The $r\,th$ terms in the expansion of ${\left( {a + 2n} \right)^n}$ is

$\frac{{n\left( {n + 1} \right) \ldots \left( {n - r + 1} \right)}}{{r!}}{a^{n - r + 1}}{\left( {2x} \right)^r}$

$\frac{{n\left( {n + 1} \right) \ldots \left( {n - r} \right)}}{{r + 1}}{a^{n - r}}{\left( x \right)^r}$

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JEE • MATHEMATICS • BINOMIAL THEOREM

Binomial theorem practice · Mixed difficulty

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331 Easy

331 of 951

Application +4 | -1

(1+i)^6+(1-i)^6=

View answer and explanation

Correct option: A

Option A, 0, is correct because when expanding

(1+i)^6

and

(1-i)^6

using the binomial theorem, the terms cancel each other out, resulting in 0.

332 Easy

332 of 951

Application +4 | -1

If the coefficients of

{x^7}

and

{x^8}

{\left( {2 + x/3} \right)^n}

are equal, then

n

is equal to

View answer and explanation

Correct option: B

Option B (55) is correct because setting the coefficients of

{x^7}

and

{x^8}

equal leads to the equation

{\frac{{\;^n{C_7} \times {2^{n - 7}}}}{{{3^7}}} = {\frac{{\;^n{C_8} \times {2^{n - 8}}}}{{{3^8}}}}

which simplifies to

n = 55

333 Medium

333 of 951

Analyzing +4 | -1

n \in N,\;n > 1

, then value of

E = a - {\;^n}{C_1}\left( {a - 1} \right) + {\;^n}{C_2}\left( {a - 2} \right) + ... + {\left( { - 1} \right)^n}\left( {a - n} \right){\;^n}{C_n}

View answer and explanation

Correct option: B

Option B (0) is correct because the given expression simplifies to 0 using binomial coefficient properties.

334 Medium

334 of 951

Application +4 | -1

The coefficient of

t^4

in the expansion of

\left(\frac{1-t^6}{1-t}\right)^3

View answer and explanation

Correct option: A

Option A (15) is correct because it is the coefficient of

t^4

in the expansion of the given expression, obtained by using the binomial theorem.

335 Medium

335 of 951

Application +4 | -1

\left(1-x+x^2\right)^n=a_0+a_1 x+a_2 x^2+\ldots+a_{2 n} x^{2 n}

, then

a_0+a_2+a_4+\ldots+a_{2 n}

is equal to

View answer and explanation

Correct option: A

\frac{3^n+1}{2}

Option A (

\frac{3^n+1}{2}

) is correct because by substituting x=1 and x=-1 into the given equation, we can eliminate the odd-powered terms and solve for the sum of the even-powered terms.

336 Easy

336 of 951

Application +4 | -1

The coefficient of

x^5

in the expansion of

\left(1+x^2\right)^5(1+x)^4

View answer and explanation

Correct option: D

Option D (60) is correct because it is the sum of the products of the coefficients that result in x⁵ when the two expansions are multiplied.

337 Medium

337 of 951

Application +4 | -1

If the expansion of

\left(x^2+\frac{2}{x}\right)^n

for positive

n

has a term independent of

x

, then

n

View answer and explanation

Correct option: B

Option B (18) is correct because for the term to be independent of x, 2n must be a multiple of 3, making n a multiple of 3/2. Among the options, only 18 satisfies this condition.

338 Medium

338 of 951

Application +4 | -1

The greatest coefficient in the expansion of

(a+b+c)^7

View answer and explanation

Correct option: D

210

Option D (210) is correct because it represents the greatest coefficient in the multinomial expansion, which occurs when the powers are as close as possible.

339 Medium

339 of 951

Application +4 | -1

\mathrm{x}

and

\mathrm{y}

are positive integers, then

{ }^{\mathrm{x}} \mathrm{C}_{\mathrm{r}}+{ }^{\mathrm{x}} \mathrm{C}_{\mathrm{r}-1}{ }^y \mathrm{C}_1+{ }^{\mathrm{x}} \mathrm{C}_{\mathrm{r}-2}{ }^{\mathrm{y}} \mathrm{C}_2+\ldots+{ }^{\mathrm{y}} \mathrm{C}_{\mathrm{r}}

is equal to

View answer and explanation

Correct option: C

{ }^{x+y} C_r

Option C (

{ }^{x+y} C_r

) is correct because the given expression is the coefficient of

a^r

in the expansion of

(1+a)^x (1+a)^y = (1+a)^{x+y}

340 Easy

340 of 951

Application +4 | -1

The middel term in the expansion of

{\left( {x - \frac{1}{x}} \right)^{18}}

View answer and explanation

Correct option: B

- _\;^{18}{C_9}

Option B (

- _\;^{18}{C_9}

) is correct because the middle term in the expansion of

{\left( {x - \frac{1}{x}} \right)^{18}}

is found by setting r = n/2 = 9 in the general term formula, and considering the negative sign.

341 Easy

341 of 951

Application +4 | -1

The digit at the unit place in the number

{19^{2005}} + {11^{2005}} - {9^{2005}}\;\mathrm{is}

View answer and explanation

Correct option: B

Option B (1) is correct because the digit at the unit place is 1, as derived from binomial expansion.

342 Medium

342 of 951

Application +4 | -1

The positive integer just greater than

(1+0.0001)^{10000}

View answer and explanation

Correct option: D

Option D (3) is correct because (1 + 0.0001)¹⁰⁰⁰⁰ approaches e, which is approximately 2.718, and the next integer greater than that is 3.

343 Hard

343 of 951

Analyzing +4 | -1

Let

a_n=(\log 3)^n \sum\limits_{{r=1}}^n \frac{r^2}{r !(n-r) !}, n \in N

. Then, the sum of the series

a_1+a_2+a_3+a_4+\ldots

View answer and explanation

Correct option: A

(1+\log 3)(9 \log 3)

Option A ((1+log 3)(9 log 3)) is correct because it is the sum of the series after applying binomial theorem and simplifying the expression.

344 Easy

344 of 951

Analyzing +4 | -1

If the magnitude of the coefficient of

{x^7}

in the expansion of

{\left( {{x^2} + \frac{1}{{bx}}} \right)^8}

,where

a,b

,are positive numbers,is equal to the magnitude of the
coefficient of

{x^{ - 7}}

in the of

{\left( {ax - \frac{1}{{b{x^2}}}} \right)^8}

,then

a

,and

b

are connected by the relation

View answer and explanation

Correct option: A

ab = 1

The correct answer is A (

ab = 1

) because equating the magnitudes of the coefficients of

x^7

and

x^{-7}

leads to the relation

ab = 1

345 Easy

345 of 951

Application +4 | -1

If the number of terms in

\left(x+1+\frac{1}{x}\right)^n\left(n \in I^{+}\right)

is 401 then

n

is greater than

View answer and explanation

Correct option: B

200

Option B (200) is correct because the number of terms in the expansion is related to n by the formula 2n+1, which equals 401 in this case.

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346 Medium

346 of 951

Analyzing +4 | -1

(1+x)^n=C_0+C_1 x+C_2 x^2+\ldots+C_n x^n

, then the value of

\sum\limits_{0 \le r} {\sum\limits_{ < s \le n} {(r.s){C_r}{C_s}} }

View answer and explanation

Correct option: C

n^2\left[2^{2 n-3}-\frac{1}{2}{ }^{2 n-2} C_{n-1}\right]

Option C is correct because the expression simplifies to the given result using combinatorial identities and properties.

347 Medium

347 of 951

Application +4 | -1

If the coefficient of

4^{\text {th }}

term in the expansion of

\left(x+\frac{\alpha}{2 x}\right)^n

is 20 , then the respective values of

\alpha

and

n

are

View answer and explanation

Correct option: D

2,6

The correct answer is 2,6 because substituting these values into the binomial expansion allows us to match the coefficient of the 4th term to the given value of 20.

348 Easy

348 of 951

Application +4 | -1

The number of dissimilar terms in the expansion of

{(a + b)^n}

n + 1

therefore no of dissimilar terms of the expansion

{\left( {a + b + c} \right)^{12}}

View answer and explanation

Correct option: D

The correct answer is D (91) because the number of dissimilar terms in the expansion of

{(a + b + c)^{12}}

is calculated using the formula

{\;^{n + r - 1}}{C_{r - 1}}

, where n is the power and r is the number of variables.

349 Medium

349 of 951

Application +4 | -1

The

r\,th

terms in the expansion of

{\left( {a + 2n} \right)^n}

View answer and explanation

Correct option: B

\frac{{n\left( {n - 1} \right) \ldots \left( {n - r + 2} \right)}}{{\left( {r - 1} \right)!}}{a^{n - r + 1}}{\left( {2x} \right)^{r - 1}}

Option B (

\frac{{n\left( {n - 1} \right) \ldots \left( {n - r + 2} \right)}}{{\left( {r - 1} \right)!}}{a^{n - r + 1}}{\left( {2x} \right)^{r - 1}}

) is correct because it correctly represents the rth term in the binomial expansion of the given expression.

350 Medium

350 of 951

Application +4 | -1

The remainder when

{5^{99}}

is divided by 13, is

View answer and explanation

Correct option: B

Option B (8) is correct because when

{5^{99}}

is divided by 13, the remainder is 8, which is obtained by breaking down

{5^{99}}

and using properties of remainders.

351 Hard

351 of 951

Application +4 | -1

The digit at unit’s place in the number

{17^{1995}} + {11^{1995}} - {7^{1995}},

View answer and explanation

Correct option: B

Option B is correct. The unit digit is determined by considering the unit digits of each term and their cyclical patterns.

352 Medium

352 of 951

Application +4 | -1

If in the expansion of

{\left( {\frac{1}{x} + x\tan x} \right)^5}

the ratio of 4th term to the 2nd term is

\frac{2}{{27}}{\mathrm{\pi }^4}

, then the value of

x

can be

View answer and explanation

Correct option: B

- \frac{\mathrm{\pi }}{3}

Option B (

\frac{\mathrm{\pi }}{3}

) is correct because it satisfies the given ratio of the 4th term to the 2nd term in the expansion.

353 Medium

353 of 951

Application +4 | -1

The greatest coefficient in the expansion of

(x+y+z+w)^{15}

View answer and explanation

Correct option: A

\frac{15 !}{3 !(4 !)^3}

Option A (

\frac{{15 !}}{{3 !(4 !)^3}}

) is correct because it represents the greatest multinomial coefficient when the power is divided as evenly as possible among the variables.

354 Medium

354 of 951

Analyzing +4 | -1

(1+x)^{2 n}=a_0+a_1 x+a_2 x^2+\ldots+a_{2 n} x^{2 n}

, then

\left(1+\frac{a_1}{a_0}\right)\left(1+\frac{a_2}{a_1}\right)\left(1+\frac{a_3}{a_2}\right) \ldots\left(1+\frac{a_{2 n}}{a_{2 n-1}}\right)

is equal to

View answer and explanation

Correct option: B

\frac{(2 n+1)^{2 n}}{(2 n) !}

Option B (

\frac{{(2 n+1)^{2 n}}}{{(2 n) !}}

) is correct because it is obtained by simplifying the product of terms involving binomial coefficients.

355 Medium

355 of 951

Application +4 | -1

The number of terms in the expansion of

\left(5^{\frac{1}{6}}+7^{\frac{1}{9}}\right)^{1824}

which are integers is

View answer and explanation

Correct option: C

102

Option C, 102, is correct because it represents the number of terms where both exponents in the binomial expansion are integers.

356 Easy

356 of 951

Application +4 | -1

The sum of rational terms in the expansion of

\left(\sqrt{2}+3^{1 / 15}\right)^{10}

View answer and explanation

Correct option: B

Option B (41) is correct because the sum of rational terms in the expansion is obtained by identifying terms where both the power of 2 and 3 are integers, which occurs only for r=0 and r=10.

357 Medium

357 of 951

Application +4 | -1

Coefficient of

{x^n}

in the expansion of

\frac{{{{\left( {1 + x} \right)}^n}}}{{1 - x}}

View answer and explanation

Correct option: B

{2^n}

Option B (

{2^n}

) is correct because the coefficient of

{x^n}

in the expansion of

\frac{{{{\left( {1 + x} \right)}^n}}}{{1 - x}}

is the sum of all coefficients in the expansion of

{(1+x)^n}

, which is

{2^n}

358 Medium

358 of 951

Analyzing +4 | -1

\left(1+x+x^2\right)^n=\sum\limits_{{r=0}}^{2 \pi} a_r x^r

, then

\sum\limits_{{r=0}}^n(-1)^r a_r{ }^n C_r={ }^n C_{n / 3}

, where

n

View answer and explanation

Correct option: C

3 k

Option C (

3k

) is correct because the given condition holds true only when

n

is a multiple of 3, which can be represented as

3k

359 Medium

359 of 951

Analyzing +4 | -1

n > 3,

then

xyz\;{C_0} - \left( {x - 1} \right)\left( {y - 1} \right)\left( {z - 1} \right){C_1} + \left( {x - 2} \right)\left( {y - 2} \right)\left( {z - 2} \right){C_2}

- \left( {x - 3} \right)\left( {y - 3} \right)\left( {z - 3} \right){C_3} + \ldots + {\left( { - 1} \right)^n}\left( {x - n} \right)\left( {y - n} \right)\left( {z - n} \right){C_n}

equals

View answer and explanation

Correct option: D

Option D (0) is correct because the given expression simplifies to zero due to the properties of binomial coefficients and alternating signs.

360 Medium

360 of 951

Application +4 | -1

The remainder when

{\left( {2021} \right)^{2023}}

is divided by 7 is:

View answer and explanation

Correct option: C

Option C (

\frac{1}{\sqrt{2}}

) is correct because it is the eccentricity of the ellipse formed by the locus of the midpoint, derived from the original ellipse's equation.

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