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Questions Related to maths

Multiple choice line of intersection of two planes line and a plane vectors, lines and planes three dimensional geometry 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.

  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
Reveal answer Fill a bubble to check yourself
D Correct answer
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 

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

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

  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. $\displaystyle 2\hat i + 2\hat j + 13\hat k$
Reveal answer Fill a bubble to check yourself
A Correct answer
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 $-2\hat{i} + 7\hat{j} + 13\hat{k}$
Multiple choice line of intersection of two planes line and a plane vectors, lines and planes three dimensional geometry maths

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.

  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
Reveal answer Fill a bubble to check yourself
D Correct answer
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. 

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

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

  1. $\dfrac{1}{\sqrt{3}}$
  2. $\dfrac{1}{2}$
  3. $1$
  4. $\dfrac{1}{\sqrt{2}}$
Reveal answer Fill a bubble to check yourself
A Correct answer
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 $\cos\alpha=\cfrac{a}{(\sqrt {(a^2+b^2+c^2)})}$
$\therefore \cos\alpha=\cfrac{1}{\sqrt3}$
Multiple choice line of intersection of two planes line and a plane vectors, lines and planes three dimensional geometry maths

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

  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
Reveal answer Fill a bubble to check yourself
A,B Correct answer
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)  } $

Multiple choice installments banking business and commercial activities economics maths

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

  1. Rs. $1000$
  2. Rs. $613$
  3. Rs. $225$
  4. Rs. $150$
Reveal answer Fill a bubble to check yourself
B Correct answer
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
$\implies1.21x+1.41x+1.07x+x=2710\ \implies 4.92x=2710\ \implies x=Rs.613$

Multiple choice installments banking business and commercial activities economics maths

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.

  1. Interest

  2. Hire Purchase

  3. Loan

  4. Finance
Reveal answer Fill a bubble to check yourself
B Correct answer
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.

Multiple choice installments banking business and commercial activities economics maths

What is hire purchase scheme?

  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.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

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

Multiple choice installments banking business and commercial activities economics maths

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

  1. Discount Sale
  2. $100\%$ Finance
  3. $0\%$ Interest
  4. $50\%$ Finance
Reveal answer Fill a bubble to check yourself
A Correct answer
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.

Multiple choice installments banking business and commercial activities economics maths

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

  1. $25\%$ Finance
  2. $75\%$ Finance
  3. $100\%$ Finance
  4. $0\%$ Interest
Reveal answer Fill a bubble to check yourself
D Correct answer
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.