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

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

^{𝑛}𝐶

_{1},

^{𝑛}𝐶

_{2},

^{𝑛}𝐶

_{3},⋯,

^{𝑛}𝐶

_{𝑛−1}must be divisible by 𝑛

**Q2.**Statement 1: 3

^{2𝑛+2}− 8𝑛 − 9 is divisible by 64,∀ 𝑛 ∈ 𝑁 Statement 2: (1+𝑥)

^{𝑛}− 𝑛𝑥 - 1 is divisible by 𝑥

^{2},∀ 𝑛 ∈ 𝑁

**Q3.**Statement 1: The coefficient of 𝑥

^{𝑛}in (1 + 𝑥 + 𝑥

^{2}/2! + 𝑥

^{3}/3! +⋯+ 𝑥

^{𝑛}/𝑛!) is 3

^{𝑛}/𝑛! Statement 2: The coefficient of 𝑥𝑛 in 𝑒

^{3𝑥}is 3

^{𝑛}/𝑛!

**Q4.**Statement 1: The value of (

^{10}𝐶

_{0}) + (

^{10}𝐶

_{0}+

^{10}𝐶

_{1})+(

^{10}𝐶

_{0}+

^{10}𝐶

_{1}+

^{10}𝐶

_{2})+⋯+(

^{10}𝐶

_{0}+

^{10}𝐶

_{1}+

^{10}𝐶

_{2}+⋯+

^{10}𝐶

_{9}) is 10⋯2

^{9}Statement 2:

^{𝑛}𝐶

_{1}+2

^{𝑛}𝐶

_{2}+ 3

^{𝑛}𝐶

_{3}+⋯+ 𝑛

^{𝑛}𝐶

_{𝑛}= 𝑛2

^{𝑛−1}

**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

**Q6.**Statement 1: In the expansion of (1+𝑥)

^{41}(1−𝑥+𝑥

^{2})

_{40}, the coefficient of 𝑥

^{85}is zero Statement 2: In the expansion of (1+𝑥)

^{41}and (1−𝑥+𝑥

^{2})

^{40},𝑥

^{85}term does not occur

**Q7.**Statement 1:

^{𝑚}𝐶

_{𝑟}+

^{𝑚}𝐶

_{𝑟−1}

^{𝑛}𝐶

_{1}+

^{𝑚}𝐶

_{𝑟−2}

^{𝑛}𝐶

_{2}+⋯+

^{𝑛}𝐶

_{𝑟}= 0, if 𝑚 + 𝑛 < 𝑟 Statement 2:

^{𝑛}𝐶

_{𝑟}= 0 if 𝑛 < 𝑟

**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 (𝑎

_{1}+ 𝑎

_{2}+⋯+ 𝑎

_{𝑚})

^{𝑛 }is

^{𝑛+𝑚−1}𝐶

_{𝑚−1}

**Q9.**Let 𝑛 be a positive integer and 𝑘 be a whole number, 𝑘 ≤ 2𝑛 Statement 1: The maximum value of

^{2𝑛}𝐶

_{𝑘}is

^{2𝑛}𝐶

_{𝑛}Statement 2:

^{2𝑛}𝐶

_{𝑘+1}/

^{2𝑛}𝐶

_{𝑘}< 1, for 𝑘 = 0,1,2,…,𝑛−1 and

^{2𝑛}𝐶

_{𝑘}

^{2𝑛}𝐶

_{𝑘−1}>1 for 𝑘 = 𝑛+1,𝑛+2,…,2𝑛