The value of ${ }^n C_3+{ }^n C_7+{ }^n C_{11}+\ldots$ is

$\frac{1}{4}\left\{2^{n+1}-2^{n / 2} \sin \frac{n \pi}{4}\right\}$

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JEE • MATHEMATICS • BINOMIAL THEOREM

Binomial theorem practice · Medium

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These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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Showing 361-390 of 468 questions

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361 Medium

361 of 468

Analyzing +4 | -1

{\left( {1 + x + {x^2}} \right)^n} = {C_0} + {C_1}x + {C_2}{x^2} + \ldots ,

then the value of

{C_0}{C_1} - {C_1}{C_2} + {C_2}{C_3} - \ldots ,

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the value of the expression is 0, which is not equal to any of the given options.

362 Medium

362 of 468

Application +4 | -1

If in the expansion of

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

, the coefficient of

x

and

{x^2}

are 3 and

- 6

respectively, then

m

View answer and explanation

Correct option: C

Option C (12) is correct because by expanding the product of the two binomials and equating the coefficients of

x

and

x^2

to the given values, we can solve for

m

363 Medium

363 of 468

Application +4 | -1

A ratio of the 5 th term from the beginning to the

5^{\text {th }}

term from the end in the binomial expansion of

\left(2^{\frac{1}{3}}+\frac{1}{2(3)^{\frac{1}{3}}}\right)^{10}

View answer and explanation

Correct option: D

4(36)^{\frac{1}{3}}: 1

Option D (

4(36)^{\frac{{1}}{{3}}}: 1

) is correct because the ratio of the 5th term from the beginning to the 5th term from the end is calculated by using the binomial theorem and simplifying the expression.

364 Medium

364 of 468

Analyzing +4 | -1

The value of

{ }^n C_3+{ }^n C_7+{ }^n C_{11}+\ldots

View answer and explanation

Correct option: A

\frac{1}{2}\left\{2^{n-1}-2^{n / 2} \sin \frac{n \pi}{4}\right\}

Option A (

\frac{{1}}{{2}}\left{2^{{n-1}}-2^{{n / 2}} \sin \frac{{n \pi}}{{4}}\right}

) is correct because by using the binomial theorem and complex numbers, the given series can be expressed and simplified to

\frac{{1}}{{2}}\left{2^{{n-1}}-2^{{n / 2}} \sin \frac{{n \pi}}{{4}}\right}

365 Medium

365 of 468

Application +4 | -1

The value of

\sum_{r=0}^{20}{ }^{50-r} C_{6}

is equal to :

View answer and explanation

Correct option: A

{ }^{51} C_{7}-{ }^{30} C_{7}

Option A (

{}^{51} C_{7}-{ }^{30} C_{7}

) is correct because it uses the Hockey Stick Identity to simplify the summation of binomial coefficients.

366 Medium

366 of 468

Application +4 | -1

If 21 st and 22 nd terms in the expansion of

(1+x)^{44}

are equal, then

x

is equal to

View answer and explanation

Correct option: C

7 / 8

Option C (7 / 8) is correct because the equality of consecutive terms in a binomial expansion leads to the equation x = 7/8.

367 Medium

367 of 468

Application +4 | -1

If for

1 \leq m \leq n, f(m, n)=C_0-C_1+C_2-

\cdots+(-1)^{m-1} C_{m-1}

, then

\frac{f(9,5)}{10}

View answer and explanation

Correct option: C

Option C (7) is correct because the alternating sum of binomial coefficients is simplified by relating it to the coefficient of a term in a binomial expansion.

368 Medium

368 of 468

Application +4 | -1

The coefficient of

x^4

\left(\frac{x}{2}-\frac{3}{x^2}\right)^{10}

View answer and explanation

Correct option: B

\frac{405}{256}

Option B is correct because by finding the general term in the binomial expansion and setting the power of x to 4, we can find the coefficient of x⁴.

369 Medium

369 of 468

Application +4 | -1

The coefficient of

x^5

\left(1+2 x+3 x^2+\ldots\right)^{-3 / 2}

View answer and explanation

Correct option: D

None of the above

Option D (None of the above) is correct because the coefficient of

x^5

in the simplified expression is 0.

370 Medium

370 of 468

Application +4 | -1

There are numerous balls of red, yellow and blue colours. In how many ways 12 balls can be chosen from them?

View answer and explanation

Correct option: C

Option C, 91, is correct because it represents the number of ways to choose 12 balls from the three colors.

371 Medium

371 of 468

Application +4 | -1

(a+b x)^{-3}=\frac{1}{27}+\frac{1}{3} x+\ldots

, then the ordered pair

(a, b)

equals

View answer and explanation

Correct option: D

(3,-9)

Option D ((3,-9)) is correct because solving the equations derived from the binomial expansion yields a=3 and b=-9.

372 Medium

372 of 468

Analyzing +4 | -1

{ }^{n+1} C_{r+1}:{ }^n C_r:{ }^{n-1} C_{r-1}=11: 6: 3

, then

n r

is equal to

View answer and explanation

Correct option: D

Option D (50) is correct because solving the given ratio equations yields n=10 and r=5, thus nr = 50.

373 Medium

373 of 468

Analyzing +4 | -1

The expression

{\left[ {x + {{\left( {{x^3} - 1} \right)}^{1/2}}} \right]^5} + {\left[ {x - {{\left( {{x^3} - 1} \right)}^{1/2}}} \right]^5}

is a polynomial of degree

View answer and explanation

Correct option: C

Option C (7) is correct because after expanding and simplifying, the highest power of x is 7.

374 Medium

374 of 468

Analyzing +4 | -1

Number of different terms in the expression

(1+x)^{300}+\left(1+x^2\right)^{200}+\left(1+x^3\right)^{150}

are

View answer and explanation

Correct option: A

385

Option A (385) is correct because it is the result of applying the principle of inclusion-exclusion to find the total number of distinct terms.

375 Medium

375 of 468

Application +4 | -1

The largest real value of

x

, such that

\sum\limits_{r=0}^4\left(\frac{5^{4-r}}{(4-r) !}\right)\left(\frac{x^r}{r !}\right)=\frac{8}{3}

View answer and explanation

Correct option: A

2 \sqrt{2}-5

Option A (

2 \sqrt{2}-5

) is correct because the largest real value of

x

satisfying the given summation can be found by recognizing the summation as a binomial expansion and solving the resulting equation.

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376 Medium

376 of 468

Application +4 | -1

The value of

\sum\limits_{{r=1}}^{10} r \cdot \frac{{ }^n C_r}{{ }^n C_{r-1}}

is equal to

View answer and explanation

Correct option: A

5(2 n-9)

Option A,

5(2n-9)

, is correct because it is the simplified form of the summation after applying the formula for combinations and simplifying the expression.

377 Medium

377 of 468

Application +4 | -1

The coefficient of

a^8 b^6 c^4

in the expansion of

(a+b+c)^{18}

View answer and explanation

Correct option: B

13063680

Option B (13063680) is correct because it is the coefficient of x³y⁴z² in the expansion of (2x-3y+4z)⁹, calculated using the multinomial theorem.

378 Medium

378 of 468

Analyzing +4 | -1

The coefficient of the term independent of

x

in the expansion of

\left( {1 + x + 2{x^3}} \right){\left( {\frac{3}{2}{x^2} - \frac{1}{{3x}}} \right)^9}

View answer and explanation

Correct option: C

\frac{{17}}{{54}}

Option C (

\frac{{17}}{{54}}

) is correct because it is the sum of the coefficients of the constant terms obtained by multiplying the terms in the first expression with the appropriate terms in the binomial expansion.

379 Medium

379 of 468

Analyzing +4 | -1

In the expansion of

\left(x^{\frac{4}{5}}+x^{-\frac{1}{5}}\right)^n

, the coeficient of the

8^{\text {th }}

and

19^{\text {th }}

terms are equal. The term independent of

x

is given by

View answer and explanation

Correct option: B

{ }^{25} C_{20}

The correct answer is B (

{ }^{25} C_{20}

) because it represents the term independent of x in the binomial expansion, found by setting the power of x to zero and using the binomial coefficient.

380 Medium

380 of 468

Application +4 | -1

Coefficient of

x^{25}

in expansion of expression

\sum\limits_{{r=0}}^{50}{ }^{50} C_r(2 x-3)^r(2-x)^{50-r}

View answer and explanation

Correct option: D

-{ }^{50} \mathrm{C}_{25}

Option D (-

{ }^{50} \mathrm{C}_{25}

) is correct because after simplification, the expression becomes (x-1)⁵⁰, and the coefficient of x²⁵ is -

{ }^{50} \mathrm{C}_{25}

381 Medium

381 of 468

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} + {C_r}{C_s})

View answer and explanation

Correct option: B

n^2 \cdot 2^n+\frac{n}{2}\left[2^{2 n}-{ }^{2 n} C_n\right]

Option B (

n^2 \cdot 2^n+\frac{n}{2}[2^{2 n}-{ }^{2 n} C_n]

) is correct because it combines the results from previous questions to find the final answer.

382 Medium

382 of 468

Application +4 | -1

\,\,\, f(x)=1-x+x^2-x^3+\ldots-x^{15}+x^{16}-x^{17}

. Then, the coefficient of

x^2

f(x-1)

View answer and explanation

Correct option: B

816

Option B (816) is correct because the coefficient of x² in f(x-1) is equal to the coefficient of x³ in 1-(x-1)¹⁸, which is calculated as

{ }^{18} C_3=816

383 Medium

383 of 468

Analyzing +4 | -1

{a_r}

is the coefficient of

{x^{r - 1}}

{\left( {1 + x} \right)^n} + {\left( {1 + x} \right)^{n + 1}} + ... + {\left( {1 + x} \right)^{n + k}}\;\left( {n < r - 1 \le n + k} \right)

, then

\sum \limits_{r = 0}^{n + k + 1} {\left( { - 1} \right)^r}{a_r}

is equal to

View answer and explanation

Correct option: A

Option A (0) is correct because the alternating sum of coefficients in the expansion equals zero.

384 Medium

384 of 468

Analyzing +4 | -1

Let

x=(8 \sqrt{3}+13)^{13}

and

y=(7 \sqrt{2}+9)^9

. If [f] denotes the greatest integer

\leq t

, then

View answer and explanation

Correct option: B

[x]+[y]

is even

The correct answer is

[x]+[y]

is even because both

[x]

and

[y]

are even, which is shown by considering the conjugate terms and their integer parts.

385 Medium

385 of 468

Application +4 | -1

T_{r+1}

is the term independent of

x

\left(\sqrt{a} x^2+\frac{1}{2 x^3}\right)^{10}

then

r=

View answer and explanation

Correct option: A

Option A (2) is correct because there are two solutions for θ in the given interval that satisfy the trigonometric equation.

386 Medium

386 of 468

Application +4 | -1

If the fourth term in the binomial expansion of

{\left( {\frac{2}{x} + {x^{{{\log }_8}x}}} \right)^6}(x > 0)

20 \times {8^7}

, then the value of

x

View answer and explanation

Correct option: D

{8^2}

Option D (

{8^2}

) is correct because when substituting x =

{8^2}

into the binomial expansion, the fourth term simplifies to

20 \times {8^7}

387 Medium

387 of 468

Application +4 | -1

Sum of coefficients of terms which contains integral powers of

x

in the expansion

(2+\sqrt{x})^{30}

View answer and explanation

Correct option: D

\frac{3^{30}+1}{2}

Option D (

\frac{{3^{30}+1}}{{2}}

) is correct because it represents the sum of coefficients of terms with integral powers of x in the binomial expansion.

388 Medium

388 of 468

Application +4 | -1

If the coefficients of 5th, 6th and 7th terms in the expansion of

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

be in AP, then the value of

n

View answer and explanation

Correct option: C

7 or 14

Option C (7 or 14) is correct because these are the values of n that satisfy the given condition where the coefficients of the 5th, 6th, and 7th terms are in AP.

389 Medium

389 of 468

Analyzing +4 | -1

If some three consecutive coefficients in the binomial expansion of

{(x + 1)^n}

of in powers of x are in the ratio 2 : 15 : 70, then the average of these three coefficients is

View answer and explanation

Correct option: C

232

Option C (232) is correct because it is the average of the three consecutive binomial coefficients after determining the values of n and r.

390 Medium

390 of 468

Application +4 | -1

The coefficient of

x^{5}

in the expansion of

\left(2 x^{3}-\frac{1}{3 x^{2}}\right)^{5}

View answer and explanation

Correct option: A

\frac{80}{9}

Option A (80/9) is correct because after applying the binomial theorem, the coefficient of x⁵ in the expansion is found to be 80/9.

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