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Maths-

General

Easy

Question

The system of equations $a x minus y minus z equals alpha minus 1$ $x minus alpha y minus z equals alpha minus 1$ $x minus y minus alpha z equals alpha minus 1$ Has no solution if $alpha$ is

Either $- 2$ or 1
$- 2$
1
Not $- 2$

The correct answer is: $negative 2$

For no solution or infinitely many solutions
$open vertical bar table row alpha cell negative 1 end cell cell negative 1 end cell row 1 cell negative alpha end cell cell negative 1 end cell row 1 1 cell negative alpha end cell end table close vertical bar equals 0$
$rightwards double arrow blank alpha open parentheses alpha to the power of 2 end exponent minus 1 close parentheses minus 1 open parentheses alpha minus 1 close parentheses plus 1 open parentheses 1 minus alpha close parentheses equals 0$
$rightwards double arrow blank alpha open parentheses alpha to the power of 2 end exponent minus 1 close parentheses minus 2 alpha plus 2 equals 0$
$rightwards double arrow blank alpha open parentheses alpha minus 1 close parentheses open parentheses alpha plus 1 close parentheses minus 2 open parentheses alpha minus 1 close parentheses equals 0$
$rightwards double arrow open parentheses alpha minus 1 close parentheses open parentheses alpha to the power of 2 end exponent plus alpha minus 2 close parentheses equals 0$
$rightwards double arrow open parentheses alpha minus 1 close parentheses open parentheses alpha plus 2 close parentheses open parentheses alpha minus 1 close parentheses equals 0$
$rightwards double arrow open parentheses alpha minus 1 close parentheses to the power of 2 end exponent open parentheses alpha plus 2 close parentheses equals 0$
$rightwards double arrow blank alpha equals 1 comma blank 1 comma blank minus 2$
But for $alpha equals 1$ , there are infinite solutions. When $alpha equals negative 2$ , wehave
$negative 2 x minus y minus z equals negative 3$
$x plus 2 y minus z equals negative 3$
$x minus y plus 2 z equals negative 3$
Adding, we get $0 equals negative 9$ , which is not true. Hence there is no solution

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The value of the determinant $open vertical bar table row 1 1 1 row cell blank to the power of m end exponent C subscript 1 end subscript end cell cell blank to the power of m plus 1 end exponent C subscript 1 end subscript end cell cell blank to the power of m plus 2 end exponent C subscript 1 end subscript end cell row cell blank to the power of m end exponent C subscript 2 end subscript end cell cell blank to the power of m plus 1 end exponent C subscript 2 end subscript end cell cell blank to the power of m plus 2 end exponent C subscript 2 end subscript end cell end table close vertical bar$ is equal to

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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

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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

parallel

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.

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