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Question

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

0
1
i


The correct answer is: i

Related Questions to study

Find the value of ‘a’ if the three equations, (a + 1)³x + (a + 2)³y = (a + 3)³, (a + 1) x + (a + 2)y = (a + 3) & x + y = 1 are consistent.

Find the value of ‘a’ if the three equations, (a + 1)³x + (a + 2)³y = (a + 3)³, (a + 1) x + (a + 2)y = (a + 3) & x + y = 1 are consistent.

The condition for the expression ax² + 2hxy + by² + 2gx + 2fy + c to be resolved into rational linear factors in the determinant form is -

The condition for the expression ax² + 2hxy + by² + 2gx + 2fy + c to be resolved into rational linear factors in the determinant form is -

If $open vertical bar table row cell 1 plus x end cell x cell x to the power of 2 end exponent end cell row x cell 1 plus x end cell cell x to the power of 2 end exponent end cell row cell x to the power of 2 end exponent end cell x cell 1 plus x end cell end table close vertical bar$ = ax⁵ + bx⁴ + cx³ + dx² + $lambda$ x + $mu$ be an identity in x, where a, b, c, d, $lambda$ , $mu$ are independent of x. Then the value of $lambda$ is

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parallel

If the following equations x + y – 3 = 0(1 + $lambda$ ) x + (2 + $lambda$ ) y – 8 = 0x – (1 + $lambda$ ) y + (2 + $lambda$ ) = 0 are consistent then the value of $lambda$ is

If the following equations x + y – 3 = 0(1 + $lambda$ ) x + (2 + $lambda$ ) y – 8 = 0x – (1 + $lambda$ ) y + (2 + $lambda$ ) = 0 are consistent then the value of $lambda$ is

If $alpha comma beta comma gamma$ are the roots of x³ – 3x + 2 = 0, then the value of the determinant $open vertical bar table row alpha beta gamma row beta gamma alpha row gamma alpha beta end table close vertical bar$ is equal to

If $alpha comma beta comma gamma$ are the roots of x³ – 3x + 2 = 0, then the value of the determinant $open vertical bar table row alpha beta gamma row beta gamma alpha row gamma alpha beta end table close vertical bar$ is equal to

If $straight capital delta$ ABC is a scalene triangle, then the value of $open vertical bar table row cell sin invisible function application A end cell cell sin invisible function application B end cell cell sin invisible function application C end cell row cell cos invisible function application A end cell cell cos invisible function application B end cell cell cos invisible function application C end cell row 1 1 1 end table close vertical bar$ is

If $straight capital delta$ ABC is a scalene triangle, then the value of $open vertical bar table row cell sin invisible function application A end cell cell sin invisible function application B end cell cell sin invisible function application C end cell row cell cos invisible function application A end cell cell cos invisible function application B end cell cell cos invisible function application C end cell row 1 1 1 end table close vertical bar$ is

parallel

Consider the system of equations-x – 2y + 3z = –1–x + y – 2z = k x – 3y + 4z = 1
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STATEMENT-2: The determinant $open vertical bar table row 1 3 cell negative 1 end cell row cell negative 1 end cell cell negative 2 end cell k row 1 4 1 end table close vertical bar$ $not equal to$ 0, for k $not equal to$ 3

Consider the system of equations-x – 2y + 3z = –1–x + y – 2z = k x – 3y + 4z = 1
STATEMENT-1: The system of equations has no solution for k $not equal to$ 3
STATEMENT-2: The determinant $open vertical bar table row 1 3 cell negative 1 end cell row cell negative 1 end cell cell negative 2 end cell k row 1 4 1 end table close vertical bar$ $not equal to$ 0, for k $not equal to$ 3

Suppose, x > 0, y > 0, z > 0 and $capital delta$ (a, b, c) = $open vertical bar table row cell x log invisible function application 2 end cell 3 cell 15 plus log invisible function application left parenthesis a to the power of x end exponent right parenthesis end cell row cell y log invisible function application 3 end cell 5 cell 25 plus log invisible function application left parenthesis b to the power of y end exponent right parenthesis end cell row cell z log invisible function application 5 end cell 7 cell 35 plus log invisible function application left parenthesis c to the power of z end exponent right parenthesis end cell end table close vertical bar$
Statement - 1 : $capital delta$ (8, 27, 125) = 0
Statement - 2 : $capital delta$ $open parentheses fraction numerator 1 over denominator 2 end fraction comma fraction numerator 1 over denominator 3 end fraction comma fraction numerator 1 over denominator 5 end fraction close parentheses$ = 0

Suppose, x > 0, y > 0, z > 0 and $capital delta$ (a, b, c) = $open vertical bar table row cell x log invisible function application 2 end cell 3 cell 15 plus log invisible function application left parenthesis a to the power of x end exponent right parenthesis end cell row cell y log invisible function application 3 end cell 5 cell 25 plus log invisible function application left parenthesis b to the power of y end exponent right parenthesis end cell row cell z log invisible function application 5 end cell 7 cell 35 plus log invisible function application left parenthesis c to the power of z end exponent right parenthesis end cell end table close vertical bar$
Statement - 1 : $capital delta$ (8, 27, 125) = 0
Statement - 2 : $capital delta$ $open parentheses fraction numerator 1 over denominator 2 end fraction comma fraction numerator 1 over denominator 3 end fraction comma fraction numerator 1 over denominator 5 end fraction close parentheses$ = 0

If $fraction numerator cos space x over denominator cos space y end fraction equals 2$ and $cos space left parenthesis x minus y right parenthesis equals fraction numerator square root of 3 over denominator 2 end fraction$ then y=

If $fraction numerator cos space x over denominator cos space y end fraction equals 2$ and $cos space left parenthesis x minus y right parenthesis equals fraction numerator square root of 3 over denominator 2 end fraction$ then y=

parallel

If $tan space left parenthesis pi divided by 4 plus theta right parenthesis plus tan space left parenthesis pi divided by 4 minus theta right parenthesis equals a$ then $t a n cubed space left parenthesis pi divided by 4 plus theta right parenthesis plus t a n cubed space left parenthesis pi divided by 4 minus theta right parenthesis equals$

If $tan space left parenthesis pi divided by 4 plus theta right parenthesis plus tan space left parenthesis pi divided by 4 minus theta right parenthesis equals a$ then $t a n cubed space left parenthesis pi divided by 4 plus theta right parenthesis plus t a n cubed space left parenthesis pi divided by 4 minus theta right parenthesis equals$

$text If end text cos left parenthesis theta minus alpha right parenthesis equals a times cos space left parenthesis theta minus beta right parenthesis equals b$ then $s i n squared space left parenthesis alpha minus beta right parenthesis plus 2 a b c o s space left parenthesis alpha minus beta right parenthesis equals$

$text If end text cos left parenthesis theta minus alpha right parenthesis equals a times cos space left parenthesis theta minus beta right parenthesis equals b$ then $s i n squared space left parenthesis alpha minus beta right parenthesis plus 2 a b c o s space left parenthesis alpha minus beta right parenthesis equals$

if $fraction numerator cos space x minus cos space alpha over denominator cos space x minus cos space beta end fraction equals fraction numerator sin squared begin display style space end style alpha cos space beta over denominator sin squared begin display style space end style beta cos space alpha end fraction text then end text cos text end text x equals$

if $fraction numerator cos space x minus cos space alpha over denominator cos space x minus cos space beta end fraction equals fraction numerator sin squared begin display style space end style alpha cos space beta over denominator sin squared begin display style space end style beta cos space alpha end fraction text then end text cos text end text x equals$

parallel

Assertion: $open vertical bar table row cell cos invisible function application left parenthesis theta plus alpha right parenthesis end cell cell cos invisible function application left parenthesis theta plus beta right parenthesis end cell cell cos invisible function application left parenthesis theta plus gamma right parenthesis end cell row cell sin invisible function application left parenthesis theta plus alpha right parenthesis end cell cell sin invisible function application left parenthesis theta plus beta right parenthesis end cell cell sin invisible function application left parenthesis theta plus gamma right parenthesis end cell row cell sin invisible function application left parenthesis beta minus gamma right parenthesis end cell cell sin invisible function application left parenthesis gamma minus alpha right parenthesis end cell cell sin invisible function application left parenthesis alpha minus beta right parenthesis end cell end table close vertical bar$ is independent of $theta$
Reason: If f( $theta$ ) = c, then f( $theta$ ) is independent of $theta$ .

Assertion: $open vertical bar table row cell cos invisible function application left parenthesis theta plus alpha right parenthesis end cell cell cos invisible function application left parenthesis theta plus beta right parenthesis end cell cell cos invisible function application left parenthesis theta plus gamma right parenthesis end cell row cell sin invisible function application left parenthesis theta plus alpha right parenthesis end cell cell sin invisible function application left parenthesis theta plus beta right parenthesis end cell cell sin invisible function application left parenthesis theta plus gamma right parenthesis end cell row cell sin invisible function application left parenthesis beta minus gamma right parenthesis end cell cell sin invisible function application left parenthesis gamma minus alpha right parenthesis end cell cell sin invisible function application left parenthesis alpha minus beta right parenthesis end cell end table close vertical bar$ is independent of $theta$
Reason: If f( $theta$ ) = c, then f( $theta$ ) is independent of $theta$ .

Statement-1 : The function f(x) = |x³| is differentiable at x = 0
Statement-2 : at x = 0, $f to the power of straight prime$ (x) = 0

Statement-1 : The function f(x) = |x³| is differentiable at x = 0
Statement-2 : at x = 0, $f to the power of straight prime$ (x) = 0

Statement-1 : The function y = sin–1 (cos x) is not differentiable at $x equals n pi comma n element of Z$ is particular at x = $pi$

Statement-2 : $fraction numerator d y over denominator d x end fraction$ = $fraction numerator negative sin invisible function application x over denominator vertical line sin invisible function application x vertical line end fraction$ so the function is not differentiable at the points where sin x = 0.

Statement-1 : The function y = sin–1 (cos x) is not differentiable at $x equals n pi comma n element of Z$ is particular at x = $pi$

Statement-2 : $fraction numerator d y over denominator d x end fraction$ = $fraction numerator negative sin invisible function application x over denominator vertical line sin invisible function application x vertical line end fraction$ so the function is not differentiable at the points where sin x = 0.

parallel

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