Maths-
General
Easy
Question
Assertion : For
the volume of the parallel piped formed by vectors
and
is maximum (The vectors form a right-handed system)
Reason: The volume of the parallel piped having three coterminous edges
and ![stack c with ‾ on top](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAAZCAYAAADjRwSLAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAY00gKyAAAAGxJREFUeNpjYICA/zjwKBgFOAAPEHcB8VMg/gXE64BYEF3BOSBuBWI+IGYD4hIg9kFW1AHEk/BZwwTEr9GNRgemQHyYkIO9oI7EC4yB+AYx3r8A9RkL1HcFQGyNrkgWiA8C8R8gvgTEySSFMAC4kxWWWazxUwAAAHd0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW92ZXIgYWNjZW50PSJ0cnVlIj48bWk+YzwvbWk+PG1pPiYjeDIwM0U7PC9taT48L21vdmVyPjwvbWF0aD6siPvzAAAAAElFTkSuQmCC)
- Statement
is true, statement
is true; statement
is a correct explanation for statement
- Statement
is true, statement
is true statement
is not a correct explanation for statement
- Statement
is true, statement
is false
- statement
is false, statement
is true
Hint:
We are given two statements. We have to tell which of the statement is true. For statement 1, we will check if for the given value of "a" the volume is maximum. We will use the formula of volume of parallelepiped. For statement 2, we have to check if parallelepiped have coterminous edges.
The correct answer is: Statement
is true, statement
is true; statement
is a correct explanation for statement ![negative 1](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABcAAAANCAYAAABCZ/VdAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAMyZLetQAAADBJREFUeNpjYKAcqAFxLRBfYKABWAzEaUD8n4GGYNRw0g3/TwQeDfMRYDipEU8dAADUJhpljLE7wQAAAFN0RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW8+LTwvbW8+PG1uPjE8L21uPjwvbWF0aD5mjCijAAAAAElFTkSuQmCC)
We are given two statements we have to check which of them are true.
Statment 1:
The first statement is about the maximum volume of parallelepiped. We will find the maximum volume of parallelepiped.
![T h e space g i v e n space v e c t o r s space a r e
A with rightwards arrow on top space equals space i with hat on top space plus space a j with hat on top
B space equals a i with hat on top space plus space j with hat on top space plus k with hat on top
C equals j with hat on top space plus space a k with hat on top](data:image/png;base64,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)
![T h e space v o l u m e space o f space a space p a r a l l e l space p i p e d space i s space g i v e n space b y space a space s c a l a r space p r o d u c t.
V space equals left square bracket stack A space with rightwards arrow on top space B with rightwards arrow on top space stack C right square bracket with rightwards arrow on top
space space space space equals open vertical bar table row 1 a 0 row a 1 1 row 0 1 a end table close vertical bar
space space space space space equals 1 left parenthesis a space minus space 1 right parenthesis space minus a left parenthesis a squared space minus space 0 right parenthesis space plus space 0
V space space equals a space minus space 1 space minus a cubed](data:image/png;base64,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We will find the maximum value of parallelepiped.
![V space equals space a space minus space 1 space minus space a cubed
fraction numerator d V over denominator d a end fraction space equals space 1 space minus space 0 space minus space 3 a squared
W e space h a v e space m a x i m a space w h e n space d e r i v a t e space o f space t h e space f u n c t i o n
i s space z e r o.
W e space w i l l space s u b s t i t u t e space t h e space v a l u e space o f space d e r i v a t i v e space t o space b e space z e r o
t o space f i n d space t h e space v a l u e space o f space a space f o r space w h i c h space w e space a r e space g e t t i n g space z e r o.
0 space equals space 1 space minus space 3 a squared
1 space equals space 3 a squared
3 a squared space equals space 1
a squared equals space 1 third
T a k i n g space s q u a r e space r o o t s space w e space g e t comma
a space equals space fraction numerator 1 over denominator square root of 3 end 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So, the first statement is true.
Statement 2:
We are getting the maximum value of parallelepiped at this value of a.
Now, we will see statment 2.
Coterminous edges means the vectors or sides end at single common point. So, yes parallelepiped has coterminous edges.
Three edges meet at one point.
As the edges end at one point, we can consider two of them as base. We can take their cross product and take them as base. And, the projection of the third vector with the cross product will give is the height.
Hence the statement 2 is true and a correct explanation of statement 1.
For such questions, we should know the formula of a parallelepiped. We should also know how to take a scalar product.
Related Questions to study
Maths-
Assertion (A): Let
. If
such that
is collinear with
and
is perpendicular to
is possible, then
.
Reason (R): If
and
are non-zero, non-collinear vectors, then
can be expressed as
where
is collinear with
and
is perpendicular to ![stack a with rightwards arrow on top](data:image/png;base64,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)
Assertion (A): Let
. If
such that
is collinear with
and
is perpendicular to
is possible, then
.
Reason (R): If
and
are non-zero, non-collinear vectors, then
can be expressed as
where
is collinear with
and
is perpendicular to ![stack a with rightwards arrow on top](data:image/png;base64,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)
Maths-General
Maths-
Statement
If
are distinct non-negative numbers and the vectors â
and
are coplanar then
is arithmetic mean of
and
.
Statement -2: Parallel vectors have proportional direction ratios.
Statement
If
are distinct non-negative numbers and the vectors â
and
are coplanar then
is arithmetic mean of
and
.
Statement -2: Parallel vectors have proportional direction ratios.
Maths-General
Maths-
If
and
is a vector satisfying
.
Assertion
can be expressed in terms of
and
.
Reason ![left parenthesis R right parenthesis colon r with ‾ on top equals 1 over 7 left parenthesis 7 i plus 5 j minus 9 k plus a with ‾ on top cross times b with ‾ on top right parenthesis](data:image/png;base64,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)
If
and
is a vector satisfying
.
Assertion
can be expressed in terms of
and
.
Reason ![left parenthesis R right parenthesis colon r with ‾ on top equals 1 over 7 left parenthesis 7 i plus 5 j minus 9 k plus a with ‾ on top cross times b with ‾ on top right parenthesis](data:image/png;base64,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)
Maths-General
Maths-
If
are non-zero vectors such that
then
Assertion
: Least value of
is ![2 square root of 2 minus 1](data:image/png;base64,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)
Reason (R): The expression
is least when magnitude of
is ![square root of 2 t a n invisible function application open parentheses fraction numerator pi over denominator 8 end fraction close parentheses end root](data:image/png;base64,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)
If
are non-zero vectors such that
then
Assertion
: Least value of
is ![2 square root of 2 minus 1](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAD8AAAATCAYAAAAnMdWSAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAAQ3ZOC+gAAAVVJREFUeNpjYKA+4AHi/xTgIQ0igHg/wwgF24E4ZSR6XASI30PpIQOsgXgNEH8C4l9AfAGIo8kwJwOI19PQfHKAGhDXQu3ECg4CcSS0sAIBLSA+ChUjBYDyeggNzScHLAbiNFILU3kgvkSCegUgfg3EHDQyn1JAck3ygwS1FUA8m4bm09XzltCkSSw4D8QeNDSfbp4HJd2T0IIKWyMGHegA8XMgZqGC+QPqeUEg3gDEbmjiKkA8H2qIDZpcOxD3E+kIXObjczQ1WoUE1SlBHaaCRa4G6nlQPX4KTe4+EJsQ4QB85g9ozGtACywuAoZkAfE/IBaH8i2A+DYRSZ5Y8+nueZBHVpGQZ78B8TIoezIQNxBQT6r5dPX8FmjMEAvmAPF3qGdeE6GXVPPp6nlSCxMBIP4LxGuB+DiFhRY9PE11Ow9DDSoYiT04U6jnZUZq3z1kKDseACyXc/zXzpfxAAAAdnRFWHRNYXRoTUwAPG1hdGggeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzE5OTgvTWF0aC9NYXRoTUwiPjxtbj4yPC9tbj48bXNxcnQ+PG1uPjI8L21uPjwvbXNxcnQ+PG1vPi08L21vPjxtbj4xPC9tbj48L21hdGg+N0mUhgAAAABJRU5ErkJggg==)
Reason (R): The expression
is least when magnitude of
is ![square root of 2 t a n invisible function application open parentheses fraction numerator pi over denominator 8 end fraction close parentheses end root](data:image/png;base64,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)
Maths-General
Maths-
Statement- 1: If
and
and
, then there exist real numbers
,
such that
![alpha stack b with rightwards arrow on top plus beta stack c with rightwards arrow on top plus gamma d](data:image/png;base64,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)
Statement- 2:
are four vectors in a 3 - dimensional space. If
are non-coplanar, then there exist real numbers
such that ![stack a with rightwards arrow on top equals alpha stack b with rightwards arrow on top plus beta stack c with rightwards arrow on top plus gamma stack d with rightwards arrow on top](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIQAAAAWCAYAAAAB6jTvAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAATRJrTQAAAA1FJREFUeNrtWF1kXEEUHlURVWHFqorYl4jIU6iIiD6EqlpVEfLQh1gRKvpUUVZFVFSoqqoVYUXEqgorrj7EyktErOhb1KqqUH2oiAhVVRG1bM/Et1zTmbv3bu7Zv8zhs3fn3t0555tvzjlzhWh+WyNcF9Ysd7AVwgFhyK7v5eLuMaHkgSKh166x5U6q/Bth0K675a4Z6mAj+xfYN9uw2R7HNmy2TjdXMKeEK3WYd4bwmXBGyBO6W6hO++a00YLpIRTqMO8LwluUT0nca8L7Fim7gThttGDGCOsh/E8pIGH7ythEAEFw+8fKqVT/AuGY8JfwiRBjdqiTkEEqzigCzID8sj0nzMHHI/xmh3CTkfBngNvyFfoqGcMrwiF4dAgRBv96sUY6i2Ln9yt+pQi/wN0GYYmQNE2wiR9EII4sFFTJ+UowWRthjxDH9Q4cFBCiujOzKGFJl49vqsgaQQTh5qCPkEM/4SUG6fcioQNxPSXcZ/IvTZhUxq5CtKPKZpf8TuG6AxusZFrjhwheKDs0zpgd5pXdN4AFF0jJt5XnDxTFCwT2m1EQcs4ufJaw2P0ez7/EpqpVyejT9ADLhFkN10mP+P4zqZ5hZWyXuWT80PQocs67CEotZ6ea/2j3IYhqs5g6ZxS7/ScWQvf8ScDycNEsK5BlR3A9Tvjgk2sTp9qjh7z+wxjMLaQ11aYJHzXOD0G0qkkRbzPtQNOcC4YsMGiIibupnMEpKIqeolO5P4CN5je+cztRvo9oaniYNmGo/RkoXlfS1jTjs0iHHISb5pQ1e1UzHkcDWWtB3ECpldzd09yXWeNdgPjO7RDp111zNhgFMa4hbwVNTx7HPbelcE9tSguMp4yUoYHcRlnTZb2vdTp2FgwiLR8ts5pxKZJHXi9f0igVMTR1OUZByGbwO45OESg1jXs5za530ADdwfcujE0xEu6A6GHXOwnHa1chZS+i05cxPnHVd05BLGM+nV0jfEHpKAt3E0nA89AwByXNI1sc+zh2XsRiqGH7yvuGhKZzlr48gCiKIJ7TN4HmcQzEnaG3SVT4TTfqdRExTNfoBdOWj7LiuOIYRXztQSaJistrbVUcZ+vp65GwxmphNYjW1xaxRAgvmGpl8tSzFMYf/QNHcxD3kV0XpwAAAW10RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW92ZXIgYWNjZW50PSJ0cnVlIj48bWk+YTwvbWk+PG1vPiYjeDIxOTI7PC9tbz48L21vdmVyPjxtbz49PC9tbz48bWk+JiN4M0IxOzwvbWk+PG1vdmVyIGFjY2VudD0idHJ1ZSI+PG1pPmI8L21pPjxtbz4mI3gyMTkyOzwvbW8+PC9tb3Zlcj48bW8+KzwvbW8+PG1pPiYjeDNCMjs8L21pPjxtb3ZlciBhY2NlbnQ9InRydWUiPjxtaT5jPC9taT48bW8+JiN4MjE5Mjs8L21vPjwvbW92ZXI+PG1vPis8L21vPjxtaT4mI3gzQjM7PC9taT48bW92ZXIgYWNjZW50PSJ0cnVlIj48bWk+ZDwvbWk+PG1vPiYjeDIxOTI7PC9tbz48L21vdmVyPjwvbWF0aD52T8m3AAAAAElFTkSuQmCC)
Statement- 1: If
and
and
, then there exist real numbers
,
such that
![alpha stack b with rightwards arrow on top plus beta stack c with rightwards arrow on top plus gamma d](data:image/png;base64,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)
Statement- 2:
are four vectors in a 3 - dimensional space. If
are non-coplanar, then there exist real numbers
such that ![stack a with rightwards arrow on top equals alpha stack b with rightwards arrow on top plus beta stack c with rightwards arrow on top plus gamma stack d with rightwards arrow on top](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIQAAAAWCAYAAAAB6jTvAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAABGJhU0UAAAATRJrTQAAAA1FJREFUeNrtWF1kXEEUHlURVWHFqorYl4jIU6iIiD6EqlpVEfLQh1gRKvpUUVZFVFSoqqoVYUXEqgorrj7EyktErOhb1KqqUH2oiAhVVRG1bM/Et1zTmbv3bu7Zv8zhs3fn3t0555tvzjlzhWh+WyNcF9Ysd7AVwgFhyK7v5eLuMaHkgSKh166x5U6q/Bth0K675a4Z6mAj+xfYN9uw2R7HNmy2TjdXMKeEK3WYd4bwmXBGyBO6W6hO++a00YLpIRTqMO8LwluUT0nca8L7Fim7gThttGDGCOsh/E8pIGH7ythEAEFw+8fKqVT/AuGY8JfwiRBjdqiTkEEqzigCzID8sj0nzMHHI/xmh3CTkfBngNvyFfoqGcMrwiF4dAgRBv96sUY6i2Ln9yt+pQi/wN0GYYmQNE2wiR9EII4sFFTJ+UowWRthjxDH9Q4cFBCiujOzKGFJl49vqsgaQQTh5qCPkEM/4SUG6fcioQNxPSXcZ/IvTZhUxq5CtKPKZpf8TuG6AxusZFrjhwheKDs0zpgd5pXdN4AFF0jJt5XnDxTFCwT2m1EQcs4ufJaw2P0ez7/EpqpVyejT9ADLhFkN10mP+P4zqZ5hZWyXuWT80PQocs67CEotZ6ea/2j3IYhqs5g6ZxS7/ScWQvf8ScDycNEsK5BlR3A9Tvjgk2sTp9qjh7z+wxjMLaQ11aYJHzXOD0G0qkkRbzPtQNOcC4YsMGiIibupnMEpKIqeolO5P4CN5je+cztRvo9oaniYNmGo/RkoXlfS1jTjs0iHHISb5pQ1e1UzHkcDWWtB3ECpldzd09yXWeNdgPjO7RDp111zNhgFMa4hbwVNTx7HPbelcE9tSguMp4yUoYHcRlnTZb2vdTp2FgwiLR8ts5pxKZJHXi9f0igVMTR1OUZByGbwO45OESg1jXs5za530ADdwfcujE0xEu6A6GHXOwnHa1chZS+i05cxPnHVd05BLGM+nV0jfEHpKAt3E0nA89AwByXNI1sc+zh2XsRiqGH7yvuGhKZzlr48gCiKIJ7TN4HmcQzEnaG3SVT4TTfqdRExTNfoBdOWj7LiuOIYRXztQSaJistrbVUcZ+vp65GwxmphNYjW1xaxRAgvmGpl8tSzFMYf/QNHcxD3kV0XpwAAAW10RVh0TWF0aE1MADxtYXRoIHhtbG5zPSJodHRwOi8vd3d3LnczLm9yZy8xOTk4L01hdGgvTWF0aE1MIj48bW92ZXIgYWNjZW50PSJ0cnVlIj48bWk+YTwvbWk+PG1vPiYjeDIxOTI7PC9tbz48L21vdmVyPjxtbz49PC9tbz48bWk+JiN4M0IxOzwvbWk+PG1vdmVyIGFjY2VudD0idHJ1ZSI+PG1pPmI8L21pPjxtbz4mI3gyMTkyOzwvbW8+PC9tb3Zlcj48bW8+KzwvbW8+PG1pPiYjeDNCMjs8L21pPjxtb3ZlciBhY2NlbnQ9InRydWUiPjxtaT5jPC9taT48bW8+JiN4MjE5Mjs8L21vPjwvbW92ZXI+PG1vPis8L21vPjxtaT4mI3gzQjM7PC9taT48bW92ZXIgYWNjZW50PSJ0cnVlIj48bWk+ZDwvbWk+PG1vPiYjeDIxOTI7PC9tbz48L21vdmVyPjwvbWF0aD52T8m3AAAAAElFTkSuQmCC)
Maths-General
Maths-
Statement-
:If
are non-collinear points. Then every point
in the plane of
, can be expressed in the form ![open parentheses fraction numerator k x subscript 1 end subscript plus l x subscript 2 end subscript plus m x subscript 3 end subscript over denominator k plus l plus m end fraction comma fraction numerator k y subscript 1 end subscript plus l y subscript 2 end subscript plus m y subscript 3 end subscript over denominator k plus l plus m end fraction close parentheses](data:image/png;base64,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)
Statement-
:The condition for coplanarity of four points
is that there exists scalars
not all zeros such that
where
.
Statement-
:If
are non-collinear points. Then every point
in the plane of
, can be expressed in the form ![open parentheses fraction numerator k x subscript 1 end subscript plus l x subscript 2 end subscript plus m x subscript 3 end subscript over denominator k plus l plus m end fraction comma fraction numerator k y subscript 1 end subscript plus l y subscript 2 end subscript plus m y subscript 3 end subscript over denominator k plus l plus m end fraction close parentheses](data:image/png;base64,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)
Statement-
:The condition for coplanarity of four points
is that there exists scalars
not all zeros such that
where
.
Maths-General
Maths-
Assertion (A): The number of vectors of unit length and perpendicular to both the vectors.
and
is zero Reason
(R):
and
are two non-zero and non-parallel vectors it is true that
is perpendicular to the plane containing
and ![stack b with ‾ on top](data:image/png;base64,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)
Assertion (A): The number of vectors of unit length and perpendicular to both the vectors.
and
is zero Reason
(R):
and
are two non-zero and non-parallel vectors it is true that
is perpendicular to the plane containing
and ![stack b with ‾ on top](data:image/png;base64,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)
Maths-General
Maths-
The value of
for which the straight lines
and
are coplanar is
For such questions, we should know the condition for two lines to be coplanar. We should also know about the scalar triple product.
The value of
for which the straight lines
and
are coplanar is
Maths-General
For such questions, we should know the condition for two lines to be coplanar. We should also know about the scalar triple product.
Maths-
The position vector of the centre of the circle ![vertical line r with rightwards arrow on top vertical line equals 5 comma r with rightwards arrow on top times left parenthesis i with ˆ on top plus j with ˆ on top plus k with ˆ on top right parenthesis equals 3 square root of 3](data:image/png;base64,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)
The position vector of the centre of the circle ![vertical line r with rightwards arrow on top vertical line equals 5 comma r with rightwards arrow on top times left parenthesis i with ˆ on top plus j with ˆ on top plus k with ˆ on top right parenthesis equals 3 square root of 3](data:image/png;base64,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)
Maths-General
Maths-
Let
where
are three noncoplanar vectors. If
is perpendicular to
, then minimum value of ![x squared plus y squared](data:image/png;base64,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)
Let
where
are three noncoplanar vectors. If
is perpendicular to
, then minimum value of ![x squared plus y squared](data:image/png;base64,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)
Maths-General
Chemistry-
Which of the following represents Westrosol?
Which of the following represents Westrosol?
Chemistry-General
Chemistry-
Which of the following sequences would yield -nitro chlorobenzene (Z) from benzene?
Which of the following sequences would yield -nitro chlorobenzene (Z) from benzene?
Chemistry-General
Chemistry-
For the preparation of chloroethane
For the preparation of chloroethane
Chemistry-General
Chemistry-
Which of the following statements is wrong about the reaction?
Which of the following statements is wrong about the reaction?
Chemistry-General
Chemistry-
In the following reaction, the final product can be prepared by two paths (I) and (II).
Which of the following statements is correct?
![](data:image/png;base64,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)
In the following reaction, the final product can be prepared by two paths (I) and (II).
Which of the following statements is correct?
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)
Chemistry-General