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JEE • MATHEMATICS • BINOMIAL THEOREM

Binomial theorem practice · Easy

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Showing 1-30 of 434 questions

1 Easy

1 of 434

Analyzing +4 | -1

a

and

d

are two complex numbers, then the sum to

\left( {n + 1} \right)

terms of the following series

a{C_0} - \left( {a + d} \right){C_1} + \left( {a + 2d} \right){C_2} - ... + ...

View answer and explanation

Correct option: C

Option C (0) is correct because the given series can be expressed as a combination of binomial coefficients with alternating signs, which sums to zero.

2 Easy

2 of 434

Application +4 | -1

The coefficient of

{x^5}

in the expansion of

\frac{{1 + {x^2}}}{{1 + x}},\left| x \right| < 1,

View answer and explanation

Correct option: D

- 2

Option D (-2) is correct because the coefficient of x⁵ in the expansion of (1+x²)/(1+x) is found by multiplying the series expansion of (1+x²) and (1+x)^-1 and identifying the x⁵ term.

3 Easy

3 of 434

Analyzing +4 | -1

Let

{\left( {1 + x} \right)^n} = \sum \limits_{r = 0}^n {C_r}\;{x^r}

and

\frac{{{C_1}}}{{{C_0}}} + 2\frac{{{C_2}}}{{{C_1}}} + 3\frac{{{C_3}}}{{{C_2}}} + \ldots + n\frac{{{C_n}}}{{{C_{n - 1}}}} = \frac{1}{k}n\left( {n + 1} \right),

then the value of

k

, is

View answer and explanation

Correct option: B

Option B (2) is correct because by simplifying the summation and comparing it to the given expression, we find that k = 2.

4 Easy

4 of 434

Application +4 | -1

The sum of the coefficients in the expansion of

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

View answer and explanation

Correct option: D

\sum {n^3}

Option D (

\sum {n^3}

) is correct because substituting x=1 into the given expression directly yields the sum of the coefficients, which simplifies to

(\sum n)^2 = \sum n^3

5 Easy

5 of 434

Analyzing +4 | -1

The coefficient of the term independent of

x

in the expansion of

{\left[ {\frac{{\left( {x + 1} \right)}}{{{x^{2/3}} - {x^{1/3}} + 1}} - \frac{{\left( {x - 1} \right)}}{{x - {x^{1/2}}}}} \right]^{10}}\mathrm{is}

View answer and explanation

Correct option: A

210

Option A (210) is correct because the coefficient of the term independent of x is found by simplifying the expression and using binomial theorem.

6 Easy

6 of 434

Application +4 | -1

If term independent of

x

in the expansion of

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

\lambda 3^5

, then find the value of

\lambda

View answer and explanation

Correct option: A

Option A (5) is correct because the term independent of x is found by setting the power of x to zero and solving for the binomial coefficient.

7 Easy

7 of 434

Application +4 | -1

a C_0+(a+b) C_1+(a+2 b) C_2+\ldots+(a+n b) C_n

is equal to

View answer and explanation

Correct option: B

(2 a+n b) 2^{n-1}

Option B (

(2 a+n b) 2^{n-1}

) is correct because the summation can be split into two parts, one involving 'a' and the other involving 'b', and then simplified using binomial coefficient identities.

8 Easy

8 of 434

Application +4 | -1

For a positive integer

n

, if the mean of the binomial coefficients in the expansion of

(a+b)^{2 n-3}

is 16, then

n

is equal to:

View answer and explanation

Correct option: A

Option A (5) is correct because the mean of the binomial coefficients is given as 16, and solving the equation derived from this information yields

n=5

9 Easy

9 of 434

Application +4 | -1

Coefficient of

x^{\prime \prime}

in the expression

\left(1-2 x+3 x^2-4 x^3+\cdots\right)^{-n}

equals

View answer and explanation

Correct option: A

{ }^{2 n} C_n

Option A (

{ }^{2 n} C_n

) is correct because the given expression simplifies to (1+x)⁽²ⁿ⁾, and the coefficient of x^n is

{ }^{2 n} C_n

10 Easy

10 of 434

Analyzing +4 | -1

The sum

\sum\limits_{i=0}^m\left(\begin{array}{c}10 \\ i\end{array}\right)\left(\begin{array}{c}20 \\ m-i\end{array}\right)

, (where

\frac{p}{q}=0

, if

p < q)

is maximum when

m

View answer and explanation

Correct option: C

Option C (15) is correct because the sum represents the coefficient of

x^m

(1+x)^{30}

, which is maximum when

m = 30/2 = 15

11 Easy

11 of 434

Application +4 | -1

The coefficient of

{x^5}

in the expansion of

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

View answer and explanation

Correct option: B

Option B (60) is correct because the coefficient of

{x^5}

can be found by multiplying the relevant terms in the expansions of

{\left( {1 + {x^2}} \right)^5}

and

{\left( {1 + x} \right)^4}

12 Easy

12 of 434

Analyzing +4 | -1

The maximum value of the term independent of

't'

in the expansion of

{\left( {t{x^{1/5}} + \frac{{{{\left( {1 - x} \right)}^{1/10}}}}{t}} \right)^{10}}

, where

x \in \left( {0,\,\,1} \right)

View answer and explanation

Correct option: B

\frac{{2.10!}}{{3\sqrt 3 {{\left( {5!} \right)}^2}}}

Option B is correct. The maximum value of the term independent of

't'

is found by using the binomial theorem and maximizing the resulting expression using calculus.

13 Easy

13 of 434

Application +4 | -1

The sum of coefficients of the expansion

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

in 6561. The coefficient of term independent of

x

View answer and explanation

Correct option: A

16\;{\;^8}{C_4}

Option A (

16{\;^8}{C_4}

) is correct because it represents the coefficient of the term independent of x in the expansion, calculated by finding the value of r for which the power of x is zero.

14 Easy

14 of 434

Application +4 | -1

x>0, y>0

then

\sum\limits_{{r=1}}^{\infty}\left(\frac{x}{2 x+3 y}\right)^{r-1}=

View answer and explanation

Correct option: B

\frac{2 x+3 y}{x+3 y}

Option B (

\frac{{2 x+3 y}}{{x+3 y}}

) is correct because it is derived from summing the infinite geometric series.

15 Easy

15 of 434

Application +4 | -1

If the

{\left( {r + 1} \right)^{\mathrm{th}}}

term in the expansion of

{\left\{ {\sqrt[3]{{\frac{a}{{\sqrt b }}}} + \sqrt {\frac{b}{{\sqrt[3]{a}}}} } \right\}^{21}}

contains

a

and

b

to one and the same power, then the value of

r

, is

View answer and explanation

Correct option: A

Option A (9) is correct because when the powers of a and b are equal in the (r+1)th term, solving the equation yields r=9.

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16 Easy

16 of 434

Application +4 | -1

If the third term in the binomial expansion of

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

- \frac{1}{8}{x^2},

then the rational value of

m

, is

View answer and explanation

Correct option: B

\frac{1}{2}

Option B (1/2) is correct because by setting the third term of the binomial expansion equal to -1/8 x², we can solve for m.

17 Easy

17 of 434

Analyzing +4 | -1

Coefficient of

x^{11}

in the expansion of

\left(1+x^2\right)^4\left(1+x^3\right)^7\left(1+x^4\right)^{12}

View answer and explanation

Correct option: C

1113

Option C (1113) is correct because it is the sum of the products of coefficients that result in

x^{11}

when expanding the given expression.

18 Easy

18 of 434

Understanding +4 | -1

Assertion: General term of the expansion

(x+2 y)^9

{ }^9 {C}_{{r}} \cdot 2^{{r}} \cdot {x}^{9-{r}} \cdot {y}^{{r}}

Reason: General term of the expansion

(x+a)^n

is given by

{T}_{{r}+1}={ }^{{n}} {C}_{{r}} \, {x}^{{n}-{r}} \cdot {a}^{{r}}

View answer and explanation

Correct option: A

Assertion is correct, reason is correct; reason is a correct explanation for assertion

Option A (Assertion is correct, reason is correct; reason is a correct explanation for assertion) is correct because the general term formula is correctly applied and stated.

19 Easy

19 of 434

Application +4 | -1

If the coefficient of

{r^{\mathrm{th}}},{\left( {r + 1} \right)^{\mathrm{th}}}

and

{\left( {r + 2} \right)^{\mathrm{th}}}

terms in the expansion of

{\left( {1 + x} \right)^{14}}

are in A.P., then the value of

r

, is

View answer and explanation

Correct option: A

5,9

Option A (

5,9

) is correct because these values of

r

satisfy the given condition that the coefficients are in A.P.

20 Easy

20 of 434

Understanding +4 | -1

The coefficient of

x^p

and

x^q

(

p

and

q

are positive integers) in the expansion of

(1+x)^{p+q}

are

View answer and explanation

Correct option: A

equal

Option A is correct because the binomial coefficients

{ }^{p+q}C_p

and

{ }^{p+q}C_q

are equal since

{ }^{n}C_r = { }^{n}C_{n-r}

21 Easy

21 of 434

Analyzing +4 | -1

The larger of

{99^{50}} + {100^{50}}

and

{101^{50}}

View answer and explanation

Correct option: C

{101^{50}}

Option C (

{101^{50}}

) is correct because

{(101)^{50}}

is greater than the sum of

{99^{50}}

and

{100^{50}}

after expansion and comparison.

22 Easy

22 of 434

Application +4 | -1

The interval in which

x

must lie so that the greatest term in the expansion of

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

has the greatest coefficient, is

View answer and explanation

Correct option: B

\left( {\frac{n}{{n + 1}},\frac{{n + 1}}{n}} \right)

Option B ((n/(n+1)),((n+1)/n)) is correct because it defines the interval for x such that the term with the greatest coefficient in the expansion of (1+x)⁽²ⁿ⁾ is also the greatest term.

23 Easy

23 of 434

Application +4 | -1

If in the expansion of

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

, the sum of the coefficients of

x^5

and

x^{10}

is 0, then the coefficient of the third term is:

View answer and explanation

Correct option: B

105

Option B (105) is correct because by finding the value of

n

using the given condition and then calculating the third term's coefficient, we get 105.

24 Easy

24 of 434

Understanding +4 | -1

The greatest coefficient in the expansion of

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

View answer and explanation

Correct option: A

{\;^{2n}}{C_n}

Option A (

{\;^{2n}}{C_n}

) is correct because the middle term in the binomial expansion of

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

has the greatest coefficient.

25 Easy

25 of 434

Application +4 | -1

The coefficient of

x^{3}

in the expansion of

3^{x}

View answer and explanation

Correct option: D

\frac{(\log 3)^{3}}{6}

The correct answer is D (

\frac{(\log 3)^{3}}{6}

). The coefficient of

x^3

in the expansion of

3^x

is found using the Taylor series expansion of

e^{x \log 3}

26 Easy

26 of 434

Application +4 | -1

If the coefficients of the second, third and fourth terms in the expansion of

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

are in AP, then

n

is equal to

View answer and explanation

Correct option: A

Option A (7) is correct because when the coefficients of the second, third, and fourth terms are in AP, n can be either 2 or 7, but n=2 is not acceptable, so n=7.

27 Easy

27 of 434

Analyzing +4 | -1

The coefficient of

x^{1012}

in the expansion of

\left(1+x^n+\right.

\left.x^{253}\right)^{10}

, (where

n \leq 22

is a positive integer), is

View answer and explanation

Correct option: B

{ }^{10} C_4

Option B (

{ }^{10} C_4

) is correct because it represents the coefficient of

x^{1012}

obtained by finding the appropriate values of r and s in the multinomial expansion.

28 Easy

28 of 434

Application +4 | -1

The coefficient of

{x^4}

in the expansion of

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

View answer and explanation

Correct option: C

\frac{{405}}{{256}}

Option C (

\frac{{405}}{{256}}

) is correct because after finding the general term and substituting the correct value of r (r=2), the coefficient of x⁴ simplifies to this fraction.

29 Easy

29 of 434

Understanding +4 | -1

n(A)=n(B)=4

then number of bijections from

A

B

View answer and explanation

Correct option: B

Option A (

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

) is correct because the number of terms in the expansion of

(x+y+z)^n

is equivalent to the number of non-negative integer solutions to the equation

p+q+r=n

, which is given by

{n+3-1 \choose 3-1} = {n+2 \choose 2} = \frac{(n+2)(n+1)}{2}

30 Easy

30 of 434

Analyzing +4 | -1

The coefficient of

{x^n}

in the expansion of

{\log _e}\left( {\frac{1}{{1 + x + {x^2} + {x^3}}}} \right),

when

n

is odd, is

View answer and explanation

Correct option: B

- \frac{1}{n}

Option B (

-\frac{1}{n}

) is correct because when n is odd, the coefficient of

{x^n}

in the expansion of

{\log _e}(\frac{1}{{1 + x + {x^2} + {x^3}}})

is given by -1/n.

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Key Takeaways

✓ This set contains 434 single-correct-answer MCQs on BINOMIAL THEOREM.
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