PERMUTATION (XI)

           PERMUTATION

            *****************

A)

i) ⁿP₂ = 12 find n.                             4

ii) ⁿP₅ = 20. ⁿP₃ then find n.            8

iii) ⁿP₅ : ⁿP₃= 2 :1.                             5

iv) ⁿP₃ : ⁿP₂=3:1.                              5

v) ᵐ⁺ⁿP₂=56 and ᵐ⁻ⁿP₂=12 find the value of m,n.                                6,2

vi) ¹⁰Pᵣ₋₁: ¹¹Pᵣ₋₂=30:11 find r.         7

vii) ⁵⁶Pᵣ₊₆ : ⁵⁴Pᵣ₊₃ = 30800 :1 find the value of r.                                       41

viii) ⁿP₁₃ : ⁿ⁺¹P₁₂ = 3:4find n.          15

ix) ²ⁿ⁺¹Pₙ₋₁ : ²ⁿ⁻¹Pₙ= 3:5.                 4

x) ⁿ⁺ʳP₂ = 110 and ⁿ⁻ʳP₂=20 find the value of n,r.                                 8,3

xi) ⁿ⁺¹⁺ʳP₂ =72, ⁿ⁻ʳP₂=12 find the value n and r.                                6,2

xii) Find H.F.C of 3!,5!,7!.             3!

xii) Find L. C. M of 3!,5!,7!            7!

xiii) Compute 8!/{(4!)(3!)}.       280

xiv) Convert in to factorial 6.7.8.9.   9!/5!

xv) (n+1)!=12(n-1)! Find n.        

xvi) 1/9! +1/10! =n/11! Find n

xvii) Is 33! Divided by 2¹⁵

xviii) True or False

a)(2+3)!=2!+3!

b) (2x3)!=(2!)x(3!)

xix) Evaluate.   n!/{(r!).(n-r)!} When
        n=15 and r=12.

xx) (n+2)!=60(n-1) ! find n

xxi) (n+2)!=2550(n!) Find n 

xxii) The value of 1.1!+ 2.2! +3.3!+4. 4! +............+n.n!               (n+1)!-1

B) There are 10 trains moving between Calcutta and Delhi. In how many ways can a man go from Calcutta to Delhi and return by a different one ?                90

C) There are 26 stations on a railway line. How many different kinds of tickets of classII must be printed in order that a passenger may go from any one station to another by purchasing a ticket.  650

D) There are four bus lines between A & B and there are 3 bus lines between B & C. In how many ways can a man take round trips by bus from A to C by way of B, if he does not want to use a bus line more than once ?                             132

1) You are given the letters of the word “MONDAY”. Find the number of arrangements in the following cases:

a) Without any restriction.        720

b) Words beginning with M.     120

c) Words beginning with Y.       120

d) Words beginning with M & ending with Y.                               24

e) Words beginning with M & not ending with Y.                             96

f) M & Y are at two extremes.    48

g) Vowels are together.             240

h) Vowels are never together.   480

i) Vowels occupy odd places.    144

j) Vowels occupy even places.   144

k) Constants are together.           144

l) How many words can be formed by taking four letters at a time. 360 

m) In how many of these (given in previous question) M is always included?            240 

n) In how many of these (given Q. no l) M is excluded ?      120

o) Number of rearrangement of the word MONDAY.          719

p) In how many words MO will be together.                             240

2) In how many ways can 8 sweets of different sizes be distributed among 8 boys of different ages, so that
a) Largest sweet goes to the youngest.                               40320

b) Smallest sweet goes to the older. 40320

c) Largest sweet goes to the youngest and smallest sweet goes to the older.                               720 

**3) Consider words formation with the letters of the word DELHI 

a) How many arrangement can be formed with the letters DELHI? 120

b) How many of them will begin with D?          24

c) How many do not begin with D.  96

d) In how many words LH will be together ?         48 

4) How many words can be formed of the letters of the word COASTING, the vowels being not separated ?                           4320 

5) In how many ways can the letters of the word"LAUGHTER" be arranged so that the vowels may never be separated?              4320

6) In how many ways can 8 examination papers be arranged in a row, so that the best and worst papers may never come together.      30240

7) How many words can be formed of the letters in the word ARTICLE so that the vowels may occupy only

a) the even positions.            144

b) the odd positions.              576 

8) Find how many words can be formed by the letters in the word FAILURE, is the four vowels

a) always coming together.      576

b) never coming together.       4464 

9) If the letters of the word JUXTAPOSED be arranged in all possible different ways, in how many of these will the vowels occur together?             120960 

10) In how many ways can the letters of the word MOBILE be arranged so that the consonants always occupy the odd places?  36 

11) In how many different orders can 8 examination papers be arranged in a row, so that the best and the papers may never come together ?                   30240

12) The Number of ways in which 16 different books can be arranged on a shelf so that two particular books shall not be together is.  14.15!

13) Six papers are set in an examination, of which two are mathematical. In how many different orders can the papers be arranged so that
a) the two mathematical papers are together.                             240

b) the two mathematical papers are not consecutive ?           480

14) In how many ways can 6 plastics beads of different colours be arranged so that the blue and green beads are never placed together?                             480 

15) In how many different ways can the letters of the vowels of the word VALEDICTORY be arranged so that the vowels are 

a) never separated.           997680

b) not together.                38949120 

16) In how many different ways can the letters of the word STRANGE be arranged so that the vowels are

a) never separated.              1440

b) not together.                     3600 

17) How many arrangements of the letters of the word COMRADE can be formed

a) If the vowels are never separated.                             720

b) if the vowels are to occupy only odd places.                            576

18) In how many ways can 3 boys and 5 girls be arranged in row so that no 2 boys are together?  14400

19) In how many ways can 5 boys and 4 girls be arranged in a row so that the boys and the girls stand alternatively.                        2880

20) In how many ways can 5 boys and 5 girls be arranged in a row so that they stand alternatively.  28800

21) In how many ways 6 boys and 4 girls be arranged in a row so that no girls are together ?             604800

22) You are given the letters of the word BALLOON. Find the arrangement

a) Without any restriction.      1260

b) Two LL will always come together.                                    360

c) Two LL and two OO will always come together.                          120

d) All the O's & the L's will come together.                                     144

e) Vowels are together.            180

f) B & N are together.                360

g) B & N are never together.    900

h) B,N & O's are together.         144

I) Two OO's together.                300

23) Find the number of arrangements that can be made out of the letters of the following words:

a) CALCUTTA.                          5040

b) ACCOUNTANT.                226800

c) CONTACT.                            1260

d) ATLANTIC.                         10800

e) MATHEMATICS.             4989600

f) INSTITUTION.                 554400

g) STATISTICS.                     50400

h) ENGINEERING.               277200

I) MISSISSIPPI.                      34650 

** 24) All different words formed by the letters of the word BHARAT. 

a) How many different words can be formed with the letters of the word BHARAT?                          360 

b) In how many of these B and H are never together.                     240 

c) How many of these begin with B and end with T ?                           12 

25) How many different words can be formed with the letters of the word CAPTAIN ? In how many of these C and T are never together ?      2520,1800 

26) In how many ways can the letters of the word ALGEBRA be arranged? In how many of these arrangements will the two A's not come together ?           2520, 1800 

27) Find how many different words can be formed from the letters of the word PEOPLE in which two P's would not remain side by side. 120 

28) In how many different ways can the letters of the word CONSTITUTION be arranged ? How many of these will have the letter N both at the beginning and at the end ?                    181000, 17600 

29) The number of ways in which the letters of the word ARRANGE can be arranged that the two R's do not come together is:.            900

30) In how many ways can be letters of the word EXAMINATION be arranged so that all the A's always come together.      907200 

31) In how many ways can the letters of the word AGARTALA be arranged ?        1680 

32) taking data from the previous question, in how many of these will the 4 A's 

a) come together.                   120 

b) not come together.            1560 

33) In how many ways can 5 dots (.) and 3 crosses (x) be arranged in a row?                                       55

34) A library has 5 copies of one book, 4 copies of each of the two books, 6 copies of each of the 3 books and single copies of eight books. In how many ways can all the books be arranged ?    39!/{(5!) (4!)²(6!)³.

35) Find the number of different Arrangements that can be made of the seven prismatic colours (violet, indigo, blue, green, yellow, orange, red) so that the violet and red shall never come together.           3600

36) There are six students of whom 2 are Indians, 2 Americans and the remaining are Russians. They have to stand in a line so that the two Indians are together, the 2 Americans are together, and also the 2 Russians are together. Find the number of ways in which they can do so ?                                6

37) How many different arrangements can be made out of the letters in the expression x²y⁴z³ when written at full length ?    1260

38) How many 5 digits numbers can be formed with the digits 1,2,3,4,5 and 6 ?                  720

39) Find the number of numbers greater than 2000 that can be formed with the digits 1 2,3,4,5.  216

40) How many numbers greater than 4000 can be formed with the digits 2,3,4,5 and 6.            192           

41) How many numbers greater than 3000 can be formed with the digits 1,2,3,4,5.                        192   

42) How many 4 digit number greater than 6000 formed by 2,4,6,7, 8.                   192 

43) How many numbers can be formed with the digits 3,4,5,67,8 

a) number of numbers formed. 720

b) divisible by 5.                         120

b) Not divisible by 5.                   600 

44) How many numbers between 4000 and 5000 can be formed by 2,3,4,5,6,7.                                60

45) How many numbers greater than 7000 can be formed with the digits 3,5,7,8 and 9                    192 

46) How many even numbers greater than 300 can be formed with the digits 1, 2,3,4,5            111 

47) How many numbers can be formed with the digits 1,2,3,4,5? How many of them are greater than 3400 ?                                      180

48) How many 3 digit numbers are there, with distinct digits, with each digit odd ?                                    60 

49) Find the total number of numbers divisible by 2 which can be formed with the digits 1,2,4,5,6 and 7.                                          360 

50) How many 5 digit numbers formed with 0,1,2,3,4.                  96

51) How many 4 digit numbers greater than 4000 formed with 0,2,3,4,6,7                                     180

52)  How many 4 digit numbers can be formed by 0,1,2,3,45,              720

53) How many odd numbers of 6 digits can be formed with 0,1,2,3,4,5                                     288

54) How many numbers lying between 100 and 1000 can be formed with 2,3,4,0,8 and 9.      100             

55) How many numbers greater than a million (10 lakhs) can be formed with 2,3,0,3,4,2,3.          360

56) How many numbers less than 1000 and divisible by 5 can be formed with 0,1,2,3,4.....9.          154            

57) How many 5 numbers between 300 and 3000 can be formed with the digits 0,1,2,3,4,5                   180

58) How many 4 digit numbers greater than 5000 can be formed with the digits 2,4,5,7,8,0.         360 

59) How many numbers greater than 7000 can be formed with 3, 5, 7, 8 and 9                                  192

60) How many numbers each lying between 100 and 1000 can be formed with 2,0,3,4,5? How many of these are odd ?                    48, 18

61) How many different numbers can be formed by using any four of the nine digits 1,2,3,.....9 such that their numbers will

a) begin with a specified digit.  336

b) begin with a specified digit and end with a specified digit?          42 

62) How many 5 digit numbers formed with 2,3,5,7,9 which are

a) greater than 30000.              96

b) less than 70000.                   72

c) lies between 30000 and 90000.  72             

63) How many 4 digit numbers can be formed with 0,1,2,3,4....9.    4536 

64) In how many ways 8 persons can be seated at a round table ? 5040 

65) In how many ways can 7 persons be arranged at a round table so that 2 particular persons can be together?               240 

66) Find the number of ways in which 5 beads can be arranged to form a necklace.                  12

67) In how many ways can 4 MBA & 4MCA be seated at the round table so that no 2 MBA students are adjacent ?                                 144 

68) In how many ways 7 Gentleman & 6 ladies sit down at a round table

a) Without restriction.                 12!

b) Male and Female sit separately

c) No two ladies are together

d) Two particular ladies sit together

e) No two Gentlemen sit together

f) Sita will sit between Radha and
    Gita.

69) In how many ways 8 stones of different colours be arranged on a ring ? In how many of these arrangements red and yellow beads being separated ?

70) A round table conference is to be held for a committee of 7 persons which includes President and Secretary. Find the number of ways the committee can be seated so that

a) The President and Secretary can sit together

b) The Secretary sits on the right side of the President

c) The President and the Secretary don't sit together.

d) Ram will be sit between President and Secretary.

71) How many ways EDUCATION will be arranged in circular with

a) No restriction

b) Vowels together

c) Consonants together

d) EDU together

e) Vowels never together

72) How many ways LOLLAABEE arrange in circular pattern with

a) without restriction

b) Vowels together

c) LLL's together

d) 3 LLL, 2 AA together

e) 3 LLL, 2 AA 2 EE together.

f) Constant never together.

73) Find the number of ways in which 8 different flowers can be strung to form garland

a) Without restriction

b) 4 particular flowers are never together

c) Using beads.

LIMITS - XI

Type-1)

1) lim ₓ→₀ (7x²-5x+1).                   1

2) limₓ→₀ (2x³+3x+4)/(x²+3x+2). 2

3) limₓ→₃√(2x+3)/(x+3).             1/2

4) limₓ→₁ √(x+8)/√x.                     3 

5) lim ₓ→₁(x²+1)/(x+1).                 1

6) lim ₓ→ₐ (√a+√x)/(a+x).        1/√a

7) limₓ→₁{1+(x-1)²}/(1+x²).         1/2

8) lim ₓ→₂ (3x²-x+1)/(x-1).            11

9) limₓ→₁ (4-x).                               3

10) lim ₓ→₀(ax²+b)/(cx+d).     b/d 

11) limₓ→_₁(x³ - 3x +1)/(x-1)       -3/2

12) limₓ→₀ (3x+1)/(x+3).          1/3

Type: 2.                                                  *******

1) lim ₓ→₁ (x²-1)/(x-1).                   2

2) limₓ→₋₅ (2x²+9x-5)/(x+5).       -11

3) limₓ→₃(x²-4x+3)/(x²-2x-3).     1/2

4) limₓ→₄ (x²-16)/(√x -2).          32

5) limₓ→₀ {(a+x)³-a³}/x.              3a²

6) lim ₓ→₁(x-1)/(2x²-7x+5).         -1/3

7) lim ₓ→₁ (x²-√x)/(√x-1).               3

8) limₓ→₃(x²-9)/{1/(x-3)+1/(x+3)}. 6

9) limₓ→₁ (x-1)/(2x²-7x+5).       -1/3

10) limₓ→₃ (x²-7x+12)/(x²-9).    -1/6

11) limₓ→₂ (7x²-11x-6)/(3x²-x-10).  17/11

12) limₓ→₂ (x³-8)/(x-2).                  12

13) limₓ→_₁(2x²+5x+3)/(x³+1).  1/3

14) limₓ→₂ x²(x²-4)/(x-2).             16

15) limₓ→₂{(x⁸-16)/(x⁴-4)+(x²-9)/(x-3)}.                                                 25

16) limₓ→₂ (x-2)/(√x -√2).      2√2

17) limₓ→₀ {(1+x)²-(1-x)²}/2x.      2

18) limₓ→₁ (x²+5x-6)/(x²-3x+2).   -7

19) lim ₓ→₁/₂{(8x-3)/(2x-1) - (4x²+1)/(4x²-1)}.                           7/2

20) limₓ→₃(x²+x-12)/(x-3).           7

21) limₓ→₁(x²+4x-5)/(x-1).           6

22) limₓ→₀ {(1+x)²-1}/x.              2

23) limₓ→₂(x²-5x+6)/(x²-3x+2).  -1

24) limₓ→₂(x²+x-6)/(x²-x-2).        5/3

25) limₓ→₂(x²-5x+6)/(x²-7x+10)  1/3

26) limₓ→₃(x²+2x-15)/(x²-2x-3).   2

27) limₓ→₁(x³-1)/(x²-1).              3/2

28) limₓ→₂{1/(x-2) - 1/(x²-3x+2)}.  1

29) limₓ→₁(x²-3x+2)/(x³-4x+3).      1

30) limₓ→₀{(4+3x)³-8x²}/{4(4-x)²}.1

31) limₓ→₂(2x²-3x+7)/(x³+5x+1) 9/19

32) limₓ→_₁(2x²+5x+3)/(x³+1).  1/3

33) limₓ→₁(2x⁴-3x+1)/(x³-5x²+4x).  -5/3

34) limₓ→₃(x³-8x²+45))(2x²-3x-9). -7/3

35) limₓ→₃(x³-6x-9)/(x⁴-81).     7/36

36) limₓ→√₂ (x⁴-4)/(x²+3x√2-8). 8/5

37) limₓ→₁(x⁴-3x³+2)/(x³-5x²+3x+1) 5/4

38) limₓ→₁{(2x-3)(√x -1)}/(2x²+x-3) -1/10

39) limₓ→₃(x²-9){1/(x+3) + 1/(x-3)} 6

40) limₓ→₂(x³-6x²+11x-6)/(x²-6x+8)  1/2

41) limₓ→₁/₂ (8x³-1)/(16x⁴-1)   3/4

42)limₓ→₄(x²-x-12)¹⁸/(x³-8x²+16x)⁹                     7¹⁸/4⁹

43) limₓ→₁{1/(x²+x-2) - x/(x³-1)}. -1/9

44) limₓ_₃(x³-7x²+15x-9)/ (x⁴-5x³+27x-27)             2/9

45) limₓ→√₂. (x⁹- 3x⁸+ x⁶- 9x⁴- 4x²- 16x+84)/(x⁵-3x⁴-4x+12).            (8√2-31)/(√2-3)

46) limₓ→₃ (x⁴- 81)/(x²-9).           18

47) limₓ→₃(x²-x-6)/(x³-3x²+x-3).  1/2

48)  limₓ→₋₂  (x³+x²+4x+12)/ (x³-3x+2).                   4/3

49) limₓ→₁(x³+3x²-6x+2)/ (x³+3x³-3x-1).                1/2

50) limₓ→₁(x⁴-3x³+2)/(x³-5x²+3x+1)  5/4

51) limₓ→₂ (x³+3x²-8x-2)/(x³-x-6).  15/11

52) limₓ→₂  (x⁴ -16)/(x-2).          32

53) limₓ→₁{(x-2)/(x²-x) - 1/(x³ -3x²+2x)}.             2

54) limₓ→₂ {1/(x-2) - 2(2x-3)/(x³- 3x² +2x)}.              -1/2

55) lim ₕ→₀  {f(1+h)-f(1)}/h, when f(x)= 1/x.                        -1

Continue.........

Type: 3.                                                  ------------

1) limₙ→₀{√(x+n) -√(x)}/n.     1/2√x

2)limₓ→₀ {√(1+x) - √(1+x²)}/x.   1/2

3) ltₓ→₀{√(1+x) -√(1+x²}/{√(1-x²)-√(1-x)}.           1

4) limₓ→ₐ{√(a+2x)-√(3x)}/ {√(3a+x) -2√(x)}, a≠ 0.                2/3√3

5) limₙ→₀ 1/n{1/√(x+n) - 1/√(x)}. -1/(2x√x)

6) limₓ→₀ {√(1+x) - 1}/x.            1/2

7) limₓ→ₐ{√(x) - √(a)}/(x-a).    1/2√a

8) limₓ→₄ {3 -√(5+x)}/(x-4).      -1/6

9) limₓ→₀{√(x+2) - √(2)}/x.      1/2√2

10) limₓ→₀ x/{√(1-x)- 1}.               2

11) limₓ→₀ {√(1+x) -√(1-x)}/2x.   1/2

12) 

13) limₓ→₀ {√(1+x+x²) -1 }/x.      1/2

14) limₓ→₀{√(1-x³) -√(1+x³)}/x².     0

15) limₓ→₄ {3-√(5+x)}/{3-√(5-x). 1/3

16) limₓ→₃{3-√(6+x)}/{√3 -√(6-x). -1/√3

17) limₓ→₀{√(1+x) -√(1+x²)}/{√(1-x²) - √(1-x)}.               1

18) limₓ→₂(x²-4)/{√(3x-2)-√(x+2)}. 8

19) limₓ→₃{√(3x+7)-√(7x-5)}/{√(5x-6) - √(2x+3)}.           - 1

20) limₓ→₁{√(x+8)-√(8x+1)}/{√(5-x)- √(7x-3)}.              7/12

21) limₓ→₁ {³√(x+7)- ³√(7x+1)}/(x-1)            1 - ³√7

22) limₓ→₂{2-√(2+x)}/{³√2- ³√(4-x)}  - 3/³√16

23) limₓ→₀  x/{√(a+x)-√(a-x)}.    √a

24) limₓ→₄ (x²-16)/{√(x²+9) -5}  10 

25) limₓ→ₐ{√(a+2x)-√(3x)}/{√(3a+x)- 2√x}            2/3√3

26) limₓ→₁{(2x-3)(√x -1)}/(2x²+x-3)  -1/10

27) limₓ→√₁₀  {√(7-2x)-(√5-√2)}/(x²-10).         (√5+√2)/6√10

28) limₓ→₂  {√(x²+1)-√5}/(x-2)   2/√5

29)  limₓ→₂ (2-√x)/(4-x).           1/4

30) limₓ→ₐ (x-a)/(√x - √a).      2√a

31) limₓ→₂  (x-2)/(√x-√2).        2√2

32) limₓ→₃ {√(x-3)+√x -√3}/√(x² -9)  -1

Continue.......

Type : 4

1) limₓ→₂(x¹⁰ - 1024)/(x-2).    5120

2) limₓ→₁ (xᵐ -1)/(x-1)                m 

3) limₓ→₃ (x⁵-243)/(x²-9).       135/2 

4)limₓ→ₐ(x⁵-a⁵)/(x³-a³).            5a²/3

5) limₓ→₅  (x⁴-625)/(x³-125).   20/3

6) limₓ→₂ (x¹⁰ -1024)/(x⁵ -32).   64

7) limₓ→₉ (x³/² -27)/(x-9).          9/2 

8) limₓ→ₐ(x³/⁵-a³/⁵)/(x¹/³-a¹/³).   9/

a⁴/¹⁵/5

10) limₓ→₁(xᵐ -1)/(xⁿ -1)         m/n

11) limₓ→ₐ (x√x- a√a)/(x-a)    3√a/2

12) limₓ→₂ (x⁷-2⁷)/(x³-2³).      112/3

13) limₓ→₀ {(1+x)ⁿ -1}/x.           - n 

14)limₓ→¹ {(1+x)⁶ -1}/{(1+x)² -1}.  3

15) lim ₓ→ₐ(x²⁾⁷- a²⁾⁷)/(x-a)   2/7a⁵⁾⁷

16) lim ₓ→₋₁/₂  (8x³+1)/(2x+1).     3

17) limₓ→ₐ{(x+2)⁵/² -(a+2)⁵/²}/(x-a)       5/2 √(a+2)³

18) lim ₓ→₂ (x-2)/(³√x - ³√2).  3(2²⁾³)

19) If lim ₓ→₂ (xⁿ - 2ⁿ)/(x-2)= 80 and n∈ ℕ find n                                    5

20) If lim ₓ→₁(x⁴-1)/((x-1)= lim ₓ→k (x³ - k³)/(x² - k²). Find k              8/3

21) If limₓ→_ₐ (x⁹+a⁹)/(x+a) = 9, then find the value of a.                ±1

22) limₓ→³ (xⁿ-3ⁿ)/(x-3) =108 and if  n is positive integer find n.             4 

Continue........

Type : 5

1)lim→₀ (e⁻ˣ -1)/x.                       -1

2) limₓ→₀ (eᵃˣ-1)/ax.                    1

3) limₓ→₀ (eᵃˣ-1)/mx.               a/m 

4) limₓ→₀ (e⁵ˣ -1)/3x.                 5/3

5) limₓ→₀ (eᵃˣ - eᵇˣ)/x.               a-b

6) limₓ→₀ (e⁷ˣ - e³ˣ -e⁴ˣ +1)/x².    12
                            
7) limₙ→₀{ ₑ(x+n)²   - ₑx²)}/n.      2x

8) lim ₓ→⁰ (eˣ- e)/(x-1).                  e 

9) limₓ→₀ (ₑlog x ₋ ₁)/ₑˣ⁻¹ ₋ ₁)        1

10) lim ₓ→₀ (eˣ - e²)/(x-2).            e²

11) limₓ→₀ (e⁷ˣ - 1)/9x                 7/9

12) lim ₓ→₀ (eˣ - e⁻ˣ)/x.                   2

13) limₓ→₀ (e¹⁵ˣ - e⁷ˣ)/x.                8

14) limₓ→₀ (e⁷ˣ + e⁵ˣ -2)/x.          12 

Type : 6

1) limₓ→₀ (3⁵ˣ - 1)/x.            5 log 3

2) limₓ→₀ (2³ˣ -1)/x.             3 log 2

3) limₓ→₀ (2ᵃˣ - 3 ᵇˣ)/x.   alog 2-b log 3

4) limₓ→₀ (12ˣ -3ˣ- 4ˣ +1)/x²  log 3. Log 4

5) limₓ→₀(aˣ - bˣ)/x.         log(a/b)

6) limₓ→₀ (10ˣ -2ˣ- 5ˣ +1)/x².   log 5. log 2

Continue......

Type : 7

1) lim ₓ→₀ {log(1+7x)}/x.               7

2) limₓ→₁ (log x)/(x-1).                   1

3) limₓ→₀ {log(6+x)- log(6)}/x.   1/6

4) limₓ→₂ {log(x) - log(2)}/(x-2).  1/2

5) limₓ→ₑ (logx  -1)/(x-e).           1/e

6) limₓ→₁(x²- x log x+ log x -1)/(x-1)   6

7) limₓ→₀ x{log(x+a) - log x}.       a 

8) limₓ→⁰ x{log(x+5) - log x}.       5 

9) limₓ→₄  (x⁷/²- 4⁷/²)/{ log(x-3)} 112

Continue.......

Type: 8

1) limₓ→∞ (4x-3)/(2x+7).            2

2) limₓ→∞(3x²+2x-5)/(x²+5x+1).  3

3) limₓ→∞ (x³+6x²+1)/(x⁴+3).      0

4) limₓ→∞(3x³+x²-1)/(x²-x+7).    ∞

5) limₓ→∞ (5x-6)/√(4x²+9).     5/2

6) limₓ→∞{√(3x²-1)-√(2x²-1)}/ (4x+3).                          (√3-√2)/4

7) limₓ→∞{√x √(x+c) -√x).        c/2

8) limₓ→∞{√(x²+x+1) - √(x²+1)}.   1/2

9) limₓ→∞{(x+1)(2x+3)}/{(x+2)(3x+4)}.             2/3

10) limₓ→∞ {x - √(x² - x)}.        1/2

11) limₓ→∞{√(x²+5x+4)- √(x²-3x+4)}.                                   4

12) limₓ→∞ 2x{√(x²+1)-x}.           1

13)  limₓ→∞{√(x²+1)-³√(x²-1)}/{ ⁴√(x⁴+1)- ⁵√(x⁴+1)}.                        1

14) limₓ→∞(5x³-3x+1)/(7x³+2x²-2). 5/7

15) limₓ→_∞(5-6x²)/(1++2x-3x²)  2

16) limₓ→∞(x√x+√x -1)/(5√x+1)  ∞

17) limₓ→∞ {(x+1)(2x+1)(3x+1)}/ {(x²+1)(5x-3)}.                             6/5

18) limₓ→∞{1²+2²+...+x²}/{(x-2)(x+3)(x-4)}.                             1/3

19) limₓ→∞{1+ 1/2 + 1/2²+.... to n terms}.                                         2

20) limₓ→∞{1+3+5+... to n terms}/(n² -1).                               1

21) limₓ→∞{2+5+8... to(2n+1)}/{1+2+3+.... to n terms}      12

22) limₓ→∞(1.2+2.3+3.4+...to n terms)/{(3-n)(n+1)(n+2).          -1/3

23) limₓ→∞{√(x²-2x+1) - √(x²-5x-3)}.              3/2

24) limₓ→∞ [³√x²{³√(x+1)- ³√x}]   1/3

25) limₓ→∞{(2x-1)³(x²+1)}²/{(x³-2x+1)(3x+1)}.                   8/3

26) limₓ→∞{(2x³-x+1)²(x²-1)³}/ {(3x+1)⁴(2x⁴-3x+1)²}.               1/81

27) limₙ→∞ (1+3+....+n)/n².   Or limₙ→∞ ∑n/n²                          1/2

28) limₙ→∞ ∑n³/n⁴                      1/4

Type: 9

1) lim ₓ→₀ sinx/x                           1

2) lim ₓ→₀ sinxº/x.                π/180 

3) lim ₓ→₀ (Sin px)/(sin qx)        p/q

4)  lim ₓ→₀ (1-cosx)/2x².             1/4

5) lim ₓ→₀ (1- cos2x)/3tan²x.     2/3

6) lim ₓ→₀(1- cos4x)/(1-cos2x).  4

7) limₓ→₀ (tanx - sin x)/x³.           1/2

8) limₓ₋₀ x tanx/(1-cos2x).           1/2

9) limₓ₋₀ (sin3x + 7x)/(4x +sin2x). 5/3

10)limₓ₋₀(sin4x+sin3x)/ (sin5x- sin4x).                                               7

11) limₓ₋₀{cos(a-x)-cos(a+x)}/x.       2sin a

12) limₓ₋₀(1-√cosx)/x².                1/4

13) limₓ₋₀sinlog(1+x)/log(1+sinx)  1

14) limₓ₋₀{(sin4x)/x}³.                64

15) limₓ₋₀ tan 4x . cosec2x.        2

16) limₓ₋₀xsinx/(1-cos2x).         1/2

17) limₓ₋₀x(sin2x - cos2x)/sin³x. 5/2

18) limₓ₋₀(cosax - cosbx)/x².  1/2(b²-a²)

19) limₓ₋₀(1-cosx)/sin²x.             1/2

20) limₓ₋₀ x(cos2x + cosx)/sin x    2

21) limₓ₋₀ (xsina - a sinx)/(x-a)    sin a - a cos a

22)limₓ₋₀ (cos5x-cos3x)/(cos3x - cosx).                              2

23) limₓ₋₀{√(1+sinx)-√(1-sinx)}/x    1

24) limₓ₋₀{cos(x+h)-cosx}/x.    - sinx

25) limₓ₋₀{tan(x+h)-tanx}/h     sec²x

26) limₓ₋₀sin(x/2)/x.                     1/2

27) limₓ₋₀ sin²x/x².                         1

28) limₓ₋₀sin(x²)/x

29) limₓ₋₀ sin3x/tan 5x.              3/5

30) limₓ₋₀ x sin(1/x).                     0

31) limₓ₋₀ {x² cos(1/x)}/sinx

32) limₓ₋₀ (1- cos²x)/{x(1+cosx)}

33) limₓ₋₀ (sinx - sina)/(x-a)

34) limₓ₋₀(sin2x-2sinx)/x

35) limₓ₋₀(3sinx -sin3x)/x³

36) limₓ₋₀ (1-cos2x)/cos²x

37) limₓ₋₀ {1-cos√(cos2x)}/x².   1/2

38) limₓ₋₀ x cot 2x. 

39) limₓ₋₀ sin²x/2x

40) limₓ₋₀ (1-cosx)/sin³x

41) limₓ₋₀(cosec²x -2)/(cotx -1)

42) limₓ₋₀ (1-cosx)/x²

43) limₓ₋₀(tan2x - sin2x)/x³

44) limₓ₋₀ x³cotx/(1-cosx)

45) limₓ₋₀(cotx - cosecx)/x

46) limₓ₋₀(sin5x - sin3x)/sinx

47) limₓ₋₀ (cos5x- cos3x)/x²

48) limₓ₋₀ (tan3x -x)/(5x - sinx)

49) limₓ₋₀ (x²- tan2x)/tanx

50)limₓ₋₀(5xcosx-3sinx)/ (2x²+tan2x)

51) limₓ₋₀(1-co2x+tan²x)/(xsin²x)

52) limₓ₋₀(sinax+bx)/(ax+sinbx)

53) limₓ₋₀(1-cosnx)/(1-cosnx)

54) limₓ₋π  sinx/(π-x)

55) limₓ₋π  sin(π-x)/π(π-x)

56) limₓ₋π/₂ (π/2 - x)tan x

57) limₓ₋π/₂ cotx/(π/2 -x)

58) limₓ₋π  sin3x/sinx

59) limₓ₋π/₂(1+cos2x)/(π-2x)²

60) limₓ₋π/₄  (cosx -sinx)/(x-π/4).   - √2

61) limₓ₋π/₂  (1-sinx)/(π/2 -x)

62) limₓ₋π/₆ (2-√3 cosx -sinx)/(6x-π)²

63) limₓ₋π   cosx/(2x -π).            -1/2

64)limₓ₋π/₃(tan²x-3tanx)/ cos(x+π/6)

65) limₓ₋π/₆(√3 sinx - cosx)/(x-π/6)

66) limₓ₋₁ (cosπx/2)/(1-x)

67) limₓ₋₁ (1-x) tan(πx/2).          2/π

68) limₓ₋π/₂ (secx - tanx)

69) limₓ₋π/₂(cos3x+3cosx)/(π/2 -x)³

70) limₓ₋π (sin3x -3sinx)/(π -x)³

71) limₓ₋π (1+cos³x)/sin²x.      3/2

72) limₓ₋π/₄(2- cosec²x)/(1 - cotx)

73) limₓ₋₀ x cos(1/x)

74) limₓ₋ᵥ  (x-v) sin {1/(x+v)}

75) limₓ₋π/₄  (sin³x - cos³x)/(x- π/4)

76) limₓ₋π/₂{√(2-sinx) -1}/(π/2 -x)²

77) limₓ₋π/₂ (2x -π)/cosx

78) limₓ₋π/₄ (1-tanx)/(1 - cotx)

79) limₓ₋₂  (tanπx)/(x+2)

80) limₓ₋π  (1+sec²x)/tan²x

81) limₓ₋ₐ  (sinx - sina)/(√x - √a)

82) limₓ₋π/₂ {√2 - √(1+sinx)}/cos²x

83) limₓ₋π/₄ (1- tanx)/(1- √2 sinx)

84) limₓ₋₀ {√2 - √(1+cosx)}/sin²x

85)limₓ₋π/₄{√2-cosx -sinx)}/(4x -π)²

86) limₓ₋₀ {(1+x)⁶-1}/{(1+x)⁵ -1}

87) limₓ₋ₐ {(x+2)⁵/³ -(a+2)⁵/³}/(x-a)

88) limₓ₋₀ {√(2x) -2}/(x-2).            1

MIXED PROBLEMS

1) limₓ₋₂ (x²-x-2)/{(x²-2x)+sin(x-2)}

2) limₓ₋₀ {tan log(1+x)}/x

3) limₓ₋₀  (eˣ +sinx -1)/log(1+x)

4) limₓ₋₀ log(cosx)/sin²x

5) limₓ₋₀  (aˣ - bˣ)/sinx

6) limₓ₋₀ (1- e⁻ˣ)/sinx

7) limₓ₋₀ {log(1+ax)}/ sin bx

8) limₓ₋₀ {⁵√(x+a) -⁵√(x)}/a

9) limₓ₋∞ √{(x -sinx)/(x+cos²x)}.   1

10) limₓ₋π/₄{4√2-(cosx+sinx)⁵}/(1-sin2x)                                             5√2

11) limₓ₋₀ (eˢⁱⁿˣ - 1)/x.                   1

12) limₓ₋₀  sin(x²+4x)/(x³-5x²+2x).  2

13) limₓ₋₀ sinlog(1+x)/log(1+sinx)

14) limᵥ₋₀{(x+v)sec(x+v) - xsecx}/v

15) limₓ₋₀ (10ˣ - 5ˣ - 2ˣ+1)/(xsinx)

16) limₓ₋₀ (15ˣ- 3ˣ - 5ˣ+1)/(xsinx)

REASONING (RELATION PUZZLE)

            RELATION PUZZLE

           *********     ********

*) A family has a man, his wife, their four sons and their wives. The family of every son has 3 sons and one daughter. Find out the total number of male members in the whole family.

*) A and B are brothers C and D are sisters. A's son is D's brother. How is B related to C ?

*) A is B's sister. C is B's mother. D is C's father. E is D's mother. Then, how is A related to D ?

*) Ravi is son of Aman's father's sister. Sahil is son of Divya who is mother of Gaurav and grandmother of Aman. Ashok is father of Tanya and grandfather of Ravi. Divya is wife of Ashok.

(I) How is Ravi related to Divya ?

(ii) How is Gaurav's wife related to
      Tanya ?

*) There are six children playing football, namely A,B,C,D,E,F. A and E are brothers. F is the sister of E. C is the only son of A's uncle. B and D are the daughters of the brother of C's father.

(i) How is C related to F ?

(ii) how many male players are
     there

(iii) how many female players are
       there

(iv) how is D related to A ?

*) Proshant Arora has three children-- Sangeeta,Vimal and Ashish. Ashish married Moni, the eldest daughter of Mr. And Mrs.Roy. the Roys married their youngest daughter to the eldest son of Mr. and Mrs. Sharma, and they had two children named Amit and Shashi. The Roys have two more children, Roshan and Vandana, both elder to Veena. Sameer and Ajay are son's of Ashish and Monica. Rashmi is the daughter of Amit.

(i) what is the surname of Rashmi?

(ii) How is Sameer related to
      Monica's father ?

(iii) What is the surname of
       Sameer

(iv) How is Mrs. Roy related to
       Ashish?

1) In a joint family, there are father, mother, 3 married son's and one unmarried daughter. Of the sons,two have 2 daughters each, and one has a son. How many female members are there in the family ?

2) If X is brother of son of Y's son,
     then how is X related to Y ?

3) Given that

(i) A is the mother of B

(ii) C is the son of A

(iii) D is the brother of E

(iv) E is the daughter of B

The grandmother of D is ?

4) A,B,C are sisters. D is the brother of E and E is the daughter of B. How is A related to B ?

5) A and B are married couple of X and Y brothers. X is the brother of A. How is Y related to B ?

6) Deepak has a brother Anil. Deepak is the son of Prem. Vimal is Prem’s father. In terms of relationship, what is Anil of Bimal ?

7) B is the husband of P. Q is the only grandson of E, who is wife of D and mother-in-law of P. How is B related to D?

8) I. F is the brother of A

II. C is the daughter of A

III. K is the sister of F

IV. G is the brother of C.

Who is the uncle of G ?

9) A is the uncle of B, who is the daughter of C and C is daughter-in-law of P. How is A related to P ?

10) P's father is Q's son. M is the paternal uncle of P and N is the brother of Q.
How is N related to M ?

11) A is the son of B. C, B's sister, has a son D and a daughter E. F is the maternal uncle of D.

(i) How is A related to D

(ii) How is E related to F

(iii) How many nephews does F
       have

12) Deepak is brother of Ravi. Reena is sister of Atul. Ravi is son of Reena. How is Deepak related to Reena ?

13) Q is the brother of R; P is the sister of Q; T is the brother of S; S is the daughter of R, Who are the causins of Q ?

14) E is the son of A. D is the son of B. E is married to C. C is B's daughter. How is D related to E?

15) A is father of C and D is son of B. E is brother of A. If C is sister of D, how is B related to E ?

16) Q's mother is sister of P and daughter of M. S is daughter of P and sister of T.
How is M related to T

17) D, the son-in-law of B, is the brother-in-law of A who is the brother of C.
How is A related to B ?

18) A is father of C. But C is not his
      son

E is the daughter of C. F is the spouse of A.

B is the brother of C. D is the son of B.

G is the spouse of B. H is the father of G.

(i) who is the grandmother of D

(ii) Who is the son of F

19) C is A's father's nephew. D is A's causin but not the brother of C. How is D related to C ?

20) P is the son of Q while Q and R are the sisters to one another. T is the mother of R. If S is the son of T, which of the following statements is correct ?

i) T is the brother of Q

ii) S is the causin of P

iii) Q and S are sisters

iv) S is the maternal uncle of P

v) R is the grandfather of P

REASONING (BLOOD RELATIONS)

         BLOOD RELATIONS

             ****** xxxxxx *****

*) Pointing towards a person, a man said to a woman, “ His mother is the only daughter of your father.” How is the woman related to that person ?

*) Pointing to a lady in the photograph, Shaloo said, “Her son's father is the son-in-law of my mother.” How is Shaloo related to the lady ?

*) Anil introduces Rohit as the son of the only brother of his father's wife. How is Rohit related to Anil ?

*) Pointing towards a person in the photograph, Anjali said, “He is the only son of the father of my sister's brother.” How is that person related to Anjali ?

*) Rita told Mani, “The girl I met yesterday at the beach was the youngest daughter of the brother-in-law of my friend's mother.” How is the girl related to Rita's friend

*) A woman going with a boy is asked by another woman about the relationship between them. The woman replied, “My maternal uncle and the uncle of his maternal uncle is the same.” How is the lady related with that boy ?

*) Pointing out to a lady, Rajan said, “She is the daughter of the woman who is the mother of the husband of my mother.” Who is the lady to Rajan

*) A man pointing to a photograph says, “The lady in the photograph is my nephew's maternal grandmother.” How is the lady in the photograph related to the man's sister who has no other sister ?

1) Pointing to a photograph, a man said, “ I have no brother or sister but that man's father is my father's son.” Whose photograph was it.

2) Pointing towards a boy, Veena said, “He is the son of only son of my grandfather.” How is that boy related to Veena ?

3) Introducing Reena, Monica said, “she is the only daughter of my father's only daughter”. How is Monica related to Reena ?

4) Pointing to a man, a woman said, “His mother is the only daughter of my mother.” How is the woman related to the mother ?

5) If X is the brother of the son of Y's son, how is X related to Y ?

6) Pointing towards Rita, Nikhil said, “I am the only son of her mother's son.” How is Rita related to Nikhil ?

7) Pointing to a lady, a man said, “ The son of her only brother is the brother of my wife.” How is the lady related to the man ?

8) Pointing to Ketan, Namrata said, “He is the son of my father's only son.” How is Ketan's mother related to Namrata ?

9) Pointing to a man on the stage, Rashi said, “He is the brother of the daughter of the wife of my husband.” How is the man on the stage related to Rashi ?

10) A woman Introduce a man as the son of the brother of her mother. How is the man related to the woman

11) Introducing a man, a woman said, “He is the only son of my mother's mother.” How is the woman related to the man?

12) Looking at a portrait of a man, Harsh said, His mother is the wife of my father's son. Brother and sisters I have none.” At Whose portrait was Harsh looking ?

13) Pointing to a man in a photograph, Asha said, “His mother's only daughter is my mother.” How is Asha related to that man ?

14) Introducing a man, a woman said, “His wife is the only daughter of my father.” How is that man related to the woman ?

15) Pointing towards a girl in the picture, Sarita said, “ She is the mother of Neha whose father is my son.” How is Sarita related to the girl in the picture?

16) Deepak said to Nitin, “ That boy playing football is the younger of the two brothers of the daughter of my father's wife.” How is the boy playing football related to Deepak?

17) Pointing to a lady on the platform, Manju said, “She is the sister of the father of my mother's son.” Who is the lady to Manju ?

18) Introducing a man to her husband, a woman said, “ His brother's father is the only son of my grandfather.” How is the woman related to this man ?

19) When Anuj saw Manish, he recalled,”He is the son of the father of my daughter's mother.” Who is Manish to Anuj ?

20) Pointing to Kapil, Shilpa said,” His mother's brother is the father of my son Ashish.” How is Kapil related to Shilpa ?

21) Showing the man receiving the prize, Saroj said,”He is the brother of my uncle's daughter.” Who is the man to Saroj ?

22) Pointing to a girl in the photograph, Amar said, “Her mother's brother is the only son of my mother's father.” How is the girl's mother related to Amar ?

23) Pointing to a photograph, Arun said. “She is the mother of my son's wife's daughter.” How is Arun related to the lady ?

24) X introduces Y saying, “He is the husband of the grand daughter of the father of my father.” How is Y related to X ?

25) If Kamal says, Ravi's mother is the only daughter of my mother,” how is Kamal related to Ravi ?

26) Rahul told Anand, “Yesterday I defeated the only brother of the daughter of my grandmother.” Whom did Rahul defeat?

27) Pointing to a woman, Naman said, “ She is the daughter of the only child of my grandmother.” How is the woman related to Naman ?

28) Pointing to a photograph, a person tells his friend, “She is the grand daughter of the elder brother of my father.” How is the girl in the photograph related to this man?

29) A man said to a lady, “Your mother's husband's sister is my aunt.” How is the lady related to the man ?

30) If Neena says, “Anita's father Raman is the only son of my

father-in-law Mahipal's,” then how is Bindu, who is the sister of Anita, related to Manipal ?

31) Pointing to the woman in the picture, Rajiv said,”Her mother only one grandchild whose mother is my wife.” How is the woman in the picture related to Rajiv ?

32) A girl introduced a boy as the son of the daughter of the father of her uncle. The boy is girl's

33) Pointing to a gentleman, Deepak said, “His only brother is the father of my daughter's father.” How is the gentleman related to Deepak ?

34) Pointing out to a lady, a girl said, “ She is the daughter-in-law of the grandmother of my father's only son.” How is the lady related to the girl ?

35) Pointing to a photograph, a lady tells Pramod, “I am the only daughter of his lady and her son is your maternal uncle.” How is the speaker related to Pramod's father ?

36) Pointing to a person, a man said to a woman, “His mother is the only daughter of your father.” How was the woman related to the person ?

37) Pointing to a man in a photograph, a woman said, “His brother's father is the only son of my grandfather.” How is the woman related to the man in the photograph

38) Arun said, “The girl is the wife of the grandson of my mother,” Who is Arun to the girl ?

39) Pointing to an old man, Kunal said, “His son is my son's uncle.” How is the old man related to Kunal

40) Pointing to a photograph, a woman says, “This man's son's sister is my mother-in-law” How is the woman's husband related to the man in the photograph ?

2nd ORDER DERIVATIVES

   2nd ORDER DIFFERENTION

       ********  +++++ *********

1) y=log(ax+b) prove
     y₂= -a²/(ax+b)²

2) log(x)/x prove x³y₂ - 2log(x)+3=0

3) y= xᵐeⁿˣ show that

y₂= {m(m-1)xᵐ⁻² +
                 2mnxᵐ⁻¹+n²xᵐ}eⁿˣ

4) x=(1-t)/(1+t) and y= 2t/(1+t) prove y₂=0

5) x=t+1/t and y= t- 1/t show that at t=2 , y₂ =-32/27

6) x²/a² + y²/b²=1 show
     y₂=-b⁴/(a²y³)

7) If ax²+2hxy+by²=1 prove that
      y₁=-(ax+hy)/(hx+by)
      y₂ = (h² -ab)/(hx+by)³

8) y(1-x)=x² show (1-x)y₂- 2y₁ =2

9) If log(√(x-2) + √(x+2))
     prove (x²-4)y₂+xy₁=0

10) log(x+√(x²+1)) show
       (x²-1)y₂+xy₁=0

11) y=√(x+1) +√(x-1) prove that
     (x²-1)y₂+xy₁=y/4

12) y= px +q/x² show
       x²y₂+2xy₁ =2y

13) u=v³log(1/v) show
       vu₂- 2u₁+3v²=0

14) if x=sin(θ), y=sin(pθ) prove that
      (1-x²)y₂ - xy₁+p²y=0

15) if x= a(θ+sin(θ)), y=a(1-cosθ) prove that 4ay₂=sec⁴(θ/2)

16) if cos⁻¹(y/b) = n log(x/n) prove
       x²y₂ + xy₁ + n²y =0

17) if y= tanx + sec x
       show (1-sinx)²y₂= cosx.

SHARE And DIVIDEND (A to Z)

          SHARE AND DIVIDEND

              *********** **********

Exercise -1

1) Find the investment in buying:

A) 850 shares of ₹10 each quoted at ₹ 21.

B) 275 shares of ₹100 each at 15 premium.

C) 300 shares of ₹10 each at 2 discount.

D) 600 shares of ₹100 each at ₹17 below par.

E) 900 shares of ₹10 each at par.

2) Find the investment in buying:

A) 300 shares of ₹10 each at ₹10 above par.

B) 700 shares of ₹10 each at ₹4 below par.

C) 400 shares of ₹100 each at 20 premium.

D) 175 shares of ₹100 each at 10 discount.

3) What money will be released on selling:

A) 300 shares of ₹10 each at 15 premium.

B) 250 shares of ₹10 each at 2 discount.

C) 225 shares of ₹100 each at 150 premium.

D) 200 shares of ₹100 each at ₹6 below par,

4) A man invests ₹5600 in the shares of a company paying 8% dividend at the time when a ₹100 share cost ₹140 in the market. Find his early income.

5) A company declares a dividend of 20% on ₹100 share which is quarted at 150.

A) What is the total cost of 50 shares?

B) What is his annual income ?

C) What is the rate of interest that he gets ?

6) Shyam invests ₹12000 in buying ₹10 shares. If the nominal value of the shares is ₹7500, what is the market price of a share?

7) A man invests ₹ 2160 in shares of ₹9 each and sells them at ₹12. Find his profit percentage from his investment ?

8) What is the percentage interest on capital invested in 18% shares, when ₹10 share costs ₹12 ?

9) A company declares a dividend of 15% on a ₹ 100 share which is quoted at ₹140.

A) Find how many shares can be purchased, out of ₹12600.

B) What is the dividend ?

C) what is the rate of interest on the investment?

10) A Man buys 100 shares in a company at ₹ 150 each. The company issues one bonus share for every two shares held by him. When the price of the shares fell to ₹120 he sold the shares. Find his percentage profit.

11) A man buys a ₹15 shares in a company which pays a 10% dividend and he buys this at such a price that his profit is 20% on his investment. At what price did he buy ?

12) A man buys a ₹20 share in a company which pays 12% dividend by investing ₹11000. If his income is ₹880, find 

A)the market value of each share.

B) the nominal value of the shares.

13) A man buys ₹10 shares which stand at ₹15 in a company which pays 9% dividend. Find

A) the market value of 200 shares.

B) the annual income.

C) the percentage return on his investment.

14) A man invests ₹30000 in 15% ₹110 shares at ₹120. When the shares rise to ₹130 he sales out enough shares to purchase a scooter for ₹3900. Find

A) the number of shares he still holds.

B) his loss of annual income.

15) A man ₹4800 in shares of a company which was company which was paying 8% dividend at the time when a ₹100 share costs ₹160. Find

A) his annual income from the shares

B) the rate of interest he gets on his investment.

16) Ravi invested ₹6250 in shares of a company paying 6% per annum. If he ₹ 25 shares for ₹31.25 each. Find his annual income from his investment.

17) Rajendra purchased 300 shares of the face value of ₹100 each from the market at ₹800 per annum. If the company paid a dividend of 40%, find Rajinder 's earning percentage on the investment.

18) Devinder invested ₹5500 in the shares of a company. At the end of the year the company declared 20% dividend which gave Devinder an income of ₹ 500. At what premium the shares were purchased?

19) Neetu invested ₹2750 when she bought ₹100 shares of a company available for ₹110 each. If the company paid 15% dividend during the year after this purchase, find Neetu's income from this dividend.

20) 100 rupee share of a company is quoted at ₹899 in the market. How much does Sanjay pay to purchase 500 such shares premium 1%. What is his gain percent on this investment. If the company pays a dividend of 25% ?

21) Leela buys 1500 shares of a company of ₹10 each quoted at ₹33. What amount of money is she investing in these shares? Also, find the profit made by the original shareholder if he had bought the shares at ₹18 each.

22) Razia invested ₹12255 in buying the shares of a company at ₹19 each. If the face value of each share be ₹10 and company paid 25% dividend at the end of the year, find the dividend earned by her.

23) Find the cost of 110 shares of ₹25 at 5 premium per share.

24) Find the cost of 25 shares of ₹100 at 5 discount per share.

25) What rate per cent will a man from his 200 ordinary shares of ₹25 each bought at 5 premium, the rate of dividend being 16%.

26) Anuj buys ₹25 shares of a company which pays 20% dividend. The market price is such that he get 25% on his money. At what price did he buy it ?

27) Find the price of 18% share of face value 100, that gives 12% income.

28) Amit invested ₹3783 in the shares of face value ₹100 each of a company. At the end of the year, the company declared dividend at 15%. He got an income of ₹450. At what price was the share quoted.

1) Mr. Mehra invested Rs. 26000 in 15% Rs100 shares quoted at a premium of 30%. Calculate

(i) The Number of shares bought.

(ii) His income from the investment.

(iii) percentage return on investment

He sold these shares when they were quoted at a premium of 50% and invested the proceeds in 10% Rs 50 shares quoted at a discount of 20%. Calculate his change of income.

2) On investing Rs12000, to buy Rs10 shares of a company, the total nominal value of the shares acquired is Rs7500. if rate of dividend is 10% p.a  Find

(i) Market value of each share

(ii) Number of shares purchased

(iii) Dividend received

(iv) Rate of Return.

3) 10% Rs15 shares of a company ensures a return of 20% on his involvement to an investor. For how much did he buy a share of the company.

4) A man invests Rs56700 in 20% Rs100 shares of a company quoted at Rs120. Find his yearly income.

He sold half of the shares he owned, when these were quoted at Rs130 and invested the money received in 10% Rs10 shares quoted at Rs8. Calculate the change in his yearly income.

5) John invested ⅓ of his savings in 20% Rs50 shares of a company A quoted at Rs60 and the remaining of his savings in 10% Rs100 shares of company B quoted at Rs110. His total Dividend from these investments is Rs4600. Find

(i) the value of his total savings.

(ii) number of shares of company A held by him.

(iii) Number of shares of company B held by him.

6) Ashok invests Rs11000 to buy 12% Rs20 shares of a company at a certain price. His income from the investment is Rs1065. Find the number of shares held by him and the market value of a share.

7) How much money is needed to buy 200, 9% Rs10 shares of a company quoted at a premium of 50% also find

(i) the market value of the Rs10 shares.

(ii) the annual income

(iii) the% return on the investment.

8) Which of the following is a better investment ?

(i) Investing in 12% Rs10 shares at par

(ii) Investing in 15% Rs100 shares at Rs120.

9) A invests Rs8000 to buy 3% Rs100 shares at Rs80. B invests Rs8000 in Rs100 shares at par. At the end of the year They receive equal dividend. Find the rate% of Dividend received by B.

10) Ravi invested Rs4800 in 6% Rs100 shares quoted at Rs120. Find
(i) Number of shares held by him.

(ii) Ravi's income from the investment.

Ravi sold these shares, when they were quoted at Rs.140 and invested the proceeds in 5% Rs10 shares quoted at Rs8. Find his income now (from the newly bought shared) and his% return now on his initial investment.

11) Ram invested Rs6840 in buying shares of nominal value Rs30 which are being sold at 20% premium. The Dividend on the shares is 12% p.a. Calculate
(i) the market value of shares.

(ii) the number of shares bought.

(iii) the Dividend He would receive at the end of the year.

12) A person invested 60% of his savings at retirement in two companies A and B as follows:

40% in company A in buying 10% Rs100 shares at a premium of 20%.
60% in company B in buying 8% Rs10 shares at a discount of 10%.

(i) If he bought 40 shares of company A, how many shares did he buy of company B ?

(ii) Work out value of his savings at retirement.

13) A person invested his savings as follows:

20% of his savings in buying 10% Rs100 shares of a company A quoted at Rs160.

60% of his savings in buying 6% Rs50 shares of a company B quoted at Rs60.

20% of his savings in buying 5% Rs100 shares of a company C quoted at Rs80.

Given that he obtained 40 shares of company A, calculate

(i) No, of shares of company B and C bought by him.

(ii) Total Dividend earned by him at the end of the year.

(iii) Overall% return on his entire investment.

14) Mr Roy invests Rs45000 in 15% Rs100 shares quoted at Rs125. When the market value of these shares rose to Rs140,he sold shares just enough to raise Rs8400. Find

(i) the number of shares he still holds.

(ii) the Dividend due to him on these reduced Number of shares.

15) A man buys 250 Rs100 shares. He received a Dividend of Rs2000 on these shares. Calculate the rate% of the Dividend declared on the shares.

16) A man purchased
50 Tata mills 14% Rs100 shares quoted at Rs116.50.
180 Share Ram mills 20% Rs10 shares quoted at Rs110
100 Hindustan motors 14% Rs1000 shares quoted at Rs1400. Find

(i) annual Dividend due to him

(ii) his total investment.

(iii) % over al return on his total investment correct to 1 decimal places.

) Find the dividend at the end of a year on 250 shares of ₹50 each, if the half yearly dividend is 4% of the value of the share. ₹1000

) A dividend of 9% was declared on ₹100 shares selling at a certain price. If the rate of return is 15/2% calculate:

a) the market value of the share. ₹120

b) the amount to be invested to obtain an annual dividend of ₹630. ₹8400

) A man buys 500, ₹20 shares at a discount of 20% and receive a return of 10% on his money. Find

a) the amount invested by him. ₹8000

b) the rate of dividend paid by the company. 8%

) A man invested ₹45000 in 15%, ₹100 shares quoted at ₹125. When the market value of these shares arose to ₹140 he sold some shares

a) the amount invested by him. 300

b) the rate of dividend paid by the company. Rs4500

) A company with 10000 shares of Rs50 each , declare an annual dividend of 5%.

a) What is the total amount of dividend paid by the company. 25000

b) What would be the annual income of a man was 72 shares in the company? 180

c) If he receives 4% on his investment, find the price he paid for each share ? 62.50

) A man invests Rs20020 in buying shares of nominal value of R26 each of 10% premium. The dividend on the shares is 15% per annum. Calculate 

a) the number of shares he buys. 700

b) the dividend he receives annually. 2730

c) the rate of interesting gets on his money. 13.64%

) A man wants to buy 62 shares available at Rs132(par value of Rs100).

a) how much should he invest ? 8184

b) if the dividend is 7.5%, what will be his annual income. 465

c) ifhe wants to increase his annual income by Rs150, how many extra shares should he buy? 20

) A man invests Rs8800 in buying shares of a company of face value of Rs100 each at a premium of 10%. If he earns Rs1200 the end of the year as dividend , find

a) the number of shares he has in the company. 80

b) the dividend percent per share. 15%

) A man invests Rs1680 in buying shares of nominal value Rs24 and selling at 12% premium. The dividend on the shares is 15% per annum. Calculate 

a) the number of shares he buys. 62.5

b) the dividend he receives annually. 225

) A company with 10000 shares of Rs100 declares an annual dividend of 5%.

a) What is the total amount of dividend paid by the company ? 50000

b) What would be the annual income of a man , who has 72 shares, in the company. 360

c) If he received only 4% on his investment, find the price he paid for each share. 125

) A lady holds 1800, Rs100 shares of a company that pays 15% dividend annually. Calculate her annual dividend. If she had bought these shares at 40% premium, what percentage return does she get on her investment ? give your answer to the nearest integer. 6720

) A man invests a sum of money in Rs100 shares, paying 15% dividend quoted at 20% premium. If his annual dividend is Rs540. calculate

a) his total investment. 4320

b) the rate of interest return on his investment. 12.5%

) What sum should a person invest in Rs25 shares, selling at Rs36, to obtain an income of Rs720, if the dividend declared is 12% ? also find.

a) the number of shares bought by him. 8640, 240

b) the percentage return on his income. 25/3%

) Rs1080 are invested in 8% Rs10 shares of a company quoted at Rs12. Calculate

a) the number of shares bought. 90

b) dividend due on the shares. 72

) By investing Rs12750 in a company, paying 8% dividend, an annual income of Rs1200 is received. What is the market value of each Rs100 share ? 85

) By investing Rs11440 in a company, paying 10% dividend , an annual income of Rs520 is received. What is the market value of each Rs 50 share. 110

) How much should a man invest in Rs 25 shares, selling at Rs36 to obtain and annual income of Rs1500, if the dividend declared is 12% ? 18000

) A company declares a dividend of 8% on Rs100 shares . Atul buys such share and gets 10% on his investment. At what price does he buy each share? Rs80

) Deepak invested in Rs25 shares of a company, paying 12% dividend. If he received 10% per annum on his investment, at what price did he buy each share? 30

) Mukul invests Rs9000 in a company paying a dividend of 6% per annum when a share of face value Rs100 stands at Rs150. What is his annual income ? If he sells 50% of his shares when the price rises to Rs200, what is his gain in this transaction. 360,1500

) Rs67200 are invested in Rs100 shares which quoted at Rs120. Find the income if 12% dividend is declared on the shares. 6720

) A man bought 500 shares, each of the face value Rs10, of a certain business concern and during the first year, after purchase, receives Rs 400 dividend on his shares. Find the rate of dividend on the shares. 8%

) Which is better investment : 12% of Rs100 at Rs120 or 8%, Rs100 shares at 90? 12% Rs100 at Rs120

) A man invests Rs11200 in a company paying 6% dividend when its R 100 shares can be bought for Rs140. Find

a) his annual income. Rs480

b) his percentage income on his investment . 30/7%

) A man invests Rs3960 in shares of a company which pays 15% dividend at a time when a Rs25 share costs Rs33. Find

a) the number of shares he bought. 120

b) The annual income from his shares. 450

c) the rate of interest which he gets on his investment. 125/11%

) A man invests Rs7770 in a company paying 5% dividend when a share of nominal value of Rs100 sells at the premium of Rs 5. Find 

a) the number of shares bought . 74

b) annual income. 370

c) percentage income . 4.76%

) A man buys Rs50 shares of a company 12% dividend, at a premium of Rs10. Find 

a) the market value of 320 shares. 19200

b) his annual income. 1920

c) his profit percent. 10%

) A man buys Rs75 shares at a discount of Rs 15 of a company paying 20% dividend. Find 

a) the market value of 120 shares. 7200

b) his annual income. 1800

c) his profit percent. 25%

) By selling at Rs92, some 5/2% share a face value of Rs100 and investing the proceeds in 5% share of face value Rs100 selling at Rs115, a person increased the income by Rs90 a year. Find 

a) the number of shares sold. 60

b) the number of shares bought. 48

c) the original income. 150

) Mr. Ghosh had 720, 12% Rs10 shares . He sold them when the shares were quoted at Rs21 and invested the sales proceed in 4% Rs5 shares quoted at Rs3.50 find

a) the sale proceeds. Rs15120

b) the number of new shares bought by Mr. Ghosh . 4320

c) the change in Mr Ghosh 's annual income. No change 

) Ravi invested Rs4800 in 6% Rs100 shares quoted at Rs120. Calculate 

a) number of shares held by him.

b) Ravi's income from the investment.

 Ravi sold these shares, when they were quoted at Rs140 and invested the proceeds in 5% Rs10 shares quoted at Rs8. Calculate Ravi's income now (from the newly bought shares) and his percentage return now on his initial investment. 40, 240, 350, 175/24%

) Ravi invested Rs6840 in buying shares of nominal value Rs30 which are being sold at 20% premium. The dividend on the shares is 12% per annum. Calculate 

a) the market value of shares. 

b) the number of shares bought by Ravi.

c) the dividend Ravi would receive at the end of the year. 36,190,684

FUNCTION(A To Z)

EXERCISE - A

1) Given f(x)= 2x +3; x belongs to R
Find i)  f(-2). 
ii) find a if f(a)= -5
iii) Is f(x)= 2x+3, a function ? Give reason.

2) given f(x)=2x² -3 , x belongs to R
Find
i) f(0). 
ii) f(-2). 
iii) f(-a)+1
iv) 2f(4) - 1/2 f(-5) ?

3) Let f(x)= (3x-1)/(x - 3) , x≠3
Find i) f(-x)
ii) f(1/x)
iii) f(5)/f(0)
iv) f(-3)/f(1/3)
v) f(a/b) x f(b/a)

4) Given f(x)= x³-,find x if f(x) is 215

5) if f(x)= 2x+1 and g(x)= x² find
i) f(-3)
ii) g(-3)
iii) gf(-3)
iv) f g (-3)

6) Let be defined by f(x)=x/(x²+1) ,
x belongs to R. Find
i) f(1/x) , x≠0
ii) f(2x)
iii) f(x - 1)
iv) f[f(-1)]
v) f[f{f(1/x)}]

7) Let the function f: N -> N be defined by.f(x)= 2x -1,.  If x is odd
                             x+2.    If x is even
Find i) f(2)=
ii) f(1)
iii) f(17)
iv) f(2)+f(7)
v) 1/2. f(3) + f(5)

8) Let the function f: R -> R be
  f(x)=.   2x +1 ,.   If x > 2
               x - 1,.      If -3≤x≤2
              3x+2,.    If x < -3.       Find
I) f(1)
ii) f(-4)
iii) f(-3) +f(3)
iv) f(6)/f(-2)

9) Given f(x)=4x²+1, where x belongs to Z
I) find domain and range of f
ii) find f(1)
iii) find the elements of domain whose images are 101
iv) find x, if f(x)=37

10) A function f is defined by
f x -> 3x+1, x belongs to W.
List the elements of f for which
x <4 and find the corresponding elements of range.

11) A function f is defined on the set real numbers as follows:
   f(x)=.   1+x ; 1≤x2
               2x-1 ; 2≤x<4
               3x -10; 4≤x<6.     Find
I) the domain of the function
ii) range of the function
iii) value of f(4)

12) If f(x)=(x²-5x+6)/(x²-8x+12) then find

a) f(2)            b)f(-5) c) f(0)   

d) f(h+1)       e) f(2+h) f) f(2/h)

13) If f(x)=2x²-5x+4 then finf

a) f(2x)          b) 2f(x)

c) if f(2x) = 2f(x) then find the value
     of x.

14) If f(x) = (x-a)(x-b)(x-c) find

a)f(a)             b) f(c)        c) f(0)

15) If f(x) =(ax+b)/(bx+a) find
     f(x).f(1/x)

16) Let y=f(x)=(x+1)/(x+2) find

a) f(y)      b) f(1/x)  c) f{f(1/x)}

6) If f(x)=10x²-13x+13 find the
     value of x when f(x)=16

17) if f(x) = (x-1)/(x+1) find the value of {f(x)+f(y)}/{1+f(x).f(y)}

18) If y=f(x)=(ax+b)/(bx-a) find f(y)

19) if f(x) = (4x-5)/(3x-4) find f{f(x)}

20) Given f(x) = x+a and g(x)=x-a
     find the value of {f(x)}² - {g(x)}²

21) If y= f(x)=(x+3)/(2x+1) and
       z=f(y) Express z=f(x).

22) Given g(x) = (x-a)/x  + x/(x-b)
      find the value of g(a+b)/2.

23) If f(x) = 1/x² find f(x+h)-f(x-h)

24) If f(x-1)=7x-5 find f(x+2)

25) If f(x)=x/(1-x) find
       {f(x+h)-f(x)}/h.

26) If f(2x-1) = (x+1)/(x+2) then
       find

a)f(x)       b) f(0) f(h+1)   c) f(2+h)

27) f{(x-2)/(x+3)} =(x-1)/(2x+1)
      find the value of f(5-2x)

28) If f(x) = (5)ˣ find

a) f(x+2)

b) f(x+y)

c) f{(2x+1)/(x+2)}

d) f(log₅x)

29) If f(x) = eᵖˣ⁺ᵗ  

    find {f(a)f(b)f(c)}/f(a+b+c)

30) If f(x) =
      a{(x-b)/a-b)} +b{(x-a)/(b-a)}    
      find the value of f(a)+f(b)

31) if f(x) = 2x²+1 and x≼2

                     1/(x-2) and 2≺x≼3

                      2x -5 and x ≻ 3
find the value of

a)f(0)

b) f√(2)

c) f(-2)

d) f(4)

e) f(2.5)

32) eʸ +e⁻ʸ = 2x find y

33) Given f(x)=ax²+bx+c and
   f(1)=3 , f(2)=7,f(3)=13 find a, b, c.

34) if f(x)=a/x +b/x +cx and
    f(1)=5,f(-2)=2,f(-1)=-3,find f(-3)

35) if f(x+3)=2x²-3x-1 findf(x+1)

36) if f(x+3)=3x²-2x+5 find f(x-1)

37) find even and odd or neither
      even nor odd of the following.

a) f(x)=x²+x⁴+x⁶

b) f(x) = 4ˣ +4⁻ˣ

c) f(x) = c

d) f(x) = x(3ˣ +1)/(3ˣ - 1)

d) f(x) = x+x³+x⁷

e) f(x) = 3x³ +x

f) f(x) = (2ˣ+1)/(2ˣ - 1)

g) f(x)= log(x+√(x²+1)).

38) Which of the following Relation are function ? Give your reasons. In case of function, determine its domain and range.
a) {(1,-2),(3,7),(4,-6),(8,11)}
b) {(1,0),(1,-1),(2,3),(4,10)}
c) {(a,b),(b,c),(c,b),(d,c)}
d) {(2,3),(1,4),(2,1),(3,2),(4,4)}
e) {(3,1),(4,2),(1,1)}
f) {(1,2),(2,2),(3,2),(4,2)}
g) {(x,y): x belongs to A, y belongs to B is surname of x} where A is the set of people in India and B is the set of surnames.
h) {(x,y): x belongs to A, y belongs to B, y is the area of a square of sidex} where A is the set of measurement of length.
i) {(x,y): x belongs to B, y belongs to P, y is a passenger on X} where B is the set of buses of a school and P is the set of pupils of some schools.
j) y= 3x+2
k) a is the capital of b where b belongs to B and B is the set all countries, a belongs to A and A is the set of capital cities of countries.
l) y< x+3.
m) y is Math teacher of x where x represents any pupil taking up Maths in a school.
n) y is Maths pupil of x, where x represents any Maths teacher in a school.

39)State the domain of following:

a) f : -> 5x
b) g : ->5x, x belongs to Z
c) h : x -> 2/(x -7)
d) F: x -> 5x, x belongs {0,1,2}
e) f: x -> x/5
f) F: x -> 6/x
g) H: x -> x²+5x-6
h) g: x -> (x -4)/{(x-3)(x+6)}
i) g: x -> x/1, x belongs to {2,4,6}
j) g: x -> 1/x, x belongs to R

40) Find whether the function are one-on-one or not.
a) f: R ->R, defined by f(x)=x³, x belongs to R.
b) f: Z -> Z, defined by f(x)=x²+5, x belongs to Z.
c) f: R {3}  ->R, defined by
f(x)=(5x+7)/(x - 3), x belongs to R-{3}
d) f: R -> R, defined by f(x)= |x|, x belongs to R.
e) f: {(2,7),(3,4),(7,9),(-1,6),(0,2),(5,3)
Is it a function, if yes it is one-one or not.
Interchange the order to find the same.
f) To each person on the earth assign the number which corresponds to his age.
g) To each country in the world assign the latitude and longitude of its capital.
h) To each book written by only one author assign the author.
i) To each country in the world which has a prime minister assign its prime minister.

41) Find whether the following are onto functions (surjections) or not.
a) f: R ->R, defined by f(x)=x³+5 for x belongs to R.
b) f: R -> R, defined by f(x)=x²+3, x belongs to R.
c) f: Z -> Z, defined by f(x)= 5x - 9, x belongs to Z.

THEORY OF QUADRATIC EQUATIONS

Exercise -1

-----------------

1) Examine the nature of the roots of the following:

A) 9x²-24x +16= 0.     rational, equ

B) x²-3√5 x -1= 0.     Irrational, uneq

C) 7x²+ 8x +4= 0.     Imaginary

D) 16x²+40x +25= 0.   Rational, eq

E) qx²+px -q= 0.         Real, uneq

F) 7x²+4x -3= 0.       Rational, uneq

2) Find the sum and Product of the roots of the following:

A) 3x²-2x +1= 0.                  2/3, 1/3

B) 2x²+x -1= 0.                    -1/2, -1/2

C) 3x²-1= 0.                                    0

D) (x-1)/(x+1)= (x-3)/2x.          0,3

3) Form the equation whose roots are:

A) 3, -8.                          x²+5x -24= 0

B) 2/7,5/7.              49x²-49x +10= 0

C) 1/2, 3/2.                    4x²-8x +3= 0

D) 2+√3, 2- √3.                x²-4x +1= 0 

E) p+q, p - q.           x²-2px +p¹- q²= 0

F) 2+3i, 2- 3i.                x²-4x +14= 0 

G) 2 - √5.                          x²-4x -1= 0 

H) √5.                                     x²-5= 0  

I) 3 - 2i.                           x²-6x +13= 0

J) 2i.                                       x² +4= 0 

     

4) A) If m and n are the roots of the equation 2x²-5x +1= 0, find the value of

a) m²+ n².                                     3/2

b) m³+ n³.                                    95/8

c) m² - n².                             ±5√17/4

B) If p, q are the roots of 2x²-5x +7= 0, find the values of:

a) 1/p + 1/q.

b) p/q + q/p.  

c) p²/q + q²/p.                          -85/28

C) If m, n are the roots of 2x²+x +7= 0 find the values of (1+ m/n)(1+ n/m).                                          1/14

D) If p, q are the roots of ax²+bx +c= 0, find the values of:

a) p²+ q².                        (b²-2ac)/a²

b) (p - q)².                      (b⁴-4ac)/a²

c) p²q+ q²p.                              -bc/a²

d) p³+ q³.                    -(3abc -b³)/a³

e) p³q + q³p.                c(b²-2ac)/a³

f) p²/q + q²/p.           (3abc - b³)/a²c

E) If the roots of 3x²-6x +4= 0 are m and n, find the value of (m/n + n/m) + 2(1/m + 1/n) + 3mn.                    8

F) If m, n are the roots of the equation ax² + bx + c= 0. Find the values of:

a) (1+ m+ m²)(1- n+ n²).     (a²+b² + c² + ab - ca +bc)/a²

b) m⁴+ n⁴.       (b⁴-4ab²c +2a²c²)/a⁴

c) 1/m⁴ + 1/n⁴.  (b²-4a²c² +2a²c²)/a⁴

d) m⁶+ n⁶.          (b⁶+9a²b²c² -6ab⁴c - 2a³c³)/a⁶

5) A) If m and n are the roots of the equation x²-4x +11= 0, find the equation whose roots are

a) m+2, n+2.                 x²- 8x +23= 0

b) 1/m, 1/n.               11x²+ 4x +1= 0

c) m/n, n/m.            11x²+6x +11= 0

B) If p, q are the roots of 2x²-6x +3= 0, form the equation whose roots are p+ 1/q and n+ 1/m 6x²-30x+25= 0.    

C) If m and n are the roots of the equation x²- 4x +11= 0. Find the equation whose roots are:

a) m+2, n+2.              x²- 8x +23= 0

b) 1/m, 1/n.            11x²+ 4x +1= 0

c) m/n, n/m.          11x² +6x +11= 0

D) If p, q are the roots of 2x²-6x +3= 0, form the equation whose roots are p+ 1/q and n+ 1/m.    6x²-30x +25= 0

6) a) Prove that the roots of the equation (x-a)(x-b)= p² are always real.

b) Prove that the roots of 3x²+22x +7= 0 can not be imaginary.

c) Find the sum and Product of the roots of x²- 12x +23= 0 and hence determine the square of the difference of the roots.    12, 23, 52

d) The sum and the product of the roots of a quadratic equation are 12 and -27 respectively. Find the equation.                  x²- 12x -27= 0

e) For what value of m the product of the roots of the equation mx² - 5x + (m+4)= 0 is 3 ?                             2

f) For what value of k will the sum of the roots of the equations x²- 2(k+3)x +21k +7 = 0.                      15

g) Find the value of m if the product of the roots of the equation x² + 21x +(m+8) = 0 be 13.                            5

h) Determine the value of p, so that the roots of the equation px² - (3p+2)x +(5p -2)= 0 are equal.   P+2, -2/11

i) Determine the value of m if the difference between the roots of the equation 2x²- 12x +m+ 2= 0 be 2.   14

j) Determine the values of p and q, so that the roots of the equation x² + px +q= 0 are p and q.    (1,2) or (0,0)

k) If the equation x²+ 2(m+2)x +9m = 0 has equal roots, find m.      4,1

l) For what values of m will roots of the equation  x²- (5+ 2m)x +(10+ 2m) = 0 be

i) equal in magnitude but opposite in sign.                                        5/2

ii) reciprocal.                             -9/2

m) For what value/s of m will be equation x²- 2(5+2m)x +3(7+10m) = 0 have

i) equal roots.                      2 or 1/2

ii) reciprocal roots                    - 2/3

n) For what values of m will the sum of the roots of the equation 2x²- 12x +m+ 2 = 0 be equal to twice their product.                         4

o) The roots m, n of the equation x² + Kx +12= 0 are such that m - n= 1, find k.                                           ±7

p) Find the values of p for which the equation x² - px +p+ 3 = 0 has

A) coincide roots.                     6, -2

B) real distinct roots.       p< -3, p> 6

C) one positive and negative root.   P < - 3.

7) If m and n are the roots of the equation x² - 4x +11= 0, find the equations whose roots are

a) m+2, n+2.                x²- 8x +23=0

b) 1/m and 1/n.          11x²+ 4x +1=0

c) m/n and n/m.      11x²+ 6x +11= 0

8) If m, n are the roots of the equation x² - px +q = 0, find the equation whose roots are:

a) m² and n².       x²- (p²-2q)x+ q²= 0

b) m/n and n/m.   qx²-(p²-2q)x+q= 0

c) m+ 1/n and n + 1/m.         qx² - p(q+1)x +(q+1)²= 0

d) 2m - n and 2n - m.     x²- px+ 9q -2p²= 0

e) m²/n and n²/m.       qx² - (p³- 3pq)x + q²= 0

f) 1/(m+n) and (1/m +1/n).        pqx² - (p² + q)x + p = 0

9) If m, n are the roots of the equation 2x²- 6x +3= 0, form the equation whose roots are m+ 1/n and n+ 1/m.              6x²- 30x+25=0

10) If m, n are the roots of the equation ax²+ bx + c= 0, form the equation whose roots are (m+ n)² and (m - n)²           a⁴x²- 2a(b² - 2ac)x+ b²(b² - 4ac) =0

11) If m, n are the roots of the equation 4x²- 8x +3= 0, form the equation whose roots are 1/(m+n)² and 1/(m- n)²             4x²- 5x+1=0

12) If m, n are the roots of the equation 2x²- 3x +1= 0, form the equation whose roots are m/(2n+3) and n/(2m+3).            40x²- 14x+1=0

13) If m, n are the roots of the equation 2x²- 6x +2= 0, form the equation whose roots are (1-m)/(1+ m) and (1- n)/(1+n).             5x²- 1=0

14)a) Find the equation whose roots are the reciprocal of the roots of the equation x² + px + q= 0.    qx²- px +1=0.

b) Find the equation whose roots are the reciprocal of the roots of the equation 2x² + 3x + 7= 0.    7x²+3x +2=0.

c) Find the equation whose roots are squres of the roots of the equation x² + 3x + 2= 0.   x²- 5x +4 =0

Exercise -2

------------------

1) If the difference of roots of x² - px +q = 0 be unity, prove p² + 4p² = (1+ 2q)².

2) If the difference of roots of ax² + bx + c= 0 be 2, prove b²= 4a(a+c).

3) If the difference of roots of x² + px +q = 0 be k, prove p² =4q+ k².

4) If one of root of x² - px +q = 0 be twice the other, prove 2p² = 9q.

5) If one of root of ax² + bx +c= 0 be four times the other, prove b² =25ac

6) If one of root of x² - px +q = 0 be thrice the other, prove 3p² = 16q.

7) If one of root of x² + px +q = 0 be r times the other, prove +r+1)² q = rp².

8) If the roots of (b²-ab)(2x -a) = (x² - ax)(2b - a) are equal in magnitude but opposite in sign, show that  a² = 2b².

9)a) If the roots of lx² + mx + m= 0 be in the ratio p : q, show that √(p/q) + √(q/p) + √m/l = 0.

b) If the roots of px² + qx + q= 0 be in the ratio m : n, show that √(m/n) + √(n/n) + √q/p = 0.

10) If the roots of x² + px + q= 0 be in the ratio m : n, show that mnp²= q(m+ n)².

11) If the roots of ax² + bx + c= 0 be in the ratio 4 : 5 , show that 20b² = 81ac.

12) If one root of x² + px + q= 0 be square of the other, show that p³ - q(3p -1)+ q² = 0.

13) If the ratio of the roots of x² + bx + c= 0 be equal to the ratio of the roots x² + px + q= 0, show that b²q = p²c.

14) If the sum of the roots of x² + px + q= 0 be three times their difference, show that 2p² = 9q.

15) If k be the ratio of the two roots of ax² + bx + c= 0 show that (k+1)²a c = kb².

16) Prove that If the ratios of the roots of x² - 2px + q²= 0 and x² - 2lx + m²= 0 be equal, show that p²m² = q²l².

17) If m and n are the roots of x² + x -1 = 0, prove m² = n+ 2.

18) The ratio of the roots of ax²+ bx + c = 0 is 3: 4. Prove 12b² = 49 ac.

19) If one root of ax²+ bx + c= 0 be the square of the other, show b³+ a²c + ac² = 3abc.

20) If the difference between the roots of ax²+ bx + c = 0 be equal to the difference between the roots of px² + qx + r= 0, show that p²(b² - 4ac) = a²(q² - 4qr).

Exercise - 3

-------------------

1) Find the value of k for which 3x² + 2kx + 2= 0 and 2x² + 3x - 2= 0 may have a common root.        7/2, -11/4

2) Find the value of m for which x²- 5x + 6= 0 and x² + mx +3= 0 may have a common root.        -4, 7/2

3) Find the value of k for which x²- kx + 21= 0 and x² - 3kx +35= 0 may have a common root.                ±4

4) If the equation x²+ p₁x+ q₁ = 0 and x² + p₂x+ q₂ = 0 have a common root, prove that it is either (p₁q₂ - p₂q₁)/(q₁ - q₂) or (q₁ - q₂)/(p₁ - p₂).

5) prove that if x²+ px +q= 0 and x² + qx + p = 0 have a common root, then either p= q or p+ q +1= 0.

6) If the equation x² - 5x + 6= 0 and x² + mx + 3 = 0 have a common root, find the value of m.      -7/2, -4

7) If the equation ax² + bx + c= 0 and bx² + cx + a= 0 have a common root, prove that, a³+ b³+ c³ = 3abc.

MIXED PROBLEM

*******************

1) If the roots of the equation (m - n) x² + (n -1)x + l = m are equal, show that l, m and n are in AP.

2) If the sum of the roots of the equation px² + qx + r = 0 is equal to the sum of the squares of their reciprocals, show that qr², rp², pq² are in AP.

3) The roots of the equation px² - 2(p +2)x + 3p= 0 are m, n. If m - n = 2, calculate the values of m, n and p.           (-1,-3,-2/3) or (3,1,2)

4) Find the condition that the roots of the equation ax² + bx + c = 0 may differ by 5.          b² - 4ac = 25a²

A) SHORT ANSWER TYPE:

1) If m, n are the roots of the equation x²+ x+1= 0, then find the value of m⁴+ n⁴+ 1/mn. 0

2) For what value of p(≠0) sum of the root of px²+2x+3p= 0 is equal to their product? -2/3

3) Form a quadratic equation whose one root is 2-√5. x²-4x-1=0

4) If 2 +i√3 is a root of x²+ px+q= 0, find p and q. -4,7

5) If one root of 2x²- 5x+k= 0 be double the other, find k. 25/9

6) If one root of x²+ (2-i)x - c= 0 be i. Find the value of c and other root of the equation. 2i, -2

7) Form a quadratic equation whose one root is 2 - 3i. x²-4x+13=0

8) If the roots of the equation qx²+ px+ q= 0 are imaginary, find the nature of the roots of the equation px²-4qx+ p=0. Real, unequal

9) If one root of x²+ px+8= 0 is 4 and two roots of x²+ px+q= 0 are equal, find q. 9

10) Construct a quadratic in x such that AM of its roots is A and GM is G. x²-2Ax+ G²= 0

11) if 5p²- 7p+4= 0 and 5q²- 7q+4= 0, but p≠ q, find pq. 4/5

12) if the equation x²+px+6= 0 and x²+4x+4=0 have a common root, find p. 5

13) if x is a real, show that the expression is always positive. Find its minimum value and the value of x for which it will be minimum. 14/5, 4/5

14) If c, d are the roots of (x-a)(x-b) - K= 0 show that a, b are the roots of (x- c)(x- d)+ K= 0.

15) If the roots of the equation x²- 4x - log₂a=0 are real, find the minimum value of a. 1/16

16) Given that m, n are the roots of x² -(a -2)x - a+1= 0. If a be real, Find the least value of m²+n². 1

17) If m, n are the roots of x²- 4x+5 = 0, form an equation whose roots are m/n +1 and n/m +1. 5x²-16x+16=0

B) CHOOSE THE CORRECT: 

1) The sum of their reciprocals of the roots of 4x²+3x+7= 0 is

A) 7/4 B) -7/4 C) -3/7 D) 3/7

2) If one root of 5x²-6x+K= 0 be reciprocal of the other, then

A) K= 6 B) K= 5 C) K= -5 D) K= 1/5

3) If x be real, the maximum value of 5+ 4x- 4x² will be

A) 5 B) 6 C) 1 D) 2

4) The roots of x²+ 2(3m,+5)x+ 2(9m²+25) = 0 will be real if 

A)m>5/3 B)m=5/3 C)m<5/3 D) m=0

5) The equation (4-n)x²+(2n+4)x +8n +1= 0 has equal integral roots, if

A) n= 0 B) n=1 C) n=3 D) none

6) The equation whose roots are reciprocal of the roots of ax²+ bx+c= 0, is

A) bx²+ cx+a= 0 B)cx²+ bx+a= 0

C) bx²+ ax+c= 0 D) cx²+ ax+b= 0

7) The value of the expression (ax)²+ bx+c, for any real x, will be always positive, if

A) b²- 4ac>0 B) b² - 4ac< 0

C) b²- 4a²c> 0 D) b² - 4a²c< 0

8) The value of m for which the equation x²-x+m²= 0, has no real roots, can satisfy

A) m>1/2 B) m>-1/2 C) m<-1/2 D) m<1/2

9)If x be real and a> 0, the least value of ax²+ bx+c will be

A) -b/a B) -b/2a C) -(b²-4ac)/2a D) -(b²- 4ac)/4a

10) The roots of ax²+ bx+c= 0 will be both negative, if 

A) a>0, b> 0, c< 0

B) a>0, c> 0 ,b< 0

C) a>0, b> 0, c>0 

D) b>0, c> 0 a< 0

11) If a, b are the roots of x² -2x +2= 0, the least integer n(>0) for which aⁿ/bⁿ = 1, is

A) 2 B) 3 C) 4.D) none

C) GENERAL QUESTIONS:

1) If the roots of 2x²+ x+1= 0 are p and q, from an equation whose roots are p²/q and q²/p. 4x²-5x+2=0

2) the equation x² - c x+d= 0 and x²- ax+b= have one root common and the second equation has equal roots.

Prove that ac= 2(b+d).

 

3) If the roots x²+ 3x+4= 0 are m,n, form an equation whose roots are (m-n)² and (m+n)². x² - 2x -63= 0.

4) If the roots of x²- px+q=0 are in the ratio 2:3, show that 6p²=25q.

5) If the roots of ax²+ bx+c=0 are m, n, form an equation whose roots are 1/(m+n), and 1/m + 1/n. bcx²+ (ac+b²)x + ab= 0.

6) If m, n are the roots of ax²+ 2b x+c= 0 and m+ + K, n+ K those of Ax²+ 2Bx+C= 0, prove that (b²- ac)/(B² - AC)= (a/A)².

7) Show that if one root of ax²+ bx+c=0 be the square of the other, than b³ + a²c + ac²= 3abc.

8) If m, n are the roots of the equation x²+ px - q= 0 and a, b those of the equation x²+ px+q=0, prove that (m- a)(m - b)= (n- a)(n- b)= 2q.

9) If the ratio of the roots of ax²+ cx+c= 0 be p: q, show that, √(p/q) + √(q/p)+ √(c/d)= 0.

10) if m be a root of equation 4x²+ 2x-1=0, prove that its other root is 4m³ - 3m.

11) If the sum of the roots of 1/(x+p) + 1/(x+ q) = 1/r be equal to zero, show that the product of root is 1/2 (p²+ q²).

12) If a, b are the roots of x²+ px+1= 0 and c, d are the roots of x²+ qx+1=0, show that q²- p²= (A-- c)(b - c)(a+ d)(b+ d).

13) Show that if x is real, the expression (x²- bc)/(2x- b - c) has no real values between b and c.

14) If one root of the equation ax²+ bx+c= 0 be the cube of the Other, show that ac(a+ c)²= (b² - 2ac)².

15) If a²= 5a - 3, b² = 5b - 3 but a≠ b, then find the equation roots are a/b and b/a. 3x²- 19x+3= 0

16) the coefficient of x in x²+ px+q= 0 is misprinted 17 for 13 and the roots of the original equation. -3, -10

17) if b³ + a²c + ac²= 3abc, then what relation may exist between the roots of the equation ax²+ bx+c= 0 ? One root is the square of the other.

18) find the maximum and minimum value of: x/(x²-5x+9). 1, -1/11

19) If m, n are the roots of ax²+ 2bx+c= 0, form an equation, whose roots are mw + nw² and mw² + nw (w= omega). (ax - b)²= 3(ac - b²)

20) If √m ± √n denote the roots of x² - px+q= 0, show that the equation, whose roots are m± n is (4x - p²)²= (p² - 4q)².

21) prove that for all real value of x, the value of p²/(1+x) - q²/(1- x) is real.

22) if x be real, prove that 4(a - x)(x - a + √(a²+ b²)) can never be greater than (a²+ b²).

23) If the quadratics x²+ px+q=0 and x²+ qx+p= 0 have a common root, prove that their other roots will satisfy the equation x²+ x+pq = 0

24) Show that if a, b, c are real, the roots of the equation (b - c) x²+ (c - a)x+(a - b)= 0 are real and they are equal if a, b, c are in AP.

25) If the the roots of the equation ax²+ 2bx+b =0 are Complex, show that the roots of the equation bx²+ (b - c)x- (a+ c - b)= 0 are real and cannot be equal unless a =b =c.

26) If a, b, c are real, show that the roots of the equation 1/(x+a) + 1/(x+ b) + 1/(x- c) = 3/x are real.

27) Show that the equation (b - c)x²+ (c - a)x+(a - b)= 0, (c - a)x²+ (a - b)x+(b - c)= 0, have a common root, find it and the remaining roots of the equations. 1, (a-b)/(b- c) and (b-c)/(c-a)

28) Prove that the roots of the equation (a - b)x²+ 2(a + b - 2c)x++ 1= 0, are real or complex according as c does not or lie between a and b.   

29) prove that if the equation ax²+ bx+ c= 0 and bx²+ cx+ a= 0 have a common root, then neither a+ b+ c= 0 or a= b= c.

30) If the equation ax+ by =1 and cx²+ dy² = 1 have only one solution, prove that, a²/c + b²/d = 1 and x= a/c, y= b/d.

31) if (a - K)x²+ b(b - K)y²+ (c - K)z²+ 2fyz+ 2gzx + 2hxy is a perfect square, show that a - gh/f = b - hf/g = c - fg/h = K

32) Prove that x²+ y²+ z² + 2ayz + 2bzx + 2cxy can be resolved into two rational factors if if a² + b² + c² - 2abc = 1.

33) find K so that the value of x given by K/2x = a/(x+ c) + b/(x- c) may be equal. If m, n are two values of K and l, p the corresponding values of x, show that m. n = (a - b)² and l² p²= c². 

     a+ b± 2√(ab)

MISCELLANEOUS-1

1) Prove that the roots of ax² + 2bx + c= 0 will be real and distinct if and only if the roots of (a+ c)(ax² + 2bx+ c)= 2(ac - b²)(x² + 1) are imaginary.

2) Form an equation whose roots are squares of the sum and the difference of the roots of the equation 2x² + 2(m+ n)+ m²+ n²= 0. x² 4mnx - (m² - n²)²= 0

3) Find the value of p if the equation 3x²- 2x + p= 0 and 6x²- 17x + 12= 0 have a common root. -15/4, -8/3

4) If the equation x²- ax + b= 0 and x²- cx + d= 0 have one root in common and second equation has equal roots, prove that ac= 2(b + d).

5) Find the values of the parameter k for which the roots of x² + 2(k - 1)x + k + 5= 0 are

A) opposite in sign. K∈(-∞,-5)

B) equal in magnitude but opposite in sign.      

C) positive. K∈(-5, -1)

D) negative. K∈(4,∞)

E) one root is greater than 3 and other is smaller than 3. K∈(-∞,-8/7) 

6) If m, n are the roots of the equation 6x² - 6x +1= 0 then prove that 1/2 (a+ bm + cm² + dm³)+ 1/2 (a+ bn + cn²+ dn³)= a/1+ b/2+ c/3 + d/4.

7) For what values of m ∈ R, both roots of equation x² - 6mx + 9m² - 2m +2= 0 exceed 3 ? M∈(11/9,∞) 

8) If the roots of the equation ax² + bx + c= 0 be (k+1)/k and (k+2)/(k +1) show (a+ b+ c)² = b² - 4ac.

9) If m, n are the roots of the equation x² - p(x +1) - c= 0, then prove that (m² + 2m+1)/(m² + 2m+c) = (n² + 2n+1)/(n² + 2n+c).

10) The condition that the equation 1/x + 1/(x + b) = 1/m + 1/(m+ b) has real roots that are equal in magnitude but opposite in sign is.

A) b² = m² B) b² = m² C) 2b² = m² D) none

11) The value of a for which one root of the equation (a -5)x² - 2ax + (a - 4)= 0 is smaller than 1 and the other greater than 2 is

A) a∈(5, 24) B) a∈(20/3,∞) 

C) a∈(5,∞). D) a∈(-∞,∞)

12) If m, n be the roots of ax²+ bx + c= 0 then the value of (am²+ c)/(am + b) + (an²+ c)/(an + b) is

A) b(b² - 2ac)/4a B) (b² - 2ac)/2a. C) b(b² - 2ac)/a²c. D) 0

13) solve:

a) (7y²+1)/(y² -1) - 4(y²-1)/(7y² +1)= -3

B) {x - x/(x+1)²} + 2x{x/(x+1)}= 3.

14) If m, n are the roots of ax² + by + cid = 0, find the equation whose roots are 1/m³, 1/n³.

15) If the equation x² - (2+ m)+ (m² - 4m + 4)= 0 in x has equal roots, then the value of m are

A) 2/3,1 B) 2/3,6 C) 0,1 D) 0,2