Binomial Theorem-Quiz-11

Binomial Theorem Quiz-11

Dear Readers,

Binomial Theorem is one of the most important chapters of Algebra in the JEE syllabus and other engineering exams. For JEE Mains, it has 4% weightage and for JEE Advanced, it has 2.42% weightage.

This section contain(s) 10 questions numbered 1 to 10. Each question contains statement 1(Assertion) and statement 2(Reason). Each question has the 4 choices (a), (b), (c) and (d) out of which only one is correct.

a)Statement 1 is True, Statement 2 is True; Statement 2 is correct explanation for Statement 1
b)Statement 1 is True, Statement 2 is True; Statement 2 is not correct explanation for Statement 1
c)Statement 1 is True, Statement 2 is False
d)Statement 1 is False, Statement 2 is True

Q1. Statement 1: If 𝑝 is a prime number (𝑝≠2), then [(2+√5)^𝑝]−2^𝑝+1 is always divisible by 𝑝 (where [.] denotes the greatest integer function) Statement 2: If 𝑛 is prime, then ^𝑛𝐶₁, ^𝑛𝐶₂, ^𝑛𝐶₃,⋯, ^𝑛𝐶_𝑛−1 must be divisible by 𝑛

•  (a)
•  (b)
•  (c)
•  (d)

Solution

Q2.Statement 1: 3^2𝑛+2 − 8𝑛 − 9 is divisible by 64,∀ 𝑛 ∈ 𝑁 Statement 2: (1+𝑥)^𝑛 − 𝑛𝑥 - 1 is divisible by 𝑥²,∀ 𝑛 ∈ 𝑁

•  (a)
•  (b)
•  (c)
•  (d)

Solution

Q3.  Statement 1: The coefficient of 𝑥^𝑛 in (1 + 𝑥 + 𝑥²/2! + 𝑥³/3! +⋯+ 𝑥^𝑛/𝑛!) is 3^𝑛/𝑛! Statement 2: The coefficient of 𝑥𝑛 in 𝑒^3𝑥 is 3^𝑛/𝑛!

•  (a)
•  (b)
•  (c)
•  (d)

Solution

Q4. Statement 1: The value of ( ¹⁰𝐶₀) + ( ¹⁰𝐶₀ + ¹⁰𝐶₁)+( ¹⁰𝐶₀+ ¹⁰𝐶₁ + ¹⁰𝐶₂)+⋯+( ¹⁰𝐶₀+ ¹⁰𝐶₁+ ¹⁰𝐶₂ +⋯+ ¹⁰𝐶₉) is 10⋯2⁹ Statement 2: ^𝑛𝐶₁+2 ^𝑛𝐶₂ + 3 ^𝑛𝐶₃ +⋯+ 𝑛 ^𝑛𝐶_𝑛 = 𝑛2^𝑛−1

•  (a)
•  (b)
•  (c)
•  (d)

Solution

Q5.Statement 1: For every natural number 𝑛≥2.
1/√1 + 1/√2 +⋯+ 1/√𝑛 > √𝑛 Statement 2: For every natural number 𝑛 ≥ 2
√𝑛(𝑛+1) < 𝑛 + 1

•  (a)
•  (b)
•  (c)
•  (d)

Solution

Q6. Statement 1: In the expansion of (1+𝑥)⁴¹(1−𝑥+𝑥²)₄₀, the coefficient of 𝑥⁸⁵ is zero Statement 2: In the expansion of (1+𝑥)⁴¹ and (1−𝑥+𝑥²)⁴⁰,𝑥⁸⁵ term does not occur

•  (a)
•  (b)
•  (c)
•  (d)

Solution

Q7.Statement 1: ^𝑚𝐶_𝑟 + ^𝑚𝐶_𝑟−1 ^𝑛𝐶₁ + ^𝑚𝐶_𝑟−2 ^𝑛𝐶₂ +⋯+ ^𝑛𝐶_𝑟 = 0, if 𝑚 + 𝑛 < 𝑟 Statement 2: ^𝑛𝐶_𝑟 = 0 if 𝑛 < 𝑟

•  (a)
•  (b)
•  (c)
•  (d)

Solution

Q7.Statement 1: The number of distinct terms in (1 + 𝑥 + 𝑥2 + 𝑥3 + 𝑥4)¹⁰⁰⁰ is 4001 Statement 2: The number of distinct terms in the expansion (𝑎₁ + 𝑎₂ +⋯+ 𝑎_𝑚)^𝑛 is ^{𝑛+𝑚−1}𝐶_𝑚−1

•  (a)
•  (b)
•  (c)
•  (d)

Solution

Q9.Let 𝑛 be a positive integer and 𝑘 be a whole number, 𝑘 ≤ 2𝑛 Statement 1: The maximum value of ^2𝑛𝐶_𝑘 is ^2𝑛𝐶_𝑛 Statement 2: ^2𝑛𝐶_𝑘+1/^2𝑛𝐶_𝑘 < 1, for 𝑘 = 0,1,2,…,𝑛−1 and ^2𝑛𝐶_𝑘 ^2𝑛𝐶_𝑘−1>1 for 𝑘 = 𝑛+1,𝑛+2,…,2𝑛

•  (a)
•  (b)
•  (c)
•  (d)

Solution

•  (a)
•  (b)
•  (c)
• (d)

Solution

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