Algebraic Identities and Fundamentals of Maths

0.0(0)
studied byStudied by 3 people
0.0(0)
full-widthCall Kai
learnLearn
examPractice Test
spaced repetitionSpaced Repetition
heart puzzleMatch
flashcardsFlashcards
GameKnowt Play
Card Sorting

1/96

flashcard set

Earn XP

Description and Tags

Flashcards made from Maths Unplugged's crash course videos. Suitable for those writing JEE. Answer only with definition (I've included cards for both ways around). There are topics not covered in MU as well, marked.

Study Analytics
Name
Mastery
Learn
Test
Matching
Spaced

No study sessions yet.

97 Terms

1
New cards

(a+b)²

a²+b²+2ab

2
New cards

(a-b)²

a²+b²-2ab

3
New cards

a²+b²+2ab

(a+b)²

4
New cards

a²+b²-2ab

(a-b)²

5
New cards

(a+b)(a-b)

a²-b²

6
New cards

a²-b²

(a+b)(a-b)

7
New cards

a^{m^n}

a^{m^n}

8
New cards

(a^m)^n

a^{mn}

9
New cards

(\Sigma a)²

\Sigma(a²)+2\Sigma(a_1a_2)

10
New cards

\Sigma(a²)+2\Sigma(a_1a_2)

(\Sigma a)²

11
New cards

\Sigma a_1a_2

\dfrac{(\Sigma a)²-\Sigma (a²)}{2} 

12
New cards

\dfrac{(\Sigma a)²-\Sigma (a²)}{2} 

\Sigma a_1a_2

13
New cards

\Sigma a_1a_2 = \dfrac{(\Sigma a)²-\Sigma (a²)}{2}

How is this derived?

By using identities like (a+b)² and (a+b+c)².

14
New cards

(1+a)(1+a²)(1+a^4)(1+a^8)…(1+a^n)

\dfrac{1-a^{2n}}{1-a}

15
New cards

\dfrac{1-a^{2n}}{1-a}

(1+a)(1+a²)(1+a^4)(1+a^8)…(1+a^n)

16
New cards

\dfrac{1-a^{2n}}{1-a}=(1+a)(1+a²)(1+a^4)(1+a^8)…(1+a^n)

How is this derived?

Multiple and divide RHS by (1-a), then you use (a+b)(a-b)=a²-b²) to cancel out all the consecutive terms.

17
New cards

(a+b+c)²

a²+b²+c²+2(ab+bc+ca)

18
New cards

a²+b²+c²+2(ab+bc+ca)

(a+b+c)²

19
New cards

(a+b+c+d)²

a²+b²+c²+d²+2(ab + ac + ad + bc + bd + cd)

20
New cards

(a+b)³

a³+b³+3ab(a+b)

21
New cards

a³+b³+3ab(a+b)

(a+b)³

22
New cards

a²+b²+c²+d²+2(ab + ac + ad + bc + bd + cd)

(a+b+c+d)²

23
New cards

a³+b³

give both forms of this.

(a+b)³-3ab(a+b)

(a+b)(a²+b²-ab)

24
New cards

(a+b)³-3ab(a+b)

a³+b³

25
New cards

(a+b)(a²+b²-ab)

a³+b³

26
New cards

(a-b)³

a³-b³-3ab(a-b)

27
New cards

a³-b³-3ab(a-b)

(a-b)³

28
New cards

a³-b³

give both forms

(a-b)³+3ab(a-b)

(a-b)(a²+b²+ab)

29
New cards

(a-b)(a²+b²+ab)

a³-b³

30
New cards

(a-b)³+3ab(a-b)

a³-b³

31
New cards

a²-ab+b²

Give all 3 forms

(a+b)²-3ab

(a-b)²+ab

\dfrac{a³+b³}{(a+b)}

32
New cards

\dfrac{a³+b³}{(a+b)}

a²-ab+b²

33
New cards

a²+ab+b²

give all 3 forms

(a+b)²-ab

(a-b)²+3ab

\dfrac{a³-b³}{a-b}

34
New cards

\dfrac{a³-b³}{a-b}

a²+ab+b²

35
New cards

a^4-b^4

(a²+b²)(a²-b²)

36
New cards

a^4+b^4

(a²+b²)²-2a²b²

37
New cards

a²+b²

(a+b)²-2ab

38
New cards

a^4+4b^4

(a²+2b²-2ab)(a²+2b²+2ab)

derive simply and simplify

39
New cards

a^4+a²+1

(a²+a+1)(a²-a+1)

40
New cards

(a²+a+1)(a²-a+1)

a^4+a²+1

41
New cards

a^8+a^4+1

(a^4+a²+1)(a^4-a²+1)

42
New cards

(a^4+a²+1)(a^4-a²+1)

a^8+a^4+1

43
New cards

ab+bc+ca

\Big(\dfrac1a+\dfrac1b+\dfrac1c\Big)abc

44
New cards

\Big(\dfrac1a+\dfrac1b+\dfrac1c\Big)abc

ab+bc+ca

45
New cards

a²+b²+c²

(a+b+c)²-2(ab+bc+ca)

46
New cards

(a+b+c)²-2(ab+bc+ca)

a²+b²+c²

47
New cards

a³+b³+c³

(a+b+c)(a²+b²+c²-ab-bc-ca) + 3abc

48
New cards

(a+b+c)(a²+b²+c²-ab-bc-ca) + 3abc

a³+b³+c³

49
New cards

If a+b+c=0, what is a³+b³+c³=?

a³+b³+c³=3abc

50
New cards

if  a³+b³+c³=3abc, then what can you infer? (2 things)

either a+b+c=0

or a=b=c

51
New cards

a²+b²+c²-ab-bc-ca

Give all 3 forms of this.

=\dfrac{1}{2}[(a-b)²+(b-c)²+(c-a)²]

=(a+b+c)²-3(ab+bc+ca)

=\dfrac{(a+b+c)³-3abc}{a+b+c}

52
New cards

if a²+b²+c²-ab-bc-ca=0, what can you infer (2 things)?

that a³+b³+c³=3abc

and (a-b)²+(b-c)²+(c-a)²=0 which means a=b=c

53
New cards

if a²+b²+c²=0, what can you infer?

a=b=c=0

54
New cards

(a²-b²)³+(b²-c²)³+(c²-a²)³=?

How do you derive this?

3\left(a^2-b^2\right)\left(b^2-c^2\right)\left(c^2-a^2\right)

This is like X³+Y³+Z³.

Since X+Y+Z=0, then X³+Y³+Z³=3XYZ.

55
New cards

What can you infer from (x-a)²+(y-b)²+(z-c)³=0?

x=a,y=b,z=c

56
New cards

What can you infer from (x-a)²+(x-b)²+(x-c)³=0?

a=b=c

if not, the equation cannot exist

57
New cards

abc+(ab+bc+ca)+(a+b+c)+1

(a+1)(b+1)(c+1)

58
New cards

(a+1)(b+1)(c+1)

abc+(ab+bc+ca)+(a+b+c)+1

59
New cards

abcd+(a+b+c+d)+(ab+ac+ad+bc+bd+cd)+(abc+abd+acd+bcd)+1

(a+1)(b+1)(c+1)(d+1)

60
New cards

(a_1+1)(a_2+1)(a_3+1)+…+(a_n+1)

1 + \Sigma a_1 +\Sigma (a_1a_2)+\Sigma (a_1a_2a_3)+…+(a_1a_2a_3…a_n)

61
New cards

1 + \Sigma a_1 +\Sigma (a_1a_2)+\Sigma (a_1a_2a_3)+…+(a_1a_2a_3…a_n)

(a_1+1)(a_2+1)(a_3+1)+…+(a_n+1)

62
New cards

(a+b+c)³

a³+b³+c³+3(a+b)(b+c)(c+a)

63
New cards

a³+b³+c³+3(a+b)(b+c)(c+a)

(a+b+c)³

64
New cards

(a+b+c)³=a³+b³+c³+3(a+b)(b+c)(c+a)

How do you derive this?

consider X=a+b and Y=c and solve (X+Y)³.

65
New cards

(a-b+c+d)(a+b-c+d)(a+b+c-d)(b+c+d-a)

How do you approach this?

(a+b+c+d-2b)(a+b+c+d-2c)(a+b+d+c-2d)(a+b+c+d-2a)

Let a+b+c+d=x)

then,

=(x-2a)(x-2b)(x-2c)(x-2d).

66
New cards

Not from MU

Every prime number greater than 3 is in the form of _____, but the converse need not be true.

6k\pm1

67
New cards

Not from MU

When given a question where a variable is a prime number, what can you do? Just like what is the first thing that should pop into your head?

Can factorise it, and the factors are always going to be 1 and itself.

68
New cards

Not from MU

If f(x)  is divided by ax-b . what form will the remainder be in?

mx+n

69
New cards
<p><strong>Not from MU</strong></p><p>How would you approach this question?</p>

Not from MU

How would you approach this question?

Group some terms, it looks like componendo dividendo, apply that.

70
New cards

Not from MU

What is the relation between |a+b| and |a|+|b|?

|a+b|\le|a|+|b|

71
New cards

Not from MU

What is the relation between |a-b| and |a|-|b| ?

|a-b|\ge|a|-|b|

72
New cards

Not from MU

What are the limitations of logarithms? \log_aN

(three limitations)

  1. a>0

  2. N>0

  3. a\ne 1

73
New cards

What is \log_{a^m}b^n ?

\dfrac{n}{m}\log_ab

74
New cards

What is \log_x a + \log_x b?

\log_x ab  

75
New cards

What is \log_x a - \log_x b ?

\log_x \dfrac{a}{b} 

76
New cards

What is \log_aa=?

1

77
New cards

What is \log_a1=?

0

78
New cards

What is \dfrac{\log_ab}{\log_ac}=?

\log_cb

79
New cards

What is a^{\log_ac}=?

c

80
New cards

What is a^{\log_bc}=?

c^{\log_ba} 

81
New cards

If in \log_ab, both a and b are on the same side of 1 (on the number line), then will the value of log be negative or positive?

positive

82
New cards

If in \log_ab, both a and b are on different sides of 1 (on the number line), then will the value of log be negative or positive?

negative

83
New cards

Under what conditions will \log_ab>1 be true?

if b>a and a>1.

84
New cards

Not from MU

What is \log_ea in terms of \log to the base 10?

\log_ea=2.303\times\log_{10}a

85
New cards

Not from MU

What is \dfrac{1}{\log_ab}?

\log_ba 

86
New cards

Not from MU

What is \log_{10}a in terms of \log to the base e?

\log_{10}a=0.434\times\log_ea 

87
New cards

Not from MU

What does the graph for y=\log_ax look like when a>1?

knowt flashcard image
88
New cards

Not from MU

What does the graph for y=\log_ax look like when a<1?

knowt flashcard image
89
New cards

Not from MU

If \log_ax < \log_ay, when is x>y? When is x<y?

x<y when a>1

x>y when a<1

90
New cards

Not from MU

when is x+\dfrac1x\ge2?

when x is a positive real number

91
New cards

Not from MU

When is x+\dfrac1x\le2?

when x is a negativ real number

92
New cards

Not from MU

What is the approximate value of \log_{10}2?

0.3010

93
New cards

Not from MU

What is the approximate value of \log_{10}3?

0.4771

94
New cards

Not from MU

What is the approximate value of \log_{e}2?

0.693

95
New cards

Not from MU

What is the approximate value of \log_{e}10?

2.303

96
New cards

Not from MU

if \log_ax>p, what happens when a>1?

x>a^p

97
New cards

Not from MU

if \log_ax>p, what happens when 0<a<1?

0<x<a^p