1) 1 2)
7.
2 3
=
3 2
13
1)
6
7 5
=
8.
12 7
2
1)
7
5
5
3)
3 2
5 5
15
16
and
is
8
27
5
10
2
1)
2)
3)
9
9
9
1 5
2
11. The value of 2 of
2 2
15
15
2
1)
2)
3) 1
2
15
10. Product of
4) All the above
4)
5
16
4)
5
6
4)
18
28
4)
7
13
Single Correct Choice Type :
1. Equivalent fraction of
1)
2.
15
16
2)
12
20
Simplest form of
1)
2
3
2)
513
503
3)
3
is
4
21
28
2012
is
2000
3)
503
500
5
m- 2
=
,then the value of m is
10
30
1) 15
2) 17
3) 20
4) 7
4. The sum of three sides of a triangle is
3
cm. If two of its sides measure
16
5
3
7
cm and 6
cm respectively, then
5
4
10
the length of the third side is
3
5
1) 4
cm
2) 2 cm
20
17
7
3
3) 3
cm
4) 4 cm
20
5
3. If
DAY-10 : SYNOPSIS
5
2)
5
2)
1
1
3 =
3
2
7
7
1)
2)
6
6
6
3)
6
11
84
3)
6
4)
13
2
19
4)
1
48
9. 2
3)
5
6
NARAYANA GROUP OF SCHOOLS
4)
5
7
Definition: If N and a 1 are any two
positive real numbers and for some real
x, such that a x N , then x is said to be
logarithm of N to the base ‘a’. It is written
x
as log a N x . Thus, a N log a N x
Note:
* It should be noted that “log”
is abbrevation of the word “logarithm”.
* Logarithms are defined only for positive
real numbers.
19
CLASS-VIII
* There exists a unique ‘x’ which satisfies
the equation a x N .
1 If x1, x 2 ,......x n are positive rational
numbers, then
So log a N is also unique.
Logarithmic Function: Functions defined
by such equations are called logarithmic
functions.
We can express exponential forms as
logarithmic forms.
Exponential form
Logarithmic form
(i) 24 = 16
4 log 2 16
(ii)
53 125
3 log 5 125
(iii)
32
2 log 3 1
1
9
9
Note:
* For any positive real number ‘a’ we have
a 1 a . Therefore
log a a 1 .
x
* If a N x log a N
‘x’
by
log a x1 , x 2 ,..x n log a x1 log a x 2 log a x 3 .... log a x n
2. If m and n are two positive rational
m
numbers, then log a log a m log a n
n
3. log a m
n
4. log a n m
5. log b n a
m
n. log
a
m
1
log a m
n
m
log b a
n
6. If m 1, b 1 are positive real numbers
i.e., The logarithm of any non - zero
positive number to the same base is unity.
Replacing
MPC BRIDGE COURSE
GENERALISATION:
log a N in a x N
log m a
then log b a log b
m
7. If a 1, b 1 are two positive real number
1
then log b a log b .
a
a log a N N
* The logarithm of unity to any non - zero
base is zero.
DAY-10: WORKSHEET
o
Recall that, if a 0, a 1 log a 1 0
LAWS OF LOGARITHMS:
1. If m, n are positive rational numbers,
then log a mn log a m log a n .
i.e., The ‘log’ of the product of two
numbers is equal to the sum of their ‘logs’.
Proof: Let log a m x
and
x
log a n y
from (1) & (2)
mn a x .a y mn a x y
Apply log to the base ‘a’ on both sides
log a mn x y
log a mn log a m log a n
NARAYANA GROUP OF SCHOOLS
1. If 24 16 , then which of the following is
true ?
1) log 2 4 16
2) log 2 16 2
3) log 4 16 4
4) log 2 16 4
2. log 0.1 0.1 = ______
y
m a ....... 1 and n a ....... 2
x log a m and y log a n
Conceptual Understanding Questions :
1) 0.1
2) 0.2
3) 0
4) 1
3) 27
4) 275
3. 5 log 5 27 = ______
1) 5
2) 55
4. log 213 1 ____
1) 1
2) 213
3)
1
213
4) 0
20
CLASS-VIII
MPC BRIDGE COURSE
9. log 3.5 = _______
5. log 24 = ____
1) log 4 log 6
2) log 2 log12
3) log 3 log 8
4) All of these
6. log 3 4 log 2 3 =
1) log 32 4 3
3) log
4
3
log
2
3
4) log 2 4
4
2) log 3 2
4
4) 44 log 3 2
3) 4 log 3 2
1) log10 5 x 2
2) 1
3) only (1)
4) Both 1 & 2
11. log10 20 log10 2 ________
7. log 34 2 ______
1) (lo g 3 2 )
log 20
1) log10
2) log10
3) log10 (20 2)
20
4) log 2 10
10
Single Correct Choice Type :
1. log 4 x 2 , then x = ___
1) 14 2) 16
3) 12
2
2. If 3
4) 11
1
, then which of the following is
9
true ?
1) log 2 3
3) log 3
1
9
2) log 3 2
1 1
9 2
4) log 3
1
9
1
2
9
1) 3
2) 2
3) 9
4. The value of log100.0001 is
1) –4
2) 4
3) –10
1) 2
4) 81
4) 10
4 is
2
2) 4
6. The value of x if log
3)
12. The value of
1) 2
2
4) 3
216 x is
6
1) 5
2) 6
3) 4
4) 3
7. Logarithm of any non-zero number to the
same base is _______
1) Same non-zero number
2) 0
3) 1
4) Negative of the non-zero number
8. Logarithm of unity to any non-zero base
is _______
1) Non-zero number 2) Unity
3) 0
4) –1
NARAYANA GROUP OF SCHOOLS
3) 5
13.The value of log 3 27 3
1)
1
2
2)
20
2
log 8
is
log 2
2) 4
3
2
3)
4) 6
is
5
2
4)
7
2
14. log b x log a b =
1) logxa
3. The value of log 9 81 =
5. The value of log
log 35
3) log10
10. log10 5 log10 2 _____
log 4 log2
2) log 3 log3
4
4
35
2) log 35 log10
10
4) Both 1&2
1) log
2) logxb
3) logax
4) logbx
15. logamn =
1) logam × logan
2) logam + logan
m
3) log a
n
4) –logam – logan
16. If log10 2 0.3010 , then logarithm of
5
32
with base 10 is
1) 9.725
2) 7.525
3) 7.725
4) 9.525
17. log10 25 log10 4 =
1) 4
2) 3
3) 2
4) 1
18. Express 2 log 5 5 3 log 2 log 50 1 as
single logarithm is
1) log 5 2) log 3
3) 1
4) log 2
21
CLASS-VIII
DAY-11 : SYNOPSIS
Sequence: It is an arrangement of numbers in a definite order according to same
rule
Example: 5, 10, 15, 20, ...................... etc.
and 4, 44, 444, ..............
i) Tn = 5n + 2 represents a sequence 7,
12, 17 ...............
ii) Tn = 3n - 4 represents a sequence -1,
2, 5 .......
iii)
Tn =
2n 3
represents a sequence
2
5 7 9
,
,
, ................
2 2 2
Series: An expression consisting of the
terms of a sequene, alternating with
the symbol “ + ” is called a “ Series ”
For example, associated with the
sequence
3 5 7
2n 1
, , .....,
........
5 7 9
2n 3
MPC BRIDGE COURSE
Arithmetic mean (A.M.): The arithmetic
mean between two numbers is the
number which when placed between
them forms with them an arithmetic
progression.
Thus the arithmetic mean between a and
b is A.M =
a b
where a and b are any
2
two positive numbers.
If there are ‘n’ A.M’s between a, b then
common difference, d
ba
n 1
Sum of first ‘n’ natural numbers,
S1 n
n n 1
2
Sum of the squares of first ‘n’ natural
numbers, S 2
n
2
n n 1 2n 1
6
Sum of the cubes of first ‘n’ natural
2
3 5 7
2n 1
........
we have series , , .....
5 7 9
2n 3
nth term: The number occuring at the nth
place of a sequence is called its nth term.
Denoted by Tn.
Progression: The sequence which obey
the definite rule and its general term is
always expressible in terms of ‘n’ is called
progression.
Example: 1) 2, 4, 6, 8, 10, ......
2) 1, 3, 9, 27 ...........
1 1 1
3) , , ,..........
5 7 9
Arithmetic progression (A.P):A Sequence
is called an arithmetic progression if its
terms continually increase or decrease
or decrease by the same number.
Example:a, a + d, a + 2d, a + 3d, ... is in A.P
nth term of A.P is tn = a + (n-1)d
Sum
of
‘n’
terms
of
A.P
Sn
n
n
2a n 1 d and S n a 1
2
2
n 2 n 1
2
numbers, S3 n
n
4
3
S n t1 t2 ...........tn 1 t n = tn ; tn sn sn 1
DAY-11: WORKSHEET
Conceptual Understanding Questions :
1. Choose a sequence from the given
1) 1,3,5,7,9............2) 1,2,4,5,7,9,10,11.....
3) -1,1,7,10,20......4) 1,7,10,20,25,....
2. Choose a series from the given
1) 1 + 2 + 4 + 5 + 7 + 9.............
2) 1 + 3 + 5 + 7 + 9.........................
3) -1 + 1 + 7 + 10 + 20..............
4) 1 + 7 + 10 + 20 + 25...................
3. 0,4,8,12,16 forms a / an
1) Arithmetic sequence
2) Geometric sequence
3) Arithmetic progression
4) Geometric progression
where l is the last term.
NARAYANA GROUP OF SCHOOLS
22
CLASS-VIII
4. The first term of an Ap is ‘a’ and its common difference is 2, then first five terms
of given AP are
1) a, a + 2, a + 4, a + 6, a + 8
2) a + 2, a + 4, a + 6, a + 8, a + 10
3) a, a - 2, a - 4, a - 6, a - 8
4) a, 2 - a, 4 - a, 6 - a, 8 - a
4. If the first term of an AP is 3 and its
common difference is 5, then its
n th
term is
1) 3 + 5n 2) 5n - 2 3) 3n - 5 4) 3 - 5n
5. If 11, 6, 1, -4, -9 forms a AP, then its
common difference is
1) -5
2) 4
3) 5
4) -4
Single Correct Choice Type :
6. The next term in the series 7, 12, 19 ....
1) 29
2) 28
3) 26
4) 24
7. 3, 7, 15, 31, 63, (..............)
1) 92
2) 115
3) 127
4) 131
8. If 5x + 2, 4x - 1, x + 2 are in A.P, then the
first term is
1) 18
2) 17
3) 19
4)16
9. If the nth term of AP is 4n - 1 then the
18th term is
1) 72
2) 73
3) 74
4) 71
10. If 8k + 4, 6k - 2, 2k + 12 are in AP then
the value of k is
1) -10
2) -11
3) 9
4) 10
th
11. The 25 term of the A.P. 10, 6, 2, -2, -6,
-10, ........ is
1) - 86
2) 106
3) 96
4) 46
12. If first term is 3 and 38th term is 114
ehn the 68th term is
1) 204
2) 202
3) 200
4) 208
rd
th
13. If the 3 and 7 terms of an A.P. are 17
and 27 respectively, then the
first
term of the A.P is
1) 9
2) 12
3) 14
4) 16
14. If a, x1, x2, x3.......xn b are in A.P then xn
1)
an b
an b
a nb
a b
2)
3)
4)
n 1
n 1
n 1
n
13.If the nth term of the A.P 23, 25, 27, 29
... and -17, -10, -3, 4 ..... are
equal, then the value of n = _________
1) n = 10 2) n = 8 3) n = 9 4) n = 12
NARAYANA GROUP OF SCHOOLS
MPC BRIDGE COURSE
DAY-12 : SYNOPSIS
Geometrical Progression:
A sequence (finite or infinite) of non-zero
number in which every term, except the
first one, bears a constant rati with its
preceding term, is called a geometric progression, abbreviated as G.P.
Illustration: The sequences given below:
i) 2, 4, 8, 16, 32,...........
ii) 3, -6, 12, -24, 48,.............
iii)
1 1 1 1
1
, , ,
,
,........
4 12 36 108 324
Note: In a G.P. any term may be obtained
by multiplying the preceding term by the
common ratio of the G.P. Therefore, if
any one term and the common ratio of a
G.P. be known, any term can be written
out, i.e., the G.P. is then completely
known.
In particular, if the first term and the common ratio are known, the G.P. is completely known. The first term and the
common ratio of a G.P. are generally denoted by a and r respectively.
General term of a G.P.: Let a be the first
term and r 0 be the common ratio of
a G.P. Let t1, t2, t3, ........, tn denote 1st,
2nd, 3rd, ...., nth terms, respectively. Then,
we have
t 2 = t1r,
t 3 = t 2 r, t 4 = t 3 r, ..., tn = t n-1r.
On multiplying these, we get
t 2 t 3 t 4 ... tn = t1t 2 t 3 ... t n-1r n-1 t n = t1r n-1, but t 1 = a.
General term = tn = arn-1.
Thus, if a is the first term and r the common ratio of a G.P. then the G.P. is a, ar,
ar2, ... ar2-1 or a, ar, ar2........ according
as it is finite or infinite.
If the last term of a G.P. consisting of n
terms is denoted by l, then l = arn-1.
Note:
* If a is the first term and r the common
ratio of a finite G.P. consisting of m
terms, then the nth term from the end is
given by arm -n.
23
CLASS-VIII
th
* The n term from the end of a G.P. with
the last term l and common ratio r i s
l/r n-1.
* Three numbers in G.P. can be taken as
a/r, a, ar, four numbers in G.P. cn b e
taken as a/r3, a/r, ar, ar3, five numbers
in G.P. can be taken as a/r2,
a/r, a, ar, ar2, etc.....
* Three numbers a, b, c are in G.P. if and
only if b/a = c/b, i.e., if and only if
b 2
= ac.
DAY-12: WORKSHEET
Conceptual Understanding Questions :
1. 1,2,22,23,24,......... forms a / an
1) Arithmetic sequence
2) Geometric sequence
3) Arithmetic progression
4) Geometric progression
2. If the first term of a GP is 3 and its common ratio is 3, then first five
terms of the GP are
1) 3,6,9,12,15
2) 3,5,7,9,11
3) 3,9,27,81,243 4) 3,7,11,15,19
3. if t7 of a GP is 128 and its common ratio
is 2, then its first term is
1) 1
2) 2
3) 3
4) 4
4. The first term of a GP is 1, and common
ratio is 2. Then the sum of its 5
terms is
1) 20
2) 25
3) 27
4) 31
5. Geometric Mean of 27 and 3 is
1) 4
2) 6
3) 7
4) 9
Single Correct Choice Type:
6. If x, 2x + 2, 3x + 3, ..... are in geometric
progression, the fourth term is
1) -27
2) 13
1
2
3) 12
4) 13
7. If the 10 th term of a G.P is 9 and 14 th
term is 4, then the 7th term is
1) 2
2) 6
3) 4
4) 8
th
th
th
8. If 5 , 8 , and 11 term of G.P. are p, q
and s respectively, then
1) p2 = qs 2) q2 = ps 3) s2 = pq 4) s = pq
9. If x + 9, x - 6, 4 are in G.P then the value
of x is
1) 8
2) 12
3) 16
4) 20
NARAYANA GROUP OF SCHOOLS
MPC BRIDGE COURSE
10. The first term of the progression G.P. is
50 and the 4th term is 1350, then the 5th
term is
1) 8050 2) 5050 3) 4050 4) 6050
11.Sum of three consecutive terms in a G.P.
is 42 and their product is 512.
Find
the largest of these numbers
1) 28
2) 16
3) 32
4) 30
12.The sum of the 8 terms of 3, 6, 12,
24,....... is
1) 765
2) 465
3) 565
4) 645
13. Number of terms of a G.P. 3,3 2,33,.....
are needed to give the sum 120 is
1) 4
2) 2
3) 8
4) 10
14. Sum to ‘n’ terms of the series 0.5 + 0.55
+ 0.555 + ....... equals to
5 10n 1
1) n
9 9.10n
1 10n 1
2) n
9 9.10n
5 10n 1
n
3)
9 9.10n
1 10n 1
n
4)
9 9.10n
15. The sum to n terms of the progression
1, -1, 1, -1, 1,...........(-1)n+1 =
1) 0 if n is even 2) -1 if n is even 3) 0 if
n is odd 4) -1 if n is odd
DAY-13 : SYNOPSIS
1. Introduction : In everyday life, we have
to deal with the collections of objects of
one kind or the other.
For Example :
i) The collection of even natural numbers less
than 12 i.e., the numbers 2, 4, 6, 8 and 10.
ii) The collection of vowels in the English
alphabet i.e., the letters a, e, i, o, u.
In each of the above collections, it is
definitely known whether a given object
is to be included in the collection or not
to be included. Each one of the above
collections is an example of a set.
However, the collection of all intelligent
students in a class of a given school is
not a set. Here it is difficult to decide
who is intelligent and who is not. The
same student may be intelligent in the
eyes of one teacher and may not be
intelligent in the eyes of another. We say
that such a collection is not well defined.
With this basic notion, we have the
following.
24
CLASS-VIII
Definition : Any well defined collection of
objects is called a set.
By ‘well-defined collection’ we mean that
given a set and an object, it must be
possible to decide whether or not the
object belongs to the set.
The objects are called the members or
the elements of the set. Sets are usually
denoted by capital letters and their
members are denoted by small letters.
We write the elements of set with in the
bracess { }.
If x is a member of the set S, we write
x S (read as x belongs to S) and if x is
not a member of the set S, we write x S
(read as x does not belong to S). If x and
y both belongs to set S, we write x, y S.
Representation of Sets : There are two
ways to represent a given set.
1) Roster or Tabular Form or list form. In
this form, list all the members of the set,
separate these by commas and enclose
these within braces (curly brackets)
For example :
i) The set S of even natural numbers less
than 12 in the tabular form is written
as S = { 2, 4, 6, 8, 10 }. Note that 8 S
while 7 S.
ii) The set S of prime natural numbers
less than 20 in the tabular form is
written as S ={2,3,5, 7, 11, 13, 17, 19 }
iii)The set N of natural numbers in the
tabular form is written as N = { 1, 2, 3,
... }, the dots indicating infinitely many
missing positive integers.
2. Set Builder or rule form : In this form,
write one or more (if necessary) variables
(say x, y etc.) representing an arbitrary
member of the set, this is followed by a
statement or a property which must be
satisfied by each member of the set.
For example :
i) The set S of even natural numbers less
than 12 in the set builder form is written
as S = { x/x is an even natural number
less than 12}.
ii) The set of prime natural number less
than 20 in the set builder form is written
as {x/x is a prime natural number less
than 20}.
NARAYANA GROUP OF SCHOOLS
MPC BRIDGE COURSE
The symbol ‘|’ stands for the words ‘such
that’ or ‘where’. Sometimes we use the
symbol ‘;’ or ‘:’ in place of the symbol ‘|’.
iii)The set N of natural numbers in the set
builder form is written as
N = { x :x is a natural number}.
Some Standard Sets : We enlist below
some sets of numbers which are most
commonly used in the study of sets :
i) The set of natural numbers (or positive
integers). It is usually denoted by N.
i.e. N = { 1, 2, 3, 4, .... }
ii) The set of whole numbers. It is usually
denoted by W. i.e. W = { 0, 1, 2, 3, ... }.
iii) The set of integers. It is usually denoted
by Z . i.e. Z = { ...., -3, -2, -1, 0, 1, 2, 3,...}
iv) The set of rational numbers. It is usually
denoted by Q i.e. Q = { x : x is a rational
number} or
m
Q x : x
, where m and n are int egers and n 0
n
v) The set of real numbers. It is usually
denoted by R. i.e. R = { x : x is a real
number } or R = { x : x is either a rational
number or an irrational number }
Note :- i
1
Types of sets :
The Empty set : A set containing no
element is called the empty set.
It is also called the null set or void set .
There is only one such set.It is denoted by
or by { }.
For example :
i) The collection of all integers whose
square is less than 0 is the empty set.
( Square of an integer cannot be negative)
ii) The collection of all girl students in a
boys college is the empty set.
iii) The collection of all the real roots of
the equation x2 + 5 = 0 is the empty set.
( The equation x2 + 5 = 0 is not satisfied
by any real number, for if ‘a’ is a real
root of x2 + 5 = 0, then a2 + 5 = 0 a2 =
– 5, which is not possible as square of a
real number cannot be negative ). The
order of the empty set is zero.
25
CLASS-VIII
Singleton set : A set is said to be a
singleton set if it contains only one
element.
The set { 7 }, { – 15 } are singleton sets. {
x : x + 4 = 0, x Z } is a singleton set,
because the set contains only one integer
namely, –4.
The set { x : x + 4 = 0, x N } is a null set,
because there is no natural number
which may satisfy the equation x + 4 = 0.
A set whose order is 1 is called a
singleton set. Thus, a singleton set is a
set which contains only one distinct
element.
For example :
i) If A = { x : x is a positive divisor of 20 },
then n(A) = 6 as A = { 1, 2, 4, 5, 10, 20 }
ii) If B = { x : x is a positive even prime },
then n(B) = 1 as B = { 2 }.
Note that B is a singleton set.
iii) If C = { x : x is an integer neither
positive nor negative }, then n(C) = 1 as
C = {0}.
C is a singleton set.
Finite and Infinite sets : A set is called finite
if the process of counting of its different
elements comes to an end; otherwise, it is
called infinite. The empty set is taken as finite.
For example : i) The set S = { 2, 4, 6, 8 } is
a finite set.
ii) The set of all students studying in a
given school is a finite set.
iii) The set N of all natural numbers is
an infinite set.
iv) The set of divisors of a given natural
number is a finite set.
v) The set of all prime numbers is an
infinite set.
Order of a finite set : The number of
different elements in a finite set S is
called order of S, it is denoted by O(S) or
n(S).
Note : The order of an infinite set is not
defined.
Equivalent sets : Two finite sets A and B
are said to be equivalent written A ~ B
(or A B), iff they contain the same
number of distinct elements i.e., iff n(A)
= n(B).
NARAYANA GROUP OF SCHOOLS
MPC BRIDGE COURSE
For example : i) The sets { 1 } and {2, 2, 2}
are equivalent.
ii) The sets { 3, 4 } and { x : x2 = 4 } are
equivalent sets.
iii) The sets { a, b, c, d, e }, { 1, 2, 3, 4, 5 }
and { a, e, i, o, u } are
equivalent sets as each of these sets
contains 5 distinct elements.
Equal sets :Two sets A and B are said to be
equal, written as A = B, iff every member
of A is a member of B and every member
B is a member of A.Remember
that equal sets are always equivalent but
equivalent sets may not be equal.
For example :i) The sets { – 1, +1 } and { x
: x2 = 1 } are equal.
ii) The sets { 0, 0 } and { 3 } are not equal,
but they are equivalent.
DAY-13: WORKSHEET
Conceptual Understanding Questions :
1. Which of the following not a well defined
collection of objects
1) The collection of days in week
2) The Colloection of all even integers
3) The Vowels in the English Alphabet
4) The collection of ten most talented in
your Class
2. Write the set {x : x is a positive integer
2
and x 40 } in the roster form
1) {1,2,3,4,5,6}
2) {1,2,3,4,5,6,7}
3) {2,3,4,5,6,7}
4) {0,1,2,3,5,6}
3. Roster form of set of ‘S’ odd natural numbers less than 15 is
1) {1,3,5,7,9,11} 2){1,3,5,7,9,11,13}
3){x/x is odd natural numbers }
4){1,2,4--------}
4. Set builder or roster form of set S of odd
natural numbers lessthan 15 is
1) {1,3,5,7,9,11} 2){1,3,5,7,9,11,13}
3){x/x is odd natural numbers }
4){1,2,4--------}
Single Correct Choice Type:
1.Which of the following collections are sets.
i)
The collection of all prime numbers
between 7 and 19.
ii) The collection of all rich persons in
India.
26
CLASS-VIII
iii) Collection of all factors of 50 which
are greater than 6.
1) (i) and (ii)
2) (i), (ii) and (iii)
3) (ii) and (iii)
4) (i) and (iii)
2. Which of the following collections are
sets.
i) The collection of all months of a year,
beginning with letter J.
ii) The collection of most talented writers
of India.
iii) The collection of all natural numbers
less than 100.
iv) The collection of most dangerous
animals of the world.
1) Only (i)
2) Only (ii)
3) Both (i) and (iii)
4) Both (ii) and (iv)
3. The solution set of the equation
x 2 x 2 0 in roster form is
1) 1, 2
4)
2) 1, 2
3) 1, 2
x : x R,1 x 2
4. If A 6,12 then set builder form is
1) A x : x R and 6 x 12
2) A x : x R and 6 x 12
3) A x : x R and 6 x 12
4) A={6,7,8,9,10,11,12}
5. Which of the following sets are singleton
sets ?
i) A = { x : 3x – 2 = 0, x Q }
ii) B = { x : x3 = 0, x R }
iii) C = { x : 30x– 59 = 0, x N }
iv) D = { x : |x| = 1, x Z }
1) (i) and (iii)
2) Only (iii)
3) (i) and (ii)
4) Only (iv)
6. Which of the following sets are null sets?
i) A = { x : x < 1 and x > 3 }
ii) B = { x : x2 = 9 and 3x = 7 }
iii) C = { x : x2 – 1 = 0, x R }
iv) D = { x : x is an even prime number }
1) (i) and (iii)
2) (i) and (ii)
3) (ii) and (iv)
4) (iii) and (iv)
NARAYANA GROUP OF SCHOOLS
MPC BRIDGE COURSE
7. Which of the following statements are not
true ?
i) { x : 3x + 1 < 10, x N } = { 1, 2, 3, 4 }
ii) { x : 2x < 50, x N } = {1, 2, 3, 4, 5 }
1) Only (ii)
2) Both (i) & (ii)
3) Only (i)
4) none of these
8. Which of the following is an empty set ?
1) { x : x R and x2 – 1 = 0 }
2) { x : x R and x2 + 1 = 0 }
3) { x : x R and x2 – 9 = 0 }
4) { x : x R and x2 = x + 2 }
1
9. If Q = x : x = , where x, y N , then
y
2
Q
3
10. Which of the following sets are finite.
i) Prime numbers set
ii) Human beings set in the world
iii) A real number set between 1 and 2.
iv) The set of multiples of 3.
1) Only (ii)
2) Only (iii)
3) (i) and (ii)
4) (iii) and (iv)
11.Which of the following is an infinite set.
i) Natural number set less than one
million.
ii) The set which can be written by using
the digit 1, repeated any number of
times.
iii) The set of elephants in the world.
iv) Spheres set passing through a given
point.
1) Only (ii)
2) (ii) and (iv)
3) Only (iii)
4) (i) and (iv)
1) 0 Q
2) 1 Q
3) 2 Q
4)
DAY-14 : SYNOPSIS
Cardinal number of set : The number of
distrinct elements contained in a finite
set is called its cardinal number and is
denoted by n(A).
Examples : If A = {1, 2, 3, 4, 5} then n(A) =
5, if B = {1, 2, 3} then n(B) = 3.
Subsets:
Let A, B be two sets such that every
member of A is a member of B, then A is
called a subset of B, it is written as A B.
Thus, A B iff (read as ‘if and only if’)
x A x B.
27
CLASS-VIII
If (read as ‘there exists’) atleast one
element in A which is not a member of
B, then A is not a subset of B and we
write it as A B.
For example:
i) Let A = {–1, 2, 5} and B = {3, – 1, 2, 7,
5}, then A B. Note that B A
ii) The set of all even natural numbers is
a subset of the set of natural numbers.
Some properties of subsets :
i) The null set is subset of every set. Let
A be any set.
A , as there is no element in
which is not in A.
ii) Every set is subset of itself. Let A be
any set.
xA xA
A A.
iii) If A B and B C , then A C . Let
xA .
x B
x C
A C.
iv) A = B iff A B and B A . Let A = B.
A B
A B . Similarly,
x B x A
A B
B A.
Conversely, let A B and B A .
x A x B A B and
x B x A
Ex 2 : A = { 1, 2, 3, 4, 5 }, B = { 2, 3, 4 }
Every element of B i.e., 2, 3 and 4 is also
an element of A.
B A
Further we note that there are two more
elements that are in A and not in B. They
are 1 and 5. Then A B . In such
circumstances we say that B is a proper
subset of A.
Ex 2 : N W Z Q R
Remark 1 : If A B then every element of
A is in B and there is a chance that A
may be equal to B i.e., every element of
B is A, but if A B , then every element
of A is in B and there is no chance that A
may be equal to B i.e., there will exist at
least one element in B which is not in A.
A B A B, A B i.e., A B, B A .
A B
B C
x A x B
MPC BRIDGE COURSE
Ex 1 : If A = { 1, 2, 3 }, then proper subsets of
A are , { 1 },{ 2 },{ 3 }, { 1, 2 }, { 1, 3 }, { 2, 3 }
B A
A=B
Note:1. Two sets A and B are equal iff A B and
B A.
2. Since every element of a set A belongs to
A, it follows that every set is a subset of
itself.
Proper subset: Let A be a subset of B. We
say that A is a proper subset of B if A B
i.e., if there exists atleast one element
in B which does not belong to A. A subset,
which is not proper, is called an improper
subset.
Observe that every set is an improper
subset of itself. If a set A is non-empty,
then the null set is a proper subset of A.
NARAYANA GROUP OF SCHOOLS
Remark 2 : If A B , we may have B A ,
but if A B , we cannot have B A .
Power Set: The set formed by all the
subsets of a given set A is called the power
set of A, it is usually denoted by P(A).
For example:
i) Let A = {0}, then P(A) = { , {0}}. Note
that n(P(A)) = 2 = 21.
ii) Let A = {a, b}, then P(A) = { , {a}, {b}, {a,
b}}. Note that n(P(A)) = 4 = 22
iii) Let A = {1, 2, 3}, then P(A) = { , {1},
{2}, {3}, {1, 2}, {1, 3}, {2, 3}, A}.
Note that O(P (A)) = 8 = 23. In all these
examples, we have observed that n(P(A)) =
2n(A).
Rule to write down the power set of a
finite set A:
First of all write .
Next, write down singleton subsets each
containing only one element of A.
In the next step write all the subsets which
contain two elements from the set A.
Continue this way and in the end write
A itself as A is also a subset of A.
Enclose all these subsets in braces to
get the power set of A.
28
CLASS-VIII
Comparable Sets : Two sets A and B are
said to be comparable iff either A B or
B A.
For example :
i) The sets A = {1, 2} and B = {1, 2, 4, 5}
are comparable as A B.
ii) The sets A = {0, 1, 3} and B = {1, 3} are
comparable as B A.
iii) The sets A = {–1, 1} and B = {x : x2 =
1} are comparable as A B and also B A.
Clearly, equal sets are always
comparable. However, comparable sets
may not be equal
Universal Set:
In any application of the theory of sets, all
sets under investigation are regarded as
subsets of fixed set. We call this set the
universal set, it is usually denoted by X or U
or .
OPERATIONS OF SETS
UNION OF SETS
The union of two sets A and B is the set
of all those elements, which are either
in A or in B (including those which are
in both)
In symbolic form, union of two sets A and
B is denoted as, A B . It is read as “A
union B”.
Clearly, x A B
And, x A B
x A or x B .
x A and x B .
It is evident from definition that
A A B; B A B
SOLVED EXAMPLES
(i) A a,e,i,o,u , B a,b,c
(ii) A 1,3,5 , B 1,2,3
Solution:- (i) We have,
A B a,e,i,o,u a,b,c
MPC BRIDGE COURSE
Here, the common elements 1 and 3 have
been taken only once, while writing A B .
UNION OF THREE OR MORE SETS
The union of n n 3
for sets A 1 ,
A2........., An is defined as the set of all
those elements which are in Ai 1 i n
for atleast one value of i. The union of
n
A1,
A 2,
A 3 ,........A n
is
denoted A i
i 1
In symbols, we write
n
A i { x : x A i for at least one value of
i 1
i, 1 i n }
SOLVED EXAMPLE :- If A = {1, 2, 3, 4}
B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and
D = {7, 8, 9, 10},
find (i) A B
(ii) A B C
(iii) B C D
Solution :(i) We have, A B 1,2,3,4 3,4,5,6
A B 1,2,3,4,5,6
(ii) We have,
A B C 1,2,3,4 3,4,5,6 5,6,7,8
A B C 1,2,3,4,5,6,7,8
(iii) We have,
B C D 3,4,5,6 5,6,7,8 7,8,9,10
B C D 3,4,5,6,7,8,9,10
INTERSECTION OF SETS
The intersection of two sets A and B is
the set of all those elements which belong
to both A and B. Symbolically, we write
A B = {x : x A and x B } and read as
“A intersection B”.
Let
x A B x A and x B
and
A B a,b,c,e,i,o,u
x A B x A or x B
Here, the common element a has been
taken only once, while writing A B .
It is evident from the definition that
A B A, A B B
(ii) We have A B 1,3,5 1,2,3
A B 1,2,3,5
NARAYANA GROUP OF SCHOOLS
SOLVED EXAMPLES
Example 1 : (i) If A = {3,5,7,9,11},
B = {7,9,11,13}, find A B .
29
CLASS-VIII
MPC BRIDGE COURSE
(ii) If A = {a, b, c}, B = , find A B .
Solution. (i) We have,
A B = {3,5,7,9,11} {7,9,11,13}
Since, 7,9,11 are the only elements which
are common to both the sets A and B.
A B 7,9,11
(ii) We have, A B a,b,c .
Since, there is no common element.
Example 2. Let A = {x : x is a natural
number} and B = { x : x is an even natural
number}.
Find A B
Solutions : We have, A = { x : x is a natural
number}. A = {1, 2, 3, 4,......}
B = {x : x is an even natural number}.
B = {2, 4, 6,.......}
We observe that 2, 4, 6,.. are the elements
which are common to both the sets A and
B. Hence, A B ={2, 4, 6...} = B.
INTERSECTION OF MORE SETS
The
intersection
of
n n 3
se ts
A1,A2,A3.........An is defined as the set of
all those elements which are in
A i 1 i n for each i.
The intersection of A 1 ,A 2 .......,A n is
n
denoted by A i
i 1
n
In symbols, we write, A i { x : x A i
i 1
for all i, 1 i n}
DISJOINT SETS : Two sets A and B are
said to be disjoint, if A B =
,
. If
A
B
then A and B are said to be inter-
secting sets or overlapping sets. e.g. Let
A = {1, 2, 3}, B = {a, b, c} A B= , hence
A and B are disjoint.
Solved example :
1. Which of the following pairs of sets are
disjoint ?
i) A = {1, 2, 3, 4},
B = { x : x is a natural number and 4 x 6 }.
ii) A = { a, e, i, o, u}, B= { c, d, e, f}
iii) A = {x : x is an even integer}, B = {x : x
is an odd integer}
NARAYANA GROUP OF SCHOOLS
Solution :i) We have, A = {1, 2, 3, 4}; B = {x : x is a
natural number and 4 x 6 }
B = { 4, 5, 6 } A B = { 1, 2, 3, 4 }
{ 4, 5, 6 } = { 4 }
Here, A B , hence A and B are not
disjoint sets, but are intersecting sets.
ii) We have, A B = { a, e, i, o, u } { c, d,
e, f } = { e }
Here, A
B ,
hence A and B are not
disjoint sets but are intersecting sets.
iii) We have, A = { x : x is an even integer}
= { ..... – 2, 0, 2, .......}
and B = { x : x is odd integer } = { ...–3, –1,
1, 3, ....}
Hence, A
B ,
hence A and B are
disjoint sets
DAY-14: WORKSHEET
Conceptual Understanding Questions :
1. _______ set is subset of every set
1) U
2)
3) finite
4) infinite set
2. A B
1) {x : x A or x B} 2) {x : x A and x B}
3) {x : x A or x B} 4){x : x A and x B}
3. If A ={1,2,3,4}, then the no.of elements of
power set of A is ____.
1)8
2)1
3)16
4)10
4. If A ={1,4,6} and B = {1,2,3,4,5,6}
then A B
1){1,4,5}
2){1,4,6}
3){1,2,3,4,5,6}
4){1,2,3,4,5}
5. If A={1,2,3}, B={2,3,4} and C = {3,4,5,6}
then A B C
1){1,2,3,4}
2){2,3,4,5,6}
3){1,2,3,4,6}
4){1,2,3,4,5,6}
6. If A={1,4,6} and B= {1,2,3,4,5,6}
then A B ____.
1){1,2,3,4}
2){1,4,6}
3){1,2,3,4,5,6}
4){2,4,6}
7. If A={1,2,3,4}, B={5,6,7}, then A B ____.
1)A
2)B
3)
4){2,3,5}
30
CLASS-VIII
MPC BRIDGE COURSE
8. The cardinal number of A = {1, 2, 3, 4, 5,
6, 7, 8} is
1) 7
2) 8
3) 6
4) 5
9. A x / x 5, x N then n(A)
10. If A 2,3, 4,8,10 , B 3, 4,5,10,12 ,
C 4,5, 6,12,14 then A B A C
2
1) 7
2) 6
3) 5
4) 4
Single Correct Choice Type:
1. If A B, then A B =
1) A
2) B
3)
2. If A = , then A B =
4) A B
1) A
2) B
3)
4) A B
3. If A={1, 2, 3, 4, 5, 6, 7} then number of
proper subsets of A is
1) 27
2) 26
3) 27 1 4) 28 1
4. Howmany elements has P(A),if A
1) 1
2) 2
3) 3
4) 0
5. A = {x / x = y + 1, y N, y < 6}, B = {x / x
= y –1, y N, 2 < y < 5} then
which of the following is correct.
1) n A n B
2) n A n B
3) n A n B
4) n A n B
6. A = {1, 2, 3}, B = {1, 2, 3, 4, 5} then which
of the following is true
1) A is subset of B 2) A is super set of B
3) A is impropersubset of B
4) A is equal set to B
7. Let A={1,2,3},B={3,4},c={4,5,6} then
A (B C )
1){3}
2){1,2,3,4} 3){1,2,5,6}
4){1,2,3,4,5,6}
8. A = { 1, 2} then which of the following is
true
1) 1 P A
3)
1,2 , P A
2) 2 P A
4) all of these
9. Given N 1, 2,3,....,100 . Then subset B
of N, whose elements are represented
by x 2 where x N
1) {3, 5, 6, ..., 100}
2) {3, 4, 5, 6, .., 100}
3) {1, 3, 4, 5, 6,....,100}
4) {2, 4, 6, 8, ..., 100}
NARAYANA GROUP OF SCHOOLS
1) 2,3, 4,5,8,10,12
2) 2, 4,8,10,12
3) 3,8,10,12
4) 2,8,10
11.Sets A and B have 3 and 6 elements
respectively. What can be the
minimum number of elements in A B ?
1) 3
2) 6
3) 9
4) 18
12. If A = {x : x = 3n, n Z } and B = {x : x =
4n, n Z }, then A B =
1) 12n
2) 15n
3) 10n
4) n2
13.If A {2, 4} and B {3,5} then A B =
1) {2, 3,4,5}
3) {2,3, 4}
14.If
2) {3, 5}
4) {3, 4}
n(u) 60, n(A) 21, n(B) 43
th en
greatest value of n A B and least
value of n A B are
1) 60, 43 2) 50, 36 3) 70, 44 4) 60, 38
15.Let A and B be two sets such that
A B A . Then A B is equal to
1)
2) B
3) A
4) A B
DAY-15 : SYNOPSIS
Difference of Sets : If A and B are two sets,
then their difference A – B is the set of
all those elements of A which do not belong to B
Let x A – B x A and x B. Similarly, the difference B – A is the set of all
those elements of B that do not belong to
A. i.e., B – A = { x : x B and x A}
x B – A x B and x A
Solved Examples :
Example - 1
If X = {a, b, c, d} and Y = { f, b, d, g }, find
(i) X – Y
(ii) Y – X
(iii) X Y
Solution :
i) We have, X – Y = {a, b, c, d} – {f, b, d, g} =
{a, c}
31
CLASS-VIII
Since, the only elements of X which do
not belong to Y are a and c
ii) We have, Y – X = {f, b, d, g} – {a, b, c, d} =
{f, g}
Since, the only elements of Y which do
not belong to X are f and g
iii) We have,X Y = {a, b, c, d} {f, b, d, g}
= {b, d}
Since, b and d are common elements of
X and Y.
Example - 2
If A = {1, 2, 3, 4, 5, 6} and B = { 2, 4, 6, 8}.
Show that A – B B – A.
Solution :- We have,
A – B = {1, 2, 3, 4, 5, 6} – {2, 4, 6, 8} = { 1,
3, 5}
........(1) ( 2, 3, 6 B)
B – A = {2, 4, 6, 8} – {1, 2, 3, 4,5, 6}= { 8 }
........(2) ( 2, 4, 6 A)
From (1) and (2), we get A – B B – A
Symmetric Difference of two sets :
Let A and B be two sets. The symmetric
difference of sets A and B is the set
(A – B)
(B
– A) and is denoted by A B
A B = (A – B)
(B
– A) or
A B A B
For example : Let A = {1, 3, 5, 7, 9}, B =
{2, 3, 5, 7, 11}
Then, A – B = {1, 9}; B – A = {2, 11}
A B = {A – B}
{B – A} = {1, 9} {2,
11} = {1, 2, 9, 11}
Complement of a set :
Let U be the universal set and A is a
subset of U. Then, the complement of A
with respect to (w.r.t) U is the set of all
elements of U which are not the elements of A. Complement of A with respect to U is denoted by A ' or Ac.
In symbolic form A’ = {x : x
U and x A}
clearly, A’ = U – A
MPC BRIDGE COURSE
Some results on complementation :
1. U’ = {x
2.
: x U} =
’ = {x U : x } = U
3. (A’)’ = {x
U : x A’} = {x U : x A } = A
4. A A’ = {x U : x A} {x U : x A } = U
VENN DIAGRAM
A Venn diagram is merely a closed figure
and the points of the interior of the closed
figure represents the elements of the set
under consideration.
Generally given below closed figures are
used to represent the sets
Note:
i) It brings out relationship in sets and
their use in simple logical problems.
ii) Universal set
is represented by a
rectangular region.
iii) Each subset is represented by a
closed bounded figure placed within these
rectangular regions.
Venn -diagr ams ill ustrati ng var ious
relationships in sets
1. To represent
A
where
is the
universal set :
The universal set is represented by a
rectangle and its subset A is represented
by a circle placed inside this rectangle
as shown.
Clearly, the shaded region (which lies
within the rectangle but outside the
circle) represents A or Ac.
For example : If U = {1, 2, 3, 4, 5, 6} and A
= {2, 4, 6}
then, A’ = U – A = {1, 2, 3, 4, 5, 6} – {2, 4,
6} A’ = {1, 3, 5}
NARAYANA GROUP OF SCHOOLS
32
CLASS-VIII
MPC BRIDGE COURSE
2. To represent two subsets A and B of a
universal set such that A
B :
The universal set is represented by a
rectangle, a circle representing the set
B is placed in this rectangle and a
smaller circle representing the set A is
placed wholly inside the circle
representing the set B.
In this Venn - diagram, we can represent
the various operations –(A B), (A B),
(A –B) and (B – A). These operations are
shown by the shaded portions in the
following figures.
4. To represent two disjoint subsets of a
universal set :
The universal set is represented by a
rectangle and its two disjoint subsets A
and B are represented by two disjoint
circles labelled as A and B respectively.
In this Venn - diagram, we can represent
the various operations – (A B), (A B),
(A – B) and (B – A). These operations are
shown by the shaded portions in the
following figures :
(A B) B
AB A
(A B) B
(B A)
3. To represent two intersecting subsets
of a universal set.
The universal set
is represented by a
rectangle and its two intersecting
subsets A and B are represented by two
intersecting circles labelled as A and B
respectively,
5. To represent three intersecting subsets
of a universal set : The universal set
is represented by a rectangle and its
three intersecting circles labelled as A,B
and C respectively.
In this Venn diagram, we can represent
the various operations (A B), (A B),
(A – B), (B – A), A B , A B etc.
These operations are shown by the
shaded portions in the following figures.
NARAYANA GROUP OF SCHOOLS
33
CLASS-VIII
MPC BRIDGE COURSE
In this Venn - diagram, we can represent
the various operations such as
(A B C), (A B C) etc., as shown by
the shaded portions in the following
figures :
2. The shaded part represents
1) A B
2) A B
3) A B
4) B A
n
3. If A = { x : x = 2 – 1, n 5, n N } and B =
{ 2, 4, 8, 16 }, then A B =
1) {1, 2, 3, 4, 7, 8}
2) { 1, 2, 3, 4, 7, 8, 15, 16, 31 }
3) {1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }
4) { 0 }
4. From the figure A B =
DAY-15: WORKSHEET
1. If A ={1,2,3,4,5} and B = {4,5} then A–B =
____.
1){1,2,3} 2){2,3,4} 3){3,4,5} 4){1,4,5}
2. If A ={1,2,3,4,5} and B = {4,5} then A B =
____.
1){1,2}
2){3,4}
3){1,2,5,6} 4){1,2,3,6}
3. If
1) {1, 2, 3, 4, 5} 2) {1, 2, 4, 5}
3) { 3 }
4) {1, 2, 3, 4, 5, 6}
5.If A, B are any two sets, then A – (A – B) =
1) A
2) B 3) A – B
4) A B
6.From the figure
A B' =
={1,2,3,4,5,6,7,8,9,10} and
A={1,2,3,4,5} then A| =_____.
1){1,2,3,4,5,6}
2){6,7,8,9,10}
3){1,2,3,6,7}
4){1,4,9,10}
4. If A B then A B =_____.
1)A
2)B
3)
4)
Single Correct Choice Type:
1. If A a, b,c,d,e ,B a,c,e, g and c a, b, g ,
then
1) A B C A B A C
2) A B C A B A C
3) A B C A B A C
4) both (2) & (3)
NARAYANA GROUP OF SCHOOLS
1) {1, 2, 4, 5}
2){1, 2, 4, 5, 6}
3){1, 2, 4, 5, 6, 7, 8} 4){1, 2, 4, 5, 6, 7}
REASONING TYPE:
7. Statement I : A = {1, 2, 3, 4}, B = {5, 6, 7,
8} , then A B {
}
StatementII : The pictorial representation
of sets is known as venn diagram
1) Both statements I and II are true 2 )
Both statements I and II are false
3) Statement – I is true but statement –
II is false
4) Statement – I is false but statement –
II is true
34
CLASS-VIII
MPC BRIDGE COURSE
LAWS OF RADICALS:
COMPREHENSION TYPE:
* If
n
a and
b are two radicals of same
n
order ‘n’ then
Example:
* If
n
4
n
3 4 2 4 32 4 6
a and
n
b are two radicals of same
n
8. In the above figure, A B is
1) { 11, 9, 7}
2) { 2, 3, 5}
3) {3, 9, 3}
4) {2, 5, 3, 7, 9}
9. In the above figure A – B is
1) { 7, 9, 11}
2) {9, 11}
3) {7, 10, 9, 11} 4) {4, 6, 8, 10}
10.
In the above figure A B ' is
1) {4, 6, 8, 10, 7, 9, 11} 2) {4, 6, 8}
3) {4, 6, 8, 10, 9, 11, 12} 4) {7, 9, 11}
DAY-16 : SYNOPSIS
order ‘n’ then
3
Example:
*
n
1
a
or a n
is
not
number(irrational)then
n
a
rational
a is called a
surd of order ‘n’
n
is called radical sign.
* ‘n’ is called the order of the surd.
* ‘a’ is called the radicand.
* n a can be read as “ nth root of a”
25
3
4
10 , etc., are surds.
Example: 3, 2, 11,
3
Note:
*
n
b
3
n
a
b
10 3
5
2
4 3
6 12 6
* If m,n are two natural numbers and ‘a’ is
any positive rational number then
p m
a
n m
a
pm
n
a
pm
m
n ap
Illustration: 6 729 23 729 2 3 93 9 3
Entire surd or pure surd: A surd,
expressed in the form a n b , where a 1 ,
is called an entire surd or a pure surd.
Illustration:
* The other name of a ‘surd’ is radical
* The symbol
10
3
a
we have n m a mn a Example:
Definition: Let ‘a’ be a positive rational
If
n
2
If ‘m’ and ‘n’ are two natural numbers,
then for any positive rational number ‘a’
n m
number and ‘n’ be a positive integer 1
a n b n ab
an a
* In general 2 a is written as a .
* Every surd is an irrational number, but
every irrational number need not be a
surd.
Example: 0.5454454.............. is not a surd
Example: 2 is a surd and also irrational,
but is only irrational and not a surd.
6, 20, 3 5, 3 25 , etc.
Simples t form of a surd: A surd,
expressed in the form a n b , where ‘b’ is
the least positive rational number, is
called the simplest form of the given surd.
Example: (i) The
entire
form
of
2 10 10 4 40
(ii) The simplest form of
32 2 16 4 2
ADDITION AND SUBTRACTION OF SURDS:
Two surds can be added or subtracted
one from the other by using distributive
law, only when they are similar surds.
We cannot add or subtract dissimilar
surds.
Example-1: (i) a c b c a b c
( distributive law)
(ii)
NARAYANA GROUP OF SCHOOLS
2 6 4 6 2 4 6 6 6
35
CLASS-VIII
MPC BRIDGE COURSE
(iii) 4 2 3 2 10 2 4 3 10 2 3 2
Example-2: Simplify 8 3 4 75 3 300
Solution:
8 3 4 3 25 3 3 100
2010
3
6. Express 4 64 as a pure surd
1)
3
2010
2)
64
8 3 4 3 52 3 3 102
7. Express
8 3 4 5 3 3 10 3
8 3 20 3 30 3
1)
8 20 30 3 18 3
Conceptual Understanding Questions :
1. An example of a surd is
1)
2) 4
3) 9
4) 3 8
2
2. A cubic surd among the following is
1)
8
2)
3)
3
4
12 4)
3. The order of the surd 7 is
1) 1
2) 2
3) 1 / 2
4. The order of
1) 1
3
2 3 1
4) - 1
6 is
2) 2
3) 3
4) 6
Single Correct Choice Type :
1.
2
1)
2.
2 3
4
10000 2) 100
3
3.
3
3)
4
100
4) 1000
3)
3
2
4) 2 2
512 =
1) 2
2)
6
2010
2)
25
125
3)
2
2
2009
=
2010
2010 4)
2010
16
201
402 2
4)
5
25 5
simplified
the
A) 14 5 B) 14 5 C) 15 5 D) 16 5
9. If 2 3 189 33 875 7 3 56 is simplified,
then the resultant answer is
A) 8 3 7
B) 6 3 7
C) 7 3 7
D) 9 3 7
10. If 54 162 74 32 + 4 1250 is simplified
,then the resultant value is
A) 63 2
B) 64 2
C) 65 2
D) 0
11. If 53 54 + 23 16 + 4 3 686 is simplified,
then the resultant answer is
A) 93 2 B) 83 2
10 2 1000 =
3
2010 402 2
as a pure surd
125 125
8. 3 45 20 + 7 5 is
resultant answer is
DAY-16: WORKSHEET
3
2010 3)
C) 83 2 D) 93 2
7 1
1
is simplified ,
×
+
3
27
3
then the resultant answer is
12. If 3 147 +
A) 198 3 B) 199 3 C) 197 3 D) 195 3
9
9
9
9
13. If 4 128 + 4 75 5 162 is simplified
,then the resultant answer is
3
2009
2009
1)
2)
2010
2010
3)
3
2009
4) 1
2010
4. Which of the following is of an Irrational
number but not surd is
22
1) 121 2) e
3)
7
4) 3.24317094237890
5. If two surds are different multiples of the
same simple surd, then the surd is
1) Compound surd 2) Dissimilar surd
3) Similar surd
4) Complex surd
NARAYANA GROUP OF SCHOOLS
A) 20 3 13 2
B) 20 3 13 2
C) 13 2 20 3
D) 13 2 20 3
1 is simplified then
6
the resultant answer is
14.If
252 + 294 + 48
A) 6 7 15 6
B) 6 7 15 6
C) 15 6 6 7
D) 6 7 15 7
36
CLASS-VIII
15.
MPC BRIDGE COURSE
Illustration-2: As,
If x =
2 363 5 243 + 192 , y =
44 5 176 + 2 99 , then x + y is
A) 15 3 12 11
16.
If x =
24 , y =
150 , B =
96 , A =
x+y
is
A +B
A) 1
B) 0
C) –1
D) 2
17. If irrational numbers have the same
irrational factors ,then they are known
as
A) similar irrational numbers
B) dissimilar irrational numbers
C) a and b
D) neither a nor b
18. Sum of 3 2 and 5 2 is
19.
B) 8 2
C) 15 2 D) 12
5 × 5 = ____________
20.
8 3 ÷ 2 3 is _______________ numbers
21. Simplified form of 4 2 + 3 32 is
A) 16 2 B) 7 32 C) 7 2
22. Simplified form
n
2
B)
3
C)
5
3
rationalising factor of
Illustration-1:
Because,
3 6 3 6
n
a is given by a1-
3,3 3 are R.F.’s of 6 3
1
n
3 3 6 3 3 6
3 3 6 3 18 R
3 3 18 3 54 R
NARAYANA GROUP OF SCHOOLS
3
4 and
4 is a R.F. of
3
3
2.
DAY-17: WORKSHEET
1. If rationalizing factor of
3
5 is multiplied
by rationalizing factor of
resultant is
5
23 , then the
A)
15
512 20
B)
15
510 26
C)
15
510 28
D)
15
510 29
1
, B=
5- 3
2. If A =
1
4
B)
1
8
3 2 -2
3 2 +2
22
4
4. If A =
B)
1
, then A × B is
5+ 3
C)
, B=
22
5
1
2
D)
3 2 +2
3 2 -2
C)
1
2
, then A + B is
21
3
D)
22
7
5+2 2
is rationalized, then the
3-2 2
simplified form is
A) 23 17 2
B) 23 18 2
C) 23 19 2
D) 23 16 2
5. If p = n a x then its rationalizing factor is
A)
2 3 4 3 2 4
The R.F. of a given surd is not
unique. A surd has infinite number of
R.F.’s.
A)
Rationalising Factor (R.F.): If the product
of two surds is a rational number, then
each of them is called a rationalising
factor (R.F.) of the other. The
7 3
Note:
3. If A =
DAY-17 : SYNOPSIS
and
3
2 is a R.F. of
A)
7
3
7-
= 2, a rational number.
D) 14 2
D)
2
a n b n ab 3 8
48 72 27 + 2 18 is
A)
2
7 73 3 4 R
7 3
Illustration-3: As,
54 , then
A) 3 2
7 + 3 is a R.F. of
7 3 is a R.F. of
B) 15 3 12 11
C) 15 3 12 11 D) 15 3 12 11
7 3
6. If
n
5
ax
B)
C)
ax
82 × 5 83 = x ,
15
n
a n-x
D) n a x-n
154 × 15 1511 = y , then
x 2 + y 2 is
A) composite number B) even prime
C) irrational
D) prime number
37
CLASS-VIII
MPC BRIDGE COURSE
7. If P 3 122 = 12 then the value of P × 3 18
is
A) 8
B) 9
C) 6
D) 4
8. If 4 a12 b16 c11 × 4 a 3 b 7 c8 × 4 abc is simplified
then the resultant is
A) a 4 b3 c 4 B) a 4 b6 c 4 C) a 4 b5 c6 D) a 4 b6 c5
9. Match the following
a) 7 2 32
1) Trinomial
b) 2 3 10
2) Monomial
d) 5 3 2 2 9 5
3) Rational
d)
4) Binomial
3
343 4 256
1) a 2
2) b 4
3) c 1
4) d 3
a 1
b2
c 1
d3
a 1
b2
c3
d4
a 1
b4
c3
d2
10. 4 1250 × 4 8 +3 75 × 3 48 is simplified
then the resultant is
A) 550
B) 560
C) 540
D) 570
Note : 1) If the rotation of the terminating
ray is anti clockwise direction, then
the angle is regarded as positive.
2) If the rotation of the terminating ray
is clockwise direction, then the angle is
regarded as negative.
DAY-18 : SYNOPSIS
INTRODUCTION
The word ‘Trigonometry’ is derived from
three Greek words ‘tri’ meaning ‘three’,
‘gonia’ meaning ‘an angle’ and ‘metron’
meaning ‘measure’. The basic task of
trigonometry is the solution of triangles,
finding unknown quantities of a triangle
from given values of other quantities.
The study of trigonometry is of great
importance in several fields, like in
Surveying, Astronomy, Navigation and
Engineering.
In
recent
times
trigonometry is widely applied in many
branches of Science and Engineering
such as seismology, design of electrical
circuits, estimating the heights of tides
in the ocean etc.
Angle : The amount of rotation of moving ray
with reference to fixed ray is called
an angle. An angle is usually denoted by
, , ... etc.
NARAYANA GROUP OF SCHOOLS
Units of measurement of angles :
For measurement of angles, there are
three systems :
They are :
i) The sexagesimal (English) system.
ii) The centesimal (French) system,
iii) The radian (or) circular measure.
The Sexagesimal system : In this system
unit of measurement of an angle is
‘degree’.
Degree: In this system one complete
rotation is divided into 360 equal parts.
Each parts is called ‘a degree’, denoted
as 10 .
Minute : A degree is further divided into
60 equal parts and each part is called
one minute, denoted as 1' .
Second: A minute is further divided into
60 equal parts and each part is called
one second, denoted as 1'' .
38
CLASS-VIII
1
NOTE: 1) 10 60' or 1
60
MPC BRIDGE COURSE
In this way, the circumference of the
circle subtends at the centre, an angle
whose measure is 360 , the we get
0
'
1
2) 1' 60'' or 1
60
2 c 360 0 i.e., one complete angle = 2 c
and c 1800 i.e., one straight angle =
'
''
3) In this system, one right angle = 900 .
The Centesimal system : In this system
unit of measurement of angle is ‘grade’.
Grade: In this system one complete
rotation is divided into 400 equal parts.
Each part is called ‘a grade’, denoted as
1g .
Minute : A grade is divided into 100 equal
parts and each part is called a minute,
denoted as 1' .
Second: A minute is divided into 100
equal parts and each part is called a
second, denoted as 1'' .
NOTE : 1) 1g 100
2) 1 100
1
or 1
100
g
1
100
or 1
3) In this system, one right angle = 100 g
4) Though we have used the same name
‘minute’ (or ‘second’) in both
‘sexagesimal system’ and ‘centesimal
system’, it can be easily
observed that are not the same.
Circular system : In this system unit of
measurement of an angle is ‘radian’.
Radian : ‘The angle subtended by an arc
of length equal to the radius of the circle
at its centre is called one radian, denoted
as 1c .
c
c
0
90
i.e., one right angle =
and 2
2
c
Approximate values of 1c and 10 :
we know that c 1800 Also 1800 c
c
1800
1c
( where 3.1415... )
180
10 0.01745c
10
\ 1c = 57o17 '45 '' (Approximately)
Note:
1) A radian is a measure of an angle.
Hence it is different from the radius of
the circle.
2) Measure of an angle is a real number.
3) When no unit of measurement is
specified for an angle, it is assumed as
radian.
4) The formula connecting the three
systems can be stated as follows:
D
G
C
o
g
90
100
2c
Where D denotes degrees, G denotes
grades and C denotes radians
Illustration:
1) Express the sexagesimal measure 300
as radian measure and centesimal
measure.
Given D 300
r
o
r
G
30
2C 30
and
100 90
90
1
1
100
G
G 100 and C
3
3 2
3
From (1) ,
1
r
Relation between degrees and radians :
The angle which is subtended by an arc
of length 2r at the centre is 2c .
NARAYANA GROUP OF SCHOOLS
D
G
2C
,
90 100
30 G
2C
....(1)
90 100
Solution : We know that,
100 g
30
3
0
C
6
c
30
6
0
39
CLASS-VIII
MPC BRIDGE COURSE
Introduction :
In the figure given below, ABC is a
triangle, right angled at B. The side
opposite to the right angle is AC and it is
called hypotenuse. Consider angle , the
side opposite
A
It is written as ‘sine ’ or ‘sin ’.
sin
side opposite to ' ' y
Hypotenuse
r
OM side adjacent to ' ' x
, This ratio is
OP
Hypotenuse
r
called cosine of an angle ’ ’. It is written
as ‘cosine ’ or ‘cos ’.
cos
C
B
Consider a system of rectangular
coordinate axes OX and OY. Draw a circle
with centre O and radius r. Choose a
point P(x,y) on the circle such that the
line OP makes an angle radians with
MP side opposite to ' ' y
OM side adjacent to ' ' x , This ratio is
called tangent of an angle ‘ ’. It is written
as ‘tan ’.
OX (positive X-axis) measured in anticlock
wise
direction.
perpendicular PM to
In
OPM
Draw
a
tan
OX
, OP= Hypotenuse=r
PM side opposite to ‘ ’ = y OM = side
side adjacent to ' ' x
Hypotenuse
r
side opposite to ' ' y
side adjacent to ' ' x
OP
Hypotenuse
r
. This ratio is
MP side oppositeto ' ' y
called cosecant of an angle . It is
written as cosec (or) csc .
adjacent to ‘ ’ = x
csc
Hypotenuse
r
side oppositeto ' ' y
OP
Hypotenuse
r
, This is called
OM side adjacent to ' ' x
x
secant of an angle . It is written as
‘sec ’.
sec
The ratios of different pairs of sides of
the right angled triangle are called
triginometrical
functions
or
trigonometrical ratios with respect to an
angle’ ’, these ratios having following
names and designations.
Hypotenuse
r
side adjacent to ' ' x
OM side adjacent to ' ' x
MP side opposite to ' ' y , This ratio is
called cotangent of an angle . It is
written as ‘cot ’.
cot
side adjacent to ' ' x
side opposite to ' ' y
MP Side oppositeto ' ' y
, This ratio is
OP
Hypotenuse
r
called sine of an angle ’ ’.
NARAYANA GROUP OF SCHOOLS
40
CLASS-VIII
MPC BRIDGE COURSE
Relations :
1) sin cos ec
MP OP
1
OP MP
sin cos ec 1
2) cos sec
OM OP
1
OP OM
cos .sec 1
3) tan cot
MP OM
1
OM MP
tan .cot 1
4)
sin MP OP MP OP MP
tan
cos OM OP OP OM OM
tan
5)
sin
cos
cos OM OP OM OP OM
cot
sin MP OP OP MP MP
cot
cos
sin
NARAYANA GROUP OF SCHOOLS
NOTE :
1. Since the six trigonometrical ratios
discussed above represent the ratios of
sides of a right angles triangle, they are
all real numbers.
2. The six trigonometrical ratios are defined
with respect to a certain angle . Hence
sine, cosine,tangent, etc., by themselves
do not have any meaning. They are
meaningful only when they are
associated with an angle like ‘ ’.
3. Sin is an abbrevation for sine and it
is not the product of sin and ‘ ’. Similar
is the case of cos and tan .
4. Cosec , Sec and cot are reciprocals
of sin , cos and tan respectively.
5. We use the notation sin2 , cos2 , tan2 ,
etc., in place of (sin )2, (cos )2, (tan )2
respectively.
6. We write cosec =(sin )-1 not as sin-1
which has a different meaning (sine
inverse ).
7. All the values of trigonometric ratios
depend just on the angles but not on the
sides.
41
CLASS-VIII
MPC BRIDGE COURSE
DAY-18: WORKSHEET
3. The
Conceptual Understanding Questions :
1. If the terminal side completes one
revolution about its vertex, then
theangle made is
2) 3600
1) 900
3)1800
4)1200
2. Express the sexagesimal measure 1350
as radian measure.
1)
4
2)
3
4
3)
3. Express the
3
4)
6
circular measure
2) 900
3) 300
6
4)1200
3
4
2)
4
3)
5
3
4)
5
5
3
5. In the ABC , B 90 0 , AB = 4 cm, BC
=3 cm, then sin A=
1)
5
2)
3
3
3)
4
3
4)
5
4
5
6. In the ABC , B 90 0 , AB = 4 cm, BC
=3 cm, then tan A=
1)
3
2)
5
5
3)
3
4
4)
3
1) 300°
3
4
1. The sexagesimal measure, = 720 in
radian measure is
1)
2
3
2)
2
5
c
3)
3
4
c
4) None
in
2) 225°
3) 270°
4) 180°
5
, and ‘ ’ is acute angle. Then
12
the value of sec cosec
1)
22
60
2)
221
60
3)
22
7
4) None
sin A cos A
sin A cos A
is
1) 7
2) 2/11
3) 1/2
4) 1
6. In a right angled triangle ABC, B 900 ,
tan A = 5/12, then
5
13
2) sin A
3) Sin A 513
5
13
4) cos A
13
5
7. If 3sinθ - 4cosθ = 0 , then the value of
3sinθ + 4cosθ is
1) 4.8
2) 0.75
3) 1.33
4) 12
8. The ‘sine’ value of an angle of a triangle
is
1
, then the sum of other two angles
2
is
1) 1500
2) 600
3) 900
4) 1200
9. If sinA : cosA = 3 : 4, then secA + cosecA
is
1)
35
12
2)
12
35
3) 1
4)
3
4
10. If sin = – 7/25 and is the third
7 cot 24 tan
quadrant, then
7 cot 24 tan
1) 17/31 2) 16/31 3) 15/31 4) none
2. 200 g =
1) 1800
5c
4
4. If tan
Single Correct Choice Type :
c
sexagerimal measure is
1) CotA
4. If 5sin 3 then tan =
1)
measure
5. Given, 4 cotA = 3, the value of
sexagesimal measure.
1) 600
circular
2) 1000
3)
c
4
NARAYANA GROUP OF SCHOOLS
4)
c
2
42
CLASS-VIII
MPC BRIDGE COURSE
DAY-19 : SYNOPSIS
To find the values of trigonometric
functions of any angle :
Quadrant
Q1
Tr. Ratios
900
cos
sin
cot
tan
cos ec
sec
Sin
Cos
Tan
Cot
Sec
Cosec
Q3
Q2
900 1800
sin
cos
cos
sin
cot
tan
tan
cot
cos ec
sec
sec cos ec
1800
sin
cos
tan
cot
sec
cos ec
Q4
2700
cos
sin
cot
tan
cos ec
sec
2700
cos
sin
cot
tan
cos ec
sec
3600
sin
cos
tan
cot
sec
cos ec
DAY-19: WORKSHEET
Conceptual Understanding Questions :
0
1. If A 30 then sin 2A is
2
1)
3
2)
3
2
3)
1
4)1
2
5. The value of sin0 0 + cos30 0 – tan45 0 +
cosec 600 + cot900 is
1)
7 3
1
6
2)
7 3
1
6
3)
7 3
6
6
4)
7 3
6
6
2. The value of sin2 30 0 cos 2 600 is
1)
1
2
2)
3
2
3)1
4)
1
2
2
6. The value of sin
3. The value of
is
cos 00 sin 900 2 sin 450 is
1)2
2)1
3)–2
sin A
1. If A 450 , B 300 , then
cos A sin A.sin B
2) 2/3
3) 1/4
4) 1/5
2. sin 30 0 cos 0 0 tan 45 0 cot 45 0 sec 60 0 cos ec 30 0
1) 0
2) 1/2
3) 3/2
4) 1/4
3. Sin 60° + cos0° - Tan 45° +cot 45° - sin
90°=
1) 2
2)
3
2
4. Sin 0° + cos 90° +
Tan 45°
1) sin 45°
3) tan 45°
1
3)
2
4)
15
13
9
11
2)
3)
4)
2
2
2
2
0
0
7. The value of Sin 45 × Cos30 + Cos 450 ×
Sin300 is
1)
4)3
Single Correct Choice Type :
1) 1/2
π
π
π
+ se c 2
+ tan 2
4
3
3
1)
3 +1
3 1
2)
3)
2 2
2 2
3 1
2
8. The value of tan245 + 2tan2 60 is
1) 5
2) 6
3) 7
4) 8
9. cosec 2 30° cot 2 30° =
3
2 cos 45° + cot 45° -
3 1
4)
2
1) 1
2) 2
3)
1
2
4)
1
4
2) cosec 45°
4) cos 45°
NARAYANA GROUP OF SCHOOLS
43
Related documents