Question 1 : For a positive integer n, if the expansion of <img style='object-fit:contain' src='https://storage.googleapis.com/teachmint/question_assets/JEE%20Advanced/5e8b0d5eafa64601c3e35f5e' height='44' width='68' > has a term independent of x, then n can be
Question 3 :
If the third term in the expansion of {tex} \left( x + x ^ { \log { 10 } x } \right) ^ { 5 } {/tex} is {tex} 10 ^ { 6 } , {/tex} then {tex} x {/tex} can be
Question 4 :
The number of values of <em>r</em> satisfying the equation <sup>69</sup><em>C</em><sub>3<em>r</em> − 1</sub> − <sup>69</sup><em>C</em><sub><em>r</em><sup>2</sup></sub> = <sup>69</sup><em>C</em><sub><em>r</em><sup>2</sup> − 1</sub> − <sup>69</sup><em>C</em><sub>3<em>r</em></sub> is
Question 5 :
The coefficients of $ x^p $ and $ x^q $ ($p$ and $q$ are positive integers) in the expansion of $ (1 + x)^{p+q} $ are<br/>
Question 6 : The value of x in the expression (x +<img style='object-fit:contain' src='https://storage.googleapis.com/teachmint/question_assets/JEE%20Advanced/5e99a8fac33dd94d23e41a5b' height='20' width='43' >)<sup>5</sup>, if the third term in the expansion is 10,00,000, is-
Question 7 :
If (1 + x + x<sup>2</sup>)<sup>n</sup> = a<sub>0</sub> + a<sub>1</sub>x + a<sub>2</sub> x<sup>2</sup> + ......+ a<sub>2n</sub>x<sup>n</sup>, then the value of a<sub>0</sub> + a<sub>3</sub> + a<sub>6</sub>+.........is -
Question 10 :
If n be a positive integer, then in the trinomial expansion of (x<sup>2</sup> + 2x + 2)<sup>n</sup>, the coefficient of
Question 11 :
The positive integer just greater than ${\left( {1 + 0.0001} \right)^{10000}}$ is
Question 12 : For a positive integer n, if the expansion of <img style='object-fit:contain' src='https://storage.googleapis.com/teachmint/question_assets/JEE%20Advanced/5e99a8f4debb574cf7016980' height='44' width='68' > has a term independent of x, then n can be
Question 13 :
In the expansion of {tex} \left( x ^ { 2 } + 1 + \frac { 1 } { x ^ { 2 } } \right) ^ { n } , n \in \mathbf { N } {/tex}
Question 14 :
Let (1+ x<sup>2</sup>)<sup>2</sup> (1+ x)<sup>n</sup> = A<sub>0</sub> + A<sub>1</sub>x + A<sub>2</sub>x<sup>2</sup> + .......If A<sub>0</sub>, A<sub>1</sub>, A<sub>2</sub> are in A.P. then the value of n is
Question 15 :
The sum of the coefficient in the expansion of (1+ax−2<em>x</em><sup>2</sup>)<sup><em>n</em></sup> is
Question 17 :
The number 51<sup>49</sup> + 51<sup>48</sup> + 51<sup>47</sup> ....... + 51 + 1 is divisible by
Question 18 :
Let n ∈N. If (1 + x)<sup>n</sup> = a<sub>0</sub> + a<sub>1</sub>x + a<sub>2</sub>x<sup>2</sup>+....+ a<sub>n</sub>x<sup>n</sup> and a<sub>n-3</sub>, a<sub>n-2</sub>, a<sub>n-1</sub> are in AP then
Question 21 :
The last digit of 3<sup>3<sup>4<em>n</em></sup></sup> + 1, <em>n</em> ∈ <em>N</em>, is
Question 23 :
If {tex} ( 1 + x ) \left( 1 + x + x ^ { 2 } \right) \left( 1 + x + x ^ { 2 } + x ^ { 3 } \right) \cdots ( 1 + x + {/tex} {tex} \left. + x ^ { n } \right) {/tex} {tex} = a _ { 0 } + a _ { 1 } x + a _ { 2 } x ^ { 2 } + \cdots + a _ { m } x ^ { m } , {/tex} then
Question 24 :
The coefficient of the $8$th term in the expansion of $(1+x)^{10}$ is
Question 25 :
If {tex} \left( 1 + 2 x + 3 x ^ { 2 } \right) ^ { 10 } = a _ { 0 } + a _ { 1 } x + a _ { 2 } x ^ { 2 } + \cdots + a _ { 20 } x ^ { 20 } {/tex} then
Question 26 :
If coefficients of r<sup>th</sup>, (r + 1)<sup>th</sup> and (r + 2)<sup>th</sup> terms in the expansion of (1 + x)<sup>14</sup> are in A.P. Then 'r' is equal to
Question 27 :
The number 51<sup>49</sup> + 51<sup>48</sup> + 51<sup>47</sup> ....... + 51 + 1 is divisible by
Question 28 :
In the expansion of (<em>x</em>+<em>a</em>)<sup><em>n</em></sup> if the sum of odd terms be <em>P</em> and sum of even terms be 𝒬, then
Question 29 :
If (1+<em>x</em>)<sup><em>n</em></sup> = <em>C</em><sub>0</sub> + <em>C</em><sub>1</sub><em>x</em> + <em>C</em><sub>2</sub><em>x</em><sup>2</sup> + ⋯ + <em>C</em><sub><em>n</em></sub><em>x</em><sup><em>n</em></sup>, then <em>C</em><sub>0</sub> − (<em>C</em><sub>0</sub>+<em>C</em><sub>1</sub>) + (<em>C</em><sub>0</sub>+<em>C</em><sub>1</sub>+<em>C</em><sub>2</sub>) − (<em>C</em><sub>0</sub>+<em>C</em><sub>1</sub>+<em>C</em><sub>2</sub>+<em>C</em><sub>3</sub>) + ⋯ + (−1)<sup><em>n</em> − 1</sup>(<em>C</em><sub>0</sub> + <em>C</em><sub>1</sub> + ⋯ + <em>C</em><sub><em>n</em> − 1</sub>), where <em>n</em> is even integer is
Question 30 :
Let n ∈N. If (1 + x)<sup>n</sup> = a<sub>0</sub> + a<sub>1</sub>x + a<sub>2</sub>x<sup>2</sup>+....+ a<sub>n</sub>x<sup>n</sup> and a<sub>n-3</sub>, a<sub>n-2</sub>, a<sub>n-1</sub> are in AP then
Question 31 :
If the middle term of {tex} \left( x + \frac { 1 } { x } \sin ^ { - 1 } x \right) ^ { 8 } {/tex} is equal to {tex} 630 / 16 {/tex}, then value of {tex} x {/tex} is (are)
Question 32 :
If coefficients of r<sup>th</sup>, (r + 1)<sup>th</sup> and (r + 2)<sup>th</sup> terms in the expansion of (1 + x)<sup>14</sup> are in A.P. Then 'r' is equal to
Question 33 :
The total number of terms in the expansion of $(x+y)^{50}+(x-y)^{50}$ is
Question 34 :
In the binomial expansion of $ (a-b)^n , n \geq{5} $, the sum of <br>5th and 6th terms is zero then a/b equal to<br><br>
Question 35 :
If P = n(n<sup>2</sup> - 1<sup>2</sup>)(n<sup>2 </sup>- 2<sup>2</sup>)(n<sup>2</sup> - 3<sup>2</sup>) .........(n<sup>2</sup> - r<sup>2</sup>), n > r, n∈N then P is divisible by -
Question 36 :
Let {tex} C _ { r } {/tex} stand for {tex} { } ^ { n } C _ { r } {/tex} and {tex} S ( n , r ) = C _ { 0 } - C _ { 1 } + C _ { 2 } - C _ { 3 } {/tex} {tex} + \ldots + ( - 1 ) ^ { r } C _ { r } {/tex}
Question 38 :
The number {tex} 101 ^ { 100 } - 1 {/tex} is divisible by
Question 39 :
If <sup>n</sup>C<sub>α</sub> = <sup>n</sup>C<sub>β</sub>, then it may be true to say that-
Question 40 :
Find the sum of coefficient of middle terms of the expansion $\left(3x-\dfrac{x^3}{6}\right)^7$:
Question 41 :
Let (1+ x<sup>2</sup>)<sup>2</sup> (1+ x)<sup>n</sup> = A<sub>0</sub> + A<sub>1</sub>x + A<sub>2</sub>x<sup>2</sup> + .......If A<sub>0</sub>, A<sub>1</sub>, A<sub>2</sub> are in A.P. then the value of n is
Question 42 :
Positive integer {tex} ( \mathrm { s } ) {/tex} which is (are) greater than {tex} ( 1 + 0.0001 ) ^ { 10000 }\ \mathrm { is } ( {/tex} are {tex} ) {/tex}
Question 43 :
For the expansion {tex} ( 1 + 2 \sqrt { x } ) ^ { 40 } , {/tex} sum of the coeffi- cients of the
Question 44 :
If <sup>n</sup>C<sub>α</sub> = <sup>n</sup>C<sub>β</sub>, then it may be true to say that-
Question 46 :
If P = n(n<sup>2</sup> - 1<sup>2</sup>)(n<sup>2 </sup>- 2<sup>2</sup>)(n<sup>2</sup> - 3<sup>2</sup>) .........(n<sup>2</sup> - r<sup>2</sup>), n > r, n∈N then P is divisible by -
Question 47 :
If n be a positive integer, then in the trinomial expansion of (x<sup>2</sup> + 2x + 2)<sup>n</sup>, the coefficient of
Question 49 : The value of x in the expression (x +<img style='object-fit:contain' src='https://storage.googleapis.com/teachmint/question_assets/JEE%20Advanced/5e8b0d6560d51c030bee35fa' height='20' width='43' >)<sup>5</sup>, if the third term in the expansion is 10,00,000, is-
Question 51 :
The coefficient of $x^{6}.y^{-2}$ in the in the expansion of $\left ( \displaystyle \frac{x^{2}}{y}-\frac{y}{x} \right )^{12}$ is<br>
Question 52 :
The middle term in the expansion of $(1 + x)^{2n}$ is
Question 53 :
The ratio of $(r + 1) ^{th}$ and $(r - 1)^ {th}$ terms in the expansion of $({a}-b)^{n}$ is<br/>
Question 55 :
If $f(x) = 1+x^2-x^3+...-x^{15}+x^{16}-x^{17}$, then the coefficient of $x^2$ in $f(x-1)$ is
Question 56 :
Let {tex} x + y = k {/tex} where {tex} x , y > 0 {/tex} and {tex} S ( k , n ) = \underset{ r = 0 } {\stackrel{ n }\sum} r ^ { 2 } \left( { } ^ { n } C _ { r } \right) x ^ { r } y ^ { n - r } {/tex} then
Question 57 :
The $6^{th}$ coefficient in the expansion of $\left (2x^2 - \dfrac {1}{3x^2}\right)^{10}$
Question 59 :
If the coefficients of the $(m+1)$th term and $(m+3)$th term in the expansion of $(1+x)^{20}$ are equal then the value of $m$ is<br>
Question 60 :
Coefficient of $\dfrac 1x$ in the expansion of $\displaystyle \left ( 1+x^{n} \right )\left ( 1+\frac{1}{x} \right )^{n}$ is<br/>
Question 61 :
In the expansion of $ \left ( a^2\sqrt{a} + \frac{\sqrt[3]{a}}{a} \right )^n $ the binomial coefficient of 3rd term is 36. The 7th term is :<br/>
Question 63 :
In the expansion of $ (a+b)^n $, the ratio of the binomial coefficients of ${ 2 }^{ nd }$ and ${ 3 }^{ rd }$ terms is equal to the ratio of the binomial coefficients of ${ 5 }^{ th }$ and ${ 4 }^{ th }$ terms, then $n = $<br/>
Question 64 :
In the binomial expansion of $ \left ( \sqrt{y} + \dfrac{1}{2\sqrt[4]{y}} \right )^8 $, the terms in the expansion in which the power of $y$ is a natural number are:<br/>
Question 65 :
The value of <sup>n</sup>C<sub>0</sub> + <sup>n+1</sup>C<sub>1</sub> + <sup>n+2</sup>C<sub>2</sub> +.....+ <sup>n+k</sup>C<sub>k</sub> is equal to-
Question 66 :
For {tex} n \in \mathbf { N } , {/tex} let {tex} S ( k ) = \underset{ r = 0 }{ \stackrel{ n }\sum} r ^ { k } \left( { } ^ { n } C _ { r } \right) ^ { 2 } {/tex}, then
Question 67 :
The coefficient of $x^{p}$ & $x^{q}$ $\left ( p,q \:\epsilon \:N \right )$ in the expansion of $\left ( 1+x \right )^{p+q}$ is<br>
Question 68 :
If the middle term of ${ \left( x+\cfrac { 1 }{ x } \sin ^{ -1 }{ x } \right) }^{ 8 }$ is $\cfrac { 35{ \pi }^{ 4 } }{ 8 } $, then value of $x$ can be<br/>
Question 69 :
In the binomial $\displaystyle \left ( 2^{1/3}+3^{-1/3} \right )^{n}$ if the ratio of the seventh term from the beginning of the expansion to the seventh term from its end is $\displaystyle 1/6$ then n is equal to
Question 70 :
Find the coefficient of ${ x }^{ 7 }$ in the expansion of ${ \left( a{ x }^{ 2 }+\dfrac { 1 }{ bx } \right) }^{ 11 }$<br/><br/>
Question 71 :
If the coefficient of $(2r+3)^{th}$ term and $(r-1)^{th} $ terms in the expansion of ${ \left( 1+x \right) }^{ 18 }$ are equal, find $r$.<br>
Question 72 :
The coefficients of ${ x }^{ n }$ in the expansion ${ \left( 2x+3 \right) }^{ n }-{ \left( 2x+3 \right) }^{ n-1 }\left( 5-2x \right) +{ \left( 2x+3 \right) }^{ n-2 }{ \left( 5-2x \right) }^{ 2 }+...+{ \left( -1 \right) }^{ n }{ \left( 5-2x \right) }^{ n }$ is
Question 73 :
Consider the expansion of $(1+x)^{2n+1}$<br>The average of the coefficients of the two middle terms in the expansion is
Question 75 :
If the second, third and fourth terms in the expansion of {tex} ( a + b ) ^ { n } {/tex} are {tex}135,30{/tex} and {tex} 10 / 3 {/tex} respectively, then
Question 76 :
Let {tex} S _ { n } ( x ) = \underset{ k = 0 }{ \stackrel { n } \sum}{ } ^ { n } C _ { k } \sin ( k x ) \cos [ ( n - k ) x ] , {/tex} then
Question 77 :
The coefficient of ${ x }^{ k }$ in the expansion of <br>$E=1+(1+x)+{ (1+x) }^{ 2 }+....+{ (1+x) }^{ n }$ is<br>
Question 78 :
Coefficient of $\dfrac 1x$ in the expansion of ${ \left( 1+x \right) }^{ n }{ \left( 1+\dfrac 1x \right) }^{ n }$ is<br/>
Question 79 :
Find $n$ in the binomial ${ \left[ \sqrt [ 3 ]{ 2 } +\displaystyle \frac { 1 }{ \sqrt [ 3 ]{ 3 } } \right] }^{ n }$ if the ratio of the $7^{th}$ term from beginning to the $7^{th}$ term from the end is $\displaystyle \frac{1}{6}$<br>
Question 80 :
If $a$ is the coefficient of the middle term in the expansion of $(1+x)^{2n}$ and $b, c$ are the coefficients of the two middle terms in the expansion of $(1+x)^{2n-1}$ then <br/>
Question 81 :
<span class="MathJax_Preview"><span class="MJXp-math"><span class="noError">}$13^{th term in the expansion of</span></span></span>\displaystyle \left ( 9x - \frac{1}{ 3 \sqrt{x}} \right)^{18}, x \neq 0$ is<br/>
Question 82 :
If the coefficient of $x^7$ in $\displaystyle \left [ ax^2 + \left ( \dfrac{1}{bx} \right ) \right ]^{11}$ equals the coefficient of $x^{-7}$ in $\displaystyle \left [ ax^2 - \left ( \dfrac{1}{bx} \right ) \right ]^{11}$, then $a$ and $b$ satisfy the relation
Question 83 :
If the second term of the expansion $\displaystyle \left [ a^{1/13}+\frac{a}{\sqrt{a^{-1}}} \right ]^{n}\: \: is\: \: 14a^{5/2}$, then the value of $\displaystyle \frac{^{n}{C}_{3}}{^{n}{C}_{2}}$ is
Question 84 :
The $4th$ term from the end in the expansion of ${ \left( \cfrac { { x }^{ 3 } }{ 2 } -\cfrac { 2 }{ { x }^{ 2 } } \right) }^{ 7 }$ is<br><br>
Question 85 :
If the fourth term in the expansion of $\displaystyle { \left( \sqrt { \frac { 1 }{ { x }^{ \log { x } +1 } } } +{ x }^{ 1/12 } \right) }^{ 6 }$ is equal to $200$ and $x>1$, then $x$ is equal to
Question 87 :
i. Three consecutive binomial coefficients cannot be in $G.P.$<br/>ii. Three consecutive binomial coefficients can be in $H.P.$<br/>Which of the above statement is correct
Question 88 :
Write the $r^{th}$ term from the end in the expansion of $(x+a)^n$
Question 89 :
In the expression of $\displaystyle \left ( 3-\sqrt{\frac{17}{4}+3\sqrt{2}} \right )^{15}$, the 11th term is a
Question 90 :
In the expansion of ${ \left( 1+x \right) }^{ n }{ \left( 1+y \right) }^{ n }{ \left( 1+z \right) }^{ n }$, then the sum of coefficients of the terms of degree $m$ is<br>
Question 91 :
Assertion (A) : The coefficient of $x^{7}$ in $(\displaystyle \frac{x^{2}}{2}-\frac{2}{x})^{9}$ is zero<br/><br/>Reason (R) : $r$ in $t_{r+1}$ that contains coefficient of $x^{7}$ is not positive integer<br/><br/><br/>
Question 92 :
If $p+q=1$, then the value of $\displaystyle \sum _{ r=0 }^{ 15 }{ { _{ }^{ 15 }{ C } }_{ r } } { p }^{ 15-r }{ q }^{ r }$
Question 93 :
If (1 + x + x<sup>2</sup>)<sup>n</sup> = a<sub>0</sub> + a<sub>1</sub>x + a<sub>2</sub> x<sup>2</sup> + ......+ a<sub>2n</sub>x<sup>n</sup>, then the value of a<sub>0</sub> + a<sub>3</sub> + a<sub>6</sub>+.........is -
Question 94 :
The ratio of 11th terms from the beginning and $11^{th}$ terms from the end in the expansion of $\displaystyle \left ( 2x - \frac{1}{x^2} \right )^{25}$ is<br/>
Question 95 :
Let {tex} ( x + 1 ) ( x + 2 ) \cdots ( x + n ) {/tex} {tex} = A _ { 0 } + A _ { 1 } x + A _ { 2 } x ^ { 2 } + \cdots + A _ { n } x ^ { n } {/tex} then
Question 96 :
If the coefficients of $r^{th}, (r+ 1)^{th}\ and \ (r + 2)^{th}$ terms in the binomial expansion of $(1+y)^{m}$ are in A.P., then $m$ and $r$ satisfy the equation
Question 97 :
If $462, 330$ and $165$ are three successive coefficients in the expansion of $(1+x)^n$, then $n =$ <br/>
Question 98 :
If in the expansion of $(1+x)^{m}\cdot (1-x)^{n}$, the coefficients of $x$ and $x^{2}$ are $3$ and $-6$ respectively, then
Question 100 :
If the coefficients of $r^{th}$ term and $(r+1)^{th}$ term in the expansion of $(1+x)^{20}$ are in the ration 1 : 2, then $r=$
Question 101 :
If sum of the coefficients of ${x}^{7}$ and ${x}^{4}$ in the expansion of ${ \left( \cfrac { { x }^{ 2 } }{ a } -\cfrac { b }{ x } \right) }^{ 11 }$ is zero, then<br>
Question 102 :
Arrange the values of $n$ in ascending order<br/>A : If the term independent of $x$ in the expansion of $\left(\displaystyle \sqrt{x}-\frac{n}{x^{2}}\right)^{10}$ is $405$<br/>B : If the fourth term in the expansion of $\left(\displaystyle \frac{1}{n}+n^{\log_{n}10}\right)^{5}$ is $1000$, <span>( $ n< 10 $)<br/>C : In the binomial expansion of $(1+x)^{n}$ the coefficients of <span> $5^{\mathrm{t}\mathrm{h}},\ 6^{\mathrm{t}\mathrm{h}}$ and $7^{\mathrm{t}\mathrm{h}}$ terms are in A.P.</span></span><br/>
Question 103 :
Coefficient of $x^{50}$ in the polynomial <br/>$\left(x+_{ }^{ 50 }{ { C }_{ 0 } }\right)\left(x+3._{ }^{ 50 }{ { C }_{ 1 } }\right)\left(x+5._{ }^{ 50 }{ { C }_{ 2 } }\right).....\left[x+(101)._{ }^{ 50 }{ { C }_{ 50 } }\right]$ is
Question 104 :
Find the coefficient of ${ x }^{ 50 }$ in the expression:<br>${ \left( 1+x \right) }^{ 1000 }+2x{ \left( 1+x \right) }^{ 999 }+3{ x }^{ 2 }{ \left( 1+x \right) }^{ 998 }+....+1001{ x }^{ 1000 }$<br>
Question 105 :
If there is a term containing $x^{2r}$ in $\left( x + \dfrac{1}{x^2} \right )^{n - 3}$, then
Question 106 :
The coefficient ${x^n}$ in the expression of ${\left( {1 + x} \right)^{2n}}$ and ${\left( {1 + x} \right)^{2n - 1}}$ are in the ratio.
Question 107 :
If the last term in the binomial expansion of <br>${ \left( { 2 }^{ 1/3 }-\cfrac { 1 }{ \sqrt { 2 } } \right) }^{ n }$ is ${ \left( \cfrac { 1 }{ { 3 }^{ 5/3 } } \right) }^{ \log _{ 3 }{ 8 } }$, then the 5th term from the beginning is<br>
Question 108 :
Let $n$ be a positive integer such that ${ \left( 1+x+{ x }^{ 2 } \right) }^{ n }={ a }_{ 0 }+{ a }_{ 1 }x+{ a }_{ 2 }{ x }^{ 2 }+...+{ a }_{ 2n }{ x }^{ 2n },$ then ${a}_{r}=$
Question 109 :
Find the coefficient of the term independent of x in the expansion of $\displaystyle\left(6x^3-\frac{5}{x^6}\right)^{12}$.
