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Question

Consider the system of equation $x plus y plus z equals 6 comma blank x plus 2 y plus 3 z equals 10$ and $x plus 2 y plus lambda z equals mu$
Statement 1 If the system has infinite number of solutions, then $mu equals 10$
Statement 2: The determinant $open vertical bar table row 1 1 6 row 1 2 10 row 1 2 mu end table close vertical bar equals 0$ for $mu equals 10$

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 not correct explanation for Statement 1

Let $increment equals open vertical bar table row 1 1 1 row 1 2 3 row 1 2 lambda end table close vertical bar equals 0 rightwards double arrow blank lambda equals 3$ . Now,
$increment subscript 1 end subscript equals open vertical bar table row 6 1 1 row 10 2 3 row mu 2 lambda end table close vertical bar$
$equals open vertical bar table row 6 1 1 row 10 2 3 row mu 2 3 end table close vertical bar equals blank mu minus 10$
$increment subscript 2 end subscript equals open vertical bar table row 1 6 1 row 1 10 3 row 1 cell blank mu end cell lambda end table close vertical bar equals open vertical bar table row 1 6 1 row 1 10 3 row 1 cell blank mu end cell 3 end table close vertical bar equals 20 minus 2 blank mu$
$increment subscript 3 end subscript equals open vertical bar table row 1 1 6 row 1 2 10 row 1 2 mu end table close vertical bar equals mu minus 10$
Clearly, for $mu equals 10$ , all of $increment subscript 1 end subscript comma blank increment subscript 2 end subscript comma blank increment subscript 3 end subscript$ are zero

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parallel

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parallel

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If $A subscript 1 end subscript comma blank B subscript 1 end subscript comma blank C subscript 1 end subscript comma blank horizontal ellipsis$ are, respectively, the cofactors of the elements $a subscript 1 end subscript comma blank b subscript 1 end subscript comma blank c subscript 1 end subscript comma blank horizontal ellipsis$ of the determinant $increment equals open vertical bar table row cell a subscript 1 end subscript end cell cell b subscript 1 end subscript end cell cell c subscript 1 end subscript end cell row cell a subscript 2 end subscript end cell cell b subscript 2 end subscript end cell cell c subscript 2 end subscript end cell row cell a subscript 3 end subscript end cell cell b subscript 3 end subscript end cell cell c subscript 3 end subscript end cell end table close vertical bar comma blank increment not equal to 0$ , then the value of $open vertical bar table row cell B subscript 2 end subscript end cell cell C subscript 2 end subscript end cell row cell B subscript 3 end subscript end cell cell C subscript 3 end subscript end cell end table close vertical bar$ is equal to

Let $stack a with rightwards arrow on top$ be a unit vector perpendicular to unit vectors $stack b with rightwards arrow on top a n d stack c with rightwards arrow on top$ inclined at an angle a, then $stack b with rightwards arrow on top cross times stack c with rightwards arrow on top$ is

Let $stack a with rightwards arrow on top$ be a unit vector perpendicular to unit vectors $stack b with rightwards arrow on top a n d stack c with rightwards arrow on top$ inclined at an angle a, then $stack b with rightwards arrow on top cross times stack c with rightwards arrow on top$ is

parallel

Let $stack a with rightwards arrow on top comma short rightwards arrow over leftwards arrow short rightwards arrow over leftwards arrow stack b with rightwards arrow on top short rightwards arrow over leftwards arrow short rightwards arrow over leftwards arrow a n d short rightwards arrow over leftwards arrow short rightwards arrow over leftwards arrow short rightwards arrow over leftwards arrow stack c with rightwards arrow on top$ be three non-coplanar vectors and $stack d with rightwards arrow on top$ be a non-zero vector which is perpendicular to $open parentheses stack a with rightwards arrow on top plus stack b with rightwards arrow on top plus stack c with rightwards arrow on top close parentheses$ and is represented as $stack d with rightwards arrow on top equals x open parentheses stack a with rightwards arrow on top cross times stack b with rightwards arrow on top close parentheses plus y open parentheses stack b with rightwards arrow on top cross times stack c with rightwards arrow on top close parentheses plus z open parentheses stack c with rightwards arrow on top cross times stack a with rightwards arrow on top close parentheses$ . Then

Let $stack a with rightwards arrow on top comma short rightwards arrow over leftwards arrow short rightwards arrow over leftwards arrow stack b with rightwards arrow on top short rightwards arrow over leftwards arrow short rightwards arrow over leftwards arrow a n d short rightwards arrow over leftwards arrow short rightwards arrow over leftwards arrow short rightwards arrow over leftwards arrow stack c with rightwards arrow on top$ be three non-coplanar vectors and $stack d with rightwards arrow on top$ be a non-zero vector which is perpendicular to $open parentheses stack a with rightwards arrow on top plus stack b with rightwards arrow on top plus stack c with rightwards arrow on top close parentheses$ and is represented as $stack d with rightwards arrow on top equals x open parentheses stack a with rightwards arrow on top cross times stack b with rightwards arrow on top close parentheses plus y open parentheses stack b with rightwards arrow on top cross times stack c with rightwards arrow on top close parentheses plus z open parentheses stack c with rightwards arrow on top cross times stack a with rightwards arrow on top close parentheses$ . Then

Let OPQR is a tetrahedron such that O is origin and $stack p with rightwards arrow on top comma short rightwards arrow over leftwards arrow short rightwards arrow over leftwards arrow stack q with rightwards arrow on top comma short rightwards arrow over leftwards arrow short rightwards arrow over leftwards arrow stack r with rightwards arrow on top$ are position vectors of P, Q, R respectively and a is the angle which OP makes with face PQR then

Hence required angle is 2.

Let OPQR is a tetrahedron such that O is origin and $stack p with rightwards arrow on top comma short rightwards arrow over leftwards arrow short rightwards arrow over leftwards arrow stack q with rightwards arrow on top comma short rightwards arrow over leftwards arrow short rightwards arrow over leftwards arrow stack r with rightwards arrow on top$ are position vectors of P, Q, R respectively and a is the angle which OP makes with face PQR then

Hence required angle is 2.

On strong heating Mgcl₂ 6H₂ Othe product obtained is

On strong heating Mgcl₂ 6H₂ Othe product obtained is

Chemistry-General

parallel

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