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Maths-

General

Easy

Question

Assertion : If $stack a with ̄ on top$ × $stack b with ̄ on top$ = $stack c with ̄ on top$ × $stack d with ̄ on top$ , and $stack a with ̄ on top$ × $stack c with ̄ on top$ = $stack b with ̄ on top$ × $stack d with ̄ on top$ then $stack a with ̄ on top$ – $stack d with ̄ on top$ is perpendicular to $stack b with ̄ on top$ – $stack c with ̄ on top$ .
Reason : If $stack r with ̄ on top$ is perpendicular to $stack q with ̄ on top$ then $stack r with ̄ on top$ . $stack q with ̄ on top$ = 0

If both (A) and (R) are true, and (R) is the correct explanation of (A).
If both (A) and (R) are true but (R) is not the correct explanation of (A).
If (A) is true but (R) is false.
If (A) is false but (R) is true.

The correct answer is: If (A) is false but (R) is true.

(Assertion false & reason is true)
 $stack a with ̄ on top$ × $stack b with ̄ on top$ = $stack c with ̄ on top$ × $stack d with ̄ on top$ … (1)
$stack a with ̄ on top$ × $stack c with ̄ on top$ = $stack b with ̄ on top$ × $stack d with ̄ on top$ … (2)
Subtract $stack a with ̄ on top$ × ( $stack b with ̄ on top$ – $stack c with ̄ on top$ ) = ( $stack c with ̄ on top$ – $stack b with ̄ on top$ ) × $stack d with ̄ on top$
( $stack a with ̄ on top$ – $stack d with ̄ on top$ ) × ( $stack b with ̄ on top$ – $stack c with ̄ on top$ ) = 0
So $stack a with ̄ on top$ – $stack d with ̄ on top$ is parallel to $stack b with ̄ on top$ – $stack c with ̄ on top$

Related Questions to study

Assertion : Vectors – ² $stack i with hat on top$ + $stack j with hat on top$ + $stack k with hat on top$ , $stack i with hat on top$ –^ $stack j with hat on top$ + $stack k with hat on top$ and $stack i with hat on top$ + $stack j with hat on top$ –² $stack k with hat on top$ are coplanar for only two values of .
Reason : Three vectors $stack a with rightwards arrow on top$ , $stack b with rightwards arrow on top$ , $stack c with rightwards arrow on top$ are coplanar if $stack a with rightwards arrow on top$ .( $stack b with rightwards arrow on top$ × $stack c with rightwards arrow on top$ ) = $stack 0 with rightwards arrow on top$ .

Assertion : Vectors – ² $stack i with hat on top$ + $stack j with hat on top$ + $stack k with hat on top$ , $stack i with hat on top$ –^ $stack j with hat on top$ + $stack k with hat on top$ and $stack i with hat on top$ + $stack j with hat on top$ –² $stack k with hat on top$ are coplanar for only two values of .
Reason : Three vectors $stack a with rightwards arrow on top$ , $stack b with rightwards arrow on top$ , $stack c with rightwards arrow on top$ are coplanar if $stack a with rightwards arrow on top$ .( $stack b with rightwards arrow on top$ × $stack c with rightwards arrow on top$ ) = $stack 0 with rightwards arrow on top$ .

Assertion (A) : If vector $stack a with rightwards arrow on top$ and $stack b with rightwards arrow on top$ are linearly dependent, then vectors $stack a with rightwards arrow on top$ , $stack b with rightwards arrow on top$ , $stack c with rightwards arrow on top$ must be dependent.
Reason (R) : If vector $stack a with rightwards arrow on top$ and $stack b with rightwards arrow on top$ are linearly independent, then vectors $stack a with rightwards arrow on top$ , $stack b with rightwards arrow on top$ , $stack c with rightwards arrow on top$ must be linearly independent, where vector $stack c with rightwards arrow on top$ is non-zero.

Assertion (A) : If vector $stack a with rightwards arrow on top$ and $stack b with rightwards arrow on top$ are linearly dependent, then vectors $stack a with rightwards arrow on top$ , $stack b with rightwards arrow on top$ , $stack c with rightwards arrow on top$ must be dependent.
Reason (R) : If vector $stack a with rightwards arrow on top$ and $stack b with rightwards arrow on top$ are linearly independent, then vectors $stack a with rightwards arrow on top$ , $stack b with rightwards arrow on top$ , $stack c with rightwards arrow on top$ must be linearly independent, where vector $stack c with rightwards arrow on top$ is non-zero.

Assertion: If in a ABC ; $Error converting from MathML to accessible text.$ = $Error converting from MathML to accessible text.$ – $Error converting from MathML to accessible text.$ and $Error converting from MathML to accessible text.$ = $Error converting from MathML to accessible text.$ ; | $Error converting from MathML to accessible text.$ |  | $Error converting from MathML to accessible text.$ |, then the value of cos 2A + cos 2B + cos 2C is – 1.
Reason: If in ABC, C = 90º, then cos 2A + cos 2B + cos 2C = – 1.

Assertion: If in a ABC ; $Error converting from MathML to accessible text.$ = $Error converting from MathML to accessible text.$ – $Error converting from MathML to accessible text.$ and $Error converting from MathML to accessible text.$ = $Error converting from MathML to accessible text.$ ; | $Error converting from MathML to accessible text.$ |  | $Error converting from MathML to accessible text.$ |, then the value of cos 2A + cos 2B + cos 2C is – 1.
Reason: If in ABC, C = 90º, then cos 2A + cos 2B + cos 2C = – 1.
Maths-General

parallel

If $stack a with ̄ on top comma stack b with ̄ on top comma stack c with ̄ on top$ are noncoplanar vectors and $stack r with rightwards arrow on top equals left parenthesis stack a with ̄ on top cross times stack b with ̄ on top right parenthesis cross times left parenthesis stack a with ̄ on top cross times stack c with ̄ on top right parenthesis$ .
Assertion : $stack r with ̄ on top$ and $stack a with ̄ on top$ are linearly dependent
Reason : $stack r with rightwards arrow on top$ is ^r to each of three $stack a with ̄ on top comma stack b with ̄ on top comma & stack c with ̄ on top$ .

If $stack a with ̄ on top comma stack b with ̄ on top comma stack c with ̄ on top$ are noncoplanar vectors and $stack r with rightwards arrow on top equals left parenthesis stack a with ̄ on top cross times stack b with ̄ on top right parenthesis cross times left parenthesis stack a with ̄ on top cross times stack c with ̄ on top right parenthesis$ .
Assertion : $stack r with ̄ on top$ and $stack a with ̄ on top$ are linearly dependent
Reason : $stack r with rightwards arrow on top$ is ^r to each of three $stack a with ̄ on top comma stack b with ̄ on top comma & stack c with ̄ on top$ .

Let $Error converting from MathML to accessible text.$ , $Error converting from MathML to accessible text.$ , $Error converting from MathML to accessible text.$ , $Error converting from MathML to accessible text.$ , $Error converting from MathML to accessible text.$ , $Error converting from MathML to accessible text.$ denote the sides of a regular hexagon.
Assertion : $Error converting from MathML to accessible text.$ × ( $Error converting from MathML to accessible text.$ + $Error converting from MathML to accessible text.$ )  $Error converting from MathML to accessible text.$
Reason : $Error converting from MathML to accessible text.$ × $Error converting from MathML to accessible text.$ = $Error converting from MathML to accessible text.$ & $Error converting from MathML to accessible text.$ × $Error converting from MathML to accessible text.$ = $Error converting from MathML to accessible text.$

Let $Error converting from MathML to accessible text.$ , $Error converting from MathML to accessible text.$ , $Error converting from MathML to accessible text.$ , $Error converting from MathML to accessible text.$ , $Error converting from MathML to accessible text.$ , $Error converting from MathML to accessible text.$ denote the sides of a regular hexagon.
Assertion : $Error converting from MathML to accessible text.$ × ( $Error converting from MathML to accessible text.$ + $Error converting from MathML to accessible text.$ )  $Error converting from MathML to accessible text.$
Reason : $Error converting from MathML to accessible text.$ × $Error converting from MathML to accessible text.$ = $Error converting from MathML to accessible text.$ & $Error converting from MathML to accessible text.$ × $Error converting from MathML to accessible text.$ = $Error converting from MathML to accessible text.$

Assertion : Let $stack a with rightwards arrow on top comma stack b with rightwards arrow on top comma stack c with rightwards arrow on top & stack d with rightwards arrow on top$ are position vectors of four points A, B, C & D and $3 stack a with rightwards arrow on top minus 2 stack b with rightwards arrow on top plus 5 stack c with rightwards arrow on top minus 6 stack d with rightwards arrow on top equals stack 0 with rightwards arrow on top$ then points A, B, C and D are coplanar.
Reason : Three non-zero, linearly dependent co-initial vectors ( $Error converting from MathML to accessible text.$ ¸ $Error converting from MathML to accessible text.$ & $Error converting from MathML to accessible text.$ ) are coplanar.

Assertion : Let $stack a with rightwards arrow on top comma stack b with rightwards arrow on top comma stack c with rightwards arrow on top & stack d with rightwards arrow on top$ are position vectors of four points A, B, C & D and $3 stack a with rightwards arrow on top minus 2 stack b with rightwards arrow on top plus 5 stack c with rightwards arrow on top minus 6 stack d with rightwards arrow on top equals stack 0 with rightwards arrow on top$ then points A, B, C and D are coplanar.
Reason : Three non-zero, linearly dependent co-initial vectors ( $Error converting from MathML to accessible text.$ ¸ $Error converting from MathML to accessible text.$ & $Error converting from MathML to accessible text.$ ) are coplanar.

parallel

Assertion : A vector $stack c with rightwards arrow on top$ , directed along the internal bisector of the angle between the vector $stack a with rightwards arrow on top$ = 7 $stack i with hat on top$ – 4 $stack j with hat on top$ – 4 $stack k with hat on top$ and $stack b with rightwards arrow on top$ = –2 $stack i with hat on top$ – $stack j with hat on top$ + 2 $stack k with hat on top$ , with | $stack c with rightwards arrow on top$ | = 5 $square root of 6$ is $fraction numerator 5 over denominator 3 end fraction$ ( $stack i with hat on top$ –7 $stack j with hat on top$ + 2 $stack k with hat on top$ ).
Reason : The vector bisecting the angle of $stack a with rightwards arrow on top$ & $stack b with rightwards arrow on top$ is given by $stack c with rightwards arrow on top$ =  $open parentheses fraction numerator stack a with rightwards arrow on top over denominator vertical line stack a with rightwards arrow on top vertical line end fraction plus fraction numerator stack b with rightwards arrow on top over denominator vertical line stack b with rightwards arrow on top vertical line end fraction close parentheses$ .

Assertion : A vector $stack c with rightwards arrow on top$ , directed along the internal bisector of the angle between the vector $stack a with rightwards arrow on top$ = 7 $stack i with hat on top$ – 4 $stack j with hat on top$ – 4 $stack k with hat on top$ and $stack b with rightwards arrow on top$ = –2 $stack i with hat on top$ – $stack j with hat on top$ + 2 $stack k with hat on top$ , with | $stack c with rightwards arrow on top$ | = 5 $square root of 6$ is $fraction numerator 5 over denominator 3 end fraction$ ( $stack i with hat on top$ –7 $stack j with hat on top$ + 2 $stack k with hat on top$ ).
Reason : The vector bisecting the angle of $stack a with rightwards arrow on top$ & $stack b with rightwards arrow on top$ is given by $stack c with rightwards arrow on top$ =  $open parentheses fraction numerator stack a with rightwards arrow on top over denominator vertical line stack a with rightwards arrow on top vertical line end fraction plus fraction numerator stack b with rightwards arrow on top over denominator vertical line stack b with rightwards arrow on top vertical line end fraction close parentheses$ .

If a, b, c are non-coplanar unit vectors such that a × (b × c) = $fraction numerator b plus c over denominator square root of 2 end fraction$ , then the angle between a and b is -

If a, b, c are non-coplanar unit vectors such that a × (b × c) = $fraction numerator b plus c over denominator square root of 2 end fraction$ , then the angle between a and b is -

If $stack a with rightwards arrow on top cross times stack b with rightwards arrow on top equals stack c with rightwards arrow on top cross times stack b with rightwards arrow on top not equal to stack 0 with rightwards arrow on top comma$ then

If $stack a with rightwards arrow on top cross times stack b with rightwards arrow on top equals stack c with rightwards arrow on top cross times stack b with rightwards arrow on top not equal to stack 0 with rightwards arrow on top comma$ then

parallel

If $stack x with rightwards arrow on top$ and $stack y with rightwards arrow on top$ are two unit vectors and  is the angle between them, then $fraction numerator 1 over denominator 2 end fraction vertical line stack x with rightwards arrow on top minus stack y with rightwards arrow on top vertical line equals$

If $stack x with rightwards arrow on top$ and $stack y with rightwards arrow on top$ are two unit vectors and  is the angle between them, then $fraction numerator 1 over denominator 2 end fraction vertical line stack x with rightwards arrow on top minus stack y with rightwards arrow on top vertical line equals$

Which of the following is a true statement ?

Which of the following is a true statement ?

Values of 'x' for which the angle between the vectors $stack a with rightwards arrow on top equals 2 x to the power of 2 end exponent stack i with hat on top plus 4 x stack j with hat on top plus stack k with hat on top$ and $stack b with rightwards arrow on top equals 7 stack i with hat on top minus 2 stack j with hat on top plus x stack k with hat on top$ is obtuse, are

Values of 'x' for which the angle between the vectors $stack a with rightwards arrow on top equals 2 x to the power of 2 end exponent stack i with hat on top plus 4 x stack j with hat on top plus stack k with hat on top$ and $stack b with rightwards arrow on top equals 7 stack i with hat on top minus 2 stack j with hat on top plus x stack k with hat on top$ is obtuse, are

parallel

Let the vectors $stack a with rightwards arrow on top comma stack b with rightwards arrow on top comma stack c with rightwards arrow on top comma stack d with rightwards arrow on top$ be such that $left parenthesis stack a with rightwards arrow on top cross times stack b with rightwards arrow on top right parenthesis cross times left parenthesis stack c with rightwards arrow on top cross times stack d with rightwards arrow on top right parenthesis equals stack 0 with rightwards arrow on top$ . Let P₁ and P₂ be planes determined by the pairs of vectors $stack a with rightwards arrow on top comma stack b with rightwards arrow on top$ and $stack c with rightwards arrow on top comma stack d with rightwards arrow on top$ respectively. Then, then angle between between P₁ and P₂ is

Let the vectors $stack a with rightwards arrow on top comma stack b with rightwards arrow on top comma stack c with rightwards arrow on top comma stack d with rightwards arrow on top$ be such that $left parenthesis stack a with rightwards arrow on top cross times stack b with rightwards arrow on top right parenthesis cross times left parenthesis stack c with rightwards arrow on top cross times stack d with rightwards arrow on top right parenthesis equals stack 0 with rightwards arrow on top$ . Let P₁ and P₂ be planes determined by the pairs of vectors $stack a with rightwards arrow on top comma stack b with rightwards arrow on top$ and $stack c with rightwards arrow on top comma stack d with rightwards arrow on top$ respectively. Then, then angle between between P₁ and P₂ is

The length of the longest diagonal of the parallelogram constructed on $5 stack a with rightwards arrow on top plus 2 stack b with rightwards arrow on top$ and $stack a with rightwards arrow on top minus 3 stack b with rightwards arrow on top comma$ if it is given that $vertical line stack a with rightwards arrow on top vertical line equals 2 square root of 2 comma vertical line stack b with rightwards arrow on top vertical line equals 3$ and $angle stack a with rightwards arrow on top comma stack b with rightwards arrow on top equals fraction numerator pi over denominator 4 end fraction comma$ is

The length of the longest diagonal of the parallelogram constructed on $5 stack a with rightwards arrow on top plus 2 stack b with rightwards arrow on top$ and $stack a with rightwards arrow on top minus 3 stack b with rightwards arrow on top comma$ if it is given that $vertical line stack a with rightwards arrow on top vertical line equals 2 square root of 2 comma vertical line stack b with rightwards arrow on top vertical line equals 3$ and $angle stack a with rightwards arrow on top comma stack b with rightwards arrow on top equals fraction numerator pi over denominator 4 end fraction comma$ is

ABCD is a quadrilateral with $Error converting from MathML to accessible text.$ and $Error converting from MathML to accessible text.$ . If its area is  times the area of the parallelogram with AB, AD as adjacent sides, then  is

ABCD is a quadrilateral with $Error converting from MathML to accessible text.$ and $Error converting from MathML to accessible text.$ . If its area is  times the area of the parallelogram with AB, AD as adjacent sides, then  is

parallel

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