Binomial Theorem | Mathematics | NDA Previous Year Questions

If the coefficients of x^r and x^{r + 1} are equal in the expansion, then r is equal to

The average of the coefficients of the two middle terms in the expansion is

The sum of the coefficients of all the terms in the expansion is

The coefficient of x⁹⁹ in the expansion of (x $$-$$ 1) (x $$-$$ 2) (x $$-$$ 3) ...... (x $$-$$ 100) is

What is $${}^{47}{C_4} + {}^{51}{C_3} + \sum\limits_{j = 2}^5 {{}^{52 - j}{C_3}} $$ equal to?

NDA 2016 Paper 2

The number of terms in the expansion of (x + a)¹⁰⁰ + (x $$-$$ a)¹⁰⁰ after simplification is

NDA 2017 Paper 2

In the expansion of (1 + x)⁵⁰, the sum of the coefficients of odd powers of x is

NDA 2017 Paper 2

The value of [C(7, 0) + C(7, 1)] + [C(7, 1) + C(7, 2)] + ..... + [C(7, 6) + C(7, 7)] is

NDA 2017 Paper 1

The expansions of (x $$-$$ y)ⁿ, n $$\ge$$ 5 is done in the descending powers of x. If the sum of the fifth and sixth terms is zero, then $${x \over y}$$ is equal to

NDA 2017 Paper 1

What is the coefficient of the middle term in the binomial expansion of (2 + 3x)⁴ ?

NDA 2018 Paper 2

Let the coefficient of the middle term of the binomial expansion of (1 + x)²ⁿ be $$\alpha$$ and those of two middle terms of the binomial expansion of (1 + x)^{2n $$-$$ 1} be $$\beta$$ and $$\gamma$$. Which one of the following relations is correct?

NDA 2018 Paper 2

In the expansion of (1 + x)⁴³, if the coefficients of (2r + 1)th and (r + 2)th terms are equal, then what is the values of r(r $$\ne$$ 1) ?

NDA 2018 Paper 1

If the coefficients of a^m and aⁿ in the expansion of (1 + a)^{m + n} are $$\alpha$$ and $$\beta$$, then which one of the following is correct?

NDA 2018 Paper 1

What is the number of non-zero terms in the expansion of $${(1 + 2\sqrt 3 x)^{11}} + {(1 - 2\sqrt 3 x)^{11}}$$ (after simplification)?

NDA 2018 Paper 1

How many terms are there in the expansion of $${(1 + 2x + {x^2})^5} + {(1 + 4y + 4{y^2})^5}$$ ?

NDA 2019 Paper 2

If the middle term in the expansion of $${\left( {{x^2} + {1 \over x}} \right)^{2n}}$$ is $$184756{x^{10}}$$, then what is the value of n?

NDA 2019 Paper 2

If the constant term in the expansion of $${\left( {\sqrt x - {k \over {{x^2}}}} \right)^{10}}$$ is 405, then what can be the values of k?

NDA 2019 Paper 2

What is the number of terms in the expansion of $${[{(2x - 3y)^2}{(2x + 3y)^2}]^2}$$ ?

NDA 2019 Paper 1

In the expansion of (1 + ax)ⁿ, the first three terms are respectively 1, 12x and 64x². What is n equal to?

NDA 2019 Paper 1

In the expansion of (1 + x)^p (1 + x)^q, if the coefficient of x³ is 35, then what is the value of (p + q)?

What is the remainder when 7ⁿ- 6n is divided by 36 for n = 100?

What is V + W equal to?

What is the value of (U + V)W?

NDA Mathematics 21 April 2024

What is the coefficient of $x^{10}$ in the expansion of $(1-x^2)^{20}\left(2-x^2-\frac{1}{x^2}\right)^{-5}$?

If the 4th term in the expansion of $\left(mx + \frac{1}{x}\right)^n$ is $\frac{5}{2}$, then what is the value of $mn$?

NDA Mathematics 21 April 2024

In a binomial expansion of $(x+y)^{2 n+1}(x-y)^{2 n+1}$, the sum of middle terms is zero. What is the value of $\left(\frac{x^2}{y^2}\right)$ ?

NDA Mathematics 21 April 2024

What is the number of real roots of the equation?

What is the sum of all the roots of the equation?

What is T1 + 2T2 + 3T3 + ... + nTn equal to ?

What is 1 - T1 + 2T2 - 3T3 + ... + (-1)nnTn equal to ?

What is T1 + T2 + T3 + ... + Tn equal to ?

What is the mean of the numbers 1, 2, 3, ... 10 with frequencies ⁹C₀, ⁹C₁, ⁹C₂ ..., ⁹C₉, respectively?

In a Binomial distribution B(n, p), n = 6 and 9P(X = 4) = P(X = 2). What is p equal to ?

If the sixth term in the binomial expansion of $\left(x^{-\frac{8}{3}}+x^2 \log _{10} x\right)^8$ is 5600, then what is the value of x ?

NDA Mathematics 16 April 2023

How many terms are there in the expansion of (3x - y)⁴(x + 3y)⁴ ?

NDA Mathematics 16 April 2023

Consider the following statements for a fixed natural number n:

1. C(n, r) is greatest if n = 2r

2. C(n, r) is greatest if n = 2r - 1 and n = 2r + 1

Which of the statements given above is/are correct ?

NDA Mathematics 16 April 2023

What is $\displaystyle\sum_{r=0}^n$2^rC(n, r) equal to ?

What is the value of q if the coefficients of x3 and x6 are equal ?

What is the ratio of the coefficients of middle terms in the expansion (when expanded in ascending powers of x)?

Under what condition the coefficients of x2 and x4 are equal ?

How many terms are there in the expansion of $\left(1 + \frac{2}{x}\right)^9\left(1-\frac{2}{x}\right)^9 ? $

Consider the following statements in respect of the expansion of (x + y)¹⁰

1. Among all the coefficients of the terms, the coefficient of the 6th term has the highest value

2. The coefficient of the 3rd term is equal to coefficient of the 9th term

Which of the above statements is /are correct ?

What is the value of

C(51, 21) - C(51, 22) + C(51, 23) - C(51, 24) + C(51, 25) - C(51, 26) + C(51, 27) - C(51, 28) + C(51, 29) - C(51, 30) ?

In the expansion of $\left(x + \frac{1}{x}\right)^{2n} $, what is the (n + 1)th term from the end (when arranged in descending powers of x) ?