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Find an approximation of ${(0.99)}^{5}$ using the first three terms of its expansions. [Write the answer up to three decimal places.]

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Important Questions on Mathematical Methods

HARD

Mathematics>Statistics>Mathematical Methods>Introduction to Binomial, Exponential and Logarithmic Series

Let

X

be a set containing

10

elements and

P (X)

be its power set. If

A

and

B

are picked up at random from

P (X)

, with replacement, then the probability that

A

and

B

have equal number of elements is:

HARD

Mathematics>Statistics>Mathematical Methods>Introduction to Binomial, Exponential and Logarithmic Series

Let

X = {(^{10} C_{1})}^{2} + 2 {(^{10} C_{2})}^{2} + 3 {(^{10} C_{3})}^{2} + \dots + 10 {(^{10} C_{10})}^{2}

, where

^{10} C_{r}, r \in \{1,2, \dots ., 10\}

denote binomial coefficients. Then, the value of

\frac{1}{1430} X

is ______ .

MEDIUM

Mathematics>Statistics>Mathematical Methods>Introduction to Binomial, Exponential and Logarithmic Series

The term independent of

x

in the binomial expansion of

(1 - \frac{1}{x} + 3 x^{5}) {(2 x^{2} - \frac{1}{x})}^{8}

MEDIUM

Mathematics>Statistics>Mathematical Methods>Introduction to Binomial, Exponential and Logarithmic Series

Let

x_{n} = {(2^{n} + 3^{n})}^{\frac{1}{2 n}}

for all natural numbers

n

. Then

MEDIUM

Mathematics>Statistics>Mathematical Methods>Introduction to Binomial, Exponential and Logarithmic Series

The positive value of

λ

for which the co-efficient of

x^{2}

in the expansion

x^{2} {(\sqrt{x} + \frac{λ}{x^{2}})}^{10}

720,

MEDIUM

Mathematics>Statistics>Mathematical Methods>Introduction to Binomial, Exponential and Logarithmic Series

If the coefficient of the three successive terms in the binomial expansion of

{(1 + x)}^{n}

are in the ratio

1 : 7 : 42

, then the first of these terms in the expansion is

HARD

Mathematics>Statistics>Mathematical Methods>Introduction to Binomial, Exponential and Logarithmic Series

The term independent of $x$ in the expansion of

{(\frac{x + 1}{x^{2 / 3} - x^{1 / 3} + 1} - \frac{x - 1}{x - x^{1 / 2}})}^{10}

MEDIUM

Mathematics>Statistics>Mathematical Methods>Introduction to Binomial, Exponential and Logarithmic Series

The value of

(^{21} C_{1} -^{10} C_{1}) + (^{21} C_{2} -^{10} C_{2}) + (^{21} C_{3} -^{10} C_{3}) + (^{21} C_{4} -^{10} C_{4}) + \dots + (^{21} C_{10} -^{10} C_{10})

MEDIUM

Mathematics>Statistics>Mathematical Methods>Introduction to Binomial, Exponential and Logarithmic Series

\sum_{r = 0}^{25} \{({}^{50}C_{r}) ({}^{50 - r}C_{25 - r})\} = K ({}^{50}C_{25})

, then

K

is equal to

HARD

Mathematics>Statistics>Mathematical Methods>Introduction to Binomial, Exponential and Logarithmic Series

The total number of irrational terms in the binomial expansion of

{(7^{\frac{1}{5}} - 3^{\frac{1}{10}})}^{60}

HARD

Mathematics>Statistics>Mathematical Methods>Introduction to Binomial, Exponential and Logarithmic Series

The fractional part of a real number

x

x - [x],

where

[x]

is the greatest integer less than or equal to

x .

Let

F_{1}

and

F_{2}

be the fractional parts of

{(44 - \sqrt{2017})}^{2017}

and

{(44 + \sqrt{2017})}^{2017}

respectively. Then

F_{1} + F_{2}

lies between the numbers

HARD

Mathematics>Statistics>Mathematical Methods>Introduction to Binomial, Exponential and Logarithmic Series

The sum of coefficients of integral powers of

x

in the binomial expansion of

{(1 - 2 \sqrt{x})}^{50}

MEDIUM

Mathematics>Statistics>Mathematical Methods>Introduction to Binomial, Exponential and Logarithmic Series

The digit in the unit place of

7^{291}

HARD

Mathematics>Statistics>Mathematical Methods>Introduction to Binomial, Exponential and Logarithmic Series

If in a regular polygon the number of diagonals is

54

, then the number of sides of this polygon is:

HARD

Mathematics>Statistics>Mathematical Methods>Introduction to Binomial, Exponential and Logarithmic Series

The coefficient of

x^{2}

in the expansion of the product

(2 - x^{2}) \{{(1 + 2 x + 3 x^{2})}^{6} + {(1 - 4 x^{2})}^{6}\}

HARD

Mathematics>Statistics>Mathematical Methods>Introduction to Binomial, Exponential and Logarithmic Series

{(\sqrt{3} + \sqrt{2})}^{4} - {(\sqrt{3} - \sqrt{2})}^{4} =

MEDIUM

Mathematics>Statistics>Mathematical Methods>Introduction to Binomial, Exponential and Logarithmic Series

If the sum of the coefficients in the expansions of

{(a^{2} x^{2} - 2 a x + 1)}^{51}

is zero, then

a

is equal to

MEDIUM

Mathematics>Statistics>Mathematical Methods>Introduction to Binomial, Exponential and Logarithmic Series

The value of

[C (7, 0) + C (7, 1)] + [C (7, 1) + C (7, 2)] + \dots + [C (7, 6) + C (7, 7)]

HARD

Mathematics>Statistics>Mathematical Methods>Introduction to Binomial, Exponential and Logarithmic Series

If the third term in the binomial expansion of

{(1 + x^{\log_{2} x})}^{5}

equals

2560,

then a possible value of

x

MEDIUM

Mathematics>Statistics>Mathematical Methods>Introduction to Binomial, Exponential and Logarithmic Series

f (x) = x^{n}

, then the value of

f (1) - \frac{f^{'} (1)}{1!} + \frac{f^{''} (1)}{2!} - \frac{f^{'''} (1)}{3!} + \dots + \frac{{(- 1)}^{n} f^{n} (1)}{n!}

Find an approximation of 0.995 using the first three terms of its expansions. [Write the answer up to three decimal places.]

Important Questions on Mathematical Methods

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Find an approximation of ${(0.99)}^{5}$ using the first three terms of its expansions. [Write the answer up to three decimal places.]