BOOSTER - A
1) If x+ 1/x = 1 and p= x¹⁰⁰⁰ + 1/x¹⁰⁰⁰ and q be digit at unit place in the number ₂2ⁿ +1, n∈ N and n > 1, then p+ q=
a) 8 b) 6 c) 7 d) none
2) For integer n> 1, the digit at unit place in the number ¹⁰⁰ᵣ₌₀∑ r! + ₂2ⁿ is
a) 4 b) 3 c) 1 d) 0
3) If (1+ x + 2x²)²⁰= a₀ + a₁x + a₂x² + ....+ a₄₀x⁴⁰, then a₀ + a₂ + a₄ .....+ a₃₈ equals
a) 2¹⁹(2²⁰+1) b) 2¹⁹(2²⁰- 1) c) 2²⁰(2¹⁹ -1) d) none
4) The coefficient of xⁿ polynomial (x + ²ⁿ⁺¹C₀)(x + ²ⁿ⁺¹C₁)(x + ²ⁿ⁺¹C₃)......(x + ²ⁿ⁺¹Cₙ) is
a) 2ⁿ⁺¹ b) 2²ⁿ⁺¹ - 1 c) 2²ⁿ d) none
5) If n ∈ N, then ⁿᵣ₌₀∑(-1)ʳ ⁿCᵣ[1/2ʳ + 3ʳ/2²ʳ + 7ʳ/2³ʳ + ..... to m terms]=
a) 2ᵐⁿ/(2ⁿ-1)2ᵐⁿ
b) (2ᵐⁿ +1)/(2ⁿ -1)2ᵐⁿ
c) (2ᵐⁿ- 1)/(2ⁿ-1)2ᵐⁿ d) none
6) Integral part of (5 √5 + 11)²ⁿ⁺¹ is
a) even b) odd
c) even or odd depending upon the value of n
d) cannot be determined
7) 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
8) If 7 divides ₃₂32³², the remainder is
a) 1 b) 0 c) 4 d) 6
9) The sum ⁿᵣ₌₁∑ r. ²ⁿCᵣ is equal to
a) n. 2²ⁿ⁻¹ b) 2²ⁿ⁻¹ c) 2²ⁿ⁻¹ +1 d) none
10) The last two digits of the number 3⁴⁰⁰ are
a) 39 b) 29 c) 01 d) 43
11) The sum ᵐᵢ₌₀ ∑(10) (20, where p)= 0
i) m- i) q)
If p< q, is maximum , when m is
a) 5 b) 10 c) 15 d) 20
12) The value {3²⁰¹³/28}, where {.}= F. F, i equals
a) 5/28 b) 17/28 c) 19/28 d) none
13) If k, n be positive integer and Sₖ= 1ᵏ + 2ᵏ+.....nᵏ then
ᵐᵣ₌₁∑ᵐ⁺¹Cᵣ Sᵣ =
a) (n +1)ᵐ⁺¹ - (n +1)
b) (n +1)ᵐ⁺¹+ (n +1) c) n d) none
14) The number of terms in (x³ +1+ 1/x³)¹⁰⁰ is
a) 100 b) 150 c) 200 d) none
15) The value of
ⁿC₁ x(1- x)ⁿ⁻¹+ 2. ⁿC₂ x²(1- x)ⁿ⁻²+ 3. ⁿC₃ x³(1- x)ⁿ⁻³+ ....+ n. ⁿCₙ xⁿ.
a) x b) nx c) n d) none
16) If (1+ x)ⁿ= ⁿᵣ₌₀∑ aᵣxʳ and bᵣ = 1+ aᵣ/(aᵣ -1) and ⁿᵣ₌₁ Π br = (101)100/100!, then n=
a) 100 b) 110 c) 120 d) none
17) If ⁿC₀/2 - ⁿC₁/3 + ⁿC₂/4 - .... + (-1)ⁿ. ⁿCₙ/(n +2)= 1/(1999 x 2000) then n=
a) 2004 b) 2006 c) 2007 d) none
18) If (1+ x)ⁿ= C₀ + C₁ x + C₂ x² + ....+ cₙ xⁿ, then ₀≤ ᵣ ≤ ₛ ≤ ₙ ∑∑(r,s)CᵣCₛ =
a) n(2²ⁿ⁻³ + ²ⁿ⁻²Cₙ₋₁)
b) n²(2²ⁿ⁻³ - ²ⁿ⁻²Cₙ₋₁)
c) n²(2²ⁿ⁻³ - (1/2)²ⁿ⁻²Cₙ₋₁ d) none
19) If (1+ x)ⁿ = c₀ + c₁ x² + ....cₙ xⁿ, then ₀ ≤ ᵣ ≤ ₛ ≤ₙ ∑∑(cᵣ ± cₛ)²=
a) (n± 1) ²ⁿcₙ ± 2²ⁿ
b) (n± 1) ²ⁿcₙ + 2ⁿ
c) (n ± 1)2ⁿ ± 2ⁿ d) none
20) If (1+ x)ⁿ = C₀ + C₁ x + C₂ x² ....Cₙ xⁿ, then for n even, C²₀ - C²₁ + C²₂ - ....+ (-1)ⁿ C²ₙ is equal to
a) 0 b) (-1)ⁿ⁾² ⁿCₙ/₂
b) ⁿCₙ/₂ d) none
21) ⁿₖ₌₀∑ ⁿCₖ/{(k+1)(k+2)}=
a) (2ⁿ⁺¹ - n -3)/{(n +1)(n +2)}
b) (2ⁿ⁺² - n- 3)/{(n +1)(n +2)}
c) (2ⁿ⁺²- n +3)/{(n +1)(n +2)} d) none
22) ᵐCᵣ + ᵐCᵣ₋₁. ⁿC₁ + ᵐCᵣ₋₂. ⁿC₂ + ....+ ᵐC₁. ⁿCᵣ₋₁ + ⁿCᵣ =
a) ᵐ⁺ⁿCᵣ₋₁ b) ᵐ⁺ⁿCᵣ c) ᵐ⁺ⁿCᵣ₊₁ d) none
23) The sum C₀/(1.2) + C₁/(2.3) + C₂/(3.4) + C₃/(4.5) + ....to (n +1) terms is
a) 1/(n +2) b) 2ⁿ/(n +2) c) (2ⁿ -1)/(n +2) d) none
24) If n is an even positive integer and k= 3n/2, then ᵏᵣ₌₁∑ (-3)ʳ⁻¹ ³ⁿC₂ᵣ₋₁ =
a) 1 b) -1 c) 0 d) none
25) The co-efficient of x³⁰¹ in the expansion of
(1+ x)⁵⁰⁰ + x(1+ x)⁴⁹⁹ + x²(1+ x)⁴⁹⁸+ ....+ x⁵⁰⁰ is
a) ⁵⁰¹C₃₀₁ b) ⁵⁰⁰C₃₀₁ c) ⁵⁰¹C₃₀₀ d) none
26) The fractional part of (√6)²ⁿ, n ∈ N is equal to
a) 1/3 b) 1/5 c) 1/6 d) none
27) The number of irrational terms in the expansion of (⁸√5 + ⁶√2)¹⁰⁰ is
a) 96 b) 97 c) 98 d) none
1b 2d 3b 4c 5c 6a 7c 8c 9a 10c 11c 12c 13a 14d 15b 16a 17d 18c 19a 20b 21b 22b 23a 24c 25a 26b 27b
BOOSTER - B
1) If the co-efficient of x⁷ in [ax² + 1/bx)]¹¹ equals the co-efficient of x⁻⁷ in [ax - (1/bx²)]¹¹ , then a and b satisfy the relation
a) a+ b =1 b) a - b =1 c) ab=1 d) a/b = 1
2) The co-efficient of a⁸b⁴c⁹d⁹ in (abc+ abd+ acd+ bcd)¹⁰ is
a) 10! b) 10//8!4!9!9! c) 2520 d) none
3) If the term independent of x in the (√x - k/x²)¹⁰ is 405, then k equals
a) 2,-2 b) 3,-3 c) 4,-4 d) 1,-1
4) The co-efficient of x²⁸ in the expansion of (1+ x³ - x⁶)³⁰ is
a) 1 b) 0 c) ³⁰C₆ d) ³⁰C₃
5) In the expansion of (5¹⁾² + 7¹⁾⁸)¹⁰²⁴, the number of integral terms is
a) 128 b) 129 c) 130 d) 131
6) If the 6th term in the expansion of (1/x⁸⁾³ + x² log₁₀x)⁸ is 5600, then x equals
a) 1 b) logₑ10 c) 10 d) x doesn't exist
7) If the last term in the binomial expansion of
(2¹⁾³ - √(1/2))ⁿ is (1/3⁵⁾³)ˡᵒᵍ₃⁸, then the 5ᵗʰ term from the beginning is
a) 210 b) 420 c) 105 d) none
8) The number of integral terms in the expansion of (√3+ ⁸√256)²⁵⁶ is
a) 33 b) 34 c) 35 d) none
9) The number of distinct terms in the expansion of (x + 1/x + x² + 1/x²)¹⁵ is/are
a) 255 b) 61 c) 127 d) none
10) The sum of the co-efficient of even power of x in the expansion of (1+ x + x²+ x)⁵ is
a) 256 b) 128 c) 512 d) 63
11) If the sum of the co-efficient in the expansion of (1- 3x + 10x²)ⁿ is a and if the sum of the co-efficient in the expansion of (1+ x)ⁿ is b.
a) a= 3b b) a= b³ c) a³= b d) none
12) The fraction part of 2⁴ⁿ/15 is (n ∈N)
a) 1/15 b) 2/15 c) 4/15 d) none
13) The value of (30) (30) + (30(30). (30)(30) (30(30)
0) (10) (1) (11) +(2) (12) +.... (20)(30=
a) ⁶⁰C₂₀ b) ³⁰C₁₀ c) ⁶⁰C₃₀ d) ⁴⁰C₃₀
14) The sum of series
²⁰C₀ - ²⁰C₁ + ²⁰C₂ - ²⁰C₃ + ....+ ²⁰C₁₀ is
a) (1/2) ²⁰C₁₀ b) 0 c) ²⁰C₁₀ d) -²⁰C₁₀
15) ⁴⁰⁴C₄ - ⁴C₁ ³⁰³C₄ + ⁴C₂ ²⁰²C₄ - ⁴C₃ ¹⁰¹C₄ is equal to
a) (401)⁴ b) (101)⁴ c) 0 d) (201)⁴
16) If (3+ x²⁰⁰⁸+ x²⁰⁰⁹)²⁰¹⁰ =
a₀ + a₁x+ a₂x² + ...+ aₙxⁿ, then the value of
a₀ - (1/2) a₁ - (1/2) a₂ + a₃ - (1/2) a₄ - (1/2) a₅ + a₆ - ....is
a) 3²⁰¹⁰ b) 1 c) 2²⁰¹⁰ d) none
17) The value of ¹⁰ᵣ₌₀∑ r¹⁰Cᵣ3ʳ(-2)¹⁰⁻ʳ is
a) 20 b) 10 c) 300 d) 30
18) The value of ¹⁵ᵣ₌₁ ∑r 2ʳ/(r +2)! is equal to
a) (17! - 12¹⁶)/(17!)
b) (18! - 2¹⁷)/(18!)
c) (16! - 2¹⁵)/(16!)
d) (15! - 2¹⁴)/15!
19) The value of ⁵⁰ᵣ₌₀ ∑(-1)ʳ ⁵⁰Cᵣ/(r +2) is equal to
a) 1/(52.51) b) 1/(52.50) c) 1/(52.51) d) none
20) 1+ x/3 + (1x4)x²/(3x6) + (1x4x7)x³/(3x6x9) + ....is equal to
a) x b) (1+ x)¹⁾³ c) (1- x)¹⁾³ d) (1- x)⁻¹⁾³
21) 1+ 1/4 + (1×3)/(4×8) + (1×3×5)/(4×8×12) + ....=
a) √2 b) 1/√2 c) √3 d) 1/√3
1c 2a 3a 4c 5b 6c 7a 8b 9b 10a 11b 12b 13b 14a 15d 16a 17a 18d 19b 20c 21b
BOOSTER - C
1) The co-efficient of x⁵ in
(1+ 2x + 3x²+....)⁻³⁾² is
a) 21 b) 25 c) 26 d) none
2) If |x|< 1, then the co-efficient of xⁿ in expansion of (1+ x + x² + x³ + ...)² is
a) n b) n -1 c) n+2 d) n +1
3) If x is positive, the first negative term in the expansion of (1+ x)²⁷⁾⁵ is
a) 5ᵗʰ term b) 8ᵗʰ term c) 6ᵗʰ term d) 7ᵗʰ term
4) The value of ¹⁰ᵣ₌₀ ∑(r) ²⁰Cᵣ, is equal to
a) 20(2¹⁸ + ¹⁹C₁₀)
b) 10(2¹⁸ + ¹⁹C₁₀)
c) 20(2¹⁸ + ¹⁹C₁₁)
d) 10(2¹⁸ + ¹⁹C₁₁)
5) The last two digits of the number (23)¹⁴ are
a) 01 b) 03 c) 09 d) none
6) If (1- x²)ⁿ = ⁿr₌₀ ∑aᵣ xʳ (1- x)²ⁿ⁻ʳ, then aᵣ is equal to
a) ⁿCᵣ b) ⁿCᵣ 3ʳ c) ²ⁿCᵣ d) ⁿCᵣ 2ʳ
7) ³⁰⁰ᵣ₌₀ ∑aᵣ xʳ = (1+ x + x² + x³)¹⁰⁰. If a= ³⁰⁰ᵣ₌₀∑ aᵣ, then ³⁰⁰ᵣ₌₀∑aᵣ is equal to
a) 300a b) 100a c) 150a d) 75a
8) The value of ⁿ⁺¹ᵣ₌₁∑(ⁿₖ₌₁∑ ᵏCᵣ₋₁) (where r, k, n ∈ N) is equal to
a) 2ⁿ⁺¹-2 b) 2ⁿ⁺¹ -1 c) 2ⁿ⁺¹ d) none
1c 2c 3c 4a 5b 6c 7b
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