JEE Advanced Level Test: Binomial Numbers- 2 - JEE MCQ

The sum of the co-efficients of all the even powers of x in the expansion of (2x² – 3x + 1)¹¹ is

The coefficient of x^r(0 < r < n – 1) in the expression (x + 2)^n-1 + (x + 2)^n-2 . (x + 1) + (x + 2)^n-3. (x + 1)² + .............+ (x + 1)^n- is

The co-efficient of x⁴⁰¹ in the expansion of (1 + x + x² + ........+x⁹)^-1, (|x| < 1) is

Let (1 + x²)² (1 + x)ⁿ = A₀ + A₁x + A₂x² + ........... If A₀, A₁, A₂ are in A.P. then the value of n is

The number 101¹⁰⁰ – 1 is divisible by

If the 6^th term in the expansion of  when x = 3 is numerically greatest then the possible integral value(s) of n can be

If  where I, n are integers and 0 < f < 1, then

Positive integers can be represented as

In the expansion of 

The co-efficient of the middle term in the expansion of (1 + x)²ⁿ is

The value of ⁿC₀ . ⁿC_n + ⁿC₁ . ⁿC_{n – 1} + ... + ⁿC_n . ⁿC₀ is

The value of r for which ³⁰C_r . ²⁰C₀ + ³⁰Cr – 1 . ²⁰C₁ +...+ ³⁰C₀ . ²⁰C _r is maximum, is

The value of r(0 < r < 30) for which ²⁰C_r . ¹⁰C₀ + ²⁰Cr – 1 . ¹⁰C₁ +...+ ²⁰C₀ . ¹⁰C_r is minimum, is

The value of r(0 < r < 30) for which ²⁰C_r . ¹⁰C₀ + ²⁰Cr – 1 . ¹⁰C₁ +...+ ²⁰C₀ . ¹⁰C_r is minimum, is

Column-I Column-II

(A) If l be the number of terms in the expansion of (P) O + T = 3

(1 + 5x + 10x² + 10x² + 5x⁴ + x⁵)²⁰ and if unit's

place and ten's place digits in 3^l are O and T, then

(B) If l be the number of terms in the expansion (Q) O + T = 7

of and if unit's place and (R) O + T = 9

ten's place digits in 7^l are O and T, then

(C) If l be the number of terms in the

expansion of (1 + x)¹⁰¹ (1 + x² _ x)¹⁰⁰ (S) T _ O = 7

and if unit's place and ten's place

digits in 9^l are O and T, then (T) O _ T = 7

subjective type

 Find the coefficients

(i) x⁷ in  (ii) x ^_7 in 

(iii) Find the relation between a and b, so that these coefficients are equal.

 If the coefficients of (2r + 4)^th , (r – 2)^th terms in the expansion of (1 + x)¹⁸ are equal, find r.

 If the coefficients of the r^th, (r + 1)^th and (r + 2)^th terms in the expansion of (1 + x)¹⁴ are in A.P., find r.

 Find the term independent of x in the expansion of

(a)  (b) 

 Given that (1 + x + x²)ⁿ = a₀ + a₁x + a₂x² + ...... +a_2nx²ⁿ, find the values of

(i) a₀ + a₁ + a₂ + ..... + a_2n ;

(ii) a₀ _ a₁ + a₂ _ a₃ ....... +a_2n ;

(iii) a₀² _ a₁² + a₂² _ a₃² + ... + a_2n²

In a Binomial Distribution, if ‘n’ is the number of trials and ‘p’ is the probability of success, then the mean value is given by

Prove that :

^{n _ 1}C_r + ^{n _ 2}C_r + ^{n _ 3}C_r + ............ + ^rC_r = ⁿC_{r + 1}^.