Binomial Theorem

01Easy64×since 2002Q2988

The sum of the series 2.²⁰C₀ + 5.²⁰C₁ + 8.²⁰C₂ + 11.²⁰C₃ + ... +62.²⁰C₂₀ is equal to :

01Easy64×since 2002Q2988

The sum of the series 2.²⁰C₀ + 5.²⁰C₁ + 8.²⁰C₂ + 11.²⁰C₃ + ... +62.²⁰C₂₀ is equal to :

02Easy64×since 2002Q2891

If (27)⁹⁹⁹ is divided by 7, then the remainder is :

03Easy64×since 2002Q2886

The positive integer just greater than

{\left( {1 + 0.0001} \right)^{10000}}

04Easy64×since 2002Q2895

The remainder when 3²⁰²² is divided by 5 is :

05Medium64×since 2002Q2887

The lowest integer which is greater than

{\left( {1 + {1 \over {{{10}^{100}}}}} \right)^{{{10}^{100}}}}

is ______________.

06Medium64×since 2002Q2888

Let

m

and

n

be the coefficients of seventh and thirteenth terms respectively in the expansion of

\left(\frac{1}{3} x^{\frac{1}{3}}+\frac{1}{2 x^{\frac{2}{3}}}\right)^{18}

. Then

\left(\frac{\mathrm{n}}{\mathrm{m}}\right)^{\frac{1}{3}}

is :

07Easy64×since 2002Q2889

The sum of all rational terms in the expansion of

\left(2^{\frac{1}{5}}+5^{\frac{1}{3}}\right)^{15}

is equal to :

08Medium64×since 2002Q2890

The remainder left out when

{8^{2n}} - {\left( {62} \right)^{2n + 1}}

is divided by 9 is :

09Medium64×since 2002Q2892

The greatest positive integer k, for which 49^k + 1 is a factor of the sum 49¹²⁵ + 49¹²⁴ + ..... + 49² + 49 + 1, is:

10Medium64×since 2002Q2893

The remainder when (2021)²⁰²³ is divided by 7 is :

11Medium64×since 2002Q2894

{1 \over {2\,.\,{3^{10}}}} + {1 \over {{2^2}\,.\,{3^9}}} + \,\,.....\,\, + \,\,{1 \over {{2^{10}}\,.\,3}} = {K \over {{2^{10}}\,.\,{3^{10}}}}

, then the remainder when K is divided by 6 is :

12Easy64×since 2002Q2896

The remainder when

(11)^{1011}+(1011)^{11}

is divided by 9 is

13Easy64×since 2002Q2897

The remainder when

(2021)^{2022}+(2022)^{2021}

is divided by 7 is

14Easy64×since 2002Q2898

The remainder when

7^{2022}+3^{2022}

is divided by 5 is :

15Easy64×since 2002Q2899

Fractional part of the number

\frac{4^{2022}}{15}

is equal to

16Medium64×since 2002Q2900

Let the number

(22)^{2022}+(2022)^{22}

leave the remainder

\alpha

when divided by 3 and

\beta

when divided by 7. Then

\left(\alpha^{2}+\beta^{2}\right)

is equal to :

17Easy64×since 2002Q2901

25^{190}-19^{190}-8^{190}+2^{190}

is divisible by :

18Medium64×since 2002Q2902

Among the statements : (S1) :

2023^{2022}-1999^{2022}

is divisible by 8 (S2) :

13(13)^{n}-12 n-13

is divisible by 144 for infinitely many

n \in \mathbb{N}

19Medium64×since 2002Q2903

The coefficients of

{x^p}

and

{x^q}

in the expansion of

{\left( {1 + x} \right)^{p + q}}

are

20Medium64×since 2002Q2904

The number of integral terms in the expansion of

{\left( {\sqrt 3 + \root 8 \of 5 } \right)^{256}}

Questions per year

131 questions in the bank

Binomial Theorem

Questions per year

131 questions in the bank