CBSE Questions for Class 11 Engineering Maths Binomial Theorem Quiz 4 - MCQExams.com

Determine the value of $$x$$ in the expression $${ \left( x+{ x }^{ t } \right)  }^{ 5 }$$, if the third term in the expression is 10,00,000 where $$t=\log _{ 10 }{ x } $$.
  • $$10\quad $$
  • $${ 10 }^{ -5/2 }$$
  • Both (A)and (B)
  • None of these
The number of terms whose values depends on $$x$$ in the expansion of $$\displaystyle\left (  x^{2}-2+\frac{1}{x^{2}}\right )^{n}$$ is
  • $$2n+1$$
  • $$2n$$
  • $$n$$
  • none of these
In the expansion of the expression $${ \left( x+a \right)  }^{ 15 }$$, if the eleventh term in the geometric mean of the eighth and twelfth terms, which term in the expression is the greatest?
  • $${T}_{6}$$
  • $${T}_{7}$$
  • $${T}_{8}$$
  • $${T}_{9}$$
The value of $$\cfrac { 1 }{ \left( n-1 \right) ! } +\cfrac { 1 }{ \left( n-3 \right) !3! } +\cfrac { 1 }{ \left( n-5 \right) !5! }+.... $$
  • $$\cfrac { { 2 }^{ n+1 } }{ n! } $$
  • $$\cfrac { { 2 }^{ n-1 } }{ n! } $$
  • $$\cfrac { { 2 }^{ n-1 } }{ n+1! } $$
  • $$\cfrac { { 2 }^{ n+1 } }{ n-1! } $$
If $${ \left( 1+x \right)  }^{ n }={ C }_{ 0 }+{ C }_{ 1 }x+{ C }_{ 2 }{ x }^{ 2 }+..........+{ C }_{ n }{ x }^{ R }$$, then the sum
$${ C }_{ 0 }+({ C }_{ 0 }+{ C }_{ 1 })+({ C }_{ 0 }+{ C }_{ 1 }+{ C }_{ 2 })+.....+({ C }_{ 0 }+{ C }_{ 1 }+{ C }_{ 2 }+.....+{ C }_{ n-1 }$$ is

  • $$n.{ 2 }^{ n+1 }$$
  • $$n.{ 2 }^{ n-1 }$$
  • $$(n-1).{ 2 }^{ n-1 }$$
  • $$(n+1).{ 2 }^{ n+1 }$$
If $$n$$ is a positive integer and $${ C }_{ k }={ _{  }^{ n }{ C } }_{ k }$$, find the value of $$\sum _{ k=1 }^{ n }{ { k }^{ 3 }{ \left( \cfrac { { C }_{ k } }{ { C }_{ k-1 } }  \right)  }^{ 2 } } $$
  • $$\cfrac { n{ \left( n-1 \right) }^{ 2 }(n+2) }{ 12 } $$
  • $$\cfrac { n{ \left( n+1 \right) }^{ 2 }(n-2) }{ 12 } $$
  • $$\cfrac { n{ \left( n+1 \right) }^{ 2 }(n+2) }{ 12 } $$
  • $$\cfrac { n{ \left( n-1 \right) }^{ 2 }(n-2) }{ 12 } $$
The sum of coefficients of $$x^{2r},r=1,2,3,...,$$ in the expansion of $$(1+x)^{n}$$ is
  • $$2^{n}$$
  • $$2^{n-1}-1$$
  • $$2^{n}-1$$
  • $$2^{n-1}+1$$
The number of terms with integral coefficients in the expansion of$$\left ( 7^{1/3}+5^{1/2}.x \right )^{600}$$ is
  • $$100$$
  • $$50$$
  • $$101$$
  • none of these
The sum of coefficients in the binomial expansion of $$\displaystyle \left ( \frac{1}{x}+2x \right )^{n}$$is equal to $$6561$$.The constant term in the expansion is
  • $$^{8}\textrm{C}_{4}$$
  • $$16\cdot^{8}\textrm{C}_{4}$$
  • $$^{6}\textrm{C}_{4}\cdot 2^{4}$$
  • none of these
If the $$4th$$ term in the expansion is of $$ \left (px+x^{-1} \right )^{m}$$ is $$2.5$$ for all $$x\epsilon R$$ then
  • $$p=\displaystyle\frac{5}{2},m=3$$
  • $$p=\displaystyle\frac{1}{2},m=6$$
  • $$p=\displaystyle-\frac{1}{2},m=6$$
  • none of these
The sum $$^{10}\textrm{C}_{3}+^{11}\textrm{C}_{3}+^{12}\textrm{C}_{3}+..........+^{20}\textrm{C}_{3}$$ is equal to
  • $$^{21}\textrm{C}_{4}$$
  • $$^{21}\textrm{C}_{4}+$$ $$^{10}\textrm{C}_{4}$$
  • $$^{21}\textrm{C}_{17}-$$ $$^{10}\textrm{C}_{6}$$
  • none of these
The sum of coefficients of all the integral powers of  $$x$$ in the expansion of $$(1+2\sqrt{x})^{40}$$ is
  • $$3^{40}+1$$
  • $$3^{40}-1$$
  • $$\displaystyle\frac{1}{2}$$ $$(3^{40}-1)$$
  • $$\displaystyle\frac{1}{2}$$ $$(3^{40}+1)$$
The absolute value of middle term in the expansion of $$\displaystyle \left ( 1-\frac{1}{x} \right )^{n}.(1-x)^{n}$$ is
  • $$^{2n}\textrm{C}_{n}$$
  • $$-\ ^{2n}\textrm{C}_{n}$$
  • $$-\ ^{2n}\textrm{C}_{n-1}$$
  • none of these
The sum of last ten coeffficients in the expansion of $$(1+x)^{19}$$ when expanded in ascending powers of $$x$$ is
  • $$2^{18}$$
  • $$2^{19}$$
  • $$2^{18}-$$$$^{19}\textrm{C}_{10}$$
  • none of these
The middle term in the expansion of $$\left ( \displaystyle \frac{2x}{3}-\frac{3}{2x^{2}} \right )^{2n}$$ is
  • $$^{2n}\textrm{C}_{n}$$
  • $$(-1)^{n}[(2n!)/(n!)^{2}].x^{-n}$$
  • $$^{2n}\textrm{C}_{n}$$.$$\displaystyle \frac{1}{x^{n}}$$
  • none of these
The sum $$^{20}\textrm{C}_{0}+^{20}\textrm{C}_{1}+^{20}\textrm{C}_{2}+^{20}\textrm{C}_{10}$$ is equal to
  • $$\displaystyle2^{20}+\frac{20!}{(10!)^{2}}$$
  • $$2^{19}-$$$$\displaystyle\frac{1}{(2)}\cdot\frac{20!}{(10!)^{2}}$$
  • $$\displaystyle2^{19}+$$ $$^{20}\textrm{C}_{10}$$
  • $$none\ of\ these$$
The number of non-zero terms in the expansion of $${ \left( 1+3\sqrt { 2 } x \right)  }^{ 9 }+{ \left( 1-3\sqrt { 2 } x \right)  }^{ 9 }$$ is
  • 9
  • 0
  • 5
  • 10
The value of $$\displaystyle \sum_{j=1}^{n}(^{n+1}C_{j}-^{n}C_{j})$$is equal to
  • $$\displaystyle 2^{n}$$
  • $$\displaystyle 2^{n}+1$$
  • $$\displaystyle 3\cdot 2^{n}$$
  • $$\displaystyle 2^{n} -1$$
The sum $$\displaystyle\frac{1}{2}$$ $$^{10}\textrm{C}_{0}-$$ $$^{10}\textrm{C}_{1}+$$ $$2\cdot  ^{10}\textrm{C}_{2}-$$ $$2^{2}\cdot  ^{10}\textrm{C}_{3}$$+...+$$2^{9}\cdot  ^{10}\textrm{C}_{10}$$ is equal to
  • $$\displaystyle\frac{1}{2}$$
  • $$0$$
  • $$\displaystyle\frac{1}{2}\cdot3^{10}$$
  • none of these
The number of integral terms in the expansion of $$\displaystyle \left ( \sqrt{3}+\sqrt[5]{5} \right )^{256}$$ is
  • $$25$$
  • $$26$$
  • $$24$$
  • None of these
If $$(1+x)^{2n}=a_{0}+a_{1}x+a_{2}x^{2}+...+a_{2n}x^{2n}$$ then
  • $$a_{n+1}>$$ $$a_{n}$$
  • $$a_{n+1}<$$ $$a_{n}$$
  • $$a_{n-3}=a_{n+3}$$
  • none of these
  • Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1
  • Statement-1 is True, Statement-2 is True; Statement-2 is Not a correct explanation for Statement-1
  • Statement-1 is True, Statement-2 is False
  • Statement-1 is False, Statement-2 is True
Let $$n\epsilon N$$.If $$(1+x)^{n}=a_{0}x+a_{1}x+a_{2}x^{2}+...+a_{n}x^{n},$$ and $$a_{n-3},a_{n-2},a_{n-1},$$ are in AP then
  • $$a_{1},a_{2},a_{3}$$ are in $$AP$$
  • $$a_{1},a_{2},a_{3}$$ are in $$HP$$
  • $$n=7$$
  • $$n=14$$
In the expansion of $$\left (\sqrt[3]{4}+\dfrac{1}{\sqrt[4]{6}}\right )^{20},$$
  • the number of rational terms $$=4$$
  • the number of irrational terms $$=19$$
  • the middle term is irrational
  • the number of irrational terms $$=17$$
The sum of the series $$\sum _{ r=0 }^{ 10 }{ _{  }^{ 20 }{ { C }_{ r }^{  } } } $$ is
  • $$\displaystyle { 2 }^{ 19 }-\frac { 1 }{ 2 } ._{  }^{ 20 }{ { C }_{ 10 }^{  } }$$
  • $$\displaystyle { 2 }^{ 19 }+\frac { 1 }{ 2 } .{  }^{ 20 }{ { C }_{ 10 }^{  } }$$
  • $${ 2 }^{ 19 }$$
  • $${ 2 }^{ 20 }$$
$$^{ n+1 }{ { C }_{ 2 }^{  } }+2\left[ _{  }^{ 2 }{ { C }_{ 2 }^{  } }+^{ 3 }{ { C }_{ 2 }^{  } }+^{ 4 }{ { C }_{ 2 }^{  } }+...+^{ n }{ { C }_{ 2 }^{  } } \right] =$$
  • $$\displaystyle \frac { n\left( n+1 \right) \left( 2n+1 \right)  }{ 6 } $$
  • $$\displaystyle \frac { n\left( n+1 \right)  }{ 2 } $$
  • $$\displaystyle \frac { n\left( n-1 \right) \left( 2n-1 \right)  }{ 6 } $$
  • None of these
If $$\displaystyle C_{0},C_{1},C_{2},....C_{n}$$ are binomial coefficient in the expansion of $$\displaystyle (1+x)^{n}$$, then value of  $$\displaystyle C_{1}+C_{4}+C_{7}+...$$ equals
  • $$\displaystyle \frac{1}{3}(2^{n}+\sqrt 3 \sin \frac{n \pi}{3})$$
  • $$\displaystyle \frac{1}{3}(2^{n}-\cos \frac{n \pi}{3}+\sqrt 3 \sin \frac{n \pi}{3} )$$
  • $$\displaystyle \frac{1}{3}(2^{n}-\sqrt 3 \sin \frac{n \pi}{3})$$
  • $$\displaystyle \frac{1}{3}(2^{n}-\cos \frac{n \pi}{3}-\sqrt 3 \sin \frac{n \pi}{3} )$$
If the third term in the expansion of $$\displaystyle (\frac{1}{x}+x^{\log_{10}x})^{5} $$ is $$1,000$$, then $$x$$-equals
  • 10$$\displaystyle ^{2}$$
  • 10$$\displaystyle ^{3}$$
  • $$10$$
  • None of these
If $$C_{0},C_{1},C_{2},.....C_{n},$$ are binomial coefficients,then $$\displaystyle\sum_{k= 0}^{n} C_{k}\:\sin kx \cos \left ( n-k \right )x$$ equals
  • $$2^{n} \sin nx$$
  • $$2^{n+1}\sin \left ( n+1 \right )x$$
  • $$2^{n-1}\sin nx$$
  • $$2^{n+1}\sin nx$$
If coefficient of $$x^{100}$$ in $$1+\left ( 1+x \right )\left ( 1+x \right )^{2}+.....+\left ( 1+x \right )^{n}\left ( if\:n \geq 100\right )$$ is $$C_{101}^{201}$$ then the value of n equals

  • $$202$$
  • $$100$$
  • $$200$$
  • $$201$$
If $$\displaystyle \left ( 1+x+x^{2} \right )^{n}=\sum_{r=0}^{2n} a_{r}x^{r}=a_{0}+a_{1}x+a_{2}x^{2}+...+a^{2n}x^{2n}$$ and
$$ \displaystyle P=a_{0}+a_{3}+a_{6}+...$$
$$ \displaystyle Q=a_{1}+a_{4}+a_{7}+...$$
$$ \displaystyle R=a_{2}+a_{5}+a_{8}+...$$
then the set of values of $$ P, Q, R $$ are respectively equals
  • $$\displaystyle (1 ,1, 1)$$
  • $$\displaystyle (3^{n},3^{n},3^{n})$$
  • $$\displaystyle (3^{n+1},3^{n+1},3^{n+1})$$
  • $$\displaystyle (3^{n-1},3^{n-1},3^{n-1})$$
The evaluated value of $$\displaystyle \sum_{i=0}^{n} \sum_{j=1}^{n}$$ $$\displaystyle ^{n}C_{j}$$ $$\displaystyle ^{j}C_{i},$$ $$\displaystyle i\leq j$$
  • $$\displaystyle 3^{n}+1$$
  • $$\displaystyle 3^{n}-1$$
  • $$\displaystyle 3^{n+1}+1$$
  • None of these
If $$\displaystyle C_{r}=^{n}C_{r}$$ and $$\displaystyle (C_{0}+C_{1})(C_{1}+C_{2})...(C_{n-1}+C_{n})=k$$ $$\displaystyle (C_{0} C_{1}C_{2}...C_{n})$$ then the value of $$ \displaystyle k$$ equals
  • $$\displaystyle \frac{(n+1)^{n+1}}{n!}$$
  • $$\displaystyle \frac{(n+1)^{n}}{n.n!}$$
  • $$\displaystyle \frac{(n)^{n}}{n!}$$
  • $$\displaystyle \frac{(n+1)^{n}}{n!} $$
If $$\displaystyle P$$ be the sum of odd term and  $$\displaystyle Q$$ that of even terms in the expansion of  $$\displaystyle (x+a)^{n}$$ , then the value of  $$\displaystyle [(x+a)^{2n}-(x-a)^{2n}]$$ equals
  • $$\displaystyle PQ$$
  • $$\displaystyle 2PQ$$
  • $$4 $$$$\displaystyle PQ$$
  • None of these
The sum of the coefficients of all odd exponets of $$\displaystyle x$$ in the product of $$\displaystyle (1-x +x^{2}-x^{3}+x^{4}+...-x^{49}+x^{50})\times(1+x+x^{2}+x^{3}+...+x^{50})$$ equals
  • $$1$$
  • $$0$$
  • $$-1$$
  • None of these
If $$C_{0},C_{1},C_{2}....,C_{n}$$ are Binomial Coefficients, such that $$\displaystyle S_{n}=\sum_{r=0}^{n}\frac{1}{{C_{r}}^{n}}$$ and $$\displaystyle t_{n}= \sum_{r= 0}^{n}\frac{r}{{C_{r}}^{n}}$$ then $$\displaystyle \frac{t_{n}}{s_{n}}$$ equals
  • $$\displaystyle \frac{n}{2}$$
  • $$\displaystyle\frac{n\left ( n+1 \right )}{2}$$
  • $$\displaystyle\frac{n+1}{2}$$
  • None of these
If$$\displaystyle\left ( 1+x \right )^{n}= \sum_{r= 0}^{n}C_{r}x^{r}$$, then the value of $$C_{0}-C_{2}+C_{4}-C_{6}+C_{8}-C_{10}+...$$ equals
  • $$\displaystyle2^{\tfrac{n}{2}}\cos \frac{nx}{4}$$
  • $$C_{1}-C_{3}+C_{5}-C_{7}+....$$
  • $$C_{0}+C_{4}+C_{8}+C_{12}+....$$
  • $$\displaystyle2^{\tfrac{n}{2}}\sin \frac{nx}{4}$$
If $$\left ( 1+x \right )^{n}= \sum_{r= 0}^{n}C_{r}x^{r}$$ then the value of $$3C_{1}+7C_{2}+11C^{3}+....+\left ( 4n-1 \right )C_{n}$$ is
  • $$\left ( 4n-1 \right )2^{n}$$
  • $$\left ( 2n-1 \right )2^{n}$$
  • $$\left ( 2n-1 \right )2^{n}+1$$
  • $$\left ( 4n-1 \right )2^{n}-1$$
If $${C_{r}}^{13}$$ denoted by $$C_{r}$$ then value of $$c_{1}+c_{5}+c_{7}+c_{9}+c_{11}$$ is equal to
  • $$2^{12}-287$$
  • $$2^{12}-165$$
  • $$2^{12}-C_{3}$$
  • $$2^{12}-C_{2}-C_{13}$$
The Coefficient of $$x^{53}$$ in $$\sum_{m= 0}^{100}{C_{m}}^{100}\left ( x-3 \right )^{100-m}2^{m}$$ is
  • $$=-\:^{100}C_{53}$$
  • $$=-\:^{101}C_{53}$$
  • $$=\:^{101}C_{47}$$
  • $$=-\:^{100}C_{47}$$
The number of terms with integral coefficient in the expansion of $$\left ( 17^{\dfrac 13} +35^{\dfrac 12}\right )^{300}$$ is
  • $$50$$
  • $$100$$
  • $$150$$
  • $$51$$
If $$A$$ is the sum of the odd terms and $$B$$ the sum of even terms in the expansion of $${ \left( x+a \right)  }^{ n }$$, then $${ A }^{ 2 }-{ B }^{ 2 }=$$
  • $${ \left( { x }^{ 2 }+{ a }^{ 2 } \right)  }^{ n }$$
  • $${ \left( { x }^{ 2 }-{ a }^{ 2 } \right)  }^{ n }$$
  • $$2{ \left( { x }^{ 2 }-{ a }^{ 2 } \right)  }^{ n }$$
  • None of these
The sum of the series $$\displaystyle\sum_{r=0}^{n}\left ( ^{n+1}C_{r} \right ) $$ equals
  • $$\left ( n+1 \right )2^{2n-1}$$
  • $$\displaystyle \frac{1}{2}\frac{2n!}{n!n!}$$
  • $$\displaystyle 2^{2n-1}\left ( n+1 \right )-\frac{1}{2}\frac{2n!}{n!n!}$$
  • $$2^{n+1}-1$$
If $$x+y= 1$$ then $$\displaystyle\sum_{r= 0}^{n} r^{n}C_{r}x^{r}y^{n-r}$$ equals
  • $$1$$
  • $$n$$
  • $$nx$$
  • $$ny$$
Number of rational term is the expansion of $$\left ( 7^{1/3}+11^{1/9} \right )^{729}$$
  • 81
  • 82
  • 730
  • None of these
The values of $$x$$ in the expansion $$\displaystyle \left ( x+x^{log_{10}x} \right )^{5}$$ , if the third term in the expansion is $$10,00,000$$
  • $$\displaystyle 10$$
  • $$\displaystyle 10^{2}$$
  • $$\displaystyle 10^{3}$$
  • None of these
If $$\displaystyle\left ( 1+x \right )^{n}=\sum_{r=0}^{n}C_{r}x^{r}$$ and $$\sum { \sum _{ 0\le i<j\le n }{ { C }_{ i }\times { C }_{ j } }  } $$ represent the products of the $$C_{i}$$'s taken two at a time, then its value equals
  • $$\displaystyle2^{2n-1}-\frac{\left (2n\right)!}{\left (n!\right)^{2}}$$
  • $$\displaystyle2^{2n-1}+\frac{2n!}{n!n!}$$
  • $$\displaystyle2^{2n-1}-\frac{2n!}{2\cdot n!n!}$$
  • None of these
Sum of the coefficients of the terms of degree $$m$$ in the expansion of
$${ (1+x) }^{ n }{ (1+y) }^{ n }{ (1+z) }^{ n }$$ is
  • $${ ({ _{ }^{ n }{ C } }_{ m }) }^{ 3 }$$
  • $$3({ _{ }^{ n }{ C } }_{ m })$$
  • $${ _{ }^{ n }{ C } }_{ 3m }$$
  • $${ _{ }^{ 3n }{ C } }_{ m }$$
The number of irrational terms in the expansion of $${ ({ 5 }^{\tfrac 16 }+{ 2 }^{ \tfrac 18 }) }^{ 100 }$$ is
  • $$96$$
  • $$97$$
  • $$98$$
  • $$99$$
Value of the expression $${ C }_{ 0 }^{ 2 }+{ C }_{ 1 }^{ 2 }+{ C }_{ 2 }^{ 2 }+.....+(n+1){ C }_{ n }^{ 2 }$$ is
  • $$(2n+1)({ _{ }^{ 2n }{ C } }_{ n })$$
  • $$(2n-1)({ _{ }^{ 2n }{ C } }_{ n })$$
  • $$\left( \cfrac { n }{ 2 } +1 \right) ({ _{ }^{ 2n }{ C } }_{ n })$$
  • $$\left( \cfrac { n }{ 2 } +1 \right) ({ _{ }^{ 2n-1 }{ C } }_{ n })$$
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