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Maths-

General

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Question

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

1
$- 1$
0
None of these

The correct answer is: 1

$open vertical bar table row 1 1 1 row cell blank to the power of m end exponent C subscript 1 end subscript end cell cell blank to the power of m plus 1 end exponent C subscript 1 end subscript end cell cell blank to the power of m plus 2 end exponent C subscript 1 end subscript end cell row cell blank to the power of m end exponent C subscript 2 end subscript end cell cell blank to the power of m plus 1 end exponent C subscript 2 end subscript end cell cell blank to the power of m plus 2 end exponent C subscript 2 end subscript end cell end table close vertical bar$
$equals open vertical bar table row 1 1 1 row cell blank to the power of m end exponent C subscript 1 end subscript end cell cell blank to the power of m plus 1 end exponent C subscript 1 end subscript end cell cell blank to the power of m plus 1 end exponent C subscript 0 end subscript plus blank to the power of m plus 1 end exponent C subscript 1 end subscript end cell row cell blank to the power of m end exponent C subscript 2 end subscript end cell cell blank to the power of m plus 1 end exponent C subscript 2 end subscript end cell cell blank to the power of m plus 1 end exponent C subscript 1 end subscript plus blank to the power of m plus 1 end exponent C subscript 2 end subscript end cell end table close vertical bar$
$equals open vertical bar table row 1 1 0 row cell blank to the power of m end exponent C subscript 1 end subscript end cell cell blank to the power of m plus 1 end exponent C subscript 1 end subscript end cell cell blank to the power of m plus 1 end exponent C subscript 0 end subscript end cell row cell blank to the power of m end exponent C subscript 2 end subscript end cell cell blank to the power of m plus 1 end exponent C subscript 2 end subscript end cell cell blank to the power of m plus 1 end exponent C subscript 1 end subscript end cell end table close vertical bar$ [Applying $C subscript 3 end subscript rightwards arrow C subscript 3 end subscript minus C subscript 2 end subscript$ ]
$equals open vertical bar table row 1 1 0 row cell blank to the power of m end exponent C subscript 1 end subscript end cell cell blank to the power of m end exponent C subscript 0 end subscript plus blank to the power of m end exponent C subscript 1 end subscript end cell cell blank to the power of m plus 1 end exponent C subscript 0 end subscript end cell row cell blank to the power of m end exponent C subscript 2 end subscript end cell cell blank to the power of m end exponent C subscript 1 end subscript plus blank to the power of m end exponent C subscript 2 end subscript end cell cell blank to the power of m plus 1 end exponent C subscript 1 end subscript end cell end table close vertical bar$
$equals open vertical bar table row 1 1 0 row cell blank to the power of m end exponent C subscript 1 end subscript end cell cell blank to the power of m end exponent C subscript 0 end subscript end cell cell blank to the power of m plus 1 end exponent C subscript 0 end subscript end cell row cell blank to the power of m end exponent C subscript 2 end subscript end cell cell blank to the power of m end exponent C subscript 1 end subscript end cell cell blank to the power of m plus 1 end exponent C subscript 1 end subscript end cell end table close vertical bar$ [Applying $C subscript 2 end subscript rightwards arrow C subscript 2 end subscript minus C subscript 1 end subscript$ ]
$equals blank to the power of m end exponent C subscript 0 end subscript blank to the power of m plus 1 end exponent C subscript 1 end subscript minus blank to the power of m plus 1 end exponent C subscript 0 end subscript blank to the power of m end exponent C subscript 1 end subscript$
$equals m plus 1 minus m$
$equals 1$

Related Questions to study

If $alpha comma blank beta comma blank gamma$ are the angles of a triangle and the system of equations $cos invisible function application open parentheses alpha minus beta close parentheses x plus cos invisible function application open parentheses beta minus gamma close parentheses y plus cos invisible function application open parentheses gamma minus alpha close parentheses z equals 0 cos invisible function application open parentheses alpha plus beta close parentheses x plus cos invisible function application open parentheses beta plus gamma close parentheses y plus cos invisible function application open parentheses gamma plus alpha close parentheses z equals 0 sin invisible function application open parentheses alpha plus beta close parentheses x plus sin invisible function application open parentheses beta plus gamma close parentheses y plus sin invisible function application open parentheses gamma plus alpha close parentheses z equals 0$ Has non-trivial solutions, then triangle is necessarily

If $alpha comma blank beta comma blank gamma$ are the angles of a triangle and the system of equations $cos invisible function application open parentheses alpha minus beta close parentheses x plus cos invisible function application open parentheses beta minus gamma close parentheses y plus cos invisible function application open parentheses gamma minus alpha close parentheses z equals 0 cos invisible function application open parentheses alpha plus beta close parentheses x plus cos invisible function application open parentheses beta plus gamma close parentheses y plus cos invisible function application open parentheses gamma plus alpha close parentheses z equals 0 sin invisible function application open parentheses alpha plus beta close parentheses x plus sin invisible function application open parentheses beta plus gamma close parentheses y plus sin invisible function application open parentheses gamma plus alpha close parentheses z equals 0$ Has non-trivial solutions, then triangle is necessarily

If $x not equal to y not equal to z$ and $open vertical bar table row x cell x to the power of 2 end exponent end cell cell 1 plus x to the power of 3 end exponent end cell row y cell y to the power of 2 end exponent end cell cell 1 plus y to the power of 3 end exponent end cell row z cell z to the power of 2 end exponent end cell cell 1 plus z to the power of 3 end exponent end cell end table close vertical bar equals 0$ , then the value of $x y z$ is

If $x not equal to y not equal to z$ and $open vertical bar table row x cell x to the power of 2 end exponent end cell cell 1 plus x to the power of 3 end exponent end cell row y cell y to the power of 2 end exponent end cell cell 1 plus y to the power of 3 end exponent end cell row z cell z to the power of 2 end exponent end cell cell 1 plus z to the power of 3 end exponent end cell end table close vertical bar equals 0$ , then the value of $x y z$ is

If $f open parentheses x close parentheses equals a plus b x plus c x to the power of 2 end exponent$ and $alpha comma blank beta comma blank gamma$ are the roots of the equation $x to the power of 3 end exponent equals 1$ , then $open vertical bar table row a b c row b c a row c a b end table close vertical bar$ is equal to

If $f open parentheses x close parentheses equals a plus b x plus c x to the power of 2 end exponent$ and $alpha comma blank beta comma blank gamma$ are the roots of the equation $x to the power of 3 end exponent equals 1$ , then $open vertical bar table row a b c row b c a row c a b end table close vertical bar$ is equal to

parallel

If $A equals open square brackets table row alpha 2 row 2 alpha end table close square brackets$ and $open vertical bar A to the power of 3 end exponent close vertical bar equals 125 comma blank$ then the value of $alpha$ is

If $A equals open square brackets table row alpha 2 row 2 alpha end table close square brackets$ and $open vertical bar A to the power of 3 end exponent close vertical bar equals 125 comma blank$ then the value of $alpha$ is

If $omega blank open parentheses not equal to 1 close parentheses$ is a cube root of unity, then value of the determinant $open vertical bar table row 1 cell 1 plus i plus omega to the power of 2 end exponent end cell cell omega to the power of 2 end exponent end cell row cell 1 minus i end cell cell negative 1 end cell cell omega to the power of 2 end exponent minus 1 end cell row cell negative i end cell cell negative i plus omega minus 1 end cell cell negative 1 end cell end table close vertical bar$ is

If $omega blank open parentheses not equal to 1 close parentheses$ is a cube root of unity, then value of the determinant $open vertical bar table row 1 cell 1 plus i plus omega to the power of 2 end exponent end cell cell omega to the power of 2 end exponent end cell row cell 1 minus i end cell cell negative 1 end cell cell omega to the power of 2 end exponent minus 1 end cell row cell negative i end cell cell negative i plus omega minus 1 end cell cell negative 1 end cell end table close vertical bar$ is

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

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

parallel

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

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.

parallel

On strong heating Mgcl₂ 6H₂ Othe product obtained is

On strong heating Mgcl₂ 6H₂ Othe product obtained is

Chemistry-General

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Which one of the following hydroxide is insoluble in water?

Chemistry-General

Which is the incorrect statement?

Which is the incorrect statement?

Chemistry-General

parallel

Which salt can be used to identify coloured cation?

Which salt can be used to identify coloured cation?

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Which will give $N subscript 2 end subscript O$ on heating?

Which will give $N subscript 2 end subscript O$ on heating?

Chemistry-General

Some of alkali metal salts are coloured. $N a subscript 2 end subscript C r O subscript 4 end subscript minus$ yellow, $K M n O subscript 4 end subscript minus$ pink, $K subscript 2 end subscript M n O subscript 4 end subscript minus$ green. It is due to

Some of alkali metal salts are coloured. $N a subscript 2 end subscript C r O subscript 4 end subscript minus$ yellow, $K M n O subscript 4 end subscript minus$ pink, $K subscript 2 end subscript M n O subscript 4 end subscript minus$ green. It is due to

Chemistry-General

parallel

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