Class notes Mathematics and science

CHAPTER

Basic Math and
1 Logarithm


(viii) Co-prime numbers: Two natural numbers (not necessarily
NUMBER SYSTEM prime) are called coprime, if their H.C.F (Highest common
(i) Natural numbers: The counting numbers 1, 2, 3, 4, ... factor) is one.
are called Natural Numbers. The set of natural numbers is e.g. (1, 2), (1, 3), (3, 4), (3, 10), (3, 8), (5, 6), (7, 8)
denoted by N. (15, 16) etc.
Thus N = {1, 2, 3, 4, ....}. These numbers are also called as relatively prime numbers.
(ii) Whole numbers: Natural numbers including zero are Note:
called whole numbers. The set of whole numbers is denoted (a) Two prime number(s) are always co-prime but
by W. converse need not be true.
Thus W = {0, 1, 2, .........} (b) Consecutive natural numbers are always co-prime
numbers.
(iii) Integers: The numbers ... – 3, – 2, – 1, 0, 1, 2, 3 .... are
(ix) Twin prime numbers : If the difference between two
called integers and the set is denoted by Ι or Z. Thus Ι
prime numbers is two, then the numbers are called twin
(or Z) = {.. – 3, – 2, – 1, 0, 1, 2, 3...}
prime numbers.
Note: (a) Positive integers Ι+ = {1, 2, 3 ....} = N e.g. {3, 5}, {5, 7}, {11, 13}, {17, 19}, {29, 31}
(b) Negative integers Ι– = {....., –3, –2, –1}. Note: Number between twin prime numbers is divisible by

(c) Non-negative integers (whole numbers) = {0, 1, 2, ...}. 6 (except (3, 5)).
(d) Non-positive integers = {......, –3, –2, –1, 0}. (x) Rational numbers: All the numbers that can be
(iv) Even integers: Integers which are divisible by 2 are called represented in the form p/q, where p and q are integers
even integers. and q ≠ 0, are called rational numbers and their set is
e.g. 0, ± 2, ± 4,....... denoted by Q. Thus Q = {p/q : p, q ∈ Ι and q ≠ 0}. It
may be noted that every integer is a rational number
(v) Odd integers: Integers which are not divisible by 2 are since it can be written as p/q. It may be noted that all
called odd integers. recurring decimals are rational numbers.
e.g. ± 1, ± 3, ± 5, ± 7...... p
Note: Maximum number of different decimal digits in
(vi) Prime numbers: Natural numbers which are divisible by 1 q
and itself only are called prime numbers. 11
is equal to q, i.e. will have maximum of 9 different
e.g. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ........ 9
decimal digits.
(vii) Composite number: Let ‘a’ be a natural number, ‘a’ is said
(xi) Irrational numbers: The numbers which can not be
to be composite if, it has atleast three distinct factors.
expressed in p/q form where p, q ∈ Ι and q ≠ 0 i.e. the
e.g. 4, 6, 8, 9, 10, 12, 14, 15 ......... numbers which are not rational are called irrational numbers
Note: (a) 1 is neither a prime number nor a composite number. and their set is denoted by Qc. (i.e. complementary set of Q)
(b) Numbers which are not prime are composite numbers e.g. 2 , 1 + 3 etc. Irrational numbers can not be
(except 1). expressed as recurring decimals.
(c) ‘4’ is the smallest composite number. Note: e ≈ 2.71 is called Napier’s constant and
(d) ‘2’ is the only even prime number. π ≈ 3.14 are irrational numbers.

, (xii) Real numbers: Numbers which can be expressed on
number line are called real numbers. The complete set of
9 2
rational and irrational numbers is the set of real numbers =
÷
and is denoted by R. Thus R = Q ∪ QC. 7 7
9 7
= ×
Negative side Positive side
7 12
3
=
–3 –2 –1 0 1 2 2 3  4
Real line
Example 2: Prove that the difference 1025 – 7 is divisible
All real numbers follow the order property i.e. if there are two
distinct real numbers a and b then either a < b or a > b. by 3.
Note: Sol. Write the given difference in the form 1025 – 7
(a) Integers are rational numbers, but converse need not = (1025 – 1) – 6. The number 1025 – 1 = 99..9
 is
be true. 25 digits

(b) Negative of an irrational number is an irrational divisible by 3 (and 9). Since the numbers (1025 – 1)
number. and 6 are divisible by 3, the number 1025 – 7, being
(c) Sum of a rational number and an irrational number is their difference, is also divisible by 3 without a
always an irrational number remainder.
e.g. 2 + 3
(d) The product of a non zero rational number and an
irrational number will always be an irrational number.
(e) If a ∈ Q and b ∉ Q, then ab = rational number, only if
a = 0. Concept Application
(f) Sum, difference, product and quotient of two irrational
numbers need not be a irrational number or we can say, 1. The product of 1.142857 and 0.63 = _____.
result may be a rational number also.
8 7
(xiii) Complex number: A number of the form a + ib is called (a) (b)
11 11
a complex number, where a, b ∈ R and i = −1 . Complex 11 8
number is usually denoted by Z and the set of complex (c) (d)
7 7
number is represented by C. Thus C = {a + ib : a, b ∈ R
and i = −1 } 2. If x = 12 − 9, y = 13 − 10, and= z 11 − 8,
then which of the following is true?
Note: It may be noted that N ⊂ W ⊂ Ι ⊂ Q ⊂ R ⊂ C.
(a) z > x > y
(b) z > y > x
Train Your Brain (c) y > x > z
(d) y > z > x
Example 1: The value of 1.285714 ÷ 1.714285 =
______.
3 7
(a) (b)
4 8
7 3 SOME IMPORTANT IDENTITIES
(c) (d)
12 7
Sol. (a) 1. (a + b)2 = a2 + 2ab + b2 = (a – b)2 + 4ab
1.285714 2. (a – b)2 = a2 – 2ab + b2 = (a + b)2 – 4ab
3. a2 – b2 = (a + b) (a – b)
= 1 + 0.285714
4. (a + b)3 = a3 + b3 + 3ab (a + b)
2 9
=1 + = 5. (a – b)3 = a3 – b3 – 3ab (a – b)
7 7
6. a3 + b3 = (a + b)3 – 3ab (a + b) = (a + b) (a2 + b2 – ab)
1.714285
7. a3 – b3 = (a – b)3 + 3ab (a – b) = (a – b) (a2 + b2 + ab)
5 12
=1 + = 8. (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca = a2 + b2 + c2
7 7
1 1 1
∴1.285714 ÷ 1.714285 + 2abc  + + 
a b c

2 JEE (XI) Module-1 PW

, 1
9. a2 + b2 + c2 – ab – bc – ca = [(a – b)2 + (b – c)2 + (c – a)2]
( 3) ( )
3
2 3
= 2 + + 3 × 2 × 3 2 + 3 + 23
10. a3 + b3 + c3 – 3abc = (a + b + c) (a2 + b2 + c2 – ab – bc – ca)
1
( 3) ( )
3
= (a + b + c) [(a – b)2 + (b – c)2 + (c – a)2] − − 3× 2 × 3 2 − 3
2
If a + b + c = 0 , then a3 + b3 + c3 = 3abc = 8 + 18 + 8 + 18 = – 50. Similarly for
11. a4 – b4 = (a + b) (a – b) (a2 + b2) 3 1
x =2 − 3, x + =52
12. a4 + a2 + 1 = (a2 + 1)2 – a2 = (1 + a + a2) (1 – a + a2) x3
1
Example 5. If x= = a , then what is the value of
1 1 x
x3 + x 2 + 3 + 2 ?
Train Your Brain x x
(a) a3 + a2 (b) a3 + a2 – 5a
3 2
(c) a + a – 3a –2 (d) a3 + a2 – 4a –2
Example 3: Show that the expression, (x2 – y z)3 + (y2 – z x)3 +
(z2 – x y)3 – 3 (x2 – y z) . (y2 – z x).(z2 – x y) is a perfect square Sol. (c)
and find its square root.e 1
Given, x + a
=
Sol. (x2 – yz)3 + (y2 – zx)3 + (z2 – xy)3 – 3(x2 – yz) x
(y2 – zx) (z2 – xy) = a3 + b3 + c3 – 3abc
Now, x 3 + x 2 + 13 + 12 =  x 3 + 13  +  x 2 + 12 
where a = x2 – yz, b = y2 – zx, c = z2 – xy x x  x   x 
= (a + b + c) (a2 + b2 + c2 – ab – bc – ca)  1 
3
1  1
2

=  x +  − 3 x +  +  x +  − 2
1  x   x   x
= (a + b + c) ((a – b)2 + (b – c)2 + (c – a)2)
2
= a3 – 3a + a2 – 2 = a3 + a2 – 3a –2.
1
= (x2 + y2 + z2 – xy – yz – zx) [(x2 – yz – y2 + zx)2
2
 + (y2 – zx – z2 + xy)2 + (z2 – xy – x2 + yz)2]

=
1 2 2 2
2
(x + y + z – xy – yz – zx) [{x2 – y2 + z(x – y)}2 Concept Application
+{y2 – z2 + x (y – z)}2 + {z2 – x2 + y (z – x)}2] 3. If x1/3 + y1/3 + z1/3 = 0, then what is (x+y+z)3 equal to?
1 2 2 2 (a) 1 (b) 3
= (x + y + z – xy – yz – zx) (x + y + z)2
2 (c) 3xy (d) 27xyz
 [(x – y)2 + (y – z)2 + (z – x)2] 4. If a + b + c = 0, then what is the value of
= (x + y + z)2 (x2 + y2 + z2 – xy – yz – zx)2 a 2 + b2 + c2
( a − b ) + (b − c ) + (c − a )
2 2 2
= (x3 + y3 + z3 – 3xyz)2
(which is a perfect square) its square roots are 1
(a) 1 (b) 3 (c)
(d) 0
3 3 3 3
± ( x + y + z − 3 xyz ) 1
5. If x + =then
p
1
x 6 + 6 equals to :
x x
Example 4. If x2 – 4x + 1 = 0, then what is the value of
(a) p6 + 6p (b) p6 – 6p
1
x3 + 3 ? 6 4 2
(c) p + 6p + 9p + 2 (d) p6 – 6p4 + 9p2 – 2
x
Sol. x2 – 4x + 1 = 0 1
6. If x + 4, then find values of
=
x
4 ± 16 − 4 × 1× 1 4 ± 2 3
⇒x= = =2± 3 1 1
2 ×1 2 2
(i) x +
3
(ii) x +
x2 x3
1 1
( ) ( )
3 3
∴ x 3 + 3 =2 + 3 + =2 + 3 + 1
( )
3
x 2+ 3
4
(iii) x +
x4
( )
3
 2 − 3 ×1  7. Prove that (1 + x)(1 + x2)(1 + x4)(1 + x8)(1 + x16)
( ) + (2 − 3)
3 3
  = 2+ 3
 (
 2+ 3 2− 3 )( ) 
 =
(1 − x 32 )
(1 − x)

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