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Statement 1:The numbers $square root of 2 comma blank square root of 3 comma blank square root of 5$ cannot be the terms of a single A.P. with non-zero common difference
Statement 2:If $p comma blank q comma blank r left parenthesis p not equal to q right parenthesis$ are terms (not necessarily consecutive) of an A.P., then there exists a rational number $k$ such that $left parenthesis r minus q right parenthesis divided by left parenthesis q minus p right parenthesis equals k$

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 True, Statement 2 is True; Statement 2 is correct explanation for Statement 1

Let $p comma blank q comma blank r$ be the $l to the power of t h end exponent comma blank m to the power of t h end exponent$ and $n to the power of t h end exponent$ terms of an A.P. Then
$p equals left parenthesis a plus open parentheses l minus 1 close parentheses d comma blank q equals a plus left parenthesis m minus 1 right parenthesis d$ and $r equals a plus open parentheses n minus 1 close parentheses d$
Hence, $r minus q equals left parenthesis n minus m right parenthesis d$ and $q minus p equals left parenthesis m minus l right parenthesis d$ , so that
$fraction numerator r minus q over denominator p minus q end fraction equals fraction numerator left parenthesis n minus m right parenthesis d over denominator left parenthesis m minus l right parenthesis d end fraction equals fraction numerator n minus m over denominator m minus l end fraction blank left parenthesis because d not equal to 0 right parenthesis$
Since, $l comma blank m comma blank n$ are positive integers and $m not equal to l comma blank left parenthesis n minus m right parenthesis divided by left parenthesis m minus l right parenthesis$ is a rational number. From (1), using $p equals square root of 2 comma blank q equals square root of 3 comma blank r equals square root of 5$ , we have
$fraction numerator square root of 5 minus square root of 3 over denominator square root of 3 minus square root of 2 end fraction equals fraction numerator n minus m over denominator m minus l end fraction blank open parentheses w h i c h blank i s blank n o t blank p o s s i b l e close parentheses$
Hence, $square root of 2 comma blank square root of 3 comma blank square root of 5$ cannot be the terms of an A.P.

Related Questions to study

Statement 1:3, 6, 12 are in GP, then 9, 12, 18 are in HP.
Statement 2:If middle term is added in three consecutive terms of a GP, resultant will be in HP.

Statement 1:3, 6, 12 are in GP, then 9, 12, 18 are in HP.
Statement 2:If middle term is added in three consecutive terms of a GP, resultant will be in HP.

Statement 1:If sum of $n$ terms of a series is $6 n to the power of 2 end exponent plus 3 n plus 1$ then the series is in AP $.$
Statement 2:Sum of $n$ terms of an AP is always of the form $a n to the power of 2 end exponent plus b n.$

Statement 1:If sum of $n$ terms of a series is $6 n to the power of 2 end exponent plus 3 n plus 1$ then the series is in AP $.$
Statement 2:Sum of $n$ terms of an AP is always of the form $a n to the power of 2 end exponent plus b n.$

Statement 1:If the arithmetic mean of two numbers is 5/2, geometric mean of the numbers is 2, then the harmonic mean will be 8/5
Statement 2:For a group of positive numbers $open parentheses G. M. close parentheses to the power of 2 end exponent equals open parentheses A. M. close parentheses cross times left parenthesis H. M. right parenthesis$

Statement 1:If the arithmetic mean of two numbers is 5/2, geometric mean of the numbers is 2, then the harmonic mean will be 8/5
Statement 2:For a group of positive numbers $open parentheses G. M. close parentheses to the power of 2 end exponent equals open parentheses A. M. close parentheses cross times left parenthesis H. M. right parenthesis$

parallel

Statement 1:The sum of $n$ terms of two arithmetic progressions are in the ratio $open parentheses 7 n plus 1 close parentheses colon open parentheses 4 n plus 17 close parentheses comma$ then the ratio of their $n$ th terms is 7: 4.
Statement 2:If $S subscript n end subscript equals a x to the power of 2 end exponent plus b x plus c comma$ then $T subscript n end subscript equals S subscript n end subscript minus S subscript n minus 1 end subscript$

Statement 1:The sum of $n$ terms of two arithmetic progressions are in the ratio $open parentheses 7 n plus 1 close parentheses colon open parentheses 4 n plus 17 close parentheses comma$ then the ratio of their $n$ th terms is 7: 4.
Statement 2:If $S subscript n end subscript equals a x to the power of 2 end exponent plus b x plus c comma$ then $T subscript n end subscript equals S subscript n end subscript minus S subscript n minus 1 end subscript$

Let $a comma blank r element of R minus left curly bracket 0 comma blank 1 comma blank minus 1 right curly bracket$ and $n$ be an even number
Statement 1: $a cross times a r cross times a r to the power of 2 end exponent midline horizontal ellipsis a r to the power of n minus 1 end exponent equals open parentheses a to the power of 2 end exponent r to the power of n minus 1 end exponent close parentheses to the power of n divided by 2 end exponent$
Statement 2:Product of $i to the power of t h end exponent$ term from the beginning and from the end in a G.P. is independent of $i$

Let $a comma blank r element of R minus left curly bracket 0 comma blank 1 comma blank minus 1 right curly bracket$ and $n$ be an even number
Statement 1: $a cross times a r cross times a r to the power of 2 end exponent midline horizontal ellipsis a r to the power of n minus 1 end exponent equals open parentheses a to the power of 2 end exponent r to the power of n minus 1 end exponent close parentheses to the power of n divided by 2 end exponent$
Statement 2:Product of $i to the power of t h end exponent$ term from the beginning and from the end in a G.P. is independent of $i$

Statement 1:Let $p subscript 1 end subscript comma p subscript 2 end subscript comma horizontal ellipsis comma p subscript n end subscript$ and $x$ be distinct real number such that $open parentheses not stretchy sum from r equals 1 to n minus 1 of p subscript r end subscript superscript 2 end superscript close parentheses x to the power of 2 end exponent plus 2 open parentheses not stretchy sum from r equals 1 to n minus 1 of p subscript r end subscript blank p subscript r plus 1 end subscript close parentheses x plus not stretchy sum from r equals 2 to n of p subscript r end subscript superscript 2 end superscript less or equal than 0$ , then $p subscript 1 end subscript comma p subscript 2 end subscript comma horizontal ellipsis comma p subscript n end subscript$ are in G.P. and when $a subscript 1 end subscript superscript 2 end superscript plus a subscript 2 end subscript superscript 2 end superscript plus a subscript 3 end subscript superscript 2 end superscript plus horizontal ellipsis plus a subscript n end subscript superscript 2 end superscript equals 0 comma blank a subscript 1 end subscript equals a subscript 2 end subscript equals a subscript 3 end subscript equals horizontal ellipsis equals a subscript n end subscript equals 0$
Statement 2:If $fraction numerator p subscript 2 end subscript over denominator p subscript 1 end subscript end fraction equals fraction numerator p subscript 3 end subscript over denominator p subscript 2 end subscript end fraction equals horizontal ellipsis equals fraction numerator p subscript n end subscript over denominator p subscript n minus 1 end subscript end fraction$ , then $p subscript 1 end subscript comma p subscript 2 end subscript comma horizontal ellipsis comma p subscript n end subscript$ are in G.P.

Statement 1:Let $p subscript 1 end subscript comma p subscript 2 end subscript comma horizontal ellipsis comma p subscript n end subscript$ and $x$ be distinct real number such that $open parentheses not stretchy sum from r equals 1 to n minus 1 of p subscript r end subscript superscript 2 end superscript close parentheses x to the power of 2 end exponent plus 2 open parentheses not stretchy sum from r equals 1 to n minus 1 of p subscript r end subscript blank p subscript r plus 1 end subscript close parentheses x plus not stretchy sum from r equals 2 to n of p subscript r end subscript superscript 2 end superscript less or equal than 0$ , then $p subscript 1 end subscript comma p subscript 2 end subscript comma horizontal ellipsis comma p subscript n end subscript$ are in G.P. and when $a subscript 1 end subscript superscript 2 end superscript plus a subscript 2 end subscript superscript 2 end superscript plus a subscript 3 end subscript superscript 2 end superscript plus horizontal ellipsis plus a subscript n end subscript superscript 2 end superscript equals 0 comma blank a subscript 1 end subscript equals a subscript 2 end subscript equals a subscript 3 end subscript equals horizontal ellipsis equals a subscript n end subscript equals 0$
Statement 2:If $fraction numerator p subscript 2 end subscript over denominator p subscript 1 end subscript end fraction equals fraction numerator p subscript 3 end subscript over denominator p subscript 2 end subscript end fraction equals horizontal ellipsis equals fraction numerator p subscript n end subscript over denominator p subscript n minus 1 end subscript end fraction$ , then $p subscript 1 end subscript comma p subscript 2 end subscript comma horizontal ellipsis comma p subscript n end subscript$ are in G.P.

parallel

Statement 1:If $3 x plus 4 y equals 5 comma$ then the greatest value of $x to the power of 2 end exponent y to the power of 3 end exponent$ is $fraction numerator 3 over denominator 16 end fraction.$
Statement 2:Greatest value occurs when $9 x equals 8 y.$

Statement 1:If $3 x plus 4 y equals 5 comma$ then the greatest value of $x to the power of 2 end exponent y to the power of 3 end exponent$ is $fraction numerator 3 over denominator 16 end fraction.$
Statement 2:Greatest value occurs when $9 x equals 8 y.$

Statement 1:There are infinite geometric progressions for which 27, 8 and 12 are three of its terms (not necessarily consecutive)
Statement 2:Given terms are integers

Statement 1:There are infinite geometric progressions for which 27, 8 and 12 are three of its terms (not necessarily consecutive)
Statement 2:Given terms are integers

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: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 $n greater or equal than 1 comma blank n to the power of 3 end exponent minus open parentheses n minus 1 close parentheses to the power of 3 end exponent plus horizontal ellipsis plus open parentheses negative 1 close parentheses to the power of n minus 1 end exponent blank 1 to the power of 3 end exponent equals fraction numerator 1 over denominator 4 end fraction open parentheses 2 n minus 1 close parentheses open parentheses n plus 1 close parentheses to the power of 2 end exponent$

parallel

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

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