If $n$ is a positive integer and ${C_k} = {\;^n}{C_k},$ then $\mathop \sum \limits_{k = 1}^n {k^3}{\left( {\frac{{{C_k}}}{{{C_{k - 1}}}}} \right)^2}$ equals

$\frac{{n\left( {n + 1} \right)\left( {n + 2} \right)}}{{12}}$

$\frac{{n\left( {n + 1} \right){{\left( {n + 2} \right)}^2}}}{{12}}$

The coefficient of $x^{25}$ in $(1+x)^5+(1+x)^6+(1+x)^7+\ldots+(1+x)^{50}$ is

$\left(\begin{array}{l}50 \\ 25\end{array}\right)$

$\left(\begin{array}{l}51 \\ 25\end{array}\right)-1$

$\left(\begin{array}{l}46 \\ 25\end{array}\right)$

The value of ${1^2}.{C_1} + {3^2}.{C_3} + {5^2}.{C_5} + ...$ is

$n{\left( {n - 1} \right)^{n - 2}} + n{.2^{n - 1}}$

$n{\left( {n - 1} \right)^{n - 2}}$

$n\left( {n - 1} \right){.2^{n - 3}}$

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JEE • MATHEMATICS • BINOMIAL THEOREM

Binomial theorem practice · Mixed difficulty

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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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541 Easy

541 of 951

Analyzing +4 | -1

\sum\limits_{r=0}^{n-1} \frac{{ }^n C_r}{{ }^n C_r+{ }^n C_{r+1}}

is equal to

View answer and explanation

Correct option: A

\frac{n}{2}

Option A (n/2) is correct because simplifying the summation using binomial coefficient identities leads to a straightforward calculation.

542 Medium

542 of 951

Analyzing +4 | -1

Let

n \geq 5

be an integer. If

9^n-8 n-1=64 \alpha

and

6^n-5 n-1=25 \beta

, then

\alpha-\beta

is equal to

View answer and explanation

Correct option: C

^n{C_3}(8 - 5){ + ^n}{C_4}\left( {{8^2} - {5^2}} \right) + \ldots { + ^n}{C_n}\left( {{8^{n - 2}} - {5^{n - 2}}} \right)

Option C is correct because it represents the difference between the series expansions of (1+8)^n and (1+5)^n, after accounting for the initial terms and dividing by 64 and 25 respectively.

543 Easy

543 of 951

Application +4 | -1

If the coefficient of

x

\left(x^2+k / x\right)^5

is 270, then the value of

k

View answer and explanation

Correct option: A

Option A (3) is correct because the coefficient of

x

\left(x^2+k / x\right)^5

being 270 leads to the equation

{ }^5 C_3 k^3=270

, which gives

k=3

544 Easy

544 of 951

Analyzing +4 | -1

The term independent of

x

in the expansion of

\left(1-\frac{1}{2 x}+3 x^7\right)\left(x^2+\frac{1}{3 x}\right)^{10}

View answer and explanation

Correct option: A

10 / 3^8

Option A (

10 / 3^8

) is correct because it is the term independent of x after expanding the given expression and finding the constant term.

545 Easy

545 of 951

Analyzing +4 | -1

The number of values of '

r

' satisfying the equation

{ }^{39} C_{3 r-1}-{ }^{39} C_{r^2}={ }^{39} C_{r^2-1}-{ }^{39} C_{3 r}

View answer and explanation

Correct option: B

Option B (2) is correct because solving the given equation using properties of combinations yields two valid solutions for 'r': 3 and 5.

546 Medium

546 of 951

Application +4 | -1

\left(2+\frac{x}{3}\right)^{55}

is expanded in the ascending powers of

x

and the coefficients of powers of

x

in two consecutive terms of the expansion are equal, then these terms are:

View answer and explanation

Correct option: A

7^{\text {th }}

and

8^{\text {th }}

Option A (7th and 8th) is correct because the coefficients of x in the 7th and 8th terms are equal after solving the binomial expansion.

547 Medium

547 of 951

Application +4 | -1

The coefficient of

{x^5}

in the expansion of

{\left( {2 - x + 3{x^2}} \right)^6},

View answer and explanation

Correct option: D

- 5052

Option D (

- 5052

) is correct because it is the result of calculating the coefficient of

{x^5}

by considering all possible combinations of terms in the multinomial expansion.

548 Medium

548 of 951

Analyzing +4 | -1

If in the expansion of

(1+x)^n, a, b

and

c

are three consecutive coefficients, then

n

is equal to

View answer and explanation

Correct option: B

\frac{2 a c+a b+b c}{b^2-a c}

Option D (None of these) is correct because the given expression for

n

does not hold true when tested with simple values.

549 Medium

549 of 951

Analyzing +4 | -1

The term independent of

x

in the binomial expansion of

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

is:

View answer and explanation

Correct option: A

400

Option A (400) is correct because it is the sum of the constant terms obtained by multiplying the given expression and considering the powers of x that cancel out.

550 Hard

550 of 951

Analyzing +4 | -1

n

is a positive integer and

{C_k} = {\;^n}{C_k},

then

\sum \limits_{k = 1}^n {k^3}{\left( {\frac{{{C_k}}}{{{C_{k - 1}}}}} \right)^2}

equals

View answer and explanation

Correct option: B

\frac{{n{{\left( {n + 1} \right)}^2}\left( {n + 2} \right)}}{{12}}

Option B (

\frac{{n{{\left( {n + 1} \right)}^2}\left( {n + 2} \right)}}}{{12}}

) is correct because, after simplifying the expression and using the formulas for the sum of first n integers, squares, and cubes, we arrive at the given result.

551 Medium

551 of 951

Application +4 | -1

The last digit of

(2137)^{754}

View answer and explanation

Correct option: D

Option D (9) is correct because the last digit is found by looking at the pattern of the last digit of powers of 7, and 754 leaves a remainder of 2 when divided by 4.

552 Medium

552 of 951

Analyzing +4 | -1

The coefficient of the

(r+1)

th term of

\left(x+\frac{1}{x}\right)^{20}

when expanded in the descending powers of

x

, is equal to the coefficient of the 6 th term of

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

when expanded in ascending powers of

x

. The value of

r

View answer and explanation

Correct option: D

None of the above

Option A (5) is the closest to the correct value of r, which is derived by equating the coefficients of the binomial expansions and solving for r.

553 Easy

553 of 951

Application +4 | -1

The coefficient of

{x^{ - 10}}\;\mathrm{in\;}{\left( {{x^2} - \frac{1}{{{x^3}}}} \right)^{10}}

View answer and explanation

Correct option: B

210

Option B (210) is correct because it is the coefficient of the term with x^-10 in the given binomial expansion.

554 Medium

554 of 951

Application +4 | -1

The coefficient of

x^4

in the expansion of

\left(1+x+x^2+x^3\right)^{11}

View answer and explanation

Correct option: A

990

Option A (990) is correct because expanding the expression and finding the coefficient of x⁴ results in 990.

555 Medium

555 of 951

Application +4 | -1

Evaluate

\frac{1}{\sqrt[3]{6-3 x}}

View answer and explanation

Correct option: B

6^{-1 / 3}\left[1+\frac{x}{6}+\frac{2 x^2}{6^2}+\ldots\right]

Option B (

6^{-1 / 3}\left[1+\frac{x}{6}+\frac{2 x^2}{6^2}+\ldots\right]

) is correct because it is the binomial expansion of the given expression.

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556 Easy

556 of 951

Application +4 | -1

Coefficient of

{x^{19}}

in the polynomial

\left( {x - 1} \right)\left( {x - 2} \right) \ldots ..\left( {x - 20} \right)

is equal to

View answer and explanation

Correct option: B

- 210

Option B (-210) is correct because it is the negative sum of the roots of the polynomial, which corresponds to the coefficient of x¹⁹.

557 Medium

557 of 951

Understanding +4 | -1

The sum

{ }^{20} C_0+{ }^{20} C_1+{ }^{20} C_2+\ldots .+{ }^{20} C_{10}

is equal to

View answer and explanation

Correct option: D

None of these

None of these is the correct answer because the sum is equal to

2^{19} + \frac{{ }^{20}C_{10}}{2}

558 Easy

558 of 951

Application +4 | -1

n

is an odd natural number, then number of zeros at the end of

99^n+1

View answer and explanation

Correct option: A

Option A (2) is correct because the last two digits of 99^n + 1 are 01 when n is odd, resulting in exactly two zeros at the end.

559 Easy

559 of 951

Application +4 | -1

If the binomial coefficient of the third term in the expansion of

\left(9 x-\frac{1}{\sqrt{3 x}}\right)^m

is 105 , then the third term in the expansion is

View answer and explanation

Correct option: A

35 \cdot 9^{13} \cdot x^{12}

Option A (

35 \cdot 9^{13} \cdot x^{12}

) is correct because it is the third term in the expansion.

560 Easy

560 of 951

Application +4 | -1

If the coefficient of

{x^2}

and

{x^3}

in the expansion of

{\left( {3 + ax} \right)^9}

be same, then the value of

a

View answer and explanation

Correct option: D

\frac{9}{7}

Option D (

\frac{9}{7}

) is correct because equating the coefficients of

{x^2}

and

{x^3}

in the expansion of

{\left( {3 + ax} \right)^9}

leads to the equation

{{\;^9}{C_2}{3^7}{a^2} = {\;^9}{C_3}{3^6}{a^3}}

, which simplifies to

a = \frac{9}{7}

561 Medium

561 of 951

Application +4 | -1

The coefficient of

x^{25}

(1+x)^5+(1+x)^6+(1+x)^7+\ldots+(1+x)^{50}

View answer and explanation

Correct option: B

\left(\begin{array}{l}51 \\ 26\end{array}\right)

Option B is correct because the coefficient of

x^{25}

is found by summing a geometric progression and then extracting the relevant coefficient.

562 Medium

562 of 951

Application +4 | -1

If the middle term in the expansion of

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

924 x^6

, then

n

is equal to

View answer and explanation

Correct option: B

Option B (12) is correct because the middle term in the expansion is used to find the value of n, which is determined to be 12.

563 Hard

563 of 951

Application +4 | -1

The coefficient of

x^3

in the expansion of

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

View answer and explanation

Correct option: D

-30

Option D (-30) is correct because it is the coefficient of x³ in the expansion of (1-x+x²)⁵, obtained by considering relevant terms in the expansion.

564 Easy

564 of 951

Application +4 | -1

The coefficient of

y

in the expansion of

{\left( {{y^2} + c/y} \right)^5},

View answer and explanation

Correct option: C

10\;{c^3}

Option C (

10\;{c^3}

) is correct because when expanding

{\left( {{y^2} + c/y} \right)^5}

, the term containing

y

has a coefficient of

10c^3

565 Medium

565 of 951

Application +4 | -1

\alpha \in[-1,1]

and

I(\alpha)

denotes the term independent of

x

in the expansion of

\left(x \sin ^{-1} \alpha+\frac{1}{x} \cos ^{-1} \alpha\right)^{10}

then

|I(\alpha)|

cannot exceed

View answer and explanation

Correct option: C

{ }^{10} C_5 \frac{\pi^{10}}{2^{20}}

Option C (

{ }^{10} C_5 \frac{\pi^{10}}{2^{20}}

) is correct because the term independent of x is maximized when

\sin^{-1} \alpha = \cos^{-1} \alpha = \frac{\pi}{4}

566 Medium

566 of 951

Application +4 | -1

The coefticient of

x^n

in the expansion of

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

View answer and explanation

Correct option: A

\frac{2^n}{n !}

Option A (

\frac{2^n}{n !}

) is correct. The coefficient of

x^n

in the expansion of the given expression is equivalent to the coefficient of

x^n

e^{2x}

, which is

\frac{2^n}{n !}

567 Hard

567 of 951

Analyzing +4 | -1

\left( {\begin{array}{cccccccccccccccccccc} {30}\\ 0 \end{array}} \right)\left( {\begin{array}{cccccccccccccccccccc} {30}\\ {10} \end{array}} \right) - \left( {\begin{array}{cccccccccccccccccccc} {30}\\ 1 \end{array}} \right)\left( {\begin{array}{cccccccccccccccccccc} {30}\\ {11} \end{array}} \right) + ... + \left( {\begin{array}{cccccccccccccccccccc} {30}\\ {20} \end{array}} \right)\left( {\begin{array}{cccccccccccccccccccc} {30}\\ {30} \end{array}} \right)

is equal to

View answer and explanation

Correct option: C

{\;^{30}}{C_{10}}

Option C is correct because the given expression is the coefficient of x²⁰ in (1-x²)³⁰, which simplifies to ^{30}C_{10}.

568 Medium

568 of 951

Application +4 | -1

n !\left(\frac{1}{n !}+\frac{1}{2 !(n-2) !}+\frac{1}{4 !(n-4) !}+\ldots \ldots+\frac{1}{n !}\right)

is equal to -

View answer and explanation

Correct option: B

2^{n-1}

Option B (

2^{n-1}

) is correct because the given expression simplifies to the sum of even-indexed binomial coefficients, which equals

2^{n-1}

569 Hard

569 of 951

Application +4 | -1

The value of

1 \times 2 \times 3 \times 4 + 2 \times 3 \times 4 \times 5 + 3 \times 4 \times 5 \times 6 + \ldots + n\left( {n + 1} \right)\left( {n + 2} \right)\left( {n + 3} \right),

View answer and explanation

Correct option: B

\frac{1}{5}n\left( {n + 1} \right)\left( {n + 2} \right)\left( {n + 3} \right)\left( {n + 4} \right)

Option C (

\frac{1}{5}n\left( {n + 1} \right)\left( {n + 2} \right)\left( {n + 3} \right)\left( {n + 4} \right)

) is correct because it represents the sum of the given series, which can be derived using summation properties and binomial coefficients.

570 Medium

570 of 951

Analyzing +4 | -1

The value of

{1^2}.{C_1} + {3^2}.{C_3} + {5^2}.{C_5} + ...

View answer and explanation

Correct option: D

None of these

The correct answer is D (None of these) because the given options do not match the derived expression for the sum of the series.

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