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MATHS True False[1 Marks ]

VECTOR ALGEBRA · Rajasthan Board (RBSE) STD 12 Science (Rajasthan - English Medium). 14 questions.

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True False[1 Marks ]

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14 questions · self-marked practice — reveal the answer and mark yourself.

Question 11 Mark
State True or False for the following:
Position vector of a point $\vec{\text{P}}$ is a vector whose initial point is origin.
Answer
True.Solution:
Since, $\vec{\text{P}}=\overrightarrow{\text{OP}}$ displacement of vector $\vec{\text{P}}$ from origin.
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Question 21 Mark
Answer the following as true or false.
Two collinear vectors are always equal in magnitude.
Answer
False $[\because\ \vec{a}\ \ \text{and}\ 2\vec{a}$ are collinear vectors but $ \Big|2\vec{a}\Big|=2\Big|\vec{a}\Big|]$
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Question 31 Mark
Two collinear vectors are always equal in magnitude.
Answer
False, Collinear vectors are parallel vector not equal vectors.
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Question 41 Mark
State True or False for the following:
If $|\vec{\text{a}}|=|\vec{\text{b}}|,$ then necessarily it implies $\vec{\text{a}}=\pm\vec{\text{b}}.$
Answer
False.Solution:
Consider $\vec{\text{a}}=2\hat{\text{i}}+2\hat{\text{j}}-2\hat{\text{k}}$ and $\vec{\text{b}}=2\hat{\text{i}}-\hat{\text{j}}-2\hat{\text{k}}$
Clearly $|\vec{\text{a}}|=|\vec{\text{b}}|,$ but $\vec{\text{a}}=\pm\vec{\text{b}}$
So, the given statement false.
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Question 71 Mark
State True or False for the following:
If $|\vec{\text{a}}+\vec{\text{b}}|=|\vec{\text{a}}-\vec{\text{b}}|,$ then the vector $\vec{\text{a}}$ and $\vec{\text{b}}$ are orthogonal.
Answer
True.Solution:
Given, $|\vec{\text{a}}+\vec{\text{b}}|=|\vec{\text{a}}-\vec{\text{b}}|$
$\Rightarrow|\vec{\text{a}}+\vec{\text{b}}|^2=|\vec{\text{a}}-\vec{\text{b}}|^2$
$\Rightarrow|\vec{\text{a}}|^2+|\vec{\text{b}}|^2+2\vec{\text{a}}\cdot\vec{\text{b}}=|\vec{\text{a}}|^2+|\vec{\text{b}}|^2-2\vec{\text{a}}\cdot\vec{\text{b}}$
$\Rightarrow2\vec{\text{a}}\cdot\vec{\text{b}}=-2\vec{\text{a}}\cdot\vec{\text{b}}$
$\Rightarrow4\vec{\text{a}}\cdot\vec{\text{b}}=0$
$\Rightarrow\vec{\text{a}}\cdot\vec{\text{b}}=0$
Hence, $\vec{\text{a}}$ and $\vec{\text{b}}$ are orthogonal.
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Question 81 Mark
State True or False for the following:
If $\vec{\text{a}}$ and $\vec{\text{b}}$ are adjacent sides of a rhombus, then $\vec{\text{a}}\cdot{\text{b}}=0.$
Answer
False.Solution:
If $\vec{\text{a}}\cdot{\text{b}}=0,$ then $\vec{\text{a}}$ and ${\text{b}}$ are perpendicular to each other.
In rhombus, adjacent sides are not necessarily perpendicular.
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Question 91 Mark
State True or False for the following:
The formula $(\vec{\text{a}}+\vec{\text{b}})=\vec{\text{a}}^2+\vec{\text{b}}^2+2\vec{\text{a}}\times\vec{\text{b}}$ is valid for non-zero vectors $\vec{\text{a}}$ and $\vec{\text{b}}.$
Answer
False.Solution:
$(\vec{\text{a}}+\vec{\text{b}})=(\vec{\text{a}}+\vec{\text{b}})\cdot(\vec{\text{a}}+\vec{\text{b}})$
$(\vec{\text{a}}+\vec{\text{b}})=\vec{\text{a}}^2+\vec{\text{b}}^2+2\vec{\text{a}}\cdot\vec{\text{b}}$
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Question 101 Mark
Two collinear vectors having the same magnitude are equal.
Answer
False, As two collinear vectors are equal only if they have same length and same sense.
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Question 111 Mark
Answer the following as true or false.
Two collinear vectors having the same magnitude are equal.
Answer
False $[\because$ Vectors $\vec{a}\ \text{and}-\vec{a}\left\{=(-1) \vec{a}=\vec{ma}\right\}$ are collinear vectors and $\Big|\vec{a}\Big|=\Big|-\vec{a}\Big|$ but we know that $\vec{a}\neq-\vec{a}$ because their directions are opposite.]
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Question 121 Mark
Answer the following as true or false.
Two vectors having same magnitude are collinear.
Answer
False $[\because\ \Big|\hat{i}\Big|=\Big|\hat{j}\Big|=1$ bat $\hat{i}\ \text{and}\ \vec{j}$ are vectors along x - axis (OX) and y - axis (OY) respectively]
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Question 131 Mark
Two vectors having same magnitude are collinear.
Answer
False, Collinear vectors may not have a same magnitude.
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Question 141 Mark
Answer the following as true or false:
$\vec{\text{a}}\text{ and }\vec{\text{a}}$ are collinear.
Answer
True, As vectors having the same and parallel support are collinear.
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