Tag: maths

Questions Related to maths

$A$ vector $\vec V$ is inclined at equal angles to axes $OX,OY$ and $OZ$. If $\vec V$ is $6units$, then $\vec V$ is

maths vectors and transformations introduction to vector algebra algebra of vectors operations on vectors

  1. $2\sqrt 3\left( \hat i+\hat j+\hat k right )$

  2. $2\sqrt 3\left( \hat i-\hat j+\hat k right )$

  3. $\sqrt 2\left( \hat i+\hat j+\hat k right )$

  4. $2\sqrt 3\left( \hat i+\hat j-\hat k right )$

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Correct Option: A

$\sum _{ i=1 }^{ n }{ \vec { ai }  } =\vec { 0 } \quad where\quad |\vec { a\quad i\quad | } =1\forall i$ then the value of $\sum _{ 1\le i }^{  }{ \sum _{ <j\le n }^{  }{ \vec { { a } _{ i } }  }  } .\vec { { a } _{ j } } $ is 

maths vectors and transformations introduction to vector algebra algebra of vectors operations on vectors

  1. -n/2

  2. -n

  3. n/2

  4. n

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Correct Option: A

If $ \vec{a} $ and $ \vec{b} $ are two non-collinear unit vectors such that $ |\vec{a}+\vec{b}| = \sqrt{3}, $ find $(2\vec{a}-5\vec{b}).(3\vec{a}+\vec{b}) $ 

maths vectors and transformations introduction to vector algebra algebra of vectors operations on vectors

  1. $ +\dfrac{11}{2} $

  2. $ -\dfrac{13}{2} $

  3. $ -\dfrac{11}{2} $

  4. $ +\dfrac{13}{2} $

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Correct Option: C
Explanation:

Given $ |\vec{a}+\vec{b}| = \sqrt{3}, $

Now squaring both sides we get,

$(\vec{a}+\vec{b}).(\vec{a}+\vec{b})=3$ [ Since$|\vec{a}|^2=\vec{a}.\vec{a}$ 
or, $|\vec{a}|^2+2\vec{a}.\vec{b}+|\vec{b}|^2=3$ [ Since 

$\vec{a}.\vec{b}=\vec{b}.\vec{a}$ ]
or, $\vec{a}.\vec{b}=\dfrac{1}{2}$.....(1). [ Since $\vec{a},\vec{b}$ are unit vectors then $|\vec{a}|=1=|\vec{b}|$ ]

Now,
$(2\vec{a}-5\vec{b}).(3\vec{a}+\vec{b}) $ 
$=6|\vec{a}|^2-13\vec{a}.\vec{b}-5|\vec{b}|^2$

$=6-\dfrac{13}{2}-5$ [ Using (1)]
$=-\dfrac{11}{2}$.

Which of the following can represent a vector?

maths vectors and transformations introduction to vector algebra algebra of vectors operations on vectors

  1. The length of the distance between the points $(0,0)$ and $(2,7)$

  2. A line segment beginning at $(2,7))$ and ending at $(0,0)$

  3. The length of the distance between the points $(2,7)$ and $(0,0)$

  4. A line segment beginning at $(0,0)$ and ending at $(2,7)$

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Correct Option: B,D
Explanation:
A vector is a quantity that can be described as having both magnitude and direction.
The length of the distance between any two points is a magnitude with no direction, so it can't represent a vector.
A line segment beginning at a certain point and ending at another can represent a vector. The magnitude of the vector is the distance between the points, and its direction is the direction from the initial point to the terminal point.
The following can represent a vector:
A line segment beginning at $(0,0)$ and ending at $(2,7)$.
A line segment beginning at $(2,7)$ and ending at $(0,0)$

Direction of zero vector

maths vectors and transformations introduction to vector algebra algebra of vectors operations on vectors

  1. does not exist

  2. towards origin

  3. indeterminate

  4. None of these

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Correct Option: C
Explanation:

As, zero vector represents a point.
Direction is indeterminate.
Hence, option C.

Which will result in a vector?

maths vectors and transformations introduction to vector algebra algebra of vectors operations on vectors

  1. Product of a scalar and a scalar.

  2. Product of a scalar and a vector.

  3. Addition of two vectors

  4. None of these

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Correct Option: B,C
Explanation:
Let two vectors
$\vec{a}=\hat{i}$ 
$\vec{b}=\hat{i}+\hat{j}$
Addition of both vector 
$\vec{a}+\vec{b}=\hat{i}+\hat{i}+\hat{j}$
$\vec{a}+\vec{b}=2\hat{i}+\hat{j}$
Here we get vector by addition of both vectors 
hence option C is correct

let two scalar $\lambda=2,\mu=1$
$\lambda\times\mu=2\times1=2$
SO from here we get a scalar quantity Hence 
Option A is not correct 

$\lambda\times\vec{a}=\lambda\hat{i}$
Here vector quantity is obtained 
hence option B is correct

What is the value of $p$ for which the vector $p\left( 2\hat { i } -\hat { j } +2\hat { k }  \right)$ is of $ 3$ units length?

maths vectors and transformations introduction to vector algebra algebra of vectors operations on vectors

  1. $1$

  2. $2$

  3. $3$

  4. $6$

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Correct Option: A
Explanation:

length of vector $a\hat { i } +b\hat { j } +c\hat { k } $ from origin is $\sqrt { a^2+b^2+c^2 } $ 

So $\sqrt { {(2p)}^2+{(-p)}^2+{(2p)}^2 } =\sqrt { 9p^2 }=3p $
Length is $3$ units given. 
$\therefore 3p=3\implies p=1$
Hence, A is correct.

If $\vec{x}$ and $\vec{y}$ be unit vectors and $\displaystyle |\vec{z}| = \dfrac{2}{\sqrt 7}$ such that $\vec{z} + (\vec{z} \times \vec{x}) = \vec{y}$ and $\theta$ is the angle between $\vec{x}$ and $\vec{z}$, then the value of sin $\theta$ is

maths vectors and transformations introduction to vector algebra algebra of vectors operations on vectors

  1. $\displaystyle \dfrac{1}{2}$

  2. $1$

  3. $\displaystyle \dfrac{\sqrt 3}{2}$

  4. $\displaystyle \dfrac{\sqrt 3 -1}{2 \sqrt 2}$

Show answer
Correct Option: C
Explanation:

$|\vec{z} + (\vec{z} \times \vec{x}) | = | \vec{y}|^2 \,\,\,\,\,\Rightarrow$
$|\vec{z}|^2+|\vec{z}|^2 |\vec{x}|^2 \,sin^2\,\theta =1$
$\displaystyle \Rightarrow \,|z| = \frac{1}{\sqrt {1 + sin^2\,\theta}} = \frac{2}{\sqrt 7} \Rightarrow sin\,\theta = \frac{\sqrt 3}{2}$

A car takes 2 hours to reach a destination by travelling at the speed of 60 km/h. when the car travels at the speed of 80 km/h it takes 

maths rule of three inverse proportion direct proportion and inverse proportion types of proportions

  1. one and half hour

  2. one hour

  3. two hour

  4. None

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Correct Option: A
Explanation:

Let the car takes x hours to reach a destination by travelling at the speed of 80 km/h. Then,

Speed (in km/hr) 60 80
Time (in hours) 2 x

Clearly, more the speed, less will be the time taken. So, it is a case of inverse proportion.
$ \therefore \,\,\, 60 \times 2 = 80 \times x  \Rightarrow x = \displaystyle \frac{60 \times 2}{80} = \frac{3}{2}$
Hence, the time taken will be $\displaystyle 1\frac{1}{2}$ hours.

Which is an example of inverse proportion?

maths rule of three inverse proportion direct proportion and inverse proportion types of proportions

  1. More amount of sweets, more total cost

  2. More length of cloth, more cost

  3. More expenditure, less saving

  4. More height of object, more length of its shadow

Show answer
Correct Option: C
Explanation:

2 quantities say it be $x , y$ are said to be in proportion when the change in value of $x$ , leads to the equal change in value of $y$.

If $x$ increases and hence  $y$ decreases proportionally, it is called Inverse proportion i.e $x\propto\dfrac{1}{y}$
For example, In option C, if our expenses are more, then we are left with lesser savings from a fixed salary or income.
hence, Expenditure and savings are in inverse proportion.
So the answer is C