Mathematics

Coordinate Geometry and Graphs

77 Questions

Coordinate geometry involves plotting linear equations, analyzing graphical data, and understanding planar graphs using vertices, edges, and faces. These concepts are essential for calculating intersection points and interpreting graphical information. Such questions frequently appear in engineering, state, and civil services examinations.

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Coordinate Geometry and Graphs Questions

Multiple choice maths mid-point and its converse application of the mid-point theorem mid point theorem mid-point theorem and its converse

Suppose $ABCD$ is a rhombus. A straight line passing through $C$ meet $AD$ which is produced at $P$ and meet $AB$ produced at $Q$. Therefore if $DP=\dfrac {1}{2}AB$, then find the ratio between $BQ$ and $AB$?

  1. $2:1$
  2. $1:1$
  3. $1:3$
  4. $3:1$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$ ABCD $ is a rhombus. $ AB =BC = CD = DA $


$ \displaystyle \frac{DP}{AB} = \frac{1}{2} \Rightarrow DP =1 AB =2 $

In rhombus  $ \angle \theta _1 = \angle \theta _2 $

$ \angle Q $ is common for the $ \triangle BCQ \, and \, \triangle APQ $

$ \because \angle APQ = \angle BCQ $

$ \because \triangle BCQ $ is similar to $ \triangle APQ $ by $ AAA $ property. 

$ \displaystyle \frac{AP}{BC} = \frac{AQ}{BQ} = \frac{AD + DP}{BC} = \frac{3}{2} $

$ \displaystyle \frac{AQ}{BQ} = \frac{3}{2} \Rightarrow \frac{AB + BQ}{BQ} = \frac{3}{2} $

$ \displaystyle \frac{AB}{BQ} = \frac{3}{2} - 1 = \frac{1}{2} $

$ \displaystyle \frac{BQ}{AB} = \frac{2}{1} $

Multiple choice maths banking and taxation reading graphs describing different situations using equations to plot lines basics of a straight line

The graph of $\dfrac {7x}{2}=18+\dfrac {4}{5}x-45$ is line____

  1. Parallel to $x-$axis at a distance of $10$ units from the origin
  2. Parallel to $y-$axis at a distance of $10$ units from the origin
  3. Parallel to $x-$axis at a distance of $20$ units from the origin
  4. Parallel to $y-$axis at a distance of $20$ units from the origin

  5. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\cfrac { 7x }{ 2 } =18+\cfrac { 4 }{ 5 } x-45\ \Rightarrow \cfrac { 7x }{ 2 } -\cfrac { 4x }{ 5 } =-27\ \Rightarrow \cfrac { 35x-8x }{ 10 } =-27\ \cfrac { 27x }{ 10 } =-27\ \Rightarrow x=-10$

Therefore graph is a straight line parallel to y-axis at a distance of $10$ units from the origin.

Multiple choice maths banking and taxation reading graphs describing different situations using equations to plot lines basics of a straight line

Draw the graph for each linear equation:
$\displaystyle y=\frac{3}{2}x+\frac{2}{3}$

  1. The line passes through $(4/9,0)$ and $m=-\dfrac32$
  2. The line passes through $(-0.4/9,0)$ and $m=\dfrac32$
  3. The line passes through $(-4/9,0)$ and $m=\dfrac32$
  4. The line passes through $(-0.9/4,0)$ and $m=-\dfrac32$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The given line equation is $\frac{3}{2}x-y+\frac{2}{3}$

The slope of given line is $-(\frac { \frac { 3 }{ 2 }  }{ -1 } )=\frac { 3 }{ 2 } $
If we put $y=0$ , then the value of $x= -\frac{4}{9}$
Therefore line passes through $(-\frac{4}{9},0)$ and slope is $\frac{3}{2}$
So the correct option is $C$

Multiple choice maths banking and taxation reading graphs describing different situations using equations to plot lines basics of a straight line

The graph of equation of the form $ax + by +c=0$ where a, b are non $-$ zero numbers,
represents:

  1. A triangle

  2. A ray

  3. A straight line

  4. a line segment
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The equation $ax+by+c=0$ represent straight line under one condition i. e, 

$|a|+|b|\neq 0$       or,     $ a\neq b\neq 0$
Also,  here
$y= \dfrac{-a}{b}x$ $  \dfrac{-c}{b}$ represent the slope interspect form where $m= (-a/b), y-m =  -c/b $ 

Multiple choice maths banking and taxation reading graphs describing different situations using equations to plot lines basics of a straight line

The graph of the function $\displaystyle \cos x.\cos (x+2)-\cos^{2}(x+1)$ is a 

  1. straight line passing through the point $\displaystyle(0,-\sin^{2}1)$ with slope $2$

  2. straight line passing through the origin
  3. parabola with vertex $\displaystyle (1,-\sin^{2}1)$
  4. straight line passing through the point $\displaystyle\left(\dfrac{\pi}2,-\sin^{2}1\right) $ and parallel to the $x-$axis
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let $y=\cos { x } \cos { \left( x+2 \right)  } -\cos ^{ 2 }{ \left( x+1 \right)  } \ =\cos { \left( x+1-1 \right)  } \cos { \left( x+1+1 \right)  } -\cos ^{ 2 }{ \left( x+1 \right)  }\$
 Using $\cfrac{1}{2} \left[\cos\left(A+B+A-B\right)+\cos\left(A+B-A+B\right)\right]\
           = \cfrac{1}{2}\left[\cos 2A + \cos 2B\right]
           = \cfrac{1}{2}\left[\cos^{2}A -1 +1 -2 \sin^{2}A\right ]
           =  \cos ^{2}\left(x+1\right)-\sin^{2}1 $
$=\cos ^{ 2 }{ \left( x+1 \right)  } -\sin ^{ 2 }{ 1 } -\cos ^{ 2 }{ \left( x+1 \right)  } \ =-\sin ^{ 2 }{ 1 } $
This is a straight line which is parallel to x-axis, it passes through $\left( \cfrac { \pi  }{ 2 } ,-\sin ^{ 2 }{ 1 }  \right) $

Multiple choice maths banking and taxation reading graphs describing different situations using equations to plot lines basics of a straight line

Complete the table, to draw the graph of line $2y=3x+2$.


$x:$ $3$ $\displaystyle \frac{7}{3}$ $-2$
$y:$ $y _1$ $y _2$ $y _3$

  1. $y _1 = \dfrac12, y _2=\dfrac72,y _3 = -2$
  2. $y _1 = \dfrac{11}2, y _2=\dfrac92,y _3 = -2$
  3. $y _1 = \dfrac12, y _2=\dfrac72,y _3 = -3$
  4. $y _1 = \dfrac{11}2, y _2=\dfrac92,y _3 = -3$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The equation of line is $2y=3x+2$

Now to complete the given table,
Substitute $x=3$ in the equation $2y=3x+2$ to fill the first column of the table
$\displaystyle 2y _1=3\times 3+2\ \Rightarrow 2y _1=9+2\ \Rightarrow 2y _!=11\ \Rightarrow y _!=\dfrac { 11 }{ 2 }$
 Substitute $x=\dfrac { 7 }{ 3 }$ in the equation $2y=3x+2$ to fill the second column of the table

$2y _2=3\times \dfrac { 7 }{ 3 } +2\\ \Rightarrow 2y _2=7+2\\ \Rightarrow 2y _2=9\\ \Rightarrow y _2=\dfrac { 9 }{ 2 }$
Finally, substitute $x=-2$ in the equation $2y=3x+2$ to fill the first column of the table
$2y _3=3\times -2+2\\ \Rightarrow 2y _3=-6+2\\ \Rightarrow 2y _3=-4\\ \Rightarrow y _3=-2$

Multiple choice maths introduction to three dimensional geometry coordinate axes and coordinate planes in 3d space coordinates in 3d introduction to 3d geometry

The graph of the equation $y^{2}+z^{2}=0$ in three dimensional space is

  1. x- axis

  2. y- axis

  3. z- axis

  4. yz-plane
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Consider the problem 


${y^2} + {z^2} = 0$

$x=0$ and $z=0$

Therefore, 
The graph of the equation 

${y^2} + {z^2} = 0$ is $x-axis$.

Hence, the correct option is $x-axis$.

Multiple choice maths introduction to three dimensional geometry coordinate axes and coordinate planes in 3d space coordinates in 3d introduction to 3d geometry

Graph $x^2+y^2=4$ in 3D looks like

  1. Circle

  2. Cylinder

  3. Hemisphere

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

The given curve is $x^2+y^2=4$ 

So $x$ coordinate and y-coordinate are connected by $x^2+y^2=4$
which is locus of a circle with radius $2$
But z-coordinate can be anything, so in three dimension the circle $x^2+y^2=4$ will be 
stretched which will be a cylinder with radius same as the radius of the circle .

Multiple choice mathematics and statistics parabola tracing of the parabola definitions related to parabola introduction to parabola

The equation of directrix from the following is,

  1. $2x - y = 0$
  2. $x + 2y = 0$
  3. $x + y = 0$
  4. $x + 3y = 0$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Let $(a,b)$ be the focus and $y=m{x}$ be directrix 

$\implies \bigg(\dfrac{m{x}-y}{\sqrt{1+m^{2}}}\bigg)^{2}=(x-a)^{2}+(y-b)^{2}$

Differentiating on both sides

$\dfrac{(m{x}-y)(m-\dfrac{d{y}}{d{x}})}{1+m^{2}}=(x-a)+(y-b)\dfrac{d{y}}{d{x}}$

$x$ axis is tangent at $(1,0)$

$\dfrac{m(m)}{1+m^{2}}=1-a\implies a=\dfrac{1}{1+m^{2}}$

$y$ axis  is tangent at $(0,2)$

$\dfrac{-2}{1+m^{2}}=b-2\implies b=\dfrac{2{m}^{2}}{1+m^{2}}$

$(1,0)$ lies on parabola

$\dfrac{(m)^{2}}{1+m^{2}}=(1-a)^{2}+b^{2}$

Substituting $a$ and $b$ values 

$\implies (4{m^2}-1)(m^{2})=0\implies m=0,\pm \dfrac{1}{2}$

For $m=0$ we get $a=1,b=0$ which means that the directrix cuts the parabola which is not possible so $m=\pm \dfrac{1}{2}$

$\implies a=\dfrac{4}{5},b=\dfrac{2}{5}$

So the focus is $\bigg(\dfrac{4}{5},\dfrac{2}{5}\bigg)$

the directrix is $2{y}+x=0$

Hence option $B$ is the answer.
Multiple choice maths congruency of triangles triangle inequality inequalities in triangle inequalities in triangles

The complex number z having least positive argument which satisfies the condition $|z - 25i| \le 15$   is:

  1. $25i$
  2. $12+5i$
  3. $16+12i$
  4. $12+16i$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
Solution:

$|z-25 i| \leq 15$

Let $z= r(cos \theta + i \, sin \theta)$

$\theta$ must be minimum

$| r \, cos \theta +i ( r\, sin \theta-25)|\leq 15$

$|\sqrt{r^2cos^2\theta+r^2 sin^2 \theta+ 625- 50 r \, sin \theta} \,|\leq 15$

square both side

$r^2 (cos^2 \theta+ sin^2 \theta)+625 - 50 r \, sin \theta \leq 225$

$r^250 r \, sin \theta \leq - 400$

$f(r)=\dfrac {400+r^2}{50 \, r}\leq sin \theta $

Find maximum value of $f(r)=\dfrac {400+r^2}{50\, r}$

$f'(r)=\dfrac {100 r^2- 50(400+r^2)}{2500 r^2}=0$

$50 r^2- 50 \times 400=0$

$r= 20$

$f(r=20)=\dfrac {800}{1000}\leq sin \theta $

$\dfrac {4}{5}\leq  sin \theta $

Least value of $sin \theta $ is $4/5$

$ tan \, \theta = 4/3 \,\,\,\,\,\,\,\,\,\, cos \theta  = 3/5$

$z= 20(3/5+4/5 \,i)$

$z= 12+16\, i$

D is correct.