PROGRESSION, RELATED INEQUALITIES AND SERIES
1) If a₁, a₂, a₃, are in AP then aₚ, aq, aᵣ are in AP if p.q,r are in
a) AP b) GP c) HP d) none
2) Let tᵣ denote the rth term of an AP. If tₘ = 1/n and tₙ = 1/m then tₘₙ equals
a) 1/mn b) 1/m + 1/n c) 1 d) 0
3) If p,q,r,s ∈ N and they are four consecutive terms of an AP then the pth, qth, rth, sth terms of a GP are in
a) AP b) GP c) HP d) none
4) If in a progression a₁, a₂, a₃, .....etc, (aᵣ - aᵣ₊₁) bears a constant ratio with aᵣ. aᵣ₊₁ then the terms of the progression are in
a) AP b) GP c) HP d) none
5) If a₂a₃/a₁a₄ = (a₂+ a₃)/(a₁+ a₄) = 3(a₂ - a₃)/(a₁ - a₄) then a₁, a₂, a₃, a₄ are in
a) AP b) GP c) HP d) none
6) Let x,y,z be three positive prime numbers. The progression in which √x, √y, √z can be three terms (not necessarily) consecutive is
a) AP b) GP c) HP d) none
7) Let f(x)= 2x +1. Then the number of real values of x for which the three unequal numbers f(x), f(2x), f(4x) are in GP is
a) 1 b) 2 c) 0 d) none
8) If a> 0, r∈ N and a₁, a₂, a₃, ....a₂ₙ are in AP then
(a₁+ a₂ₙ)/(√a₁+ √a₂) + (a₂ + a₂ₙ₋₁)/(√a₂ + √a₃) + (a₃ + a₂ₙ₋₂)/(√a₃ + √a₄) + ...+ (aₙ + aₙ₊₁)/(√aₙ + aₙ₊₁) is equal to
a) n -1 b) n(a₁ + a₂ₙ)/(√a₁ + √aₙ₊₁) c) (n -1)/(√a₁ + √aₙ₊₁) d) none
9) If a₁, a₂, a₃, ....a₂ₙ₊₁ are in AP then
(a₂ₙ₊₁ - a₁)/(a₂ₙ₊₁ + a₁) + (a₂ₙ - a₂)/(a₂ₙ + a₂) + .... + (aₙ₊₂ - aₙ)/(aₙ₊₂ + aₙ) is equal to
a) n(n +1)/2 . (a₂ - a₁)/aₙ₊₁
v) n(n +1)/2
c) (n +1)(a₂ - a₁) d) none
10) a₁, a₂, a₃, .... be in AP and aₚ, aq, aᵣ be in GP. Then aq : aₚ is equal to
a) (r - p)/(q - p) b) (q - p)/(r - q) c) (r - q)/(q - p) d) none
11) If a,b,c are in GP then a+ b, 2b, b + c are in
a) AP b) GP c) HP d) none
12) If a,b,c,d are non-zero real numbers such that
(a² + b² + c²)(b²+ c²+ d²) ≤ (ab+ bc+ cd)² then a,b,c,d are in
a) AP b) GP c) HP d) none
13) If 4a²+ 9b² + 16c²= 2(3ab + 6bc + 4ca) , where a,b,c are non-zero numbers, then a,b,c are in
a) AP b) GP c) HP d) none
14) If a,b,c are in AP then a/bc, 1/c, 2/b are in
a) AP b) GP c) HP d) none
15) If in an AP, t₁ = log₁₀a, tₙ₊₁ = log₁₀b and t₂ₙ₊₁ = log₁₀c then a,b,c are in
a) AP b) GP c) HP d) none
16) If n!, 3 x n! and (n+1)! are in GP then n!, 5x n! and (n +1)! are in
a) AP b) GP c) HP d) none
17) In an AP, the pth term is q and the (p + q)th term is 0. Then the qth term is
a) -p b) p c) p+ q d) p - q
18) In a sequence of (4n +1) terms the first (2n +1) terms are in AP whose common difference is 2, and the last (2n +1) terms are in GP whose common ratio is 0.5. if the middle terms of the AP and GP are equal then the middle term of the sequence is
a) (n. 2ⁿ⁺¹)/(2ⁿ -1) b) (n. 2ⁿ⁺¹)/(2²ⁿ -1) c) n. 2ⁿ) d) none
19) If x² + 9y² + 25z² = xyz (15/x + 5/y + 3/z) then x,y,z are in
a) AP b) GP c) HP d) none
20) If a,b,c,d and p are distinct real numbers such that
(a²+ b²+ c²)p² - 2(ab+ bc+ cd)(b²+ c² + d²)≤ 0 then a,b,c,d are in
a) AP b) GP c) HP d) none
21) The largest term common to the sequence 1,11,21,31,....to 100 terms and 31,36,41,46,....to 100 terms is
a) 381 b) 471 c) 281 d) none
22) The interior angles of a convex polygon are in GP, the common difference being 5°. If the smallest angle is 2π/3 then the number of sides is
a) 9 b) 16 c) 7 d) none
23) The minimum number of terms of 1+ 3+5+7+.... that add up to a number exceeding 1357 is
a) 15 b) 37 c) 35 d) 17
24) In the value of 100! the number of zeros at the end is
a) 11 b) 22 c) 23 d) 24
25) The sum of all the proper divisors the 9900 is
a) 33851 b) 23952 c) 23951 d) none
26) The sum of all odd proper divisors of 360 is
a) 77 b) 78 c) 81 d) non
27) In the sequence 1,2,2,3,3,3,4,4,4,4,..., where n consecutive terms have the value n, the 150th term is
a) 17 b) 16 c) 18 d) none
28) In the sequence 1,2,2,4,4,4,4,8, 8, 8, 8, 8, 8, 8, 8,.... where n consecutive terms have the value n, the 1025th term is
a) 2⁹ b) 2¹⁰ c) 2¹¹ d) 2⁸
29) Let [tₙ] be a sequence of integers in GP in which t₄ : t₆ = 1: 4 and t₂ + t₅ = 216. Then t₁ is
a) 12 b) 14 c) 16 d) none
30) If log(5c/a), log(3b/5c) and log(a/3b) are in AP, where a,b,c are in GP, then a,b,c are the lengths of sides of
a) an isosceles triangle
b) an equilateral triangle
c) a scalene triangle d) none
31) Let S be the sum, P be the product and R be the sum of the reciprocals of n terms of a GP. Then P²Rⁿ : Sⁿ is equal to
a) 1:1 b) (common ratio)ⁿ : 1 c) (first term)² : (common ratio)ⁿ d) none
32) If the pth, qth and rth terms of an AP are in GP then the common ratio of the GP is
a) (p+ q)/(r+ q) b) (r - q)/(q - p) c) (p - r)/(p - q) d) none
33) The number of terms common between the series 1+2+4+8+....to 100 terms and 1+4+7+10....to 100 terms is
a) 6 b) 4 c) 5 d) none
34) The 10th common term between the series 3+7+11+... and 1+6+11+...is
a) 191 b) 193 c) 211 d) none
35) Three consecutive terms of a progression are 30,24,20. The next term of the progression is
a) 18 b) 120/7 c) 16 d) none
36) If the numbers are in GP then the numbers obtained by adding the middle number to each of the three numbers are in
a) AP b) GP c) HP d) none
37) If a₁, a₂, a₃ are in AP, a₂, a₃, a₄ are in GP and a₃, a₄, a₅ are in HP then a₁, a₃, a₅ are in
a) AP b) GP c) HP d) none
38) If a,b,c,d are four numbers such that the first three are in AP while the last three are in HP then
a) bc= ad b) ac= bd c) ab= cd d) none
39) If the first two terms of an HP be 2/5 and 12/23 then the largest positive term of the progression is the
a) 6th term b) 7th term c) 5th term d) 8th term
40) If x, 2y, 3z are in AP, where the distinct numbers, x,y,z are in GP then the common ratio of the GP is
a) 3 b) 1/3 c) 2 d) 1/2
41) If x>1, y> 1, z> 1 are three numbers in GP then
1/(1+ lnx) , 1/(1+ ln y), 1/(1+ logz) are in
a) AP b) HP c) GP d) none
42) If a, a₁, a₂, a₃,....a₂ₙ₋₁, b are in AP a, b₁, b₂, b₃,....b₂ₙ₋₁, b are in GP and a, c₁, c₂, c₃,,.....c₂ₙ₋₁, b are in HP, where a,b are positive, then the equation aₙx² - bₙx + cₙ = 0 has its roots.
a) real and unequal
b) real and equal
c) imaginary d) none
43) If a, x, b are in AP, a, y, b are in GP and a,z, b in HP such that x = 9z and a> 0, b >0 then
a) |y|= 3z b) 3|y|= x c) 2y = x + z d) none
44) If there numbers are in HP then the numbers obtained by subtracting half of the number from each of them are in
a) AP b) GP c) HP d) none
45) a,b,c,d , e are five numbers in which the first three are in AP and the last three are in HP. If the three numbers in the middle are in GP then the numbers in the odd places are in
a) AP b) GP c) HP d) none
46) Let a₁, a₂, a₃,....a₁₀ be in AP and h₁, h₂, h₃,....h₁₀ be in HP. If a₁ = h₁ =2 and a₁₀ = h₁₀ = 3 then a₄h₇ is
a) 2 b) 3 c) 5 d) 6
47) If in an AP, Sₙ = p. n² and Sₘ = p. m², where Sᵣ denotes the sum of r terms of the AP, then Sₚ is equal to
a) p³/2 b) mnp c) p² d) (m + n)p²
48) If Sᵣ denotes the sum of the first r terms of an AP then (S₃ᵣ - Sᵣ₋₁)/(S₂ᵣ - S₂ᵣ₋₁) is equal to
a) 2r -1 b) 2r+ 1 c) 4r + 1 d) 2r +3
49) Sᵣ denotes the sum of the first r terms of a GP. Then Sₙ, S₂ₙ - Sₙ, S₃ₙ - S₂ₙ are in
a) AP b) GP c) HP d) none
50) If (1- p)(1+ 3x + 9x²+ 27x³+ 81x⁴+ 243x⁵)= 1- p⁶, p ≠ 1 then the value of p/x is
a) 1/3 b) 3 c) 1/2 d) 2
51) If the sum of the series 1+ 2/x + 4/x² + 8/x³ + .....to ∞ is a finite number then
a) x< 2 b) x > 1/2 c) x > -2 d) x< -2 or x > 2
52) Let Sₙ denotes the sum of the first n terms of an AP. If S₂ₙ= 3Sₙ then S₃ₙ : Sₙ is equal to
a) 4 b) 6 c) 8 d) 10
53) In a GP of even number of terms, the sum of all terms is 5 times the sum of the odd terms. The common ratio of the GP is
a) -4/5 b) 1/5 c) 4 d) none
54) In an AP, Sₚ = q, Sq = p and Sᵣ denotes the sum of the first r terms. Then Sₚ₊q is equal to
a) 0 b) -(p+ q) c) p+ q d) pq
55) The co-efficient of x¹⁵ in the product
(1- x)(1- 2x)(1- 2²x)(1- 2³x)...(1- 2¹⁵x) is equal to
a) 2¹⁰⁵- 2¹²¹ b) 2¹²¹ - 2¹⁰⁵ c) 2¹²⁰ - 2¹⁰⁴ d) none
56) The co-efficient of x⁴⁹ in the product (x -1)(x -3)...(x - 99) is
a) -99² b) 1 c) -2500 d) none
57) If a,b,c are in AP then a+ 1/bc, b+ 1/ca, c+ 1/ab are in
a) AP b) GP c) HP d) none
58) The AM of two given positive numbers is 2. If the larger number is increased by 1, the GM of the numbers becomes equal to the AM of the given numbers. Then HM of the given numbers is
a) 3/2 b) 2/3 c) 1/2 d) none
59) Let a,b be two positive numbers, where a> b and 4x GM = 5x HM for the numbers. Then a is
a) 4b b) b/4 c) 2b d) b
60) If a, a₁, a₂, a₃,......a₂ₙ, b are in AP and a, g₁, g₂, g₃, .....g₂ₙ, b are in GP and h is the HM of a and b then
(a₁+ a₂ₙ)/(g₁g₂ₙ) + (a₂ + a₂ₙ₋₁)/(g₂g₂ₙ₋₁) + ....+ (aₙ + aₙ₊₁)/(gₙgₙ₊₁) is equal to
a) 2n/b b) 2nh c) nh d) n/h
61) Let a₁ =0 and a₁, a₂, a₃,....aₙ be real numbers such that |aᵢ|= |aᵢ₋₁ +1| for all i then the AM of the numbers a₁, a₂, a₃,.....aₙ has the value A where
a) A< -1/2 b) A< -1 c) A≥ -1/2 d) A= -1/2
62) Let there be a GP whose first term is a and the common ratio is r. If A and H are the arithmetic mean and the harmonic mean respectively for the first n terms of the GP, A. H is equal to
a) a²rⁿ⁻¹ b) arⁿ c) a²rⁿ d) none
63) If the first and the (2n -1)th terms of an AP, a GP and an HP are equal and their nth terms are a, b and c respectively then
a) a= b = c b) a≥ b ≥ c c) a+ c = b d) ac - b²= 0
64) (aⁿ + bⁿ)/(aⁿ⁻¹ + bⁿ⁻¹) is the HM between a and b if n is
a) 0 b) 1/2 c) -1/2 d) 1
65) If the harmonic mean between P and Q be H then H(1/P + 1/Q) is equal to
a) 2 b) PQ/(P + Q) c) (P+ Q)/PQ d) 1/2
66) Let x be the AM and y,z be two GMs between two positive numbers. Then (y³ + z³)/xyz is equal to
a) 1 b) 2 c) 1/2 d) none
67) If HM : GM= 4: 5 for two positive numbers then the ratio of the numbers is
a) 4:1 b) 3:2 c) 3:4 d) 2:3
68) In a GP of alternatively positive and negative terms, any term is the AM of the next two terms. Then the common ratio is
a) -1 b) -3 c) -2 d) -1/2
69) If a,b, c are in AP and p, P' are the AM and GM respectively between a and b, while q, q' are the AM and GM respectively between b and c, then
a) p²+ q²= p'²+ q'² b) pq= p'q' c) p²- q²= p'²- q'² b) none
70) If -π/2< θ< π/2 then the minimum value of cos³θ + sec³θ is
a) 1 b) 2 c) 0 d) none
71) If a> 1, b > 1 then the minimum value of logᵥa+ logₐv is
a) 0 b) 1 c) 2 d) none
72) The minimum value of 4ˣ + 4¹⁻ˣ, x ∈ R, is
a) 2 b) 4 c) 1 d) none
73) If x= log₅3+ log₇5 + log₉7 then
a) x≥ 3/2 b) x ≥ 1/³√2 c) x ≥ 3/³√2 d) none
74) If aₙ > 1 for all n ∈N then
logₐ₂a₁ + logₐ₃a₂ +.....+ Logₐₙaₙ₋₁ + log₁aₙ has the minimum value
a) 1 b) 2 c) 0 d) none
75) The product of n positive numbers is 1. Their sum is
a) a positive integer
b) divisible by n
c) equal to n + 1/n
d) greater than or equal to n
76) If x,y,z are three real numbers of the sum sign then the value of x/y + y/z + z/x lies in the interval
a) [2, +∞) b) [3, +∞) c) (3, +∞) d) (-∞,3)
77) The least value of 2 log₁₀₀a - logₐ0.0001, a> 1 is
a) 2 b) 3 c) 4 d) none
78) If 0< x<π/2 then the minimum value of (sin x + cos x + cosec 2x)³ is
a) 27 b) 13.5 c) 6.75 d) none
79) If x,y,z are positive then the minimum value of
xˡᵒᵍʸ ⁻ˡᵒᵍᶻ + yˡᵒᵍᶻ⁻ˡᵒᵍˣ + zˡᵒᵍˣ⁻ˡᵒᵍʸ is
a) 3 b) 1 c) 9 d) 16
80) a,b,c are three positive numbers and abc² has the greatest value 1/64. Then
a) a=b =1/2, c= 1/4
b) a=b =1/4, c= 1/2
c) a=b = c= 1/3 d) none
81) if a> 0, b >0, c > 0 and the minimum value of
a(b² + c²) + b(c² + a²) + c(a² + b²) is λabc then λ is
a) 2 b) 1 c) 6 d) 3
82) The value of ¹⁰ₙ₌₁ ∑ ⁿ₀∫ x dx is
a) an even integer
b) an odd integer
c) a rational number
d) an irrational number
83) The sum of (0.2 + 0.004+ 0.00006+ 0.0000008+... to ∞) is
a) 200/891 b) 2000/9801 c) 1000/9801 d) none
84) If (2n + r)r, n ∈ N, r ∈ N is expressed as the sum of k consecutive odd natural numbers then k is equal to
a) r b) n c) r+1 d) n+1
85) ⁿᵣ₌₁ ∑r² - ⁿₘ₌₁ ∑ ᵐᵣ₌₁∑ r is equal to
a) 0 b) (1/2) (ⁿᵣ₌₁ ∑ r²+ ⁿᵣ₌₁ ∑ r)
c) (ⁿᵣ₌₁ ∑ r² - ⁿᵣ₌₁ ∑ r) d) none
86) If (1+ x)(1+ x²)(1+ x⁴)....(1+ x¹²⁸)= ⁿᵣ₌₀ ∑ xʳ then n is
a) 225 b) 127 c) 63 d) none
87) The value of ᵐₙ₌₁∑ log(a²ⁿ⁻¹/bᵐ⁻¹) (a≠ 0, 1; b ≠ 0, 1) is
a) m log(a²ᵐ/bᵐ⁻¹)
b) log(a²ᵐ/bᵐ⁻¹)
c) (m/2) log(a²ᵐ/b²ᵐ⁻²)
d) (m/2) log(a²ᵐ/bᵐ ⁺¹)
88) The sum of the product of the ten numbers ±1, ±2, ±3, ±4, ±5 taking two at a time is
a) 165 b) -55 c) 55 d) none
89) The sum of the series 1/log₂4 + 1/log₄4 + 1/log₈4 +....+ 1/log₂ⁿ 4 is
a) n(n +1)/2 b) n(n +1)(2n +1)/2 c) 1/n(n +1) d) n(n +1)/4
90) If ⁿₙ₌₁∑n, √10/3. ⁿₙ₌₁∑n², ⁿₙ₌₁∑n³ are in GP the value of n is
a) 2 b) 3 c) 4 d) none
91) The value of ⁿᵣ₌₁ ∑ {(2r -1)a + 1/bʳ} is equal to
a) an²+ (bⁿ⁻¹ -1)/(bⁿ⁻¹(b -1))
b) an² + (bⁿ -1)/(bⁿ(b -1))
c) an³ + (bⁿ⁻¹ -1)/(bⁿ(b -1)) d) none
92) If aₙ = ⁿₙ₌₁∑ (1+ 2+ 2²+.... To n terms)/2ⁿ then sₙ is equal to
a) 2ⁿ - (n +1) b) 1 - 1/2ⁿ c) n - 1 + 1/2ⁿ d) 2ⁿ -1
93) Let Sₙ denote the sum of the cubes of the first n natural numbers and sₙ denote the sum of the first n natural numbers. Then ⁿᵣ₌₁ ∑ Sᵣ/sᵣ is equal to
a) n(n+1)(n+2)/6
b) n(n+1)/2
c) (n²+ 3n+2)/6 d) none
94) It is known that ∞ᵣ₌₁ ∑ 1/(2r -1)² = π²/8. Then ∞ᵣ₌₁ ∑ 1/r² is equal to
a) π²/24 b) π²/3 c) π²/6 d) none
95) It is given that 1/1⁴ + 1/2⁴ + 1/3⁴+......to ∞= π⁴/90. Then 1/1⁴ + 1/3⁴ + 1/5⁴+......to ∞ is equal to
a) π⁴/96 b) π⁴/45 c) 88π⁴/90 d) none
96) If in a series tₙ ²⁰ₙ₌₁∑tₙ is equal to
a) (20! -1)/20! b) (21! -1)/21! c) 1/2(n -1)! d) none
97) If tₙ denotes the nth term of the series 2+3+6+11+18+.... then t₅₀ is
a) 49² -1 b) 49² c) 50²- 1 d) 50²+2
98) 2¹⁾⁴. 4¹⁾⁸. 8¹⁾¹⁶ ....is equal to
a) 1 b) 2 c) 3/2 d) none
99) The sum of n terms of the series
1²+ 2.2² + 3² + 2.4²+ 5² + 2.6²+... is n(n+1)²/2 when n is even. When n is odd, the sum is
a) n²(n +1)/2 b) n(n²-1)/2 c) 2(n +1)²(2n +1) d) none
100) If n is an odd integer greater than or equal to 1 then the value of n³ - (n -1)³ + (n -2)³ - .....+ (-1)ⁿ⁻¹. 1³ is
a) (n+1)²(2n -1)/4
b) (n -1)²(2n -1)/4
c) (n +1)²(2n +1)/4 d) none
101) Observe that
1³= 1, 2³= 3+5, 3³= 7+9+11, 4³= 13+15+17+19.
Then n³ as a similar series is
a) [2{n(n -1)/2 +1} - 1] + [2{n(n +1)/2 +1} + 1]+ .... [2{n(n +1)/2 +1} +2n -3]
b) (n²+ n +1)+ (n²- n +3)+ (n²- n +5)+ ....+ (n²+ 3n -1)
c) (n²- n +1)+ (n²- n +3)+ (n²- n +5)+ ....+ (n²+ n -1) d) none
102) Let tᵣ= 2ʳ⁾² + 2⁻ʳ⁾². Then ¹⁰ᵣ₌₁∑t²ᵣ is equal to
a) (2²¹ -1)/2¹⁰ + 20
b) (2²¹ - 1)/2¹⁰ + 19
c) (2²¹ -1)/2¹⁰ - 1 d) none
103) Let Sₖ = lim ₙ→∞ ⁿᵢ₌₀∑ 1/(k +1)ⁱ. Then ⁿₖ₌₁∑ k Sₖ equals
a) n(n +1)/2
b) n(n -1)/2
c) n(n +2)/2 d) n(n +3)/2
104) Let tₙ = n. (n!). Then ¹⁵ₙ₌₁∑tₙ is equal to
a) 15! -1 b) 15! + 1 c) 16! -1 d) none
105) The sum of
3/(1.2). 1/2 + 4/(2.3) . (1/2)² + 5/(3.4) . (1/2)³+ ....to n terms is equal to
a) 1- 1/(n+1)2ⁿ
b) 1- 1/(n. 2ⁿ⁻¹)
c) 1+ 1/(n +1)2ⁿ d) none
106) Let f(n)= [1/2 + n/100] where [x] denotes the integral part of x. Then the value of ¹⁰⁰ₙ₌₁∑ f(n) is
a) 50 b) 51 c) 1 d) none
107) Aᵣ ; r= 1,2,3,....n are n points on the parabola y²= 4x in the first quadrant. If Aᵣ = (xᵣ, yᵣ), where x₁, x₂, x₃, ....xₙ are in GP and x₁= 1, x₂= 2, then yₙ is equal to
a) (-2)⁽ⁿ⁺¹⁾/² b) 2ⁿ⁺¹ c) (√2)ⁿ⁺¹ d) (2)ⁿ⁾²
108) In the given square, a diagonal is drawn, and parallel line segments joining points on the adjacent sides are drawn on both sides of the diagonal. The length of the diagonal is n √2 cm. If the distance between consecutive line segment be 1/√2 cm then the sum of the lengths of all possible line segments and the diagonal is
a) n(n +1)√2 cm b) n² cm c) n(n +2)cm d) n²√2 cm
109) ABCD is a square of length a, a ∈ N, a > 1. Let L₁, L₂, L₃, ...be points on BC such that BL₁ = L₁L₂ = L₂L₃ = ....= 1 and M₁, M₂, M₃, ....be points on CD such that CM₁ = M₁M₂ = M₂M₃ = ....= 1. Then ᵅ⁻¹ₙ₌₁∑ (AL²ₙ + LₙM²ₙ) is equal to
a) (1/2) a(a -1)² b) (1/2) a(a -1)(4a -1)
c) (1/2) a(a -1)(2a -1)(4a -1) d) none
110) The sum of infinite terms of a decreasing GP is equal to the greatest value of the function f(x)= x³+ 3x - 9 in the interval [-2,3] and the difference between the first two terms is f'(0). Then the common ratio of the GP is
a) -2/3 b) 4/3 c) 2/3 d) -4/3
111) The lengths of three unequal edges of a rectangular solid block are in GP. The volume of the block is 216 cm³ and the total surface area is 252 cm². The length of the longest edge is
a) 12 cm b) 6cm c) 18cm d) 3 cm
112) ABC is a right angled triangle in which angle B =90° and BC= a. If n points L₁, L₂, ....Lₙ in AB are such that AB is divided in n+1 equal parts and L₁M₁ , L₂M₂, ....LₙMₙ are line segments parallel to BC and M₁, M₂, ....Mₙ are on AC then the sum of the lengths of L₁M₁, L₂M₂, ....LₙMₙ is
a) a(n +1)/2 b) a(n -1)/2 c) an/2 d) impossible to find from the given data
113) If AM of the numbers 5¹⁺ˣ and 5¹⁻ˣ is 13 then the set of possible real values of x is
a) {5,1/5} b) {1,-1} c) {x : x²-1= 0, x ∈ R} d) none
114) If the AM of two positive numbers between three times their geometric mean then the ratio of the numbers is
a) 3± 2√2 b) √2±1 c) 17+ 12√2 d) (3- 2√2)⁻²
115) If a,b,c are in HP then 1/(b - a) + 1/(b - c) is equal to
a) 2/b b) 2/(a+ c) c) 1/a + 1/c d) none
116) Sᵣ denotes the sum of the first r terms of an AP. Then S₃ₙ : (A₂ₙ - Sₙ) is
a) n b) 3n c) 3 d) independent of n
117) If aˣ= vʸ = cᶻ and x, y,z are in GP then log꜀v is equal to
a) logᵥa b) logₐv c) z/y d) none
118) The value of ⁿᵣ₌₁∑ 1/[√(a+ rx) + √{a + (r -1)x}] is
a) n/{√a + √(a+ nx)}
b) {√(a+ nx) - √a}/x
c) n/{√(a+ nx) - a}/x d) none
119) Let ⁿₙ₌₁∑r⁴ = f(n). Then ⁿᵣ₌₁ ∑ (2r -1)⁴ is equal to
a) f(2n) - 16 f(n) for all n ∈N
b) f(n) - 16 f{(n-1)/2} when n is odd
c) f(n) - 16 f(n/1) when n is even d) none
120) If 1. ⁿP₁, ⁿP₂, ⁿO₃ are three consecutive terms of an AP then they are
a) in GP b) in HP c) equal d) none
121) In a GP the product of the first four terms is 4 and the second term is the reciprocal of the fourth term. The sum of the GP up to infinite terms is
a) 8 b) -8 c) 8/3 d) -8/3
122) If ⁿ ₖ₌₀ ∑(ᵏₘ₌₀ ∑ m¹)= an⁴ + bn³+ cn²+ dn + e then
a) a=1/12 b= 1/6 c) d= 1/6 d) e=0
123) If a,b,c, d are four positive numbers then
a) (a/b + b/c)(c/d + d/e)≥ 4 √(a/e)
b) (a/b + c/d)(b/c+ d/e)≥ 4 √(a/e)
c) a/b + b/c + c/d + d/e + e/a ≥ 5
d) b/a+ c/b + d/c + e/d + a/e ≥ 1/5
124) Let f(x)= +1- xⁿ⁺¹)/(1- x) and g(x)= 1- 2/x + 3/x² - ...... + (-1)ⁿ (n+1)/xⁿ. Then the constant term in f'(x) x g(x) is equal to
a) n(n²-1)/6, when n is even
b) n(n+1)/2, when n is odd
c) -n(n+1)/2, when n is even
d) -n(n -1)/2, when n is odd
125) Let aₙ = product of the first n natural numbers. Then for all n∈N
a) nⁿ ≥ aₙ b) {n+1)/2}ⁿ≥ n! c) nⁿ ≥ aₙ₊₁ d) none
126) Let the set A={2,4,6,8,....} And B={3,6,9,12,....} and n(A)= 200, n(B)= 250. Then
a) n(A∩B)= 67 b) n(A∪B)= 450 c) n(A∩B)= 66 d) n(A∪B)= 384
127) Let a, x, b be in AP; a, y, b be in GP and a, z, b in HP. If x= y +2 and a= 5z then
a) y²= xz b) x> y > z c) a= 9, b= 1 d) a= 1/4, b= 9/4
128) S₁, S₂, S₃, ......be squares such that for each n ≥ 1, the length of a side of Sₙ equals the length of a diagonal of Sₙ₊₁. If the length of a side of S₁ is 10cm the pn for which of the following values of n is the area of Sₙ less than 1 cm² ?
a) 7 b) 8 c) 9 d) 10
129) Three positive numbers form a GP. If the middle number is increased by 8, the three numbers form an AP. If the last number is also increased by 64 along with the previous increase in the middle number, the resulting numbers form a GP again. Then
a) common ratio= 3
b) first number= 4/9
c) common ratio= -5
d) first number= 4
130) If a, b, c are in GP and a, p, q are in AP such that 2a, b+ p, c+ q are in GP then the common difference of the AP is
a) √2 a b) (√+1)(a - b) c) √2(a+ b) d) (√2-1)(b - a)
131) If x, y, z are positive numbers in AP then
a) y²≥ xz b) y≥ 2√(xz)
c) (x + y)/(2y - x) + (y+ z)/(2y - z) has the minimum value 2
d) (x + y)/(2y - x) + (y+ z)/(2y - z)≥ 4
132) Between two unequal numbers, if a₁, a₂ are two AMa; g₁, g₂ are two GMs and h₁, h₂ are two HMs then g₁. g₂ is equal to
a) a₁h₁ b) a₁h₂ c) a₂h₂ d) a₂h₁
133) The number 1,4,16 can be three terms (not necessarily consecutive) of
a) no AP
b) only one GP
c) infinite number of APs
d) infinite number of GPs
SAP- 1
1) If α and β are the roots of ax²+ bx + c= 0, find the values of following:
a) 1/(aα + b) + 1/(aβ + b). b/ac
b) β/(aα + b) + α/(aβ + b). -2/a
c) (aα + b)⁻³ + (aβ + b)⁻³. (b³- 3abc)/a³c³
d) (aα + b)⁻²+ (aβ + b)⁻². (b²- 2ac)/a²c²
2) If α and β are the roots of the equation ax² + bx + c=0, find the equation whose roots are as given below:
a) 1/(α + β), 1/α + 1/β. bcx²+ (b²+ ac)x + ab=0
b) α/β, β/α. acx² - (b²+ 2ac)x + ac=0
c) α + 1/β, β + 1/α. acx²+ b(a+ c)x + (a+ c)²=0
d) α²+ β², 1/α² + 1/β². a²c²x²- (b²- 2ac)(a²+ c²)x + (b²- 2ac)²=0
e) 1/(aα + b), 1/(aβ + b). acx²- bx +1= 0
3) α ≠ β, but α²= 5α -3, β²= 5β- 3, find the equation whose roots are α/β and β/α. 3x²- 19x +3=0
4) If α, β are the roots of x²+ ax + b = 0, then show that α/β is a root of the equation bx² + (2b - a²)x + b = 0.
5) If α and β are the roots of x² - p(x +1) - c = 0, show that (α +1)(β+1)= 1+ c. Hence show that (α²+ 2α +1)/(α²+ 2α +c) + (β² + 2β +1)/(β² + 2β +c) = 1.
6) In a triangle PQR, angle R=π/2. If tan(P/2) and tan(Q/2) are the roots of the equation ax² + bx + c = 0, (a≠0), the pn
a) a+ b = c b) b+ c = a c) a+ c = b d) b= c. a
7) If sinθ and cosθ are the roots of the equation lx²+ mx + n = 0, then show that l²- m²+ 2ln = 0
8) If one root of the equation ix² - 2(i + 1)x + (2- i)= 0 is 2- i, then show that the other root is - i.
9) Find the roots of the equation 8sec²x - 6 secx + 1= 0. No roots, because 2 or 4 not possible
10) Find the roots of the equations a(b - 2c)x² + b(c - 2a)x + c(a - 2b)= 0 if ab + bc + ca = 0. 1, c(a - 2b)/a(b - 2c)
8a) If the roots of the equation (x - a)(x - b) - k = 0 be c and d, then show that the roots of the equation
(x - c)(x - d)+ k = 0 are a and b.
b) If α, β are the roots of the equation (x - a)(x - b)+ c= 0
Find the roots of the equation
(x - α)(x - β)= c.
9) a) Ramesh and Mahesh solve an equation. In solving Ramesh commits a mistake in constant term and finds the roots 8 and 2. Mahesh commits a mistake in the coefficient of x and finds the roots -9 and -1. Find the correct roots. 9,1
b) Two candidates attempt to solve a quadratic of the form x² + px + q = 0. One starts with wrong value of p and finds the roots to be 2 and 6. The other starts with a wrong value of q and finds the roots to be 2, -9. Find the correct roots. -3,-4
c) The coefficient of x in the quadratic equation x² + px + q=0 was taken as 17 in place of 13, its roots were found to be -2 and -15. Find the roots of the original equation. -10, -3
10a) Show that A. M of the roots of x² - 2ax + b² = 0 is equal to the geometric mean of the roots of the equation x² - 2bx + a²= 0, and vice versa.
b) Let p and q be roots of the equation x²- 2x + A=0 and let r and s be the roots of the equation x²- 18x + B=0.
If p< q < r < s are in arithmetic progression, then A= ..... and B= ... -1,3,7,11
11) a) Given that α, γ are roots of the equation Ax²- 4x +1= 0 and β, δ the roots of the equation Bx²- 6x +1=0, find the values of A and B such that α, β γ and δ are in HP. 3,8
b) The number of quadratic equations which are unchanged by squaring their roots is
a) 2 b) 4 c) 6 d) none. 4
12a) If α and β are the roots of the equation, 2x²- 3x - 6=0, find the equation whose roots are α²+2, β²+2. 4x²- 49x + 118=0
b) The roots of the equation 8x²- 10x +3 =0 are α and β² where β²> 1/2 then the equation whose roots are ( α + iβ)¹⁰⁰ and ( α - iβ)¹⁰⁰ is
a) x²- x +1=0 b) x²+ x +1=0 c) x²- x -1=0 d) x² + x -1=0 b
13) If α and β are the roots of the equation x²- 2x +3=0, find the equation whose roots are
a) α+2, β+2. x²- 6x+11=0
b) (α-1)/(α+1) , (β-1)/(β+1). 3x²- 2x +1=0
14)a) If α be a root of the equation, 4x²+ 2x -1=0, show that 4α³- 3α is the other root.
b) Form a quadratic equation whose roots are a/(√(a) ± √(a - b)). x²- 3x +2=0
c) α, β are the roots of the equation x²- 2x +3=0. Determine the equation whose roots are P= α²- 3α²+ 5α -2 and Q= β²- β²+ β+5. x²- 3x +2=0
15) Let a, b, c be real numbers with a≠ 0 and let α, β be the roots of the equation ax²+ bx + c=0.
Express the roots of a³x²+ abcx + c³= 0 in terms of α, β. α'= α²β, β'= αβ²
16 a) Let α, β be the roots of ax²+ bx + c=0 and γ, of lx²+ mx + n=0, then find the equation whose roots are αγ+ βδ and αδ + βγ. x²- Sx + P= 0
b) If α + β= 3 and α³+ β³=7, then α and β are the roots of 9x²- 27x +20=0.
c) Find a quadratic equation whose roots α and β are connected by relation α +β= 2 and (1- α)/(1+β) + (1- β)/(1+ α) = 2{(4λ²+15)/(4λ²-1)}. x²- 2x - (4λ²+11)/4 = 0.
17) a) If α, β are the roots of the equation x²- px + q=0, then find the equation the roots of which are (α²- β²)(α³- β³) and α³β²+ α²β³. t²- St + P= 0
b) If α, β are the roots of the equation x²- bx + c=0 then find the equation whose roots are (α²+ β²)(α³+ β³) and α⁵ β³+ α³ β⁵- 2 α⁴β⁴. x²- Sx+ P= 0, where S= p+ q and P= pq
c) Find the equation whose roots are (α+ β)² and (α-β)², where α and β are the roots of 2x²+ 2(m + n)x + (m²+ n²)= 0. x²- 4mnx - (m²- n²)²= 0.
18) a) If α, β be the roots of x²- px + q=0 and α', β' be those of x²- p'x + q' =0. Find the value of
(α - α')²+ (β - α')² + (α - β')¹+ (β -β')². 2[p²- 2q + p'²- 2q' - pp']
b) If α, β are the roots of the equation 6x²- 6x +1=0 then show that (1/2) (p+ qα + rα²+ sα³)+ (1/2) (p + qβ+ rβ²+ sβ³) is p/1 + q/2 + r/3 + s/4.
19) a) If α, β be the roots of ax²+ 2bx + c = 0 and α+ δ, β+ δ be those of Ax²+ 2Bx + C= 0, then show that (b²- ac)/(B²- AC)= (a/A)².
Another form
The two quadratic equation ax²+ bx + c= 0 and lx²+ mx + n = 0 have roots α, β and δ, γ respectively. If α, β, δ, γ be in AP and ∆₁ and ∆₂ be the discriminants of these quadratics, then show that ∆₁/∆₂ = a²/l².
b) The ratio of the roots of the equation ax²+ bx + c=0 is same as the ratio of the roots of the equation Ax²+ Bx + C=0. If D₁ and D₂ are the discriminants of ax² + bx + c= 0 and Ax² + Bx + C=0 respectively, then show that D₁ : D₂ = b² : B².
20) Let α, β are the roots of x²- x + p=0 and δ, γ be the roots of x²- 4x + q=0. If α β, δ, γ are in GP then the integral values of p and q respectively, are
a) -2,-32 b) -2,3 c) -6,3 d) -6,-32 a
21) a) If α, β be the roots of x²+ px + q=0 and γ, δ the roots of x²+ px + r=0, show that (α - γ) (α- δ)= (β- γ)(β- δ)= q+ r.
b) If If α, β be the roots of x²+ px + 1=0 and γ, δ the roots of x²+ qx + 1 =0, show that (α - γ) (β- γ)(α + δ)(β+ δ)= q² - p²
22)a) If the roots of the equation px²+ qx +2=0 are reciprocals of each other, then
a) p=0 b) p= -2 c) p= ±2 d) p= 2. d
b) Let P, Q, R be defined as
P= a²b + ab² - a²c - ac²,
Q= b²c + bc² - a²b - ab²
R= a²c + c²a - c²b - cb²
Where a, b, c are all +ve and the equation Px²+ Qx + R=0 has equal roots then a, b, c are in
a) AP b) GP c) HP d) none. c
23) Find the condition that the roots of the equation ax²+ bx + c be such that
a) one root is n times the other. nb²= ac(n +1)²
b) one root is three times the other. 3b²= 16ac
c) both roots are equal. b²= 4ac
24) If the roots of the equation ax²+ bx + c=0 are of the form (k +1)/k and (k+2)/(k+1), show that (a+ b + c)² - 4ac.
25) Let a, b, c, d are real numbers in GP. If u, v, w satisfy the system of equations u+ 2v + 3w = 6, 4u + 5v + 6w = 12, 6u + 9v = 4
Then show that the roots of the equation (1/u + 1/v + 1/w)x²+ [(b - c)²+ (c - a)²+ (d - b)²]x + u+ v + w = 0 and 20x² + 10(a - d)²x - 9=0 are reciprocals of each other.
26) a) If one root of the equation ax²+ bx + c =0 be the square of the other, then show that b³+ ac²+ a²c = 3abc.
b) If one of the equation x²+ px + q=0, is square of the other then show that p³ - q(3p -1) + q²= 0.
c) For the equation 3x²+ px + 3 =0, p> 0, if one of the roots is square of the other, then p is equal to
a) 1/3 b) 1 c) 3 d) 2/3 c
27)a) If x= 2+ 2²⁾³ + 2¹⁾³, then the value of x³- 6x² + 6x is ____ 2
b) If one root of the equation ax²+ bx + c=0 is equal to the nth power of the root, then show that
(acⁿ)¹/⁽ⁿ⁺¹⁾ + (aⁿc)¹/⁽ⁿ⁺¹⁾ + b =0
28a) If the roots of the equation 1/(x + p) + 1/(x + q) = 1/r are equal in magnitude but opposite in sign show that p+ q = 2r and that the product of the roots is equal to -(1/2) (p²+ q²).
b) If the roots of the equation 3x²+ 2(k²+1)x + (k²- 3k +2)= 0 be of opposite signs, then show that 1< k < 2.
29) a) Solve the equation:
{(x - b)(x - c)}/{(a- b)(a - c)} + {(x - c)(x - a)}/{(b- c)(b - a)} + {(x - a)(x - b)}/{(c- a)(c - b)} = 1. -(1/2)(p²+ q²)
b) If the equation (k²- 5k + 6)x²+ (k²- 3k +2)x + (k²- 4)= 0 is satisfied by more than two values of x, then determine the value of k. 2
30) a) If the sum of the roots of ax²+ bx + c =0 be equal to sum of their squares show that 2ac = ab+ b².
b) If the sum of the roots of the equation ax²+ bx + c= 0 is equal to sum of the squares of their reciprocals, then show that bc², ca², ab² are in AP or c/b, b/a, a/c are in HP.
31) a) α, β are the roots of the equation λ(x²- x)+ x + 5 = 0. If λ₁ and λ₂ are the two values of λ for which the roots α, β are connected by the relation α/β + β/α = 4/5, find the value of λ₁/λ₂ + λ₂/λ₁ . 254
b) If α, β be the roots of the equation λ²(x²- x)+ 2λx +3=0 and λ₁, λ₂ be the two values of λ for which α and β are connected by the relation α/β + β/α = 4/3 then find the equation whose roots are λ₁²/λ₂ and λ₂²/λ₁. λ²- 4λ- 6=0, (λ≠ 0)
32) a) If the ratio of the roots of the equation x²+ px + q=0 be equal to ratio of the roots of x²+ lx + m =0, then show that p²m = l²q.
b) If the ratio of the roots of a₁x²+ b₁x + c₁= 0 be equal to the ratio of the roots of a₂x²+ b₂x + c₂ = 0, then show that a₁/a₂, b₁/b₂, c₁/c₂ are in GP.
33) a) If α, β are the roots of the equation x +1= λx(1- λx) and λ₁, λ₂ be the two values of λ determined from the equation α/β + β/α = π -2, show that λ₁²/λ₂²+ λ₂²/λ₁²+2= 4{(π+1)/(π-1)}².
b) If the ratio of the roots of the equation lx²+ nx + n =0 be p: q, then show that √(p/q) + √(q/p) + √(n/l)= 0
34) a) Find the value of p for which x+1 is a factor of x⁴+ (p -3)x³ - (3p -5)x²+ (2p -9)x + 6. Find the remaining factors for this value of p. 4, (x+1)(x-1)(x+3)(x-2)
b) If x²- 3x +2 is a factor of x⁴- px² + q=0. Prove p=5, q= 4.
35) a) The roots x₁ and x₂ of the equation x² + px + 12=0 possesses the property x₁ - x₂ = 1. Find the value of p. ±7
b) Knowing that 2 and 3 are the roots of the equation 2x³+ mx²- 13x + n=0, determine m and n and find third root of the equation. -5,30,
36) a) If x²+ x²/(x +1)²= 3 and x be real, then show that x= (1±√5)/4.
b) If x²+ x+ 1 is a factor of ax³+ bx²+ cx + d, then show that the real root of ax³+ bx²+ cx + d=0 is -d/a.
37) If x= 2+ i √3 then find the value of
a) 4x²+ 8x +35. 55+24√3 i
b) If 2+ i √3 is a root of x²+ px + q=0 where p, q are real then (p,q)= (.....). -4,7
38) a) If x= 1+ 2i then show that x³+ 7x²- 13x +16= - 29.
b) Find the equation one of whose roots is 2+ √3 and hence find the value of expression x³- 7x²+ 13x -2 for x= 2+ √3.
c) Find all the roots of the equation 4x⁴- 24x³+ 57x²+ 18x - 45=0 if one of them is 3+ i √6. 3± i√6, ±(√3/2)
39)a) Let α + iβ, α, β ∈ R, be a root of the equation x³+ qx + r=0, q, r ∈R. Find a real cubic equation, independent of α and β, whose one root is 2α. t³+ qt - r =0
b) Find a quadratic equation whose one root is square root of -47 + 8 √3. x²±2x +49=0
40) Solve:
a) x³- 13x²+ 15x +189= 0 if one root exceeds other by 2. 7,9,-3
b) x⁴- 2x³+ 4x²+ 6x -21=0 if two of its roots are equal in magnitude but opposite in sign. ±√3, 1± i √6
c) If the sum of two roots of the equation 4x⁴- 8x³- 13x²+ 2x +3=0 is zero, find all its roots. ±1/2,3,-1
41) Solve: x⁴- 2x²+ 8x - 3=0. -1±√2, -1±√2 i
42) If 1, a₁, a₂, .....aₙ₋₁ are the n, nth roots of unity, then show that
(1- a₁)+1- a₂)+1- a₃).....(1- aₙ₋₁)= n.
43) a) If α and β be the roots of the equation x² - ax + b=0 and Vₙ = αⁿ + βⁿ, then show that Vₙ₊₁ = aVₙ - bVₙ₋₁. Hence obtain the value of α⁵ + β⁵. a⁵- 5a³b + 5ab²
b) If α, β are the roots of x²+ px + q=0 and also of x²ⁿ + pⁿxⁿ + qⁿ =0 if α/β, β/α are the roots of xⁿ + 1+ (x +1)ⁿ= 0, then show that n must be an even integer.
44) Let f(x)= Ax² + Bx + C where A, B, C are real numbers. Show that if f(x) is an integer whenever x is an integer, then the numbers 2A, A+ B and C are all integers. Conversely, show that if the numbers 2A, A+ B and C are all integers then f(x) is an integer whenever x is an integer.
45)a) Let a, b, c be real. If ax²+ bx + c=0 has two real roots α and β, where α< -1 and β> 1, then show that
1+ c/d + |b/a|< 0
b) If the roots of the equation x²- 2ax + a²+ a -3=0 are real and less than 3, then
a) a< 2 b) 2≤ a ≤3 c) 3< a ≤ 4 d) a > 4. a
c) Find the values of real parameter 'a' for which the equation
(tan²θ+1)²+ 4a(tan²θ +1)tanθ + 16 tan²θ =0 has four distinct roots in (0,π/2). (-5/2,-2)
46) a) if a+ b+ c=0, then the equation 3ax²+ 2bx + c=0 has atleast one root in (0,1). 3ax²+ 2bx + c
b) If 2a+ 3b + 6c=0 (a,b, c ∈R) then show that the equation ax²+ bx + c =0 has atleast one root in [0,1].
47) a) If b> a, then the equation (x - a)(x - b) -1=0, has
a) both roots in [a,b]
b) both roots in (-∞,a)
c) both roots in (b, + ∞)
d) one root in (-∞, a) and other in (b, +∞). d
b) If α and β (α < β) are the roots of the equation x²+ bx + c=0, where c< 0 < b, then
a) 0<α<β b) α<0< β |α| c) α< β< 0 d) α<0<|α|<β
48) a) show that the value of λ for which 2x² - 2(2λ+1)x + λ(λ+1)= 0 may have one root less than λ and other root greater than λ are given by λ > 0 or λ < -1.
b) For the equation
x²- (k +1)x + (k²+ k -8)= 0 if one root is greater than 2 and other is less than 2. Then show that k lies between -2 and 3.
49) a) If a,b,c are real numbers, a≠ 0. If α is a root of a²x²+ bx+ c= 0, β is a root of a²x²- bx - c= 0 and 0<α<β, then the equation a²x²+ 2bx + 2c= 0 has a root γ that always lies between α and β.
b) If 1 lies between the roots of the equation 3x²- 3 sinαx - 2 cos²α = 0 then α lies in the interval
a) (0,π/2) b) (π/12,π/2) c) (π/6,5π/6) d) (π/6,π/2) U (π/2,5π/6)
50) Let -1≤ p ≤ 1. Show that the equation 4x³- 3x - p= 0 has a unique root in the interval [1/2,1] and identify it.
Sap-2
1) Find the condition that in the equations ax²+ bx + c=0 and a'x²+ b'x + c'= 0
a) one root be common. (ca' - ac')/(ab' - a'b)
b) A root of first be reciprocal of a root of the second. (cc' - aa')²= (ba' - cb')(ab' - bc')
c) both have the same pair of roots. a/a' = b/b' = c/c'
2) a) If the equations x²+ px + q=0 and x²+ p'x + q' =0 have a common root show that it must be equal to (pq' - p'q)/(q - q') or (q - q')/(p' - p)
b) If the equations f(x) ≅ ax³ + 3bx² + 3cx + d= 0 and g(x)≅ ax² + 2bx + c=0 have a common root, then show that (bc - ad)² = 4(ac - b²)(bd - c²).
3a) Find k if the equations 4x² - 11x + 2k = 0 and x² - 3x - k = 0 have a common root and obtain the common root for this value of k. 17/6
b) Determine a such that x²- 11x + a and x²- 14x + 2a may have a common factor. 0,24
4a) Find the value of a solid that the equations (2a -5)x²- 4x - 15= 0 and (3a - 8)x² - 5x - 21= 0 have a common root. 4,8
b) If the equations x² - x - p= 0 and x² + 2xp - 12=0 have common root, find it. 2
c) Find the condition on the complex constant α, β if z²+ αz + β=0 has real roots. (β - conj β)²= (conj of α - α)(α. Conj of β - conj of α.β)
5) If a, b, c are in GP, then the equations ax² + 2bx + c=0 and dx² + 2ex + f =0 have a common root if d/a, e/b, f/c are in
a) AP b) GP c) HP d) none a
Another form:
If ax² + 2bx + c=0 and px² + 2qx + r=0 have a common root a/p, b/q, c/r in AP, then show that p, q, r are in GP
6) a) If α, β are the roots of x² + px + q=0 and δγ are the roots of x² + rx + s=0, evaluate (α-γ)(α-δ)(β-γ)(β- δ) in terms of p, q, r, s.
Deduce the condition that the equations have a common root. (q - s)²= (p - r)(qr - ps)
b) Eliminate x from the equations a+ c = (b/x) - dx , a - c = (d/x) - bx.
7) α and β are the roots of ax² + bx + c= 0 and f(x)= a₁x² + b₁x + c₁ then show that f(α). f(β)= 1/a² [(c₁a - ca₁)² - (ab₁ - a₁b)(bc₁ - b₁c)]²
8) If α, β are the roots of ax²+ bx + c =0. α₁, - β are the roots of a₁x² + b₁x + c₁ =0, show that α, α₁ are the roots of x²/{b/a + b₁/a₁)} + x + 1/{b/c + b₁/c₁} =0
9)a) If the equations x² + bx + ca =0 and x² + cx + ab =0 have a common root, then their other roots are the roots of the equation, x² + ax + bc=0.
b) If the equations x² + abc + c=0 and x² + acx + b =0 have a common root, then establish that their other roots are the roots of the equation x² - a(b + c)x + a²bc = 0.
10) If one root of the equation x² + ax + b=0 is a root of the equation x² + cx + d= show that the other roots satisfy the equation
x² + x(2a - c)+ (a² - ac + d)= 0.
11) a) If each pair of the three equations x² + p₁x + q₁= 0, x² + p₂x + q₂ = 0 and x² + p₃x + q₃ = 0 have a common root, then show that
p₁² + p₂² + p₃² + 4(q₁ + q₂ + q₃)= 2(p₁p₂ + p₂p₃ + p₃p₁).
b) If every pair of equations x² + ax + bc=0, x² + bx + ca=0, x² + cx + ab=0 have a common root, then find the sum and product of these common roots. -(1/2) ∑ a, abc
c) If the three equations x² + ax + 12=0, x² + bx + 15=0 and x² + (a+ b)x + 36=0 have a common possible root then find a and b and the roots. 7,-7 and 8,-8
12) if each pair of equations a₁x² + b₁x + c₁ = 0, a₂x² + b₂x + c₂ = 0 and a₃x² + b₃x + c₃ =0 have a common root, then show that
i) (c₁a₂ + c₂a₁)/(c₁a₂ - c₂a₁) + a₁a₂a₃/a₃+a₁b₂ - a₂b₁) = 0
ii) {(a₁b₂ - a₂b₁)/(a₁c₂ - a₂c₁)}² = a₁a₂c₃/c₁c₂a₃.
13) If the equations ax² + 2bx + c= 0, a'x² + 2b'x + c'=0 have a common root, show that the equation
(b² - ac)x² + (2bb' - ac' - a'c)x + (b'² - a'c')= 0 has equal roots.
14) a) If the equations x² + ax + b = 0 and x²+ bx + a=0 have a common root, then the numerical value of a+ b is ....-1
b) If the equation x² + bx + c =0 and bx² + cx +1=0 have a common root, then either b+ c+1=0 or b² + c² +1= bc + b + c
15) If the equations ax² + bx + c=0 and x³ + 3x² + 3x +2=0 have two common roots, then show that a= b= c.
16) a) If the roots of the equation, (c² - ab)x² - 2(a² - bc)x + (b² - ac)=0 be equal show that either a=0 or a³+ b³ + c³ = 3abc.
b) For what values of m the roots of the equation
x² - 2x(1+ 3m)+ 7(3+ 2m)=0 will be equal? 2, -10/9
17) a) If the two roots of the equation (λ-1)(x² + x +1)² - (λ+1)(x⁴ + x² +1)=0 are real and distinct then show that λ lies in the interval (-∞, -2) U (2,∞).
b) The equation (6- x)⁴ + (8- x)⁴= 16 has
i) sum of roots 28.
ii) product of roots 2688
iii) two real roots
iv) two imaginary roots.
c) Determine the nature of the roots of the equation
(x - a)³+ (x - b)³+ (x - c)³= 0. Ono real root and two imaginary roots
18) a) Determine the values of m for which the equation
5x²- 4x + 2+ m(4x² - 2x -1)=0 will have
i) equal roots. 1,-6/5
ii) product of roots as 2. -8/9
iii) sum of the roots as 6. -13/11
b) The product of the roots of the equation x² - 3λx + 2e²ˡᵒᵍλ -1=0. If the roots be real, then λ= 2.
19) a) If the roots of the equation, (b - c)x² + (c - a)x + (a - b)= 0 be equal, then show that a, b, c are in arithmetic progression.
b) If a(b - c)x² + b(c - a)x + c(a - b)=0 has equal roots, show that a, b, c are in harmonical progression.
20) a) Show that the roots of (x - a)(x - b)+ (x - b)+x - c)+ (x - c)+x - a)=0 are always real and they will be equal if and only if a= b = c.
b) Examine the nature of the roots of the equation (b - x)² - 4(a - x)(c - x)=0 where a,b, c are real. Real
21) a) The equation ax² + bx + c=0 where a, b, c are real numbers connected by the relation 4a+ 2b + c=0 and ab > 0 has real roots.
b) If a, b, c are positive and are in AP. Show that the roots of the equation ax² + bx + c =0 are real for|c/a -7|≥4√3.
22) a) Show that the roots of
(a² + b²)x² - 2b(a+ c)x + (b² + c²)=0 will be real if a, b, c are in GP and in this case these roots will be equal as well.
b) If a, b, c be distinct real and +ve numbers which are in HP, then the roots of the equation ax² + 2bx + c=0 are real and distinct. Is this statement true or not? No it is imaginary
23) a) Discuss the nature of the roots of the equation 4ax² + 3bx + 2c = 0 where a, b, c ∈ R and are connected by the relation a+ b + c= 0. Real
b) If the roots of the equation
[a²+ 2(1- b)]x²+ 2a(1+ b)x + 2b(b -1)+ a²=0
be equal then show that a²= 4b.
24)a) If the roots of the equation (a² + b²)x² -2(bc + ad)x + (c² + d²)= 0, be real, then show that they will be equal as well and then a/b = d/c.
b) Show that if the roots of the equation (a⁴ + b⁴)x² + 4abcdx + (c⁴ + d⁴)=0 are real then they can't be unequal.
c) If a, b, c be real then show that the roots of the equation 1/(x + a) + 1/(x + b) + 1/(x + c)= 3/4, x≠ 0 are real.
25) Discuss the nature of the roots of the equations:
a) (a+ c - b)x² + 2cx + (b + c - a)=0.
b) (b + c)x² - (a+ b + c)x + a=0.
c) 2(a² + b²)x² + 2(a+ b)x +1=0.
26) a)Show that the roots of the equation
(b + c - a)x² + (c + a - b)x + (a+ b - c)=0 are rational if a, b, c be all rationals such that a+ b+ c=0.
b) Show that the roots of the equation (a+ 2b - 3c)x² + (b + 2c - 3a)x + (c + 2a - 3b)=0 are rational, if a, b, c are rational. Hence determine the roots of the equation.
c) If (ax²+ bx + c)y + (a'x² + b'x + c')=0 and x is a rational function of y then show that (ac' - a'c')²= (ab' - a'b)(bc' - b'c)
27) If the roots of the equation x² - 2cx + ab=0 be real and unequal, then show that the roots of x² -2(a+ b)x + (a² + b²)+ 2c² =0 will be imaginary.
b) Consider the equations ax² + 2bx + c= 0 and (a+ c)(ax² + 2bx + c) - 2(ac - b²)(x² +1)=0.
If the roots of one are real (complex) then the roots of the other are complex (real).
28) a) If the roots of the equation x² - ax + b =0 are real and differ by a quantity which is less than c(c> 0), then b lies between (a² - c²)/4 and a²/4.
b) If the roots of the equation
(a-1)(x² + x +1)² = (a+1)(x⁴ + x² +1) are real and distinct then show that a² - 4> 0.
29) a) Show that if p,q,r,s are real numbers and pr= 2(q+ s), then atleast one of the equation x² + px + q=0, x² + rx + s =0 has real roots.
b) If P(x)= ax² + bx + c and
Q(x)= - ax² + dx + c where ac≠ 0 then P(x) Q(x)=0 has atleast two real roots.
30) If a< b < c < d, then the roots of the equation (x - a)(x - c)+ 2(x - b)(x - d)= 0 are real and distinct. T/F
31) a) The equation 2(log₃x)² - |log₃x|+ k = 0 has four solutions, determine the interval in which k lies. (0,1/8)
b) If α and β are the roots of x² + px + q =0 and α⁴, β⁴ are the roots of x² - rx + s=0, then the equation x² - 4qx + 2q² - r=0 has always two real roots.
32) a) If the roots of the equation x² + a² = 8x + 6a be real then show that a lies between -2 and 8.
b) Show that if the roots of 9x² + 4ax +4=0 are imaginary, then a must lie between -3 and 3.
c) The equation x² +2(m -1)x + m +5=0 has atleast one+ve root. Determine the range for m. (-∞, -1]
33) a) If a≠ 1 and a≠ -2, show that the roots of the equation
(a² + a -2)x² + (2a² + a +3)x + a² -1=0 are rational. Hence solve the equation. -(a-1)/(a+2), -(a +1)/(a -1)
b) Let A, B, C be three angles such that A=π/4 and tanB tanC= p. Find all possible values of p such that A, B, C are the angles of a triangle.
34) a) If p, q, r are real and p≠ q then roots of the equation
(p- q)x² + 5(p+ q)x - 2(p - q)= 0 are
a) real and equal b) complex c) real and unequal d) none. c
b) The roots of px² + 2qx + r=0 and qx² - 2√(pr) x + q=0 are simultaneously real then
i) p=q, r≠0 ii) p/q= q/r iii) 2q= ± √(pr) iv) none. ii
35) a) If the roots α, β if ax² + bx + c=0 be real then establish the relation between the coefficients under the following conditions:
i) roots are equal and opposite
ii) roots are of opposite signs.
iii) roots are both -ve
iv) roots are both+ve.
b) Let a> 0, b> 0, then both roots of the equation ax²+ bx + c=0
i) are real and negative
ii) have negative real parts
iii) none. i, ii
36) a) Show that the roots of
(a- b)²x² + 2(a + b - 2c)x +1= 0 are real or imaginary according as c does not or does lie between a and b, a< b.
b) If the roots of the equation
(m -3)x² - 2mx + 5m=0 are real and+ve then show that m∈ ]3,15/4]
c) If the equation x² +2(a +1)x + 9a -5=0 has only negative roots then show that a≥ 6.
d) If both the roots of the equation x² - 6ax + 2 - 2a + 9a² =0 exceed 3, then show that a> 11/9.
37) Find for what real values of x the expressions
a) x² - 2x -3. -1,3
b) 2x² - 5x -3 are positive or negative. -3,1/2
c) If x² + 2ax + 10 -3a > 0, for all x∈ R then
i) a< -5 ii) -5< a < 2 iii) a> 5 iv) 2< a <5. ii
38) a) For real values of x, show that the value of the expression (11x² +12x +6)/(x² + 4x +2) cannot lie between -5 and 3.
b) x² + (a - b)x + (1- a - b)=0, a, b∈ R.
Find the condition on a, for which both roots of the equation are real and unequal.
c) Determine the values of x which satisfies the inequalities
x² - 3x +2> 0 and x² -3x -4≤ 0.
39)a) If x be real show that the expression (x +2)/(2x² + 3x +6) takes all values in the interval [-1/13, 1/3].
b) Show that the value of
tanx/tan3x or (sinx cos3x)/(cosx sin3x)
Whenver defined never lies between 1/3 and 3.
40) a) Show that for real values of x the expression (x-1)+x+3)/(x -2)(x+4) cannot lie between 4/9 and 1
b) If x is real, the experiment(x² +2x -11)/(x -3) takes all values which do not lie between 4 and 12.
41) a) If x is real, find the maximum and minimum values of (x² + 14x +9)/(x² + 2x +3).
b) Show that for any real values of x the expression (x + a)/(x² + bx + c²) will have any value if b²> 4c² and a² + c²< ab.
42) a) If x is real show that (x² - x +1)/(x² + x +1) takes values from 1/3 to 3.
b) Show that (x² - 3x +4)/(x² + 3x +4) can never be greater than 7 nor less than 1/7 for real values of x.
c) Find out the range in which the value of the function (x² + 34x -71)/(x² +2x -7) lies for all real values of x. Justify your answer.
43) a) Find the values of x for which the following inequality holds
(8x² + 16x -51)/{(2x -3)(x +4)}> 3.
b) Find the values of x, which satisfy the inequality
(x -2)/(x +2) > (2x -3)/(4x -1).
44) Find the set of all x for which 2x/(2x²+ 5x +2)> 1/(x+1).
45) a) Show that if x is real, the expression (x² - bc)/(2x - b - c) has no real value ps between b and c.
4ˣ - 3. 2ˣ⁺³ + 128=0 are
15) If f(x)= x²+ 2bx + 2c² and g(x)= - x²- 2cx + b² are such that minimum f(x)> maximum g(x), then relation between b and c, is
16) if a, b are the roots of x²+ px +1= 0, and c, d are the roots of x²+ qx +1= 0, the value of
21) Let a> 0, b> 0 and c> 0. Then both the roots of the equation
24) If a, b,c are positive real numbers which are in GP , then the equation ax²+ 2bx + c= 0 and dx² + 2ex + f= 0 have common root if a/d, b/e, c/f are in
25) If P(x)= ax²+ bx + c and Q(x)= - ax²+ dx + c, where ac ≠ 0, then P(x) Q(x)= 0 has
29) let α, β be the roots of the equation (x - a)(x - b)= c with c≠ 0. then the roots of the equation (x - α)(x - β)+ c= 0 are
33) Let p and q be the roots of x²- 2x + A= 0 and r and s be the roots of x²- 18x + B= 0. If p< q < r< s are in AP, then ordered pair (A, B) is equal to
34) In a triangle PQR, angle R= π/2. If tan(P/2) and tan(Q/2) are the roots of the equation ax² + bx + c=0 where a≠ 0, then
36) For the equation 3x²+ px + 3= 0 , p> 0, if one of the roots is square of the other, than p is equals to
37) If the roots are the equation x² - 2ax + a²- 3 = 0 are real and less than 3, then
d) one root in (-∞,a) and other in (b, ∞).
39) Let α, β be the roots of x² - x + p = 0 and γ, δ be the roots of x² - 4x + q = 0 . If α, β, γ, δ are in GP then the integral value of p and q respectively, are
40) If a, b, c are not all equal and α and β be the roots of the equation ax² + bx + c = 0, then value of (1+ α+ α²)(1+ β+ β²) is
43) two complex numbers α and β are such that α + β = 2 and α⁴+ β⁴= 272, then the quadratic equation whose roots are α and β is
44) The equation (cos p -1)x² + (cos p)x + sin p = 0 in variable x has real roots, if p belongs to the interval
1/(x + a) + 1/(x + b) = 1/c are equal in magnitude but opposite in sign, then their product is
46) If the quadratic equations x²-11x + a= 0 and x²-14x + 2a= 0 have common root, then the values of a are
47) If α, β are the roots of the equation ax² + bx + c = 0, then the value of α³+ β³ is
48) If the sum of the roots of the quadratic equations ax² + bx + c = 0 is equal to the sum of the squares of their reciprocals, then
49) If the ratio of the roots of the equation x² + bx + c = 0 is the same as that of the ratio of the roots of x² + qx + r = 0, then
50) If a, b are the non zero distinct roots of x² + ax + b = 0, then the least value of x² + ax + b is
53) For real x, the function (x - a)(x - c)/(x - b) will assume all real values provided
54) Let a,b,c ∈R and a≠ 0. If α is a root of a²x²+ bx + c= 0, β is a root of a²x²- bx - c= 0 and 0< α < β, then the equation a²x²+ 2bx + 2c= 0 has a root γ that always satisfies
55) Suppose p,q,r,s ∈R and α, β be the roots of x²+ px + q= 0 and α⁴, β⁴ be the roots of x²- rx + s= 0, then the equation x²- 4qx + 2q² - r= 0 has always
57) Let f(x) be a quadratic expression which is positive for all x, if g(x)= f(x)+ f'(x) then for all real x,
58) If α, β are the roots of ax²+ bx + c= 0, then the quadric equation whose roots are 2α+3 and 2β+3 is
59) If α, β are the roots of the equation ax²+ bx + c= 0, then the equation whose roots are α³, β³ is
61) Let P(x) be a polynomial with integral coefficients . If there exist two integers a and b such that P(a) - P(b)= 1, then
f(x)= a²{(x - b)(x - c)}/{(a- b)(a - c)} + b²{(x - c)(x - a)}/{(b- c)(b - a)} + c²{(x - a)(x - b)}/{(c- a)(c - b)} is identically equal to
65) If 0< a < b< c < d, then the quadratic equations ax²+ {1+ a(b + c)}x + abc - d = 0 has
