INTEGRATION By SUBSTITUTION

INTEGRATION By SUBSTITUTION
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1) ∫ (3x+5)⁷ dx

2) ∫ (4-9x)⁵dx

3) ∫ dx/(2-3x)⁴

4) ∫ √(ax+b)dx

5) ∫√(1+x)dx

6) ∫ 2x dx/(2x+1)²

7) ∫ (2+3x)dx/(3-2x)

8) ∫ dx/(3- 8x)⁵

9) ∫ x √(x+a)dx

10)  ∫ x/√(x+a)dx

11)   ∫ x(x² +1)¹² dx

12) ∫ xdx/(x² -1)⁴.

13) 

14)

15)  ∫ (x -2) dx/√(2x²-8x+5)

16)  ∫ (x -2) dx/³√(x² - 4x +5)

17)  ∫ (2x+3)√(x²+3x-1) dx

18) ∫ (3x+5)/(3x²+10x+2)²/³ dx

19) ∫ ⁵√(5+3x)dx

20) ∫ (4+5x)⁷ dx

21) ∫ xdx/(x² -1)⁴

22) ∫  dx/√(3-4x)

23) ∫ 6xdx/(4 - 3x²)²

24) ∫ x³dx/(x²+1)

25) ∫ x⁸dx/(1-x³)¹/³

26) ∫ (√x + ³√(x²)²) dx

27)  ∫  x²/(x+1) dx

28) ∫ x³/(x-1)dx

29) ∫ x³/(x+2)dx

30) ∫ (1-x⁴)/(1-x) dx

31) ∫ (2x-1)/(x+1)   dx

32) ∫ (4+5x)/(3-2x)   dx

33)  ∫ (x³+5x²-3)/(x+2)  dx

34)  ∫(x+2)/√(x+2)   dx

35)  ∫ (1 - x)/√(1+x)  dx

36)  ∫ (4x +3)/√(2x +1)  dx

37)  ∫ (3x +2)/√(x+1)  dx

38) ∫ (2x-1)√(x+3) dx

39) ∫ (5x - 1) √(2 - x) dx

40)  ∫ x dx/(√x +1)

41) ∫ (3+2x)/(2-3x) dx

42) ∫ (x+3)√(2x-1)dx

43) ∫ (3x -1)√(x+2)dx

44) ∫   (2x - 1)/(x+1) dx

45) ∫ (4 + 5x)/(3 - 2x)dx

46) ∫ (x³+5x² -3)/(x+2) dx

47) ∫ (x-2)dx/√(2x² -8x +5)

48) ∫ (x-2)dx/³√(x²-4x+5)

49) ∫ (2x+3)√(x²+3x -1)dx

50) ∫ (x+1)dx/√(x+2)

51) ∫ (1-x)dx/√(1+x)

52) ∫ (4x+3)dx/√(2x+1)

53) ∫ (3x+2) dx√(x+1)

54) ∫ (2x-1)√(x+3)dx

55) ∫ (5x-1)√(2-x)dx

56) ∫ xdx/√(x+1)

57)  ∫ dx/{(√(ax +b) - √(ax- c)}

58) ∫  dx/{√(2x+3) - √(2x+1)}

59) ∫ dx/{√(2x+5)- √(2x - 3)}

60)  ∫ dx/{√(3x+4) + √(3x-10)}

61) ∫ dx/{√(x+3) - √(x+1)}

62) ∫ dx/{√(4x+3) - √(x+3)}

63) ∫ (logx)dx/x

64) ∫ dx/(xlogx)

65) ∫ log(x²) dx/x

66) ∫ (log x)²dx/x

67) ∫ (x+1)(x+log x)² dx/x.

68) ∫ log√(x) dx/ 3x

69)  ∫ {(x+1)(x+logx)²} dx/ x

70) ∫ (x+1)(x+log x)³/x dx

71) ∫ eˣdx/(eˣ - 1)

72)  ∫ dx/(eˣ -1)

73)  ∫ dx/{(1+eˣ)(1+e⁻ˣ)}

74) ∫ dx/(eˣ +1)

75) ∫  dx/{(eˣ+1)(e⁻ˣ+1)}

76)   ∫ cosx√(1+cos2x)dx

77) ∫ sinx/(1+sinx) dx

78) ∫ cosecx/(cosecx - cotx) dx

79) ∫(cos 2x + 2sin²x)/cos²x   dx

80)  ∫ sec²2ax dx

81) ∫ (tan 2x + sec 2x)² dx

82) ∫ secx dx/(secx + tanx)

83) ∫ cos⁴x sin⁴x dx

84) ∫ (7cos³x + 8 sin³x)/
                           (3sin²xcos²x)

85) ∫ sec²(log x) dx /x

86)  ∫ (1+cosx)/√(x+sinx) dx

87) ∫ (x²+ cos3x +1)/(x³+sin3x +3x)

88)  ∫ cotx eᵐ ˡᵒᵍ ⁽ˢⁱⁿ ˣ⁾ dx

89)  ∫ sin x /√(1+ cis x)  dx

90)  ∫ 2√(tan x)/ sin2x   dx

91)∫(sin2x+4sin x)/
                      (sin²x-4cosx +1)⁴

92) ∫ (secx cosecx)/log(tanx) dx

93)  ∫ {(3tan²x+2sec²x)sec²x}/
                 (tanx +2tanx +3)³ dx.

94) ∫ 3x√(2x²+5) dx
                  ₃
95) ∫ x² eˣ   dx

96) ∫ Sin 2x cos²2x dx

97) secᵐ⁺¹x tanx dx

98) ∫ sin√(x)/√(x) dx

99) ∫ dx/(x+√x)

100) ∫ tan(log x)dx/x  

101) ∫ dx/{(1+x²)√(tan⁻¹x+c)}

102) ∫ (1+cosx)dx/(x+sinx)

103) ∫ sin2x dx/(a²sin²x + b²cos²x)

104) ∫ √(tanx) dx/sin2x)

105) ∫ √{(sinx/cos⁵x)}  dx

106) ∫ e²ˣdx/(eˣ +1)

107) ∫ dx/(eˣ +e⁻ˣ) dx

108) ∫ sin2x dx /(a + b cosx)²

109) ∫ log√x/(3x)

110) ∫ (1+sinx)dx/{sinx(1+cosx)

111) eˣ(1+x)dx/cos²(xeˣ)

112) {(x+1)(x+logx)²}dx/ x

113) ∫ (eˣ -1)/(eˣ +1)   dx

114) ∫ secx log(secx + tan x) dx

115)  ∫ x³(1+x²)¹/³  dx

116)  ∫ (3x +4)/(4x+5) dx

117)  ∫ (1- x)/√(1+x)  dx

118) xdx/(ax +b)²/³

119)  ∫ dx/{x²(3+4x)²}

120)  ∫ dx/{√(x) - √(x- 1)}

121) ∫ dx/(xlogx)

122)  ¹₀∫ x³√(1+3x⁴) dx

123)  ²₁∫ sin(log x) dx/x

124)  ∫ x²dx/(x²+5)

125) ∫ 5ˡᵒᵍ₅ˣ dx

126) (3sin x + 4cosx)/
                     (4sinx - 3cosx) dx
                      ₋₁
127)  ∫eᵐᵗᵃⁿ    ˣ  dx/(1+x²)

128)  ∫ x sin(4x²+7) dx

129)  ∫ x√(x-5) dx

130)  ∫ (8x+13)dx/√(4x+7)

131) ∫ x³dx/√(1+2x)

132)  ∫ (x²+1)dx /(x²-1)²

133) ∫x² dx/(a+bx)²

134) ∫ (x² - 1)dx/
              {(x⁴+3x²+1)tan⁻¹(x +1/x)}

135)  ∫ d(x² +1)/√(x²+2)

136)  ∫ (cos 5x + cos 4x)/
                     (1-2cos3x)   dx

137) ∫(cos7x - cos8x)/(1+2cos5x)

138) ∫ (x+ 1/x)ⁿ⁺⁵   {( x² -1)}/x²   dx

139)  ∫ aˡᵒᵍˣ ⁺ ˢⁱⁿ ²ˣ (1+2xcos2x)/x
  
140)  ∫ x¹³/²(1+x⁵/²)¹/² dx

141) ∫ dx/(1+⁵√(x+1) 

141) ∫(7x²+1) dx/(3x+4)²/⁷

142)  ∫ xdx/{√(x+2) +√(7x+2)}

143)  ∫ dx/√(eˣ -1)

144)  ∫ (cos4x- cos2x)/
                     (sin4x - sin2x)  dx

145) ∫ √(tanx)/(sinx cosx)  dx

146)  ∫ cis xdx /(a + bsinx)²

147)   ∫ dx/{x²√(1-x²)}

148)  ∫ dx/{x²-1)√(1-x²)}

149)  ∫ secxdx/{log (secx+ tan x)}

150) ∫ sec²x tanxdx/(a²+b²tan²x)²

151) ∫ sec⁴xdx/√(tanx)

152)  ∫ cosx dx/ cos(x+4)

153) {cos(x+4)} dx/cosx

154) ∫cos (x+α)dx/cosx  

155) ∫ cos xdx/cos(x+a)

156)  ∫ x⁷dx/(x⁴ - 1)

157) ∫ x⁵dx/√(x³+1)

158)  ∫ (2x -1)/(x+1)²  dx

159)  ∫ dx/(1+eˣ/²)

160) ∫ dx/{√(ax +b) + √(ax -b)}

PRACTICE PAPER For (JOINT ENTRANCE EXAMINATION)

      PRACTICE PAPER  For IIT
( COMPETITIVE EXAMINATION)

1) Three dice are thrown simultaneously. The probability of getting a sum of 15, is

a) 1/72  b) 5/36  c) 5/72  d) none

2) Two dice are thrown simultaneously to get the coordinates of a point on x - y plane. Then the probability that this point lies inside or on the region bounded by | x |+|  y | =3, is

a) 3/14   b) 2/3   c) 1/12    d) 4/14.

3) Let (x) denotes the probability of the occurence of event x. Then all those points (x, y) = P(A), P(B), in a plane which satisfy the cinditiins, P(A) ≥ 3/4 and 1/8 ≤ P(A∩B) ≤ 3/8 imply.

a) P(A)+P(B) < 11/8 
b) P(A)+P(B)>11/8
c) 7/8 ≤ P(A)+ P(B) < 11/8
d) none of these.

4) Two points are taken at random on the given straight line segment of length a. Then the probability for the distance between them to exceed a given length c,
where  0 < c < a, is

a) (1+c/a)².               b) (1 - c/a)².
c)(1+a/c)².                d)  None

5) If a ∈  [20,0], then the probability that the Equation
16x² +8(a+5)x - 7a - 5 =0 has imaginary roots, is

a)13/20.                  b) 20/13.
c) 13/24.                 d) 15/20.

6) If the sides of a triangle are decided by the row of a single die thrice, the probability that triangle is of maximum area given that its an isosceles triangle, is

a) 1/7. b) 1/27. c) 1/14. d) none.

7) Four cards are drawn from a pack of 52 playing cards. Then the probability of drawing at least one pair is

a) 1 - {(4⁴ x ¹³C₄)}/⁵²C₄
b) 1 -  ¹³C₄/⁵²C₄
c) 1 -  {(4⁴x 13!)}/52!
d) none

8) If P(A/B)=P(B/A). A and B are two non- mutually exclusive events, then

a) A and B are necessarily same events.
b) P(A) = P(B)
c) P(A∩B) = P(A) P(B)
d) All if the above.

9) The probability that
sin⁻¹(sin x) + cos⁻¹ (cosy) is an Integer x, y ∈ {1,2,3,4}, is

a) 1/16   b) 3/16  c) 15/16  d) none

10)  A and B throw a dice each. The probability that A's throw is not greater than B's throw, is

a) 7/12  b)  5/12   c) 1/12    d)1/2

11) If A and B are square metrices of order ' n' such that
A² - B²=(A - B)(A+B), then which of the following will be true ?

a) Either of A or B is zero metrix.
b) A=B.    c) AB=BA    d) Either of A or B is an identity metrix.

12)If Metrix A= x  2. And | A³ |=125
                            2  x.  Then x= ?

a) ±1    b) ±2       c) ±3.     d) ±5

13) If Determinants A= x    1   1
        B =  x   1                  1   x    1
                1   x                  1   1    x
Then d A/d x =

a) 3B+1.    b) 3B.   c) - 3B.  d) 1-3B

14) If the Determinant of the adjoint of a (real) matrix of order 3 is 25, then the Determinant of the inverse of the metrix is -

a) 0.2.   b) ±5.  c) 1/⁵√625. d) ±0.2

15) A tangent is drawn to the circle to the circle 2x²+2y²-3x+4y=0 at the point 'A' and it meets the line x+y=3 at B(2,1), then AB = ?

a)√10.     b) 2.       c)2√2.        d) 0

16) The sum of the squares of the eccentricities of the conics
x²/2+ y²/3=1 and x²/4 - y²/3=1 is

a) 2.     b) √(7/8)    c) √7.    d) √3

17) If x+y=tan⁻¹ and
d²y/dx² = f(y)dy/dx, then f(y) = ?

a) -2/y³   b) 2/y³  c) 1/y     d) -1/y

18) The coefficient of x⁴ in the expansion of (1+x+x²+x³)ⁿ is

a) ⁿC₄                  b) ⁿC₄ + ⁿC₂
c) ⁿC₄ + ⁿC₂+ⁿC₄ . ⁿC₂
d) ⁿC₄ + ⁿC₂+ⁿC₁. ⁿC₂

19) Range of the function
f(x)=(x²+x+2)/(x²+x+1) , x ∈ R is -

a) (1,∞)                    b) (1,11/7)
c) (1,7/3)                  d) (1,7/5)

20) If α, β are the roots of 8x²-3x+27=0, then the value of
{(α²/β)¹/³   + (β²/α)¹/³} is

a) 1/3  b) 1/4   c) 1/5   d) 1/6

21) Let A={a,b,c}. which of the following is not an equivalence relation in A .

a) R₁ ={(a,b),(b,c),(a,c),(a,a)}
b) R₂= {(c,b),(c,a),(c,c),(b,b)}
c) R₃= {(a,a),(b,b),(c,c),(a,b)}
d) None of these.

22) If   ∫ x dx/(x²-4x+8) =
  k(x²-4x+8) + tan⁻¹{(x-2)/2} +c,
  find k.

a) 1/2   b) 1   c) 2    d) 3.

23) Given two events A and B. if odds against A are as 2/1 and those in favour of A∪B are as 3/1, then .

a) 1/2< P(B)≤3/4
b) 5/12≤P(B)≤3/4
c) 1/4≤P(B)≤3/5         d) None.

24) If sinα,sin²α,sin⁴α,sin⁵α,
       (-π<α <π) are in A.P.,
       then α lies in the interval.

a) (-π/2, π/2)            b) (-π/3,π/3)
c) (-π/6, π/6)            d) None .

25) The point on the curve y=(x-3)², where the tangent is parallel to the chord joining (3,0) and (4,1) is-

a) cosx   b) tan x   c) x²    d) | x- 1|

26) cos[2cos⁻¹1/5 + sin⁻¹ 1/5] =

a) 1/5   b)-2√6/5  c) -1/5  d)√6/5.

27) ∫ cosⁿ⁻¹x/sinⁿ⁺¹x  dx, n≠ 0

a) cotⁿx/n + C   b) -cotⁿ⁻¹x/(n-1)+C
c) -cotⁿx/ n  + C  d) cotⁿ⁻¹x/(n-1) +C

28) The maximum area of a rectangle that can be inscribed in a circle of radius 2 units is-

a) 8π sq.units.     b) 4 sq.units
c) 5 sq.units.       d) 8 sq units.

29) The 100th term of the sequence 1,2,2,3,3,3,4,4,4,4....is

a) 12.  b) 13.  c) 14   d) 15.   e) 16.

30) Five Numbers are in H.P The middle term is 1 and the ratio of the second and the fourth terms is 2:1. Then the sum of the first three terms is-

a) 11/2  b) 5. c) 2.   d) 14/3

31) Find x
      sin[2cos⁻¹{cot(2tan⁻¹x)}] =0

a) 1   b) -1   c) 1± √(2)  d) -(1±√2)

32) Cos4x + (10tanx)/(1+tan²x) =3
       in the interval (-π/2 , π/2) has

a) no solution       b) one solution
c) two solution    d) three solution

33) The largest interval for which 
        x²² - x¹⁹ + x¹⁴ - x⁵ + 1 > 1 is
a) (0,1)                              b) (-∞,1]  c)[1,∞)                            d) (-∞,∞)

34) 100 students appeared for a test comprising 5 subjects: Hindi, English, Mathematics, Physics and Chemistry. The number of them passed in these subjects are 99,85,91,89,87 respectively. The number of students who passed in all the 5 subjects can be
a) 45.   b) 56.    c) 67.     d) 78

Continue.........

For solution
or for ANSWER
call/message.

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GEOMETRIC PROGRESSION (A to Z)

GEOMETRIC PROGRESSION
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EXERCISE - A

1) FIND
a) 3,6,12,24....... 9th term.               768

b) 3,6,12,24....... nth term.            3.2ⁿ⁻¹

c) 12, 4, 4/3, 4/9,....10 th term.   4/6561

d) 12, 4, 4/3, 4/9,....10 th term.      4/3ⁿ⁻²

e) 0.4,0.8, 1.6.....7 & n th term.      25.6 , 2ⁿ/5

f) 1,1/2 , 1/2²,..... 9 th term.          1/256

g) 2, ,4, 8,.......7 th term.                     128

h) 2, 2√2,4, 8√2 ,.....n th term.           ₂1/2 (n+1)
i) 15/8, 3/8,3/40, ....nth term.         15/8 (1/5)ⁿ⁻¹

j) 16, 8, 4, 2,.  10th and nth term.      1/32, 2⁵⁻ⁿ
k) 4, 2√2, 2, √2, .... 10th & (2n+1)th term.      √2/8, 2²⁻ⁿ

l) x, xy², xy⁴,xy⁶....25th & pth term.     xy⁴⁸, xy²ᵖ⁻²

EXERCISE - B

1)  Find x if series in GP..
a) 3x, 2x+3, 3x+8.                -3, 3/5

b) 3x+1, 7x, 10x+8.             2, -4/19

c) x+9, x-6, 4.                           0,16 

d) -2/7, x, -7/2.                        ±1

EXERCISE - C 

1) Which term of the G. P.

a) 4+12+36+108+.... is 2916.               7th

b) 3, -9, 27, ...is 2187.                           7th

c) 1+3+9+.....is 6561                            9th

d) √6, 2√3,2√ 6,....  ......64√3.            12th

e) 3, 6, 12, 24...... is 3072?                11th

f) 1/4, -1/2, 1, ......is -128.                 10th

g) √3, 3, 3√3,......is 729.                    12th 

h) 4, -8,16,-32......is 1024 ?                9th

2)a) Is 1218 a term of the GP 5, 15, 45,.....?                                                     No 

b) Is 3000 a term of the GP 3, 15, 45, 135, ...?                                                    No

c) Is 640 a term of the GP of 5, -10, 20,....?                                                     No

EXERCISE - D

1)a) Find the 4th term from the end of the G. P. 2/27, 2/9, 2/3, ....162.

b) Find the 4th term from the end of the G. P. 1/2,1/6,1/18,1/54,.....1/4374

c) Find the 6th term from the end of the GP 8, 4, 2,...,1/1024.                          1/32

d) Find the 4th term from the end of 2/27, 2/9, 2/3, ......162.                          6

e) Find the 8th term from the end of the GP 3,6, 12, 24,....12288.                       96

EXERCISE - E

1)a) The first and the fifth terms of a GP are 4.2 and 67.2 respectively, find common ratio.                                        2

b) The 10th term and common ratio of a GP are -512 and -2 respectively. Find the first term.                                                1

c) The 5th and 13th terms of a GP are 32 and 8192. Find its 10th and nth terms.                     1024,  2ⁿ 

d) Find the 7th term of the GP., for which the 5th and 12th terms are 48 and 6144 respectively.                        192

e) If the sum of the 3rd and 4th terms of a GP be 60 and that of the 6th and 7th terms be 480, find the 10th term of the GP.                                        2560

f) The 4th term of a GP is 32 and the common ratio is (-1/2), find the 10th term.                                   1/2

g) The 4th, 7th terms of a GP are -135 and 3645 respectively. Find the progression. What is the 6th term of the GP?                                      -1215

h) The 1st and 4th term of GP are 64, 216 respectively. Find the 7th term.  729 

i) The 2nd and 5th terms of GP are 9/16, 1/6 respectively, find the 7th term.  2/27 

j) The 9th and 21st terms of GP are 36, 972 respectively, find the 5th term.      12

k) The 5th and 8th terms are 80 and 640 respectively. Find series.     5,10,20,40....

l) The 4th and 7th terms are 1/18 and -1/486 respectively. Find the series.                -3/2, 1/2, -1/6....

m) The 5th, 8th and 11th terms of a GP are a, b, c respectively. Show that b²= ac.

n) The 1st of a GP is -3 and the square of the 2nd term is equal to its 4th term. Find its 7th term.                              -2187

o) The 7th term of a GP is 8 times the 4th term and 5th term is 48. Find GP series.

p) The 4th term of a GP is square of its second term, and the first term is -3. Find its 7th term.

q) The 2nd term of a GP is b and the common ratio is r. If the product of the first three terms of this GP is 64, find b.

r) If 5,x,y,z,405 are in GP then find the values of x,y,z.

EXERCISE - F

1) a) The sum of first three terms of a GP is 13/12 and their products is -1. Find the numbers.           3/4, -1, 4/3 or 4/3, -1, 3/4

b) Find three Numbers in GP. Whose sum is 13 and the sum of whose squares is 91.                  1,3,9 or 9,3,1

c) If the sum of three Numbers in GP is 52 and the sum of whose products in pairs is 624.                 36,12,4 or 4,12,36

d) Three Numbers are in GP. Whose sum is 70. If the extremes be each multiplied by 4 and the means by 5, they will be in A. P. Find the numbers.

e) If the continued product of three Numbers is 21 and products is 64, find the numbers.                    1,4,16 or 16,4,1

f) The sum of three numbers in GP is 39/10 and their product is 1. Find the numbers.                   5/2,1,2/5 or 2/5,1,5/2

g) The sum of three numbers in GP is 21 and the sum of their squares  is 189. Find the numbers.            3,6,12 or 12,6,9

h) The sum of three numbers in GP is 30 and their product is 216. Find the numbers.                   6(2-√3),6,6(2+√3) or 6(2+√3), 6, 6(2-√3).

i) The sum of three numbers in GP is 7 and sum of their squares is 21. Find the numbers. Determine the first five terms of the GP.                           4,2,1, 1/2, 1/4 

j) The sum of three numbers in GP is 19 and their product is 216. Find the numbers.                                4,6,9 or 9,6,4

k) The sum of three numbers in GP is 13/12 and their product is -1. Find the numbers.       4/3,-1,3/4 or 3/4,-1,4/3

l) The product of first three terms of a GP is 1000. If we add 6 to its second term and 7 to its third term, the resulting three terms forms an AP. Find the terms of the GP.                   20,10, 5 or 5, 10, 20

m) The sum of three numbers in GP is 35 and their product is 1000. Find the numbers.                      20,10,5 or 5,10,20

n)   The sum of three numbers in GP is 26 and the sum of the product taking two parts at a time is 156. Find the numbers.                          2,6,18 or 18,6,2

o) The sum of three numbers in AP is 21. If 4,5 and 8 are added respectively to the numbers, the results are in GP. Find the numbers.                                         4,7,10

p) A number consists of three digits which are in GP and the product of the digits is 216. If 495 be added to the number, the digits will be reversed. Find the original number.                            469

q) Three numbers are in GP, whose product is 216. If 4 be added to the first term and 6 to the second term, then the resulting numbers and the third number are in AP. Obtain the numbers in GP.              2,6,18 or 18,6,2

r) The sum of three numbers in GP is 70. If the two extremes be multiplied each by 4 and the mean by 5, the product are in AP. Find the numbers.         10,20,40 or 40,20,10

s) The sum of three numbers in GP is 14. If the first two numbers are each increased by 1 and the third number decreased by 1, the resulting numbers are in AP. Find the numbers in GP.    2,4,8 or 8,4,2 

t) Three numbers are in AP and their sum is 21. If 1,5,15 be added to them respectively, they form a GP. Find the numbers.                             5,7,9

u) Find three numbers a, b, c between 2 and 18 such that 

(i) their sum is 25.

(ii) the numbers 2, a, b are consecutive terms of an AP.

(iii) the numbers b, c, 18 are consecutive terms of a GP.                                  5,8,12

v) divide 65 to 3 parts which will be in GP and such that the third part shall be 9/4 times the sum of the first two parts.          5,15,45 or 80, -60,45

w) If, of the three consecutive terms of a GP the middle term is 6 and the first and the third terms are together equal to 15, find the terms.                3,6,12 or 12,6,3

x) The sum of three numbers in AP is 18. If 2,2,6 be added respectively to 1st, 2nd and 3rd numbers, the resulting numbers are in GP. Find the numbers.       2,6,10 or 14,6, -2

y) Three numbers whose sum is 18 are in AP.,if 2, 4 and 11 are added to them respectively, the results are in GP. Determine the numbers.       3,6,9 or 18, 6, -6

z) Three numbers whose sum is 15 are in AP., if 1, 3 and 9 are added to them respectively, the results are in GP. Determine the numbers.             3,5,7 or 15, 5, -5

a') Three numbers whose sum is 15 are in AP.,if 1, 4 and 19 are added to them respectively, the results are in GP. Determine the numbers.   2,5,8 or 26, 5, -16

b') Three numbers whose sum is 15 are in AP.,if 1 is substracted from the second term, the terms are in GP. Determine the numbers.   2,5,8 or 8, 5, 2

c') Three numbers form an increasing GP. If the third number is decreased by 16, we get an AP. If then the second number is decreased by 2, we again get a GP. Find the three numbers.    1,5,25 or 1/9, 13/9, 169/9

2)a) sum of four numbers in GP is 180, sum of their two extreme numbers is 108. Find the numbers.     12,24,48,96 0r 96,48,24,12

b) The product of 4 positive numbers in GP is 729 and the sum of two intermediate terms is 12. Find the numbers.                   1,3,9,27 or 27, 9, 3, 1

c) The sum of four terms in GP is 60 and the arithmetic mean of the first and the last numbers is 18. Find the numbers.   32,16,8,4 or 4,8, 12,32

d) find four numbers in GP in which the third term is greater than the first by 9 and the second term is greater than the fourth by 18.

e) Divide 20 into 4 parts which are in AP such that the product of first and fourth is to the product of second and third in the ratio 2:3.

EXERCISE - G

1) Find the GM between the numbers..

a) 5 and 125.                                         25

b) 1 and 9/16.                                      3/4

c) 0.15 and 0.0015.                          0.015

D) a³b and ab³.                                   a²b²

2) Insert 2 GM between..

a) 9 and 243.                                    27, 81

b) 5 and 135.                                  15,45

3) Insert 3 GM between..

a) 1/3 and 432.        2,12,72 or -2,12,-72

b) 1 and 256.             4,16,64 or -4,16,-64 

c) 16 and 256.                         32,64,128 

d) 3 and 3/16.                         3/2,3/4,3/8

e) 16 and 1.                      8,4,2 or -8, 4, -2

f) 2 and 162.              6,18,54 or -6,18, -54

4) Insert 4 GM between..

a) 6 and 192.                         12,24,48,96

b) 4 and 972.                     12,36,108,324

5) Insert 5 GM between..

a) 1/3 and 243.                   1,3,9,27,81

b) 3 and 81.            3√3, 9, 9√3, 27, 27√3, or -3√3,9, -9√3, 27, -27√3

6) Insert 6 GM between..

a) 8 and 1/16.               4,2,1,1/2,1/4,1/8

7)a) The arithmetic mean of two positive numbers is 40 and their geometric mean is 24. Find the numbers.

b) If the sum of the 3rd and 4th terms of a GP be 60 and that of the 6th and 7th terms be 480, find the 10th term of the GP.

EXERCISE - H

16) Find the sum of.

a) 81+27+9+3+....to 8 terms.

b) 2+6+18+54+..10th terms.           59048

c) 2+√2+1+1/√2+....to 13 terms.

d) 1-1/2+1/4-1/8,.....12 terms.    4095/2048

e) (a-b)²,(a-b),(a-b)/(a+b),...n terms.

f) 4+44+444+......up to nth term.

g) 0.6+0.66.+0.666+.... to nth term.

h) 7+0.7+0.77+0.777+... to nth.

i) 0.7+0.77+0.777+... n terms.

j) 9.15+0.015+0.0015+...to 8 terms

k) 2/9-1/3+1/2-3/4+....to 5 terms.

l) 3/5+4/5²+3/5³+4/5⁴+.. to 2n

m) 11+103+1005+..... n terms.

EXERCISE - I

1) How many terms of the G. P 3,3/2,3/4.... be taken together to make 3069/512.

2) How many terms of 2+6+18 must be taken to make the sum equal to 728 ?

3) The sum of n terms of the of the G.P 3,6,12,.....381.

4) The common ratio of a GP is 3 and the last term is 486. If the sum of these terms be 728, find the first term.

5) The sum of first three terms of a GP is 16 and the sum of the next three terms is 128. Find the sum of n terms of the G. P.

6) Find a G. P for which the sum of the first two terms is -4 and the fifth term is 4 times the third term.

7) How many terms of the Geometric series 1+4+16+64+... will the sum 5461 ?

8) The sum of some terms of a GP is 315 whose first term and the common ratio are 5 and 2, respectively. Find the last term and the number of terms.

9) In an increasing G. P, the sum of the first and last term is 66, the product of the second and the last but one is 128 and the sum of the terms is 126. How many terms are there in the progression.

10) * Find the sum of the product of the corresponding terms of the sequence 2,4,8,16,32, and 128,32,8,2,1/2.

11) The ratio of the sum of first three terms is to that of first 6 terms of a GP is 125:152. Find the common ratio.

12) Find the least value of n for which the sum 1+3+3²+... n terms is greater than 7000.

13) The product of first three terms of a GP is 1000. If 6 is added to its second term and 7 added to its third term, the terms become in A. P. Find the G. P.

14) The sum of three Numbers in GP is 56. If we substracted 1,7,21 from these numbers in that order, we obtain A. P. Find the numbers.

EXERCISE - J

1) Find the sum of infinity
a) 1-1/3+1/3²-1/3³+1/3⁴......
b) 2/5+3/5²+2/5³+3/5⁴+ ...

2) Find the sum of the terms of an infinite decreasing GP in which all the terms are positive, the first term is 4, and the difference between the third and fifth term is equal to 32/81

3) The sum of first two terms of an infinite G. P is 5 and each term is three times the sum of the succeeding terms find the G. P.

4) Express the recurring decimal
0.125125125 ...... as a rational number.

EXERCISE - K

1) a,b,c are in GP, prove that log a,
log b , log c in AP.

2) If a,b,c are in A. P and a,b,c are in GP., Then prove p - q, q - r, r - s are in GP.

3) Find k such that k+9, k- 6 and 4 form three consecutive terms of a GP.

4) Three Numbers are in A. P and their sum is 15. If 1,3,9 be added to them respectively, they form a GP. Find the numbers.

5) The sum of three Numbers which are consecutive terms of an AP is 21. If the second Number is reduced by 1 and the third is increased by 1, we obtain three consecutive terms of a GP. Find the numbers.

6) The sum of three Numbers a,b,c in A. P is 18. If a and b are each increased by 4 and c is increased by 36, the new numbers form a GP. Find a, b, c.

7) The p-th term of GP is q and the q-th term is p. Find the (2p-q)th term

8) If the first and the nth terms of a GP are a and b respectively and if P is the product of the first n terms, prove that P²=(ab)ⁿ.

9) The (m+n) th and (m-n)th term of a GP are p and q respectively. Show that the mth and nth terms are (√pq) and p(q/p)ᵐ/²ⁿn.

10) If the p th, q th and r th terms of a GP are a,b,c respectively, prove aʸ⁻ᶻ. bᶻ ⁻ˣ. cˣ ⁻ʸ=1

EXERCISE - L

57) a,b,c are in GP., Prove that:
a) a(b²+c²)= c(a²+b²)

b) a²b²c²(1/a³ +1/b³+1/c³) = a³+b³+c³

c) (a+b+c)²/(a²+b²+c²) = (a+b+c)/  (a- b +c)    

d) (a+2b+2c)(a - 2b + 2c)= a²+4c²

ⁿ⁺ ⁻ ¹ ² ³ ⁴ ⁵ ⁶ ⁷ ˣ ⁱ ᵅ ₓ ₊ ₋ ₐ ₙ ₁ ₂ ₃ ₄ ₅ ₆ ₇ ₈ ₉

MIXED QUESTIONS

1) Find the 9th and n-th terms of the series 4, -8, 16, -32,.......       1024, (-2)ⁿ⁺¹

2) The 4th and the 7 th terms of a GP are  -135 and 3645 respectively. Find the progression. What is the 6th term.      5,-15,45,-135, -1215

3) Which term of the GP 2√3,2√6,4√3,... is 64√3?                                        12th.

4) If x, y, z be the p-th,  q-th and r-th terms respectively both of an AP and of a GP, then prove that xʸ⁻ᶻ. yᶻ⁻ˣ. zˣ⁻ʸ= 1.

5) Insert three geometrical means between 2 and 162.          6,18,54 or -6, 18, -54

6)  Find the sum :

A) 1- 1/2 +1/2² -1/2³ +...of the 20 terms.    2/3  (1- 2⁻²⁰)

B) 5+ 555+ 55555 +....up to n terms.     50/891  (10²ⁿ -1) - 5n/9

7) Without assuming the summation formula of GP. Find the sum to n terms of the GP series: 1+ 1/2+ 1/2²+ ...   2 -1/2ⁿ⁻¹

8) if the 6th and 9th terms of a GP are 64 and 512, find the sum of 9 terms of the GP.                       1022

9) If the sum of first n terms of a GP is p and the sum of the first 2n terms is 3p, then show that the sum of the first 3n terms is 7p.   

10) if in a GP first term is a, n- term is b, and the product of the first n terms is P, then prove that P² = (ab)ⁿ.

11) if S be the sum, P the product and R the sum of the reciprocal of n terms of GP. Prove that P² =(S/R)ⁿ.

12) Find three numbers in GP. Whose sum is 35 and product is 1000.     20, 10, 5

13) The sum of the three numbers in AP is 15. If 1, 4, and 19 are added to them respectively, the results are in GP. Find the numbers.          26,5,-16 or 2,5,8

14) The ratio AM and GM of two numbers is m: n; show that the ratio of the number is m+ √(m²- n²): m - √(m² - n²).

15) If one Arithmetic mean A and two geometric means p, q be inserted between two given quantities then prove that, p²/q + q²/p = 2A.

16) Show that the Arithmetic mean of two positive quantities can never be less than their geometric mean.

17) Find the sum: 1+ (1+3)+(1+3+3²)+.. up to n terms.          1/4(3ⁿ⁺¹ -2n -3).

18) find the n-term and the sum up to n-th term of the series 1+3+7+15+...   2ⁿ⁺¹ - n - 2

19) find the sum 1+ 3/2 +5/2²+7/2³+... up to n terms.                 6- (2n+3)/2ⁿ⁻¹ 

20) Prove that in a GP, the product of any two terms equidistant from the beginning and the end is constant.

21) In a GP, first term is 7, last term is 448 and the sum of the terms is 889; find the common ratio and number of terms of the GP.                2, 7

22) A, B and C have together ₹5700 and the amount of money possessed by them from a GP. If B had ₹150 more, the amounts should form an AP. Find the amounts they pocesses.          2700, 1800, 1200 

23) A bouncing tennis ball rebounds each time to a system highest equal to one half the height of the previous bounce. If it is dropped from a height of 16 m . Find the total distance it has travelled when it hits the ground for the 10th time.                                    767/16

24) The second term of a GP is b the and the common ratio is r. If the product of the first three terms of the GP is 64, find b.                            4

25) The n-th term of a series is 4.3ⁿ⁻¹. show that the series is a GP.

26) If the (p+ q)th term of a GP is m and p- q term is n, then find its p-th term.       ±√(mn)

27) If a, b and c are in GP and a, 2b, 4c are in AP. then find the common ratio of the GP.                1/2

28) Find the indicated terms of the following series:

a) the 10th and nth term of the series 16, 8, 4 2,.....              1/32, 2⁵⁻ⁿ

b) The 8th and 2n-th terms of the series 6, -12, 24, -48,.... (p is an integer).   -768, -3.2²ⁿ.

c) the 10th and (2n + 1)-thterms of the series 4, 2√2, 2, √2,....        √2/8, 2²⁻ ⁿ 

29) The 4-th term of a GP is 32 and common ratio is (-1/2), find the 10-th term.               1/2

30) The first term of a GP 64 and 4th term is 216; find the 7th term.    729

31) The second term of a GP is 9/16 and 5th term is 1/6, find the 7th term.     2/27

32) The 9 term of a GP 36 and 21st term is 972, find the 5th term.      12

33) the p-th term of a GP is q and q-th term is p, find (2p-q)-th term.     q²/p 

34) Which term of the series 4, -8, 16, -32,..... is 1024 ?             9th term 

35) Can 640 be any term of the GP 5, -10, 20, .....?             9th

36)If a, b, c are an AP and x,y,z are in GP then show that xᵇ⁻ᶜ. yᶜ⁻ᵃ. zᵃ⁻ᵇ = 1.

37) If p,q,r are in AP, then show that p-th, q-th and r-th terms of any GP are in GP.

38) If the p-th, q-th and r-th terms of a GP are a, b, c respectively, then prove that aᑫ⁻ʳ.bʳ⁻ᵖ.cᵖ⁻ᑫ =1.

39) If aᵖ = bᑫ =cʳ and a, b, c are in GP then show p⁻¹, q⁻¹, r⁻¹ will be in AP.

40) If a¹⁾ˣ = b¹⁾ʸ = c¹⁾ᶻ and a, b, c are in GP, then show that x,y z will be in AP.

41) Insert two Geometric mean between 5 and 135.                             15,45

42) if 5, x, y ,z, 405 are in GP, then find the value of x, y, z.               ±15,45,±135

43) find the sum of the following series 139 27 29 terms 2 to 12 terms 33003 and terms 184 12 13 14 197 up to 1100315 in the n terms 999 and after and turns 444 up to terms 666 ad up to n terms without ascending the summation formula of GP find the sum of n terms of the series 13 how many terms of the series 36 12 24 must of the added from the first so that the sum maybe 1023 GP is 247 Terminus 192 find the sum of his first end up the sum of the first in the second term is 16 in the sum of the fourth and 5th terms is 432 find the sum of the first system of the GP the sum of the first three terms of the GP is 84 and the sum of the first 6 term 6 7 56 find the sum of the first 8 terms of the GP the ratio of the first three terms and the first 6 terms of a GP is 125 150 to find the combination of the GP be respectively the sums of first terms of GP then prove that some pizza product not the sum of the reciprocal of the first 5 terms of the GP then prove that the sum of two and terms of the cities in GPS to hour and the sum of the reciprocallers of those answer are find the continued product of the terms find three terms in 21 and product 64 the sum of the three consecutive numbers in GP 21 and the sum of the square is 189 find the numbers divide 26 into 3 parts such that the parts from a GP and the sum of the product taking to parts of the time is 119 some of the phone numbers in GPS 180 numbers is 108 find the numbers positive numbers in AP is 21 in 4518 respectivali to the number the result I am the GP final numbers the first 8 and 22nd the terms of an apr 3 consecutive terms of a GP find the common ratio of the GP a number consist of 3 digits which are in GP in the product of the digit is 2006 if 495 added to the number the digit will be reverse find the original number 615 and the geometric mean is 9 find the numbers if I am a two numbers between so that the numbers are if two quantities are in the ratio 322 30022 then so that automatically thrice of their geometric mean the difference of two positive number is 54 in the difference of the Arithmetic Geometric mean is 9 find the numbers arithmetic means between two given quantity then so that ABC and gpn x y z a m a b c l respectively so that ABC and AP and b c a r in GB prove that are in IP x y p and AP cxy the are injp so that GB the positive Geometric mean to unqual position quantities prove that ringtones find the in terms and some of the n terms 3 5 9 7 1 4 13 14 find the sum of 28 terms 1283 a4a if the number of terms of the GP be odd prove that the product of the first last term is equal to the square of the middle term the first in the last term of the GP respectively 362 and the sum of the terms is 1440 find the combination the number of terms of the GP three numbers in GPS product is 2164 be added to the first time and 6 to the second term resulting number in the third number AP obtain the numbers in GP the sum of the three numbers in GP 70 the two exchange be multiplied each by 4 and the mean by 5 the product in AP find the numbers the sum of the three numbers in GPS 14 if the first two numbers are each increased by one in the third number decrease by 1 the resulting number in AP find the numbers in GP answer two numbers between 6 and 16 such that the first three numbers in AP in the last three numbers are in GP answer two numbers between 4 and 12 such that the first three numbers are in GP in the last 3 members are in AP the first term of an ap and the of a GPR such 4 and their third terms are same if the second term of the AP succes that are the GP / 6 find AP and the GP if ABC are in APN ABD are in GP then prove that a - b - C are in GP is your equal numbers that ABC are in AP rhgp then so that if find and the list value for which is the geometric mean between a b and the find the value of n solve for x x y z i n g p and then so that when are the combination of the GP 190 without preceding installment find the amount of the first in the last installments a man borrows in 19682 repay the money in 9 and monthly installment it installment being thrice the preceding one after the 7th installment has been paid he wants to repair the balance in lump how much has he pay now what a certain tennis ball is drop from my height is to a height of the distance if the ball is drop from the height of 125 M how far has it travel when you hit the ground for 6th time the product of the first three terms of the GPS 278 find the midter the first term of a GP is 5 and 4th term is 300 25 in the common ratio the second term of a GP is - 24 and 5th term is 81 find its first term the first two term of the GP is 9 and 2 which term of the series is 1243 a GPA is the first term 6 in the last term is the 96 which term is twice is present in terms find the number of terms GP find the common ratio are GP find the value of a the third term of the GP is equal to the square of the first term of 4th term is 32 find the common ratio the sum of the first two terms of a GP 16 and the sum of the next two terms is 144 find the combination of the series 43 so that the series in GP the sum of the first find Geometric mean between 34 and 1627 at 3 positive numbers ABC and GP then so that x y z such that the prove that xyz in GP if pqr in the GP find the value of can any term of gpb zero give reason for your answer is there any two different real number such that the am and GMR equal give reason for your answer what type of series will be found by reciprocal of the terms of AGP give reason for your answer can 3D print is be in AP and GPR the same time give reason GPS for the product of the first five terms is the sum of the percent terms of the GP denoted by then what is the value of

47) If S be the sum, P the product and R the sum of the reciprocals of n terms of a GP, prove (S/R)^n= P².

RANGE, QUARTILES, PERCENTILE, DECILES

        QUARTILES, DECILES, 
               PERCENTILES
****.               ******.        ******

1) Value: 30 40 50 60 70 80 90
           F:  5   15 12 18 30 25  20
      C. F : 5   20 32 50 80 105125
Find
a) lower quartile (Q₁)
b) Upper quartile (Q₂)
c) Inter quartile (Q₃)
d) semi-inter quartile
d) D₁
e) 5th Decile (D₅)
f) D₉
g) P₁
h) P₅₀
i) P₉₉
j) Median

2) X :11-15 16-20 21-25 26-30
     f  :   7          9        15      23     
find
a) Q₁
b) Q₂
c) Q₃
d) D₁
e) D₅
f) D₉
g) P₁
h) P₅₀
i)  P₉₉
j) median
k) mode

3) which quartile is median.

4) The point of intersection of less than and more than type gives the result of-

5) If the relation between two variables x and y be 3x+3y= 50, and median of y is 5.
Then median of x is.

6) Find M. D about median

a)16,19,29,36,46,55,62,69,75,89.

b) 8,15,53,48,19,62,7,15,77,95.

c) 13,84,68,24,96, 139,84,27

d) 8,15,53,49,19,62,7,15,94,77.

e) X: 10   11  12  13  14  Total
     F:   3    12  18  12  3      48

f)   7,9,16,24,31,39.

g) C. I: 2-4  4-6  6-8  8-10
       F:   3     4      2 b   1.

h) C. I: 2-6  6-10  10-14  14-18
      F:    6      8        4           2

i) 46,79,26,85,39,65,99,29,56,72 .

j) 10,20,40,60,70,100also find the coefficient of M. D

7) Find mean deviation of mean:
a) 1,2,3,4,5,6,7

b) 31,35,29,63,55,72,37.

c) 27,33,49,61,76,104,126.

d) 29,35,51,63,78,106,128.

e) 31,35,29,63,55,72,37.

f) X: 10  11  12  13  14 Total
    F:   3   12   18 12   3    48

g) X:  5  10  15  20.    total
     F:   8   2    4    8         20

h) 7,9,16,26,31,39.

i) First ten natural numbers.

j) X: 8-11 12-15 16-19 20-23 24-27
   F:  5          6       20      10        4

k) 487, 508, 620, 382, 408, 266,
    186,218 also find coefficient of
     M. D

8) Find Quartile deviation:

a) X: 15  30  45  60  75  90
     F:  4    7    15   8    7    2

b) 12,7,15,10,19,17,25.

c) 350,840,650,710,980,575,290.

d)X:49-53 53-56 56-59 58-62 62-65
    F:   2         7         24      7         13

9) The coefficient of variation for the two groups are 58% and 69% . Their standard deviation are 21.2 and 15.6.Compute their arithmetic mean.

10) Find Range and coefficient of
       Range of following:
a) 12,7,15,10,19,17,25.

b) 9,7,25,18,38,12,30,35.

STANDARD DEVIATION

     STANDARD DEVIATION
           ***                    ***

1)Find S. D of
a)1,2,3,4,5,6,7,8,9

b) 46,63,46,59,52,60,54.

c) 20, 8,120,60,40.

d) 4,5,6,6,7,8

e) 3,5,11,7,8,10,9,12,14,11.

f) 240.12, 240.13, 240.15, 240.12, 240.17, 240.15, 249.17, 240.16, 240.22, 240.21.

g) 1,2,3,4,5,6,7,8,9,10

h) 1,2,3,4,5,6,7,8,9,10,11.

2) Calculate the variance of

a)1,5,6

b) first n positive integers ?

3) Find the coefficient of variation-

a)5,10,30,40,65

b) 40,30,80,60,50,90,70.

d) X: 1-10 11-20 21-30 31-40 41-50

     F:  5        12      20       25      40

4) The S. D of 1,2,3…,n is√14 find n.

5) Find standard deviation of:

a) C. I:   2-4    4-6    6-8    8-10
     F:       3.       4         2      1

b) X: 10  20  30  40  50  60
    F:.  9   18  25  27  14   7

c)    X: 25     36    45       55      65
        F: 60    130  150    130     30

d) X:  50    55    60    65      70
    F:   30   40     65    50      15

e) Age:  30    40   50    60     70
       F:.   64   132 153  140   51

f) X: 59-61 61-63 63-65 65-67
    F:    4         30      45       15  

g) X: 20-24 25-29 30-34 35-39
     F:    16       28       14     12

h) X: 120-124 125-129 130-134
     F:    12             25             28

6) The mean of two samples of sizes 50 and 100 respectively are 54.4 and 50.3; and their SD are 8 and 7 . What is mean and SD of the combined set of observations?

7) A sample size 15 has mean 3.5, and SD is 3.0. another sample of size 22 has mean 4.7 and SD 4.0. If two samples are pooled together, find the mean and standard deviation of the combined sample.

8) The mean and SD of 20 items is found to be 10 and 2. At the time of checking, it was found that one item 8 was incorrect. Calculate the SD if the wrong item is replaced
by 12.

9) The mean and SD of a sample size 10 were found to be 9.5 and 2.5. later on an additional observation become available. This was 15, and it was included in the original sample. Find the SD of the 11 observations.

10) For a group containing 100 observations, the mean and SD are 8 and√10.5. For 50 observations selected from those 100 observations the mean and SD are 10 and 2. Find the mean and SD of the other half.

11) The mean and SD of 20 items is found to be 10, and 2 respectively. At the time of checking, it was found that one item 8 was incorrect. Find the SD if the wrong item is omitted.

12) The first two samples has 100 items with mean 15 and SD 3. If the whole group has 250 items with mean 15.6 and SD √13.44, find the SD of the second group.

13) In a distribution of 200 boys, where 0-5,5-10 etc. are groups, mean and SD are 40 and 15. On checking it was found that the marks obtained by one student was wrongly taken as 53 in place of 43. Find the correct SD.

14) A student obtained the mean and SD of 100 observations as 40 and 5.1 respectively. It was later on found that he had wrongly copied one observation as 50, the correct figure being 40. Find correct mean, SD.

15) Prices of a particular commodity in five years in two cities are given:

City A: 20   22 19 23  26
City B: 10   20 18 12  15
Find the city which had more stable prices.

16) Which factory has greater variability in industrial wages ?
FACTORY:          A             B
SD:                    34.5       28.5
EMPLOYEES:   476        524

17) The coefficient of variation for the two series are 58% and 69% respectively. Their Standard deviation are 21.2 and 15.6. compute their arithmetic means.

18) Wages paid last year to the cleaning and maintenance staff of a shopping mall had a mean of Rs30000 with a SD of Rs2400. What will be the new mean and SD if all wages are increased by
a)Rs 3000
b) 10%

19) The following facts were gathered regarding an industrial dispute:

Information       Before After
No.of workers.   516  508
Mean wages.   49.50     51.75
Median Wages 52.70                50
Variance wages 100                 121
Compare the position before and after the dispute in terms of modal wages and coefficient of variation.

19) The mean and SD of height reading of a group of a employees of a firm are found to be 172cm and 18cm, while the same measure for their weight reading are 65 kg and 9kg. Compare the variability of the height reading with that of the weight readings.

20) The performance of two batsman S and R in five successive one-day cricket matches are :
S:    32 48   71 05 40
R:   19 47   67 06 101
Identify the batsman who is more consistent.

CENTRAL TENDENCY (MODE)

                   (MODE)
                 ***********

1) Following is the data of the heights of a group of students: 61,62,63,64,60,65,63,64,64,66,64. Now suppose that a group of students whose heights are 60,96, 59,68,67, and 70 inches is added to the original group.
a)What is the mode of the
     combined group ?

2) Mode of 5,3,27,5,9,3,8,5.

3) Mode of 4,3,2,5,3,4,5,3,7,3,2,6.

4) Mode and median of 25, 1275, 162,967,162.

5)Mode of 2, 1, 4, 2, 6, 4, 3, 2, 8,2,
      2, 1,4,6

6) mode of 1, 2, 2, 3, 3, 3 ,3, 4, 4,
      4,4,5,5,

7) Find Mean, Median, Mode of
       4,6,10,12,14,14,20,22.

8) The median of a set of observations is 12.Their mean is 13. Find their Mode.

9) The mode of 3,4,5,3,6,5,x,3,4,3.

10) If mean and mode is 26 and 24
       find  MEDIAN

11) For a moderately symmetric distribution the difference between mode and median is 24. Find the difference between median and the mean of the distribution.

12) For a moderately symmetric distribution the difference between mode and median is 24. Find the difference between median and the mean of the distribution.

13) X:   0     1      2    3     4     5
       F:   0      3     6     4    7      5
Find mode.

14) X: 1   2.  3.   4    5   6.   7   8   9
       F:  7  11 16 17  26  31 11 1  1
           Find mean and mode.

15) Marks     Frequency
         1-5        7
        6-10     10  Find mode
       11-15.   16
       16-20.   32
       21-25.   24
       26-30.   18
       31-35.   10
       36-40.    5
       41-45.    1
16)
Le10. Le20. Le30. Le40. Le50 Le60
 F: 4     20       35     55.      62      67
        Find median, mode

17) Wages.     No of workers
        00-10         4
        10-20        16   Total workers is
         20-30        a     230.
         30-40         b   If median=33.50
         40-50         c       mode=34 find
         50-60         6       a, b and c
         60-70         4

18) Class.    Frequency
       00-10             14
       10-20              a    If mode is 24
       20-30             27     Find a,b
       30-40              b
       40-50            15
              Total     100

CENTRAL TENDENCY ( MEDIAN)

                  MEDIAN

                     ******

A) Find the median of following:

1) 37,31,42,43,46,25,39,43,32.

2) 31,38,27,28,39,35,49.

3) 83,37,70,29,45,63,41,70,30,54.

4) 79,82,36,38,51,72,68,70,64,63.

5) Median of 7,2,5,9,6

6) 50,53,61,49,45,63,48

7)10,75,3,81,18,27,4,48,12,47,9,15

8) Find median of the  first 9 consecutive natural numbers.

9) The median of the first 49 natural numbers is…

10) Median 8,3,11,7,12,6,9.

11) Median of 33,86,68,32,80,48,7

12) 31,35,27,29,32,43,37,41, 34,28,36,44,45,42,30

13) 3,1,5,6,3,4,5,3,7,2

14) 11,12, 14,18, x+2, x+4, 30, 32,
       35,41 arranged in ascending
      order is 24, find x.

15) If the weights of 10 students
(in kg) arranged in a ascending order are: 53,47,34,36,x -5, x+ 5, 54,37,44,32 with median 45 kg find value of x.

16) Find the median of the following data: 41,43,127,99,61, 92,71,58,57. If 58 is replaced by 85, what will be the new median.

B) Find the median

1) X:  3  6  10  12  7   15
     F:  3  4    2    8  13  19

2)   X: 5  7  9  12  14  17  19  21
       F:  6 5  3   6     5    3    2    4

3) S. No.  Mar(Eng). Marks(math)
       1              53              55
       2              54              41
       3              32              48
       4              30              49
       5              60              27
       6              46              24
       7              28              23
       8              25              20
       9              48              28
      10             72              60
      11             33              43
      12             65              66
Indicate in which subject the level of achievement is higher ?

4) X: 4  4.5  5  5.5  6  6.5  7 7.5  8
     Y:10 18  22 25  40 15 10 8    7

5) X:  8  5   6   10   9   4    7
     Y:  6  4   5    8    9   6    4.    

6) Mid values.       frequency   
         115                      6
         125                     25
         135                     48
         145                     72
         155                    116
         165                      60
         175                      38
         185                      22
         195                       3

C) Find median of the following:

1)X: 0-10 10-20 20-30 30-40 40-50
    F:   5        7        10       10       8

2) 
X: 1-10 11-20 21-30 31-40 41-50
F:    3        5        8          6         3

3) Marks     Number of students 
      15-20              4
      20-25              5
      25-30              1
      30-35              6
      35-40              5
      40-45              8
      45-50              9
      50-55              6
      55-60              4
      60-65              2

4) Age(in yrs.)           Class
       5-14                      447   
     15-24                      307
      25-34                     279   
      35-44                     220
      45-54                     157
      55-64                      91
      65-74                      39

5) Age(years).    No of persons
      Below 10                2
      Below 20                5
      Below 30                9
      Below 40              12
      Below 50              14
      Below 60              15
      Below 70            15.5
70 and above          15.6

6)Mark more than  No of students
        0                                 50     
       10                                46
       20                                40
       30                                20
       40                                10
       50                                  3

7) Find the missing Numbers

  C .I         Frequency
300-325         5    median is 413.11
325-350       17
350-375       80
375-400.        ?
400-425       326
425-450.        ?
450-475        88
475-500         9
Total           10008

8) Find missing Numbers if

       N=100 and median is 30

Marks       frequency
0-10              10
10-20.            ?
20-30            25
30-40            30
40-50.            ?
50-60            19

 

COMPOUND ANGLES(XI)

      COMPOUND ANGLES
            ****************

FORMULAE 

*********

1) sin(A+B)= sinA cosB+ cosA sinB 

2) sin(A--B)= sinA cosB- cosA sinB 

3) sin(A+B+C)= sinA cosB cosC+ sinB cosA cosC+ sinC cosA cosB + sinA sinB sinC 

   = ΣsinA cosB cosC - Π sinA

4) cos(A+B)= cosA cosB-sinA sinB 

5) cos(A-B)= cosA cosB+ sinA sinB 

6) cos(A+B+C)= cosA cosB cosC -sinA sinB cosC - sinA cosB sinC - cosA sinB sinC.

= Π cosA - ΣsinA sinB cosC 

= CosA cosB cosC[1- tanA tanB - tanB tanC - tanC tanA]

 

7) tan(A+B)= (tanA+tan B)/(1- tanA tanB)

8) tan(A-B)= (tanA-tan B)/(1+ tanA tanB) 

9) tan(A+B+C)= (tanA+tan B+ tanC - tanA tanB tanC)/(1- tanA tanB - tanB tanC - tanC tanA) 

= (S₁ - S₃)/(1- S₂)

10) cot(A+B)= (cotA cot B-1)/(cotB + cotA)

11) cot(A-B)= (cotA cot B+1)/(cotB - cotA).

12) sin(A+B).Sin(A-B)= sin²A- sin²B

                                    = cos²B - cos²A

13) cos(A+B)cos(A-B)=cos²A- sin²B

                              = cos²A - sin²B = c

MAXIMUM & MINIMUM VALUES OF TRIGONOMETRIC EXPRESSIONS:

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

1) a cosx + b sinx will always lie in the interval [- √(a²+b²), √(a²+b²)] i.e. the maximum and minimum values are √(a²+b²), - √(a²+b²)

2) Minimum value of a² tan²x+ b² cot²x= 2ab, where a,b > 0.

3) - √(a²+b²+ 2ab cos(x-y) ≤ a cos(x+m) + b cos(y+m) ≤ √(a²+b²+ 2ab cos (x-y) where x, y are known angles.

4) Minimum value of a² cos²x + b² sec²x is either 2ab or a²+ b², if for some real x equation a cosx= b secx is true or not true. (a, b > 0)

5) Minimum value a² sin²x + b²cosec²x is either 2ab or a²+ b², if for some real x equation a sinx= b cosecx is true or not true (a,b > 0)

               EXERCISE -1

               ---------------------

1) Find the expansion of

a) sin(A+B-C)

b) cos(A-B-C)

c) tan(A-B+C)

d) cot(A+B+C)

e) sec(A-B+C)

f) cosec (A- B- C) 

               EXERCISE-2 

              -------------------

1) Find the values of

a) sin75°                         (√3+1)/2√2 

b) cos105°                   -(√3-1)/2√2 

c) cot 15°                                 2+√3 

d) cosec(-105°)                      √2-√6

e) sec (-75°)                           √2+√6

2) PROVE: 

a) cos15° - sin15°= 1/√2

b) sin15°+ sin75°= √(3/2).

c) tan75°- cot 75°= 4 sin60°

d) sin15°+ tan 30°cos15°= √6/3.

e) sin165° + cos165°= -1/√2

f) sin105° + cos105°= cos45°

g) cos80 cos 20 + sin 80 sin 20=½

h) cos84º25′ cos24º25′ + cos5º35′ cos65º35 ′= ½

i) cos80°40'. Cos39°20' - sin80°40'. sin39°20'= -1/2.

j) cos(30°-x) cos(60°-y)- sin(30°-x) sin(60°-y)= sin(x+y).

k) cos(45- x)cos(45-y) - sin(45-x) sin(45 -y) = sin (x+y)

l) cos(π/4 - x) cos(π/4 -y)- sin(π/4 - x) sin(π/4 -y)= sin(x+y)

m) sin(A+B)sin(A- B) + cos(A+B) cos(A -B) + 2 sin²(B) = 1.

n) sin(n+1)A. cos(n-1)A - cos(n-1)A. sin(n-1)A = sin2A.

o) cosA sin(B-C) + cosB sin(C-A) +  cos C sin(A -B) =0.

p) sin(A+B). sin(A-B) +sin(B+C). sin(B-C) +sin(C+A) sin(C-A) = 0.

q) sinA sin(B- C)+ sinB sin(C-A)+ sinC sin(A-B)= 0.

r) cosA sin(B-C)+ cosB sin(C-A)+ cosC sin(A-B)= 0.

s) sin(B+C) sin(B-C)+ sin(C+A) sin(C-A) + sin(A+B) sin(A-B)= 0.

t) sin(A- B) sin(C- D) + sin(C-B) sin(D- A) + sin(D- B) sin(A- C)= 0.

u) sin2x cosx + cos2x sinx = sin4x cosx - cos4x sinx.

v) cos(A- B)- sin(A+B)= (cosA - sinA)(cosB - sinB).

w) cos(A+ B)+ sin(A- B)= (cosA + sinA)(cosB - sinB).

                 EXERCISE --3

                  **************

PROVE:

a) sec(x-y)=(secx secy)/(1+tanx tany)

b) cosec(x+y) = (cosec x cosec y)/(coty + cot x).

c) sec(x+ y)=(secx secy)/(1- tanx tany)

d) sin(B- C)/(cosB cosC) + sin(C- A)/(cosC. cos A) + sin(A- B)/(cosA cos B) = 0.

e) sin(B- C)/(sinB. sinC) +sin(C- A)/(sinC. sin A) +sin(A- B)/(sinA. sin B) = 0.

f) tan 20+tan 25 + tan 20 tan25 =1

g) tan 3x - tan2x - tanx =tan3x tan2x tanx.

h) tan(y-z) +tan(z-x)+tan(x-y)= tan(y-z) tan(z-x) tan (x-y).

i) tan(B+C-2A)+ tan(C+A-2B)+ tan(A+B- 2C)= tan(B+C-2A) tan(C+ A - 2B) tan(A+B- 2C).

j) √3(tan 170 - tan 140) =1+tan 170 tan 140.

k) tan 75 - tan30 - tan 75 tan30= 1

l) If tan10+ tanx= 1- tan10 tanx find the value of x.                             35° 

m) (cosA+sinA)/(cosA- sinA) =tan(π/4 + A).

n) (cos7+sin7)/(cos7- sin7)= tan52

o) (cos10- sin10)/(cos10+sin10)= tan35

p) (cos15+sin15)/(cos15-sin15)= √3

q) (cos11+sin11)/(cos11-sin11)= tan56.

r) tan(45°+A)= (cosA+ sinA)/(cosA- sinA).

s) √2 sin(45+A)= sinA+ cosA.

t) (cos8+ sin8)/(cos8 - sin8)=tan53 

u) tan(45+A) tan(45-A)= 1. 

v) cot(45+x)= (cotx -1)/(cotx+1) = (cosx- sinx)/(cosx+ sinx).

                   EXERCISE-4

                   ***********

PROVE:

a) 1+tan2α. tanα =sec2α

b) cos(A-B)/2 - sinA. sin (A+B)/2 = cosA. cos (A+B)/2

c) cot θ - cot 2θ = cosec2θ.

d) cosA+ cos(120+A)+ cos(120-A)= 0.

e) tan(x+y) + tan(x-y)=  sin2x/(1- sin²x - sin²y).

f) tan(x+y) + tan(x-y)= sin2x/(cos²x - sin²y).

g) cot(A+B) + cot(A-B) = Sin2A/(sin²A - sin²B)

h) tan²A - tan²B = {sin(A+B) sin(A-B)}/cos²A cos²B. 

i) cotA/(cotA- cot3A) - tanA/(tan3A - tanA)= 1

j) 1/(tan3A - tanA) - 1/(cot3A - cotA) = cot2A.

k) tan(A+B) tan(A-B)= (sin²A- sin²B)/(cos²A - cos²B).

l) {tan(A+B)- tanA}/{1+ tan(A+B) tanA)}= tan B.

m) sin(x+y)}/sin(x-y)= (tanx+ tany)/(tanx - tany).

m) sinx + cosx =√2 sin(π/4 +x)= √2 cos (π/4 - x).

n) cos 15 + √3 sin15=√2 

o) {sin(2A+B)}/sinA - 2 cos (A+B)= sinB/sinA 

                EXERCISE-5

              **************

1) Given sinα=4/5, sin β=3/5 find cos(α+β).                                        0

2) Given sinα=3/5, tan β=5/12 find sin(α+β).                                    56/65

3) Given secα=5/4, cosec β=13/12 find sec(α - β).                        65/56

4) If sinα= 1/√10, cos β=2/√5 find the value of (α+β).                    π/4

5) Show that Sin(A-B)=16/65, cos(A+B)= 33/65 if sinA= 2/5, cosB= 12/13.

6) sec(A-B)=85/84, if secA= 17/8, cosecB= 5/4.

7) If sinx= 3/5, sin y= 8/17, find 

A) sin(x± y).                77/85,13/85   

B) cos(x± y).              36/85, 84/85

C) tan(x± y)               77/36, 13/84 

8) if sinx= 3/5, cos y= 5/13, x and y in Quadrant I. Then find the value of

A) sin(x+y).                             63/65

B) cos(x+y).                           -16/65

C) tan(x+y).                          -63/16

9) if cosx= -12/13, cot y= 24/7, x in II and y in Quadrant III. Then find the value of

A) sin(x+y).                           -36/325

B) cos(x+y).                         323/325

C) tan(x+y).                         -36/323

10)  if sinx= 8/17, tan y= 5/12, x and y in Quadrant I. Then find the value of

A) sin(x-y).                           21/221

B) cos(x-y).                         220/221

C) tan(x-y).                         21/220

11) if sinx= -12/13, cot y= 24/7, x in I quadrant and y in Quadrant III. Then find the value of

A) sin(x-y).                         204/325

B) cos(x-y).                        -253/325

C) tan(x-y).                        -204/253

12) If tanA= 5/6, tanB= 1/11 then find the value of (A+B).            45°

13) If tanA= m/(m+1), tanB= 1/(2m+1) then find the value of (A+B).                                          π/4

14) A, B, C are three positive acute angles and tanA= 1/2, tanB= 1/5, tanC= 1/8, show that A+B+C= π/4.

15) sin (x+y)= 4/5, cos(x-y)= 12/13 and 0< x< π/4, 0< y< π/4; find the value of tan 2x.                      33/56

16) If tan(A+B)= 1, tan(A-B)= 1/7 then find tanA and tanB.       1/2 or-2 and 1/3 or -3

 

             EXERCISE --6

            ****************

PROVE:

1) If 2 tan B + cot B= tan A, show that cotB= 2 tan(A- B).

2) cos²A + cos²(120°+A)+cos²(120-A) = 3/2.

3) If tanA= 2 tanB, find the value of {sin(A+B)}/{sin(A-B)}.                     3

4) If tan x - tan y =a, cot y - cot x=b Show that1/a + 1/b = cot(x - y)

5) If tan3A- tanA= x and cot3A - cotA= y, show cot2A= 1/x - 1/y.

6) If sin(A+B)= n sin(A-B), then show that tanA= {(n+1)/(n-1)} tanB. 

7) If tan(A+B)= n tan(A-B) then show (n+1) sin2B= (n-1) sin2A.

8) If m tan(A-30)= n tan(A+120), then show cos2A=(m+n)/{2(m-n)}.

9) If tan m= (K sin a. cos a)/1- Ksin² a), show tan(a-m)= (1-K) tan a.

10) If sin a= A sin(a+b), show that tan(a+b)= sin b/(cos b - A).

11) If a cos(m+n)= b cos(m -n), show that (a+b) tan m= (a-b) cot n.

12) If tan a= x sin b/(1- x cos b) and tan b= y sin a/(1- y cos a) prove that sin a: sin b= x: y.

13) if cos (A-B) sin(C- D)= cos(A+B) sin(C+D), show that tan D= tan A tanB tan C. 

14) If sin a sin b - cos a cos b +1= 0, then show that 1 + cot a tan b= 0.

15) If tan m= (asinx + bsiny)/(a cosx-+ b cosy), then show that a sin(m -x) + b sin(m-y)= 0.

16) If cotA= cos(x+y) and cotB= cos(x-y), then show tan(A-B)=(2sinx siny)/(cos²x + cos²y).

17) An angle a is divided into two parts m and n such that sim m: sin n= K: 1, show that tan m= K sin a/(1+ K cos a) and tan m= sin a/(K+ cos a).

18) An angle m is divided into two parts, a, b such that tan a: tan b= x: y; prove sin(a-b)= {(x-y)(sin m}/(x+y)

19) If A+ B= π/4, show that (1+ tanA)(1+tanB)= 2.

20) If A+ B= 45°, show that tan A+ tanB + tanA tanB= 1.

Hence otherwise, express tan22°30' in surd form.                         √2 - 1.

21) Given that A= B+ C, show that tanA - tanB - tan C=tanA tanB tanC. 

22) If A+ B+C= 2π and sinA= sinB cosC, prove 2tanB + tanC= 0.

23) If A+ B= 135, show that (1+ tanA)(1+ tanB)= 2 tanA tanB.

24) If A, B, C are in AP and B=π/4; what is the value of tanA tanB tanC.                                                  1

25) If tan y= 2 sin z sin x cosec(z+x), show that cotx, coty, cotz are in AP. 

26) If A+ B+C=π, Prove that,

A) tanA+ tanB+ tanC= tanA tanB tanC.

B) tanA/2 tanB/2+ tanB/2 tanC/2 + tanC/2 tanA/2= 1.

C) cotA/2 + cotB/2+ cotC/2=  tanA/2 + cotB/2 cotC/2. 

27) If A+ B+ C=π/2, show that cotA+ cotB+ cotC= cotA cotB cotC.

28) If A+ B= 225°, show tanA+ tanB= 1 - tanA tan B.

29) In a ∆ ABC If cosA= cosB cosC, prove that

A) tanA= tanB+ tan C 

B) cotB. cotC= 1/2

30) If cos(B-C)+ cos(C-A)+ cos(A-B) =-3/2, then show that ∑ cos a= 0 and ∑sin a = 0.

31) If cosx + cosy+ cos z=0 and sin x + sin y+ sin z=0, show that,

A) cos(x-y)= cos(y-z)= cos(z-x)=-1/2

B) cos(x-y)+ cos(y-z)+cos(z-x)=-3/2

32) Express a cosx + b sinx in the form of r sin (θ+α) or r cis(θ -α) from which find  the values of r, α,θ

33) Find Maxumum and minimum value of 5 cos x + 12 sin x -12.

34) Find Maxumum and minimum value of 3cos x+ 4sin x +5.     10, 0

35) Eliminate A and B:

A) tanA+ tanB= a, cotA+ cotB= b, A+B= k 

B) sinA+ sinB= a, cosA+ cosB= b, A-B= π/3.