Tag: mathematics and statistics

Questions Related to mathematics and statistics

For the ellipse 4x2+y2−8x+2y+1=04x2+y2−8x+2y+1=0 which of the following statements are correct:

mathematics and statistics conic sections standard equation of ellipse introduction to ellipse standard equation of an ellipse

  1. Foci are $\displaystyle \left ( -1, -1\pm \sqrt{3} \right ),$ Directrices are  $\displaystyle y=-1\pm \frac{4}{\sqrt{3}}$

  2. Foci are $\displaystyle \left ( 1, 1\pm \sqrt{3} \right ),$ Directrices are  $\displaystyle y=1\pm \frac{4}{\sqrt{3}}$

  3. Foci are $\displaystyle \left ( 1, -1\pm \sqrt{3} \right ),$ Directrices are  $\displaystyle y=-1\pm \frac{4}{\sqrt{3}}$

  4. Foci are $\displaystyle \left ( 1, -1\pm \sqrt{3} \right ),$ Directrices are  $\displaystyle y=1\pm \frac{4}{\sqrt{3}}$

Show answer
Correct Option: C
Explanation:

$\displaystyle 4x^{2}+y^{2}-8x+2y+1=0$
$\displaystyle 4(x^2-2x)+(y^2+2y)=-1$
$\displaystyle 4(x^2-2x+1)+(y^2+2y+1)=-1+5=4$
$\displaystyle 4(x-1)^2+(y+1)^2=4$
$\displaystyle \frac{\left ( x-1 \right )^{2}}{1}+\frac{\left (

y+1 \right )^{2}}{4}=1$
$\displaystyle \Rightarrow a^2 =1, b^2 = 4$ Clearly here $a^2< b^2$ so the axis of the ellipse is parallel to y-axis
Now,  $\displaystyle e=\sqrt{1-\frac{a^2}{b^2}}=\frac{\sqrt{3}}{2} \therefore $ Foci $\displaystyle \left

( 1, -1\pm \sqrt{3} \right ),$ Directrices  $\displaystyle y=-1\pm

\frac{4}{\sqrt{3}}.$

Find the length of latus rectum of the ellipse $4x^2\, +\, 9y^2\, \,+ 8x\, \,+ 36y\, +\, 4\, =\, 0$.

mathematics and statistics conic sections standard equation of ellipse introduction to ellipse standard equation of an ellipse

  1. $\displaystyle \frac{1}{3} $

  2. $\displaystyle \frac{2}{3} $

  3. $\displaystyle \frac{4}{3} $

  4. $\displaystyle \frac{8}{3} $

Show answer
Correct Option: D
Explanation:

$4x^2+9y^2+8x+36y+4=0$
$\Rightarrow 4(x^2+2x)+9(y^2+4y)=-4$
$\Rightarrow 4(x^2+2x+1)+9(y^2+4y+4)=-4+4+36$
$\Rightarrow 4(x+1)^2+9(y+2)^2=36$
$\Rightarrow \dfrac{(x+1)^2}{9}+\dfrac{(y+2)^2}{4}=1$
$\therefore a^2=9,b^2=4$
Hence length of latus rectum is $=\dfrac{2b^2}{a}=\dfrac{8}{3}$

Find the the length of the major axis of ellipse: $12x^2+4y^2+24x-16y+25=0$

mathematics and statistics conic sections standard equation of ellipse introduction to ellipse standard equation of an ellipse

  1. $ \sqrt{3}$

  2. $ 2 \sqrt{3}$

  3. $ 2 \sqrt{\dfrac32}$

  4. $ 2 \sqrt{6}$

Show answer
Correct Option: A
Explanation:

$12x^{ 2 }+4y^{ 2 }+24x-16y+25=0$


$\Rightarrow 12(x^2+2x+1)+4(y^2-4y+4)=3$

$\Rightarrow 12(x+1)^2+4(y-2)^2=(\sqrt 3)^2$

$\Rightarrow \displaystyle \dfrac { { \left( x+1 \right)  }^{ 2 } }{ { \left( \dfrac { 1 }{ 2 }  \right)  }^{ 2 } } +\dfrac { { \left( y-2 \right)  }^{ 2 } }{ { { \left( \dfrac { \sqrt3 }{ 2 }  \right)  }^{ 2 } } } =1\ $


Length of major axis $=2a=\sqrt3.$

Arrange the following ellipses in the ascending order of their lengths of major axis:
$\mathrm{A}:x^{2}+2y^{2}-4x+12y+14=0$
$\displaystyle \mathrm{B}:\frac{(x-1)^{2}}{9}+\frac{(y-1)^{2}}{16}=1$
$\mathrm{C}:4x^{2}+9y^{2}=1$
$\mathrm{D}:x=3+6\cos\theta,y=5+7\sin\theta$

mathematics and statistics conic sections standard equation of ellipse introduction to ellipse standard equation of an ellipse

  1. C, A, B, D

  2. C, A, D, B

  3. A, B, C, D

  4. C, D, A, B

Show answer
Correct Option: A
Explanation:
For A:
${ x }^{ 2 }+2{ y }^{ 2 }−4x+12y+14=0\\ \Rightarrow{ x }^{ 2 }-4x+4+2\left( { y }^{ 2 }+6y+9 \right) -8=0\\ \Rightarrow { \left( x-2 \right)  }^{ 2 }+2{ \left( y+3 \right)  }^{ 2 }=8\\ \\ \Rightarrow \cfrac { { \left( x-2 \right)  }^{ 2 } }{ 8 } +\cfrac { { \left( y+3 \right)  }^{ 2 } }{ 4 } =1$

$\therefore$ Length of major axis= $2\sqrt { 2 } $

For B:
$\\ \\  \cfrac { { \left( x-1 \right)  }^{ 2 } }{ 9 } +\cfrac { { \left( y-1 \right)  }^{ 2 } }{ 16 } =1$

$\therefore$ Length of major axis= 4

For C:
$4{ x }^{ 2 }+9{ y }^{ 2 }=1\\ \Rightarrow\cfrac { { x }^{ 2 } }{ \frac { 1 }{ 4 }  } +\cfrac { { y }^{ 2 } }{ \frac { 1 }{ 9 }  } =1$

$\therefore$ Length of major axis= $\cfrac{1}{2}$

For D:
$\\ x-3=6\cos \theta\\ y-5=7\sin \theta$

$\therefore$ Length of major axis= 7

$\therefore$ Option [A] C,A,B,D

lf $ax^{2}+by^{2}+2gx+2fy+c=0$ represents an ellipse, then

mathematics and statistics conic sections standard equation of ellipse introduction to ellipse standard equation of an ellipse

  1. its major axis is parallel to $x-$axis

  2. its major axis is parallel to $y-$axis

  3. its axes (i.e. major axis and minor axis) are neither parallel to $x-$axis nor parallel to $y-$axis

  4. its axes are parallel to co-ordinate axes

Show answer
Correct Option: D

The abscissa of the focii of the ellipse $25(\mathrm{x}^{2}-6\mathrm{x}+9)+16\mathrm{y}^{2}=400$ is:

mathematics and statistics conic sections standard equation of ellipse introduction to ellipse standard equation of an ellipse

  1. $ (4,-ae), (4,ae)$

  2. $ (3,-ae), (3,ae)$

  3. $ (5,-ae), (5,ae)$

  4. None of these

Show answer
Correct Option: B
Explanation:

The equation of the ellipse can be written as $\dfrac{(x-3)^{2}}{4^{2}}+\dfrac{y^{2}}{5^{2}}=1$
$\therefore $ Minor axis is along the line $x-3=0$ and major axis is $y=0$ 
$(\because a^{2}=4^{2}< b^{2}=5^{2})$
$\therefore S\,$ and $ S^{'}$ are $(3,3),(3,-3)$
$\because 4^{2}=5^{2}(1-e^{2})\Rightarrow e=\dfrac{3}{5}\left ( \because ae=5\dfrac{3}{5}=3 \right )$ and $S\equiv(3,ae )$, $S^{'}\equiv(3,-ae).$
Ans: B

The eccentricity of the curve with equation ${ x }^{ 2 }+{ y }^{ 2 }-2x+3y+2=0$ is

mathematics and statistics conic sections standard equation of ellipse introduction to ellipse standard equation of an ellipse

  1. $0$

  2. $\sqrt { 2 }$

  3. $1/2$

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

Show answer
Correct Option: A
Explanation:

Given ${x}^{2}+{y}^{2}-2x+3y+2=0$

$\Rightarrow \left({x}^{2}-2x\right)+\left({y}^{2}+3y\right)+2=0$
$\Rightarrow \left({x}^{2}-2x+1-1\right)+\left({y}^{2}+2\times 1\times \dfrac{3}{2}+\dfrac{9}{4}-\dfrac{9}{4}\right)+2=0$
$\Rightarrow {\left(x-1\right)}^{2}-1+{\left(y+\dfrac{3}{2}\right)}^{2}-\dfrac{9}{4}+2=0$
$\Rightarrow {\left(x-1\right)}^{2}+{\left(y+\dfrac{3}{2}\right)}^{2}=-2+1+\dfrac{9}{4}=\dfrac{5}{4}$
Divide both sides by ${\left(\sqrt{\dfrac{5}{4}}\right)}^{2}$ we get
$\dfrac{{\left(x-1\right)}^{2}}{{\left(\sqrt{\dfrac{5}{4}}\right)}^{2}}+\dfrac{{\left(y+\dfrac{3}{2}\right)}^{2}}{{\left(\sqrt{\dfrac{5}{4}}\right)}^{2}}$  is an ellipse where $a=\sqrt{\dfrac{5}{4}}$ and  $b=\sqrt{\dfrac{5}{4}}$
Eccentricity $e=\sqrt{1-\dfrac{{b}^{2}}{{a}^{2}}}=\sqrt{1-\dfrac{{\left(\sqrt{\dfrac{5}{4}}\right)}^{2}}{{\left(\sqrt{\dfrac{5}{4}}\right)}^{2}}}=\sqrt{1-1}=0$


If the interest on $1700$ rupees is $340$ rupees for $2$ year the rate of interest must be

mathematics and statistics banks and simple interest introduction to interests introduction to interest introduction to interest payments

  1. $12\ \%$

  2. $15\ \%$

  3. $4\ \%$

  4. $10\ \%$

Show answer
Correct Option: D
Explanation:

Principle$=Rs1770\quad\quad Time=2years$

$SI=Rs340\quad\quad Rate=?\ \cfrac{P\times R\times T}{100}=340\Rightarrow \cfrac{1770\times R\times 2}{100}=340\ \Rightarrow R=\cfrac{340\times100}{1770\times2}=\cfrac{340\times5}{177}=9.6\%$

The simple interest on a sum money is 4/9 of the principal and the number of years is equal to the rate percent per annum. The rate per annum is :  

mathematics and statistics banks and simple interest introduction to interests introduction to interest introduction to interest payments

  1. $5$%

  2. $6\dfrac{2}{3}\%$

  3. $6$%

  4. $7\dfrac{1}{5}\%$

Show answer
Correct Option: B
Explanation:
Let the principal be $P$.
Rate of interest be $R\%$
According to the question, Time$=R$
Simple interest $=\dfrac{4P}{9}$.
$SI =\dfrac{\left(PTR\right)}{100}$
$\Rightarrow \dfrac{4P}{9} =\dfrac{\left(PTR\right)}{100}$
$\Rightarrow \dfrac{4P}{9} =\dfrac{\left(P\times R\times R\right)}{100}$
$\Rightarrow \dfrac{4P}{9} =\dfrac{\left(P\times {R}^{2}\right)}{100}$
$\Rightarrow \dfrac{4}{9} =\dfrac{{R}^{2}}{100}$
$\Rightarrow {R}^{2}=100\times\dfrac{4}{9}$
$\Rightarrow R= 10\times \dfrac{2}{3}=\dfrac{20}{3}$
Therefore, rate of interest is $\dfrac{20}{3}\%$ or  $6\dfrac{2}{3}\%$.

A sum of money at simple interest amounts to Rs. 815 in 3 years and to Rs. 854 in 4 years. The sum is :

mathematics and statistics banks and simple interest introduction to interests introduction to interest introduction to interest payments

  1. Rs. 650

  2. Rs. 690

  3. Rs. 698

  4. Rs. 700

  5. Rs. 715

Show answer
Correct Option: C
Explanation:

S.I. for $1$ year $= Rs. (854-815) = Rs. 39$


S.I. for $3$ years = Rs. $39 \times 3=Rs. 117$


Therefore,

Principal $= Rs. 815 - Rs. 117 = Rs. 698$