Have a query? Ask a Tutor!!

Access to best educators and exclusive paper discussions

Offsite Campus Turito Championship

Can’t find what you’re looking for?

Our tutors are from top schools around the world, and they are waiting for you 24/7, anytime, anywhere.

Homework- One to One Solutions
Understand concepts to solve better
Get your doubts solved instantly

Maths-

General

Easy

Question

Statement 1:Sum of the series $1 to the power of 3 end exponent minus 2 to the power of 3 end exponent plus 3 to the power of 3 end exponent minus 4 to the power of 3 end exponent plus horizontal ellipsis plus 11 to the power of 3 end exponent equals 378$
Statement 2:For any odd integer $n greater or equal than 1 comma blank n to the power of 3 end exponent minus open parentheses n minus 1 close parentheses to the power of 3 end exponent plus horizontal ellipsis plus open parentheses negative 1 close parentheses to the power of n minus 1 end exponent blank 1 to the power of 3 end exponent equals fraction numerator 1 over denominator 4 end fraction open parentheses 2 n minus 1 close parentheses open parentheses n plus 1 close parentheses to the power of 2 end exponent$

Statement 1 is True, Statement 2 is True; Statement 2 is correct explanation for Statement 1
Statement 1 is True, Statement 2 is True; Statement 2 is not correct explanation for Statement 1
Statement 1 is True, Statement 2 is False
Statement 1 is False, Statement 2 is True

The correct answer is: Statement 1 is False, Statement 2 is True

For odd integer $n$ , we have
$S subscript n end subscript equals n to the power of 3 end exponent minus open parentheses n minus 1 close parentheses to the power of 3 end exponent plus horizontal ellipsis plus open parentheses negative 1 close parentheses to the power of n minus 1 end exponent 1 to the power of 3 end exponent$
$equals open square brackets 1 to the power of 3 end exponent plus 2 to the power of 3 end exponent plus 3 to the power of 3 end exponent plus horizontal ellipsis plus open parentheses n minus 1 close parentheses to the power of 3 end exponent plus n to the power of 3 end exponent close square brackets minus 2 open square brackets 2 to the power of 3 end exponent plus 4 to the power of 3 end exponent plus 6 to the power of 3 end exponent plus horizontal ellipsis plus open parentheses n minus 1 close parentheses to the power of 3 end exponent close square brackets$
$equals fraction numerator n to the power of 2 end exponent open parentheses n plus 1 close parentheses to the power of 2 end exponent over denominator 4 end fraction minus 2 cross times 2 to the power of 3 end exponent open square brackets 1 to the power of 3 end exponent plus 2 to the power of 3 end exponent plus horizontal ellipsis open parentheses fraction numerator n minus 1 over denominator 2 end fraction close parentheses to the power of 3 end exponent close square brackets$
$equals fraction numerator n to the power of 2 end exponent open parentheses n plus 1 close parentheses to the power of 2 end exponent over denominator 4 end fraction minus 2 to the power of 4 end exponent fraction numerator open parentheses fraction numerator n minus 1 over denominator 2 end fraction close parentheses to the power of 2 end exponent open parentheses fraction numerator n minus 1 over denominator 2 end fraction plus 1 close parentheses to the power of 2 end exponent over denominator 4 end fraction$
$equals fraction numerator n to the power of 2 end exponent open parentheses n plus 1 close parentheses to the power of 2 end exponent over denominator 4 end fraction minus fraction numerator open parentheses n minus 1 close parentheses to the power of 2 end exponent open parentheses n plus 1 close parentheses to the power of 2 end exponent over denominator 4 end fraction$
$equals fraction numerator open parentheses n plus 1 close parentheses to the power of 2 end exponent over denominator 4 end fraction open square brackets n to the power of 2 end exponent minus open parentheses n minus 1 close parentheses to the power of 2 end exponent close square brackets$
$equals fraction numerator 1 over denominator 4 end fraction open parentheses 2 n minus 1 close parentheses open parentheses n plus 1 close parentheses to the power of 2 end exponent$
Now, putting $n equals 11$ in above formula, $S subscript 11 end subscript equals 756$ . Hence statement 1 is false and statement 2 is correct

Related Questions to study

Statement 1:In a G.P. if the $open parentheses m plus n close parentheses to the power of t h end exponent$ term be $p$ and $open parentheses m minus n close parentheses to the power of t h end exponent$ term be $q$ , then its $m to the power of t h end exponent$ term is $square root of p q end root$
Statement 2: $T subscript m plus n end subscript comma blank T subscript m end subscript comma blank T subscript m minus n end subscript$ are in G.P.

Statement 1:In a G.P. if the $open parentheses m plus n close parentheses to the power of t h end exponent$ term be $p$ and $open parentheses m minus n close parentheses to the power of t h end exponent$ term be $q$ , then its $m to the power of t h end exponent$ term is $square root of p q end root$
Statement 2: $T subscript m plus n end subscript comma blank T subscript m end subscript comma blank T subscript m minus n end subscript$ are in G.P.

Statement 1:If $open vertical bar x minus 1 close vertical bar comma blank vertical line x minus 3 vertical line$ are first three terms of an AP, then its sixth term is 7< third terms.
Statement 2: $a comma blank a plus d comma blank a plus 2 d comma...$ are in AP $open parentheses d not equal to 0 close parentheses comma$ then sixth term is $open parentheses a plus 5 d close parentheses.$

Statement 1:If $open vertical bar x minus 1 close vertical bar comma blank vertical line x minus 3 vertical line$ are first three terms of an AP, then its sixth term is 7< third terms.
Statement 2: $a comma blank a plus d comma blank a plus 2 d comma...$ are in AP $open parentheses d not equal to 0 close parentheses comma$ then sixth term is $open parentheses a plus 5 d close parentheses.$

Statement 1:If sum f $n$ terms of a series $2 n to the power of 2 end exponent plus 3 n plus 1 comma$ then series is an AP.
Statement 2:Sum of $n$ terms of an AP is always of the form $p n to the power of 2 end exponent plus q n.$

Statement 1:If sum f $n$ terms of a series $2 n to the power of 2 end exponent plus 3 n plus 1 comma$ then series is an AP.
Statement 2:Sum of $n$ terms of an AP is always of the form $p n to the power of 2 end exponent plus q n.$

parallel

Let $a comma blank b comma blank c$ be three positive real numbers which are in HP.
Statement 1: $fraction numerator a plus b over denominator 2 a minus b end fraction plus fraction numerator c plus b over denominator 2 c minus b end fraction greater or equal than 4.$
Statement 2:If $x greater than 0 comma$ then $x plus fraction numerator 1 over denominator x end fraction greater or equal than 4.$

Let $a comma blank b comma blank c$ be three positive real numbers which are in HP.
Statement 1: $fraction numerator a plus b over denominator 2 a minus b end fraction plus fraction numerator c plus b over denominator 2 c minus b end fraction greater or equal than 4.$
Statement 2:If $x greater than 0 comma$ then $x plus fraction numerator 1 over denominator x end fraction greater or equal than 4.$

Statement 1:If $x to the power of 2 end exponent plus 9 y to the power of 2 end exponent plus 25 z to the power of 2 end exponent equals x y z open parentheses fraction numerator 15 over denominator x end fraction plus fraction numerator 5 over denominator y end fraction plus fraction numerator 3 over denominator z end fraction close parentheses$ , then $x comma blank y comma blank z$ are in H.P.
Statement 2:If $a subscript 1 end subscript superscript 2 end superscript plus a subscript 2 end subscript superscript 2 end superscript plus horizontal ellipsis plus a subscript n end subscript superscript 2 end superscript equals 0$ , then $a subscript 1 end subscript equals a subscript 2 end subscript equals a subscript 3 end subscript equals horizontal ellipsis a subscript n end subscript equals 0$

Statement 1:If $x to the power of 2 end exponent plus 9 y to the power of 2 end exponent plus 25 z to the power of 2 end exponent equals x y z open parentheses fraction numerator 15 over denominator x end fraction plus fraction numerator 5 over denominator y end fraction plus fraction numerator 3 over denominator z end fraction close parentheses$ , then $x comma blank y comma blank z$ are in H.P.
Statement 2:If $a subscript 1 end subscript superscript 2 end superscript plus a subscript 2 end subscript superscript 2 end superscript plus horizontal ellipsis plus a subscript n end subscript superscript 2 end superscript equals 0$ , then $a subscript 1 end subscript equals a subscript 2 end subscript equals a subscript 3 end subscript equals horizontal ellipsis a subscript n end subscript equals 0$

The harmonic mean of the roots of the equation $open parentheses 5 plus square root of 2 close parentheses x to the power of 2 end exponent minus open parentheses 4 plus square root of 5 close parentheses x plus 8 plus 2 square root of 5 equals 0$ is

The harmonic mean of the roots of the equation $open parentheses 5 plus square root of 2 close parentheses x to the power of 2 end exponent minus open parentheses 4 plus square root of 5 close parentheses x plus 8 plus 2 square root of 5 equals 0$ is

parallel

The sum to 50 terms of the series $fraction numerator 3 over denominator 1 to the power of 2 end exponent end fraction plus fraction numerator 5 over denominator 1 to the power of 2 end exponent plus 2 to the power of 2 end exponent end fraction plus fraction numerator 7 over denominator 1 to the power of 2 end exponent plus 2 to the power of 2 end exponent plus 3 to the power of 2 end exponent end fraction plus horizontal ellipsis$ is

The sum to 50 terms of the series $fraction numerator 3 over denominator 1 to the power of 2 end exponent end fraction plus fraction numerator 5 over denominator 1 to the power of 2 end exponent plus 2 to the power of 2 end exponent end fraction plus fraction numerator 7 over denominator 1 to the power of 2 end exponent plus 2 to the power of 2 end exponent plus 3 to the power of 2 end exponent end fraction plus horizontal ellipsis$ is

If $a comma blank x$ and $b$ are in A.P., $a comma blank y comma$ and $b$ are in G.P. and $a comma blank z comma blank b$ are in H.P. such that $x equals 9 z$ and $a greater than 0 comma blank b greater than 0$ , then

If $a comma blank x$ and $b$ are in A.P., $a comma blank y comma$ and $b$ are in G.P. and $a comma blank z comma blank b$ are in H.P. such that $x equals 9 z$ and $a greater than 0 comma blank b greater than 0$ , then

Let $a subscript 1 end subscript comma a subscript 2 end subscript comma horizontal ellipsis comma a subscript 10 end subscript$ be in A.P. and $h subscript 1 end subscript comma h subscript 2 end subscript comma horizontal ellipsis comma h subscript 10 end subscript$ be in H.P. If $a subscript 1 end subscript equals h subscript 1 end subscript equals 2$ and $a subscript 10 end subscript equals h subscript 10 end subscript equals 3$ , then $a subscript 4 end subscript h subscript 7 end subscript$ is

Let $a subscript 1 end subscript comma a subscript 2 end subscript comma horizontal ellipsis comma a subscript 10 end subscript$ be in A.P. and $h subscript 1 end subscript comma h subscript 2 end subscript comma horizontal ellipsis comma h subscript 10 end subscript$ be in H.P. If $a subscript 1 end subscript equals h subscript 1 end subscript equals 2$ and $a subscript 10 end subscript equals h subscript 10 end subscript equals 3$ , then $a subscript 4 end subscript h subscript 7 end subscript$ is

parallel

If $x comma blank y comma blank z$ are real and $4 x to the power of 2 end exponent plus 9 y to the power of 2 end exponent plus 16 z to the power of 2 end exponent minus 6 x y minus 12 y z minus 8 z x equals 0$ , then $x comma blank y comma blank z$ are in

If $x comma blank y comma blank z$ are real and $4 x to the power of 2 end exponent plus 9 y to the power of 2 end exponent plus 16 z to the power of 2 end exponent minus 6 x y minus 12 y z minus 8 z x equals 0$ , then $x comma blank y comma blank z$ are in

Let $alpha comma blank beta element of R$ . If $alpha comma blank beta to the power of 2 end exponent$ be the roots of quadratic equation $x to the power of 2 end exponent minus p x plus 1 equals 0$ and $alpha to the power of 2 end exponent comma beta$ be the roots of quadratic equation $x to the power of 2 end exponent minus q x plus 8 equals 0$ , then the value of ‘ $r$ ’ if $fraction numerator r over denominator 8 end fraction$ be arithmetic mean of $p$ and $q$ is

Let $alpha comma blank beta element of R$ . If $alpha comma blank beta to the power of 2 end exponent$ be the roots of quadratic equation $x to the power of 2 end exponent minus p x plus 1 equals 0$ and $alpha to the power of 2 end exponent comma beta$ be the roots of quadratic equation $x to the power of 2 end exponent minus q x plus 8 equals 0$ , then the value of ‘ $r$ ’ if $fraction numerator r over denominator 8 end fraction$ be arithmetic mean of $p$ and $q$ is

If $a comma fraction numerator 1 over denominator b end fraction comma blank c$ and $fraction numerator 1 over denominator p end fraction comma blank q comma fraction numerator 1 over denominator r end fraction$ form two arithmetic progressions of the same common difference, then $a comma blank q comma c$ are in A.P. if

If $a comma fraction numerator 1 over denominator b end fraction comma blank c$ and $fraction numerator 1 over denominator p end fraction comma blank q comma fraction numerator 1 over denominator r end fraction$ form two arithmetic progressions of the same common difference, then $a comma blank q comma c$ are in A.P. if

parallel

If $open vertical bar a close vertical bar less than 1$ and $open vertical bar b close vertical bar less than 1$ , then the sum of the series $1 plus open parentheses 1 plus a close parentheses b plus open parentheses 1 plus a plus a to the power of 2 end exponent close parentheses b to the power of 2 end exponent plus open parentheses 1 plus a plus a to the power of 2 end exponent plus a to the power of 3 end exponent close parentheses b to the power of 3 end exponent plus horizontal ellipsis$ is

If $open vertical bar a close vertical bar less than 1$ and $open vertical bar b close vertical bar less than 1$ , then the sum of the series $1 plus open parentheses 1 plus a close parentheses b plus open parentheses 1 plus a plus a to the power of 2 end exponent close parentheses b to the power of 2 end exponent plus open parentheses 1 plus a plus a to the power of 2 end exponent plus a to the power of 3 end exponent close parentheses b to the power of 3 end exponent plus horizontal ellipsis$ is

The value of $not stretchy sum from r equals 0 to n of open parentheses a plus r plus a r close parentheses open parentheses negative a close parentheses to the power of r end exponent$ is equal to

The value of $not stretchy sum from r equals 0 to n of open parentheses a plus r plus a r close parentheses open parentheses negative a close parentheses to the power of r end exponent$ is equal to

The sum of series $1 plus fraction numerator 4 over denominator 5 end fraction plus fraction numerator 7 over denominator 5 to the power of 2 end exponent end fraction plus fraction numerator 10 over denominator 5 to the power of 3 end exponent end fraction plus horizontal ellipsis infinity$ is

The sum of series $1 plus fraction numerator 4 over denominator 5 end fraction plus fraction numerator 7 over denominator 5 to the power of 2 end exponent end fraction plus fraction numerator 10 over denominator 5 to the power of 3 end exponent end fraction plus horizontal ellipsis infinity$ is

parallel

card img

With Turito Academy.

card img

With Turito Foundation.

card img

Get an Expert Advice From Turito.

Turito Academy

card img

With Turito Academy.

Test Prep

card img

With Turito Foundation.