1) The number of terms in the expansion of (1+3x+3x²+x³)⁶ is:
A) 8 B) 9 C) 19 D) 24
2) The number of terms in the expansion of:
(1+5x+10x²+10x³+5x⁴+x⁵)²⁰ is:
A) 100. B) 101. C) 120 D) N
3) The number of terms in (1+x)¹⁰¹(1+x²-x)¹⁰⁰ is:
A) 302. B) 301. C) 202. D) 101
4) If 7 divides ₃₂32³², the reminder is :
A) 1 B) 0 C) 4 D) 6
5) The number of terms free from the radical sign in the expansion of (1+ 3¹⁾³+ 7¹⁾⁷)¹⁰ is:
A) 1 B) 6 C) 11 D) none
6) The number of irrational terms in the expansion of (⁸√5 + ⁶√2)¹⁰⁰ is:
A) 97 B) 98 C) 96 D) 99
7) The number of rational terms in the expansion of (1+√2 +³√3)⁶ is:
A) 6 B) 7 C) 5 D) 8
8) The total number of terms in the expansion of (x+y)¹⁰⁰+(x-y)¹⁰⁰ after simplification is:
A) 50 B) 51 C) 202. D) N
9) The number of integral terms in the expansion of ((√5+⁶√7)⁶⁴² is:
A) 106 B) 108 C) 103 D) 109
10) The number of non zero terms in the expansion of (1+3√2x)⁹+ (1-3√2x)⁹ is:
A) 9 B) 0 C) 5 D) 10
11) The number of terms in the expansion of [a+4b)³(a-4b)³]² is:
A) 6 B) 7 C) 8 D) 32
12)(√3+1)⁴ +(√3 -1)⁴ equals.
A) a negative integer
B) a rational number
C) an irrational number. D) N
13) The expression: [x +√(x³ -1)]⁵ + [x - √(x³ -1)]⁵ is a polynomial of degree.
A) 5 B) 6 C) 7 D) 8
14) If x+ y=1, then
ⁿᵣ₌₀ ∑ r² ⁿCᵣ x y ⁿ⁻ʳ equals :
A) nxy B) nx(x+ny)
C) nx(nx+y) D) none
15) If n is a positive integer, which of the following two will always be integers:
A) (√2+1)²ⁿ +(√2 -1)²ⁿ
B) (√2+1)²ⁿ - (√2 -1)²ⁿ
C) (√2+1)²ⁿ⁺¹+(√2 -1)²ⁿ⁺¹
D) (√2+1)²ⁿ - (√2 -1)²ⁿ
(a) only A and B
(b) only A and C
(c) only A and D
(d) only B and C
16) If the coefficient of mth, (m+1)th and (m+2)th terms in the expansion of (1+x)ⁿ are in AP, then
A) n² + 4(4m+1)+ 4m² - 2= 0
B) n² + n(4m+1)+ 4m² + 2= 0
C) (n - 2m)² = n+2
D) (n + 2m)²= n+2
17) The value of x for which the 6th term in the expansion of:
A) [₂log₂(√9ˣ⁻¹+7) + 1/₂1/5 log₂(3ˣ⁻¹ +1)]⁷ is 84, is
A) 4 B) 3 C) 2 D) 1
18) If the third term in (x + ₓlog₁₀x)⁵ is 10⁶, then x maybe
A) 1 B) 10 C)1/√10⁷ D) 10²
19) The 5th term from the end in the expansion of : (x³/2 - 2/x)¹² is:
A) 7920/x⁴ B) -7920/x⁴
C) 7920x⁴ D) - 7920x⁴
20) The unit digit of 17¹⁹⁸³ + 11¹⁹⁶³ - 7¹⁹⁸³ is:
A) 1. B) 2. C) 3. D) 0
21) If the rth term in the expansion of (x/3 - 2/x²)¹⁰ contain x⁴, then r is equal to.
A) 2 B) 3 C) 4 D) 5
22) If the 6tg term in the expansion of[1/³√x⁸ + x² logx]⁸ is 5600, then x is equals:
A) 1 B) log 10 C) 10
D) x does not exist.
23) If three consecutive cofficient in the expansion of (1+x)ⁿ are in the ratio 1:3:5, then the value of n is :
A) 6 B) 7 C) 8 D) 9
24) If the coefficient of x⁷ in the expansion of (ax² +1/bx)¹¹ is equal to the coefficient of 1/x⁷ in (ax - 1/bx²)¹¹, then ab equals:
A) 1 B) 2 C) 3 D) 4
25) The coefficient of x¹⁷ in : (x-1)(x-2)(x-3)......(x-18) is:
A) 342 B) 171/2 C) -171 D) 684
26) if the coefficient of (2r+4)th and (r-2)th terms in (1+x)¹⁸ are equal, then r equals :
A) 12. B) 10. C) 8 D) 6
27) If a₁ , a₂ , a₃ , a₄ are coefficients of any four consecutive terms in the expansion of (1+x)ⁿ, then a₁/(a₁+ a₂) + a₃/(a₃+ a₄) equals:
A) a₂/(a₂+ a₃) B) 1/2 a₂/(a₂+a₃)
C) 2a₂/(a₂+ a₃) D) 2a₃(a₂ + a₃)
28) The greatest integers which divides the number 101¹⁰⁰ - 1 is:
A) 100. B) 1000
C) 10000. D) 100000
29) For integer n> 1, the digit at unit's place in the number ¹⁰⁰ ᵣ₌₀∑ r! + ₂2ⁿ is:
A) 0 B) 1 C) 2 D) none
30) If {x} denotes the fractional part of x, {3²⁰⁰/8} equals:
A) 1/8 B) 3/8 C) 5/8 D) none
31) The digit at unit's place in ₂9¹⁰⁰ is:
A) 2 B) 4 C) 6 D)8
32) The coefficient of x⁵ in the expansion of (x+3))⁶ is:
A) 18 B) 6 C)12. D) none
33) If in the expansion of (1+x)ᵐ (1-x)ⁿ , the coefficient of x and x² are 3 and - 6 respectively. then m is:
A) 6 B) 9 C) 12 D) 24
34) The sum of the coefficients in (1+x - 3x²)²¹⁴³ is:
A) 2²¹⁴³. B)) 0. C) 1. D) -1
35) The coefficient xᵏ in the expansion of:. 1+ (1+x) + (1+x)² +.. +(1+x)ⁿ is:
A) ⁿCₖ B) ⁿ⁺¹Cₖ C) ⁿ⁺¹Cₖ₊₂ D)none
36) The coefficient of 1/x⁷ in. (ax - 1/bx²)¹¹ is:
A) - ¹¹C₅(a⁵/b⁶). B) ¹¹C₆(a⁶/b⁵)
C) - ¹¹C₆(a⁶/b⁵) D) ¹¹C₆(a⁵/b⁶)
37) Given positive integers r> 1, n > 2 and the coefficients of (3r)th and (r+2)th terms in Binomial expansion of (1+x)²ⁿ are equal, then :
A) n= 2r B) n= 3r
C) n= r+1. D) none
38) The coefficient of x⁵³ in the expansion of ¹⁰⁰ₘ₌₀∑ ¹⁰⁰Cₘ(x-3)¹⁰⁰⁻ᵐ 2ᵐ is:
A) ¹⁰⁰C₄₇ B) ¹⁰⁰C₅₃
C) - ¹⁰⁰C₅₃ D) - ¹⁰⁰C₅₄
39) 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. ⁿ₂
40) In the expansion of (1+x+x³+x⁴)⁴, the coefficient of x⁴ is :
A) ⁴⁰C₄ B) ¹⁰C₄ C) 210 D) 310
41) The coefficient of x⁹⁹ in:
(x+1)(x+3)((x+5).....((x+199) is:
A) 1+2+3+......+99
B) 1+3++5+.......199
C) 1.3.5.......199. D) none
42) If in the expansion of (1+x)ⁿ. a, b , c are three consecutive Coefficient, then n equals:
A) (ac+ab+bc)/(b²+ac)
B) (2ac+ab+bc)/(b²-ac)
C) (ab+ac)/(b²-ac). D) none
43) if A and B are the coefficient of x ⁿ in the expansion of (1+x)²ⁿ and (1+x)²ⁿ⁻¹ respectively, then:
A) A= B. B) A= 2B. C)2A= B. D) N
44) if (1+ ax)ⁿ= 1+8x + 24x²+... then the value of a and n is:
A) 2, 4 B) 2,3 C) 3,6 D) 1, 2
45) Sum of all coefficient in the binomial expansion of (x² + x -3)³¹⁹ is
A) 1 B) 2. C) -. D) 0
46) If the coefficients of x⁷ and x⁸ in (2+ x/3)ⁿ are equal, then n is :
A)56 B) 55. C) 45 D) 15
47) Sum of coefficients of even powers of x in the expansion of (1+x+x²+x³)⁵ is
A) 256. B) 128 C) 512 D) 64
48) If n is a positive integers and Cₖ= ⁿCₖ, then the value of. ⁿₖ₌₁ ∑ k³ (Cₖ/Cₖ₋₁)² equals :
A) {n(n+1)(n+2)}/12
B) {n(n+1)²}/12
C) {n(n+2)²(n+1)}/12 D) none
49) In a Binomial expansion (1+x)ⁿ , n is a positive integer, the coefficient of 5th ,6th and 7th terms are in AP, then the value of n is:
A) 7 B) 5 C) 3 D) 10
50) in the expansion of (1+x)⁵⁰, the sum of the coefficient of odd powers of x is:
A) 0. B) 2⁴⁹. C) 2⁵⁰. D) 2⁵¹
51) If in the expansion of (a+x)ⁿ , P and Q represent the sum of odd and even terms respectively, then P² - Q² equals:
A) (a² - x²)ⁿ B) (a² - x²)²ⁿ
C) (a² + x²)²ⁿ D) (a² + x²)ⁿ
52) if the sum of the cofficient in the expansion of (a²x² - 2ax +1)⁵¹ vanishes, then a equals:
A) 2. B) -1. C) 1. D) -2
53) If (3√3+5)²ⁿ⁺¹ = p+f, where p is an integer and f is a proper fraction, then f(p+f) equals:
A) 5ⁿ⁺¹. B) 3²ⁿ⁺¹ C) 2²ⁿ⁺¹ D) 3²ⁿ⁺¹
54) If (1-x+x²)ⁿ= a₀+a₁x+ a₂x²+ ... + a₂ₙx²ⁿ. Then a₀+ a₂+ a₄+ a₂ₙ equals :
A) (3ⁿ+1)/2 B) (3ⁿ-1)/2
C) (1- 3ⁿ)/2 D) 3ⁿ + 1/2
55) 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
56) The larger of 99⁵⁰ + 100⁵⁰ and 101⁵⁰ is:
A) 99⁵⁰ + 100⁵⁰ B) both are equal
C) 101⁵⁰ D) none
57) which one is correct ?
A) 1999²⁰⁰⁰ > 2000¹⁹⁹⁹
B) 1998¹⁹⁹⁹ < 2000¹⁹⁹⁸
C) 100⁴¹ < 101⁴⁰
D) 26²⁵ < 25²⁶
58) find the value of x in the expansion [x+xˡᵒᵍ ˣ]⁵, if the third term of the expansion is 1000000
A) 10 B) 11 C) 12 D) none
59) the value of ¹⁵C₀² - ¹⁵C₁²+ ¹⁵C₂² - ...... - ¹⁵C₁₅² is:
A) 15 B) -15 C) 0 D) none
60) The value of C₀ + 3C₁ + 5C₂ + 7C₃ + .....+ (2n+1)Cₙ is:
A) 2ⁿ B) 2ⁿ+n. 2ⁿ⁻¹
C) 2ⁿ.(n+1). D) none
61) the value of C₁+4C₂+7C₃+....+(2n-2)Cₙ is :
A) (3n-4)2ⁿ⁺¹ B) (3n-4) 2ⁿ⁻¹+2.
C) (3n-4)2ⁿ D) (3n-4) 2ⁿ⁻¹ + 1.
62) C₀ - C₁+ C₂ - C₃+...+(-1)ⁿCₙ is equal to:
A) 2ⁿ B) 2ⁿ - 1 C) 0 D) 2ⁿ⁻¹.
63) value of 2C₀ + 2²/2 C₁+2³/3 C₂ + + .......+ 2¹¹/11 C₁₀ is
A) (3¹¹-1)/11 B) (2¹¹-1)/11
C) (11³-1)/11 D) (11²-1)/11
64) sum to (n+1) terms of the series :
C₀/2 - C₁/3 + C₂/4 - C₃/5 +... is:
A) 1/(n+1) B) 1/(n+2)
C) 1/{n(n+1)} D) none
65) if n is an integer, then: C₀²- C₁² + C₂² - C₃² ......+ (-1)ⁿCₙ² is:
A) ²ⁿCₙ B) (-1)ⁿ²ⁿCₙ
C) (-1)ⁿ ²ⁿCₙ₋₁ D) None
66) if (1+x)ⁿ= C₀+ C₁x + C₂x²+ ...Cₙxⁿ, then the value of:
C₁+ 2C₂+ 3C₃+ .....+ nCₙ is
A) n. 2ⁿ⁻¹ B) (n+1)2ⁿ
C) (n+1)2ⁿ⁻¹ D)(n+2)2ⁿ⁻¹
67) If (1+x)ⁿ= 1+ C₁x+ C₂x²+ .....+ Cₙxⁿ, then:
C₁² - 2C₂² + 3 c₃² - ....... - 2nC₂ₙ² is:
A) n² B) (-1)ⁿ⁻¹n
C) (-1) ⁿ⁻¹. n. ²ⁿ⁻¹Cₙ D) - n²
68) If C₀ , C₁ , C₂ , ......, Cₙ are binomial coefficient in the expansion of (1+x)ⁿ, then:
lim ₓ→∞[Cₙ -(2/3)Cₙ₋₁ +(2/3)²Cₙ₋₂ + ......+(-1)ⁿ (2/3)ⁿC ₀] is
A) 0 B) 1 C) -1 D) 2
69) m, n ,r are positive integers such that r< m,n , then:
ᵐCᵣ + ᵐCᵣ₋₁ⁿC₁+ ᵐCᵣ₋₂ ⁿC₂+.....+ ᵐC₁ⁿCᵣ₋₁+ ⁿCᵣ equals:
A) (ⁿCᵣ)² B) ᵐ⁺ⁿCᵣ
C) ᵐ⁺ⁿCᵣ+ ᵐCᵣ+ ⁿCᵣ D)none
70) 1/{1!(n-1)!} + 1/{3!(n-3)!} + 1/{5!(n-5)!} + ..... Equal:
A) 2ⁿ/n! B) 2ⁿ⁻² /n! C) 0. D) none
71) The middle term in the expansion of (1+x)²ⁿ is:
A) (2n)!xⁿ/n! B) (2n)! xⁿ⁺¹/{n!(n-1)!}
C) (2n)!/(n!)². D) (2n)!/{(n+1)!(n-1)!}
72) if in expansion of (2a -a²/4)⁹ the sum of the middle terms in S, then the following is true:
A) S= (63/32)a¹⁴(a+8)
B) S= (63/32)a¹⁴(a-8)
C) S= (63/32)a¹³(a-8)
D) S= (63/32)a¹³(a+8)
73) The middle term in the expansion of (x+ 1/x)²ⁿ, is:
A) {1.3.5...........(2n-3)}/n!
B) {1.3.5...........(2n-1)}/n!
C) {1.3.5...........(2n +1)}/n! D) none
74) middle term in the expansion of (1+3x+ 3x² + x³)⁶ is:
A) fourth B) third C)10th D)none
75) coefficient of the term independent of x in (2x- 3/x)⁶ is:
A) 4320 B) 216 C) -216 D) -4320
76) The fourth term in Binomial expansion of (x² - 1/x³)ⁿ is independent of x, when n is equal to
A) 2 B) 3 C) 4 D) none
77) in the expansion of (x² +2/x)ⁿ for positive integer n has a term independent of x, then n is:
A) 23. B) 18 C)16 D) 0
78) The term independent of x in the expansion of (2x + /3x)⁶ is:
A)160/9 B) 80/9 C)160/27 D)80/3
79) If the expansion of (x² +2/x)ⁿ for positive integer n has 13th term independent of x, then the sum of the divisors of n is:
A) 36 B) 38 C) 39 D) 32
80) the term independent of x in the expansion of (1+x)ⁿ(1+1/x)ⁿ is:
A) C₀²+ 2C₁² + ...(n+1)Cₙ²
B) (C₀+ C₁ + C₂+ ...Cₙ)²
C) C₀²+ C₁² + C₂x²+ ...Cₙ² D) none
81) If x= 1/3, then the greatest in the expansion of (1+4x)⁸ is the:
A) 4th term B) 5th term
C) 6th term D) third term
82) The largest term in the expansion of (3+2x)⁵⁰ where x= 1/5 is:
A) 5th B) 51st C) 7th D) 6 th
83) The greatest Coefficient in the expansion of the (1+x)²ⁿ⁺² is:
A) (2n)!/(n!)² B)(2n+2)!/[(n+1)!]²
C) (2n+2)!/[n!(n+1)!]
D) (2n)!/[n!(n+1)!]
84) The greatest value of the term independent of x in the expansion of (x sina + 1/x. Cos a)¹⁰, a belongs to R, is:
A) 2⁵ B) 10!/(5!)²
C) 1/2⁵. 10!/(5!)² D) none
85) if the largest interval to which x belongs so that the greatest term in (1+x)²ⁿ has the greatest coefficient is (10/11, 11/10), then n equals:
A) 9 B) 10 C) 11 D) none
86) Let aₙ = 1000ⁿ/n! for n ∈ ℕ. then aₙ is greatest when
A) n= 997. B) n= 998
C) n= 999 D) n= 996
87) if x is nearly equal to 1, then (mxᵐ - nxⁿ)/(m-n) equal:
A) xᵐ⁺ⁿ B) xᵐ⁻ⁿ C) xᵐ D) xⁿ
88) let R=(5√5+1)²ⁿ⁺¹ and f= R --[R], where [ ] denotes the greatest integer function, then R f is equal:
A) 4²ⁿ⁺¹ B) 4²ⁿ. C)4²ⁿ⁻¹ D) none
89) Remainder when 4¹⁰³ is divided by 125 is:
A) 1. B) 25. C) 18. D) 19
90) If the coefficient of the middle term of (1+x)²ⁿ⁺² is p and the coefficients of the middle term in the expansion (1+x)²ⁿ⁻¹ are q and r, then:
A) p+q=r. B) p+r= q
C) p= q+r. C) p+q+r= 0
91) ⁿᵣ₌₁∑(ʳ⁻¹∑ₚ₌₀ ⁿCᵣ ʳCₚ2ᵖ) is equal to
A)4ⁿ- 3ⁿ+1 B) 4ⁿ - 3ⁿ - 1
C) 4ⁿ - 3ⁿ +2 D) 4ⁿ - 3ⁿ
92) The term independent of x in the expansion of [(t⁻¹-1)x + (t⁻¹+1)⁻¹x⁻¹]⁸ is:
A) 56{(1-t)/(1+t)}³
B) 56{(1+ t)/(1- t)}³
C) 70{(1-t)/(1+t)}⁴
D) 70{(1+ t)/(1- t)}⁴
93) The 4th term in the expansion of [ ₓ1/(log x +1) ₊ ₓ 1/12]⁶ is 200 and x > 1, then x equals:
A) ₁₀√2 B) 10 C) 10⁴ D) none
94) Value of (18³+7³+3.18.7.25)/(3⁶ + 6.243.2+ 15.81.4 + 20.27.8 + 15.9. 16 + 20.27.8) is:
A) 1 B) 5 C) 25 D) 100
95) in the expansion of the following expression: 1+(1+x) +(1+x)²+.....+(1+x)ⁿ . the coefficient of xᵏ (0≤ k ≤ n) is:
A) ⁿ⁺¹Cₖ₊₁ B) ⁿCₖ
C) ⁿCₙ₋ₖ₋₁ D) none
96) Let n and k be positive integers such that n≥ {k(k+1))}/2. The number of solutions (x₁ , x₂, ...xₖ), x₁ ≥ 1 > x₂≥ 2, ....xₖ ≥ k,all integers satisfying (x₁ + x₂+ ...+xₖ= n, is
A) ᵐCₖ₋₁ B) ᵐCₖ₊₁
C) ᵐCₖ D) where m= 1/2 ( 2n - k² +k -2)
97) let n be an odd integers. If sin nx =ⁿᵣ₌₀∑bᵣ sinʳ x for every values of x, then:
A) b₀ = 1, b₁= 3 B)b₀ = 1, b₁= n
C) b₀= -1, b₁=n
D) b₀ =0, b₁= n²-3n+3
98) The coefficient of x⁵⁰ in the expression:(1+x)¹⁰⁰⁰ + 2x(1+x)⁹⁹⁹ + 3x²(1+x)⁹⁹⁸+ .....+ 1000 x¹⁰⁰⁰ is:
A) ¹⁰⁰⁰C₅₀ B) ¹⁰⁰¹C₅₀
C) ¹⁰⁰²C₅₀. D) ¹⁰⁰⁰C₅₁
99) The last term in the binomial expression of (³√2 - 1/√2)ⁿ is (1/3. 1/³√9)^log₃⁸
A) ¹⁰C₆ B) 2.¹⁰C₄ C) 1/2 ¹⁰C₄ D) N
100) The coefficient of x³y⁴z in the expansion of (1+x+y-z)⁹ is:
A) 2.⁹C₇.⁷C₄. B) -2. ⁹C₂. ⁷C₄
C) ⁹C₇. ⁷C₄ D) none
101) Let n be an odd natural number greater than 1. Then the number of zeros at the end of the sum 99ⁿ+ 1 is:
A) 3. B) 4. C) 2. D) none
102) If (1+x)¹⁰ = a₀+ a₁x + a₂x² + ...+ a₁₀x¹⁰, then (a₀ - a₂+ a₄- a₆+ a₈ - a₁₀)² + (a₁ - a₃ + a₅ - a₇ + a₉)² equals:
A) 3¹⁰ B) 2¹⁰ C) 2⁹ D) none
103) If a and d are two complex numbers, that the sum to (n+1) terms of the series:
aC₀ - (a-d)C₁+ (a+2d)C₂ - (a+3d)C₃ + ....is :
A) a/2ⁿ B) na C) 0 D) none
104) The term independent of x in the expression (1+x+2x³)(3x²/2 - 1/3x)⁹ is:
A) 7/18. B) 2/27
C) 7/18+ 2/17. D) 7/18 - 2/27
105) 101¹⁰⁰ -1 is divisible by:
A) 100. B) 101. C) 99. D) 1001
106) If 1/(! 11!) + 1/3!9! + 1/5!7! = 2ᵖ/q! and f(x+y) = f(x). f(y) for all x and y. f(1)= , f'(0)= 10, then
A) f'(p)=q B) f'(q)=p
C) f'(p)≠ f'(q) D) none
107) If (1+x)ⁿ= ⁿᵣ₌₀ ∑aᵣ xʳ and bᵣ = 1+ aᵣ/(aᵣ -1) and ⁿᵣ₌₁ ∑ br = (101)¹⁰⁰/100! then n equals.
A) 99 B) 100 C) 101. D) none
108) If (1+x++x²)ⁿ = a₀+ a₁x + a₂x² + a₂ₙ x²ⁿ, then
aₙ₋₃ aₙ₋₁ aₙ₊₁
aₙ₋₆ aₙ₋₃ aₙ₊₃ is
aₙ₋₁₄ aₙ₋₇ aₙ₊₇
A) equal to 0 B) greater than 0
C) less than 0 D) none of these
109) The coefficient of x²y², yzt² and xyzt in the expansion of (x+y+z+t)⁴ are in the ratio :
A) 4:2:1 B) 2:4:1 C)1:2:4 D) 2:3:4
110) The sum of the series:
1/(1!(n-1)!) + 1/3!(n- 3)!) + 1/(5!(n-5)!)+......+ 1/((n-1)!1!) is:
A) 2ⁿ⁻¹/(n-1)! B) 2ⁿ/((n-1)!
C) 2ⁿ⁻¹/(n!). D) 2ⁿ/n!
111) If the 7th terms from the beginning and the end in the expansion of (³√2 + 1/³√3)ⁿ are equal, then n equals:
A) 9 B) 12 C) 15 D) 18
112) If x denotes the fractional part of x, then {3²ⁿ/8}n belongs to N is:
A) 3/8 B) 7/8 C) 1/8 D) none
113) if n is even positive integer, then the condition that the greatest term in the expansion of (1+x)ⁿ may have the greatest Coefficient also is
A) n/(n+2) < x < (n+2)/n
B) (n+1)/n < x < n//(n+1)
C) n/(n+4)< x < (n+4)/n ..d) none
114) The number of terms whose values depends on x in the expansion of (x² - 2 + 1/x²)ⁿ is
A) 2n+1. B) 2n. C)) n. D) none
115) the coefficient of x³ in the expansion of (1-x+x²)⁵ is:
A) 10 B) - 20 C) -50 D) -30
116) the coefficient of x⁶ in ([(1+x)⁶ +(1+x)⁷+....+(1+x)¹⁵ is:
A) ¹⁶C₉ B) ¹⁶C₅ - ⁶C₅ C) ¹⁶C₆ -1 D) n
117) if the rth term is the middle term in the expansion (x² - 1/2x)²⁰, then the (r+3) th term is:
A) ²⁰C₁₄ .1/2¹⁴ . x B) ²⁰C₁₂ 1/2¹². x²
C) 1/2¹³ ²⁰C₇. x. D) none
118) If Cᵣ stands for ⁿCᵣ and ⁿᵣ₌₀ ∑ r. Cᵣ/Cᵣ₋₁ = 210 then n equal
A 19. B) 20 C) 21 D) none
119) the sum of the last term coefficients in the expansion of (1+x)¹⁹ when expanded in ascending powers of x is
A) 2¹⁸. B) 2¹⁹. C) 2¹⁸- ¹⁹C₁₀ D) n
120) the sum 1/2 ¹⁰C₀ - ¹⁰C₁ + 2¹⁰C₂ - 2² ¹⁰C₃ + .....+ 2⁹ ¹⁰C₁₀ equal
A) 1/2 B) 0 C) 1/2 3¹⁰. D) n
121) for a positive integer n, let a(n) = 1+ 1/2+ 1/3+ 1/4+....+ 1/2ⁿ -1 then
A)a(100)≤100 B) a(100)>100
C) a(200)≤100 D) a(200)>100
122) Let n belongs to N if +1+x)ⁿ= a₀ + a₁x + a₂x²+....+ aₙxⁿ and aₙ₋₃, a ₙ₋₂, a ₙ₋₁ are in AP., Then
A) a₁, a₂, a₃ are in AP
B) a₁, a₂, a₃ are in HP
C) n= 6. D) n= 14.
) The number of terms in the expansion of (1+3x+3x²+x³)⁶ is
A) 18. B) 9. C) 19. D) 24
) The number of distinct terms in the expansion of (x+y-z)¹⁶ is
A) 136. B) 153. C) 16 D) 17
) The number of irrational terms in the expansion of (⁸√5+⁶√2)¹⁰⁰
A) 97. B) 98 C) 96 D) 99
) The number of terms whose values depends on x in the expansion of (x² - 2+ 1/x²)ⁿ is
A) 2n+1 B) 2n C) n D) none
) The number of real negative terms in the binomial expansion of (1+ix)⁴ⁿ⁻², n belongs to N, x> 0, is:
A) n. B) n+1. C) n-1. D) 2n
) In the expansion of {x+√(x²-1)}⁶ + {x- √(x² - 1)}⁶, the number of terms is:
A) 7 B) 14 C) 6 D) 4
) The number of terms in the expansion of (x² +1+1/x²)ⁿ, n belongs to N, is
A) 2n B) 3n C) 2n+1 C) 3n+1
) The number of rational terms (1+√2+ ³√3)⁶ is.
A) 6 B) 7 C) 5 D) 8
) The number of terms with Integral coefficient in the expansion of (7¹⁾³ + 5¹⁾². x)⁶⁰⁰ is
A) 100 B) 50 C) 101. D) none
) The sum of the rational terms in the expansion x (√2+⁵√3)¹⁰ is:
A) 32 B) 9 C) 41 D) none
) The last term in the binomial expansion of (³√2 - 1/√2)ⁿ is 1/3.³√9)^log₃8 Then the 5th term from the beginning is:
A) ¹⁰C₆ B) 2.¹⁰C₄ C) 1/2.¹⁰C₄ D) n
) If the 4th term in the expansion of (px +1/x)ᵐ is 2.5 for all x belongs to R then
A) p= 5/2, m= 3 B) p= 1/2, m= 6
C) p= -1/2, m= 3 D) none
) In the expansion of (1+ax)ⁿ, n belongs to N, the coefficient of x and x are 8 and 24 respectively. Then
A) a= 2, n= 4 B) a= 4, n= 2
C) a= 2, n= 6. B) a= -2, n= 4
) In the expansion of (x³ - 1/x²)ⁿ, n belongs to N, if the sum of the coefficients of x⁵ and x¹⁰ is 0 then n is:
A) 25. B) 20. C) 15. D) none
) The coefficient of x²⁰ in the expansion of:
(1+x²)⁴⁰(x²+2+1/x²)⁻⁵ is:
A) ³⁰C₁₀ B) ³⁰C₂₅. C)1. D) none
) The coefficient of x⁸b¹⁰ in the expansion of (a+b)¹⁸ is:
A) ¹⁸C₈. B) ¹⁸C₁₀. C) 2¹⁸. D) none
) If the coefficients of the (m+1)th and the (m+3) term in the expansion of (1+x)²⁰ are equal then the value of m is:
A) 10. B) 8. C) 9. D) none
) The coefficient of x³ in the expansion of (1-x+x²)⁵ is:
A) 10 B) - 20 C) - 50 D) - 30
) If the coefficient of the 2nd, 3rd and 4th terms in the expansion of (1+x)ⁿ , n belongs to N, are in AP then n is:
A) 7 B) 14 C) 2 D) none
) The coefficient of x⁶ in [(1+x)⁶+ (1+x)⁷ + ...+ (1+x)¹⁵] is:
A) ¹⁶C₉ B) ¹⁶C₅ - ⁶C₅ C) ¹⁶C₆ -1 D) n
) The coefficient of x³y⁴z in the expansion of (1+x+y-z)⁹ is:
A) 2. ⁹C₇. ⁷C₄ B) -2. ⁹C₂. ⁷C₃
C) ⁹C₇. ⁷C₄. D) None
) The coefficient of x¹³ in the expansion of (1-x)⁵ (1+x+x²+x³)⁴
A) 4. B) -4. C) 0. D) none
) The coefficient of x⁶. y⁻² in the expansion of (x²/y - y/x)¹² is:
A) ¹²C₆ B) - ¹²C₅ C) 0 D) none
) The greatest value of the term independent of x in the expansion of (x sina + x⁻¹cos a)¹⁰, a belongs to R, is
A) 2⁵ B) 10!/(5!)²
C) 1/2⁵ . 10!/(5!)². D) none
)In the expansion of (x³ - 1/x²)¹⁵, the constant term is:
A) ¹⁵C₆ B) 0 C) - ¹⁵C₆ D) 1
) The constant term in the expansion of (1+x)¹⁰.(1+ 1/x)¹² is
A) ²²C₁₀ B) 0 C) ²²C₁₁ D) none
) The term independent of x in the expansion of (1-x)² (x+1/x)¹⁰
A) ¹¹C₅ B) ¹⁰C₅ C) ¹⁰C₄ D) N
) The middle term in the expansion of (2x/3 - 3/2c²)²ⁿ is
A) ²ⁿCₙ B) (-1)ⁿ (2n)!x⁻ⁿ/(n!)²
C) ²ⁿCₙ . 1/xⁿ D) none
) The middle term in the expansion of (1- 1/x)ⁿ.(1-x)ⁿ is
A) ²ⁿCₙ B) - ²ⁿCₙ
C) - ²ⁿC ₙ₋₁ D) none
) If the rth term is the middle term in the expansion of (x² - 1/2x)²⁰ then the (r+3)th term is
A) ²⁰C₁₄ . x/2¹⁴ B) ²⁰C₁₂. x²/2¹²
C) - 1/2¹³. ²⁰C₇. x. D) none
) Let n belongs to N and n < (√2+1)⁶. Then the greatest value of n is
A) 199 B) 198 C) 197 D) 196
) If the Coefficient of the 5th term be the numerically greatest coefficient in the expansion of (1-x)ⁿ then the positive Integral value of n is:
A 9 B) 8 C) 7 D) 10
) The greatest Coefficient in the expansion of (1+x)²ⁿ is.
A) {1.3.5....(2n-1)}n!/n!ⁿ B)²ⁿCₙ₋₁
C) ²ⁿCₙ₊₁. D) none
) Let n be odd natural numbers greatest than 1. Then the number of zeros at the end of the sum 99ⁿ +1 is:
A) 3 B) 4 C) 2 D) none
) Let f(n) = 10ⁿ + 3. 4ⁿ⁺²+5, n belongs to N. The greatest value of the integer which divides f(n) for all n is:
A) 27. B) 9. C) 3. D) none
) 2⁶⁰ when divided by 7 leaves the remainder
A) 1. B) 6. C) 5. D) 2
) If [x] denotes the fractional part of x then [3²ⁿ/8], n belongs to N,is
A) 3/8. B) 7/8. C) 1/8. D) none
) The sum of the coefficients in the binomial expansion of (1/x+2x)ⁿ is equal to 6561. The constant term in the expansion is
A) ⁸C₄ B) 16.⁸C₄ C) ⁶C₄ 2⁴. D) none
) The sum of the numerical coefficient in the expansion of (1+x/3 +2y/3)¹² is
A) 1. B) 2. C) 2¹² D) none
) The sum of the last ten coefficients in the expansion of (1+x)¹⁹ when Expanded in ascending powers of x is:
A) 2¹⁸ B) 2¹⁹ C)2¹⁸- ¹⁹C₁₀D) non
) The sum of the coefficients of x²ʳ, r= 1,2,3,..., in the expansion of (1+x)ⁿ is:
A) 2ⁿ B) 2ⁿ⁻¹ - 1
C) 2ⁿ - 1. D) 2ⁿ⁻¹ + 1
) The sum of the coefficients in the polynomial expansion of (1+x- 3x²)²¹⁶³ is:
A) 1. B) -1. C) 0. D) none
) The sum of the coefficients of all the Integral powers of x in the expansion of (1+2√x)⁴⁰ is
A) 3 ⁴⁰ +1 B) 3⁴⁰ - 1
C) 1/2 (3⁴⁰- 1). D) 1/2 (3⁴⁰ +1)
) If (1+x - 2x²) ⁸= a₀ +a₁x + a₂x² + .... + a₁₆ x¹⁶ then the sum a₁+ a₃+ a₅ +....+a₁₅ is equal to
A) -2⁷ B) 2⁷ C) 2⁸. D) none
) The sum of ²⁰C₀+ ²⁰C₁+ ²⁰C₂ + .... + ²⁰C₁₀ is equal to
A) 2²⁰+ 20!/(10!)²
B) 2¹⁹ - 1/2 . 20!/(10!)²
C) 2¹⁹+ ²⁰C₁₀. D) none
) The sum of ¹⁰C₃ +¹¹C₃+ ¹²C₃+ .... + ²⁰C₃ is equal to..
A)²¹C₄ B) ²¹C₄+ ¹⁰C₄
C) ²¹C₁₇ - ¹⁰C₆ D) none
) If (1+x)¹⁰=a ₀+a₁x+ a₂x²+...+ a₁₀x ¹⁰ then ( a₀-a₂+₄-a₆+a₈-a₁₀)²+(a₁ - a₃ + a₅ -a ₇+ a ₉)² is equal to
A) 3¹⁰ B) 2¹⁰ C) 2⁹ D) none
) The sum of 1/2 ¹⁰C₀ - ¹⁰C₁ +2.¹⁰C₂ - 2². ¹⁰C₃ +...+ 2⁹ ¹⁰C₁₀ is equal to
A) 1/2 B) 0 C) 1/2. 3¹⁰ D) none
) 1.ⁿC₁ + 2.ⁿC₂ + 3.ⁿC₃ +...+ n.ⁿCₙ is equal to...
A) n(n+1) 2ⁿ/4. B) 2ⁿ⁺¹- 3
C) n.2 ⁿ⁻¹ D) none
) If aₙ ⁿ ᵣ₌₀∑ 1/ⁿCᵣ then ⁿᵣ₌₀ ∑ r/ ⁿCᵣ equal to
A) (n-1) aₙ. B) naₙ
C) 1/2 naₙ D) none
) The sum of the series. ⁿᵣ₌₁∑(-1) ʳ⁻¹. ⁿCᵣ (a-r) is equal to
A) n. 2ⁿ⁻¹ +a B) 0
C) a D) none
) Let (1+x)ⁿ= ⁿ ᵣ₌₀∑ aᵣ. xʳ. Then (1+ a₁/a₀)(1+ a₂/a₁)...(1+aₙ/(aₙ₋₁) is equal to
A) (n+1)ⁿ⁺¹/n! B) (n+1)ⁿ/n!
C) nⁿ⁻¹ /(n-1)! D)(n+1)ⁿ⁻¹/(n-1)!
) The value of ⁿᵣ₌₁∑r. ⁿCᵣ/ⁿCᵣ₋₁ is equal to
A) 5(2n-9). B) 10n
C) 9(n-4) D) none
) The sum ⁿᵣ₌₁∑ r. ²ⁿCᵣ is equal to
A) n. 2²ⁿ⁻¹ B) 2²ⁿ⁻¹
C) 2ⁿ⁻¹+ 1. D) none
) The sum 1.²⁰C₁ - 2.²⁰C₂ + 3.²⁰C₃ - ...... - 20. ²⁰C₂₀ is equal to
A) 2¹⁹ B) 0 C) 2²⁰ -1. D) none
) Let f(x)=(√(x²+1)+ √(x²-1) + {2/√(x²+1) + √(x² -1)}⁶. Then
A) f(x) is a polynomial of the sixth degree in x.
B) f(x) has exactly two terms
C) f(x) is not a polynomial in x
D) coefficient of x⁶ is 64
) The coefficient of a⁸ b⁶ c⁴ in the expansion of (a+b+c)¹⁸ is..
A) ¹⁸C₁₄ .¹⁴C₈ B) ¹⁸C₁₀ .¹⁰C₆
C) ¹⁸C₆ .¹²C₈ D) ¹⁸C₄ .¹⁴C ₆
) The term independent of x in the expansion of (1+x)ⁿ.(1-1/x)ⁿ is
A) 0, if n is odd
B) (-1) ⁽ⁿ⁻¹⁾/² .ⁿ C(ₙ₋₁)/₂ , if n is odd
C) (-1)ⁿ⁾² .ⁿCₙ/₂, if n is even
D) none
) The coefficient of the (r+1)th term of (x + 1/x)²⁰ when Expanded in the descending powers of x is equal to the coefficient of the 6th term of (x² + 2 + 1/x²) ¹⁰ when Expanded in ascending powers of x, the value of r is
A) 5. B) 6. C) 14. D) 15
) If (1+x)²ⁿ = a₀+ a₁x + a₂x²+ ...+ a₂ₙ x²ⁿ then
A) a₀ +a₂ +a₄ +...= 1/2(a₀+a ₁+₂+a₃ + ....)
) aₙ₊₁ < a ₙ
C) a ₙ₋₃ = aₙ₊₃ D) none
) In the expansion of ³√4 + 1/⁴√6) ²⁰.
A) the number of rational terms= 4
B) the number of irrational terms= 18
C) the middle term is irrational
D) the number of rational terms= 17
) Let n belongs to N. If (1+x)ⁿ = a₀ + a₁x+ a₂x² +......+ aₙxⁿ, and aₙ₋₃ , aₙ₋₂ , aₙ₋₁ are in AP then
A) a₁ , a₂, a₃ are in AP
B) a₁ , a₂ , a₃ are in HP
C) n= 7 D) n= 14
) Let R= (8+ 3√7)²⁰ and [R]= the greatest integer less than or equal to R. Then.
A) [R] is even. B) [R] is odd
C) R - [R]= 1 - 1/(8+ 3√7)²⁰
D) none
) 1/{1!(n-1)!} +1/{3!(n-3)!} + 1/{5!(n-5)!} + .... is equal to
A) 2ⁿ⁻¹/n! For even values of n only
B) (2ⁿ⁻¹+1)/n! For odd values of n only
C) 2ⁿ⁻¹ /n! For all n belongs to N
D) none
) In the expansion of (x+y+z)²⁵
A) every term is of the form ²⁵Cᵣ. ʳcₖ. x²⁵⁻ʳ . y ʳ⁻ᵏ . zᵏ
B) the coefficient of x⁸ y⁹ z⁹ is 0
C) the number of terms is 325
D) none
1) Find the sum of all the coefficients in the expansion of (2x-3y)¹⁵ . -1
2) Prove that, ⁿC₁ + ⁿC₂ + ⁿC₃+....+ ⁿCₙ = 2ⁿ - 1
3) Find the relation between coefficients of xᵐ and xⁿ in the expansion of (1+x)ᵐ⁺ⁿ. Coeff. of xᵐ= coeff. Of xⁿ .
4) Which term in the expansion of √(1+2x)⁵ is the first negative term ? 5th term
5) Prove that the Coefficient of xⁿ (1+x)²ⁿ is double the Coefficient of xⁿ in (1+x)²ⁿ⁻¹.
6) If y= 2x + 3x²+ 4x³+ .....∞, then express x in ascending powers of y. x= y/2- 3y²/8+5y³/16....
7) Determine the larger of the following two
99⁵⁰ + 100⁵⁰ and 101⁵⁰. 101⁵⁰> 99⁵⁰ + 100⁵⁰
8) Find the value of √(1+ 1/2 + 1/2² + ......∞). √2
9) The sum of the Coefficients in the expansion of (a+2b)ⁿ, where n is a positive integer, is
A) 2ⁿ B) ⁿ⁻¹ C) 3ⁿ. D) 3ⁿ⁻¹
10) The expression (1+x)⁶(1+2x)¹² is Expanded in ascending powers of x. The total number of terms is:
A) 13 B) 18 C) 19. D) 20
11) The expansion of √(9-4x) will be valid for
A) all x B) |x|<9/4 .
C) |x|<2/3 D) |x|<4/9
12) y= x+x² + x³ +.....∞, then x equals
A) y/(y+1). B) (y+1)/y
C) y/(y-1) D) (y-1)/y
13) If (1-x +x²)ⁿ = a₀ + a₁x+ a₂x² + ....+ a ₂ₙ x ²ⁿ , then a₀ + a₂ + a₄ + ......a₂ₙ equals
A) (3ⁿ +1)/2. B) (3ⁿ-1)/2
C) 3ⁿ + 1/2. D) none
14) The Coefficient the xⁿ in (1+ 2x + 3x² + 4x³ +....)¹⁾² is.
A) -1 B) (-1)ⁿ C) 0 D) 1 .
15) The Coefficient of x⁶ in the expansion of 1+ (1+x) + (1+x)² + ....+ (1+x)¹² is
A) ¹²C₆ B) ¹³C₆. C)¹²C ₇ D) ¹³C₇.
16) Show that the Coefficient of middle term in the expansion of (1+x)²ⁿ is equal to the sum of the Coefficients of middle terms in the expansion of (1+x)²ⁿ⁻¹ .
17) Find the term independent of x in (1+x)ᵖ (1+ 1/x)ᑫ, where p and q are positive integers. (p+q)!/p!q!
18) If the sum of all the Coefficients in the expansion of (x² + 1/x)ⁿ is 1024, find n and the Coefficient of x⁸. 10
19) If p be positive integer, then show that the expression xᵖ⁺¹- (p--1)x + P is Divisible by (x-1)².
20) Prove that ⁿC₁x (1-x)ⁿ⁻¹ +2. ⁿC₂ x² (1-x)ⁿ⁻² + 3.ⁿC₃ x³ (1-x)ⁿ⁻³ + ......+ r. ⁿCᵣ xʳ (1-x)ⁿ⁻ʳ + ..........n.ⁿCₙ xⁿ = nx, where n is any positive integer
21) If n is an integer greater that , show that
a - ⁿC₁(a-1)+ ⁿC₂(a-2) - ...+(-1)ⁿ.(an)= 0
22) if C₀ , C₁, C₂, ... Cₙ be the cofficients in the expansion of (1++x)ⁿ , prove that C₀C ₁ + C₁C₂ +C₂C₃ + .......+ Cₙ₋₃ Cₙ = (2n))!/{(n-1)!(n+))!}} .
23) If t₀ , t₁ , t₂,......tₙ are the successive terms of (a+x)ⁿ, show (t ₀ - t₂ + t₄ - ....)² + (t₁ - t ₃ + t₅ - ..)² = (a² + x²)ⁿ.
24) If Pₙ denotes the product of all the Coefficients in the expansion of (1+x)ⁿ , show that Pₙ₊₁/ Pₙ = (n+1)ⁿ/n!
25) For what values of x, the expansion of (1-2x) ⁻¹⁾² will be valid? What will be the fourth term of the expansion? |x|<1/2, 5x³/2
26) If c is such a small quantity that c⁴ can be neglected in comparison to l⁴, show that √{ll/((l+c)} + √{l/(l--c)}= 2 + 3/4. c²/l² (nearly)
27) Find the sum of: 1/2+ 1.3/2..4 (1/4) + 1.3.5/2.4.6 (1/4)² + ....... To ∞,. 4/3(2√3--3)
28) If a, b, c, d are the Coefficients of any four consecutive terms in the expansion of (1+x)ⁿ, n being positive integer, prove that a/(a+b)+ c/(c+d) = 2b/(b+c)
29) Find the sum of 2 + 5/(2!.3)+ 5.7/(3!3²) +5.7.9/(4!3³)+ ...∞, 3√3
30) If C₀, C₁, C₂, ....Cₙ are the Coefficients in the expansion of (1+x)ⁿ prove that C₁/C₀ + 2.C₂/C₁ + 3. C₃/C₂+.......Cₙ/Cₙ₋₁= n(n+1)/2
31) if (1+x)ⁿ = C₀ + C₁x+ C₂x²+... + Cₙxⁿ, then show that the sum of all the product of the quantities C₀ , C₁ , C₂ , ....C ₙ taken two at a time is 2²ⁿ ⁻¹ - (2n-1)!/{n!(n-1)!}
32) If (1+x)ⁿ = C₀ + C₁x+ C₂x² + .... + Cₙ xⁿ, prove that (C₀+C₁)(C₁+C₂) ...(C ₙ₋₁+ C ₙ)= (n+1)ⁿ/n! C₁C₂C₃.......Cₙ
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