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Mathematical Induction and Binomial Theorem
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$$ \text { If } \sum\limits_{k=1}^{10} K^{2}\left(10_{C_{K}}\right)^{2}=22000 L \text {, then } L \text { is equal to }$...
JEE Main 2022 (Online) 29th July Evening Shift
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Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of $$\left(\...
JEE Main 2022 (Online) 29th July Morning Shift
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If $$1 + (2 + {}^{49}{C_1} + {}^{49}{C_2} + \,\,...\,\, + \,\,{}^{49}{C_{49}})({}^{50}{C_2} + {}^{50}{C_4} + \,\,...\,\,...
JEE Main 2022 (Online) 28th July Evening Shift
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Let the coefficients of the middle terms in the expansion of $$\left(\frac{1}{\sqrt{6}}+\beta x\right)^{4},(1-3 \beta x)...
JEE Main 2022 (Online) 28th July Evening Shift
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Let for the $$9^{\text {th }}$$ term in the binomial expansion of $$(3+6 x)^{\mathrm{n}}$$, in the increasing powers of ...
JEE Main 2022 (Online) 27th July Evening Shift
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If the coefficients of $$x$$ and $$x^{2}$$ in the expansion of $$(1+x)^{\mathrm{p}}(1-x)^{\mathrm{q}}, \mathrm{p}, \math...
JEE Main 2022 (Online) 26th July Morning Shift
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If the maximum value of the term independent of $$t$$ in the expansion of $$\left(\mathrm{t}^{2} x^{\frac{1}{5}}+\frac{(...
JEE Main 2022 (Online) 25th July Morning Shift
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The remainder on dividing 1 + 3 + 32 + 33 + ..... + 32021 by 50 is _________.
JEE Main 2022 (Online) 24th June Evening Shift
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Let Cr denote the binomial coefficient of xr in the expansion of $${(1 + x)^{10}}$$. If for $$\alpha$$, $$\beta$$ $$\in$...
JEE Main 2022 (Online) 25th June Morning Shift
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If the sum of the co-efficient of all the positive even powers of x in the binomial expansion of $${\left( {2{x^3} + {3 ...
JEE Main 2022 (Online) 25th June Evening Shift
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If $$\left( {{}^{40}{C_0}} \right) + \left( {{}^{41}{C_1}} \right) + \left( {{}^{42}{C_2}} \right) + \,\,.....\,\, + \,\...
JEE Main 2022 (Online) 26th June Evening Shift
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Let $$A = \sum\limits_{i = 1}^{10} {\sum\limits_{j = 1}^{10} {\min \,\{ i,j\} } } $$ and $$B = \sum\limits_{i = 1}^{10} ...
JEE Main 2022 (Online) 26th June Morning Shift
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If the coefficient of x10 in the binomial expansion of $${\left( {{{\sqrt x } \over {{5^{{1 \over 4}}}}} + {{\sqrt 5 } \...
JEE Main 2022 (Online) 27th June Morning Shift
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If the sum of the coefficients of all the positive powers of x, in the Binomial expansion of $${\left( {{x^n} + {2 \over...
JEE Main 2022 (Online) 27th June Evening Shift
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The number of positive integers k such that the constant term in the binomial expansion of $${\left( {2{x^3} + {3 \over ...
JEE Main 2022 (Online) 28th June Morning Shift
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Let the coefficients of x$$-$$1 and x$$-$$3 in the expansion of $${\left( {2{x^{{1 \over 5}}} - {1 \over {{x^{{1 \over 5...
JEE Main 2022 (Online) 29th June Evening Shift
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If the sum of the coefficients in the expansion of (x + y)n is 4096, then the greatest coefficient in the expansion is _...
JEE Main 2021 (Online) 1st September Evening Shift
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If the coefficient of a7b8 in the expansion of (a + 2b + 4ab)10 is K.216, then K is equal to _____________....
JEE Main 2021 (Online) 31st August Evening Shift
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If $$\left( {{{{3^6}} \over {{4^4}}}} \right)k$$ is the term, independent of x, in the binomial expansion of $${\left( {...
JEE Main 2021 (Online) 31st August Morning Shift
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3 $$\times$$ 722 + 2 $$\times$$ 1022 $$-$$ 44 when divided by 18 leaves the remainder __________.
JEE Main 2021 (Online) 27th August Evening Shift
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Let $$\left( {\matrix{ n \cr k \cr } } \right)$$ denotes $${}^n{C_k}$$ and $$\left[ {\matrix{ n \cr ...
JEE Main 2021 (Online) 26th August Evening Shift
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If the co-efficient of x7 and x8 in the expansion of $${\left( {2 + {x \over 3}} \right)^n}$$ are equal, then the value ...
JEE Main 2021 (Online) 25th July Evening Shift
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Let n$$\in$$N and [x] denote the greatest integer less than or equal to x. If the sum of (n + 1) terms $${}^n{C_0},3.{}^...
JEE Main 2021 (Online) 25th July Evening Shift
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The term independent of 'x' in the expansion of $${\left( {{{x + 1} \over {{x^{2/3}} - {x^{1/3}} + 1}} - {{x - 1} \over ...
JEE Main 2021 (Online) 25th July Morning Shift
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The ratio of the coefficient of the middle term in the expansion of (1 + x)20 and the sum of the coefficients of two mid...
JEE Main 2021 (Online) 25th July Morning Shift
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The number of elements in the set {n $$\in$$ {1, 2, 3, ......., 100} | (11)n > (10)n + (9)n} is ______________....
JEE Main 2021 (Online) 22th July Evening Shift
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If the constant term, in binomial expansion of $${\left( {2{x^r} + {1 \over {{x^2}}}} \right)^{10}}$$ is 180, then r is ...
JEE Main 2021 (Online) 22th July Evening Shift
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The number of rational terms in the binomial expansion of $${\left( {{4^{{1 \over 4}}} + {5^{{1 \over 6}}}} \right)^{120...
JEE Main 2021 (Online) 20th July Morning Shift
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Let $${}^n{C_r}$$ denote the binomial coefficient of xr in the expansion of (1 + x)n. If $$\sum\limits_{k = 0}^{10} {(...
JEE Main 2021 (Online) 18th March Evening Shift
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The term independent of x in the expansion of $${\left[ {{{x + 1} \over {{x^{2/3}} - {x^{1/3}} + 1}} - {{x - 1} \over {x...
JEE Main 2021 (Online) 18th March Evening Shift
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Let the coefficients of third, fourth and fifth terms in the expansion of $${\left( {x + {a \over {{x^2}}}} \right)^n},x...
JEE Main 2021 (Online) 17th March Evening Shift
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If (2021)3762 is divided by 17, then the remainder is __________.
JEE Main 2021 (Online) 17th March Morning Shift
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Let n be a positive integer. Let $$A = \sum\limits_{k = 0}^n {{{( - 1)}^k}{}^n{C_k}\left[ {{{\left( {{1 \over 2}} \right...
JEE Main 2021 (Online) 16th March Evening Shift
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Let m, n$$\in$$N and gcd (2, n) = 1. If $$30\left( {\matrix{ {30} \cr 0 \cr } } \right) + 29\left( {\matrix{...
JEE Main 2021 (Online) 26th February Morning Shift
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The total number of two digit numbers 'n', such that 3n + 7n is a multiple of 10, is __________.
JEE Main 2021 (Online) 25th February Evening Shift
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If the remainder when x is divided by 4 is 3, then the remainder when (2020 + x)2022 is divided by 8 is __________....
JEE Main 2021 (Online) 25th February Evening Shift
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For integers n and r, let $$\left( {\matrix{ n \cr r \cr } } \right) = \left\{ {\matrix{ {{}^n{C_r},} &am...
JEE Main 2021 (Online) 24th February Evening Slot
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The coefficient of x4 in the expansion of (1 + x + x2 + x3)6 in powers of x, is ______....
JEE Main 2020 (Online) 5th September Evening Slot
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The natural number m, for which the coefficient of x in the binomial expansion of $${\left( {{x^m} + {1 \over {{x^2}}}} ...
JEE Main 2020 (Online) 5th September Morning Slot
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Let $${\left( {2{x^2} + 3x + 4} \right)^{10}} = \sum\limits_{r = 0}^{20} {{a_r}{x^r}} $$ Then $${{{a_7}} \over {{a_{13}}...
JEE Main 2020 (Online) 4th September Morning Slot
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For a positive integer n, $${\left( {1 + {1 \over x}} \right)^n}$$ is expanded in increasing powers of x. If three conse...
JEE Main 2020 (Online) 2nd September Evening Slot
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If Cr $$ \equiv $$ 25Cr and C0 + 5.C1 + 9.C2 + .... + (101).C25 = 225.k, then k is equal to _____....
JEE Main 2020 (Online) 9th January Evening Slot
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The coefficient of x4 is the expansion of (1 + x + x2)10 is _____.
JEE Main 2020 (Online) 9th January Morning Slot
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If the sum of the coefficients of all even powers of x in the product (1 + x + x2 + ....+ x2n)(1 - x + x2 - x3 + .........
JEE Main 2020 (Online) 7th January Morning Slot
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MCQ (Single Correct Answer)
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$$\sum\limits_{r=1}^{20}\left(r^{2}+1\right)(r !)$$ is equal to
JEE Main 2022 (Online) 29th July Evening Shift
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The remainder when $$7^{2022}+3^{2022}$$ is divided by 5 is :
JEE Main 2022 (Online) 28th July Morning Shift
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The remainder when $$(2021)^{2022}+(2022)^{2021}$$ is divided by 7 is
JEE Main 2022 (Online) 27th July Morning Shift
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$$\sum\limits_{\matrix{ {i,j = 0} \cr {i \ne j} \cr } }^n {{}^n{C_i}\,{}^n{C_j}} $$ is equal to
JEE Main 2022 (Online) 26th July Evening Shift
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The remainder when $$(11)^{1011}+(1011)^{11}$$ is divided by 9 is
JEE Main 2022 (Online) 25th July Evening Shift
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For two positive real numbers a and b such that $${1 \over {{a^2}}} + {1 \over {{b^3}}} = 4$$, then minimum value of the...
JEE Main 2022 (Online) 30th June Morning Shift
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The remainder when 32022 is divided by 5 is :
JEE Main 2022 (Online) 24th June Morning Shift
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If $${1 \over {2\,.\,{3^{10}}}} + {1 \over {{2^2}\,.\,{3^9}}} + \,\,.....\,\, + \,\,{1 \over {{2^{10}}\,.\,3}} = {K \ove...
JEE Main 2022 (Online) 25th June Morning Shift
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The coefficient of x101 in the expression $${(5 + x)^{500}} + x{(5 + x)^{499}} + {x^2}{(5 + x)^{498}} + \,\,.....\,\, + ...
JEE Main 2022 (Online) 25th June Evening Shift
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The remainder when (2021)2023 is divided by 7 is :
JEE Main 2022 (Online) 26th June Morning Shift
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If $$\sum\limits_{k = 1}^{31} {\left( {{}^{31}{C_k}} \right)\left( {{}^{31}{C_{k - 1}}} \right) - \sum\limits_{k = 1}^{3...
JEE Main 2022 (Online) 28th June Morning Shift
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The term independent of x in the expansion of $$(1 - {x^2} + 3{x^3}){\left( {{5 \over 2}{x^3} - {1 \over {5{x^2}}}} \rig...
JEE Main 2022 (Online) 28th June Evening Shift
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Let n $$\ge$$ 5 be an integer. If 9n $$-$$ 8n $$-$$ 1 = 64$$\alpha$$ and 6n $$-$$ 5n $$-$$ 1 = 25$$\beta$$, then $$\alph...
JEE Main 2022 (Online) 29th June Evening Shift
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If the constant term in the expansion of $${\left( {3{x^3} - 2{x^2} + {5 \over {{x^5}}}} \right)^{10}}$$ is 2k.l, where...
JEE Main 2022 (Online) 29th June Morning Shift
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$$\sum\limits_{k = 0}^{20} {{{\left( {{}^{20}{C_k}} \right)}^2}} $$ is equal to :
JEE Main 2021 (Online) 27th August Morning Shift
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If $${{}^{20}{C_r}}$$ is the co-efficient of xr in the expansion of (1 + x)20, then the value of $$\sum\limits_{r = 0}^{...
JEE Main 2021 (Online) 26th August Morning Shift
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If the coefficients of x7 in $${\left( {{x^2} + {1 \over {bx}}} \right)^{11}}$$ and x$$-$$7 in $${\left( {{x} - {1 \over...
JEE Main 2021 (Online) 27th July Morning Shift
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The lowest integer which is greater than $${\left( {1 + {1 \over {{{10}^{100}}}}} \right)^{{{10}^{100}}}}$$ is _________...
JEE Main 2021 (Online) 25th July Evening Shift
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If the greatest value of the term independent of 'x' in the expansion of $${\left( {x\sin \alpha + a{{\cos \alpha } \ov...
JEE Main 2021 (Online) 25th July Evening Shift
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The sum of all those terms which are rational numbers in the expansion of (21/3 + 31/4)12 is :...
JEE Main 2021 (Online) 25th July Evening Shift
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A possible value of 'x', for which the ninth term in the expansion of $${\left\{ {{3^{{{\log }_3}\sqrt {{{25}^{x - 1}} +...
JEE Main 2021 (Online) 27th July Evening Shift
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If b is very small as compared to the value of a, so that the cube and other higher powers of $${b \over a}$$ can be neg...
JEE Main 2021 (Online) 25th July Morning Shift
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For the natural numbers m, n, if $${(1 - y)^m}{(1 + y)^n} = 1 + {a_1}y + {a_2}{y^2} + .... + {a_{m + n}}{y^{m + n}}$$ an...
JEE Main 2021 (Online) 20th July Evening Shift
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The coefficient of x256 in the expansion of (1 $$-$$ x)101 (x2 + x + 1)100 is :...
JEE Main 2021 (Online) 20th July Morning Shift
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Let (1 + x + 2x2)20 = a0 + a1x + a2x2 + .... + a40x40. Then a1 + a3 + a5 + ..... + a37 is equal to ...
JEE Main 2021 (Online) 18th March Morning Shift
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The value of $$\sum\limits_{r = 0}^6 {\left( {{}^6{C_r}\,.\,{}^6{C_{6 - r}}} \right)} $$ is equal to :
JEE Main 2021 (Online) 17th March Evening Shift
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If the fourth term in the expansion of $${(x + {x^{{{\log }_2}x}})^7}$$ is 4480, then the value of x where x$$\in$$N is ...
JEE Main 2021 (Online) 17th March Morning Shift
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If n is the number of irrational terms in the expansion of $${\left( {{3^{1/4}} + {5^{1/8}}} \right)^{60}}$$, then (n $$...
JEE Main 2021 (Online) 16th March Morning Shift
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The maximum value of the term independent of 't' in the expansion of $${\left( {t{x^{{1 \over 5}}} + {{{{(1 - x)}^{{1 \o...
JEE Main 2021 (Online) 26th February Morning Shift
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If $$n \ge 2$$ is a positive integer, then the sum of the series $${}^{n + 1}{C_2} + 2\left( {{}^2{C_2} + {}^3{C_2} + {}...
JEE Main 2021 (Online) 24th February Evening Slot
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The value of -15C1 + 2.15C2 – 3.15C3 + ... - 15.15C15 + 14C1 + 14C3 + 14C5 + ...+ 14C11 is :...
JEE Main 2021 (Online) 24th February Morning Slot
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If the constant term in the binomial expansion of $${\left( {\sqrt x - {k \over {{x^2}}}} \right)^{10}}$$ is 405, then...
JEE Main 2020 (Online) 6th September Evening Slot
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If {p} denotes the fractional part of the number p, then $$\left\{ {{{{3^{200}}} \over 8}} \right\}$$, is equal to :
JEE Main 2020 (Online) 6th September Morning Slot
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If for some positive integer n, the coefficients of three consecutive terms in the binomial expansion of (1 + x)n + 5 ar...
JEE Main 2020 (Online) 4th September Evening Slot
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The value of $$\sum\limits_{r = 0}^{20} {{}^{50 - r}{C_6}} $$ is equal to:
JEE Main 2020 (Online) 4th September Morning Slot
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If the term independent of x in the expansion of $${\left( {{3 \over 2}{x^2} - {1 \over {3x}}} \right)^9}$$ is k, then 1...
JEE Main 2020 (Online) 3rd September Evening Slot
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If the number of integral terms in the expansion of (31/2 + 51/8)n is exactly 33, then the least value of n is :...
JEE Main 2020 (Online) 3rd September Morning Slot
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Let $$\alpha $$ > 0, $$\beta $$ > 0 be such that $$\alpha $$3 + $$\beta $$2 = 4. If the maximum value of the term ...
JEE Main 2020 (Online) 2nd September Morning Slot
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In the expansion of $${\left( {{x \over {\cos \theta }} + {1 \over {x\sin \theta }}} \right)^{16}}$$, if $${\ell _1}$$ i...
JEE Main 2020 (Online) 9th January Evening Slot
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If $$\alpha $$ and $$\beta $$ be the coefficients of x4 and x2 respectively in the expansion of $${\left( {x + \sqrt {{x...
JEE Main 2020 (Online) 8th January Evening Slot
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The coefficient of x7 in the expression (1 + x)10 + x(1 + x)9 + x2(1 + x)8 + ......+ x10 is:...
JEE Main 2020 (Online) 7th January Evening Slot
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The greatest positive integer k, for which 49k + 1 is a factor of the sum 49125 + 49124 + ..... + 492 + 49 + 1, is:...
JEE Main 2020 (Online) 7th January Morning Slot
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The term independent of x in the expansion of $$\left( {{1 \over {60}} - {{{x^8}} \over {81}}} \right).{\left( {2{x^2} -...
JEE Main 2019 (Online) 12th April Evening Slot
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If 20C1 + (22) 20C2 + (32) 20C3 + ..... + (202 ) 20C20 = A(2$$\beta $$), then the ordered pair (A, $$\beta $$) is equal ...
JEE Main 2019 (Online) 12th April Evening Slot
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The coefficient of x18 in the product (1 + x) (1 – x)10 (1 + x + x2)9 is :...
JEE Main 2019 (Online) 12th April Morning Slot
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The smallest natural number n, such that the coefficient of x in the expansion of $${\left( {{x^2} + {1 \over {{x^3}}}} ...
JEE Main 2019 (Online) 10th April Evening Slot
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If the coefficients of x2 and x3 are both zero, in the expansion of the expression (1 + ax + bx2 ) (1 – 3x)15 in power...
JEE Main 2019 (Online) 10th April Morning Slot
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If some three consecutive in the binomial expansion of (x + 1)n is powers of x are in the ratio 2 : 15 : 70, then the av...
JEE Main 2019 (Online) 9th April Evening Slot
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If the fourth term in the binomial expansion of $${\left( {{2 \over x} + {x^{{{\log }_8}x}}} \right)^6}$$ (x > 0) is...
JEE Main 2019 (Online) 9th April Morning Slot
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If the fourth term in the binomial expansion of $${\left( {\sqrt {{x^{\left( {{1 \over {1 + {{\log }_{10}}x}}} \right)}}...
JEE Main 2019 (Online) 8th April Evening Slot
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The sum of the co-efficients of all even degree terms in x in the expansion of $${\left( {x + \sqrt {{x^3} - 1} } \right...
JEE Main 2019 (Online) 8th April Morning Slot
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The sum of the series 2.20C0 + 5.20C1 + 8.20C2 + 11.20C3 + ... +62.20C20 is equal to : ...
JEE Main 2019 (Online) 8th April Morning Slot
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The total number of irrational terms in the binomial expansion of (71/5 – 31/10)60 is :
JEE Main 2019 (Online) 12th January Evening Slot
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A ratio of the 5th term from the beginning to the 5th term from the end in the binomial expansion of $${\left( {{2^{1/3}...
JEE Main 2019 (Online) 12th January Morning Slot
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Let Sn = 1 + q + q2 + . . . . . + qn and Tn = 1 + $$\left( {{{q + 1} \over 2}} \right) + {\left( {{{q + 1} \over 2}} \...
JEE Main 2019 (Online) 11th January Evening Slot
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Let (x + 10)50 + (x $$-$$ 10)50 = a0 + a1x + a2x2 + . . . . + a50x50, for all x $$ \in $$ R; then $${{{a_2}} \over {{a_0...
JEE Main 2019 (Online) 11th January Evening Slot
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The sum of the real values of x for which the middle term in the binomial expansion of $${\left( {{{{x^3}} \over 3}...
JEE Main 2019 (Online) 11th January Morning Slot
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The value of r for which 20Cr 20C0 + 20Cr$$-$$1 20C1 + 20Cr$$-$$2 20C2 + . ...
JEE Main 2019 (Online) 11th January Morning Slot
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The positive value of $$\lambda $$ for which the co-efficient of x2 in the expression x2 $${\left( {\sqrt x + {\lambda...
JEE Main 2019 (Online) 10th January Evening Slot
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If the third term in the binomial expansion of $${\left( {1 + {x^{{{\log }_2}x}}} \right)^5}$$ equals 2560, then a possi...
JEE Main 2019 (Online) 10th January Morning Slot
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If $${\sum\limits_{i = 1}^{20} {\left( {{{{}^{20}{C_{i - 1}}} \over {{}^{20}{C_i} + {}^{20}{C_{i - 1}}}}} \ri...
JEE Main 2019 (Online) 10th January Morning Slot
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The coefficient of t4 in the expansion of $${\left( {{{1 - {t^6}} \over {1 - t}}} \right)^3}$$ is :
JEE Main 2019 (Online) 9th January Evening Slot
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If the fractional part of the number $$\left\{ {{{{2^{403}}} \over {15}}} \right\} is \, {k \over {15}}$$, then k is equ...
JEE Main 2019 (Online) 9th January Morning Slot
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The coefficient of x2 in the expansion of the product (2$$-$$x2) .((1 + 2x + 3x2)6 + (1 $$-$$ 4x2)6) is :...
JEE Main 2018 (Online) 16th April Morning Slot
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The coefficien of x10 in the expansion of (1 + x)2(1 + x2)3(1 + x3)4 is equal to :...
JEE Main 2018 (Online) 15th April Evening Slot
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If n is the degree of the polynomial, $${\left[ {{2 \over {\sqrt {5{x^3} + 1} - \sqrt {5{x^3} - 1} }}} \right]^8} + $$...
JEE Main 2018 (Online) 15th April Morning Slot
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The sum of the co-efficients of all odd degree terms in the expansion of $${\left( {x + \sqrt {{x^3} - 1} } \right)^5} +...
JEE Main 2018 (Offline)
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The coefficient of x−5 in the binomial expansion of $${\left( {{{x + 1} \over {{x^{{2 \over 3}}} - {x^{{1 \over 3}}} + ...
JEE Main 2017 (Online) 9th April Morning Slot
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If (27)999 is divided by 7, then the remainder is :
JEE Main 2017 (Online) 8th April Morning Slot
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The value of $$\left( {{}^{21}{C_1} - {}^{10}{C_1}} \right) + \left( {{}^{21}{C_2} - {}^{10}{C_2}} \right) + \left( {{}^...
JEE Main 2017 (Offline)
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If the coefficients of x−2 and x−4 in the expansion of $${\left( {{x^{{1 \over 3}}} + {1 \over {2{x^{{1 \over 3}}}}}} \r...
JEE Main 2016 (Online) 10th April Morning Slot
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If the number of terms in the expansion of $${\left( {1 - {2 \over x} + {4 \over {{x^2}}}} \right)^n},\,x \ne 0,$$ is 28...
JEE Main 2016 (Offline)
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The sum of coefficients of integral power of $$x$$ in the binomial expansion $${\left( {1 - 2\sqrt x } \right)^{50}}$$ i...
JEE Main 2015 (Offline)
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If the coefficints of $${x^3}$$ and $${x^4}$$ in the expansion of $$\left( {1 + ax + b{x^2}} \right){\left( {1 - 2x} \ri...
JEE Main 2014 (Offline)
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The term independent of $$x$$ in expansion of $${\left( {{{x + 1} \over {{x^{2/3}} - {x^{1/3}} + 1}} - {{x - 1} \over...
JEE Main 2013 (Offline)
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If $$n$$ is a positive integer, then $${\left( {\sqrt 3 + 1} \right)^{2n}} - {\left( {\sqrt 3 - 1} \right)^{2n}}$$ is ...
AIEEE 2012
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The coefficient of $${x^7}$$ in the expansion of $${\left( {1 - x - {x^2} + {x^3}} \right)^6}$$ is
AIEEE 2011
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Let $${s_1} = \sum\limits_{j = 1}^{10} {j\left( {j - 1} \right){}^{10}} {C_j}$$, $${{s_2} = \sum\limits_{j = 1}^{10} {} ...
AIEEE 2010
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The remainder left out when $${8^{2n}} - {\left( {62} \right)^{2n + 1}}$$ is divided by 9 is :
AIEEE 2009
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Statement - 1 : $$\sum\limits_{r = 0}^n {\left( {r + 1} \right)\,{}^n{C_r} = \left( {n + 2} \right){2^{n - 1}}.} $$ Stat...
AIEEE 2008
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The sum of the series $${}^{20}{C_0} - {}^{20}{C_1} + {}^{20}{C_2} - {}^{20}{C_3} + .....\, - \,.....\, + {}^{20}{C_{10}...
AIEEE 2007
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In the binomial expansion of $${\left( {a - b} \right)^n},\,\,\,n \ge 5,$$ the sum of $${5^{th}}$$ and $${6^{th}}$$ term...
AIEEE 2007
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For natural numbers $$m$$ , $$n$$, if $${\left( {1 - y} \right)^m}{\left( {1 + y} \right)^n}\,\, = 1 + {a_1}y + {a_2}{y...
AIEEE 2006
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If the expansion in powers of $$x$$ of the function $${1 \over {\left( {1 - ax} \right)\left( {1 - bx} \right)}}$$ is $$...
AIEEE 2006
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If the coefficients of rth, (r+1)th, and (r + 2)th terms in the binomial expansion of $${{\rm{(1 + y )}}^m}$$ are in A...
AIEEE 2005
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If $$x$$ is so small that $${x^3}$$ and higher powers of $$x$$ may be neglected, then $${{{{\left( {1 + x} \right)}^{{3 ...
AIEEE 2005
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If the coefficient of $${x^7}$$ in $${\left[ {a{x^2} + \left( {{1 \over {bx}}} \right)} \right]^{11}}$$ equals the coe...
AIEEE 2005
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If $$A = \left[ {\matrix{ 1 & 0 \cr 1 & 1 \cr } } \right]$$ and $$I = \left[ {\matrix{ 1 & 0 ...
AIEEE 2005
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The value of $$\,{}^{50}{C_4} + \sum\limits_{r = 1}^6 {^{56 - r}} {C_3}$$ is
AIEEE 2005
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If $${S_n} = \sum\limits_{r = 0}^n {{1 \over {{}^n{C_r}}}} \,\,and\,\,{t_n} = \sum\limits_{r = 0}^n {{r \over {{}^n{C_r}...
AIEEE 2004
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The coefficient of $${x^n}$$ in expansion of $$\left( {1 + x} \right){\left( {1 - x} \right)^n}$$ is
AIEEE 2004
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The coefficient of the middle term in the binomial expansion in powers of $$x$$ of $${\left( {1 + \alpha x} \right)^4}$$...
AIEEE 2004
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Let $$S(K)$$ $$ = 1 + 3 + 5... + \left( {2K - 1} \right) = 3 + {K^2}.$$ Then which of the following is true
AIEEE 2004
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The number of integral terms in the expansion of $${\left( {\sqrt 3 + \root 8 \of 5 } \right)^{256}}$$ is
AIEEE 2003
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If $$x$$ is positive, the first negative term in the expansion of $${\left( {1 + x} \right)^{27/5}}$$ is
AIEEE 2003
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If $${a_n} = \sqrt {7 + \sqrt {7 + \sqrt {7 + .......} } } $$ having $$n$$ radical signs then by methods of mathematica...
AIEEE 2002
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If the sum of the coefficients in the expansion of $$\,{\left( {a + b} \right)^n}$$ is 4096, then the greatest coefficie...
AIEEE 2002
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The positive integer just greater than $${\left( {1 + 0.0001} \right)^{10000}}$$ is
AIEEE 2002
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The coefficients of $${x^p}$$ and $${x^q}$$ in the expansion of $${\left( {1 + x} \right)^{p + q}}$$ are
AIEEE 2002
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$$r$$ and $$n$$ are positive integers $$\,r > 1,\,n > 2$$ and coefficient of $$\,{\left( {r + 2} \right)^{th}}$$ t...
AIEEE 2002
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