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Questions Related to maths

Consider the planes $\displaystyle 3x-6y-2z=15$ and $\displaystyle 2x+y-2z=5.$ 


Assertion: The parametric equations of the line of intersection of the given planes are $\displaystyle x=3+14t, y=1+2t, z=15t.$ because  

Reason: The vector $\displaystyle 14\hat{i}+2\hat{j}+15\hat{k}$ is parallel to the line of intersection of given planes.

line of intersection of two planes line and a plane vectors, lines and planes three dimensional geometry maths

  1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

  2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

  3. Assertion is correct but Reason is incorrect

  4. Both Assertion and Reason are incorrect

Show answer
Correct Option: D
Explanation:

$3x-6y-2z=15$

$2x+y-2z=5$
For $z=0$ we get $x=3,y=-1$
Direction vectors of plane are $<3 -6 -2 >$ and $<2, 1 ,-2>$
Then the dr's of line of intersection of planes  is $<14, 2 ,15>$
$\therefore \dfrac{x-3}{14}=\dfrac{y+1}{2}=\dfrac{z-0}{15}=\lambda$
$\Longrightarrow x=14\lambda+3y=2\lambda-1z=15\lambda$
hence Both Assertion and Reason are incorrect 

The line of intersection of the planes $\displaystyle \bar r (3\hat i - \hat j + \hat k) = 1$ and $\displaystyle \bar r (\hat i + 4\hat j - 2\hat k) = 2$ is parallel to the vector

line of intersection of two planes line and a plane vectors, lines and planes three dimensional geometry maths

  1. $\displaystyle -2\hat i + 7\hat j + 13\hat k$

  2. $\displaystyle 2\hat i - 7\hat j - 13\hat k$

  3. $\displaystyle 2\hat i + 7\hat j + 13\hat k$

  4. $\displaystyle 2\hat i + 2\hat j + 13\hat k$

Show answer
Correct Option: A
Explanation:
We know that $\vec{r} = {x}\hat{i} +{y}\hat{j}+{z}\hat{k}$
Hence
The equations of the planes can be written as $3x - y + z - 1 = 0$ and $x + 4y - 2z - 2 = 0$
We find the coordinates of a point lying on their intersection line.
Let $z = t$
$\therefore 3x - y = 1 - t \ ...(1)$
$x + 4y = 2t + 2 \ ...(2)$
Multiplying equation (1) by 4 and adding that to equation (2), we get 
$13x = 4 - 4t + 2t + 2 = 6 - 2t$
$\therefore x = \cfrac{6 - 2t}{13}$
Substituting this in equation (1), we get 
$y = 3x - 1 + t = \cfrac{18 - 6t - 13 + 13t}{13} = \cfrac{5 + 7t}{13}$
Thus, the point can be written as $\left ( \cfrac{6 - 2t}{13}, \cfrac{5 + 7t}{13}, t \right )$
i.e. $\left ( \cfrac{6}{13}, \cfrac{5}{13}, 0 \right ) + \left ( \cfrac{-2}{13}, \cfrac{7}{13}, 1 \right ) t$
$\Rightarrow$ the line of intersection is parallel to $-2\hat{i} + 7\hat{j} + 13\hat{k}$

Consider three planes$P _1: x-y+z=1$$P _2: x+y-z=-1$$P _3: x-3y+3z=2$Let $L _1, L _2, L _3$ be the lines of intersection of the planes ${P} _{2}$ and ${P} _{3},\ {P} _{3}$ and ${P} _{1}$, and ${P} _{1}$ and ${P} _{2}$, respectively.
STATEMENT-$1$ : At least two of the lines ${L} _{1},\ {L} _{2}$ and ${L} _{3}$ are non-parallel.
and 
STATEMENT -$2$ : The three planes do not have a common point.

line of intersection of two planes line and a plane vectors, lines and planes three dimensional geometry maths

  1. Statement-1 is True, Statement -2 is True; Statement-2 is a correct explanation for Statement-1

  2. Statement -1 is True, Statement -2 is True; Statement-2 is NOT a correct explanation for Statement-1

  3. Statement -1 is True, Statement -2 is False

  4. Statement -1 is False, Statement -2 is True

Show answer
Correct Option: D
Explanation:

Given three planes are 
${ P } _{ 1 }:x-y+z=1$   ...$(1)$
${ P } _{ 1 }:x+y-z=-1$    ....$(2)$
and ${ P } _{ 1 }:x-3y+3z=2$    ...$(3)$
Solving Eqs. $(1)$ and $(2)$, we have 
$x=0,z=1+y$
which does not satisfy Eq. $(3)$
As, $x-3y+3z=0-3y+3\left( 1+y \right) =3\left( \neq 2 \right) $
$\therefore$ Statement II is true. 
Nest,since we know that direction ratio's of line of intersection of planes ${ a } _{ 1 }x+{ b } _{ 1 }y+{ c } _{ 1 }z+{ d } _{ 1 }=0$ and
${ a } _{ 2 }x+{ b } _{ 2 }y+{ c } _{ 2 }z+{ d } _{ 2 }=0$
$b _{ 1 }{ c } _{ 2 }-{ b } _{ 2 }{ c } _{ 1 },{ c } _{ 1 }{ a } _{ 2 }-{ a } _{ 1 }{ c } _{ 2 },{ a } _{ 1 }{ b } _{ 1 }$
Using above result, we get
Direction ratio's of lines ${ L } _{ 1 },{ L } _{ 2 }$ and ${ L } _{ 3 }$ are $o,2,2;0,-4,-4;0,-2,-2$ respectively. 
$\Rightarrow $ All the three lines ${ L } _{ 1 },{ L } _{ 2 }$ and ${ L } _{ 3 }$ are parallel pairwise. 
$\therefore $ Statement I is false. 

Let L be the line of intersection of the planes $2x + 3y + z = 1$ and $x + 3y + 2z = 2$. If L makes an angle $\alpha$ with the positive x-axis, then $\cos \alpha$ equals

line of intersection of two planes line and a plane vectors, lines and planes three dimensional geometry maths

  1. $\dfrac{1}{\sqrt{3}}$

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

  3. $1$

  4. $\dfrac{1}{\sqrt{2}}$

Show answer
Correct Option: A
Explanation:
We have 
$2x+3y+z=1$
$\Longrightarrow 2x+3y=1-z$      ...(i)
$x+3y+2z=2$
$\Longrightarrow x+3y=2-2z$    ...(ii)
$(i)-(ii)$
$\Longrightarrow 2x+3y-x-3y=1-z-2+2z$
$\Longrightarrow x=z-1$
$\Longrightarrow z=\cfrac{x+1}{1}$
Putting values of $z$ in (ii), we get,
$z-1+3y=2-2z$
$\Longrightarrow 3y=2-2z-z+1$
$3y=-3(z-1)$
$\Longrightarrow y=-(z-1)=-z+1$
$\therefore z=\cfrac{y-1}{1}$
Hence, we have $\cfrac{x+1}{1}=\cfrac{y-1}{1}=\cfrac{z}{1}$
thus $\cos\alpha=\cfrac{a}{(\sqrt {(a^2+b^2+c^2)})}$
$\therefore \cos\alpha=\cfrac{1}{\sqrt3}$

Find the angle between the line of intersection of the planes $\overrightarrow { r } .\left( i+2j+3k \right) =0$ and $\overrightarrow { r } .\left( 3i+2j+3k \right) =0$ with coordinate axes

line of intersection of two planes line and a plane vectors, lines and planes three dimensional geometry maths

  1. with $x$-axis $\displaystyle \dfrac { \pi  }{ 2 } $

  2. with $y$-axis $\displaystyle \cos ^{ -1 }{ \left( \dfrac { 3 }{ \sqrt { 13 }  }  \right)  } $

  3. with $y$-axis $\displaystyle \cos ^{ -1 }{ \left( \dfrac { 2 }{ \sqrt { 13 }  }  \right)  } $

  4. all of these

Show answer
Correct Option: A,B
Explanation:

Direction ratios: $\displaystyle \begin{vmatrix} i & j & k \ 1 & 2 & 3 \ 3 & 2 & 3 \end{vmatrix}=\left( 0,6,4 \right) $ or $(-,3,-2)$

Therefore angle with $x$-axis: $\displaystyle \dfrac { \pi  }{ 2 } $
with $y$- axis $\displaystyle \cos ^{ -1 }{ \left( \dfrac { 3 }{ \sqrt { 13 }  }  \right)  } $
with $z$-axis $\displaystyle \cos ^{ -1 }{ \left( \dfrac { -2 }{ \sqrt { 13 }  }  \right)  } $

What annual instalment will discharge a debt of Rs. $2710$ due in $4$ years at $7\%$ simple interest?

installments banking business and commercial activities economics maths

  1. Rs. $1000$

  2. Rs. $613$

  3. Rs. $225$

  4. Rs. $150$

Show answer
Correct Option: B
Explanation:

Let the annual installment be $x$

Now, $ x$ accumulates an interest of $7.1$ every year
$\implies$
$x$ deposited in first year becomes $1.21x$ in next $3$ years
$x$ deposited in second year becomes $1.11x$ in next $2$ years
$x$ deposited in first year becomes $1.07x$ in next $4$ years
$\implies1.21x+1.41x+1.07x+x=2710\ \implies 4.92x=2710\ \implies x=Rs.613$

Under the __________ scheme, the article will not be owned by the buyer until and unless he/she has paid the complete purchasing price of the article.

installments banking business and commercial activities economics maths

  1. Interest

  2. Hire Purchase

  3. Loan

  4. Finance

Show answer
Correct Option: B
Explanation:

$\Rightarrow$  Under the $Hire\, Purchase$ scheme, the article will not be owned by the buyer until and unless he\she has paid the complete purchasing price of the article.

$\Rightarrow$  Hire-purchase system is a special system of purchase and sale. When goods are purchased on hire purchase system, the purchaser pays the price in instalments, these instalments may be Monthly, Quarterly or yearly etc. 
$\Rightarrow$   Goods are delivered to the purchaser at the time of Hire Purchase Agreement but the purchaser will become the owner of goods only on payment of last instalments.
$\Rightarrow$  All instalments paid are treated a hire until the last instalment paid off.

What is hire purchase scheme?

installments banking business and commercial activities economics maths

  1. A method of buying goods through making instalment payments over time.

  2. A method of trading goods through making instalment payments over time.

  3. A method of buying goods through making interest payments over time.

  4. A method of buying goods through making money payments over time.

Show answer
Correct Option: A
Explanation:

A hire purchase is a method of buying goods through making instalment payments over time.

The interest scheme in which companies reduce the amount of the instalments to promote sales is called as __________.

installments banking business and commercial activities economics maths

  1. Discount Sale

  2. $100\%$ Finance

  3. $0\%$ Interest

  4. $50\%$ Finance

Show answer
Correct Option: A
Explanation:

$\Rightarrow$  The interest scheme in which companies reduce the amount of the installments to promote sales is called as $Discount\,Sale.$

$\Rightarrow$  Discounts  are reductions to a basic price of goods or services.
$\Rightarrow$  The purpose of discount sale is to increase short-term sales, to move out-of-date stock, to reward valuable customers, to encourage distribution channel members to perform a function, or to otherwise reward behaviors that benefit the discount issuer.

The instalment scheme in which companies take $4$ or $5$ instalments in advance is called as:

installments banking business and commercial activities economics maths

  1. $25\%$ Finance

  2. $75\%$ Finance

  3. $100\%$ Finance

  4. $0\%$ Interest

Show answer
Correct Option: D
Explanation:

The installment scheme in which companies take $4$ or $5$ installment in advance is called as : $0\%$ Interest

For $0$ percent loans, you pay no interest. That means you're borrowing money from a bank but paying no fee for the privilege of doing so. Essentially, $0$ percent interest gives you the chance to pay the same amount of money as a cash buyer, even though you're spreading your payments over a longer term.
It is the total cost of the article paid as loan to the buyer.
The interest collect $3-5$ months installments in advance.