If ${\left( {1 + x} \right)^n} = {C_0} + {C_1}x + {C_2}{x^2} + ... + {C_n}{x^n}$, then $\begin{array}{*{20}{c}} {\sum \;}\\ {0 \le r < s \le n} \end{array}\;\;\sum \left( {r + s} \right){C_r}{C_s}\;$ is equal to

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

The coefficient of ${\lambda ^n}{{\rm{\mu }}^n}$ in the expansion of ${\left[ {\left( {1 + \lambda } \right)\left( {1 + {\rm{\mu }}} \right)\left( {\lambda + {\rm{\mu }}} \right)} \right]^n}$ is

$\mathop \sum \limits_{r = 0}^n C_r^3$

$\mathop \sum \limits_{r = 0}^n C_r^2$

$\mathop \sum \limits_{r = 0}^n C_{r + 2}^2$

$\mathop \sum \limits_{r = 0}^n C_{r + 3}^2$

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JEE • MATHEMATICS • BINOMIAL THEOREM

Binomial theorem practice · Mixed difficulty

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781 Medium

781 of 951

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}

782 Medium

782 of 951

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.

783 Hard

783 of 951

Understanding +4 | -1

\left(4-5 x^2\right)^{-1 / 2}

can be expanded as a power series of

x

View answer and explanation

Correct option: B

|x| < 2 / \sqrt{5}

Option B (

|x| < 2 / \sqrt{5}

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

784 Medium

784 of 951

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.

785 Medium

785 of 951

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.

786 Medium

786 of 951

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.

787 Easy

787 of 951

Application +4 | -1

If the fifth term in the expansion of

\left(x^{1 / 3}+\frac{1}{x}\right)^n

does not depend upon

x

, then

{ }^n C_2+{ }^n C_{14}

equals

View answer and explanation

Correct option: A

240

Option A (240) is correct because the value of n is determined by the condition that the fifth term is independent of x, and then the expression is calculated.

788 Medium

788 of 951

Application +4 | -1

The number of terms which are free from fractional powers in the expansion of

\left(a^{1 / 5}+b^{2 / 3}\right)^{45}, a \neq b

View answer and explanation

Correct option: C

Option C (4) is the correct answer because there are four terms in the expansion that are free from fractional powers.

789 Medium

789 of 951

Application +4 | -1

The coefficient of

x^{n}

in expansion of

(1+x)(1-x)^{n}

View answer and explanation

Correct option: B

(-1)^{n}(1-n)

Option B ((-1)^n(1-n)) is correct because it is the result of calculating the coefficient of x^n in the given expansion.

790 Easy

790 of 951

Application +4 | -1

The term independent of

x

in the expansion of

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

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the term independent of x does not exist as the power of x cannot be zero.

791 Medium

791 of 951

Application +4 | -1

x

is positive, the first negative term in the expansion of

(1+x)^{27 / 5}

View answer and explanation

Correct option: C

8 th term

Option C (8 th term) is correct because it is the first term where the factor (27/5 - r + 1) becomes negative, leading to a negative term in the expansion.

792 Easy

792 of 951

Analyzing +4 | -1

The total number of terms which depend on the value of

x

in the expansion of

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

View answer and explanation

Correct option: B

2 n

Option B (

2 n

) is correct because the terms independent of x are removed, leaving 2n terms that depend on x.

793 Medium

793 of 951

Application +4 | -1

(1+x)^n=\sum_{r=0}^n{ }^n C_r \cdot(x)^r

, then

\left(1+\frac{{ }^n C_1}{{ }^n C_0}\right) \cdot\left(1+\frac{{ }^n C_2}{{ }^n C_1}\right) \cdots\left(1+\frac{{ }^n C_n}{{ }^n C_{n-1}}\right)=

View answer and explanation

Correct option: C

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

Option C (

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

) is correct because the expression simplifies to a product of terms that results in this form.

794 Easy

794 of 951

Application +4 | -1

Constant term in the expansion of

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

View answer and explanation

Correct option: C

-252

Option C (-252) is correct because it is the constant term obtained by finding the term independent of x in the binomial expansion.

795 Medium

795 of 951

Application +4 | -1

Sum of coefficients of all terms containing integral power of

x

in expansions of

(1-2 \sqrt{x})^{20}

View answer and explanation

Correct option: D

\frac{3^{20}+1}{2}

Option D is correct. The sum of coefficients of terms with integral powers of x is found by substituting x=1 into the expression derived from adding (1-2√x)²⁰ and (1+2√x)²⁰ and dividing by 2.

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SEQUENCE OF SERIES 1764 MCQs DEFINITE INTEGRATION 1443 MCQs MATRICES 1363 MCQs INDEFINITE INTEGRATION 1179 MCQs PROBABILITY 1146 MCQs COMPLEX NUMBERS 980 MCQs

796 Easy

796 of 951

Application +4 | -1

\left(1+x+x^2\right)^8=a_0+a_1 x+a_2 x^2+\cdots+

a_{16} x^{16}

, then

a_5

equals

View answer and explanation

Correct option: A

504

Option A (504) is correct because the coefficient of x⁵ in the expansion of (1+x+x²)⁸ is 504.

797 Easy

797 of 951

Application +4 | -1

The coefficient of

a^3 b^4

in the expansion of

(a+2 b)^3(2 a-b)^4

View answer and explanation

Correct option: B

-15

Option B (-15) is correct because it is the coefficient of

a^3b^4

obtained after expanding and multiplying the two binomial expressions.

798 Medium

798 of 951

Analyzing +4 | -1

a_{r}

is the coefficient of

x^{10-r}

in expansion of

(1+x)^{10}

then

\sum\limits_{{r=1}}^{10} r^{3}\left(\frac{a_{r}}{a_{r-1}}\right)^{2}

is:

View answer and explanation

Correct option: B

1210

Option B (1210) is correct because after simplifying the summation, it can be expressed in terms of sums of squares and cubes of integers, which have known formulas.

799 Hard

799 of 951

Analyzing +4 | -1

The coefficient of

{x^n}

in the polynomial

\left( {x + {\;^n}{C_0}} \right)\left( {x + \;3{\;^n}{C_1}} \right)\left( {x + \;5{\;^n}{C_2}} \right) \ldots \left[ {x + \left( {2n + 1} \right){\;^n}{C_n}} \right]

View answer and explanation

Correct option: C

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

The correct answer is C ((

n + 1

)

2^n

) because the coefficient of x^n is the sum of the roots of the polynomial, which can be simplified using binomial properties.

800 Medium

800 of 951

Application +4 | -1

The coefficient of

{x^3}{y^4}{z^5}

in the expansion of

{(xy + yz + xz)^6}

View answer and explanation

Correct option: B

Option B (60) is correct because after expanding the expression and solving the system of equations for the exponents, the coefficient simplifies to 60.

801 Easy

801 of 951

Application +4 | -1

If,

{a_n} = \sum \limits_{r = 0}^n \frac{1}{{{\;^n}{C_r}}}

, then

\sum \limits_{r = 0}^n \frac{r}{{{\;^n}{C_r}}}

equals

View answer and explanation

Correct option: C

\frac{1}{2}n{a_n}

Option C (

\frac{1}{2}n{a_n}

) is correct because by using the property of binomial coefficients and summing the series in reverse, we can derive the given expression.

802 Easy

802 of 951

Application +4 | -1

{49^n} + 16n - 1

is divisible by

View answer and explanation

Correct option: C

Option C (64) is correct because the expression can be rewritten using the binomial theorem, and the resulting terms are divisible by 64.

803 Easy

803 of 951

Application +4 | -1

The coefficient of

{x^7}

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

is equal to

View answer and explanation

Correct option: A

62640

Option A (62640) is correct because it is the coefficient of

{x^7}

obtained by considering all possible combinations of powers of the terms in the expansion of

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

that result in

{x^7}

804 Medium

804 of 951

Application +4 | -1

If the polynomial

f(x)=1-x+x^2-x^3+\cdots+x^{19}+x^{20}

is expressed as

g(y)=a_0+a_1 y+a_2 y^2+\cdots+a_{20} y^{20}

, where

y=x-4

, then the value of

a_0+a_1+a_2+\cdots+a_{20}

View answer and explanation

Correct option: D

\frac{1+5^{21}}{6}

Option D (

\frac{{1+5^{21}}}{{6}}

) is correct. The value of the polynomial is found by substituting y=1 into the expression g(y) after converting f(x) to a geometric series.

805 Easy

805 of 951

Analyzing +4 | -1

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

, then

\begin{array}{cccccccccccccccccccc} {\sum \;}\\ {0 \le r < s \le n} \end{array}\;\;\sum \left( {r + s} \right){C_r}{C_s}\;

is equal to

View answer and explanation

Correct option: A

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

Option A (

n[2^{2n-1} - ^{2n-1}C_{n-1}]

) is correct because it is derived by manipulating the sum of products of binomial coefficients.

806 Medium

806 of 951

Analyzing +4 | -1

Coefficient of

x^m

in the expansion of

S=(1+x)^{2 m}+x(1+x)^{2 m-1}+x^2(1 +x)^{2 m-2} +\ldots .+x^{2 m}

View answer and explanation

Correct option: B

{ }^{2 m+1} C_m

Option B,

{ }^{{2 m+1}} C_m

, is correct because the given series is a geometric progression that simplifies to

(1+x)^{2m+1} - x^{2m+1}

, and the coefficient of

x^m

{ }^{{2m+1}} C_m

807 Medium

807 of 951

Application +4 | -1

The greatest integer which divides the number

101^{100}-1

View answer and explanation

Correct option: C

10000

Option C (10000) is correct because the greatest integer that divides

101^{{100}}-1

is 10000, which is found using the binomial theorem.

808 Easy

808 of 951

Analyzing +4 | -1

{C_r}

stands for

{\;^n}{C_r},

then the sum of first

\left( {n + 1} \right)

terms of the series

a\;{C_0} - \left( {a + d} \right){C_1} + \left( {a + 2d} \right){C_2} - \left( {a + 3d} \right){C_3} + \ldots ,

View answer and explanation

Correct option: C

The correct answer is C (0) because the given series can be expressed as a sum involving binomial coefficients with alternating signs, which evaluates to zero.

809 Easy

809 of 951

Analyzing +4 | -1

The coefficient of

{\lambda ^n}{\mathrm{\mu }^n}

in the expansion of

{\left[ {\left( {1 + \lambda } \right)\left( {1 + \mathrm{\mu }} \right)\left( {\lambda + \mathrm{\mu }} \right)} \right]^n}

View answer and explanation

Correct option: D

\sum \limits_{r = 0}^n C_r^3

Option D (

\sum \limits_{r = 0}^n C_r^3

) is correct because it represents the coefficient of

\lambda ^n \mu^n

in the expansion of the given expression, which simplifies to the sum of cubes of binomial coefficients.

810 Easy

810 of 951

Application +4 | -1

If the

r

th term in the expansion of

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

contains

{x^4},

then

r

is equal to

View answer and explanation

Correct option: B

Option B (3) is correct because the value of r that makes the term contain x⁴ is 3, as found by solving the equation 13 - 3r = 4.

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